Vincent Selhorst-Jones combines a scientific background, acting training, and years of teaching experience to help you fully comprehend concepts rather than just “plug 'n' chug” equations. Vincent is passionate about sharing his love of physics and will help you understand how a problem comes together, what it means, and why you should care. Topics cover everything in High School Physics from Motion, Force, and Energy, to Waves, Thermodynamics, and Electricity. Vincent has been teaching over 8+ years and double-majored in Mathematics and Theater at Pomona College, as well as received an M.F.A. in Acting from Harvard University.

expand all   collapse all

Hi, welcome to educator.com. This is the beginning of Physics.0000

We are going to do first up, before we get into the Physics, we are going to do a quick Math review. 0004

So, even if you feel really strong at Maths, just make sure to real quickly skin through the section, because we want to make sure you understand all these concepts clearly,0008

because they might come up in this course, and they might also come up in whatever course you are taking.0015

This is a supplement to your other Physics courses.0020

So, it is a good idea to make sure you have definitely got the background and the skills inside of this Math review.0021

Let's get started: First off, the metric system. Also, called the S.I. units, which is from the French System Internationale, which is the people who first created the metric system and first propagated its use.0026

So, the metric system is created in 1800's to, actually may be the sub 1800's, I should know that, anyway, I am sorry, so, anyway the metric system was created to standardize the measurements and it has done a great job at that.0038

Almost all the countries in the world, the exception, the United States of America are completely standardized on it, and even in the United States, everybody in Science, in Physics, they all use the metric system.0052

The metric system is great, and it's the way to do things. 0063

So, the basic units we work with in Physics are, distance, just the metre denoted by a small 'm' mass, which is the kilogram, denoted by 'kg'.0066

I would like to point out it's the kilogram, so it's not actually the gram that we consider our basic unit of mass, we consider the kilogram our basic unit of mass, just an interesting thing to point out. 0076

The volume, volume which comes in litres denoted by a small 'l' or sometimes a 'cursive l', if it gets confused as a '1' sometimes, and finally time, the second, which is denoted by 's'.0085

Scientific notation. 0101

What if we had a problem involving the number say 47 billion or 0.00000002?0103

If you had to write down that number, more than two or three times, I think you would be unhappy, and I think you should be unhappy.0111

That is a lot of times that you have to write a number.0117

That many digits is just a pain, and you are not really putting in much information as you feels like in those zeros.0120

So we wouldn't want to write that all those times. 0124

So, how else could we write it. The trick here is scientific notation. The idea here is, that you can convert it by using powers of ten.0126

So, 47000000000 is the same as 47 × 109, because we got 1,2,3..4,5,6..7,8,9 , so times ten to the 9th, and if we wanted to have it so we only had one digit at the very front, we could push it over for one more and we could have 4.7 times ten to the tenth.0135

Same idea if we want to move a digit up, we go back 1,2,3..4,5,6,7,8 spaces, so that would be 2 × 10-8.0154

We are able to compact information this way because, ten to the one is equal to 10, so ten to the two is equal to hundred and so on and so forth.0165

We can also go negative, so ten to the negative one is 0.1, so ten to the negative two is 0.01.0171

This allows us to slide digits around, so we don't have to write really really long numbers, because when we are dealing with say number of atoms or the charge of an electron or the distance from here to the sun, we are going to be dealing with very large and very small numbers depending. 0177

So Physics deals some extreme values of numbers, and we don't want to get cramps, because we are going to write 30 zeroes every time a number comes up.0191

Significant figures. 0199

Also called sig figs. Significant figures are a way of showing, how precise our information is.0200

Since all information are susceptible to some amount of error, even if you look at a really fine ruler, it's hard tell what is the difference between one hundredth of a millimetre to the left or the right.0206

There is always some amount where it's a judgment call, and you might be slightly wrong.0216

There is always uncertainty in every measurement, there is always some little bit of possible error.0220

So significant figures expresses how certain we are with the measurement, or what the uncertainty in the measurement is.0226

It says how much we can trust our info.0232

Significant figures give us a way in letting us know how much we should rely on the information we have.0236

Which figures are significant? That requires a little thought.0241

Significant digit is any of the following: Any digit that is not a zero. 0246

The zeroes between non-zero digits, and zeroes to the right of significant digits.0250

The only digits that are not significant are the digits to the left of significant digits, which makes a lot of sense.0255

If I wrote two, and if I wrote a bunch of zeroes in front of it, well, that is the exact same thing.0260

There is no way to measure the difference between two and a two with a bunch of zeroes in front of it.0266

It means the exact same thing, and there is no way you can measure the difference, and there is no significance in all of these zeroes.0270

We would not care about them, knock them out. 0277

This would only have a significant digit of 'one', one significant digit, one sig fig. 0279

Let's try some other examples: Our first one, we got 1,2,3,4.0284

So this one would have four significant figures.0289

This one has 1,2,3, what about that times 10 to the fourth. 0292

Well, that times ten to fourth if we would have 10 to the fourth, well that would be, 1-0-3-0-0.0297

So, 10300. 0307

But if we had 10300, what we would be saying is, we have precisely measured 10300. 0309

Precisely measured 10300 metres.0315

But what we really did, we only managed to precisely measure the first 10300, but it might be up or down a little bit.0317

It could be 10349, or 10251.0325

It could be something that is close to what we could round, we are only sure up to that 10300.0330

So that is the point of the sig fig here.0335

So, that scientific notation also gives us the ability to show the information that we have measured for sure, but there might be some hash to just the zero. 0338

So let us not multiply out to say the scientific notation and then find the sig figures, you find the sig figs before you multiply the scientific notation. 0347

So this one would have three significant digits.0355

Here we have 1,2 and all of these are zeroes, so it just has two significant digits. 0358

Here we have 1,2,3,4,5, so it has five significant digits because these ones don't count, but these ones do count because they are to the right0365

It means that you have measured something precisely.0375

There is a difference if I say, I weigh about 75 kilos. I weigh about 75 kilograms, or if I say I weigh 75.000 kilograms.0377

That means I have managed to get a really precise reading.0388

I am down to within a grams certainty of my weight. 0391

So, it is a very precise reading of my weight, very precise reading of my mass.0394

So, that 000 at the end matters, but in the front once again there is no extra information there. 0398

Finally, if we had 4.700, we would have 1,2,3,4.0404

We count from the right end, or the left in this case because there is no zeroes to the left.0410

So, we count zeroes on the right, here we would have four.0415

How do significant figures interact with one another? 0420

If you add, subtract, multiply or divide numbers, we have to pay attention to how the significant figures interact. 0423

The resulting numbers are only going to have as many significant figures as the lesser of the two numbers of significant figures.0429

The smallest number of sig figs in the number you start with becomes the number of sig figs the result has. And this makes sense.0434

If I know I weigh precisely, I have the mass of 75.000 kilograms but then I get on a boat with somebody else who weigh about 80 kilograms, I can't say, together we weigh precisely 80 plus 75, precisely 155.000.0440

I cannot do that because, I don't know, maybe they weigh 83 kilograms.0458

They were unsure when they told me their mass.0462

So, I cannot be certain of that.0464

It means we have to go to the least significant digits we have, which is two, which is those two digits of 80 kilograms.0466

We only got two significant figures. 0475

That is the case, we wind up actually having two, round up because we have 155, it could become 160.0479

Here is some great examples: We have two 2 kilograms here, and 0.0803 kilograms here.0487

Mathematically if we add them together, we get the number 2.0803 kilograms.0492

But, this guy has one significant figure. 0497

This guy has 1,2, THREE significant figures.0500

It does not matter, he is the smaller one, so we have to cut off after just one, and we round here, we wind up getting just two kilograms, because we only had that significant figure of 2 kilograms in the first one.0503

Over here, we know that we are going 6.083 metre per second so we got 1,2,3,FOUR.0516

Here we got 1,TWO.0523

So just two significant figures over here. It is the smaller one, it wins out. 0524

So we have to round to here. This guy will manage to cause it to round up, and we will wind up going to 13 metres.0529

Just a quick trig review, if you do not remember your trig, that is going to really matter with time.0539

So brush up on that.0541

Pythagorean theorem, a2 + b2, the two smaller sides of a right triangle, equals the other side, the hypotenuse squared.0544

a2 + b2 = c2. 0553

Then we have also got the trigonometric functions to relate those sides together. 0555

The sine of θ is equal to the side opposite over the hypotenuse.0558

So, this would equal 'b' over 'c'.0563

Cosine of θ is equal to the side adjacent over the hypotenuse. 0566

So this would be 'a' over 'c', and finally tan θ is equal to the side opposite divided by the side adjacent.0573

So this would be 'b' over 'a'.0584

Definitely important thing to remember. 0587

Inverse trigonometric functions. What if we know what the sides of the triangle is, and we want to find the angle.0590

Then we use an inverse trigonometric function.0594

The arcsine or the sin-1, however you want to say it, because it is measuring what is the arc of that, right?0597

The arc that goes along with a given ratio. arcsine of sin θ equals θ.0603

Allows us to reverse it.0611

If we look this up, if we use a calculator, it gives us an answer. If we look it up in a big book, with just a look up table, it gives us an answer. 0612

Same basic idea.0619

We are able to figure out all of those ratios beforehand through clever thought, and then at any time if you want to figure out what the angles are being, we just look at the book we created, look at the table, look at the calculator.0621

If sin θ = b/c, we could find θ with sin-1.0631

So θ would be sin-1(b/c), the arcsin(b/c).0635

We plug in numbers, and we get what the angle is.0641

Vectors: Vectors are a way to think about movement.0645

In another sense, they are a way to simultaneously consider the distance and the angle.0647

'v' here has gone some distance and it is up some angle.0651

'u' has managed to go some distance and it is up some angle.0657

But alternatively we could think of it as 'v' went over to the right by 4 and it went up by 5.0660

'u' went to the left, so it went negative two (-2), and it went up by 2.0668

That is the idea of a vector. We can expand this. We can do a vector addition.0675

If we got two vectors we can add them, we can put them head to tail, numerically you will add their components. 0679

So, you have got 'v' and you have got 'u', v+u is just the sum of the numbers involved.0684

This one is 4 and -2. 4 and -2 becomes 2.0690

5+2 becomes 7. 0695

There you go. As simple as that. 0701

Subtraction is just adding by the negative version of the number. 0703

If you want to know what the negative version of 'u' was, -u, we just put a negative sign in front of what it was originally. 0707

We apply that in, we get (2,-2). We add 'v' to the negative version of 'u'.0713

2 plus, it was 4 before, we get 6.0720

-2 plus, it was 5 before, we get 3. Simple as that.0725

Scalar: Vector is a distance and a direction. 0731

Scalars in a way, are just a number. It is a way to scale a vector.0735

It is a multiplication thing. You scale the vector, you change how much it grows or shrinks.0740

You can the length, and even flip the direction of a vector by using scalar. 0745

You just multiply each element of a vector with it. Vectors are multi-dimensional, scalars are just one dimensional. 0750

If s=3, then if we had 'v' as (4,5), what we have been using so far, then 3 would just be 1,...2, ...3 out.0755

So, 'sv' is just 3 v's stacked on top of one another.0767

Makes sense. v+v+v. 3 × v.0771

If we had -2, then we have to flip to the negative version. Here is where -v would show up. 0775

We stack it twice, and we have got -2v.0780

If we want to do it numerically, we just wind up multiplying it by each component involved. 0783

3 times (4,5) becomes (12,15), -2 times (4,5) becomes (-8,-10).0790

If we want to break a vector in to its components, we just do it.0798

We know what each of the components are, so we can see how much should we move in the 'x', how much should we move in the 'y'.0801

So, vx and vy. 0807

If v = (4,5), then we see that the x side must be length 4, and the y side must be length 5.0809

Also, if we wanted to, we can say, v = (4,5), which is the same thing as (4,0) plus (0,5), which is basically what we see right here.0817

We added here, to here, and we get to the same spot.0830

If we want to find the length of a vector, we use the Pythagorean theorem.0839

We know what those sides are, because we know what the x-component is, we know what the y-component is.0842

How does the Pythagorean theorem work?0846

Square root of the two smaller sides, 42 + 52 equals the square of the other side.0850

We call it the absolute value, the magnitude, that is how we denote it.0860

In this case, square root of (16+25), does not wind up coming out to be a nice round number, we get the square root of 41, that is as simple as it is going to be.0864

But that is what its length is. If you want, you can change it into decimal using a calculator. 0879

There is a relationship between length, angle and coordinates. 0886

In general, if we wanted to know what arc, what it was if we knew that our vector had a length 5 and an angle 0f 36.87 degrees above the horizontal, what would be the vector, let's just make a triangle. 0889

Our angle here is 36.87 degrees, and this here is 5, this look like a perfect time to use sin θ.0904

This side over here, let's call it 'y'. 0916

So, sin θ equals y/5.0918

We multiply both sides by 5, so we get 5sin θ equals y.0927

We plug in what that θ was, 5sin(36.87).0932

Punch that into a calculator, multiply by 5, and we are going to wind up getting 3.0938

So, that side is equal to 3. Same basic idea over here. 0944

We call this side x, and any time we are doing this, it is going to be the hypotenuse divided by the other.0948

Cosine equals adjacent divided by hypotenuse. 0954

So, any time we want to know the adjacent side, it is just going to be hypotenuse times cosine of the angle. 0957

Or if we want to know the opposite side, it is going to be hypotenuse times sine of angle. Simple as that. 0960

'x' is going to be 5cosine(36.87),0966

Toss that into a calculator, and we get 4.0975

The vector 'v' would be its two components put together, (4,3). 0977

If you want to check that out, 42 + 32 = 25, which is the square of 5.0985

Checks out by the Pythagorean theorem. We got the answer.0991

That is basically all the Math that we got to have under our belt if we want to get started in this Physics course. 0994

Hope all that made sense. If it did not make sense, go back, check some of the stuff that you do not remember from trigonometry. 0999

Just get back up to speak, because we are going to wind up using a lot of this, especially when we are talking about multi-dimensional stuff.1003

Alright, see you at the next...1010

Hi, welcome back to educator.com, today we are going to be talking about Newton's third law.0000

Put your hand, just as a quick experiment, on the edge of the table, or something hard that is fixed in place, and push for a while.0005

Just push for 10 seconds or so, and then take a look at what the bottom of your hand looks like.0014

If you look at it, you will notice that there is actually a mark on your hand, there will be some sort of line, or some sort of indentation.0020

So what happened?0030

You were pushing on the table, but what happened to your hand?0031

Your hand got pushed back on by the table.0034

Every force is going to have a force that resists it.0037

There is going to be an equal and opposite reactionary force.0040

If you push a certain amount of force into the table, the table is going to push a certain amount of force back into your hand.0042

You push on the table, it pushes back on you.0048

Newton's third law, one way to put it, (Newton put it in a slightly different way himself), but in modern language, 'Whenever an object a force on another object, the second object exerts a force of equal magnitude in the opposite direction, on the first object'.0052

If you put in a force of some amount of newtons, say, x N this way, the object is going to put a force of x N this way.0067

If you put it symbolically, the force of A on B = - the force of B on A, but negative, i.e. FA-B = - FB-A, so same magnitude, but opposite directions.0075

Just some basic everyday examples..0085

How about the hammer hitting a nail?0088

We have got a nail like this, and a hammer comes down on it, what happens?0090

The hammer hits the nail, and bounces back off, stops in place, the nail gets driven down into the wood, so there is definitely a force on the nail, but the hammer stops, for the hammer to stop, there has to be a force on the hammer.0099

The amount of that force is the amount of force that the hammer puts in to the nail, the same amount of force that the hammer gets from the nail.0112

They are going to be in opposite directions, but they are going to be equal, and that is what stops the hammer, and moves the nail, and then friction arrests the movement of the nail, we will talk about that later when we get to friction.0119

What if you are swimming?0129

If you are swimming, you are swimming along, you got yourself in the water, and you push on the water this way.0131

For example, if you do a breaststroke, and when you are swimming like this, your hands are cupping the water, they are catching the water, and they are pushing against the water.0140

So, you push against the water, and because you are pushing against the water, the water pushes back on you, which pushes you forward.0148

If you are driving along in a car, the car, it has wheels, (we have not talked about friction yet, we will very soon), the wheels, they push on the earth, they spin , and because of friction, them trying to rotate, pushes on the earth.0155

They are going to try to rotate this way, the wheels try to rotate this way, because of their friction, they have got friction, they stick to the earth, they are going to push the earth this way, the response of which is going to push the car this way.0178

Friction once again, combined with Newton's third law, is what gives our car motion.0189

When we walk on the earth, when you walk along, your foot pushes into the earth at a certain amount, which is going to cause you to be pushes in the opposite direction.0196

You push on the earth, if you want to jump, you bend your knees, and you push down really hard, and that is going to result in you getting pushed up, which is going to cause you to fly up into the air.0210

That is how you move, you move by pushing on something, and it responds because of Newton's third law, every force has an equal and opposite reaction force.0223

You push a certain amount , and that thing is going push back on you with that same amount in the opposite direction.0232

A careful thing to pay attention to is that, Newton's third law is always true, but often when we are working on our problems, it will not come into play.0239

It is still there, it is still happening, but this is because our problems are not normally going to have to include Newton's third law in the way we think about it.0247

It is still there if you want to think about it, but it will not have any effect on what we are doing in our work.0255

One reason why it often happens is that the force in the problem is external, it is guaranteed by the problem, we do not care what happens to the thing causing the force.0260

Very often we have talked about a block on a frictionless table or a block on a table.0269

We put some force into that block, what is putting that force in the block?0275

We do not know, it could be rockets on the back of the block, or it could be a person just standing there and pushing on the block, and so let us say it is a person standing there pushing on the block.0279

What happens to the person?0293

The person is going to wind up getting pushed back by the block by the amount that he push on the block.0294

But, we have been guaranteed by the problem, we have been told that a constant force is being applied, so we know that the person is somehow dealing with the force from the block, probably by putting it into the earth, by pushing against feet, and they are managing to handle the force that they are getting back from the block, and manage to keep up a constant force.0300

However the force is handled, we are guaranteed by the problem that there is a constant force, so with that point we can just deal with the constant force.0318

We do not have to worry about where the other part of it is going, because we have been given this external force that is guaranteed.0325

We do not have to care about where the forces are coming from, because it is given to us in the problem statement, it is given.0331

The other thing is, equal and opposite forces are often translated into the earth.0338

For example, with the person pushing on the block, which cause the person to be pushed back by a certain amount.0341

What does that person do?0354

The person then translates that force by pushing into the earth with his legs, which then cancels it out, so the amount that they get pushed by this becomes zero, because they get a net force of zero.0356

They put a push into the ground, (push by the person are in 'blue'), they push a blue up here, and a blue down here with their legs.0368

So the reactionary force is, from the person's point of view, they wind up experiencing the reaction forces, so the two cancel out, the person stands still, they can manage to keep up that constant force.0382

What happens to the force that gets us into the earth?0393

The equal and opposite force, if they get translated into the earth (or any planet), it is normally expected that the planet's mass is so large, so incredibly large compared to the force put into it, that it is going to experience this miniscule acceleration, we do not have even have to care about it, because the acceleration is so small, it is of little consequence.0396

Mass is giant for the earth, compared to these forces we are dealing with, they are comparatively puny, so there is no effect on the earth as far as we are concerned.0415

Let us start looking at some examples.0425

Let us say that the earth is sitting here, and up in the sky, we have got the moon.0427

If the earth exerts a pull of force of gravity Fg on the moon, how much does the moon pull on the earth?0436

So, Fg has been pulled by the earth.0448

We can turn these into vectors if we want, so, Fg is pulled by the earth on the moon, how much does the moon pull back on earth?0452

It is just going to be the equal and opposite force.0461

Gravity is a two way street, and so, - Fg is the exact amount that the moon is going to pull on earth, because it is going to pull with the exact same magnitude, but in the opposite the direction, they are pulling towards one another.0463

Why is the moon not falling to the earth?0481

We will talk about that later, when we talk about how gravitation works, the fact that moon is basically in continuous free fall, and the pull is what keeps it in that free fall as opposed to just slinging off into space, but we will talk about that in a future section.0483

For now, we have to understand the gravity is equal between the two things.0495

Same if you are sitting wherever you are, standing wherever you are, the earth is pulling on you, but you are also pulling on the earth by a certain amount.0500

From the earth's point of view, it is hardly going to notice you because your mass is so slow compared to the earth's mass.0508

But, you are still exerting a force on the earth, and it is equal to your own weight.0514

Second example: Say there is an astronaut in deep space, and he has got no external forces acting on him.0519

The astronaut has a wrench.0531

Now, what happens if the astronaut throws that wrench, the astronaut has a mass of 100 kg, and over here, we have got, mass of wrench = 2 kg .0536

The astronaut throws the wrench by applying a force of 50 N on the wrench.0552

He applies it for time = 0.2s.0562

What is the wrench's velocity?, what is the astronaut's velocity?0568

Normally when we would have talked about this being on earth, the person would have translated the force from the throw through their legs, and made it so that they had no force.0572

But this person is currently in deep space.0580

He has no where he can translate this force, so the throw is going to affect him equally.0581

So what is the force on the person?0586

From Newton's law, we know that they have to be equal and opposite, so the astronaut is going to also experience a 50 N force.0589

It is going to be going in the opposite direction.0595

So, what is the wrench's velocity?0598

We just use F = ma , 50 N = (2 kg) × a , a = 25 m/s/s, is the acceleration of the wrench.0600

What is the acceleration of the astronaut?0619

Fon the astronaut = maon the astronaut , - 50 = 100 × a (negative because it is in the opposite direction, and since we took the other as the positive direction), a = (1/2) m/s/s .0622

Once again, keep attention to where your units are, although we have been dropping them for ease.(it will save you from making a really simple foolish mistake.)0659

Now, if we want to know what the velocity is, well, how long do they experience the acceleration?0680

They experience the acceleration for 0.2 s, so velocity = a × t .0685

So, for the wrench, velocitywrench = 20 × 0.2 = 5.0 m/s.0694

Velocityastronaut = (-1/2)m/s/s × 0.2 s = -0.1 m/s.0713

So they are going to wind up experiencing very different velocities because of their very different masses, just like the experience between you and the earth, you notice the pull off the earth, but the earth hardly notices the pull of you, because of your very different masses.0732

It is the same thing with the astronaut and the wrench.0745

He will throw the wrench, and the wrench will move away from him 50 times faster than him moving in the opposite direction, because his mass is 50 times more. He currently weighs nothing, because he is in deep space.0747

But the inertia is going to be 50 times more.0760

So we have go the answer here, we know that it is going to move at 5 m/s, and the astronaut is going to move in the opposite direction at 0.1 m/s.0763

Third example: For this one, we have got a woman sitting in a bosun's chair, (bosun's chair is designed to help to make it easier for you to lift yourself up, used in sailing, used in window washing things like this).0772

It is hanging from a mass-less that runs over a mass-less frictionless pulley.0786

The chair and the woman have a combined mass of 75 kg.0789

With what force does she need to pull on the rope to have a constant velocity?0797

What force does she need to pull for it to have an acceleration of 1 m/s/s?0803

Let us think about what happens when she pulls.0808

She pulls down with some force, and that is going to be the force that becomes the tension.0810

So she pulls down with some T, now this force is translated into the rope, so we have got this pull coming along, and it is going to pull up here, with a force of T.0814

But, we have not paid attention to Newton's third law.0828

If you try to yank on something, if you are climbing, if you are pulling a rope, you put a tension into it, which cause you to rise, because the amount of tension that you put into the rope is the amount of force resultant o you, going in the opposite direction.0831

The resultant force from her pull is going to be T.0845

So, what is the total force that the woman is going to experience from the pull?0849

She is going to experience 2T going up.0852

What other forces are on her?0855

Her weight = Mg, and she has got a 2T going.0857

So, if she wants to have a constant velocity, we need the sum of the forces = mass × acceleration, if we want constant velocity, that is going to be equal to zero.0865

So (making up the positive direction), 2T - Mg = 0 (no acceleration).0874

So, 2T = Mg, T = 75 × 9.8 / 2 , T = 367.5 N, is the tension that she needs to put into the rope.0890

What if she wants to have an acceleration of 1 m/s/s?0911

Very similar, 2T - Mg = M × 1, T = (Mg + M)/2 , T = ((75 × 9.8) + 75) / 2 , T = 405 N .0916

Notice, she is able to do this at a very different rate, she is able to put way less force into pulling herself up using the bosun's chair because she is taking advantage of the Newton's third law.0952

She knows that by putting a tension into the rope, she is going to get lifted by that tension doubly, by the resultant force from pulling, the tension put into the rope will wind up pulling her up, but so will the resultant force.0960

She is making Newton's third law work for her, it is helping her out in this case.0980

So she is able to pull with considerably less than what her weight is.0984

She is going at a constant velocity, she only has to pull at half her weight.0987

So that is a really clever way to be able to use less effort on our part, to be able to go up.0991

We are able to take advantage of the way Physics works.0996

That is the end for Newton's third law.0999

Hi, welcome back to educator.com, today we are going to be talking about friction.0000

At this point, you have got a really strong grasp on the basics of Mechanics.0005

Force = mass × acceleration, we have talked about it in two dimensions, you have got a really good idea of how Newton's laws work.0009

But so far, we had to pretend that friction does not exist, as if something that we could not really deal with.0015

But no more, now we are finally going to tackle friction.0022

You have got enough understanding about mechanics, you will be able to understand how to use friction in our work.0024

First, let us get a sense of how friction works in two dimensions.0032

Imagine you have got a plank of wood that you are pushing along at a constant speed.0037

Here is some floor, here is some plank of wood on that floor, and we are pushing it along at a constant speed.0042

First thing to notice, is that in real life, we are used to the idea that if we want something to move, (since everything experiences friction), you have to push on it if you want to keep a constant speed.0047

It is not going to have that constant speed unless you push on it, because friction is going to sap the energy out of it.0057

So, for the first time, we are saying that we need a constant force to keep that constant speed.0064

Up until now, if we had any force at all, we would have had an acceleration automatically, because we have been talking about being on a frictionless surface.0071

It would be a small acceleration, but we would have had some acceleration because we would have had some force, unless all the forces are cancelling out.0079

Now, we are going to have all the forces cancel out, because we have friction cancelling out the forces we are putting in, so we can have a constant velocity.0086

With that out of the way, we have got this plank moving along at a constant speed, because we are putting in some force into it.0092

Now, which would be easier to push, which would take less push, which would take less force for pushing on the plank?0098

The plank was on a floor that is made of wood, or the plank on a floor made of rubber, which one of these will stick together more, which one will have more friction?0104

Just like you would expect, the wood.0113

The wood is going to stick less, and the rubber is going to stick more.0116

If we want to make it easy for ourselves, we are going to want that wood floor.0120

What if we were to put the plank on a piece of ice?0123

It is going to make it even easier.0127

Different pairs of materials have different connections.0128

They behave differently with one another because of material science and chemistry and stuff that we are not going to really talk about, but friction is a pretty complicated idea that we will experience in lots of further courses and there is lots of cool interesting things to learn about it.0132

But in our case, we just know that, if we have different materials, we are going to have different frictions.0146

Different PAIRS of materials, that is an interesting thing to keep in mind, it is not just one material, it is the pair of it.0151

If we had a rubber plank on top of that rubber floor, we would have experienced even more friction.0157

An ice plank on top of an ice floor, it would have been the least of all.0161

It is the pair together that gives us the friction between them.0165

Let us talk about another thing for our intuition to deal with.0170

Imagine that, that same plank of wood is on a wood floor, but this time, we are going to put some sack of sand on top of it.0172

So, we have got some sack on top of it, and there is going to be some amount of sand, it is either going to have 10 kg of sand, or 20 kg of sand.0180

Which one is going to be easier to push along?0191

The 10 kg sack or the 20 kg sack, which one is going to have more friction reacting with that?0193

More friction force for us to overcome, the 10 kg sack push on the plank or the 20 kg sack push on the plank?0201

Which one would you expect?0210

It is just like you would expect, it is the 10 kg sack.0212

More pressure means more friction, harder we push on something, the more the friction that we have to overcome.0214

The lighter something is, the lesser friction that we have to overcome.0222

Assuming the same object, and the same material for the incline, which of the following three situations will be the easiest to push?0227

It is similar to the pressure idea, now we are going to start talking about the normal force.0234

In all of them, gravity ('g') is pointing straight down.0240

Which one of them would be easier, which one of them would you expect to?0246

Just like you would expect, the steepest incline.0248

Why is that?0250

We can explore that idea by looking at two extreme scenarios.0252

The exact same object and the exact same surface, but very different orientations.0256

One of them is a horizontal orientation, the other one is a vertical orientation.0260

Which one of these is going to have more friction going this way?0264

Here is our friction force, here is our friction force, which one is going to experience more?0267

The one that is sitting on it.0274

Why is that?0276

That is because, this one has 'mg' down here, so it has got the pressure (the normal force) pushing that amount.0277

How much does this one on the right, how much is the force normal?0289

We have got 'mg' down here, but there is nothing this way, so our normal force, FN = 0, because there is no pressure, no interaction, nothing holding it against the wall to cause friction to happen.0293

If you push really hard on something, it is going to have more friction.0312

If you do not have any push between the things at all, there is no way for the materials to interact, there is no friction between them.0315

If we have no normal force, we have no friction.0321

If we have a lot of normal force, if we push really hard on it, we are going to have more friction.0324

If we were to instead, come along and push crazy hard on this, then we are going to have a resultant normal force that is equal and opposite, we are going to have this normal force because it is not going to blow through that wall, assuming the wall is able to withstand that much force, we might actually to able to arrest the power of gravity, arrest the acceleration due to gravity, the force due to gravity will be canceled out because we will be able to make a really large friction by pushing really hard.0329

You can test this out in a real quick demonstration.0356

If you take just a normal book, and you go up to a flat wall, and you just put the book up the wall, and you take your hand away, of course the book falls to the ground.0359

It is like you would expect.0366

If you were to put the book up against the wall, and push really hard with the flat of your hand, not under it, because then you would be holding it up, it would not be friction, it would be just direct force applied through your finger tips.0370

But instead, if you were to push really hard against it, you will be able to keep it in place, because you put so much pressure on it, the friction of the book against the wall is going to be able to overcome the pull of gravity.0380

It is going to beat out gravity, and it is just going to stay still.0392

Just like you would expect, from all this talking, friction is not just weight, it is about how hard the object is pushed, it is about the pressure between the object and the surface, the two materials, the interaction, it is the normal force.0397

For those of you having trouble with calculating the normal forces on inclines, I would recommend you to refer to the 'Newton's second law in multiple dimensions lecture', it will give a good explanation.0410

You need to calculate just how much of the gravity is perpendicular and parallel to the surface.0429

To sum up, friction is based on the interaction between the materials involved in it, and the normal force of the object on the surface.0445

What kind of materials do we have, how hard the pressure is, the two things, the normal force.0451

If you want to turn that into an equation, that's going to become the friction = μ × FN...0458

μ is a Greek letter, and it is the coefficient of friction between the two materials, and it is spelled 'm-u', it will change depending on what the materials are, and it is going to vary a lot depending on specifics, and we have to determine it experimentally.0471

There is no easy formula for determining what it is going to be. You just have to go into a lab, get it, or look it up in a table. 0493

Even in looking up a table, it is going to vary, because depending on the specific condition of the object, whether it is dirty, clean, if it is wet, if it has grease on it, if there is a layer of air, if it is operating in vacuum -- what things are happening between it, it is going to vary a lot. 0499

So, it is basically up to you to figure it out in a lab, or to be able to look it up in a table where it has some very, very similar situations to the way you are doing it. 0515

Or, it is given to you precisely in the problem statement.0522

So, figuring out μ can be a little difficult, but normally that's what the problems will be about, or it will be given to us in the problem.0525

Once again, going back to the equation, friction = μ, the coefficient that represents the interaction between the two materials, times the force, the normal force, fn. So μ × fn.0532

One thing to keep in mind, is that we do not have to worry about the area touching it.0548

If we had a block of mass 'm', and we had a table of mass, 'M', but the same material on the bottom.0556

Same material here, same material here, same surfaces, it is not about the cross-section, the area touching the ground, it is just about the pressure.0567

Why is that?0575

That has to do with the way friction works, it is what is happening on a really microscopic thing.0577

If we have a lot of area, the pressure per square area, the force per square area, is going to wind up being much smaller in the case when we have got that large surface.0583

So, same pressure, but it is going to be extended over a large area, whereas in the table example, where we have got just the little weak contacting, it is going to be the same pressure, but it is going to be over a small area, so the total effect is going to be the same, either a small force per area, but over large area, or a high force oer area, but over a small area, the total effect of the pressure is going to be the same.0600

So you do not have to worry about the cross-section, you just have to worry about the interaction between the materials.0628

One last thing: Friction is a force.0635

We know forces come in vectors, so what direction does friction come in?0637

It is not going to go in the direction of the normal fore, that is why our equation in our previous page was not in vectors.0641

Because it is upto us to figure out what direction friction is going to go in.0647

Friction always opposes the movement.0651

Whatever direction it is moving in, keep in mind that it is the velocity , not the acceleration, whatever direction it is currently moving in, it is the opposite direction that the friction is going to point.0653

Friction always is fighting current motion, so the velocity, whatever the direction of velocity is, the opposite of that direction, is the direction that our friction is going to move in.0665

So with this point, we have got a pretty good understanding of how force works.0676

We have got this interaction between μ and the normal force.0680

Let us consider these two diagrams here:0682

We have got, the block is the same in both diagrams, and the surface it is resting on is the same on both diagrams.0685

Let us assume that F1 and F2 are both big enough to move the block.0692

But also that, F1 and F2 are equal in magnitude, they are the same number of newtons.0696

If F1 and F2 have different orientations, but same magnitude, which block will accelerate faster?0707

If we break down our forces into components (we can do that since force is a vector), we look at the vertical amount in F1 and the horizontal amount in F1, and over here, the vertical amount of F2, and the horizontal amount of F2.0714

We see that the thing that is actually do the motion here, is this right here, it is going to be the actual horizontal motion is going to stem from the horizontal component of our force.0736

If we were instead looking at what the normal force is now, we need to figure out what the normal force is going to be.0752

Both these cases, we still have gravity to contend with.0759

We have not dealt with gravity.0761

So there is the force of gravity, and over here, it is going to be the exact same force of gravity, so force of gravity on both of them.0762

How much does the normal force has to be to cancel these things out.0769

Before when we were talking about the force of gravity and the normal force, they were going to be equal to one another (in the horizontal case), because the only thing creating the normal force is gravity.0774

But in this case, if you push through an object, and the object does not blow through the table, then that means that the table has to resist both the object's force of gravity, and in addition, the force that you put into the object.0789

So, the table, the surface has to resist both the forces, that has been put into it by ourselves, by the problem, and the force that is put into it by gravity.0803

In the first case on the left, it is going to have to fight both gravity, and the amount of the force, the normal force is going to be FN over here.0813

In this case, we have already got this component over here, is going to cancel out this component over here, so the normal force over here, is just going to be this little smidgen, down here.0825

In F2's case, we lift off some of the effective weight, what the normal force has to be is much smaller.0836

So which one of these is going to have a higher friction, this one is going to have a much smaller friction because it has got a much smaller normal force.0845

But over here, we have got this huge normal force in comparison, so we have got this giant friction.0862

We have got the same equal force horizontally, so we know that the giant friction is going to wind up sapping more of the acceleration and so, F2 is the more efficient, easier way, it is going to cause more acceleration.0866

F2 will accelerate the block faster, because it will have the smaller FN.0881

So it is really important to pay attention to the interaction between the force of gravity, then also the forces that we are putting into our object.0886

One more thing to talk about, is the idea of, an object being still, at rest on a surface, and an object moving along on a surface.0894

Which one of these will take more effort, more force from us?0902

Just start a refrigerator moving, sliding on a floor, just start that refrigerator up, or keeping an already sliding refrigerator go away.0907

If we want to just, just start it moving up in addition to creating motion requiring some amount of force from us to get that started, there is actually going to be this little thing, if you have to sort of like, unstick it, we have to pop it off of where it was already located.0915

It might seem like a trick question, but it really is not, it really cannot take more force to start something moving than to just to fight kinetic friction.0932

Kinetic friction is going to be different from static friction.0941

The friction of when it is moving, is going to be different from the friction when it is still.0944

Why does this happen?0948

That is a really complicated thing, it is something for future classes in chemistry, more physics, friction is something there is still doing lots of research into, so it is really complicated for right now, but it is definitely something interesting, but we do not have time to talk about it right now.0949

The exact reason is lots of complicated, but it suffices to say that on a microscopic level, the two surfaces interact differently between one another.0969

They are going to wind up interacting in a different way when they are going to be still, and when they are already moving against one another, slight differences happening microscopically , and sometimes major differences as we will see in some of the numbers that we are going to see soon.0976

Static versus kinetic, if we are going to be able to talk about two different kinds of friction, kinetic- the moving kind, and static- the still kind, we are going to have to use a different coefficient for each one.0993

So, μ is now going to split into two different categories: static is going to be μs, kinetic is going to be μk.1006

So, we have got μstatic and μkinetic.1017

One thing to keep in mind: In almost all cases, μs is greater than μk,1020

There are a very few special cases where this is not going to be true, but as far as we are going to deal with in our course, it is almost always true, sometimes they will be equal and there is really freaky materials where μk is larger, but it is beyond this course, it is not something we are going to have to worry about.1029

If you get really interested in material science, it might be the kind of thing you have to deal with in graduate school, but not something that you have to worry about in high school physics.1052

Applying kinetic friction is pretty easy.1061

If we just want to have friction on an object, it is just going to be, μk × FN, until the object stops moving it is going to be in the direction opposing the current motion.1064

That is a little bit different.1078

If we have an object sitting still, and we push on that object, we have got an object like this, and it is giant, and a guy comes up, and he pushes on it, lightly.1080

It is going to be able to defeat him, but it is not going to go back with all of the friction, you know, if you have to push this lightly, if it is going to be able to cancel out this lightly, and this lightly, and say it is able to cancel out all the way up till this big, it is not going to react with the static friction force in the opposite direction of this big every time.1091

It is going to cancel out whatever is put into it.1110

Static friction is going to be able to cancel out up to the amount of force, up to it is maximum amount.1113

So the maximum static friction, static friction resists an object starting to move it, until it gets surpassed.1119

Until we get to that really extreme case, we are always going to have the case that static friction is going to oppose however much force is put into it.1125

It is not going to put in more than that, it is just going to oppose the amount put into it, until we suddenly get to the point where we are able to equal and then surpass static, just that equal point is the razor's edge of flipping over into kinetic friction, at which point the object lurches forward, unsticks, starts to move, and then kinetic friction comes into play, and in almost always μk will be less than μs.1136

So we have some slight acceleration, if we kept up a constant force.1157

The static friction cancels out the force that would cause acceleration, but it never exceeds them.1160

That gives us, the maximum static friction = μs × FN, but keep in mind that it is the maximum static friction, not more than that, but just the maximum.1166

It is the top amount that it can be, we are not going to see that every time we put any small force into it, it is going to be the top amount, that is μs × FN.1179

What is some basic examples of μs and μk?1193

These are some approximate values, this table here, keep in mind that these can vary depending on the specific situation, the condition of the materials involved, wet, greasy, air between them, perfect vacuum, there is certain material properties that can happen.1196

For the most part, these are going to remain the system, but it about the whole system interacting together, so it is really something that has to be experimentally determined, or given to us in the problem statement, or something we are solving for from the problem statement.1214

Take a look at these, these give us some idea how these things work.1226

Notice, μs and μk can change very greatly, the difference between cast iron when it is moving and when it is static, is vast, it is almost a tenth of what it start off as.1234

But rubber on concrete, it is not much of a change, it is still a change, but it is not giant.1248

Ice on ice, once again, pretty large change there.1255

Teflon on Teflon, Teflon starts off with a very low friction coefficient, but it stays the same whether it is moving or whether it is still, Teflon is the stuff that goes on to non-stick frying pans. (Teflon is actually a brand name, no one ever recognizes the chemical name, unless they learned it before in chemistry.)1258

This gives us some idea of what it is, we start to see that μs is almost always larger than μk, sometimes they are equal, and like I said before, there are few freaky cases where μk is larger.1277

It really can vary what it is, we see massive changes from 1.1 to 0.04, we can have even higher than 1.1, grip of a rock climbing shoes on rock is going to be even larger than 1.1, μk can get very small, μs can get very small, really depends on the situation.1301

You have to get it in the, either the problem statement, for most part we see numbers between 0.2 and 1 as the very highest, but for very slippery objects, we will see even lower, it has to do with what we are getting in the problem, and the specific materials we are working with in our case.1324

One special thing to talk about, is wheels.1341

How do wheels work!1343

So, you might think at first at wheel are going to have kinetic friction between the road and themselves, because they are moving.1345

Not actually true.1353

One special thing to note is that, when a wheel rolls along a surface, it is going to use its static friction, not the kinetic.1356

Now, why is that?1362

When the wheel is rolling, at the moment of contact, consider this sort of like flash forward thing, you have got some point here, and then that point is here, and then that point is here.1366

At the moment of contact, when it is right here, when it is on the ground, it is actually still because it gets laid down, and then it gets picked back up, it does not move relative to the ground until it is off of the ground.1382

If we have got this perfect circular wheel rolling, the wheel is not actually going to wind up having any friction on the ground.1396

In reality, the contact patch just moves slightly, but we are talking extremely small rolling resistance.1403

For instance, 0.001, that sort of scale, very small.1408

So for our purposes we can pretend that there is no friction from a rolling wheel, if it is able to stick to the ground.1413

Static friction is what you use for a wheel.1420

Notice, this does not mean that a car is being slowed by friction to the wheel.1425

Static friction can be very large, numbers like 1.0 for a wheel on concrete in dry conditions, but that does not mean that the car is taking all that out.1429

In fact, because it is being put down , and then it is moving off, it never moves, it is never trying to be moved around, when it is on ground, it is like it is practically still.1440

It is perfectly still from the point of view of the tyre at that moment.1452

That piece, that dot, does not start to move away until it is off of the ground.1455

Once it is off the ground, it can move around, because it is not going to have any friction.1459

So the only thing that creates friction is that tiny contact patch, and because that tiny contact patch is picked up before it moves relative to the earth, relative to the road, it is not going to give us any frictional force on our car.1463

So on the contrary, the fact that it is the static friction is what is going to allow the car to move smoothly, and experience practically no friction.1476

I have included bearings, and good oil, it being able to have a good wheel system, you are going to be able to have a almost frictionless motion, and you will be able to have all the motion to the car translated easily as it is running frictionless.1484

At least, that is what we would hope.1497

In reality, there is going to be some slight friction, because nothing is perfect.1500

But, it is going to be pretty darn good.1504

It is going to be way better than if we just had a metal body on the ground, that we are shoving along.1506

So, we will be able to experience effectively no friction, while it rolls along the ground in a straight line.1511

When the car turns, and tries to change its velocity, either by accelerating, so it is going to have those contact patches spinning up, because they are going to be moving faster than they were before, and this is a little complicated to think about.1516

But the acceleration, the force, it is the frictional force that allow the car to get that traction, which is why you sports cars, racing cars have really big flat large wheels, because they want a big contact patch, so they can get lots of force into the earth, where as cars that are trying for efficiency tend to have much thinner wheels.1530

They are going to have less contact patch.1551

If you want to be able to get a car that gets better fuel efficiency, you pump up the wheels a little bit heavier, because that will make them firmer, tighter, and will be able to have a less contact patch on the ground, which means they will have a little bit less friction.1553

Remember these are very small numbers, if you are driving at 100 miles, it can have an effect.1565

Or if you were to turn, that is when friction is going to come into play, normally you would have the wheel running like this, but then if we want to turn, the wheel is going to turn like this, but the motion of the car is going to be like this.1570

So, normally your wheels are going like this, and we have effectively no friction.1591

If instead we turn, the car has two choices, if it were to keep going in this path, then all of a sudden, friction will be breaking its contact patches with the earth, because that is not the direction wheel wants to roll in.1594

Instead, it is going to go this way.1610

So, if the car were to keep going this way, it would break friction, friction would fight it.1612

So instead, it goes this way, which means that friction is going to wind up actually pulling this way.1618

This is a little bit complicated to think about, but the force of the wheels, friction is the only thing that connects the car to the earth.1622

The car and the road are connected through the friction of the tyres.1630

So when you go into a turn, the thing that pulls you into the turn, is going to be the friction of the wheels on the ground, and it is going to be μs.1634

This is a lot of explanation for something that does not seem to make sense, but if you want to be able to understand how a car rounds a corner, like we will in the section when we talk about uniform circular motion and force, we are going to be actually understand this.1641

So this stuff actually matters, it is a little complicated to think about at first, but it will make sense.1654

If something is going to be rolling, it effectively has, static friction, it effectively has no friction, because it is going to be putting in its contact patch, and lifting it up.1659

But if it wants to have an acceleration, that contact patch only has to move relative to the ground, otherwise, the rotation movement of the wheel is going to change the speed that the wheel is moving along.1667

So its going to require friction to be the interaction, the interplay between those two things.1678

Let us finally start talking about examples for the normal basic friction.1683

We have got a block of mass 10 kg, resting on a flat surface, horizontal force acting on it.1687

It just barely begins to move, unsticks at the force F = 60 N .1694

What is μs?1699

First let us do a free body diagram.1700

What forces are acting on it?1702

There is Mg, pulling down, FN = Mg, (flat surface, nothing else pulling down, so they are going to cancel one another out.)1704

Now we are going to work to figure out what the friction is, and we know that friction is going to pull this way.1716

If μs, (we are dealing with static friction, the moment of unsticking, that razor's edge between staying still and just beginning to move, is going to be the maximum static friction).1721

The force is going to be equal to the maximum friction, for it to unstick at 60 N.1746

What is the maximum static friction?1751

That is μs × FN = μs × Mg.1753

What is F?1764

F = 60 N.1766

For it to just unstick, we know that the maximum static friction had to just be, barely on that razor's edge, where they are just equal.1770

As soon as you surpass it, you flip it to motion, you switch over to kinetic.1778

So, that precise moment when they are equal, is the moment of unsticking.1782

Now, plug in our numbers, so, μs = 60 / Mg , we know 'M' and 'g', so, = 60/(10 × 9.8) = 0.61.1787

So for this case, between this block and this block and this surface, we have got μs = 0.61.1810

Once it starts to move, it still has that force of 60 N on it, and now it has an acceleration of 1 m/s/s.1816

We know that the sum of the forces, = mass × acceleration.1823

We know what 'a' is, we know the forces operating on it.1830

The force, up here, = 60 N - μk × Mg = Ma = 10 × 1 .1835

We get, 60 - 10 = μk × Mg, 50/Mg = μk , μk = 0.51.1865

There is our answer for what μk is.1893

Next example: For this example, we have got a block resting on a surface that can be tilted.1896

We have got some tilt, θ on our surface, μs = 0.35.1902

What angle θ will the block barely begin to slide, what is that instantaneous, that razor's edge, that break over point between staying still relative to the incline and suddenly starting to move along the incline?1912

Notice, for this problem, we do not have the mass of the block, but it turns it we are not actually going to need it.1920

So, we have got a block, it is going to weigh some 'm', so mg is the pull of gravity on it.1931

How much of this is going to be perpendicular?1938

The perpendicular force, is going to depend on what θ is.1942

How much is the parallel force?1950

That is also going to depend on θ.1953

What is θ , we can figure it out by referring to that old lecture that I had on Newton's second law in multiple dimensions.1954

We can also see that in the extreme case (90 degrees), then we have that this is going to be sin(90), so we would have all parallel.1963

In the case of 0 degrees (other extreme), we would have cos(0) =1, so all perpendicular.1974

So we see that θ has to go here, but we can also figure that out by other geometrical means. (This can be a real confusion down the road, so it is good to refer to.)1981

And there is lots of problems that involve incline, so it is really important to have a good understanding of how this works.1995

This is mg, so we have, the perpendicular force of gravity = mg × cos θ .2000

How much is the normal force?2013

That is going to keep it from bursting through the incline, so FN = Fg(perpendicular) = mg cos θ .2015

What is the parallel force?2026

The parallel force = mg sin θ .2028

(we are able to do this because we have a right triangle, so these two things are perpendicular, so we use basic trig.)2036

At this point, what are the forces acting on it?2044

At this point, we have the parallel force going this way, mg sin θ = gravity parallel.2047

What other thing is operating on it?, friction!2057

Friction is going to be pulling backwards.2061

We are going to be using static fricton, because the block starts off at rest.2063

What is the moment of flip, is going to be when that maximum static friction, is just equal to Fg (parallel), that is the razor's edge, the moment of flipping.2068

What is the maximum static friction?2086

That is μs × FN = Fg (parallel) = sin θ × mg.2088

What is FN, μs × mg × cos θ = sin θ × mg, (we get mg on both sides, that is why we do not need to know the mass, cancel on both sides.)2101

μs × cos θ = sin θ , collapse it into one trig function by dividing by θ2122

Since sin θ / cos θ = tan θ , we get, μs = tan θ.2131

Now, we have done this in general, if you want to know what the angle is, we plug in numbers.2137

We get, 0.35 = tan θ , taking arctan on 0.35, θ = 19.3 degrees.2143

That tells us what this specific angle is.2166

But it also tells us in general, if we want to do this for anything, that is a really easy to find out what μs is for any pair of objects.2168

For any material on some surface, you measure the angle and just keep very slowly tilting it until it just begins to move.2176

Really easy way to experimentally derive what μs is going to be.2185

Example 3: We have a block against a wall, and it is sliding down.2192

Between the block and the wall, μk = 0.2.2196

How hard do we have to push against the block, to cancel out gravity, to give it a constant velocity?2202

If we push in with some force here, what is the normal force?, it is just going to push back with the exact same amount, so FN = whatever force we put in, in terms of magnitude.2209

What is going to be operating on this in addition?2223

We got gravity pulling down by mg, friction pulling up by some amount that is going to be connected to μ, and the normal force.2225

Are we moving, or are we not moving?2242

In this case, we started off knowing that the block is sliding down, that means we will be using μk.2244

What is the force of friction?2252

Friction = μk × FN .2253

That means, 0.2 × FN (FN is the amount that we push in, that is the amount the wall has to resist.)2261

We get, μk × F .2272

If we want those two things to cancel out, we want an acceleration = 0.2280

That means that sum of the forces, is going to have to be equal to 0, because, ma = 0.2287

That is the way we are doing it right now, is we know that we are in equilibrium, because there is no acceleration.2304

It is going to have a velocity, it is sliding down, but we know that there is going to be no acceleration.2309

The net of the forces, we have gravity, friction; we also have in the horizontal direction, the force that we are pushing, and the normal force, but they cancel each other out.2316

We do not have to worry about that, because it is just staying parallel to that wall, so all we have to worry about is the things that can have an effect, in this case gravity and friction.2327

Let us say that up is positive, so, frictional force - mg = 0.2339

So, what is the frictional force?2350

It is μkF - mg = 0, μkF = mg, so, F = mg / 0.2.2352

In this case, we get, F = 5 × mg, so the amount of force that we need to push it, to keep a constant velocity or keep it still (then we use μs) is going to be dependent on the coefficient of friction.2375

In this case, 5 × (force of gravity).2413

Last example: This one will definitely require some thinking.2417

We will start off thinking about the problem and then actually approaching it.2424

It is a great way to approach a problem in general, think about it, then approach it, then actually do the math.2427

We have got 2 blocks of masses, M1 = 2 kg, and M2 = 1 kg.2432

They are sitting atop each other, they have μs = 0.7 between them.2438

The bottom block is resting on a horizontal frictionless surface.2444

What is the minimum force to keep the top block from slipping?2447

First of all, if they are moving at the same rate, what does that mean?2452

That means we have got some a1 acceleration, we have got some a2 acceleration, and they are both going to be moving in the direction of force.2459

These are our accelerations, going this way.2468

If they had different accelerations, then one of them is either going to be sliding off the other, or sliding behind the other, there is going to be a difference in their relative velocities, which means that they cannot be staying together anymore, they have to be slipping, by the definition of slipping.2477

That means, just to begin with, we know that the acceleration of the first block , has to equal the acceleration of the second block, so we can call them in general, 'a'.2494

What keeps block 2 on top of block 1?2508

Ther is nothing, no forces we are putting in externally, the only force that is keeping it there, is the force of friction.2514

M1 is moving this way, that means that for them to stay attached, static friction wants them to stay in place, M2' friction is going to pull this way.2522

So, we got friction moving this way.2532

So, friction is going to be pulling block 2 over, what about block 1?2541

Resultant force, so M1 is going to be reduced by that same friction, these will be equal in terms of magnitude, not direction.2545

So, M2 is going to be accelerated by friction, M1 is going to be decelerated, or at least lose some fo its force to friction.2558

That gives us an idea of what we are actually doing here.2565

We have got these 2 blocks, they are pulled along, only the bottom one is being pulled along, and the way it is able to communicate with the top one, the way that it is able to cause it to move, is by using friction.2568

The bottom one and the top one, they only communicate by friction, so friction has to be the way here.2579

If we are able to put so much force, this makes sense, I am sure you have seen it, if you have got 2 books on top of one another, we yank the bottom book really hard, the top book will just fall down, whereas if you yank the bottom book really slowly, they will both slide along easily together.2584

So it is going to be connected to the coefficient of friction, and the masses of the books.2598

What is going to be the minimum force to cause the top block to slip?2602

What is the maximum force to keep it in place?2611

It is going to be the same thing, that razor's edge once again between slipping and not slipping.2613

But, now we have got the understanding to actually approach this problem.2617

Final thing, now we can actually do the math.2622

Let us start looking at the two free body diagrams.2626

In this case, we have got some force.2629

What else is operating on it?2632

We have got friction, from the top, what about its own friction?2634

Does it have its own friction from the ground?2640

No!, remember, we said that it is on a frictionless surface, so in this case, there is only one friction, there is just the friction between the blocks.2643

What is mu;s?2656

We know μs; what is normal force?2658

How hard does M2 push in!2660

M2 is going to push in with M2g, so, FN = M2g, because it does not burst through the box, it does not move through it, it stays on top of it, so the normal force has to equal M2g.2663

With this in mind, we can start coming up with our formulae.2682

Net force is, we know that, F - friction = M1 × a . (We can use vectors, but we do not have to be, because we understand the directions, because they have been dealt with.)2685

So, F - friction = M1 × a .2717

What is the forces on M2?2720

Just friction, = M2 × a .2725

With that in mind, we can start to figure out what is F going to have to be equal to.2730

F - M2a = M1a, so, F = (M1 + M2) × a .2734

That is how much force is necessary to give an acceleration of this, because it has to move both the objects, the whole system.2755

We can sub that back in, we can now figure out what is friction.2763

F/(M1 + M2) = a , plug that into this formula right here,2768

We get, the friction = M2 × F/(M1 + M2), so, friction = M2 × F / (M1 + M2).2780

Now we could sub these things in, we could figure out what are the actual numbers, what also is friction.2807

In this case, friction = F / (1+2) = F/3, because that is the amount that it has to get, it is the share that the top block has to get, because it has one third the total mass of the system, so it has to get equal share for its mass, to be able to move it with the same acceleration.2813

So friction has to be equal to F/3, for the acceleration to be the same between both the objects.2841

What is the maximum amount of friction?2847

Remember, that it is going to have maximum friction, (static friction, because we are static here), is going to be the maximum velocity without slipping.2849

Once again, it is that razor's edge, so the minimum velocity for slipping is going to be that flip over point, the maximum velocity without slipping is going to be the same thing as the minimum velocity of slipping, if we just go an infinitesimal amount over, we are going to start to slip.2863

So, the maximum static friction = the maximum velocity that we can move at.2886

The maximum force, maximum velocity, maximum acceleration,2891

ACTUALLY I SHOULD NOT HAVE SAID THE VELOCITY, THE MAXIMUM ACCELERATION, my apologies, you can of course have any velocity, it could be whizzing along in space, a million miles per hour, it does not matter.2894

From its point of view, it is not experiencing any force, so it is about the maximum acceleration.2906

So, maximum force.2912

With all that in mind, it will slip at the moment, when, F/3 = max. static friction.2915

What is max. static friction?2928

μs × M2g, so, F = 3 × μs × M2g.2930

Plug in numbers, 3 × 0.7 × 1 × 9.8 = 20.58 N.2951

That is how much it is, to finally get the thing to just start moving, if we just barely see 20.58, that is the razor's edge, slightest bit f difference off 20.58, and it will just start to slip, because it will just exceed this maximum static force.2976

Hope friction made sense, if you got difficulty in understanding how an incline works, definitely refer to Newton's second law in multiple dimensions, it will give you an understanding of how to deal with parallel and perpendicular forces, it is important to understand that when you are dealing with friction.2992

Hope you enjoyed it, see you later.3009

Hi, welcome back to educator.co, today we are going to be talking about force and uniform circular motion.0000

Last time we talked about objects moving in a circle, we talked about uniform circular motion, we realized that for something to be able to continue moving in a circle, if an object is going here, here, and keep up the uniform speed, it has got to have an acceleration pointing to the middle of the circle at all times.0007

We realized that we needed this centripetal acceleration, an acceleration pointing in, centripetal - towards the centre.0024

From Newton's second law, we now know that for any acceleration to exist, we have to have a force creating that acceleration.0032

This implies that there has to exist some sort of centripetal force.0041

What is that centripetal force?0045

How much is it?0048

From previous work, we know that, force = ma (Newton's second law), and we learned that centripetal acceleration's formula was, acceleration = (speed)2 / radius . (radius of the circle)0049

Remember that the acceleration always points to the centre, it is the centripetal acceleration, to the centre of the circle, wherever your object is on the circle.0065

Putting these together, we get that the magnitude of the centripetal force has to be equal to, mv2 / r ,and it always points towards the centre of the circle.0074

Here is some examples of centripetal force in action:0088

When do we see it in real life!0090

Anytime we see rotational movement, centripetal force has to be in place.0092

If a rock is spinning on a string, what is keeping it in that string?0096

If we have got a rock, and it is spinning, and the thing that is keeping it, that force, is the tension in the rope, if that it is horizontal, if it is vertical, it is going to be combination of tension and gravity, sometimes working together, sometimes working against each other.0100

If a car is rounding a corner, then the force of friction on the tyres, is going to be the friction of the tyres, has to be pulling in to the centre of the circle, throughout the corner.0113

A roller coaster doing a loop, what keeps you at the top?0124

It keeps on the top, because the normal force of the car, at the speed that you are going through, during that top of the circle, and that radius, that is what is going to hold you at the top.0128

An airplane banking in the air, turning, banking like this, and pulling up, if it is turning in the air, similar to the car rounding the car, but now it is the force of pressure, the air pressure, and the fact that the winds are making a certain shape, allowing it to force itself around.0136

It is able to use air pressure and friction, through that it is able to turn it through the air.0153

Where does centripetal force come from?0160

Remember, in all the examples that we talked about, centripetal force was not just inherently present.0162

To have that circle, we needed centripetal force.0167

To have that rock go in a circle, we needed that rope to provide tension.0171

To have that car turn in a circle, we needed those tyres to have friction.0175

To have that roller coaster stay atop, we need the normal force keep it pushing in.0180

Centripetal force is created by other things, centripetal force does not get added in, we just know that the sum of our forces has to equal the centripetal force we are dealing with.0186

So all those previous examples, there is always some force keeping the object moving in a circle, there is always some force satisfying that centripetal force relationship.0197

Centripetal force will not show up, we will not see it in our free body diagrams directly, we see it as the sum of the forces will have to be this centripetal force, otherwise you cannot have a circle.0204

You have a circle, it means that you need a constant centripetal acceleration, to have a constant centripetal acceleration, you need to have a constant centripetal force.0214

But, that does not guarantee that you are going to have that.0221

It is just the qualification.0224

If you know something is in a circle, then you know it is a centripetal force.0225

If you have a centripetal force, then you know it is going to be a circle.0228

But it does not create the centripetal force, it has to be given by something else.0231

So, it is a relationship that must be upheld by the forces acting on the object, to maintain the circle.0236

We do not get centripetal force, it is created form what we already have.0241

Centrifugal force:0246

You probably have heard of the idea of centrifugal force, at some point.0248

Centripetal points into the middle of the circle, so centripetal points in, whereas centrifugal points out.0251

So, centrifugal, and centripetal, like that.0263

Centripetal points in, centrifugal points out, centripetal meaning centre seeking, centrifugal meaning centre fleeing.0271

You may have heard before that centrifugal force is not real, that it is a fictitious force.0278

Sort of!, the reason why people call that a fictitious force is, say you are in a circle, and you might have seen these at a fair.0285

You are in a circle, and they pt you up against the wall, and you are spinning really fast, you get pushed up against it, because of centrifugal force, sort of.0292

What is really going on, is the combination of friction and the centripetal force, is keeping you, the fact that your body wants to move in a certain path, and then the wall is there, and it is spinning, all these things combined to give you a centripetal force created by other things, centripetal force pulling into the centre, so you get pushed against the wall.0303

But, when we experiences it, it feels from our point of view that our back is being pushed by a constant pressure, that we have got this pressure of the wall pushing on us, constant, and just that it is down.0322

Because from our point of view, we are not moving, the circle is spinning, but it is like we are still, and the world around us is spinning.0335

And that is to do with the reference frames.0341

So this is an example of a non-inertial reference frame.0343

A reference frame where laws of inertia, laws of Newton's laws, they do not hold like they used to.0346

So, it is important to pay attention, we have to step outside and we have to observe the circle moving from outside, otherwise things get to start really funky.0351

That said, centrifugal force is not necessarily false, it is just, this is not quite the right way to think about it.0359

It is not that the force is constant, and just pushing down, and created out of nowhere, it is created out of a variety of other things, it is created out of the centripetal force, it is the reaction force, it is Newton's third law in action.0365

Remember Newton's third law?0377

Equal and opposite force stuff?!0378

That is exactly where this is coming from.0380

If we have a centripetal force keeping an object in a circle, say we have got that circle, and we know that there is a centripetal force, of some amount pulling in.0381

Based on Newton's third law, there is also a force of this much, pulling the other way.0390

Which one do we experience in our bodies?0397

We experience centrifugal force.0401

So, why does your body feel centrifugal force, and not centripetal?0404

Human body is built to feel pressure, the way our skin cells, the way our nerve cells combines together to work, is one, something pushes on you, you feel the push, you do not feel the other force involved.0410

When you push on a book, you do not feel the book pulling away from you, you do not feel the push on the book, you feel the push on your hand, you feel the reaction force.0423

When you turn a corner in a car, you feel the car, the side of the car push on you, even though you are really being pulled to stay in the circle.0432

There is two forces at hand always, but you only feel one of them, because of the way your body is designed to sense things.0440

So, that is just why we feel it.0446

But the centrifugal force is there, centrifugal force is just the reaction to the centripetal force, it is the reaction force.0449

It is Newton's third law in action.0455

So, centrifugal force is not false.0457

It is a real thing, but the way that it gets explained and talked about, sometimes, that is not always true.0461

Sometimes, people talk about it as if it is its own thing, but really it is because it is a matched pair, the centripetal force.0469

We talked about centripetal force as really a force in itself, so it can get a little bit confusing, but we understand that centripetal force is created by other forces, and centrifugal force is the equal and opposite half of that, that is the other side of the coin, iti allows us to see what the other part is, it allows us to see what our bodies feel.0473

So we have got plenty of understanding to start doing some problems.0493

We got a car of mass, m = 1500 kg, and it makes a quarter circle turn of radius = 200 m, while maintaining a constant speed of 20 m/s.0496

What is the centripetal force on the car?0509

We just pump it all into our equation, F = mv2 / r = 1500 × (20)2 / 200 = 3000 N, to maintain a constant speed, that car has to experience 3000 N of force pulling it in.0512

Where does that force come from?0537

That force has to come from friction of the tyre on the road.0540

The only interaction that the car can make with the world around it, is through the road, through friction, through what is touching it through the wheels.0543

So, the friction of the wheel son the road is what creates it.0551

Given that the coefficient of friction between the tyres and the road is, μs = 0.8, because remember, we talked about in our section on friction, μs is what we use for tyres, because they are in constant contact.0555

The contact patch is effectively still on the ground.0568

So we use μs.0572

What is the maximum speed the car can take the turn out without slipping?0574

This will take a little bit more thinking.0577

Here is all our important stuff, we got the radius, the coefficient of friction (static), what speed will it slip at?0580

In this case, we are not going to need mass, we will see in a few seconds.0588

So, let us look at it in a top-down perspective.0592

Here is our car driving along, and the car needs to have a curved path.0596

That means, for the whole time, it needs to have static friction pulling in, like this, at some certain amount.0600

The static friction, otherwise the car will just slip, because if all of a sudden it were not able to do that, its wheels would not be turned at some angle, for the turn.0608

The wheels, here is one wheel of the car, if instead it were to just escape, it would not be rolling anymore, the tyre would slide along the ground, it will start going to a skid, a bad thing of course.0620

But it does not want to go to skid, it wants to stay at rest, friction wants to keep two materials together, wants to keep them from sliding.0634

It does not want things, it is force, but friction is going to keep it together as opposed to letting it slide.0648

As long as we do not exceed the maximum static friction, will be able to keep that turn.0651

How much is it?0657

The maximum static friction is going to tell us what the maximum centripetal force is.0658

The maximum static friction is the most centripetal force we can get.0667

The most centripetal force, assuming that our mass and radius stays the same, which they do, is going to be, the maximum velocity.0673

What we have to do, is we have to solve for what is the maximum static friction going to say about that maximum centripetal force.0682

Maximum static friction, μs × FN , this case, car is flat on the ground, so, FN = mg , since the car is flat on the road, when it turns it is going to still experience its full normal force.0691

So, the maximum static friction has to be equal to the maximum centripetal force.0713

What is the maximum centripetal force?0718

It is going to be, m × v2 / r .0720

We get, μs × mg = mv2 / r .0727

Cancel those m's, and now we get, μs × g × r = v2 .0735

Taking square root, we get, v = sqrt( μs × g × r) = the maximum velocity.0747

So, what is that maximum velocity?0755

Just when it is on that razor's edge of slipping, what is it?0757

Punching in numbers, v = 39.6 m/s = the maximum velocity that it can take without starting to slip.0760

At the same time, that is also where it starts to slip, because it is hard to stay precisely on 39.6, you might accidentally go to say, 39.6000000001 m/s, you cannot stay precisely.0785

So, that is the moment of slipping, just when you pass 39.6, that is when the car suddenly lose traction, and that is really bad, because we have been relying on the fact that, if we had static friction, once we exceeds static friction, (as we know from our talk about friction), you flip into using kinetic friction.0796

Kinetic friction for a tyre is considerably less than static friction, that means you had a certain amount of control, and then you have even less control, once you start to slide.0815

So, if things start to slide, you are on a slippery slope, things are going to get worse than worse.0825

Also, something to keep in mind, we are assuming that μs = 0.8, that is a reasonable amount for a car tyre on a dry, clean road.0830

What happens if all of a sudden, you hit a patch of wet water, or it has been raining recently, or there is an oil slick on the ground?0839

Your μs could drop to something really low.0847

If it was just wet, it is perfectly reasonable for it to be 0.4, and if you hit like, a patch of grease, or hydroplane (causing almost no friction), there is a large pool of water and you go at high speed, suddenly your friction force is going to drop way down all of a sudden.0850

That means you have got that much less control.0868

Yu \$mu;s drops to 0.4, all of a sudden, your top speed, you can take that corner at without starting to slide out, without skidding and fish-tailing, is much lower.0870

This is an important reason, this is why you have to drive carefully when it has been raining, is because you could rely on a dry safe road and go fast, when it is dry and safe.0880

But if it is wet, all of a sudden, the top speed that you can take a turn at drops massively just because of the laws of Physics.0890

There is no way around it, it is more dangerous, because the maximum forces that you can attain with your car becomes much less once it is wet.0897

There is less friction to go around, so less forces to around, which means that speeds that you can go at, have to go down.0905

Example 2: We have a bucket full of water, and it is being spun around in a vertical circle on a rope that is 1 m long.0914

How fast does the bucket need to spin to keep that water from sloshing out?0921

We have almost all certainly seen this demonstration.0927

You have some bucket on a rope, and you spin it really fast, and the water inside the bucket will not fall out.0929

If you spin it fast enough, it gets held in.0935

One way to talk about it being held in, is the centrifugal force.0938

We can say that it is the centrifugal force that is holding it in.0941

But we can also think about this problem, we do not even actually need force to solve it, it just helps us to have force to think about it.0943

Let us say that the total centripetal force needs to be this long a vector, pointing in the top.0951

We are going to need to make the actual distance be our, the actual distance is going to be this length here.0961

The distance is the length, would be the magnitude for the vector.0971

Say you need this much centripetal force, to stay in that circle.0975

If you need this much centripetal force and gravity is going to pull down by this much, then we need, so here is 'mg', 'mg' is definitely guaranteed, as you are going to the top, there is going to be some weight pulling you in.0983

You are going to have it, because you are on Earth, you are in a vertical circle, so throughout the circle, you are going to have some 'mg' pulling you.0998

But if you are at the top, then we need more.1006

This is longer, so we need extra bit and we get that from tension.1010

The tension in the rope is what is going to keep it, extra, so that is going to make out for that extra amount of centripetal force that we need.1015

So we are going to have some tension in the rope.1023

What happens all of a sudden if we need less centripetal force?1024

If we are going at a lower speed, and we only need this much centripetal force.1030

Say this is the amount of centripetal force we need.1035

If we need this much centripetal force, then we do not need any tension at all.1038

There is no tension whatsoever.1043

But, we still have all this gravity.1046

So this gravity suddenly, we have got this much 'mg' and this much 'mg', so 'mg' for the centripetal force, and then 'mg' left over.1050

We cannot get rid of it, if it is not being, you know, used to keep it in the circle, it is still going to have an effect on it, so all of a sudden, this bucket is going to fall out.1066

You do not take the top of it at a fast enough speed, and your spin decays, your circle decays, and you fall out of your orbit, you fall out of going around that centre point, and the bucket is going to fall out, if we do not have enough.1078

What we need is, we need to have enough centripetal force, so that all of 'mg' gets used up.1092

We need to have the centripetal force be greater than or equal to the weight.1100

Because if it is less than 'mg', then that means we have some 'mg' left over, and we do not, we still have to use 'mg', it is the Physics, it does not get like, "well, we have this remainder, we do not need it, we will not use it", No!, weight is still going to have an effect, it is still going to pull the bucket down, so it is going to pull it out of a circle.1108

We have to need all of the weight, we can also go over the weight, and then we will make up the extra with the tension in the rope.1123

But for it to stay in it, we are going to have to have the centripetal force be greater than or equal to 'mg'.1133

Smallest centripetal force that we are allowed to use will give us the smallest speed.1138

The lowest, how fast the bucket has to go at a minimum, to keep the water from sloshing out.1144

Basic idea that we come up from all of this, is we got the fact that the centripetal force has to be greater than or equal to mg, or it sloshes out.1150

Otherwise, we will have left over gravity, and that left over force of gravity will still have an effect, it will pull out the water, it will pull out the bucket from the circle.1168

So, to keep the water in the bucket, to keep the bucket in the circle, centripetal force has to be greater than or equal to mg.1177

We want to find out what the smallest speed we can go is, then we consider one centripetal force equals mg.1188

Centripetal force is, mv2 / r = mg1191

One thing to see is, we actually did not have to do this, by using forces.1205

As we can see now, we can easily cancel out those m's, because really what we are saying is, we need the centripetal acceleration, to be greater than the acceleration due to gravity.1209

Centripetal acceleration is less than the acceleration of gravity, gravity is still going to accelerate you, so it is going to make up the extra, no matter what, so you have to use all that centripetal acceleration.1219

Same basic idea, we could have approached it by just talking about acceleration, but little bit easier to see it as forces, because we have a better understanding of how force works.1229

Just intuitively, as humans, we are more used to working in forces than just the acceleration.1238

At this point, if we want to solve for this, we know the radius is, we got v, take the square root, is just, rg.1243

Plug in numbers, sqrt(rg) = sqrt(1 m × 9.8)1251

So we have got that speed has to go at the minimum speed it has to go at, is 3.13 m/s, or greater.1260

If we go at that speed or more, for a 1 m radius, it will be able to keep it in the bucket.1272

Keep in mind, if the radius changes, the minimum speed that you have to go at, keep the water in the bucket will change.1281

But for this case, the 1m long rope, we have to have 3.13 m/s or greater to be able to keep the water from sloshing out.1288

Last example: Say we have got a rock of mass 1 kg, attached to a string of length 0.5 m.1297

So, r = 0.5 m, and we have got the rock up here, 1 kg.1304

And the string snaps at a tension = 75 N, is our snap, that is when the string all of a sudden will snap.1312

So, we very slowly increase the speed of the rock traveling in a vertical circle, until the string snaps.1321

Starts off, going through slowly, then faster and faster, then snap!, all of a sudden it snaps.1329

Let us think about, at what point on the circle will the string snap.1335

Where does it snap?1339

Let us do a quick free body diagram.1341

At the top, we have got mg, and say for ease that (just to understand it graphically), we have to have a full length that is equal to the length of the radius.1343

In this case, we put these two vectors together, we get some tension pulling into the centre, some gravity pulling in to the centre, when it is at the top.1358

What is at the side, we have got gravity not really having an effect, because it is going perpendicular, so it is not doing much to us right now.1367

But we still got that tension, so now the tension has to go all the way in.1375

Clearly, when you are on the side, you are going to have more tension, then when you are at the top.1380

What happens when you are at the bottom?1384

Now, we have got mg pulling down, but we also got to have a tension, that is able to make up for mg, so we can still maintain the same centripetal force.1386

To maintain the same centripetal force at the top, gravity is working with us, it means we need less tension in our string.1397

When we are on the side, gravity does not have any effect, it means that we need just the same amount of tension that we needed as our centripetal force.1405

When we are at the very bottom, gravity is going to work the most against us, and so the tension is going to be the maximum.1412

So, string snaps at bottom.1419

String snaps at the bottom of the circle, because that is when gravity, and tension are going to be butting heads.1424

The tension has to be the biggest, because that is when gravity is working against us.1431

What speed we will have to go?1435

Now we have that centripetal force, = the net force.1437

The sum of the things, because centripetal force is only created from other things, centripetal force = net of the forces.1449

What forces are acting on the rock?1455

We have got gravity pointing down, and we have got the tension pointing up.1458

In this case, we will make up positive, so, tension - mg .1464

That is what the net force is.1469

Centripetal force = mv2 / r1471

We know pretty much what all these numbers are, except, what is tension?1482

We are looking at the instant of snapping, because we very slowly sped up to this, if we sped up to this suddenly, we might have accidentally put on a 75 N tension at a different spot.1485

That is what we had to speed up slowly, so we can be sure that it would snap at the bottom.1494

We have got that tension = 75, sub things in.1498

So, at the instant of snapping, the tension 75, we know what mass is, we know what gravity is, we know what the radius is, now we just have to figure out what v2 is.1503

So, v2m / r = 75 - (1 × 9.8) , move things around, we get, v2 = 0.5 × (75-1.98) / 1 , v2 = 32.6, take square root of both sides, we get that the speed of snapping is going to be, v = 5.71 m/s.1514

Once we get faster than 5.71 m/s, it will snap at the bottom of our circle.1585

As soon as we get to 5.71 m/s, and the rock passes through the bottom, that is the, just enough tension to create 75 N pull in the string, and snap it.1590

Hope everything made sense, hope you learned a lot.1600

Hi, welcome back to educator.com, today we are going to be talking about work.0000

This is going to be our first introduction to energy.0004

Introduction to energy, but we are talking about work, you thought that this is going to be an introduction to energy, so what is going on here!0008

Do not worry, both these things are actually really deeply connected.0015

To be able to talk about energy, we are going to wind up talking about work, and to talk about work, we are going to have to talk about energy.0018

But, first we can tackle the idea of work on its own, it is going to really help us understand energy.0022

Bear with me, and let us just learn about work as its own idea, and then we will move on to using it as part of energy.0026

Before we can rigorously define what work really means, let us look at a couple of scenarios before we try to figure out what we want, how we want to use work, what we want to find it as.0034

Let us say we have got some 10 kg block sitting on a frictionless table.0045

We are going to have a whole bunch of different scenarios, we have four different scenarios in the next page, and each one of those, we are going to slide the block some distance with some force, but these will vary in the scenarios, and we will talk about each one.0051

In our first one, we have got 'f' and 'd'.0062

We push it from here, to here, and we do that with that much force.0068

So, that is one thing, but what if we did it for this much distance?0073

Same force, but a much larger distance.0077

If that is the case, by the end of it, because it is frictionless, it is going to pick up speed.0081

Remember, the longer the force is going, the longer the acceleration, so it is going to have more speed in it.0085

So, it makes more sense to think that, the longer the distance, the more energy, the more work that we put into it.0090

You push for a longer period of time, it makes more sense that you are putting more work.0095

If you had to do something farther, that is more work than having to do it for less distance, if you are putting in the same force every moment.0100

What if instead, if we had a really big force and a small distance?0107

Clearly, between this box up here, and this box here, we are going to have a much bigger amount of work put in to the yellow box, the bottom box, the big force and the small distance, than the small distance and the small force.0110

It is not going to go as far, it is not going to move as far, at the end of that distance we will put way more work into it, because we are pushing so much harder.0131

If we push lightly on an object for a little distance, versus we put our entire body against it and push as hard as we can for a little distance, at the end of it, the thing is going to be moving a lot faster, makes sense to think of it as putting in more work.0139

What if we put both of them together, if we had a really big force, and a really big distance!0150

Clearly, that is going to be the one that has the most work.0155

We push really hard, for a really long time, that is going to be the most work.0158

So, it is about force and distance together.0162

More force means more work, more distance of pushing that force means more work.0164

Put both of them together, and that is even more work.0169

What if we consider two different blocks with masses that are different?0173

Same force in both cases.0177

So we have got the same force, and the same distance, but in this one, we have got a really big mass, whereas in this one, we have got a really small mass.0179

Notice that, because F = ma, the amount of acceleration is relative to the masses, so one vision: we have got a low mass object traveling really quickly at the end of its distance.0192

The other one, we got a high mass object traveling really slowly at the end of its distance.0202

But, we had to push the same force for the same distance.0207

So from the point of the view of the pusher, it is the same effort that we had to put into it.0212

They changed the system of mass, having to do with the speed that it is going at.0216

The two things are connected.0221

But, from the point of view of the pusher, it is the same work, it is the same amount of push for the same distance.0222

These are the same thing, so even though the result is different and how it comes out, it is a different mass with a different speed, we are going to have the same pushing, the same work in it.0231

So, we are going to think of work as just force and distance connected together.0242

The mass is not going to have direct effect.0246

These are different results, but it is the same force that is put into the system.0249

With this idea in mind, we have got a notion that work is force involved multiplied by the distance.0253

Way more force and way more distance stacks up hugely.0257

More force means more work, more distance means more work, these things that make good intuitive sense.0261

If we had to push a car for 1 foot versus push a car for 1 mile, or push a car for a metre versus pushing the car for a km, clearly the really bug distance is going to be the one that is going to take more work, if you are pushing it with the same force the whole time.0266

But there is one last thing that we have to consider before we really define what work means.0282

Consider the idea that there is a giant 20 tons semi-truck, which I will illustrate with an incredible box as my figure, and you are standing in front of it, and you are pushing as hard as you can, you push so hard on that, but it is this huge 20 tons semi-truck, it does not move at all, it does not budge even a mm, so did you do any work?0287

In one way, you definitely strained, and you put a lot of effort into it, you tried really hard, so the idea of looking from the point of view of the pusher is one thing, but what we really want to define is, we want to look at work as the way you change the world around you.0311

Even in the case of boxes with different masses, we were changing the world, we were putting a velocity into it, which was not there previously.0325

We created acceleration, we created change in velocity, by putting that work in.0330

In this case, you put a lot of effort in, you tried really hard, you push and strain, but nothing happens.0335

So we want to define work as 'change in the world'.0342

You have expended a lot of effort, but you did not change anything.0346

So, we are to going to define work as change in the world, so no distance, if you did not make any distance, even if you had a huge force, no distance means no work.0349

It is force × distance, zero distance means no work, even if it is a giant force.0357

With this idea in mind, there is one thing to consider.0363

We have got a block, and this lock moves in this direction.0368

But the entire time, we have got a force moving this way, perfectly perpendicular to the motion of the block.0373

Does the force do any work on the block?0382

We talked about the fact that, if you did not change anything, then you did not put in any work.0385

If there is no change from the force, then the force does no work.0393

Force has to be connected to the distance.0397

So in this case, no work is done by the force, because it is perpendicular.0400

That motion to the side is going to happen whether or not the perpendicular force does anything or not.0404

It is not able to change its distance, because it is perpendicular.0410

The only way it would be able to change its motion, is if the motion was going somewhere like this, but the entire time it slides along, the effect of the force has no acceleration, because its motion is this way only the entire time.0414

So, this one does not happen.0427

Force and distance are perpendicular, so from that we see that the force does no work, it does not change the motion of the object, no change means no work.0433

Forces perpendicular to displacement, contribute no work.0443

Force has to be at least partially in the direction, the amount that is perpendicular will contribute no work.0447

With all this thinking, we got things down pretty well.0454

It is the length of the displacement times the amount of force parallel to the displacement, the amount that is perpendicular has no effect.0457

Now we can finally create a formula.0465

If we have got some object, it does not matter what the mass is, remember the mass has an effect on the outcome of what happens in the world, but the work that is put in, is going to be the same whether it is a tiny mass or a really large mass.0468

The work = the size of the force × the size of the displacement × cos θ , θ is the angle between the two.0480

Why is that?0490

Remember, basic trick, since this is the hypotenuse, and this is the side adjacent, that side = force × cos θ .0491

So, the amount that is parallel, is going to be F × cos θ , so the amount that is the sin θ , the side opposite has no effect, so we can completely get rid of it.0505

So the only one we have to care about is the force × cos θ and that appears here, and here, and then we take the amount f the displacement in here, so the work = force × the distance × the cosine of the angle between the two, and that tells us what the work is.0515

That is the formal definition of work, it allows us to look at all the ideas that we have talked about so far and make sense of them.0534

One alternate formula you can use, in addition to force × distance × cos θ , we can also formulate it as a dot product from math.0541

The work = (force).(distance), (as vectors).0549

So the dot product is, if you take, a.b = (the x components multiplied with one another) + (the second two components multiplied with one another)0554

It might seem surprising at first, but it turns out actually having the exact same effect.0579

If you look at a formulation where, if one of them is lying on the x axis, then you can actually quickly see why if this is here, then the amount that this is out is the x axis amount here, x-y, well, it is going to wind up being F, if this is F again, then Fcos θ = its x component, because that is how much it is, because we can see that the way the vector breaks down.0582

We can break the vector into its constituent perpendicular and parallel pieces.0608

It will be a little bit more complicated to prove this in a different angle, but you can trust me on this, work = (force vector).(distance vector)0614

We can also use just force × distance × cos θ if we know the magnitudes and the angle.0623

Sometimes one is going to be more useful than the other, it depends on the specific conditions, and what you need to do.0628

As always you got to pay attention to what you are trying to solve fro in Physics, and figure out what is the best thing for you to use right there.0634

Finally, what units does work use?0642

From our formula, work = force × distance × cos θ .0644

cos θ , θ comes in angles, angles do not really have a unit, they are radians, but they are unitless, cos θ is just a scalar.0648

The only things that come with units are our force and our distance.0660

Force's unit is newtons, and distance's units is metres, if we are working in the S.I. system.0664

That means, work = N m , for ease we call this a joule, which we shorten as J.0672

J on its own, that is what we do for work, and later we will find out it is also what we do for energy when we talk about how those two are connected.0679

We call it a joule in honor of James Joule, who did pioneering work in heat and energy, in the 1800's.0685

We are ready for our examples.0694

Real simple, real easy one to start off with.0695

We got a bus, and we push it a distance of 10 m with a parallel force of 20 N.0697

If it is parallel, what is the angle?, the angle = 0, so cos θ = 1.0707

So, work = force × distance × cos θ = 20 N × 10 m × 1 = 200 J, and that is our answer.0713

If we wanted to, we could have also done that in dot product form, because we know that they are parallel, so the force would just have a vector of (20 N,0 N) , and the displacement vector would be (10 m,0 m).0738

So, we wind up getting 200, because 20 × 10 = 200, and 0 × 0 = 0, so we get 200, the exact same answer.0757

Second example: Ball of mass 0.25 kg is dropped at a height of 12 m.0769

When it hits the ground, how much work has the force of gravity done on the ball, what if the mass was 'm,, and the height was 'h'.0775

First off, we have got some ball, and we drop it 12 m, so the ball is up here, and mass = 0.25 kg, what is the force pulling on that ball?0781

Force pulling on that ball is the force of gravity, we assume no air resistance for ease, actually in this problem we can have air resistance, but we are paying attention to just the work done by the force of gravity, if we have to look at the energy later on, we would have to take the air resistance into account, but the force of gravity is going to do the same work no matter what, as far as it does move those 12 m.0796

In any case, force of gravity = mg, so in this case, if it travels 12m, and what is cos θ?, θ = 0 (since parallel).0822

So, work = force × distance × cos θ = mg × 12 × 1 = 0.25 × 9.8 × 12 × 1 = 29.4 J.0841

What if we wanted to solve this in the general case, what if we want to talk about, what if we were dealing with an arbitrary mass 'm', what if the height was just an arbitrary 'h'?0872

If that is the case, the work once again, the fall is parallel to the force of gravity (same direction), so work = F × d = mg × h = mgh, is the work done by the force of gravity for an object dropping.0883

Third example: A box moves a distance 5 m to the right, a force of 10 N pushes on it in the opposite direction.0912

How much work does the force do on the box?0922

The first thing to think about is, what is the angle.0924

Here is the 5 m, what is the angle between those two vectors?0929

5 m is a vector, 10 N is a vector, so what is the angle between them.0935

They are parallel, but they are pointing in opposite directions, we got to pay attention to the fact that they are going opposite.0939

So, 180 degrees.0947

So, if θ = 180 degrees, what is cos(180)?0951

cos(180), remember in your unit circle, it is pointing in the opposite direction, so it is going to be -1.0955

We drop this all in our formula for work, we got, work = force × distance × cos θ = 10 N × 5 m × (-1) = -50 J.0961

This is a totally new idea we have not encountered before.0994

We got the idea that we can actually take work out of a system.0997

If we had it going with it, that means we would be making it go faster, you put in work into it, because you would be making it going with it.1002

Bu this time, we are actually resisting the motion that it has.1009

It is going to move forward 5 m, but this time we are pushing in the opposite direction.1012

If we push in the opposite direction, this means that we are actually resisting it.1016

We are using our work to take the total work in the system out, the total energy in the system out.1022

We will talk more about the connection between energy and work, but right now, before, work was contributing to the distance it was moving, it was contributing to motion.1027

In this case, our force was going against the motion, so it is actually taking away from the motion, so it is a negative work.1035

If we want to do this with the dot product, work = F.d, if we make this the positive direction, then what is our distance vector?1042

It is going to be equal to (+5m,0m), what is the forces?1052

It is going to be going in the opposite direction, so (-10 m,0 m).1059

We put these two together with our dot product, and we got, 5× (-10) + 0 × 0 = -50 J.1064

Two different ways of doing it, both equally valid, gives you the same answer, the idea is the fact that, one you work against the motion of it, you have negative work, you are taking work out.1076

Fourth example: We have got a box traveling a distance of 50 m, 30 degrees South of East.1088

So it is moving South of East by 30 degrees, and it travels 50 m.1094

The box is acted upon during that motion by a force of 50 N, in a direction of 30 degrees North of East.1101

Even though there is a force moving on it, it does not change the displacement.1107

We know the displacement vector beforehand, it is given to us, we can be sure of it.1113

For some reason, there is something keeping it on that track, we are just worried about what the work that force does is.1116

We do not have to worry about the displacement changing, displacement is given to us in the beginning.1123

The work is going at an above direction by another 30 degrees, so it is 30 degrees North of this.1129

How much work does the force do?1134

We are looking at this from above, so it is flat, so we do not have to worry about the gravity, that is what the North and East and South all tell us.1136

In this case, what is θ?1142

The angle between the two is not just 30, it is the total between the two, so it is 60 degrees.1144

So, θ = 60 degrees.1151

Use our formula for work, work = force × distance × cos θ = 60 × 50 × cos(60) = 1500 J, is the answer.1154

Now, in this case, we could also do this in vector mode.1183

For this one, we were given the angles and magnitudes, so it is less useful, but I want to show you how to use the dot product, because sometimes, you are going to get things in vectors, and it is way more useful not have to convert into angles and magnitudes and see if you can do the problem, it is useful to go, "We have got vectors, let us use the dot product!"1186

We will convert this first into vectors.1205

For the top one, if it is 60 N on the hypotenuse, and 30 degrees angle here, then over here, it is going to be 30 N vertical, and what is its horizontal going to be, it is going to be 51.96, which is what we get when we take cos(30) × 60.1209

What about for the triangle representing the displacement?1232

It has got 50 m on its hypotenuse, so what is its vertical, its vertical = 25 (remember, we got 30 degrees here, and in 30-60 triangle, the side opposite to 30 is 1/2), and up here, cos(30) = sqrt(3)/2, is going to be 43.30.1236

In this case, this means that we know our force vector, what is the horizontal component?, it is 51.96, and let us say this (right) is positive, and up is positive, and vertical is +30 N .1265

We look at the displacement vector, and that is going to be equal to, (43.30 m, -25m).1293

This one is pointing down.1306

We take the dot product of these two, force dotted with distance, f.d, that is going to wind up equaling (51.96 × 43.30) + (30 × -25) = (2249.9) - (750) = 1499.9 J .1309

The only reason this wound up being any different from this answer which they are approximately equal, is because of rounding errors.1347

Rounding errors when we wound up figuring out what these two horizontal components were, we got slight answers off, because when we are using our calculator we wound up having to round it, because we did not use the entire thing, which we should, because remember we want to take some care about how significant digits work when you use an entire 10 digit long expansion.1356

Because that is the kind of extreme amount of accuracy to have, so we wound up having a slight rounding error, but when we consider the fact that this is 5 digits long, and we are only off by the very last digit, that is really close to 1500 J, that is as precise as we will be able to measure anything, in any lab we do.1374

And probably any lab you would wind up doing very long time, unless you are working in seriously experimental Physics.1390

Example 5: This is our last example.1397

Similar to example 2, remember in example 2, we talked about dropping a ball from a height 'h' or 12 m, we have got the same mass and the same height as we did in example 2.1400

We drop a ball 12 m, it is going to be the mass of the ball times gravity because that is the force of gravity times the height that it falls and cos θ just winds up going away turning into 1, because cos(0), because they are parallel is just 1, so we can just pay attention to the force × distance, so mgh.1412

What about this case, a ball of mass 0.25 kg is tossed out of a window at a height of 12 m.1432

Now, it travels up, and then travels down.1439

So, we do not know how high it gets, and I did not tell you precisely what it was.1444

It could be that it winds up getting really tall or it could be practically flat and then falling immediately.1449

It could be either one of these, I did not tell you how it is going to look.1455

So how could we figure out what it is going to be.1459

Remember, we know about how negative work works.1461

Notice that for the amount up here, for this portion right here above the height of the window, no matter what happens, the ball is going to wind up going positive.1466

Let us call it 'H', so there is the h that it falls to the ground, it should be 'h', that is the amount that is guaranteed, and then there is H which is the variable amount that it winds up going depending on how we throw it.1480

It has to go up by H, but then to be able to make it down, it has to also go down by H.1491

We have got a positive H and a negative H.1498

What happens if we look at the work done over the positive H and the negative H section?1500

It is traveling to the side, gravity is only going to be caring about the component, about up and down because everything else is going to be perpendicular, so we can just toss it out.1505

We only have to care about the up and down components.1515

Positive H is the amount that it travels up, that is the parallel amount, the perpendicular amount, the motion sideways we can just get rid off because it does not matter, that is the amount that is perpendicular and we can throw it away because we only care about the parallel amount.1518

In this case, we only have to care about the up by H and down by H, and then h.1534

We already know that the amount of work done in the h is going to be mgh.1542

What about the amount of work done by H?1547

This one, g is negative number.1550

So, +H, if we go this way, they are actually going to have an angle, θ = 180 degrees.1558

This is going to give us, -mgH.1566

What about the direction where it goes down?1572

That is going to be θ = 0, because now they are going in the same direction.1574

So this is going to be, +mgH.1578

We have got the idea that you go up by some amount of height, but you are fighting against gravity, so gravity is taking work out of the system, that means a negative work.1583

-mgH, the amount that you travel up, but then we wind up having travel the exact same amount down if we are going to make it to the ground, so that means, that amount that we just lost in work is going to be regained in work.1592

The H, mgH is going to wind up cancelling the -mgH, the two are just going to hit each other, and they are going to disappear, and we are going to wind up getting, these two things, just cancel each other out, and in the end, the only thing that we are left having to care about, is the h that we got right here.1604

That is the important part, mgh.1626

No matter how crazy a throw we have, if we go way up, or it is really flat, it does not really matter because the amount of work done by the extra arc from where it, the amount that is not just where it starts falling to 12 m we are guaranteed, is going to be canceled out.1630

The two works go opposite to one another, so it just gets canceled out.1647

In the end, it is mgh, the end of our answer is going to be the exact same answer we got for example 2.1651

The work = mgh, which is going to be, 0.25 × 9.8 × 12 = 29.4 J, is what we got.1656

The reason why is because the amount of arc that goes above where we started gets canceled out, because it winds up having to travel up, but then it travels that same amount back down, before it can do the real fall of h.1671

So H is, wind up cancelling one another out, because they do the same thing, they are doing with the same gravity , but they go in opposite directions, so they just cancel one another.1686

All we have to worry about is where we started, if we want to figure out the amount of work that gravity is going to put into it when it hits the ground.1701

Hope you enjoyed this, hope you got a good understanding of work.1708

This is going to be really useful in the next section when we talk about energy.1714

Hi, welcome back to educator.com, today we are going to be talking about kinetic energy.0000

Just to begin with, you have definitely been exposed to the idea that there is a lots of different types of energy out there.0005

There is lots of different kinds of motion, energy, and ways for things to be energetic out there.0016

Some examples: there is kinetic motion energy, gravitational potential, if you lift something up and drop it, or lift it even higher, it is going to wind up having more energy.0022

Chemical energy, say hydrocarbons like gasoline inside of your car, light energy: the sun casting all that light energy at us, heat energy: if you are near fire, there is heat energy in the air.0032

Nuclear energy: radioactive Uranium is going to be able to put out energy in a form, electricity: moving electrons, and even more things than what I am talking about right now.0046

There is lots of different ways to have energy out there.0058

It is really useful to us as physicists to be able to understand how energy works, and get a really strong grasp on it, if you are going to be able to solve a lot of things, and model stuff, to be able to have control over environment.0060

Another idea you have probably been exposed to is this one:0073

Energy cannot be created or destroyed, only changed from one type to another.0076

The conservation of energy, you have almost certainly heard this.0081

It is not possible to get rid of energy in a system, it is only possible to change it from one form to another.0083

If you have a car driving along and hits the breaks, it is not that its kinetic energy goes into nothing, the kinetic energy goes into the squeal of the breaks.0089

The friction of the brake pads, the friction of the tyres on the road.0097

There is various different kinds of heat, the heat energy is what is going to come out of it instead.0101

Or you could have that chemical energy burning, it is not that the chemical energy goes away, it gets turned into heat, it gets turned into sound, it gets turned into kinetic energy, if say it is a rocket or a space ship.0106

All these sorts of things.0116

We can think of energy through this metaphor of a bucket full of sand.0118

We have got a bunch of different buckets, and each one has an amount of sand.0122

Say we are looking at a rocket on a launch pad.0126

It has some amount of chemical energy, it has got lot of chemical energy.0130

All of the fuels in these rockets, it has got no kinetic energy, it is just right at the bottom, no kinetic energy, and it has got some amount of gravitational energy, it is currently sitting on a pad, and it is also, maybe we are at some high place, wherever we are launching it from.0133

So, it is slightly higher up.0149

What happens when we ignite those rockets?0150

When we ignite the rockets, we use that chemical energy up, so we wind up burning all that chemical energy.0153

And that energy instead cause the rocket to move upwards, we causes an increase in kinetic energy, it is now moving up.0160

Now we have got an increase in kinetic energy, and the gravitational energy will go up as it moves up.0171

But the total amount of sand, the amount that the chemical goes down by, is the total amount that the kinetic and gravitational go up together.0179

If we look at the total amount of sand in all the buckets, all use the same.0187

It is just we change the location from one bucket to the other.0191

We either want our sand or energy in our chemical bucket or we want it in our kinetic bucket, or we want it in our gravitational bucket, but these are only different ways of swapping it around.0194

Or may be what will happen is, the rocket would run into a satellite, now we are having two system interact.0203

Now, swapping from the kinetic energy of the rocket, it is getting that kinetic energy swapped into the kinetic energy of the satellite, because the satellite gets knocked out of the orbit, and it hurdles away.0209

These sorts of things, it is not that we destroy energy or we change it inherently, it just gets changed from form to form, or from one object to another object, it gets transferred from systems, or gets transferred from types with in a system, or it gets transferred from types between systems.0218

But it does not get lost, does not get destroyed, it is always there, it is moved around in different ways.0235

Now, we will talk a little bit more about what specifically, why we can guarantee this, why we believe in it, why we think that this is one of the fundamental laws of the universe, but that is for thermodynamics, and that is while down the road.0241

But for now we can trust in the fact that the conservation of energy works, and that is going to be incredibly key to be able to solve all sorts of different things.0252

It gives us a lot of power over the understanding of how the world around us works.0260

What is energy?0264

We have talked about it having these different forms, we have not really talked about what it is.0266

It is hard to come with a truly rigorous definition that we can actually make use of, and be able to both play with it, and do cool things and be able to talk about it in a deep rigorous way, all the same time.0270

That is really hard, so we are going to sidestep that, we will be able to think of energy as the capacity to do work.0282

Being able to put work in, being able to make a change in the world.0287

Energy is your ability to change the system around you to build a change in environment, or change yourself with in the environment, it is to do work.0291

Work is change in energy.0299

So, what is work?0300

If work is what energy can do, work is a lot easier to define in fact.0303

Work is just the transfer of energy from one type to another.0309

Doing work is just the act of transferring energy, whereas the quantity if work is the amount of energy transferred.0311

If we had two buckets, and we wind up moving a whole bunch of stuff from one bucket to the other bucket, the amount that we move over, the quantity, is the quantity of the work, and doing work, this is the act of doing work, it is the moving it over.0317

Work is just moving energy from bucket to bucket.0341

That makes sense.0344

Wait!, that was a circular definition.0346

I defined energy with work, and work with energy, but in fact I did not really define either, that is not allowed, you cannot just make circular definitions and expect anyone to trust you.0349

You should not do that, you want to be able to have something where you can stand in, and at least make a definite claim.0358

You got me, that is a really good point, we really should be having something that is a meaningful statement that we can really push against and test and try out.0364

We do not want circular definitions, because they are not scientific statements.0373

But, in my defence, we have actually defined work.0376

Work is the transfer of energy, work is the motion of energy form one of our theoretical bucket to another theoretical bucket, not theoretical but metaphorical.0380

You move energy from one type to another, form one system to another, and that is work.0389

But what we did not really define is, we did not really define energy.0395

We did not do that rigorously.0398

To be honest, to finding energy precisely, in a really rigorous meaningful way, like push against and really scientifically talk about, and it to really mean something, is kind of outside the scope of this course.0400

It is actually kind of tough to talk about what it really means to be able to have a really deep understanding of that, and there is a whole bunch more to learn about energy before we can really have a very strong perfect meaning, to be able to truly say that.0414

But that is okay, we know enough intuitively, we are able to talk about the idea of energy, you know about having rocket fuel, you know about something moving very fast, you know about something being in a high location or stretching out a string.0426

Any of the different ways of storing energy, or seeing a system that has a lot of energy, you got a really good intuition, and that is enough.0438

You will be able to rigorously define and being able to talk about things in a really specific way, for a later advance Physics course, that starts to matter.0445

For now, we are going to be still be able to do plenty of stuff with what we have got,0453

We have got plenty to work with.0457

Being able to define it rigorously, it is something to do down the road, when you study a whole bunch more Physics.0459

With all that behind us, let us talk about how to derive a formula, useful mathematical formulae, we can talk about things quantitatively.0466

How can we get the amount of energy in something's motion.0474

Kinetic energy.0477

From the theoretical point of view, we have seen that putting work into a system is the same as putting energy in.0478

But we have not quantified it yet.0483

Let us fix it.0485

Consider a simplified picture of work, where the force F is parallel to the displacement.0488

That means that θ = 0, so from here on we are just going to pretend that cos θ = 1, we are just going to get rid of it, so for this case, work = force × distance. (normally, work = force × distance × cos θ)0493

We have got a block of mass 'm', and it is on some magical frictionless surface and it starts at rest.0513

This is how things are to begin, let us talk about from here.0520

Consider the following three formulae from all of our previous work.0524

Definition of how we got the force, force = ma, also from kinematics, vf2 = vi2 + 2ad (d is distance), and what we just defined in our last section, work = fdcos θ = fd, because we are looking at a simplified picture.0526

From this, we get, we can talk about the acceleration, because we want to be able to slope things in to our velocity, so we can talk about what velocity means, we need to be able to get a relationship between 'fd' and velocity.0548

What we are going to be doing is to mathematically massaging this, until we are able to get an expression that shows us something that can connect work and this new formula would become kinetic energy.0561

We are going to say that is work, because work is the amount of transferred energy, if we put work into something that starts off with no kinetic energy, that will tell us what the formula for kinetic energy is.0573

If we want to get acceleration, we get that, a = F/m .0581

Also, vi = 0, we said that this thing started at rest, so we can knock that out.0590

For ease, vf = v.0596

From there, v2 = 2ad, we have 'a' up here, plug that in, v2 = 2F/m × d, we are really close, we got an 'F' and a 'd' already.0607

We move that m and 2 over, multiply both sides by m, we get, mv2, and divide both sides by 2, we get, (1/2)mv2 = Fd = work.0629

So the work, if we start off with no kinetic energy, if we out work into the system, that is all going to become kinetic energy, because we have got nothing resisting it, no friction, it is just freely moving, then all of that just turns into (1/2)mv2.0642

The amount of kinetic energy in the system, is (1/2)mv2.0657

Now, from our derivation, we get this formula, the amount of energy in a moving object, Ekinetic = (1/2)mv2, notice this is using speed, we cannot square velocity, squaring a vector does not mean anything, but we can square a magnitude.0662

So we use the magnitude, its energy is a scalar value, it is not a vector, direction does not affect the energy, you could be going a 100 miles/h to the north, to the east, to the south, up, down, does not matter, 100 km/h, 100 km/s, they are all going to wind up being directionless, it is all about the speed that they are going at.0681

Ekinetic = (1/2)mv2 .0704

Now we have got a formula, now we can really get our hands dirty.0712

Units: We talked about work being the quantity of energy moved, so if work is the quantity of energy moved, if that is the amount that we move, we are going to need energy to be the same as work, in terms of units.0715

So, at the moment, we are just hoping that work and energy are going to wind up having the same units.0728

Let us see if that is the case.0731

Ekinetic = (1/2)mv2, so what is m?, (1/2) is just a scalar, so we get rid of it, we got m is kg, velocity is m/s, but squared, m2/s2, now Fdcos θ , cos θ is a scalar, just a number, changing the amount, but it does not change the units.0733

Force is newtons, and we got metre, we need to figure out what newtons are!0761

F = ma, 'a' is m/s/s, mass is kg, that means, m/s/s × kg, this is the same as, m2/s2kg, everything checks out, we are happy!0765

So the units make sense, so the world is safe, so we can just shorten things easily, we will call it a joule, a joule is both the measurement for energy and for the amount of work moved.0788

Work is joules, energy is joules, joules is energy.0800

Conservation of energy: To be able to really do anything with kinetic energy, we are going to need to talk about the fact that it stays the same, unless it is affected by its environment.0806

We talked about the fact that energy cannot be made or destroyed, it is only transferred from different systems to different systems, or from different types to different types.0815

Energy in a system must stay constant unless some of its energy is transferred, work, that is what work is, to another system, or another type in another system.0823

But, if we look at all the types within a system, the energy of the system, so this is not just kinetic, but all the energies within the system at start, plus the work, the amount of that change, and that is equal to the energy of the system at the end.0834

All the types of energy at once.0850

Notice that energy transferred into the system, if you put work into the system, to increase its energy, that is represented as a positive value, positive work, environment does work on the system, force is going with motion, chemical energy being contributed, things like that.0852

Energy that is transferred out is negative work.0868

If work is taken out of it, if system does work on its environment, or the environment takes energy out of the system, say friction, then that is going to be negative work.0870

Work is positive if it is going into the system, negative if it is coming out of it, it is going to be up to us to pay some attention to what is going on here.0880

Energy of the system at the start + the work = energy of the system at the end, that is the conservation of energy, that guarantee is going to give us so much power in solving an entire new set of problems, allows to really get a good understanding of how the world works.0888

One last thing to talk about: How does friction work!0903

It is really easy in fact, form an intuitive stand point, friction is the environment taking energy out of it, or we can think of as the energy putting energy in the environment, in either case, the object's motion is going to be slow, it is going to lose its kinetic energy, it is not really going to gain anything, we are just going to get heat dispersal.0906

Friction reduces the energy in the system.0922

By how much?0925

It is as simple as using our formula for work, and our knowledge about friction, remember, if we have got something sliding along, friction always works this way.0927

If it goes this distance, then the work is just going to be equal to, (-) × (magnitude of friction) × (distance), that is it.0936

The work in the system will be negative because its losing it, it is just the friction force times the distance that the object travels.0947

Friction always points backwards, and always gives a negative value, and always saps the energy out of the system we are looking at.0953

First off, real easy one: Block of mass 10 kg is at rest on a frictionless surface.0962

It has a horizontal force of 20 N, applied to it for a distance of 25 m.0973

What is its velocity afterwards?0980

Notice, if we were doing it the old way of kinematics, and F= ma, it will be a little bit more difficult, we will have to figure out what the acceleration is, and we will have to use that complicated, vf2 = vi2 + 2ad, whereas in this case, we just figure out the work, we plug into our formula, BOOM!, we have got it.0982

How much work is put into the system?0998

The work = force × distance × cos θ = fd (since parallel) = 20 N × 25 m = 500 J, going into the system.1000

The system starts at rest, we have got the block starting at rest on a frictionless surface, so no energy is lost, so, (energy of the system at the beginning) + (work) = (energy of the system at the end).1021

At the beginning, this is zero, there is no energy, it is sitting there still, 0 + 500 J = (1/2)mv2, since we know it is all going to be velocity in the end.1036

From here, we just solve for what 'v' is, we know what 'm' is.1051

2 × 500 = 1000, 1000/m = v2, sqrt(1000/m) = v, m is 10 kg, so, v = sqrt(1000/10) = sqrt(100) = 10 m/s .1058

Do not even need a calculator for this one, because it is so easy, 10 m/s, and we are able to do it by just figuring out the work, and figuring out what the kinetic energy is, what the energy connected to that speed is.1086

We know it starts off with no energy, and then we know that all the energy goes into its kinetic energy, so it is as simple as figuring out how much energy is put into the system, and we are done.1099

Example 2: Two identical cars of mass 1500 kg are driving directly towards one another at 15 m/s, and -15 m/s.1109

They crash into one another in a horrible screaming trash of metal, they come to a complete rest after impact.1118

How much energy is released during the crash?1123

At the end, we have got 0.1127

So, whatever energy beginning is, that is going to be our answer.1132

How much energy is in the system in the beginning?1138

How much is it in each car?1140

For one car, the Ekinetic = (1/2)mv2, and we are going to assume that all the energy is in its kinetic energy, because it is just its motion here, so that makes sense.1142

(1/2)mv2, we know it is 1500 kg, we know what the mass is, we plug these things in, (1/2) × 1500 × 152, and the energy of one of the cars is going to be equal to 168750 J.1157

That is a pretty fair amount.1180

But we got to remember that, also the car is going to have that released, because it is in each one of the case, we do not just have that +15, we have that -15 velocity.1182

Since it is speed that we are looking at, we know that they are each going to have the same kinetic energy, so for the crash, it is going to be , the energy of the car × 2, so 2 ×, Ekinetic(of one car), which is = 337500 J.1196

So that is how much energy gets released during the crash.1220

That is a fair bit of energy.1227

Now, what would happen if we double their velocities, instead of driving at 15 m/s, and -15 m/s and hitting one another, they were at 30 m/s each when they crash into one another in completely opposite directions.1228

If that is the case, then we are going to have to change what the velocity is.1244

We know that the crash at double speed (in red, since it is going to be a lot more dangerous), is going to be, 2 × (1/2)mv2 = 1500 × 302 = 1350000 J.1249

Notice that 1350000 is way more than double 337500 J, in fact it is quadruple, because it is going up with the squared of the velocity, the squared of the speed here.1282

It is not just about going double the speed, that means you have quadruple the energy, and that is why freeway accidents are so dangerous.1295

Because everyone is traveling at such a high speed, it is more dangerous, it is 4 times more dangerous in terms of the energy given out, to have an impact when you are driving at 40 m/s than at 20 m/s.1302

Those are reasonable freeway speeds, and those are like normal city speeds.1317

So, it is much more dangerous to get in a collision on the freeway, it is simply because the energy involved to stop those cars, at any reasonable rate is going to take a whole lot of more energy, so the energy is a lot more dangerous for the freeway impact.1322

And that is why it is so important to be careful on driving on the freeway, it is because it is way more dangerous, potentially, if you get into an accident.1335

Example 3: What if we got a block of mass 20 kg, once again sliding on a frictionless table, no air resistance, nothing like that, so it is just about the energy being put into it, with vi = 4 m/s.1343

Then acted upon by a force of 35 N at an angle of 20 degrees above the horizontal for a distance of 15 m.1360

What is its speed afterwards?1366

We do not have to worry about the force of friction, so we do not have to worry about the normal force, so we do not have to break down that force of 35 N into its components, we just have to figure out how much work does it put into our system.1367

The work = fdcos θ = 35 × 15 × cos(20) = 493.3 J.1378

Conservation of energy formula, we know that the energy at the beginning in our system + work put in = the energy at the end in our system.1407

What is the energy in the beginning?1421

Remember, it did not start at rest this time, this time, it had an initial velocity of 4 m/s.1422

So, we have to include that.1428

(1/2)mvi2 + 493.3 = (1/2)mvf2 (work is positive because the environment acted on the object, not the object losing energy to the environment), (also, these are actually the magnitudes, since we are talking about speeds.)1429

Plug things in, (1/2) × 20 × 42 + 493.3 = 653.3 J.1465

So, that means our energy at the end, is equal to just the motion in the velocity, just the energy in the velocity, Ekinetic, is going to be, (1/2)mvf2 = 653.3 J.1490

We solve the algebra, we get, vf = sqrt(2 × 653.3 / 20) = 8.08 m/s.1517

We know what the starting energy is, we know how much work goes into the system, we put those things together, that gives us the ending energy, and then we figure out what energies are going to be being used in the end, it is just the kinetic energy, that is the only energy that is going to be in our system at this point, so we know that the total ending energy is equal to the kinetic energy at the end.1559

And we just solve for it.1580

Final one: This one is going to take a bunch of ideas.1583

Block of mass 3 kg, is resting on a surface, where it does have friction.1586

Friction coefficients are, μs = 0.8, and μk = 0.4.1591

It is acted upon by a force of 100 N, at an angle of 40 degrees below the horizontal, so it is pushing down on to the block.1598

Does it move?1607

The force acts on it for a distance of 50 m, what is its speed after 50m, how far will it slide afterwards?1608

First things first, we need to figure out if it is able to move.1614

How do we do that?1618

To be able to figure out if it is able to move, we need to compare the force acting on it horizontally, to the maximum static frictional force, that it has.1619

First, we are going to break this down, into its component pieces.1629

Actually, we probably would be better off by looking at it in the other point of view, because we know what the 40 degrees are, so, we look at it, over here, so, 100 N is the hypotenuse, 40 degrees here, so cos(40) × 100 will give us what the horizontal action is.1634

That is going to be 76.6 N, and the vertical is going to 64.3 N, pointing down, pointing to the right.1652

If we want to split this into a vector, we know that the force vector is going to be 76.6 N, and -64.3 N.1665

Before we are able to figure out what the normal force acting on the block is, we need to figure out what the force of gravity is, because this block is being supported by the table.1677

If the block is not falling through the table, then that means all the vertical forces on it are in equilibrium.1685

The normal force, the force contributed by the table is going to have to beat out, not just the force of gravity now, but also the force of pushing down into block.1689

What is the force of gravity?1699

Force of gravity = mg = 3 × 9.8, plug that in later.1701

If we want to know what the normal force is, we know that the normal force is going to have to be equal to cancelling out both of those.1718

It is going to have to be positive, pointing in the up direction, it is going to be equal to 63.3, so it can cancel out the force pushing down on the block, and then also, plus mg, so it can cancel out gravity.1725

We put those together, and that gives us 93.7 N.1740

So, the normal force is 93.7 N.1744

Now we are ready to calculate what our maximum static friction is.1747

Maximum friction static = μs × FN = 0.8 × 93.7 = 75.0 N.1751

That is the maximum friction static.1773

Now, we need to compare that to the force of our pushing on the block, what is the force pushing on the block?1776

Its horizontal force is 76.6 N, is greater.1785

That means, YES!, it moves.1798

That means , we can actually pay attention, we do have to care about the rest of the problem, because we are able to beat out static friction.1805

Now, let us work on everything else.1813

If friction static is able to be defeated, then we can now figure out how much work gets put into it by the force.1816

The work of the force = magnitude of force × magnitude of displacement × cos θ = 100 × 50 × cos(40) = 3830 J.1824

Now, remember, this entire time it is moving, it is also being hit by friction.1863

We have got friction fighting it.1870

The whole time it is moving that 50 m, it is being fought by friction.1873

So, it moves along, it slides along, but as it is sliding, it is also continually losing energy to friction.1877

The block is giving energy to its environment, it is gaining energy from the force that the environment is putting into it, but it is also winding up giving heat energy through friction.1884

So, it is going to have that much work coming out if it.1894

It moves along that 50 m, and it is going to continue moving, because the force is able to beat out the force of friction.1897

But, once the force stops acting on it, it is going to slide.1904

How far will it slide?1908

It is going to take all of the friction to sap all of the kinetic energy out of it.1909

Once it has lost all of its kinetic energy, it will be still, if we can figure out how far that distance is, we will know what the slide is.1914

What is the speed at 50 m?1921

We need to first figure out how much work has friction done over those 50 m.1924

That is going to be, force × distance × cos θ = μk × FN × 50 m, (cos θ = 1, since perfectly parallel.)1931

One last thing to figure out: which direction is friction working?, it is working negative, it is pulling away from this the whole time.1963

Our work value for the friction is going to be negative.1970

So, -0.4 × 93.7 × 50 = -1874 J.1973

It gains 3830 J from the force acting on it, but it loses 1874 J through the force of friction acting on it.1983

What is its starting kinetic energy?1992

Its starting kinetic energy is nothing, it sits still at first.1994

(Energy at the beginning = 0) + (the work) = (Energy at the end).1999

The work and now our ending snap shot is at this 50 m line, we want to figure out what is the velocity there.2007

The work that goes in, 3830, the work that comes out is 1874, and that is = the kinetic energy at that moment, because all of our energy, all of our work, is going into kinetic energy, (1/2)mv2, where we are talking about the speed really.2013

We figure out what that is, we get, 1956 = (1/2)mv2, solve for v (speed, this whole time we know that v is single dimension, so that is not a problem), v = sqrt(2 × 1956 / 3), solving, we get that it is sliding at a very fast 36.1 m/s, at the end of it push.2037

Now we know how fast it is sliding at the end of the push, how far will it slide after that?2079

Now we need to figure out, the only thing working on it now, is we have got, E(beginning) + Work = E(end).2085

So, for the first one, the one on the left, over here, we had a beginning snap shot, the beginning snap shot was still, and the ending snap shot was the very end of the push.2095

For this one over here, we are talking about, beginning snap shot is the end of the push, when it already has all that energy stored in it, and its ending snap shot is just as it comes to rest, once it is still again.2108

We know that, E(beginning) + Work = E (end), E(beginning) = 1956, we can also figure that out by doing (1/2)m(36.1)2 because that is how much energy is, but that is going to wind up being 1956, because we already solved for that).2119

So, 1956 + work = 0, (when still, it has no energy.)2150

That means, our work = -1956.2160

What is the work in this case?2165

The work is just friction, remember?2166

We know this is friction, because the only thing acting on it now is the work of friction.2169

What is the distance that is slides?2174

We know that friction is negative, so, μk × FN, but now the normal force is actually a different number.2176

FN is equal to just gravity at this point, because now we no longer have it pushing on it, so, -μk × mg × d = -1956.2187

We know what μk and FN are, we do not know what the distance is, that means we only have one variable to solve for, so we can make these positive on both sides.2213

d = 1956 / (mg μk) = 1956 / (3 × 9.8 × 0.4) = 166.3 m, this time it slides a very long distance.2225

It gets pushed for 50 m, and that whole time it has got a very large additional frictional force contributed by the force pushing down.2282

But once we remove that, it is going to be able to slide longer, because it does not have to deal with the force pushing down on it the whole time now.2291

The normal force change between the two different worlds we are looking at, in the second world, we did not have that force pushing down on it now, so we now have a different normal force, and that is a definite trap that you could fall into.2299

If you forget to change that, at which point you get a different answer.2310

Bu tin this case, we caught that, we know that it is going to be 'mg' is just the normal force at this point.2313

μk × FN × d is the work done by friction, so now we need to figure out how much distance it goes, and once it completes that slide, it will manage to take out all of the kinetic energy, and it will come to a rest or come to a zero kinetic energy.2319

So, we get 166.3 m, and there is our answer, that is how far it slides after you stop pushing it.2337

Hope you enjoyed it!2343

Hi, welcome back to educator.com, today we are going to be talking about gravitational potential energy.0000

In our last section, we mentioned that there is a huge variety of forms that energy can take.0005

Then we explored specifically the idea of kinetic energy.0010

Now we are going to investigate the idea of gravity giving energy to objects.0013

When you lift an object up above the ground, you put gravitational potential energy into it.0017

Why do we call it potential energy?0022

Potential energy is stored energy that can be later released.0024

So, it is something that will allow us to do work later.0027

But, that is kind of true of kinetic energy, right?0030

The very least, it is able to store, in a way, store the energy as kinetic energy, and then release it as frictional work.0032

So, what is the difference between potential energy and kinetic energy?0039

If we really want to get rigorous about this, potential energy is energy associated with position/arrangement of a system of objects, and the forces interacting between them.0043

That is a little hard to use in our work though, so it is more than enough for us to think of it as storing energy for later use, something that we can hold on to indefinitely, or at least for a while.0052

We have got a pretty good understanding intuitively, once again we can appeal to our intuition, and we do not have to worry about having a really rigorous understanding until later Advanced Physics stuff, so for now, we can think about potential energy as just something that, just we are storing energy for a while, and also, we will know about it because we are going to talk about each kind of potential energy very specifically in length.0062

Let us start off by thinking about how potential energy works.0082

If we have got an object dropped from ever increasing heights, the higher the height it is dropped from , the faster it would be moving when it gets to the ground.0086

This means that a higher height implies higher speed at the ground.0096

Higher speed means more energy, that means that more height must be more energy.0104

If we have got a little mass with a little distance, that is going to be less energy than that same little mass, and a much larger distance.0110

Bigger the distance, the greater the energy.0117

If we want to talk about things specifically, quantitatively, how much energy does it have?0123

If we drop an object of mass 'm' from a height 'h', how much work does the force of gravity do on it?0127

The object is going to be pulled down, by a force mg, right?0132

The force of gravity is mg, mg always operates down, it is always pulling directly down, just like our height is directly down.0137

So, the motion down is going to be straight down, that means, work = force × distance × cos θ = mgh, since they are parallel, cos θ knocked down.0145

So, the amount of work is just going to be mgh.0168

If gravity does a work of mgh on our box, that is how much energy we must have stored in it by lifting it to that height.0173

Therefore, the energy of gravity = mgh.0180

Now, think about the fact that, no matter what path we take to get an object up to, say the top of this pillar, if we go directly up, and put it here, or we go like this, and then put it here, once it is still and sitting up there, it is going to have the exact same energy either direction.0190

No matter how we get it there, it winds up having the same energy, once they get to the same height.0211

The only important thing is the height that it has.0216

It matters if it is moving, but once it still, it is just the height.0218

It is completely based on the height.0221

So, mgh gives us potential gravitational energy.0224

Two important things to notice about the formula that we just made:0229

It is up to us to set what that base height means.0231

For that pillow that we just talked about, it is up to us to know that, that is the zero.0234

We could have the zero have up here, and it does not matter, it is all based on a relative idea, so it is comparing the two heights, if we want to talk about what the energy of the ball is here, what the energy of the ball is here, it does not matter if they both have negative, it is still going to have a positive difference, because it would have gone up to get up here.0239

Whereas, if it is down here, it is going to be positive, it does not matter, it is positive either way, because this one is less negative from the point of that zero, and this one is more positive from the point of that zero.0257

But either way, it is the relative difference, it is the change in 'h', not just the starting height on its own.0268

Being at the top of Mt. Everest, and going down 10 feet, is not more energy that starting at the top of Empire State Building, and going all the way to the ground floor.0274

There is way more energy in that tall drop from the top of the building down, even though Mt. Everest starts off higher from the centre of the Earth.0282

Formula also relies on the idea that 'g' is constant.0291

We can do this, because 'g' is reasonably constant near the surface of the Earth.0295

We can trust that we can have 9.8, really close to 9.8, remember, if you are on a really high location, depending on where you are on the Earth, specific loads of, like mineral deposits, so there is going to be minor changes in 'g', but more the most part, gravity is really constant on the surface of Earth, we can trust it to be 9.8.0298

We can use this formula, but if we went into space, or a different planet, the formula would change as 'g' changes, and if we were to go really far from Earth, we would not even be able to use this anymore.0315

It is going to rely on the fact that we can rely on the fairly constant gravity of Earth, within a near area from the surface of the Earth.0325

That is what we are going to trust on, whenever we are working on problems, we trust on the fact that we know what the gravitational constant is, and we know that it is going to be fairly, with the acceleration of gravity is not going to change much, whatever distance we go above the Earth, as long as it is a reasonable distance compared to the size of the Earth.0332

Finally, to be able to use anything in our energy formulae usefully, we are going to have the fact that energy is conserved.0352

We keep talking about hits, and we are going to keep talking about it, energy cannot be created or destroyed, it is always going to be conserved, it is only transferred.0357

This means that energy in a system stays constant, unless it is transferred out of the system.0367

The energy in the system at the start + the work = energy of the system at the end.0370

Positive work means energy is put into the system, work is put into the system, and the energy increases, negative work means energy is taken out, say friction.0376

That is what work is, work that is positive, is energy going in, work that is negative, is energy coming out.0386

We are ready to tackle the examples.0392

If we have a rock dropped from a height of 20 m, and we ignore air resistance, how fast will it be traveling 10 m above the ground?0394

Then how fast we will be traveling just before the impact?0400

Say we have got this rock, and there is 20 m, and there is the 10 m that it falls first.0403

From here to here, what is the change in height?, the change in height = 10, and the change in height here = another 10.0411

How fast is it traveling 10 m above the ground?0420

At beginning, it starts off at rest, so energy at the beginning, kinetic energy (K.E.) = 0, so there is no energy in it other than potential gravitational energy.0422

No work happens, because it does not lose anything to friction, it does not lose anything to air resistance, it is just direct transfer between gravitational potential energy moving directly into K.E.0431

For the first one, there is two ways of approaching this one:0441

We could look at this as either being, mgΔh = (1/2)mv2 (we are talking about speed, since this is single dimension.)0444

mgΔh, we do not know 'm', that might be a problem, but that we can take it for a bit, we do know 'g' and Δh, and we have got 'm' on both sides, so we can strike out our m's, we do not have to worry about that.0458

Remember, gravity works on everything, no matter what the mass is, it has a uniform acceleration, barring air resistance.0478

A uniform acceleration caused by gravity happens in our potential gravity to kinetic transform, because of the fact that, that 'm' constant shows up on both sides of the equation.0486

That means, gΔh = (1/2)v2, sqrt(2gΔh) = v.0499

Plug in numbers, sqrt(2 × 9.8 × 10) = v = 14 m/s, at 10 m above the ground.0512

One other way that we could have done this though, is we could have said, Esystem(beginning) + work = Esystem(end), mghi = mghend + (1/2)mv2, since work = 0, and also it has got some speed at the end.0534

If we move this over, we got, mghi - mghend, which is why we wind up pulling it out, and we look at, mgΔh.0570

It is the relative change that tells us how much energy is put into the system from gravitational potential.0583

That equals our (1/2)mv2, and that is why we did not have to put on both sides.0588

We could though, some case it is going to be useful to do that, we want to keep that tool in our tool box, but for this case, we are just looking at the change in height, gives us how much energy gets put in.0592

If we want to consider what happens when it moves it to all 20, we have got, mgΔh = (1/2)mv2, 2gΔh = v, 2×9.8×20 = velocity = 19.8 m/s.0602

That is the velocity at 0 m above the ground.0638

The things are, when we are at 10 m above the ground, we got 14 m/s, and we are at 0 m above the ground, when we are just touching the ground, just the instant before the impact, just that split second before it lands on the ground, 19.8 m/s.0642

A 100 g is thrown from a height of 1.4 m above the ground at a speed of 10 m/s.0659

At a height of 2.5 m, it has a speed of 7.9 m/s.0664

How much of its energy has been lost to air drag?0668

This is a case where it is actually going to be useful for us to have height on both sides, I think it is easier that way, you might not, but you can definitely do the other way if you felt like it.0670

In this case, we will do my way though.0679

Energysystem(beginning) + work = Energysystem(end), we want to figure out what the work is, how much work does the air drag do, and we should get a negative number, because the air drag is going to sap the energy out of it, it is going to suck energy out.0681

Esystem(beginning) has two things, it has a height above the ground, and it also has an initial speed.0704

What is its initial height?, mghinitial + (1/2)mv2 + work = mghfinal + (1/2)mvfinal2 (again, v is speed, note that it could be a vector as well, the important thing is to understand what you are writing, a useful diagram could be really helpful here.)0711

This frisbee, it starts off flying at some speed, and with some height, hinitial, and then later in time, it is at higher height, but less speed, and we want to figure out how much work is done between those two moments.0751

Those are our two snap shots.0777

At the beginning, we have got, in S.I., 0.1 kg, (sometimes we can use other unit system, but if we use 100 g instead of 0.1 kg, we would not be working with joules anymore, we would be working with something else, and we will completely screw up our ability to do this problem properly, because we have got m/s which is S.I., and we have got grams in there, which is not S.I., and things are going to get really funny, it is going to get bad.)0779

So, vinitial = 10, then, hinitial = 1.4, hfinal = 2.5, and vfinal = 7.9.0822

Plug everything in, we get, 0.1 kg×9.8 m/s/s×1.4 m + (1/2)×0.1×102 + work = 0.1 kg×9.8 m/s/s × 2.5 m + (1/2)×0.1 kg × (7.9)2. 0840

We cannot make it easier by canceling 'm', because the work does not involve 'm', but it is important to be able to think about what you are doing and catch them when they happen.0860

Start calculating everything out, eventually, work = -0.802 J.0919

We plug everything in, move things around, put in a calculator, and we get the fact that we get -0.802 J at the end.0931

That is what our answer is, it loses that much energy to air drag, which is not very much, but remember, there is not all that much energy in a very light weight frisbee being thrown, not that fast.0937

Example 3: Block of unknown mass is slid down a frictionless slide of height 5 m.0952

Then it slides along a table where it has a friction coefficient of μk = 0.3, we only have to know the kinetic friction, because it is already starting on motion.0957

It starts on a frictionless incline, but very still, it starts to slide down it, and how far does the block slide on the table?0968

What is the initial energy in it?0977

Ei + work = Eend, so how much energy does it have at the end?0981

It is still, so it has no energy.0994

The work is going to have to suck out, is going to have to cancel out all the initial energy.0996

What energy does it have in the very beginning?, is it moving in the very beginning?1001

No!, it just starts, and then slides on a frictionless slide.1006

So, how much energy does it have?, it has mgh, its starting height worth of energy.1010

We have got, mgh = -W, work here is by friction.1018

How much is friction?, force of friction = μk×FN.1026

Remember, we do not have to worry about it once it is on the slide, because there is no friction on the slide, so we only have to worry about the flat plane.1032

On the flat plane, the only thing pulling down on it is mg, and mg is also equal to FN, so, friction = μkmg.1039

We go back to this, we get, mgh = -fdcosθ, θ = 180 degrees, because its motion is this way, but friction is pulling this way, so a 180 degrees.1055

So, mgh = -fdcos(180) = fd (cos(180) = -1).1093

So, mgh = μk×mg×d.1116

Look!, mg gets canceled on both sides.1132

So, h = μk×d, d = h/μk = (5 m)/(0.3) = 16.7 m, is the distance that it slides.1135

All the energy that it starts with, is just going to be its gravitational potential energy, that is all the energy in it, so it slides down the slide, once it gets to the bottom, it then starts to be zapped by friction.1167

So, the distance, starting here, once it gets to the friction table, is going to 16.7 m, because we know that, Ei + W = Eend, (gravitational potential energy) + [(friction force)×(distance it slides)] = 0, since still in the end.1181

Of course it might be still higher, but we know are looking at the change in height from start end to the bottom, and we said our base height as being the table's height, that is a clever thing we did, we did not even point out while we were working on it.1208

One thing I would like to point out: When we worked on this problem, for almost all of it, we did not put in our numbers, we do not need to put in numbers until the very end.1226

It is so much easier to be able to work with μk and sub in at the very end, that is a very useful trick, when you have got a lot of non-sense going around, you only need to substitute in the values until you get to the very end, because you might get the chance to eliminate some of them, it is easier to write letters, than complicated sets of numbers, because you might wind up making a simple error when you are trying to put down five 4 digit numbers, as opposed to 5 letters.1240

It is easier to watch what you are doing and have an understanding of what is going on, have an intuitive understanding of what you are doing, what this represents mathematically, by moving things with letters than just these meaningless numbers that you might not understand as well.1273

It is really helpful to be able to move things around as letters, and at the very end substitute in.1286

Final one: An acrobat of mass, m= 55 kg, swings from a trapeze of length 15 m, starts at rest, and descends 5 m during the sway.1293

The trapeze rope has been sabotaged by an evil villain, and it will snap at 800 N of tension.1303

Will our acrobat make it across safely?1309

This is a question that is important to his life.1312

If it is going to snap, where would be the point it would experience the most tension, when is the most tension going to happen?1316

If we are able to figure out that it is going to not snap at the point it is experiencing the most tension, then we know he will make it across safely, because the entire time the swing occurs, the acrobat is able to swing across without having to worry about the tension snapping it, because if the tension is maximum at one point, then we know that if we can figure out that one point's maximum tension, it will tell us whether or not it snaps.1326

Because everything is going to be at least less than that.1355

If everything is less than that, all we figure out is what the maximum tension put on it is.1360

Where will the maximum tension occur?1365

Now we have to think about, where do we have to get tension from!1367

The tension has to keep it in a circle.1372

So, the tension is going to be connected to keeping it in a circle, so force centripetal for it to be able to go in a circle, forcecentripetal = m×v2/r (again, v is speed)1375

At what point on the circle will he be traveling the fastest?1402

It is going to be the point, where he has converted all of his initial gravitational potential into motion, which is going to be the lowest point on the circle.1411

That is going to have the largest 'v'.1421

The largest 'v' occurs at the bottom, also the largest 'v' is when there is going to be the most fight between the tension of the rope, and the gravity pulling him down.1425

So, mg is going to be pulling down, and the rope is going to have to also give the centripetal force necessary to keep it in the circle, and also beat the gravity force that is trying to pull him down.1435

These two things combined will cause the tension to be highest at the bottom, so that is the point that we have to look at.1447

We know that, Ei + W = Eend, so the moment we have to look at, the snap shot, is going to be the moment of most tension, that is the point we want to check out and figure out if he is going to survive.1454

If he is going to survive, we are going to have to look at that moment of maximum tension.1473

So we are going to make that ending snap shot be, when it is pointing directly down.1478

There is no work, since there is no friction, everything is smooth.1481

So, mgh = (1/2)mv2, m's cancel out, so we do not have to worry about the mass.1491

So, gh = (1/2)v2, v = sqrt(gh) = 9.9 m/s, is the 'v' at the bottom, that is the speed our acrobat would be traveling if the rope were definitely safe.1505

If the rope snaps, it will snap somewhere between here, and here; after this it will wind up being safe, so all we have to do is to test the most extreme point, and we will be able to figure out if it snaps before it gets to the bottom.1528

If the bottom tension is greater than 800 N, we know the rope snaps, somewhere on the way to getting to the most extreme tension.1545

We do not know precisely where yet, we could figure it out, that would be a more difficult question.1553

We know that, vbottom = 9.9 m/s, what is the centripetal force necessary to keep him in a circle?1557

That is what it has to be, on the acrobat, forcecentripetal = m×v2/r, and we know m and r.1564

(55 kg)×(9.9)2/(15 m) = 359 N, is the necessary centripetal force, the necessary sum of forces to be able to keep him in that circle.1577

The centripetal force is pulling this way.1597

We have got tension, is pulling this way, and has to also be able to be some of mg.1602

We know that, T - mg = 359 N = centripetal force.1609

T = 359 + mg, at the moment of maximum tension that that rope would experience if it were a safe healthy rope.1617

Remember, if it is not a safe healthy rope, it might snap before we even get there, so the actual rope might not even ever experience that tension, because the most it is going to experience is 800 N before it snaps.1628

We do not know if he will make it yet, we got to finish this problem.1638

359 + mg = 359 N +(55 kg) × (9.8 m/s/s) = 898 N, SNAP!, he does not make it!1641

But, on the bright side, there is a net underneath it, let us draw a net, the net catches him, and he works later, and he finds the villain who is responsible for it, and he puts him to justice.1670

Hope you enjoyed this, hope you will come back, and we will learn about more about energy, and have an even stronger understanding of how energy works.1684

Hi, welcome back to educator.com, today we are going to be talking about elastic spring energy.0000

Spring energy or elastic potential energy, and we are also going to use this as a combination to talk about everything we have learned in energy so far.0005

Let us dive right in.0011

Just to start off, when we are talking about springs or any elastic object, elastic object is something that tend to return to its original shape when deformed.0013

Lots of objects have this behavior: springs, rubber bands, metal (when you deform metal to a slight amount, it wants to spring back into its original shape).0021

Of course there is extremes, if you deform something past a certain point, its plasticity fails it, and it is no longer able to return to what it was before.0031

But assuming it do not go past that really extreme case, you will be fine, and the rules that we are going to talk about will work out.0039

We are going to be able to talk about anything elastic, but in this case, we will just use a spring for all our examples, but this holds true for anything that has this really strong elastic property.0045

If you have any problems about bungee cords or rubber bands, where you know that it is going to involve some sort of elastic constant, you will be able to use the stuff that we are going to be talking about today.0055

For right now, we are going to refer to everything as springs, but it does not mean it is only limited to using for springs.0065

Let us go back to looking at springs very carefully now.0072

If we fix one end of a spring in place, and the spring is just sitting there, in will be in a relaxed position and no force will be exerted, you have a spring sitting on a desk, it does not do anything, it just sits there.0075

But, if we compress the spring, if we push the spring in, it will resist it, it has a force against it, and the amount that we compress it, is going to cause it to have more force.0086

If we compress a little bit, it is going to give us a little bit of force back.0097

But if we compress with a lot, it is going to give us a lot of force in response.0101

The more you compress it, or the more you stretch it, the more force you will get.0105

Exact same thing happens if we are stretching it.0109

Similarly, if we pull the string, we get a resistance of another force going in the opposite direction.0111

So, if we pull a little bit, we get a little bit in response.0115

Small pull results in a small force, but a large pull results in a large force.0119

The more you deform the spring, the more force you are going to get back, the harder it is going to try to get back to its original position.0126

The force will always point in the direction opposite to displacement.0134

If you displace the spring, if you compress the spring, it will resist compression by going in the opposite direction.0139

If you pull it, if you displace it apart, it is going to resist by effectively pushing in.0144

If you push, it pulls, of you pull, it pushes.0153

Of course it becomes kind of meaningless, because you think of pull as something that goes away, but what we really mean is that the force is going to be the opposite to that force that you put on it.0156

Not equal forces, but the opposite direction, magnitudes will depend on how far it has been compressed or pulled.0165

Our displacement is based on how far we do it, but each spring is going to have a different amount of force.0171

It stands to reason that some springs, and some objects are going to be more forceful than another.0180

Consider we had this spring in a mechanical pencil, versus the spring in a car garage door, or the springs for keeping up the suspension in a truck, these are going to be totally different things that we are looking at.0184

We are going to have to deal with in some way, with different kinds of springs having different forces connected.0195

We know we have to use the displacement quantity somehow, and we are also going to have to use something intrinsic to the spring that we are working with.0201

If we put this all together, we will be able to get this idea empirically verified.0207

We are not going to do this here, but it has been empirically verified, which is how we know it is true.0213

The fact that scientists have gone out and tested it, it was a good idea that seemed likely, and people went out and actually tested it, and that is how we know that Hooke's Law works.0219

We call it Hooke's Law after Robert Hooke who was a pioneering English scientist who studied springs in the late 1600's and he described it first, so we named the law after him.0228

From this we got the fact that, the force of the spring will be equal to, -kx.0238

The negative is there, because 'x' is how much we compressed it from equilibrium, so 'x' is the displacement that the spring undergoes from its relaxed state.0243

'x' is how much, if we start off with some base amount, then 'x' is the amount that we wind up compressing it to.0252

So, this amount here, would be 'x'.0264

But there is also 'k', which is something that is inherent to the spring.0266

If we are dealing with a big, hard, powerful spring, then it is going to have a spring constant that is much larger, it is N/m .0270

How much the compression results in force, newtons.0280

Big 'k' means that we are going to have a really strong spring, small 'k' means flimsy spring.0282

The negative there is to make sure that we flip the direction of our vector, because 'x' is the displacement vector, so to put a negative tells us, go in the opposite way.0289

-kx, something that includes the intrinsic nature of the spring, and also says go in the opposite direction, of how far you have pulled.0298

We know how far is the displacement, but we also know the direction.0305

-kx, is the force exerted by a spring on an object.0309

Say we had a vertically aligned spring, and we already tested it and found out that k = 200 N/m, so our spring is like this.0316

On top of it, we place a mass of 1.2 kg.0327

What is going to happen next is, this is going to go to equilibrium, and the spring is going to wind up getting compressed some amount.0331

Same mass, but now the spring is compressed.0342

The spring is resisting it, it pushes down, and then it stops.0348

What do we have in equilibrium?0352

We know in equilibrium, the forces involved have to be canceling one another out.0353

Force of gravity is still there, so we definitely know that there has to be a spring force involved.0357

What is pulling down on the box?0361

The box is being pulled down by mg, so 1.2×g.0363

What is resisting that pull, is the force of the spring, which is, = -kx.0369

The negative in this case is negative to mg, but we already got that, because we drew this in.0378

Since we drew in the vector, we can think of this as a positive vector, it is now up to us to deal with the vectors.0384

In this case, we know that, mg - kx = 0, but we can also see that those two vectors have to equal length, they have to have equal magnitudes, and they point in the opposite direction.0389

We can also think of this as just, mg = kx, because we know that to be in equilibrium, they have to be the same magnitudes, and they are pointing in opposite directions.0409

So, mg = kx, we know m,g,k; plug in numbers, we get, 1.2×9.8 = 200x, x = 0.059 m.0417

But in this case, we have to pay attention to, which way did we go?0446

We went this way, so x is considered positive, because for some reason we decided to make gravity positive up here, we made it positive, so unlike usual, we got x being positive, it is really just saying, it went down by a 0.059 m.0449

You could also think of this being a negative value, if we were going in the normal direction as positive-negative, but once again, like usual, it is up to us to pay attention to how the coordinates are.0466

If we pay attention to what we mean by the diagrams thought, our problems we will be able to work out, the important thing is that you know what you are doing.0475

In this case, if we look carefully at the diagrams, we see that it is compressed, so it has gone down by 'x' amount.0480

So, we made mg positive for this case, and we made kx the negative one, but if we look at it as arrows and vectors, we know that those two arrows are pointing the opposite directions, and they have equal magnitudes, we can start off with mg = kx, and we just have to know that our 'x' involves this arrow pointing down.0489

Now we have got Hooke's law.0508

Hooke's law, we can definitely understand at this point.0510

Force of the spring = -kx, where 'x' is how much it has been compressed, and 'k' is the spring constant.0512

Now, we want to be able to turn this into energy, if you compress the spring, and then hold it in place, you have done work, you put energy into the system.0520

If you compress the spring and hold it in place, and then you let go down the road, it is going to pop out, and you are going to be able to have energy stored, you will be able to cause a ball to be shot off of a launcher, you would be able to have a piece of lead pushed forward in your mechanical pencil.0530

Anything that you can wind up storing energy in a string for, all sorts of co-uses you can do for this, bungee jumping for example.0544

We are going to need a formula to be able to talk about elastic potential energy.0551

Sadly, we do not have mechanical tools to do that yet.0555

We need calculus.0557

But we can still use this formula.0558

The formula is that the energy in our spring = (1/2)kx2, where 'x' is the amount of displacement off of equilibrium.0561

Remember, it has to be measured from what the spring would be if it were at rest.0575

Also, it needs to be noted that just like with energy of gravity, we have to compare two different spring values.0579

We start off with a compressed spring, and then we compress it even more, what we wind up doing is, we put in energy, but we have not put in energy form rest to our ending compression, we put in energy from our beginning compression to our ending compression.0585

So, we will have to pay attention to starting compression versus ending compression.0600

If our ending compression is rest, we are doing well, we know we just have to do (1/2)kx2, but if you compare two compressed values, then you are going to have to actually compare them by coming up with two different values using the formula.0604

If we are going to be able to use anything in energy, we have to rely on conservation of energy, the fact that we can trust energy to always stay the same in the universe.0617

So, Esystem (start) + work = Esystem (end), we have said this repeatedly , and it still remains true.0625

This case, it means that the work, positive work implies that the energy in the system has gotten larger, work that is positive means energy going into the system.0633

Work that is negative means energy taken out.0643

So friction would always be negative, but if we were to have a spring and then push down, we do work into it, which means that the system have more energy stored in it.0646

Remember, this equation continues to hold true, as long as you are learning about energy.0656

Any form of energy that you learn about, it is going to wind up still having this true.0660

The important thing is to pay attention to all of the energy changes that are involved.0663

All the energy changes, all the work that is involved, and then you are going to be able to use this formula.0668

If you come up with some of it, but forget to put in other parts, say if you put in your start in height, but you forget to put in your ending height, it is not going to work.0672

If we know we are on a flat surface, we can forget about both if them because it is the same amount of energy stored, on both the left side and right side.0681

But, if it changes, you are going to have to make sure you account for it in your equation.0688

That is the only way conservation of energy works, we have to pay attention to what changes.0692

We have assumed that all the particles have stayed there, all the particles have stayed constant, nothing changed forms, nothing has been burned, no chemical have been used to create energy or take away energy, so we can trust that chemical energy on both sides is the same in this case, so we never had to pay attention to it.0697

We only had to attention to what you know is going to change.0714

So, it is important to pay attention to what does change.0717

Otherwise you will not be able to trust using this formula.0720

But, this is always going to be true for everything you use, so it is a great formula, it is great to pay attention to it.0723

Just think, what we started with, plus the changes is what we end with; makes sense!0728

Finally, I just want to mention that, we have definitely learned a lot of useful formulae, and we have got a really good chance of understanding how energy interacts with physical world, but you should not think that it is all of it.0732

Like I was talking about chemical energy, there is a whole bunch of kinds of energy that we have not talked about, lot of them are not going to have real simple formulae that we can plug into equations, so there is a lot more complicated stuff going on here.0743

This is far from the end of what you can learn about energy.0755

There is lots to learn, and I encourage you to go and learn more, because there is all sorts of interesting stuff out there, we are setting the stage for future exploration, later Physics courses, if you decide to continue to learn more about energy.0758

But, at this point, we have got a real good start, we have got a real good understanding of how the world works around us in terms of energy, and there is a huge amount of stuff that we can do with the stuff we have learned so far.0768

We have a real chance to tackle the hardest problems involving energy, andn we are going to see some really interesting applications in the set of examples.0777

These are going to be the most advanced energy examples we have seen yet, and will be a chance to really stretch our muscles.0784

First example, first big strong example: We have got a mass less spring launcher, (the reason why we are talking about all these springs being mass less, is because the spring had some mass, and it is going to take some its energy to move around.)0792

So we have to pretend, we have to be able to assume that we can treat the spring as mass less, and in most cases, the mass of the spring is very low compared to the mass of the object dealt with.0809

It is not that unreasonable.0819

But, if it was a really big heavy spring, and the object was not that heavy, it is something that we have to take into account.0820

The mass of the spring is going to have some effect on what is happening.0825

That is why all our problems are mass less, to be able to just jump into being able to using energy conservation.0828

But if you knew that the spring was not mass less, it is an even harder kind of example that you have to pay attention to, you really have to think about it0833

Back to our example, a mass less spring launcher, with spring constant, k = 900 N/m, located above the ground.0841

If we put a ball of mass 0.8 kg into the launcher, compress the spring by 0.2 m, and then the ball also starts at 8 m after compression.0859

Disregarding air resistance, what is the ball's speed on impact with the ground?0860

We are going to put it in a launcher, we are going to compress it by this amount, and we know that once it is compressed, it is going to have a height of 8 m above the ground, and then we are going to let it lose, and it is going to shot off of the launcher, and it is going to take an arc, then come down to the ground, and hit the ground.0865

First question: What about, will the orientation of the launcher affect the impact speed?0882

What would happen if we had the orientation like this?0887

If we had the orientation like this, the ball might shoot up, get to a different height, and then fall back down.0891

But, remember, what is the energies that we are going to be dealing with.0899

Energy we are going to be dealing with, is the height involved in the launcher, the amount of spring energy involved in the launcher, does it have any velocity in the beginning?0902

No, it does not have any velocity in the beginning.0910

Do we have to worry about work being lost?0912

Do we have to worry about friction taking place?0914

No, we do not have to worry about that, because we have been told that we can disregard air resistance, so we know that what begins in the beginning, is going to be the height, and the energy in the spring.0915

The angle has nothing to do with it.0924

The speed, because we are going to solve for speed, using (1/2)mv2 is going to wind up being the same in any case.0926

We should pay attention to the fact that, we are going to get totally different velocities at the end.0934

If we had an even flatter one, it would have, we would have totally different velocities depending on the orientation.0939

But, the amount of speed in the object, is going to be the same, it is just that the angle will change.0947

We can trust in the fact that the speed, the impact speed will be the same no matter how we do it.0954

If it were the impact velocity, then we would have to pay attention to the orientation, then we would have to pay attention to the arc being made, we would have to start caring about these sorts of things.0959

But, in our case, we know that we are only being asked for the impact speed, so it is enough to solve for it using energy real easy.0967

What do we know here?0974

We know that, Energy(start) + Work involved = Energy(end).0977

Right off the bat, we know that, work = 0, because we know that there is no friction, and the ball is not going to do anything, and no force is being applied to the ball, once it is already in flight.0983

So, we can just knock out work.0992

What is the energy at the start?0994

Energy at the start is, (potential energy of gravity) + (potential energy of spring) = (its kinetic energy when it is about to impact), mgh + (1/2)kx2 = (1/2)mv2.0996

We know m, g, h, k, x; so the only thing left to find out is what our speed is.1021

We get, (0.8 ×9.8×8) + (1/2)×900×(0.2)2 = (1/2)×0.8×v2.1031

80.72 J is L.H.S, multiply by 2, 2×80.72 J / 0.8, v = sqrt(2×80.72 J / 0.8) = 14.2 m/s.1058

The important thing is, we knew what the energies involved at the beginning were, we knew what the potential energy was, we knew what the spring energy was.1089

We know no work was done to it, we know that at the end we are only going to have to be looking at the speed energy, the kinetic energy, because we are going to have its impact speed, so we know it is at a height of zero when it is impacting, and there is no spring around it, so we do not have to worry about compression, we know everything else remains the same.1095

So, that is it.1112

Example 2: Block of mass, m = 0.5, sitting on a horizontal table, which has a friction constant of μk = 0.55.1115

It begins at rest, goes against a mass less spring, (since mass less, we do not have to worry about how much energy is put into moving the spring around, how much force/work is put into moving the spring around, so we make it easy for ourselves by making it a mass less spring.)1125

We compress 'x' m, with spring constant, k = 470 N/m.1140

After we release the block, it slides 7 m before coming to a stop.1145

Right now, this is the moment, we are holding in place and let go.1149

Once we let go, it slides forward 7 m, at which point it comes to a dead stop, no speed, no velocity.1152

It goes along, and cause it to slow down.1168

Once it is moving, what force is going to be able to slow it down?1173

Friction, so friction is the force moving against it.1176

Remember, work = fd, our force here is friction, we know, d = 7 m that it slides.1179

That 'x' is included in that 7 m, because once we let go, it slides 7 m.1188

That 'x' is just the beginning of the 7 m, the 'x' between equilibrium, and where we compressed the spring, it is just the first part that it moves through of the 7 m that it slides.1196

At this point, we have enough to solve for 'x'.1206

We know, E(start) + work = E(end), what is the energy at end?1209

It is still at the same height, so mgh = 0 throughout.1219

Is it moving anymore?1224

It is not moving.1225

It is not moving.1226

Are there any springs involved anymore once it is out over here?1227

No, there is nothing out here, it is just sitting over here, so we know, E(end) = 0.1230

What is the energy at the start?1236

E(start) is just the energy that we put into the spring: (1/2)kx2 + work involved.1238

Is work positive or negative out here?1244

We know that it is going to have to be negative, because we are going to have to take this positive number, x2 is always going to be positive.1247

We are going to have to lower this down to zero.1254

How does it happen?1256

We know that friction is going to be going this way, so we have got, friction×d, distance is positive, friction is negative in terms of vectors.1257

Or we can think of it as, distance is negative and friction is positive, but that is weird, because normally we are going to think, going to the right is positive, so in this case we are going to have works as a negative number.1272

So, this is going to be, -(the size of the friction force)×d.1283

Or, we could think of this as, +work as friction vector, dotted with the distance vector, or + (work of the friction)×d×cos θ (angle between them, θ = 180 degrees).1292

In any case, the important thing to know that our work is going to have to be a negative number, and it going to be, friction×d for the magnitude, so at this point, we are ready to do it.1306

We got how much the friction force is, normal force = mg (only thing applied to it is gravity, table is canceling out gravity), so, friction(kinetic) = μk×FN.1315

So the work involved = (1/2)kx2 - μkFN×d = 0.1338

We know what 'k' is, we do not what 'x' is, we know μk, we know FN, we know 'm', 'g' and 'd'.1354

So, at this point, we know everything but one of them, so we solve for it.1374

x2 = 2×μkFNd / k, x = sqrt (2 × 0.55 × 0.5 × 9.8 × 7 / 470) = 0.283 m.1378

It is important to pay attention to the fact that, because 'x' was squared, this actually was +/-, it is up to us to know which direction does it go.1426

If we want to make this into a vector, it is going to be, -0.283, because remember, we know that our 'x' was going to the left.1435

So we know that we had a displacement in the direction that we decided to call negative.1443

However, in this case, what we know is that 'x', the length of x', the amount that it was shoved over, is 0.283 m.1448

We can think of it as the magnitude and the direction that you moved in, or you can just think of it as a vector, with a positive or negative in front of it, depending on it.1454

And if it were a two or three dimensional vector, it will be getting a little bit more complicated, but in our case it is one dimension, so it is enough to just have just +/- in front.1461

Important part though is , 0.283 m, and to know what direction it went in.1468

Next example: We start off with a block of mass 5 kg, and it is dangling from a mass less spring.1474

At rest, the block stretches the spring form its relaxed position by 0.2 m.1481

We start off knowing, we have been pulled 0.2 m.1486

If we wanted to, we can figure out what 'k' is.1491

That is very good, because that does not show up anywhere, and we will need it later.1493

At this point, we know that if we pull it down by 0.2 m, the pull of the spring is able to equally resist the pull of gravity, so that they are in equilibrium, they are perfect equal opposite.1496

At this point, at 0.2 m down, we got that gravity is canceled out by the spring force, that is what it means if the block stretches from its relaxed position by 0.2 m when it is at rest.1511

After it being at rest, the block is then grabbed and pulled a further 0.3 m, so the total here would be, it starts at 0.2, and we take it to 0.5 m.1523

We hold it at rest, so we know that it has no motion at that moment, and we let go, and it begins to move up.1533

You pull the spring down and you let go, and you are used to seeing that it is going to go up before falling back down, and then it will start to oscillate.1538

How high will it make it, what is the highest point it will make it to above its resting location, how high is it make it before it winds up being canceled out and pulled back down?1545

At the beginning, we do not know 'k', but that is what we can use this piece of information for.1556

We can solve for 'k', if we got this information about what it is like at rest, when it is at equilibrium.1562

Once we have got 'k', we can use energy on this, to figure out how high it is going to make it up.1569

Let us get to work.1574

First, we are going to solve for what 'k' is.1577

It goes down 0.2 m, so we know that kx - mg = 0 (up positive, down negative, kx fighting mg)1585

And this is at rest.1608

We do not 'k' yet, but we do know mg and x, we get, k = mg/x at rest, k = 5×9.8/0.2 = 245 N/m, which makes sense, because mg is in N, d is in m, so 245 N/m.1613

With that, we can go on to figure this out using energy.1647

If we pull it down, we have got some energy stored in it.1650

Any work be done on it over the course of its movement, any energy taken out of the system?1655

Is there any friction involved?, No, there is no friction involved.1660

Is there anything else, any other external forces acting on the system?1664

No, there is no other external forces, just the internal forces of the system, of energy in the form of gravity, which we can consider an internal force because it is dealt with your potential gravity, and the force of the spring, which is another internal force we are dealing with through the form of spring energy.1667

E(start) + work = E(end).1685

We do not lose any energy, or gain any energy from the environment around us, so we can knock out work.1693

So, E(start) = E(end).1700

What is E(start)?, we start off with some amount of potential energy, in this case, it is actually going to be a negative amount of potential energy, because the box is below what it becomes later.1703

Later on, it is going to go up, so that is going to have actually going to gain potential energy over the course of its movement up.1715

If you take a box and lift it up, it has to gain potential energy to get up there.1726

Either at the beginning we are going to have a negative potential energy, or at the end we are going to have a larger potential energy, a positive potential energy.1730

The important part is that the difference between the two is going to be, that it is going to have to gain the potential energy, the ending potential energy will be greater than the starting potential energy.1737

Also at the start, we are going to have the amount of energy we put into the spring.1748

What about the end, will there be energy in the spring?1752

Of course! If we are up here, we are not in equilibrium, now we have got a compressed spring, a tightly compressed spring, if we have gone past equilibrium.1755

It makes sense, if we pull it down and let go, we are going to get a little bit past equilibrium if we pull it far enough.1763

We are going to have to deal with the amount of energy stored in the spring, and worst case, if we still put (1/2)kx2 on the ending amount, and it turns out that the x is zero, that we wound being at equilibrium, it will just get rid of itself, and we do not have to worry about the fact that it is there.1770

It is better to have a little more information then to turn out not needing it, than to forget to use it and lose the problem because of that.1784

One last thing: How are we going to decide height? How are we going to decide equilibrium?1791

We know that we already have equilibrium at zero.1796

We can have 'x' be, up will be positive, down will be negative.1798

But, what about height? What are we going to do about height?1802

Let is make height, for ease, also be zero there.1805

That means that we can set, x = h for this problem, it will make things a lot easier on us.1810

It is up to say our coordinate system, with the coordinate system defined by us, we have nowhere.1817

We have to impose a way of looking at the picture.1828

We have to impose an orderly system so that we can think about it, without that orderly system, we are lost in the wilderness.1831

We have to impose, 0 = height somewhere, we have to say what that base height is.1838

And why not put it at the equilibrium!1843

That means that the amount that we vary in our 'x' is going to wind up being exactly equal to the amount that we varied in our height, we can wind up tying those two things down, without having any way to transfer between the two, it is not like, height is going to be the maximum height that it achieves, so we have to figure that out, it is going to be really hard.1845

It will still be possible, but it will make the algebra so much more difficult.1861

Right now, it makes it really easy to say x = height, because we have established them as being, x = 0 here, and 0 = height there, so they are both the same amount, because they are both going to move by the same distance.1864

Now we are ready to get into it.1877

(1/2)kx2(start) + (how much energy is stored as P.E. in the beginning) = (1/2)kx2 (end) + mgh (end).1879

Substitute in numbers, (1/2)×245×(-0.5)2 + 5×9.8×(-0.5) = (1/2)×245h2(end) + 5×9.8×h.1917

x in the beginning is going to be negative since we are pulling it 'down', and greater on the right side since it is going up.1929

At this point, we can calculate everything on the left side.2025

Move everything to the right side, and solve, we get, 0 = 122.5h2 + 49h - 6.125.2033

That is enough to know what it is, but it is not enough to know directly, this is not like an easy algebra problem, where we just have height on its own, and we solve for both sides.2051

In this case, we have got a parabola, so we have to find out what the zeros of this parabola are.2063

We got a couple of choices: one , we can factor it, but it looks like a real beast to factor it by hand, since you have got 6.125, 49, 122.5 in there, this is not going to be just easy to come up with factors out of hand, so that option, pretty much gone.2068

We could use the quadratic formula, great to apply it where we have numbers that are very difficult to factor, because you plug in, it will come out, it might take a little bit longer, but it will definitely work, in this case, factoring is going to take way longer than the quadratic formula would, because it is going to be really hard to factor.2082

Final choice: If we had a graphing calculator, you could punch it into your graphing calculator, look at the parabola, and just find the zeros.2097

But, remember, in any of these cases, you are going to wind up getting two answers, so what are the two answers that we get for height?2104

We get, height = -0.5 m or 0.1 m, so which one of these is our answer?2110

If you look at -0.5, that is where we started, of course that is an answer.2123

If we go back to this very first line, it makes perfect sense that if we plug in 0.5 here, and -0.5 here, we have got the exact same expression on the left side and right side of our equation.2128

Of course they are going to be equal, that is going to be a solution because it is trivially a solution, it is obviously a solution, so -0.5 is no surprise, that is one of the places it will work out, it is one of the extreme cases.2142

What is the other extreme case?2155

The other extreme case is 0.1 m.2157

So, the 0.1 m is where it will get up to before it has to start falling back.2160

It is the other solution to our parabola, it is the other solution to our equation, so 0.1 m is the maximum height that it attains.2165

Final example: A block is placed at the top edge of a frictionless half pipe.2174

On the other side of the half pipe is an uncompressed, yet again, mass less spring, the block has a mass of 20 kg, and starts at a height of 3.7 m.2180

At rest, the bottom part of the spring is at a height of 2 m.2192

When released, the block will slide down the half pipe, compress the spring additionally by x = 0.5 m, so and then it will be forced back, because that is the maximum amount that it makes to.2197

If that is the maximum amount that it makes it, what do we know about just before it being forced back?2224

If you get to the very apex of your movement, and then you have to fall back down, what do we know about that snap shot, that exact instant when you are at the very apex?2228

We know at the moment, when you are forced back, your velocity is zero.2240

If you take a quick example, that is not quiet the sane as this, we toss a ball into the air.2244

At some point, that ball is going to fall back to the earth.2250

If it is moving along, it is going to continue having that 'x' moving this way.2253

But, its y vector is going to get smaller, until eventually it is zero at the instant it begins to fall back down.2257

So, at the very top, we know it is zero.2268

So, at the apex, at the very switch over between moving in one direction, and coming back in one direction, you have to have a zero velocity.2273

So that is going to be a key point for us to understand in this problem.2281

We know that, m = 20 kg, height that it starts at is 3.7 m, the height that the bottom of the spring is at, is 2 m, and the x that it gets compressed, is 0.5 m, now we want to know what s the 'k' involved.2283

We know that, E(start) +work involved = E(end).2304

What is the work involved?2313

We are on a frictionless half pipe, it is a mass less spring, we can assume that there is no air resistance, because it is not moving that fast, or we did not mention it, we will make it easy for ourselves, we are going to remove all the cases that does work.2315

In a real engineering situation, we will have to start taking into account, what kind of friction is involved, how dense is the air, is it an area that has a lot of air pressure, it may be taking place in a vacuum, but it will have some amount of friction involved.2335

There are things that we have to care about in engineering, but we got a specific example problem where we know that it is frictionless, we know that there is no air resistance, we know that the spring is mass less, so we do not have to worry about work going into anything else than just our equation, so we can knock out work.2354

What is the energy that it starts with?2371

It starts off at a certain height.2372

Is it moving at height? Is energy stored in its motion?2374

No, it starts at rest the instant before it is let go, and starts to move.2378

At the beginning, we just have potential energy of gravity.2384

At the end, what are the things going to be involved?2388

We have compressed the spring by an amount, so we have got (1/2)kx2 + mgh(end) (there is no energy in velocity, since it is not moving, and total height winds up being 2.5 m up here.)2395

At this point, we are ready to do it, so we got, mgh(start) - mgh(end) = (1/2)kx2, we know what x, m, g, h(start), h(end) are, we just plug in.2422

So, [2×mgh(start) - mgh(end)]/x2 = k = 2×20×9.8×[h(start) - h(end)]/x2 = 2×20×9.8[3.7 - 2.5]/(0.5)2 (h(end) is, (where the bottom of the spring is) + (the 0.5 that it went up beyond that) because it compressed the spring 0.5 past equilibrium, so that to make it equal to equilibrium, 2 m, and a little bit further to 2.5 m off the bottom.)2443

Our final answer is, 1882 N/m = k.2544

Now, I left everything in the variable form very long time, because I think that the easiest way to do a problem is to get all of your thinking down with variables, because variables are easier to think about what does it mean in a general way.2562

Once you start throwing in numbers, you must wind up getting this mass of numbers that often, you cannot separate what the ideas are here.2575

But in this case, we are able to see that it is the difference between the starting potential energy, minus the ending potential energy, is going to be how much energy we have left over for our spring, and then 2x2 is what we got for, for how much the springs other components are, to get the 'k' that we trying to get to.2582

To get to 'k', we have to multiply by 2, and divide by x2, so I left it as that, and substituted in.2605

There is no reason you could not substitute things as you find, if it is easier for you, and you feel comfortable with, it will certainly work.2610

I recommend trying it this way every so often though, I certainly think it is easier.2615

To keep all your variables together at the end, and then do one long session of computations at the end, but sometimes that is going to wind up being a trouble for you.2619

If you wind up having difficulty with the order of operations, like if you did not do this operation first, you would be in hot water.2626

You should pay attention to what makes it easier for you, but I think this is the best way to do it.2633

In any case, we have gotten to our answer, 1882 N/m.2638

This is going to end the session on energy, this ends our set of lessons on energy.2643

I hope you learned a lot about energy, hope you got a much better understanding, there is a whole lot more that you can do with it.2647

But, that is for our future courses.2651

Hi, welcome back to educator.com, today we are going to be talking about power and simple machines.0000

Let is just start off with, what would you say is the difference between a go-kart, a family car and a race car is?0006

I will be honest, there are a lot of differences, but I would say their main difference is their top speed.0013

It is high fast they can go.0020

How quickly they can get in going a certain speed.0022

You are going to get a lot of difference in how fast a race car can get from 0 to 60, ad how fast the family van can get from 0 to 60.0025

The major issues here are, how fast they can go at their maximum, how much power can they put out.0032

What is the idea of power?0037

So far we have talked about speed and its connection to energy, but we have not talked about different rates of gaining that kinetic energy, we have just talked about it being there.0040

We have not talked about the difference between getting it to going fast quickly, we have only been talking about going fast versus going really fast.0049

There has been, the speed that you are going at, there has been no talk about how fast you can get to go in that speed, what is your acceleration has had no effect on this.0057

That is where power is going to come in, we are going to start talking about how quickly an object or system gains energy.0066

Consider the idea that we are climbing a flight of stairs.0074

Let us assume that we weigh 50 kg, for me 50 kg is fairly well under my weight.0077

The stairs are 5 m high.0090

There are two scenarios.0092

In one of them, you climb the stairs in 5 s you really hustle.0093

But in the other one, it take you 30 s.0097

In both these scenarios, we are going to have the exact same amount of energy at the end, the same amount of potential energy, (not develop the same power.)0099

We climb the same height, we are dealing with the same gravity, we have the same mass.0108

But, very different scenarios.0113

How fast you climb those stairs, that is something we should talk about, and care about.0115

In both cases, we have that same gain of energy of gravity, 50×9.8×5, so in both cases we have 2450 J, when we make it to the top of the stairs.0120

But they are clearly very different scenarios.0129

So we need a way to talk about the interaction between work and energy, and time.0131

Work and energy, and how fast we are able to put work and energy into a system.0138

How quickly we are able to change the work and energy in a system.0142

This is going to really matter for some applications.0145

For that race car, we want to be in a race car that can put massive quantity of energy into its system, really fast it can get off the starting line and win the race.0147

With this idea, we make a really simple creation to call this power, just, power = work/(amount of time it takes), work/time, that will give us a way to talk about how much work we are able to deal with in how much time.0157

Just like velocity was how much distance we have gone, divided by how much time to do it, we het a very similar idea with work/time for power.0172

With power defined as work/time, we can easily create a few equivalent formulae.0180

First, since work is a measure of how much energy is being shifted around, we know work = change in energy, always.0185

Another formula is, power = Δenergy/time.0191

There is another interesting formula we can create.0199

Alternately, we can look back to how we originally formulated work.0201

Work = F.d = Fdcosθ, in this case it is going to help us to use that dot product.0206

This allows us to use velocity.0222

Power = work/time = F.d/time.0224

We pull off that force, and we get, F.d/time, but distance/time = velocity, so, we get, F.v, so, Power = F.v, which also, if we do not want to use the dot product, Fv×cosθ.0229

That same idea that worked with work, works with power.0253

So, F.v, or ΔEnergy/time, or work/time, these are all same way to say power.0257

What unit is power in?0266

Work and energy are both in joules, time is in seconds, so, power = work/time, implies, J/s is the unit for power.0268

For ease, we can call 1 J/s a watt, watt is in honour of James Watt who has done a lot of work with energy in 1800's.0287

A watt is a measure of weight.0295

Just like 1 m/s is the rate that you are moving at, 1 J/s, 1 watt is the rate that you are putting energy into a system.0298

One moment, you could have a totally different power, the next moment, just like your velocity can change.0307

watt gives us an instantaneous measure of how much energy is going into a system, that is the definition of power.0314

In other unit systems, there is also the horse power, you probably heard cars referred to in terms of horse power, and the kilowatt hour, another way of saying watts×time, watt×1000×time.0324

Horse power, kilowatt hour, all ways of saying power, that is why energy bills involve these things, cars involve these things, you see these things any time you want to talk about how quickly we can get energy into, or out of the system.0343

We are going to make a little tangent here.0356

This does not directly have to do with power, but I know, it has been driving you crazy that we have not discussed block and tackle system more.0359

I know you remember block and tackle problems extremely well, we talked about them in advanced uses of Newton's second law, and they seemed like magic, they seemed absolutely incredible, and it has been blowing your mind, you keep thinking about it, how is it possible, Physics must be lying, fret no more, I am going to make it better for you.0368

Finally we have the understanding of work and energy to see how those systems make perfect sense.0391

There is nothing magical about them, there is nothing insane about them, the world is not coming apart as it seems, it makes perfect sense when you look at it in terms of energy.0394

I know you love thinking about it, but once again, let us talk about it briefly, quick reminder about how the block and tackle system works.0401

Let us say we have got the force of gravity pulling down on this block, Fg, it is pulling down on this block.0411

If we want to keep it still, or move it up, say we want to keep it still, we are going to have to pull with the force of gravity here, put that much tension into it, so we have got canceling it out, the force of gravity over here.0417

But over here, something weird happens.0430

If we pull with a certain tension, then that tension is going to get pulled in here, but it is also going to get pulled in here.0433

So we still got that same mg, that same force of gravity, but over here, we are going to have the tension = (1/2)Fg, because if this is (1/2)Fg, and this is (1/2)Fg, then when we put them together, we are going to wind up combining two, one whole force of gravity.0440

With the block and tackle system, we are able to distribute our forces over multiple pulleys.0460

We are able to distribute the same tension force in multiple places.0466

It seems crazy, we are able to get more force for the same original cost.0470

How is this possible! This seems like madness.0474

The thing to notice here, is that the block and the single pulley system, say we wanted to raise this block 1 m.0479

If we wanted to raise the block by 1 m, how much rope would we have to pull?0488

We would have to pull 1 m of rope.0491

We have to pull, say some force F.0494

Over here, we know that we only have to pull at half of that force, F.0497

To be able to get that raising happening.0501

But, if we want this block to raise up 1 m, we do not just have to get this move by 1 m, if this moves 1 m, we would be lopsided, we have to get this move 1 m as well.0503

Both sides of our rope system have to move up a metre, that means we have to get 2 m of motion in our rope.0514

Over on the left side, we are able to use 1 m of motion for 1 m of motion.0524

Over here, if we want to get that 1 m of lift, we have to put 2 m of distance into our rope.0530

Even if we can use half the force, we have to pull double the distance.0536

So, our work, the amount of energy in our system is preserved, the force×distance.0542

This is force × 1 = F, for work.0547

Over here though, we have got, work = (1/2)F × 2 = F, so they wind up being the exact same things, checks out.0555

For us to be able to manipulate how the system works, we still have to maintain that conservation of energy, that conservation of work.0568

It is equivalent work, because the change in the system, the real change in the system, is how high up we are able to change that block's height.0576

If you want to do that, we are going to have to put in the same amount of work, no matter how we go about it.0585

Work that goes in, is equal to F in both cases, because you have to pull double the rope.0590

And if we had a multiply pulley system, where we were able to have four pulleys, we only have to pull with a quarter of force, we would wind up having to pull 4 times the distance.0597

Everything works out, there is nothing magical about it, it makes perfect sense.0605

It is the same idea in place with all of our machines.0610

Let us look at ramps and levers.0613

First we will look at levers.0617

If we want to get some object to move up here, traditionally you have a lever, you stick it under, you got a fulcrum, you got a long lever arm, and you pull, you pry.0618

You put a low pressure here, and you get a really strong force here.0627

High pressure here, low pressure here.0632

How is it being done, it is being done based on work.0634

You got that small force over here, small force, but it covers a really long distance.0637

On the other side, we get this small distance covered, which means, to be that work to be preserved, it is going to be have to put up a massive force.0643

The reason why a lever works, is that the energy is conserved.0651

If you put a slight force over a long distance, and the other side has a slight distance, then it is going to need a big force to compensate.0654

Force×distance has to be equal in any case.0663

Over the lever, if you put that fulcrum really close to one side, we will be able to get a giant force with little distance.0667

So, the amount of work is preserved, it makes perfect sense.0674

You see the exact same thing with a ramp.0677

If we want to get this box up this ramp, then we can push with this little force over this really big distance.0679

But, if we want to get this box directly up, then we will have to lift a whole lot harder, but we wind up going a smaller distance.0685

The ramp works, because we are able to get a small force over a long distance, whereas if you just want to lift it up with brute strength, we need a really powerful force, but we will be able to save some distance.0694

The same idea, force×distance, they are always equal.0706

The way that you are going to distribute, how you are going to put it in, what ratio you want to put it in, that is up to you.0709

But, it is going to have to come out equal when you multiply the two.0715

However you put it in, work is going to be conserved, energy is going to be conserved, the energy that goes into it is going to be the same however you do it.0719

Machines do not allow us to break the rules of Physics, they just allow us to take advantage if the resources that we have on hand.0726

They allow us to use the rules of Physics on our side.0732

If we have a little force, but a lot of distance or time, we can figure out an alternative rather than needing a really big force.0736

Like with the ramp, like with the lever, like with the pulley system, you can have that slight force, and then figure out the way to multiply by using more distance, or more force, or both , so you can take advantage of what you have, by being able to use the same amount of energy.0742

Same amount of energy will go into the system, same work, but it is up to us to figure how to get that work into it, and that is where the cleverness of machines comes into being, at least simple machines.0757

Now we are ready for our examples.0769

How much work is going to be involved here?0771

50 kg block is pushed along a horizontal surface at a constant velocity by a parallel force 47 N.0773

It covers 10 m in 5 s.0779

What is the power of the force?0781

Lets us draw a quick diagram.0783

Do we have to care about the mass?0801

We do not have to care about the mass.0802

Our power formula is, work/time.0804

We can figure out the work, work = F.d = Fdcosθ = Fd (since parallel).0807

So, 47×10 = 470 J of work.0820

What is the time? 5 s, so , Power = 470 J / 5s = 94 J/s = 94 W, that is the power of the force.0828

It does not change, the power remains the same, because we got a constant velocity.0850

Remember, we could have also used, F.v.0855

If you wanted to, we could figure out, it travels 10 m in 5 s, means that we got, 2 m/s = v, and F = 47 N, so, power = 47×2 = 94 W, two different ways.0861

Example 2: 2 kg watermelon starts at rest, and it is lifted vertically, 9 m.0887

It takes time 20 s, and ends at rest.0898

Over that lifting, what was the power developed in lifting that watermelon?0900

You started at some height, you reached another height, how do we deal with that?0905

Potential energy.0909

What is the change in energy?0911

ΔE = mgΔh = 2×9.8×9 = 176.4 J of work.0913

We know that, power = ΔE/t = 176.4 J/20 s = 8.82 W.0933

There you go, the change in energy divided by how much time it takes to put it i n there, just like velocity, just like acceleration, it is what you have got already, distance or speed divided by how much it is altered, how much it is being changed by, how much it is being increased by, that tells us how much the power being developed is.0952

The power of the system is the change of energy, how much work is going into that system.0970

Just like velocity is how much distance is going into an object, whereas the acceleration is how much velocity is going into an object, just a way of thinking how much velocity is going into an object, a way of thinking how much you are putting into a thing.0978

S0, 8.82 W is what is put in, in that 20 s.0987

Example 3: 20 kg block is initially at rest on a flat frictionless surface.0992

Parallel force of 10 N acts on the block.0997

What is the work done on the block in (A) the first second (B) the second second (C) the third second (D) the instantaneous power at the end of 3rd second.0999

First thing to think about, work = F.d.1009

Is this object accelerating?1015

It is on a flat frictionless surface, it has got a force acting on it, of course it is accelerating.1016

The amount of distance it is going to cover is going to change the entire time.1020

It is also going to have a change in velocity.1023

Now we have got two different ways of looking at this.1025

We can approach this by wither thinking about the distance that it has changed, it is going to be able to give us our work, and from the work, we will be able to get our power.1027

But we can also think that the velocity that it has at each time, would be a way to tell us what is the change in energy, and from the change in energy, we can get the amount of power.1040

These are both perfectly good ways to do it, and we will do both of them just to be able to understand two different ways to approach this problem.1055

First way, we are going to go with distance.1060

What is the formula for distance?1066

We got, F = ma, 10 N = 20 kg×a, a = (1/2) m/s/s.1068

What is the other formula for distance?1082

From basic kinematics, d(t) = (1/2)at2 + v0t + d(0) = (1/2)at2 = (1/4)t2, is our distance, (d(0) and v0 are zero).1084

Now we have got a distance formula.1107

Now we want to find out where is it at 1 s, 2 s and 3 s.1112

Plug things in, d(0) = 0, d(1) = (1/4), d(2) = (1/4)×22 = 1, d(3) = (1/4)×32 = (9/4).1122

If you wanted to see what the distance covered in that period of time is, that change in distance, we can say what is the distance between 0 and 1?1149

That is clearly (1/4).1157

What is the distance between 1 and 2? That is (3/4).1159

What is the distance between 2 and 3? That is (5/4).1165

These three changes in distance, if we want to figure out what the work involved is, we know, work = Fd.1171

Then, work that happened from 0 to 1st second is, 10×(1/4) = 2.5 J of work.1180

What was the work done between 1 and 2? That is, (3/4)×10 = 7.5 J.1200

What is the work from 2 to 3 s? That is, (5/4)×10 = 12.5 J.1209

Now, if we want to know how much power, what the average power was, over each one of these, we just go, 2.5 J/1 s = 2.5 W, was the average power in that first second.1218

In the 2nd second, 7.5 J/1 s = 7.5 W.1228

In the 3rd second, 12.5 J/1 s = 12.5 W.1237

Remember, these are going to be the average powers, because this is not going to give us the instantaneous.1244

Clearly, the amount of power is changing the faster it goes, because it is getting the chance to cover more distance, and that is how that force being applied more and more, since work = Fd.1248

If we want to know what the instantaneous power is, we are going to need to know what its speed is at a given moment.1257

Remember, power = F.v, v(3 s) = at = (1/2)×3 = 3/2 is the velocity at the 3rd second.1264

If we want to figure out the instantaneous power, Power at the third second, power (3 s) = 10 N × (3/2) = 15 W. (Dot product becomes multiplication because of one dimension, in dot product we are dealing with more dimensions, in more dimensions, we will have to know how to use the dot product, we will have to multiply the first components together, add them to the second components, multiply and add them to the third components, so on and so forth, for as many components you have. You probably will do with only 2 or 3 since you dealing with Physics, but it will work with any.)1304

So, the instantaneous power is 15 W for that third second, which makes sense, we see that our average, 2.5, 7.5, 12.5, it continues to go up, so the at the very end of the third second, we got 15 W of instantaneous power.1344

Remember, the instantaneous power can change just like your instantaneous velocity can change.1358

If you are driving in a car that is accelerating, every instant that you move along, your velocity is getting larger and larger.1364

So, if the car is accelerating, you got a larger velocity instant by instant.1370

In this case, we got a larger and larger power instant by instant.1374

If we have got an alternate method, let us start using it, how will that alternate method work?1378

So, we know, we did distance already, we can figure this out using distance, but we can also use velocity to tell us changes in energy.1382

Now we are going to do this using velocity.1390

If we want to see what its velocity is at 1 s, at 2 s, at 3 s, what is the formula for velocity?1394

We know, v(t) = at = (1/2)t, (since previous values still work), makes sense.1403

In this case, what will be velocity at 0 s? Zero, it is still.1424

v(1 s) = (1/2) m/s, v(2 s) = 1 m/s, and v(3 s) = (3/2) m/s.1431

If you want to see what the energy in its movement at that time is, we know, E(0) = (1/2)mv2 = 0, E(1 s) = (1/2)×20×(1/2)2 = 2.5.1443

E(2 s) = (1/2)×20×1 =10, and E(3 s) = (1/2)×20×(3/2)2 = 22.5.1473

So, we have got, 2.5 J, 10 J, 22.5 J.1495

If we want to figure out what the change in energy is, because we are working towards figuring out what the work is in each of these seconds, change in energy = work, work = ΔE.1500

ΔE(0 to 1) = 2.5 - 0 = 2.5 J, ΔE(1 to 2) = 10 - 2.5 = 7.5 J, ΔE(2 to 3) = 22.5 - 10 = 12.5 J, it is the exact same thing that we saw by doing it the other way.1514

Figuring out through the work, figuring out through the change in energy, of course they are going to give the same answer because they are the same thing.1543

If we want to figure what the power developed at the 3rd second was, we just do the exact same thing we did previously, so we can skip that, because we are just figuring out instantaneous power, and we discussed that on the last slide.1548

But, it is kind of cool to be able to see that we have got two different ways of approaching it.1559

So, whichever way that makes more sense to you, is the way you want to do it.1563

The important thing to think is, "Okay, great! I have got lots of methods, I have got lots of ways I can attempt a problem." Figure out what is the best for you, what is the best possible way to approach a problem, and then do it.1567

There are lots of tools for any given job, and it is up to you to figure out what tool to use.1577

Last example: This one is a fun one.1583

We have got a race car of mass 1500 kg, and it has an engine capable of putting out 700 hp ~ 5.22×105 W of power.1585

Neglecting air friction, and friction on the ground, the air drag, the car begins at rest, and we assume that the car puts out its maximum amount of power, how long will it take the car to accelerate to 50 m/s on flat ground?1595

In this case, what do we want to use, what power formula are we going to use?1608

Do we know what the work is? Do we know what the forces involved are?1612

We do not really.1615

Do we know what the change in energy is?1616

We do know what the change in energy is.1618

It starts at a stop, it stops going at 50 m/s.1619

Do we know what its instantaneous velocity is?1623

No, because that is going to change depending.1625

So, we do not really want to go with that one, because that will give us the force, and the force is not really useful, so the best choice for this one is, power = ΔE/t.1629

We know what the time is, can we figure out what is the change in energy?1639

We do not know what the time is, we are solving for the time.1681

But we do know what the change in energy is, we do know what the power is, so we are good to go.1686

That gives us, t= ΔE/power = (1/2)mv2/power = (1/2)×1500×(50)2/(5.22×105) = 3.59 s, that tells us how long it will take us for that car to accelerate form a dead stop, to going at 50 m/s, and that is equivalent to 110 miles/h.1702

3.59 s to get to 110 miles/h, or 50 m/s, that is pretty darn good, that explains why race cars are so powerful.1725

Hope you enjoyed this lesson, hope power made sense, just think of it as the change in work, the change in energy, and how long it took to get there.1732

There is a great analog between speed and velocity and energy and power, it is the same thing.1739

How fast are we changing, how much are we changing from moment to moment.1752

Hope you enjoyed this.1756

Hi, welcome back to educator.com, today we are going to be talking about centre of mass.0000

Before we try to explain what the centre of mass is, lets us just consider, if we tossed a ball into the air.0006

As we have seen before and learned before, the ball is going to fall a parabolic path, why?0010

Because we know that the horizontal movement is constant, it is just going to be a constant linear rate of movement, whereas the vertical motion is going to be based on a parabolic motion, it is going to be based on time squared, t2.0015

It is going to wind up giving us this nice curved parabola of motion.0028

You have seen this before, and if you see anything in real life, you toss the ball into the air, you can see it move as a parabola, you see a jet of water, you are going to see a parabola, parabolas are around us because of this.0032

But what is the thing that is moving in that parabola?0042

In this case, it is the ball that is moving along.0044

What if we threw something else, what if we threw a stick?0047

We grab one end of the stick, and we chuck the stick.0050

As it goes through the air, the stick is going to twirl.0053

But the stick as a whole, is still going to fall that parabolic motion.0056

If we look at this diagram, it cuts through the same location, a little bit before every point here, it is the same location that falls the parabola.0060

What do we call that location, what is that?0072

That is the 'centre of mass', this stick is still following a parabolic path.0076

It does rotate in the air, it has this centre, this centre of mass.0080

That is going to perfectly fall in the path.0084

We call this centre the centre of mass, for any system of objects, the centre of mass is going to be the point that move as if all the system's mass was located there, and all forces on the system were applied there.0087

This is really powerful, this means that we can have a complicated system that is having multiple things happening to it, so we might have little adjustments like, say, a car driving along on the free way.0102

Lots of things are happening to the car.0112

If you move around inside the car, and there will be little things happening inside of the car, we can look at the whole entity and say what is the centre of mass of the car, and its motion is going to be effected only by what is happening between the external environment and it.0114

We do not have to worry about the internal things.0129

So, centre of mass is a really powerful concept.0131

Here is an example of centre of mass: A fire works rocket is launched into the sky, and there is no air resistance, and this is important because if there were air resistance, it would completely break this example up, and we will explain why in a few moments.0134

The centre of mass of the system will take the same path, whether or not the rocket explodes.0149

Here, we shoot the rocket up, it gets to the top of its arc, it starts to fall down, it falls down, unimpressive!0153

But over here, we shoot it up, it explodes, we are used to thinking an explosion as, it goes in all different ways.0161

It does, but the centre of the mass of the system still continues to being the same.0169

Even as the arcs move outward as it falls, the centre of mass of that set of arcs, is going to still wind up being in the same place.0177

The centre of the system is always the same, because the explosion is internal to the system.0187

Why does it matter if there is no air resistance?0192

Because, the air resistance of the rocket on its own is a totally different set of air resistance, that all of the fragments of the rocket.0194

The fragments of the rocket are going to experience way more air resistance, and that is why it is going to have a shifting parabola, its arc is going wind up changing because of the change in air resistance once it makes that transition to be exploded.0201

But the centre of mass is still going to fall as being the unified thing, it will wind up changing the arc, because now the centre of mass is experiencing a new set of forces, because of the increased air resistance.0215

But this stuff about the centre of mass is still true.0226

It is easier to understand the idea when you think of no air resistance, because there is clear difference of exploding and non-exploding, having no difference in the way they fall.0228

That is surprising, it is not necessarily intuitive, but that is because of the way the centre of mass works.0235

Once we include air resistance, it starts to get more complicated, but the important thing to remember, is that the centre of mass, tells us how the system moves as the whole thing.0240

No matter what happens to the system internally, so explosion, you moving around inside the car, on a train, you throw up a ball, centre of mass of the whole system is going to be the same thing.0249

Let us figure out a way to get what the centre of mass is, as a useful mathematical formula.0260

Start off with these two pictures.0267

In this one, we have got the same mass here, and here, and so, it is obvious.0269

If you got two masses, they are the same thing, the centre of mass of the system is going to be in the middle.0273

If you had a stick, that was just equal density throughout, you put the finger to the middle of the stick to hold it up, right?0277

It makes sense, the middle is going to be there.0283

But what happens if one end of the stick was really heavy.0286

You got a big knot on one end of your stick, and at the end we got long and thin and tapered.0288

Over here, we are going to have a heavy mass, and over here, we are going to have a small mass, so what is that going to mean?0294

The centre of mass of the system is going to be way closer to the heavy part.0299

We have to take into account, both the mass of the objects, and how far they are, in terms of some location.0302

The distance apart matters, the distance from some reference point is what we will wind up going with, and how heavy each point mass in our system is.0308

Centre of mass in a system where we are using point masses, we use point masses, because we know that centre of mass of a circle is just the centre of the circle.0316

Centre of mass of the system, we consider the mass and distance together, the weighted average.0329

The location of the centre of mass, x = m1x1, (mass of the first one × location of the first one).0334

Then, m2x1, and that gives us a way to be able to unify both the idea of mass and its location, which we are to be able to do, to be able to talk about the location of mass.0345

Then, in addition to that, we are to be able to deal, with how much each one of those, put into the system.0357

If we have two giant masses, but they are equal masses, and we have two small masses, and equal masses, both these cases, we are going to wind up having the same centre of mass between these two systems.0363

So, we are going to have a way of dealing with not just, they are going to have very different m's, but those m's will get divided out, so it is what they are in relation to the rest of their system.0379

So, that is why m1+m2 is on the bottom, because we have to have a way of cancelling out what the mass we are dealing with is.0390

So, it i snot just about what the mass and location is, it is what the total masses of the system is, it is weighted average.0398

What if we were to do this with an arbitrary number of point masses?0405

The exact same method, we just need to expand it.0408

Now the centre of mass is going to be, (m1x1 + m2x2 + .... + mNxN)/(m1+m2+....+mN), for N masses.0410

One important thing to notice is that it is up to us to define our position system.0435

It is going to make things easier often, if we are able to define one of our points as a zero point, to be able to say, "Okay, let us make x1 effectively zero", and we will measure everything in reference to that x1, because we will gets distance from object to object when we are measuring things in the real world, but it is up to us, as usual, to impose a coordinate system.0440

An easy thing to do is often to say, we will just make one of these zero, and work from there.0459

Sometimes, that is not what you want to do, it is going to depend on the problem.0466

It is also going to depend on if the coordinate system has already been imposed for you.0470

We will have to take it in a case to case basis, but as a general rule, it is going to work some times, if you put one as a zero point.0473

If you already have position system, you have to use that one, if you get the chance to impose one of your own, you might want to consider making one of your points the zero point.0480

But, you do not always want to do that, as we will see in our fourth and final example for this lesson.0490

What happens if we are doing this in multiple dimensions?0496

Working with point masses in multiple dimensions, it is just the same, we just throw on vectors.0498

The formula holds true in each dimension on its own, the x axis dimension is going to hold true, the y axis dimension is going to hold true, the z axis dimension is going to hold true, so we just expand our formula to vectors.0502

x = (m1x1 + m2x2 + .... + m2x2)/(m1 + m2 + ....+ mN).0514

And, x is (x,y) or (x,y,z), so x is not just the x axis, it is now meaning all of the axes as one, we are now saying, our location is called x.0529

This just becomes, what your location is for a given thing.0543

Another idea: If we have an object that has a uniform (or homogeneous, meaning same throughout) distribution of mass, we can use symmetry to find the centre of mass.0546

For example, this square, the centre of this side is here, and the centre of this side is here.0559

You put those through, and BOOM!, we see through symmetry, that the centre has to be there.0565

We do the same sort of thing here, the centre of this side is here, the centre of this side is here, BOOM!, we find the next one.0572

Triangles are little bit more complicated, but here if you want to get very clever, we can figure out that the amount of area above is the same as the amount of area below.0582

So, we know that this is our dash line.0598

How do we figure out the width?0600

That is very easy, we just drop a vertical line from the very centre, since in this case we got an equilateral triangle, and that is how we figure it out.0605

That might be a little bit harder to do on our own, but we can at least get an idea of where it approximately is going to be.0614

What if we had a torus (torus is a special word for a donut), or a circular disc, that is missing the inner disc?0619

In this case, where is the centre of the big circle?0627

Centre of the big circle is just the centre of the circle.0629

Where is the centre of the small circle?0632

Centre of the small circle is also going to be the centre.0633

So, we have got this removed centre out of the middle, it is not going to affect the large thing, because the centre from all these points is going to wind up still being here.0637

One centre out of another centre, it is still going to wind up being the same thing.0647

So, our centre is still just in the middle, because we have got this sort of negative mass, taken out of a solid disc, and both of them wind up having the same centre of mass, so we have got it.0651

What do we do if we were given a non-homogeneous thing?0662

Something with the mass is not evenly distributed through the object, or what if we have an evenly distributed object, but it is a weird, complicated, non-symmetric shape, like this, how do we find out the centre of mass?0665

Well, you do not! We are screwed, I am sorry.0683

At the moment, we cannot do that, but that is just for now.0686

The reason why we cannot do it, is just because we do not have calculus in our tool box.0691

To be honest, the stuff that you will learn in calculus to do this, actually is not that hard, it is totally in the range of doing, once you got little bit of calculus under your belts.0694

You will not find it that hard, once you take calculus, but right now we just do not have the ability to talk about it, we have to refer to integrals, and right now, we cannot refer to integrals, because we do not know what calculus is, for this course.0703

But, once we do get to calculus in the future, you will totally be able to understand it, but right now we cannot, so we will not be discussing any hard centre of mass problems.0712

We still got lots of things that we can do.0726

Finally, what is the centre of gravity?0730

Often, the centre of mass, the centre of gravity, they get thrown about, completely interchangeably.0732

But, there is a very minor distinction between them, what is that distinction?0737

Centre of mass is where the aggregate mass of the system is located.0741

We can think of that as, where the average, the place where we can consider all the mass effectively is.0745

The car might be in different places, but we can treat it as a point mass, by being able to figure out where is its mass located around the centre.0751

Like this stick, like the rocket, where is everything sort of located around, that is where the centre of mass is, the aggregate, the sort of average where we can locate the mass as a specific point, even if that specific point is hollow, like in the case of the donut or the torus.0761

Centre of gravity on the other hand, is a little bit different.0775

Centre of gravity is the place where the force of gravity acts on a system in aggregate, where the average of the centre of gravity is.0777

Consider, if we had a super tall building.0784

A super tall building is going to have a different gravity here, a than up here.0788

They are going to be very similar, but the farther you get away from earth, the lesser the effect of gravity of earth is.0793

On a really tall building, hundreds of storeys tall, that building is going to experience slightly less gravity than when you are on the top floor, you will weigh ever so slightly less than when you are on the bottom floor.0799

This means that what is up here, is going to be affected less by gravity, than what is down here, which means that the centre of mass might be in the very centre of the building, the centre of gravity is going to be a little bit below that, because more of the bottom is going to be affected by gravity than the top.0812

More of the gravity contributes towards where the centre of gravity is located.0829

Now, keep in mind, for a small object, these are going to be the same thing.0833

These are going to be so minisculely different that we can effectively treat them as the same thing.0837

For vey large, very tall objects, the pull of gravity will be slightly different.0841

Keep in mind, slightly different in this case means that for the tallest of skyscrapers in the world, we are talking about a matter of 1 mm, and that is like the tallest of skyscrapers.0847

For our purposes, we can pretend they are the same thing.0859

In result, we will be able to consider that the centre of gravity, and the centre of mass effectively same thing.0861

In reality, they are not precisely the same thing, but for our purposes, they are going to be so close, we might as well treat them as synonyms, we might as well treat them as the same thing.0867

Newton's second law and the centre of mass:0876

We got his great, really useful formula from Newton's second law: F = ma.0879

We can expand this idea, to work on an entire system by using the centre of mass.0885

The net force on a system, Fnet = (total mass of the system)×(acceleration of the centre of mass) = MaCM.0889

F, if it is internal, so if we have some system, and some F occurs inside of the system, what do we know from the Newton's third law?0909

We know that there is also resisting inner force, that is going to be the same thing.0916

If there is an internal force that happens inside of the system, the internal system force is also going to result in another internal system force, equal in magnitude, opposite in direction, equal and opposite reaction force.0920

We know that from Newton's third law.0933

So, if our force is internal, it is going to get canceled out by its reaction force, so we do not have to worry about having a net force in the end.0935

The only forces that we are going to have to worry about, are external forces.0941

If it is an external force, we have to care.0946

Finally, acceleration of the centre of mass, is just the acceleration of the centre of mass.0949

We do not necessarily know anything about the specific objects that make up our system.0954

Our system is made up of many different things, many disparate objects.0959

It might be the case that the acceleration of the centre of mass is going one way, but one piece of our system, is going in the opposite direction.0963

Just because we know something about the centre of mass, does not mean we know everything about the system.0970

It just gives us one more piece of information to understand our system.0974

It is really useful in many cases, but it will not necessarily tell us everything.0978

Once again, you have to pay attention, you have to understand what you are looking at, and really be thinking about the problem you are working on, not just blindly follow formulae.0981

We are ready for our examples.0990

Nice one to start off with: We have got three balls on a mass less rod.0992

Mass less means we do not have to worry about it for our centre of mass, all we have to worry about are the balls on the rod.0996

The first ball, is right here, and it weighs 4 kg, the next ball is 1 m away, and it weighs 1 kg, and the next ball is 2 m away, and it weighs 2 kg.1002

How do we figure out where the centre of mass is?1025

Do we have a coordinate system yet? We do not have a coordinate system yet.1028

We know what the distance is, but we have to impose a coordinate system.1031

The first thing is to impose a coordinate system.1034

I like the 4 kg ball, because it is right on the left, we are used to going right as positive, and the 4 kg, since that is the biggest, most massive, the centre of mass is going to be the closest to 4 kg, it is probably going to suck up most of the centre of mass towards it.1036

We will call this (1), x1, and it is going to be zero, it also means m1 = 4 kg.1056

What is this going to be located at? It will be at a location of 1, because it is 1 m away.1063

So, m2 is 1 m, m2 = 1 kg.1068

Finally, x3 will be at 3 m, and correlates to m3 = 2 kg.1070

xCM = m1x1 + m2x2 + m3x3)/(m1 + m2 + m3).1082

So, xCM = (4×0 + 1×1 + 2×3)/(4+1+2) = 7/7 = 1 m away, which means that the centre of mass is actually right there, right under that second ball, that is our precise centre of mass.1095

Next up: We got three point masses, and this time we are going to do it multi dimensionally.1128

m1 = 1 kg, m2 =5 kg, m3 = 2 kg, we got located at point, x1 = (0,10,0), x2 = (2,1,3), x3 = (3,-2,-4).1142

What do we have for vectors?1155

x = (m1x1 + m2x2 + m3x3)/(m1 + m2 + m3).1159

So, (1×(0,10,0) + 5×(2,1,3) + 2×(3,-2,-4))/(1+5+2) = [(0+10+6),(10+5-4),(0+15-8)]/8 = (16,11,7)/8 = (2,11/8,7/8) m, which tells us the position of the centre of mass.1174

There is nothing specifically there, but we know that, that is where the aggregate mass is, we will be able to do a force that worked on the whole system, worked on any part of the system in fact, that would effectively work on the whole system, because that is working on the system.1287

So we know that the centre of mass is going to be adjusted based on this, so that is where our starting location is.1301

Example 3: Now things are starting to get [?].1306

Now we are going to have to do some real thinking as opposed to following just formulae.1309

To begin with, let us look at a problem: A circle of uniform density with radius 2r has a circle of radius r removed from its far left side.1312

Where is the centre of mass?1320

This thing is not simply symmetric.1322

What we can do, is we can think of it as a trick, we can come up with a trick to think of this, we know what the centre of mass of the whole circle is.1325

It is just the centre of the circle.1334

We can consider the large circle as a whole circle.1336

So, our big circle would have its centre of mass at its centre, at x = 0, let is make here the very centre, x = 0.1344

We know what that centre is, we can figure out what its area is, we have a uniform density, we do not know what the mass is precisely, but that is going to work out in the end, because we are going to divide by the mass amount in the end.1354

But, we have to also deal with this part that is missing.1367

Here is the trick: We can think of the part that is missing not being there, but being there as a negative, we can think of a big circle that had a smaller negative circle, an anti-circle added to it.1371

If we put these together, then it is going to equal, a big circle with the negative circle cut out off it, now it kind of look like the Death Star, but that does not matter.1397

So, we have got, (big circle) + (negative circle) = (this thing right here).1412

How do we do this in terms of math?1418

What is the formula for the area of a circle? A = πr2, so for the big circle, A = 4πr2, and the location of its centre is zero.1427

The centre of big circle is, xbig = 0.1448

What about the nega., what is the area of this one?1451

A = πr2, and xnega = -r, making going to the right as positive, so now we have got a location for this.1454

Now, we could figure out what the mass is hypothetically.1477

For some amount of area, we get some amount of mass, there is going to be some conversion factor.1480

We wind up using this conversion factor both on the top and the bottom because it will be divide by the total mass, so ultimately what we really care about, is the area, because it is going to be a constant conversion factor throughout for both our negative circle, (which is going to be the negative version of that conversion factor), and the big circle.1486

So, in the end, we can just turn out our xCM to paying attention to where the area is distributed around these centre points.1504

So instead, we can make this xCM = (m1x1 + m2x2)/(m1 + m2).1510

In this case, (1) is the big circle, and (2) is the negative circle.1521

So, xCM, what is the mass, what is the effective mass of the big circle? The effective mass of the big circle is just how much area it has to sling around.1537

That is, 4πr2.1546

But, multiplied by its location, i.e. zero, so it is going to get knocked out.1550

How much area does the negative circle has to sling around? It has got -πr2, negative area because we have to be removing it, we know that the total area of this thing is going to be (big circle) - (negative circle), so it has got to be -πr2 is what its real area is, because what it contributes, is how much it takes away, it is a really important thing, I do not want you to be forgetting about that.1557

So, 0 - πr2×(-r) / (4πr2 - πr2) = πr3/3πr2 = r/3.1612

Let us think about that before we quickly assume that it is right.1661

That makes sense, we got a big circle which is missing a chunk, so where do you think the centre of mass is going to be?1664

Is the centre of mass wind up begin here, closer to the non-existent part?1671

No, there is no mass there to pull it there, so it is going to wind up showing over here.1677

The centre of mass is going to get shown at r/3, it is going to wind up being pulled over just a little bit, because it is missing that chunk, so it is going to be a little bit over.1684

A circle of uniform density, radius 2r, has a circle of radius r removed from its far left side, and we find out that the centre of mass has to be located at r/3 to the right of this place that has not been removed of the larger circle.1696

Final example, Example 4: A boy of mass mb = 40 kg, is standing on the right edge of a raft, that is 2 m long, with mass mr = 20 kg.1712

A camp counselor comes to hand in a piece of mail from the edge of the dock, but he can only reach out 1 m.1723

The boy can reach out 0.5 m, but the left edge of the boat is 0.5 m from the dock.1728

Will the boy be able to walk over and get the mail without rowing the raft?1733

The first thing to think is, yes, of course he can walk over, he walks over here, and grabs his mail, right?1735

Wrong! It just cannot happen like that, he cannot just walk over, because if he walks over, what happens is, there is no friction between the water and boat effectively, it is going to wind up getting moved.1743

So there is no external forces over here, we do not expect the water to resist, it does not give us much friction.1758

So the boat is going to wind up sliding this way, the centre of mass if the system will be preserved, so by the time the boy gets to the left edge of the boat, he might be only be here, and it might be the case that the boat edge is like this.1763

The boat and the boy represent the system, and we know that the centre of mass of that system will be preserved, because there is nothing to keep it in place, there is nothing to hold it in place, there is no external forces acting on the system.1782

As the boy moves around, the boat will have to move in response to that.1792

The centre of mass of the system has to be preserved.1796

If you are an astronaut in space, and you threw a wrench, and the wrench would fly away from you, but in result, you would wind up floating backwards, because (1) we can think about it as the response force of the wrench on you, but we can also think of it, as the centre of mass of the system is preserved.1799

They are the same thing, equivalently, the idea that forces come in pairs that cancel one another out, is the same idea as the centre of mass of the system is not affected by the forces, they are equivalent.1821

Now, we are going to have to get down and do some maths.1836

What is the centre of mass of the system to begin with?1840

To begin with, centre of the mass of the raft, it is reasonable to assume that the raft is homogeneous, or at least that it is symmetrical, so we can assume that the raft's centre is in the middle of it.1843

The boy is at the edge at 2 m, and we know that xr is going to be at 1.5 m, but we have not discussed yet what our coordinate system is.1855

Let us make, to the right positive, but where are we going to make zero?1864

In this case, we are going to have shifting motion between our before and after pictures, before the boy starts moving, after he walks over.1867

So, it is up to us to figure out what is a good stationary reference frame that we can use.1874

In this case, I think the pier is a good place to make x = 0.1882

The pier is going to stay there, and that is what we care about.1886

We care if the boy can get 1.5 m away from the edge of the pier.1888

Camp counselor can reach out 1 m, the boy can reach out 0.5 m, so if the boy can get to x = 1.5 m, or less, he is good, he can get his mail without having to row.1893

If he cannot, he has to use the row, and actually move the boat, or swim over.1902

At this point, we can solve for what the centre of mass of the system is.1908

So, xr = 1.5 m, xb = 2.5 m, the edge of the boat was 0.5, and the boat itself was 2 m long.1913

What is xCM going to be?1924

xCM = (m1x1 + m2x2)/(m1+m2) = (20×1.5 + 40×2.5)/(20+40) = 130/60 = 2.17 m, is the centre of mass of the boat-boy system at the beginning.1929

If you think about where that is, that winds up being somewhere over here.1978

Now, without even doing the rest of the math to figure out where the boy can get to, we can figure out he is not going to be able to do it.1981

How can we figure out? We know symmetry is going to work on our end here.1989

The boy, if he moves here, is going to be directly over the centre of mass of the system.1993

That means that when the boy gets to here, the boat is going to wound up getting here as well, the centre of mass of each of the objects is going to have to line up with the centre of mass of the system.1997

Each of the objects is moved to the centre of mass of the system, then the centre of mass of the system is going to be, where the objects are, if the object moves to the centre of mass, then the other object will also has to move to the centre of mass since it is a two object system.2007

That means, if the boy is at one extreme here, then he is only going to be able to make that much further and jump out afterwards.2022

The boat will wind up making out its centre of mass, will also wind up getting up over here, so the boy to get to the other maximum is only going to here.2030

So, the difference between that was less than 0.5 m, to get to here, and then it will be less than 0.5 m, to get here, so this is going to be greater than 1.5 m.2039

We do not know the precise numbers right now, but we can at least know that he is not going to make it.2046

But, let us figure out exactly what it is, for practice.2050

Now we have got our after picture, the counselor is standing here, still, at x = 0.2055

But now, the boat, is there, and the boy, is on the left edge.2062

Do we know where the boy is located?2070

No we do not, that is the whole point if this.2072

So, xb = ?2074

Do we know where the raft is located?2077

If we know where the raft is located, we know where the boy is located, exactly because of that, we know that the two are related, so we can think that, xr = xb + 1, because he is going to be 1 m from the end, because he is going to walk as far as he possibly can.2079

We have got, xr = xb + 1, xb is unknown, but we know that xCM is going to remain 2.17 m from what we solved before.2095

If we know that xCM = (m1x1 + m2x2)/(m1+m2).2104

2.17 m = (20×(xb+1) + 40×xb)/60.2116

130 = 20xb+20+40xb, 100 = 60xb, xb = 1.83 m, is the boy's location at the end, so we know he cannot get the mail.2138

He cannot get mail without rowing.2172

1.83 is where he finishes getting up to, and we know this because we know that the centre of mass of the system is going to have to be the same before and after, because there is no external forces on the system.2177

We know that there is a connection between where the boy's location is, and the centre of mass of the raft is, and we can treat that as a point mass for the raft, so that gives us enough information for us to be able to solve for where the boy must land.2186

And there you go, we just do the math, and we will be able to figure it out.2197

Hope you understood this, hope it made a lot of sense.2200

It will be really useful when we will be talking about momentum.2204

We will be able to talk about collisions, which will give us pretty much all the understanding we need about kinematics and force.2209

Hi, welcome back to educator.com, today we are going to be talking about linear momentum.0000

From our work and energy, we already know that the mass and the speed of an object is able to determine its kinetic energy.0006

But, when we were dealing with energy, we only had speed as the way of determining energy.0012

There was not anything talking about direction.0017

Kinetic energy was great for telling u slots of stuff, but it did not tell us if we were going to the north, south, up or down.0019

To capture that, we are going to introduce a new idea: Linear Momentum, what your motion is along a line.0026

We want the linear momentum to talk about an object's motion in a given direction, just like energy gave us an idea of speed and mass for an object, linear momentum will give us something that tells us about the motion of an object.0032

Consider two scenarios: We got a box moving at the same speed but in opposite direction.0045

To capture the difference, we will not be able to just use the speed, because it is moving in the speed in both the cases.0052

But we will need to also capture its direction.0057

To do that, we need to use vector.0060

We are going to have to use v, not just 'v', the speed, we need to have its actual velocity.0062

Also, what if the boxes had different masses?0071

If we had two different boxes, that were both moving in the same speed, one of them was say 1 kg, and the other a 100 kg, we probably want to think of them as being different things.0073

It will take a whole lot more effort to stop a 100 kg box than the 1 kg box.0083

So, direction is part of it, but we are also going to have to take into account, the mass.0088

Clearly, it is going to a good idea to include mass in our idea of linear momentum.0093

So, we are going to have to be able to deal with the velocity vector, not just the speed, and also mass.0097

Put in that together, we get, linear momentum, and we define it as, mass×(velocity vector), mv.0106

Notice that, since this is a vector quantity that we are dealing with, it is going to be, how much we are moving in the x coordinate, how much we are moving in the y coordinate, if you are also moving in the z coordinate, it is going to be, mvx + mvy + mvz.0116

We are going to break it up as a vector, and m will just scale the vector.0135

These two characteristics, they define, what we are going to create as momentum.0138

Momentum, p = mv.0142

Why do we use a p?0146

I honestly do not have a good answer, I wish I did, there are possibilities, it has a Latin root, but I was not able to figure it out, sometimes there are mysteries in the world.0148

An important thing to notice here is that, this p is not just a scalar quantity, it is not just a single number, it is a vector.0166

If v comes in (x,y,z), our p is also going to have to come in (x,y,z).0174

Units of linear momentum are, mv, so kg×(m/s).0183

We are going to consider a new idea: Impulse.0191

What if we want to talk about how much an object's momentum changes, that is important.0193

If we got a box moving along, and if we put a force on the box, we are going to change the momentum of the box, because we will change the speed that it is moving at.0198

So we are going to define the idea of impulse.0205

What really changes the velocity of the thing?0207

It is just going to be the fore involved, but not just the force.0212

The same force is going to be very different if you put a 100 N on an object for 0 s, 1 s, 10 s, 100 s, totally different things are going to happen depending on the amount of time that the force is acting on it.0217

The objects mass remains constant, pretty reasonable.0227

The object's velocity, and thus its momentum, is going to change based on the force applied, and how the long that force lasts.0232

We define impulse as the letter 'j' (no particularly good reason here, just making sure we are using letters that have not already been taken by somebody else), j= force×time = Ft, just like before, j is a vector because force is a vector.0238

Makes sense, because we are talking about change in a vector quantity.0255

Note that impulse is a vector, and its units are going to be, Ft, so N s.0259

At this point, we have created some definitions, and we can see that linear momentum and impulse are connected because we wanted impulse to represent a way of shifting around momentum.0268

You put force into an object for a certain object of time, it is going to change the momentum that the object has, because we will be changing the speed that it is changing at.0279

But, what is the precise mathematical relationship?0286

Let us figure it out.0288

We look more closely at the formula for impulse.0289

j = Ft, if we expand that out, we can get this.0292

F = ma, so, j = mat = mΔv, (since 'a' is how much your velocity is changing with time).0299

Since mass is not changing, we can pull that change outside, and we get, Δv, because we do not have to worry about, since velocity is the only thing that can change, we are assuming that the mass is constant, so, mΔv is the same thing as, Δ(mv), because mass is just a constant, and velocity is the variable, at this point.0328

Remember, we defined, p = mv, so it has to be the case that, Δ(mv) is the same as, Δp.0350

So, in the end, j = Δp, so impulse is simply the change in the momentum.0362

Note that impulse and momentum have the same units, 'N s' is the same thing as 'kg m/s', because N comes from, if F = ma is kg m/s/s, so, N is kg m/s/s, and multiply with s, so kg m/s, is what we had for momentum.0371

It makes a lot of sense, our units wind up working out, so, j = Δp.0404

In this section, we have talked about what linear momentum is, but why have we talked about 'linear' momentum, when we have not heard about any other kinds of momentum, why is it called linear momentum if it is the only momentum that we are concerned with?0417

The thing that is going on, is there are other kinds of momentum, there is angular or rotational momentum.0438

Spinning objects, objects that are spinning, if you take a wheel and you spin it really fast, it will keep spinning, right?0445

It has a momentum, it is not moving anywhere, it is just sitting there in space and spinning, but it takes effort to start it spinning, and it takes effort to stop it spinning, so there are torques involved, we have not talked about rotational mechanics.0452

The entire thing in Physics you cannot talk about, but we just do not have quite enough math to really feel comfortable handling it, we are almost there, this is definitely close to being within our grasp, but a little too much math for us to tackle there, so we are holding off on it, that is why we have not talked about angular momentum, which is also similar to rotational momentum.0465

We have been talking about linear momentum, because we want to make sure that this is kept clear, as this is linear momentum as opposed to this other kind of momentum, but often when we are talking about linear momentum, we will also just refer to it simply as momentum, because that is the thing that is more common, but it is important to keep in mind that there are other kinds of momentum.0485

Just because we are talking about one of them, does not mean that there is nothing else out there.0501

A skate board of mass 4 kg is rolling along at 10 m/s.0508

What is its momentum? This one is pretty easy.0512

What is the basic definition for momentum? p = mv = 4 kg × 10 m/s = 40 kg m/s. (since one dimension, velocity becomes a single number).0513

Same skateboard, m = 4 kg, is rolling along with an initial velocity of 10 m/s, just like before.0538

A force of F = -6 N is applied to it, for t = 6 s.0547

At the beginning of this problem, we got some skateboard rolling along on the ground, and it is moving this way.0551

However, as time moves on, there winds up being a force applied to it in this direction, so later on, this skateboard is going to be rolling along with a much smaller velocity vector.0558

It is still going to be moving forward, potentially, depends on how long that force is actually, may be that force is going to push it so hard that it winds up going in the other direction, we are going to have to do some math to figure it out.0569

But, the force is acting on it in the direction opposite of current travel, we are travelling in the positive direction (right), and now force is going to wind up in the negative direction.0583

That is the importance of using vectors, we know how positive and negative direction, even if we are still on one dimension.0596

So, What is the impulse vector?0602

Impulse, j = Ft = -6×6 = -36 N s, so what is the final velocity that it is going to have?0604

In this case, we know that change in momentum is equal to the impulse.0629

We already figured out what the initial momentum is.0635

In the last problem, it wound up being, 4×10 = 40 kg m/s, so the final one is going to be, pi + j = pf, since pf - pi = j.0638

pi = 40, and the change is -36, so in the end we get, 4 kg m/s = pf, is the final momentum.0670

Final momentum does not quite tells us the final velocity.0684

But we can figure that out pretty easily from there.0687

pf = mvf, 4 = 4vf, so, vf = 1 m/s, is the final velocity.0688

It is still moving in the positive direction.0709

It would be possible to figure this out without using momentum and impulse, but momentum and impulse may dissolve pretty simple things that we have to do, very direct, multiplying and then adding, and then doing some really simple algebra.0711

but, we could go back to doing this with Newton's second law.0725

If we want to do this in Newton's second law, we have got, F = ma, -6 = 4a, a = -1.5 m/s/s.0729

What does the change in velocity wind up being?0753

Δv = -1.5×t = -1.5×6 = -9 m/s, and so, if you started with initial velocity of 10 m/s, then, vf = 10 + (-9) = +1 m/s.0755

So, if we wanted, we could do this in terms of basic fundamental, Newton's second law, but in this case it is pretty easy.0780

Remember, the way momentum works, the way we have defined it, is really just sort of jumping off the point of using Newton's second law.0789

That is how we got impulse.0796

Impulse was based around the fact that, the reason why impulse is equal to the change in momentum is because we used F = ma, at what point we got, Ft, so in the end, they are deeply interconnected.0797

So, we can decide to go with Newton's second law, but in lots of problems, it is going to wind up being the case that it is actually a little bit easier the way of linear momentum, especially when we are dealing with momentum problems.0812

In the next lesson, we are going to wind up seeing why it is really useful to have momentum when we get to the conservation of momentum, and that is why this stuff really matters.0821

Example 3: A ball of mass m = 0.5 kg is moving horizontally with vi = 10 m/s.0835

It bounces off a wall, after which it moves with vf = -7 m/s.0842

What is the change in linear momentum, what is the magnitude of the impulse?0848

We got this ball, moving along the positive direction, and it hits the wall, and afterwards it changes, rebounds, and it is moving in the negative direction.0851

What is the initial momentum? mv = 0.5×10 = 5 kg m/s.0862

What is the final momentum? 0.5×(-7) = -3.5 kg m/s.0885

So, the change is, final - initial, Δp = -3.5 - 5, seems a bit weird, but makes sense, we started off in the positive direction, and ended up going in the negative direction, so the entire change has got to be one of negative momentum occurring.0898

So, -8.5 kg m/s is the change in the linear momentum.0937

The magnitude of impulse, remember, the magnitude is the size of something, so the size of this, j = Δp = magnitude(-8.5 kg m/s) = 8.5 kg m/s.0945

In the end, when we are dealing with magnitude, it does not care about direction, it does not care about positive or negative, it just cares what [unclear] like the thing we are dealing with.0972

In this case, we had -8.5 as the change in momentum, but the magnitude of the change in momentum was just the total moment, 8.5.0983

This is sort of similar to what we saw in energy before, instead of using velocity, it was the length of the velocity vector that we cared about, its speed.0993

It did not matter if it was pointing flat, it was pointing straight up , pointing at a 45 degree angle, all that mattered was what the total length was.1001

That is what we are seeing here, when we ask what the magnitude of the impulse, we are asking for what is the length of that thing.1010

In multiple dimensions, we take, sqrt(x2+y2+z2), because that is how we take the magnitude of a vector.1017

Last example: A ball of putty with a mass of 1 kg is about to fall on your head.1027

The velocity initially is -5 m/s, makes sense since it is falling down.1032

Which one is going to hurt worse, it lands and sticks on your head, so it lands and sticks, so the final velocity is zero, or, it lands and bounces off your head, with a final velocity of 4 m/s.1037

Let us call these cases, A and B.1048

In case A, it hits your head, and it sticks in place.1052

Which one of these is going to hurt worse?1062

Normally we think of things being bouncy as good, they are easy right?1065

But what is really going on, which one is taking more force, the super ball that hits your head and just sort of sits there, or the ball that hits your head and ricochets right off?1075

We can break this down with Physics and Math.1083

In both cases, what is really going to be defined as hurting?1086

The way that we define as 'hurt' is probably the amount of force that it exerts on you.1092

So, you would rather have 2 N of force shoved on you than a billion newtons of force being shoved on you, a billion newtons force applied on you, your body is going to look like a pancake.1097

But 2 N of force, a small amount of force is going to hurt less than large amounts of force.1108

What we are looking for is, which one of these cases is going to produce less force on impact.1115

In both of these cases, it is going to be important to know how long the balls contacting your head, we will be able to figure out what average force is going to be.1121

Impact time for both of these cases will be 0.25 s, so let us figure out how much force is involved.1128

To do that, we need to figure out what the impulse in both cases are.1135

To do that, we need to know what the initial velocities are, what the initial momentum is, and what the final momentum is.1139

From there, we will be able to figure out the impulse, and from impulse, we will be able to figure out pretty easily what the force is.1145

In both them, we need to know what is their initial momentum.1150

Initial momentum = m×vi = 1×(-5) = -5 kg m/s.1154

Case A, it lands and sticks on you head, so its vf = 0.1168

If that is the case, what is the final momentum?1174

Still mass of 'm', but now it sticks, so it now has no velocity, so it has got a momentum of zero in the end, 0 kg m/s.1177

Compare that to B, where we have got a final momentum = 1×4 = 4 kg m/s.1189

It might make good sense at this point to go, 'Okay, the one with less momentum is case B, because we go from 5 to 4'.1207

But that is not the case.1215

The whole case is, we go from -5 to 4.1217

So which is a bigger change, going from -5 to 0, or -5 to 4?1220

We check that out, we get, Δp = (final) - (initial) = +5 kg m/s, is in A.1224

In B, Δp = 4 - (-5) = 9 kg m/s.1240

So, there is more change in momentum in case B than in case A.1254

So it is going to make more sense for B to hurt, because that force, to change that momentum, has to come from somewhere.1257

We have got a constant time, so it has got to be, the amount of force involved is going to have to be more in case B.1264

We will finish this out, at this point we can see that the answer is going to be case B.1272

So, j = Ft, we also know that j = Δp, so for case A, we got, 5 = F×0.25 s, F = 20 N.1276

So, 20 N is how much force you wind up undergoing for A.1301

Now, how much force does you wind up undergoing in B?1306

j =Ft = Δp, so, 9 = F×0.25, so, F = 36 N, in part B.1310

So, part B winds up putting more force on your head, in both cases, they are pretty small, so worst case you are going to have a little bit of headache, but 36 N is more force you have to suffer than 20 N.1328

So, it is actually better to have an object that lands on your head and just sticks there, something that goes 'splat!', than something that goes, 'boing!', because 'boing!', that force to make it bounce off is going to have to come from somewhere.1340

It is coming from your head, that is going to make it hurt more.1351

It is better to have it land and splat, it has less force, because it has to change its momentum if it is going to be able to bounce off your head.1354

I hope this lesson made sense, I hope you got a good understanding of linear momentum, because the next thing that is coming is conservation of momentum, and that is where the real point of momentum is.1361

Alright, good day!1368

Hi, welcome back to educator.com. Today we are going to be talking about one-dimensional kinematics. 0000

Kinematics in a single dimension. What does kinematics mean first of all?0004

Kinematics just means study of motion.0009

Something that is going to talk about how does something move.0012

That is what we are going to be learning about today. 0015

How do we talk about motion! How do we talk about things moving around! That is clearly an important part of Physics.0017

First idea, position. Position is simply the location of an object at a given moment in time. 0022

For this one, we could have something like this, and BOOM!, it would be a 3. Whatever 3 means. That is a really important point to bring up. 0027

We are the ones who have to impose the coordinate system.0035

Nature does not come with an inherent coordinate system already told to us. 0038

We have to say, okay, this place is zero and then we are going to measure metres this way.0042

So, we get 3 metres, or 3 whatever it is that we have measured.0048

But it is up to us to decide where we are going to put that zero and how we are going to measure off it. 0052

We impose the coordinate system. It is us who are assigning position value ultimately. 0057

It is coming from nature, but where our starting place is, that is on us, it is on us to figure out how we are going to orient things.0062

That is a really important thing in Physics sometimes.0069

Next idea, distance. Distance is just a measurement of length.0073

For example, let us say you start at your house.0076

And then you walk a 100 m to the North. 0079

And then after that, you feel like walking a bit more, so you walk another 50m to the North. How far would you have traveled? 0082

Well, you just put the two together, and BOOM!, you have got a 150 m of travel to the North. 0088

Displacement is change in position. Displacement is very similar to distance, but sometimes they can be very different. 0097

Consider the following idea: Once again, you start at your house. And you walk a 100 m to the North.0103

But then, you keep feeling like walking, but you do not feel like walking to the North anymore, you decided to walk to the South. 0109

Now, you walk 50 m to the South. What is your displacement? 0115

What is your change in position from where you originally were. 0120

Well, your change in position from where you originally were is, that much.0123

You are now 50 m to the North of your house. So, your change in displacement is 50m North. 0127

But what is the distance that you traveled!0135

The whole distance that you traveled, you wound up going up here, and then you turned back and then you walked another. 0137

So, you walked a 100, and then you walked a 150 total.0142

You walked a 100 up, and then you walked a 50 down, so the total amount of distance you traveled was a 150 m. Very different numbers here. 0146

How did we get from where we are now to where we end up, versus distance, which is what was the total length we took.0159

Just to compare them once again, distance: how far we traveled, how far an object travels, where displacement is the change between the start and the end points. 0166

Notation. When we are talking about these ideas, we want some kind of shorthand variable to denote them.0176

We are going to solve for this stuff eventually. 0180

It might seem confusing at first, but we are going to wind up denoting them all with just simply 'd', 'd' would be the letter we use to denote it. 0182

Location, distance and displacement, they are all going to get denoted with 'd'. 0189

And it is going to wind up getting obvious that, over time, we know what we talking about.0194

We know we are talking about distance, sometimes we know we are talking about displacement. We are going to get it contextually. 0198

We are going to get it based on how we are working with things. 0203

We will know what we mean by 'd', do not sweat over the fact that it currently means three different things. 0205

It will make sense when we are actually working on problems.0210

Speed. Speed is an idea I am sure you are all already familiar with. It is just how fast something moves. 0214

If we want to know how fast something moves, it is going to be the distance that it traveled, divided by the time it took to travel it.0218

You can travel a 100m, but there is a big difference in it if you travel it in 1 s, or if it took you a 1000 hours to travel it.0224

That is where speed comes in.0230

If a car travels 100 m in 5 s, What would its speed be? 0232

Well, speed, simply the distance traveled, 100 m divided by 5s, and that is going to get us 20 m/s.0236

It travels 20 m/s, because every second it goes 20m.0246

This is an important point. m/s is the standard S.I. metric unit for motion, the 'm' and 's' being the units, which brings this into begin with.0252

Velocity. Velocity, is very much like speed, but instead we are going to be talking about displacement.0264

Velocity is equal to the displacement divided by the time.0269

So, it's how much the position or the location has changed from beginning to end, not the total distance that it travels over.0272

A race car can travel around a circular track 200 times in a couple of hours, but what was its velocity on the average!0280

Well, it did not leave the track, it just eventually parked in the same spot it left from, so it had no average velocity.0287

But it would have had a very great average speed because it whipped around that track very quickly.0293

We denote velocity with a 'v'.0298

New idea. Delta, Δ.0302

It is the Greek letter Δ. Δ is a capital Greek letter pronounced Delta.0305

It is used to indicate 'change in'. 0310

If we want to indicate a change in location, we can say Δ d, which would be the starting location subtracted from the ending location.0313

Final location, dfinal - dinitial, which is what displacement would be, right?0321

To figure out what your displacement would be, it is going to be where you ended minus where you started.0327

So, Δ, the idea of 'change in', lots of time, we are going to have to consider not what the absolute value is, what its whole value is, but what was the change from the beginning to the end.0333

Really important idea.0342

Get this in just a simple, this nice Δ symbol.0344

For velocity, v = the Δ of the distance.0350

Δ (Displacement) divided by time, Δ (Displacement) divided by time for velocity. 0354

If you want to talk about speed, it would be Δ (Distance) divided by time. 0359

Acceleration is the rate at which velocity changes. 0364

It is a big difference to be in a truck that accelerates really slowly, it gets from 10m/s to 30 m/s over a course of 10s, versus an incredibly fast sports car, that gets from 10m/s to 30 m/s in 2 s.0367

So, acceleration is how fast we manage to change velocity, or in our new letter, our new formation, that we can use Greek letters, new way of talking, we have got acceleration = Δ v / t .0383

Change in velocity divided by time. 0396

If we had a car moving with a velocity of 20m/s, and accelerated at a constant rate of 40 m/s over 10s, what would be the car's acceleration?0400

Well, acceleration, is equal to Δ v, so Δ v/t, 0408

Δ v is vfinal - vinitial, divided by time, we plug in the numbers we have, vf = 40, ended at 40 m/s, it started at 20 m/s, and it is divided by 10s.0415

So, that is going to get us, 20/10 or 2.0438

And the units we get out of this is, we had m/s on top, and we divided by seconds on the bottom, so it is going to be m/s/s.0443

Gravity is a constant acceleration that always points to the centre of the Earth.0453

It is always -9.8 m/s/s.0457

So, if you are anywhere on Earth, you are going to experience an acceleration of -9.8 m/s/s.0461

The reason that you are not currently falling through the floor to the centre of the Earth is, you got something resisting that.0467

If you jumped to the air, for that brief moment, before you eventually landed on the ground again, you would be accelerated down towards the centre of the Earth at -9.8 m/s/s. 0471

That negative sign denotes that we are always pointing down, that we are always going down with it, because generally we talk about up as the positive direction.0480

So, -9.8, that negative is there to denote we are going to go down.0489

As a special note, some people denote acceleration as m/s/s, others denote it as m/s2, as we are dividing by second twice.0494

I prefer m/s/s, because it gives you a little bit more of an idea of what is going on with the acceleration, it is how many are m/s changing for every second.0502

A 5 m/s/s means that 1s later you are going 5 m/s more.0511

m/s2, same meaning mathematically, but we lose a little bit of the inherent sense of what we are getting at. 0517

Formulae. So, the most important formula for all kinematics is what shows us all of these different ideas interact.0525

The distance, your location, location at some time, t is equal to 1/2 times the acceleration the object is experiencing times t squared plus the initial velocity of the object times t plus the initial location di. ( dt = 1/2 at2 + vit + di)0531

We cannot actually explain where this formula is coming from, because it comes out of calculus.0546

The way we get this formula, it does not take much difficulty, if we had a little bit of calculus under our belt, we will be able to pick it up fast, but at the moment we do not have calculus.0553

So, We cannot understand where this is coming from quite yet, but we can just go ahead and accept this on a silver platter, so this is a really important formula, we will be taking it for right now, and we will be running with it.0560

We will be using it a lot.0571

Next idea is one that allows us to relate velocity, acceleration and distance without having to use time. 0573

And this one we can actually derive.0578

First of, remember, acceleration is equal to, change in 'v' over time. 0580

Well, that is the same thing as vf - vi over time. 0587

We could solve this for time if we felt like it.0592

Time is equal to vf - vi divided by acceleration. 0595

We can take this idea right here, and we could plug it into this.0601

Now, first thing is m let us state first this is actually dfinal because that is where it is at the time t that we are looking at, the final location for our purposes.0607

1/2 at2 + vit + di, because that is the starting location where the object was.0615

We plugged that into there, and we are going to get, just box this out, so we have little bit of room to work.0625

df = 1/2 a (vf - vi/a)2 + vi(vf - vi)/a + di.0631

That di, we are not going to need it, we are going to move it over to the other side for now, so we get df - di. 0653

And then, while we are at it, let us multiply in that 'a', on that one, so we got, 1/2 times, well, it is going to cancel out, the squared of the a2 on bottom, so we are going to get, vf, let us change this up, so we have a fraction that split now.0659

It is going to be just 'a' on the bottom, because it used to be a2, but we hit it with an 'a'.0675

vf - vi, but this part is still squared, plus vi × (f - i)/a .0679

At this point, we got an 'a' on both of our sides, we got this 2 showing up here.0696

So let us multiply the whole thing by 2a, and let us also realise, df - di is another way of saying that it is just 'change in', Δ.0700

So, we got 2a Δ = (vf-i)2 + 2 vi(f- vi) .0707

Let us take all this, move it over here, for ease of work.0745

So, we have got 2a Δ D = (vf)2 - 2vfvi + (vi)2 + 2vi vf - 2 (vi)2.0755

So, at this point we see, this shows up here, and here, so we cancel them out, -2v (i)2 is going to cancel out.0785

So, we are going to get 2a Δ D = (vf)2 - (vi)2 . 0790

So, we can move this over, and we have got (vi)2 + 2a Δ D = (vf)2. 0800

That is exactly what we see up here. 0816

So, if you know the final velocity, and you know the initial velocity, and you know the acceleration, then you can find the change in distance. 0819

Or if you know three of these four elements, you can do it.0824

We can do this without having to work around with 't'. Sometimes, that is a nice thing for us.0827

Remember, acceleration will have to be constant for this, and also for the problem above this, but everything else works out great. 0830

Final couple of formulae to point out. Just definitional formula.0840

So, we defined acceleration = change in velocity/time and velocity = change in distance/time. 0844

With this point, we are ready to hit the examples.0852

If we have a rock, and we throw it directly down from a very tall cliff, the initial velocity is -7.0 m/s.0854

We ignore air resistance. What would be the speed of the rock 4 s later? 0866

Well, gravity is equal to -9.8 m/s/s.0869

What is acceleration? Acceleration = change in velocity / time .0875

Since there is nothing else acting on it, it is just a rock falling, we do not even have to worry about air resistance, acceleration is just equal to gravity, so we got -9.8 = change in velocity / time.0881

What is the time? 4 s . 0895

We multiply both sides by 4, we are going to get -39.2 m/s = change in velocity. That is not quite enough, we started at something. 0897

This is our change in velocity. We have to take -7, our initial velocity, and add it to our change in velocity.0910

So, that is going to be equal to -7 + (-39.2) = -46.2 m/s .0921

Next problem. Second example: 0936

We got that same cliff, but now we are going to say, it is precisely 200 m tall.0939

Once again, we chuck a rock down it, at a speed -7 m/s .0944

Ignoring air resistance, how long will it take that rock to hit the ground below?0953

Gravity = -9.8 m/s/s, and what formula we are going to use, we are going to use location, based on time, the final location at some time, 't', is equal to 1/2 at2 + vit + the initial location.0958

For this one, what is our initial location, 200 m above the ground.0979

Let us make the ground, zero.0983

That is our base location. 0986

So, our initial location is, positive 200 m (+200 m) above that ground.0989

Our initial velocity is -7 m/s towards the ground.0993

And our initial acceleration, the only acceleration we have throughout, the constant acceleration we have throughout is, -9.8 m/s/s, the acceleration given by gravity. 0996

We plug all those things in, what are we going to want to solve for, we are going to want to solve for the time one were at zero.1006

So, d(t) = in general, for any time it is going to be 1/2 t2 × (-9.8) = -4.9 t2 + (-7t) + 200 .1012

So, we want to know our location at any time 't', we just chuck in that time, and we will find out what our location is.1040

At least until we hit the ground, or if we go back before zero.1046

Because this equation is just Math, it is supposed to tell something, assuming we are using it right. 1050

But if we go past the time it hits the ground, it does not know where the ground is, it is just a Math equation.1054

So it starts giving us negative numbers.1059

It is going to operate like a problem, it is up to us to be careful of how we use some of these equations. 1061

But, we know what we are doing here.1066

So, what final location do we want to look for? We want to look for the zero. 1068

We want to find out when is that rock at the ground.1072

So that is the case, -4.9 t2 -7t + 200 . 1075

We want to solve this equation for what time, it will give us the location, the final location of zero.1082

How do we solve something like this?1088

We got a couple options. One, we could look to factor, but that does not look very easy to factor to me.1090

Two, we could put it into a numerical solver or some sort of good calculator, and we could figure it out.1095

We could graph it with a graphing calculator, and look for the zeroes, or we could put it into the good old quadratic formula: (-b +/- (b2 - 4ac)1/2)/2a. 1100

In this case, what is our b?1113

It's -7. So, we got (7 +/- ((-7)2 - 4 × (-4.9) × (200))1/2) / 2 × (-4.9) .1116

It does not look very friendly, that is not super friendly and easy, but we can definitely punch that out if we work through. 1147

So, equals (7 +/- (49+3920)1/2)/(-9.8) = (7 +/- (3969) 1/2)/(-9.8), 1153

We are going to wind up getting two different answers out of this, the two possible answers for our time, this quadratic formula would give us, are -7.1 s and 5.7 s, because of that plus/minus. 1176

We are going to get two optional answers depending on if we work with plus, or if we work with the minus.1190

Because it is two different answers that will work. Both of these answers are going to solve this equation right here. 1194

But which one is the right one? Well, this equation does not apply negative time.1200

We start the time, we want to start the clock when we throw it, that is when we set our initial distance, that is when we set our initial velocity, that is the moment of throwing.1206

So that is when we need to set our time as zero.1214

So that is our zero time, just as we throw the rock. 1216

So, that is the case, we are going to have to go with the positive answer because it is the only one that make sense.1220

-7.1 s, that works because what we have described here is, we have described a parabola.1225

We are looking for when does that parabola cross the zero.1232

When does it hit the x-axis. What are the zeroes, the roots to this parabola.1236

Of course, there is two answers to this.1240

But, it is up to us to pay attention and to go, 'Oh yea, that would not make sense for that to be a..'. 1242

Math, it is a really useful tool when we are solving Physics, but it is up to us to keep it rained in.1249

It is just going to give us the answers, it is just going to do its own thing, because we are using it to model real world phenomena.1254

It is up to us to pay attention to how we are applying it.1260

So, in this case, we will get two answers out of it, but only one of them makes any reasonable sense. So, that is the answer we have to choose. 1263

Third example: We have got a person on a bike, traveling 10 m/s.1270

The person begins braking, and then comes to a stop 10 m later. What was its acceleration then?1274

In this, did we say what the time involved was? NO, We did not!. We could probably work it out.1279

We could solve for it. But, it would be easier if we did not have to. 1285

So instead, we have got that formula (vf)2 = (vi)2 + 2a × (change in distance) .1288

Let us say, what he started out was just at zero, because we are setting it arbitrarily.1299

It does not mean to us where we start, and our change in distance is that 10 m.1303

Final can be 10, but the important thing is, wherever he started, he stops 10 m later, so the change in distance is 10. 1307

The acceleration, we do not know the acceleration, that is what we wanted to know. 1314

We know what he started at, he started at 10 m/s .1317

Do we know what he stopped at? Yes, we know that he came to a full stop, so final velocity must be zero. 1320

We plug these in, we get 02 = 102 + 2a ×10 , 0 = 100 + 20a . 1325

So, we have got -100 = 20a, we have got -5 m/s/s , because that is what acceleration comes in, equals our acceleration, which makes a lot of sense, because if this person is moving forward, for them to come to a stop, they are going to have to have a negative acceleration, they are going to have to be slowing themselves down.1345

They are going to have to be opposing the movement they already had.1360

Final example: We have got a UFO that is currently 500 m away from the surface of the earth.1364

Here is the Earth, and here is our little UFO, hovering in space, and it is currently 500 m above the Earth.1370

And at this moment, it has velocity of 50 m/s, so it is not really sitting there.1384

We take a snapshot and in that first snapshot, it is 500 m away from the surface of the Earth.1389

And in that snapshot, it currently has a velocity of 50 m/s.1394

It also has a constant acceleration, of a 100 m/s/s.1402

So, what would be the velocity of the UFO in 10 s? 1411

If we want to know the velocity of the UFO, the acceleration = change in velocity / time.1413

We know that the acceleration is 100, change in velocity divided by, we are looking 10s later, so we have got 1000 m/s = change in velocity, so our final velocity, is going to be, (our initial velocity, let us denote it as vi).1418

vf is going to be those two added together, so we have got 1050 m/s.1439

It's moving pretty fast at the end there.1443

If we want to know what its location is, how far is the UFO from the surface of the Earth in 10 s, we need to set up that same equation we used before, distance (t) = 1/2 at2 + vit + initial starting location.1446

We plug those in, we are looking for its distance at 10 s.1467

Is equal to 1/2 times, what is the acceleration?, interesting point to look at, is gravity affecting this, yes, gravity is affecting this, but we do not need to subtract for it, because we are told in the problem that it has a constant acceleration of 100 m/s/s, away from the Earth.1471

So, whoever gave us the problem, however we got this information, we know the acceleration.1486

There is some forces involved, gravity is pulling, and it probably has got thrusters or something, causing it to move away from the Earth.1491

But, we do not have to worry about that, because we are told what its acceleration is precisely.1496

So, we are good with that. So, 100, (and its positive because it is moving up), times the time squared, we do know the time, plug that in, 102 plus, what is initial velocity, 50 times the time, 10, plus its initial location, 500.1500

After plugging things in, 50 × 100, 1/2 times 100 becomes 50, 102 becomes a hundred, plus 500, plus 500, we get 50 × 100 becomes 5000, plus 500 + 500 becomes 1000.1523

So, we get in the end, it is 6000 m above the Earth.1546

Hope you enjoyed that. Hope it made sense. 1553

Next time we will look at how multi-dimensional kinematics works, when we are looking at more than one dimension.1555

See you at educator.com later.1560

Hi, welcome back to educator.com, today we are going to be talking about collisions and linear momentum.0000

One really great importance of Physics is, being able to understand the interactions between the objects, the way things work between one another in the world.0006

One of the most basic forms of interaction is collision.0015

When two things hit together, it is useful to be able to know how things are going to play out at that moment.0018

If you can model it, we have done all sorts of useful things.0024

Think of all the things that we can do with it.0026

We can investigate a car crash, you could dock a shuttle with a space station, you could look at the collisions between atomic particles.0027

There is all sorts of stuff, stuff bounces off one another, where stuff hit one another and sticks together, you will be able to understand it a lot better if you have momentum and how it is conserved, how it works for momentum.0034

Investigating collisions!0046

If you wanted to talk more about collisions, what ideas would be useful?0048

We can all figure out at this point , momentum is going to be what we are going to be talking about.0052

We just introduced the idea of momentum, and the reason why we did that was to set ourselves up to be talking about conservation of momentum, and how it works with collisions.0058

If we are going to investigate collisions, if we are going to investigate how motion changes, and what is happening as things go on, we want to be able to talk about how motion is working.0066

One of the best things we have for talking about motion is momentum, p = mv.0076

If we got that under our belts, that is going to really help us.0080

But that is not quite enough, we need one more thing.0084

The other thing we need, is the centre of mass.0087

Talking about the centre of mass allows us to explore the motion of a system, how things work together, and that is what we are going to need to be able to talk about collision.0090

Two objects working together, hitting each other, that is the system, you look at those two objects and the way they interact together, they are going to have to interact as a system because we have decided to make them a system.0097

If you look at them as a system, we will be able to know things, the way their momenta are working, and the way their centre of mass is doing things.0106

We are going to use to use centre of mass to learn indirectly more about momentum.0113

We are going to derive this.0117

We can extend the idea of momentum to a system, by looking at the total of the momenta in the system, makes up the momentum of the system.0118

This symbol right here mean, ADD things together. (Might be a little confusing if you have not seen it before.)0126

That says, add up all the momenta in the system.0135

Say we are looking at a pretty basic system that has some p1, and p2, a two object system.0139

Then the momentum of that system is adding them together, and that makes sense, the final momentum of the system is just all of them added together.0146

Just like we said that the energy of the system is all of the energies involved, you look at the energy for, potential gravity, kinetic energy, spring energy, you added them all to find out what the energy of the system was.0153

Exact same thing here: To find out the momentum of a system, you just add all the momenta together.0167

How do we know for sure though, that the mass of the system, (all masses summed up) times the velocity of the centre of mass, is equal to the momentum of the system (all added together)?0173

We can actually prove that.0185

MsysvCM = (m1+m2)(m1v1+m2v2)/(m1+m2) (in 2 particle system, what happens when two objects interact with one another, this works with an arbitrary number of things in our system, but we are going to be looking at the collision between just two things, to understand how this came to be, also note that velocity of centre of mass is the weighted average.)0186

We have got something that cancels off pretty easily, and from there, (m1v1+m2v2) = p1+p2 = Σp.0255

So, all the momenta in the system added up together, is equal to the momentum of the system, which turns out to be, = MsysvCM.0290

We look at the thing as a whole, it is the mass of the system times the movement of the system of a whole, or we look at the pieces that make it up, it is all those pieces added together, great intuition, works out perfectly.0299

Similarly, we are not going to show this mathematically, but we can do this with impulse.0312

The change in momentum of the system, Δpsys = jsys = Fextt.0316

That makes sense, if you are going to have a change in the system, you are going to put an impulse on the system, just like we did when we were working with single object at a time, but it is also going to be the case that the force on it is external, because remember, form our work with centre of mass, internal forces, because of newton's third law, every action has an equal and opposite reactionary force,0322

So, if it is an internal force in the system, sure, stuff is going to happen inside of the system, but the ultimate change to the centre of mass, is going to get a force going one way, but it is going to get the exact same force going the other way, they will cancel out.0343

Centre of mass is going to experience nothing.0355

So, form our work in centre of mass, we know that internal forces have no effect on the system's centre of mass, so no effect on the system's momentum.0356

So, Δpsys = jsys = Fextt.0362

Now we are ready to make something really useful.0372

Conservation of linear momentum: If there are no external forces acting on the system, there is no net external force acting on the system, and there should not be external forces acting on the system, but as long as they are going to cancel out, at static equilibrium, we are going to have the change in momentum of the system = zero.0375

That means, system's momentum at the beginning = system's momentum at the end.0394

If we know that there is no net external force, we know that we are going to have a constant momentum, a conserved linear momentum.0401

That is really handy, because that is going to give us a whole bunch of power, when we are looking at new kinds of problems.0409

Once again, notice that there can be internal forces, it does not matter if there are internal forces, they will have no effect on the system's momentum as a whole.0416

For example, if we had a bomb, or some explosive things, and we blew it up into a bunch of pieces.0422

Each one of those pieces would fly away.0436

But, since it is all internal forces in that system, the system's momentum is going to wind up being that same steady state when it was at rest, it is going to be zero.0439

Of course, there is also gas in there, so it is pretty complicated thing to model, you might have to deal with the gas moving in every which way as well, but the total system's momentum, when we take into account the pieces flying every way, the gas in the centre expanding out rapidly, the end of it is going to wind up also being zero momentum.0450

All sorts of things are going in opposite directions, but when we add them all up together, they ultimately cancel each other, and we get back to rest.0470

Conservation and external force:0477

This means, if we have an external force acting on our system, we require to have no external force to allow the conservation of momentum, so fi there are external forces, we fling up our hands, and go, 'Oh, we cannot do this problem'?0479

No, we do not have to do that, because we can still examine a collision, even when there is an external force. Why?0491

External forces are normally pretty small compared to collision forces.0496

Think about, if a car was crashing into another car, there is going to be little bit of friction on the ground, but for the amount of time that the crash occurs, friction on the road, like small amount of rolling resistance, small amount of internal friction inside the engine, it is going to be very little compared to the massive change in forces involved in two cars crashing together.0501

Those massive forces are going to be so large compared to all the other forces in effect, that we can basically forget about those other forces, we can forget about those external forces, because we are going to be looking at a brief period of time.0527

We can take a snap shot just before the incident of impact, a snap shot just after the incident of impact, because the impact is going to be more than an instant, but it will be something very brief, order of (1/10) of a second or less.0542

We can take a look just before the impact happens, and just after the impact is complete, the external forces are likely to be pretty small compared to the forces involved in the collision, so we can still effectively use conservation of momentum.0555

It is not perfectly correct, but it is so close as to be plenty good.0570

We are going to be totally be able to use it, and it will still make a lot of sense.0576

As long as we consider the instant just before and just after the collision, we can treat it as the momentum is conserved during the collision.0578

It will not be perfectly conserved, but it will be such a small change, that it is as if it was conserved, so it is good enough for us.0584

Finally, momentum versus energy:0593

It is really important to notice that momentum and energy are totally different.0596

mv is very different from (1/2)mv2, conservation of momentum does not imply conservation of the other.0599

In fact, in almost all every day collisions, some energy is lost to heat via friction.0607

When, for example, you drop a ball, the ball hits the ground and rebounds.0611

If it had the same exact amount of energy that it started with, the amount of energy in that ball would have to be the same as what it was when it was dropped in the beginning.0617

That means, that it would be able to bounce and come up to the same height, again and again, but it does not happen.0625

For anybody who has dropped a ball, if you drop a ball, it does not keep bouncing forever, it eventually stops moving.0632

If it stops moving, the energy is being sapped.0642

Momentum is conserved, when you consider the earth and the ball interacting with one another, but the energy is not conserved in the ball-earth system.0644

The ball-earth system winds up dissipating energy.0654

That energy does get dissipated, conservation of energy does still work, but it winds up being dissipated into things that we cannot really watch, like heat energy, sound energy, so we do not really have a good way of doing it.0658

So from our point of view, mechanical energy is not conserved, the way that we know how to deal with energy, just is not conserved.0669

So we cannot use it, we can only keep momentum in that case.0675

The system's momentum will remain constant through a collision, but the energy of the system has no such restrictions.0678

When we are dealing with collisions, we know for sure, that momentum is going to stay the same.0685

But, energy might change.0689

Similarly, when we throw an object, we know that the energy in the object is going to remain the same through its motion.0693

However, since there are external forces, like gravity acting on the object, we know that the momentum of the object is not going to be the same.0699

Sometimes energy will be conserved, with momentum not being conserved, sometimes momentum will be conserved with energy not being conserved.0706

It depends on what you are looking at, you want to think about it as how short is the time frame, what external forces are acting on it, think in terms of collisions, means momentum you can trust, and no external forces means that we can trust in energy.0712

And external forces and energy, remember, gravity is not an external force for energy, because we have taken care of it within our energy formula.0730

But for momentum, there is no such formulae for these other things, like gravity or friction.0737

Still, finally, there are some collisions where energy is not lost.0742

So, we are going to have to categorize our collisions into different types.0747

When momentum is kept, when energy is kept; when momentum is kept, when energy is not kept.0749

Fist one, elastic: Collision is called elastic when energy is conserved in addition to momentum.0754

The change in the energy of the system = zero, and the change in the momentum of the system = zero.0760

These are really uncommon in everyday activities.0764

Stuff that we are used to in reality, in real life, it pretty much never happens, if not completely never.0767

Inelastic: Collisions that we do see in everyday life.0774

A collision is called elastic when energy is not conserved during the collisions.0777

So, we cannot trust the energy of the system to be equal to the energy of the system at the end.0780

Beginning system energy, not equal to ending system energy.0784

But, the change in the momentum of the system, is trustable, we can keep that.0790

The beginning of the collision, the end of the collision, they are going to wind up having the same momentum on both sides.0794

Those are called inelastic collisions.0799

When energy is not conserved, inelastic.0802

Finally, in addition, when object involved sticks together after the collision, when they hit each other and they just keep moving as just one thing, it is called completely inelastic.0805

They just do not hit each other, and lose some energy, they hit each other, and get turned into one object, or something that can be treated as one single object.0814

Finally, almost all everyday collisions are inelastic, sometimes it is useful to approximate some collisions as elastic.0823

However, in the atomic world, particles often have elastic collisions.0830

In the atomic world, elastic collisions are actually really common and standard.0833

But, form the sort of thing that we see in everyday life, we are almost never going to see an elastic collision.0836

However there are certain things that we can approximate as elastic collisions, and learn some more stuff about.0841

On to the examples.0847

Two cars are being driven directly towards one another.0849

They are remote controlled, so nobody is hurt in the process of this example!0851

First car has mass, m1 = 1000 kg, and velocity, v1 = 30 m/s.0855

Second car has mass, m1 = 2000 kg, and traveling in the opposite direction, so, -20 m/s.0861

On impact, they stick together, so we are having a completely inelastic collision.0866

What is the velocity of the twisted hulk of metal they now make just after they collide?0870

We know that, psys(beginning) = psys(end).0875

p1(initial) + p2(initial) = p1(final)+p2(final).0886

What about the things on the right, we no more have two separate objects, we have one, stuck together.0900

In reality, this is not going to be that, it is going to the momentum of the twisted hulk of metal, momentum of the wreck, momentum of the whole.0911

So, m1v1 (initial) + m2v2(initial) = (m1+m2)vhulk.0930

So, 1000×30 + 2000×(-20) = 3000vhulk, so, -10000 kg m/s / 3000 = vhulk = -3.33 m/s, approximately.0957

That is what is left, that is what the speed that is left, actually velocity, since we know that it is going to move in the direction of the heavier car, heavier car was going to slower, it had more momentum, so it winds out in the end.1000

We are going at a -3.33 m/s velocity.1013

Example 2: Two billiard balls of mass m1 and m2, same mass of 0.16 kg, are on a pool table.1020

One of them is moving horizontally with speed of v1 = 5 m/s towards the other.1029

After the collision, the second ball is moving with a speed of v2 = 3 m/s, at 30 degrees below the horizontal.1033

What is the first's velocity?1040

In this case, we need to figure out, how much is going to the right, and how much is going up.1042

Now, we can figure out what the momentum is for the first ball, the ball that is moving.1052

We can figure out, the second one, we know what the momentum in the second one is.1059

We can figure out how much it is moving this way, and how much it is moving this way, and once we know that, we can go back and we can figure out, in this one, these two added up, are going to have to wind up equaling this one's vertical motion, which turns out to be nothing.1062

And we know that these two added up, are going to have to wind up equaling the horizontal motion.1092

We are now going to start working on it.1104

Velocity for the ball that is first moving before impact is 5 m/s to the right, o m/s up and down.1111

v2 = 3 m/s, 30 degrees below the horizontal.1119

Let us figure out what is v2(final) in terms of component vector.1124

v2(final) = cos(30)×3 m/s (since this side is adjacent), and the other side will be, sin(30)×3 m/s.1132

So, (2.598, -1.5).1157

That is an important thing that we have to catch specifically.1166

It is not just sin(30), it is actually, sin(-30), because it is below the horizontal, and we probably want to make going up positive, and going to the right positive, because that is what we are used to as standard.1168

So, it is not just sin(30), it is sin(-30) because it is below the horizontal, it is up to us to pay attention to the signs, we have to be understanding what we are saying with this.1181

So, (2.598,-1.5) m/s.1194

Now we know what the velocity of the second thing is.1198

From there, we can figure out what its momentum is.1200

First off, we know that the momentum at the beginning of the system is going to be equal to the momentum at the end of the system.1203

So, m1v1 = m1v1(after impact) + m2v2(after impact).1249

Remember, m1 and m2 are the same thing, so divide everything by that m1, m2 combination, since m1 = m2 = 0.16 kg.1275

So, velocity = v1(final) + v2(final), (5,0) = (v1x,v1y) + (2.598,-1.5).1292

At this point, we can add things together, so, remember, we are dealing with vectors, each one of the pieces of vector, x component is totally separate from the y component, so now we can break this into two separate equations.1335

We got, 5 = v1x + 2.598, and, 0 = v1y + (-1.5).1346

So, 1.5 = v1y, and , 5-2.598 = 2.402 = v1x.1362

Putting these together, v1 = (2.402,1.5) m/s, and that makes good intuitive sense.1380

If you hit the ball, the other ball winds up moving away as well, you wind up getting something that is moving slower now.1399

If it is going down, if it was going horizontally first, and the other thing bounces down, it is going to have to get rebounded upwards.1405

So, it makes good intuitive sense, passes the sanity check, and works out.1413

We are able to start these things moving, because of conservation of momentum, and we can make it a little bit easier by canceling out the masses, because we know that they all have the same mass, so it is a little bit easier for us, and from there we break one of them into its component pieces, and from there we are able to work in factors, because vectors, we can just deal with the component pieces.1415

Example 3: Two rubber balls are on a flat surface.1434

The first one has a mass, m1 = 2 kg, and the second has a mass, m2 = 1 kg.1437

The first one is moving at a velocity 5 m/s.1442

The first ball collides with the second ball, and we are going to be able to treat this collision as elastic.1445

I know, normal everyday collisions do not wind up being elastic, they are almost all inelastic, but it is good for us to get some practice with an elastic one, and rubber balls do manage to keep a lot of their energy.1449

What is the velocity of each ball after the collision?1459

Notice, this ball is going to be moving, and this ball is going to be moving after the collision.1462

One thing to notice is, which ball has to be moving faster.1467

The ball on the right is moving slower than the ball on the left, that is not going to make sense, because that means that somehow collision happened and then this ball managed to move through the other ball.1471

That does not make sense, we know that these are solid objects that are moving in one direction, it cannot move through it, so that is going to guarantee the fact that the velocity of the second ball is going to have to be greater than the velocity of the first ball.1483

In this problem, we are also going to wind up ditching the vector notation on the top, because we are working only in one dimension.1496

It is little bit easier than normal , because we are only working in one dimension, and we got a whole lot of writing coming up to do in an elastic problem, so we are going to make it a little easier by ditching the arrows, but it is important to remember that we are working on the basis of the idea of vectors, we are still going to care about direction and all those things.1503

For the beginning, we know that v2 > v1.1525

We know that, psys(beginning) = p1(final) + p2(final), (one dimension, so ditching arrows on top, but all the ideas of vectors are still there, we really care about direction of momentum.)1532

m1v1 (initial) = m1v1(final) + m2v2(final).1584

So, 2 kg×5 m/s = 2×v1 + 1×v2.1617

Sp, 10 = 2v1 + v2.1630

Now, that i snot enough to solve, we got one equation and two unknowns, so we are not able to do it.1637

But luckily, this is an elastic collision, so now we can bring energy.1642

We know, Energy(initial) = Energy(final), and the final energy of the system, since one thing is moving at the beginning, it is just that one thing's motion.1645

The final energy of the system is going to be both the things moving together, so, E1 + E2.1655

What is the energy at the beginning? (1/2)×2×(vi)2 = (1/2)m1v12+(1/2)m2v22.1663

We cancel (1/2)'s all the way across, substitute things in, 2×52 = 2v11+v22.1680

So, 50 = 2v12+v22.1699

At this point, we have got two unknowns over in this one.1708

How do we do this? Go back, and substitute one equation into the other one.1711

We get, 10 - 2v1 = v2.1717

Move that over here, so, 50 = 2v12+(10-2v1)2.1721

Simplify that out, 50 = 2v12+100-40v1+4v12.1736

If you are really good at factoring, you might be able to factorize this, but, most of us are not that great at factoring, so what do we use now?1770

We got two options: (1) calculator, graph it and see the roots, because we will be able to treat this as a single function and when that function equals zero, you would have found the answer to this equation.1776

Alternately, we could just throw down with the quadratic formula, and we will be able to figure out what it is.1789

What is the quadratic formula? (if you want to do advanced mathematics, or even physics, you could buy a graphing calculator, but it is not necessary.)1794

[-b +/- sqrt(b2 - 4ac)]/2a.1821

Plug things in, [40 +/- sqrt(402 - 4×6×50)]/(2×6) = (40 +/- 20)/12, gives us two possible answers.1829

The two answers are, 60/12 = 5 m/s, and 20/12 = 1.67 m/s, approximately.1861

We have got two answers, so it is up to us to figure out which one makes sense, which one is going to allow v2to be greater than v1.1881

If it is 5 m/s for v1, was it not 5 m/s beforehand?1887

That means that it hits the ball and then it keeps going at the same speed, that does not make any sense.1889

It manages to just go through the ball, remember, math is your friend, but it is not something that just spits out your answer.1905

It is up to you to keep it tamed, you have to understand what you are doing with it.1911

We brought this quadratic equation to bare, but it is up to us to understand how to interpret the results that it gives us.1915

It gives us two answers that work mathematically, the energy will be conserved, and the momentum will be conserved if that other ball was never there, it just manages to pass through it.1921

But, we know that it cannot be the truth, it did not just pass through it, it has to contact that ball.1929

So, the answer that has got to be, is this one tight here.1934

So we have got, v1 = 1.67 m/s.1938

If that is the case, we can then plug this back in, and we get, 10 - 2×(1.67) = v2, so, 6.66 m/s = v2, approximately.1943

There we go, we are able to figure out what the other velocity has got to be, because we are able to bring all these equations bare.1964

We bring what we know about energy formula, what we know about momentum, and we are able to do both of them, because we got two unknowns and two equations, we are able to solve it.1971

We know that v1 is 1.67 m/s, still in the positive direction, and v2 is positive at 6.66 m/s, and it turns out it is going to be faster than the initial object actually was.1980

That is the stuff that happens in elastic collisions, things do not behave necessarily quite like you expected, because energy is conserved, so we got that v2 thing going on, so some weird stuff can happen sometimes.1997

Well, not weird, because it is physics, it is reality, so it cannot be weird in that way, but it is certainly not what we necessarily expect.2008

Example 4: A ballistic pendulum is one way to measure the velocity of a bullet fired from a gun.2028

Ballistic pendulum is a big heavy thing sitting on a rod, and there is this thing here, and a bullet will come in here, and it is going to impart its momentum, so the thing will swing up.2032

It winds swinging up, just like we got here, bullet goes in here, launches itself in, and put some momentum into it, and the ballistic pendulum will continue to swing up.2050

It is one way to measure the velocity of a bullet fired form a gun.2061

A bullet is fired into a large mass at the end of a pendulum where it launched itself.2062

By measuring how high the pendulum rises, we will be able to figure out the velocity of the bullet before impact. How do we do that?2068

Denote the bullet's mass as mb, the pendulum's mass as mp, assume the pendulum's rod's mass is negligible, so all we have to worry about is the mass of the block.2074

The height that the block achieves will be 'h'.2086

Using all these things, (in real life, we could build something like this, and using a pencil we could mark how high it went up, and we will be able to weigh the mass of the bullet and the pendulum, we will be able to find a fairly light weight strong rod that was able to hold it in place without being much compared to the mass of the pendulum, and we will actually be able to do this, this is what people did in the 1700's).2091

What is the trick that we will be able to use to solve for this?2123

Momentum is not going to be conserved throughout the entire thing.2126

The momentum of the bullet, as it goes up, we have got gravity that is dealing against it.2130

So, gravity is an external force from the point of view of the momentum, if it is going up, it is going to have to have some force moving it up, so we have got external forces on the bullet.2135

But at the same time, energy is not going to be conserved.2147

We got this impact, we never said it was elastic, this is an inelastic collision, this is definitely an inelastic collision.2149

We cannot conserve energy throughout, we cannot conserve momentum throughout, how do we do this?2160

We can conserve them one piece at a time.2165

Notice, the instant, if we break this into three things, the bullet goes in, and at this moment, when it is just after the collision occurs, before it has the chance to swing up, momentum is conserved.2168

So, pi = pf, then from here, once it is swinging up, there is no more collisions, it is just a basic pendulum rod going up, so we can use energy.2187

So, for here, Ei = Ef.2199

We are able to break this into two pieces: the momentum is going to be conserved for just the collision, and after the collision, energy will be conserved.2203

We break it into two pieces: beforehand and afterwards.2212

If pi = pf, what is in the initial momentum?2215

mbvb, there is nothing else moving, we got the pendulum at rest.2222

Once again, we are going to nick those vectors, since we are dealing with linear quantities.2227

Wait a second! I hear you ask, is it not going to move up and to the right when it swings up!2237

Yes, that is true, but we will be dealing with that as energy, so when we get to that point, we only have to worry about speed.2242

For the collision, as far as we are concerned, it hits and goes in, and it is still basically moving in just one dimension, because it launches itself very quickly, and the collision happens very quickly compared to the movement of the swinging.2247

So, mbvb = (mb+mv)vsys, for ease, we will denote, vsys after collision as, vp.2258

That makes it a little bit easier to write, because we are going to have to be talking about two very different world, the world of the impact, and the world after impact where it is swinging.2297

This will make it a little bit easier, vp, it is important to pay attention to what wee are doing here.2303

We can figure out what the velocity of the bullet is, in terms of, (mb+mp)/mb, but are not able to solve for this, we still need to figure out what the velocity of the pendulum is, we do not know what the velocity of the pendulum is.2309

Remember, we only have the height.2328

Now, start looking in terms of energy.2330

Ei = Ef.2332

The energy of the system at the beginning of the swing, does it have any height?2336

No, we can consider that as its base line height. Is it moving?2340

Yes, it is definitely moving, there are no spring involved, and the only thing that we got there is the kinetic mechanical energy, kinetic energy.2342

(1/2)(mb+mp)vp2 = Mgh (at the top of its arc, it has got to just finish moving, then the only thing it has is stored gravitational potential energy.)2349

'M' in this case, is not just 'M', it is the mass of the whole thing, mb+mp, so, mb+mpgh.2356

Cancel out mb+mp on both sides, so, vp2 = 2gh, vp = sqrt(2gh).2387

Substitute that back in, vb = (mb+mp)/(mb×sqrt(2gh)), there we are.2402

By just measuring the height that the thing goes up to, by already knowing the constant of gravity, by weighing the mass of the bullet, weighing the block, we are able to figure out the velocity of the bullet, which is pretty great.2427

These are all the things that we were able to do in the 1700's.2437

Even without fancy technologies like LASER, and all these other cool stuff, we can actually come up with a really solid way to measure the velocity of a bullet by what we know about momentum and what we know about energy.2440

Hope you enjoyed this.2452

Hi. Welcome back to Educator.com. Today we’re going to be talking about gravity and orbits. This will also complete our first section, how we have now covered all of the basics of mechanics.0000

We’ll start to move on to other things in future sections, but this is it for mechanics.0010

We got a good strong understanding of mechanics, you should be proud of yourself.0013

So, basic introduction to gravity. What’s holding you to the Earth right now? What’s holding me to the Earth right now? Gravity, gravity is holding me down, keeping me on Earth.0018

What causes the Earth to orbit the sun? Gravity. The reason why the Earth goes around the sun. Gravity. The reason why Jupiter goes around the sun. Gravity. The reason why everything is moving around everything? Gravity.0026

Gravity is one of the basic forces of the universe, really, really important.0035

Gravity pulls massive objects together. The more mass you have, the more pull you exert on other massive objects you pull.0039

Two objects, if they’re both really massive will pull together more than if one object has the same mass but the other one has a small mass.0046

Also, the distance between them affects it. If you’re really close, you’ll wind up having more gravity than if you are very far away.0054

Now, this idea of gravity, we’re used to it. We accept it right now, but keep in mind, the idea of gravity, was actually really once stridently fought against.0060

People did not believe in gravity at all, did not accept the fact that the heavens, that the stars above us would wind up undergoing the same pulls that we were used to in our finial normal, normal day life existence just on Earth.0068

The fact that humans are pulled on the same thing as on the stars seems ridiculous to some people, but it’s the true.0082

Everything ends up having the same set of rules.0088

Gravity, it’s actually a real thing and we’re used to that but keep in mind people didn’t always think that.0091

Law of universal gravitation, formula for how gravity works.0101

To derive a formula for the force of gravity, that’s kind beyond the scope of this course, so we’re just going to start by plucking it out of thin air.0105

The force of gravity, the magnitude of the force of gravity is equal to G, some constant, times the mass of first object, times the mass of the second object, divided by the square of distance between those objects.0114

It’s an inverse square. If you’re farther away, it’s not just the distance divided, it’s the distance squared divided.0126

Let’s talk about things more specific. Force of gravity, the size of force of gravity is equal to GxM1xM2/r2.0134

M1 and M2, two mass of the objects involved, that’s pretty simple. R, is the distance between the two objects.0143

If you really want to be specific, it’s more accurate to say it’s the center of the mass for each object.0149

Now, keep in mind, we’re normally going to be dealing with very large distances and comparatively very small objects.0155

Like the distance between the Earth and the Sun is considerable larger than either the size of the Earth or even the size of the Sun.0162

We can worry about the center of mass but for the most part we’re going to be dealing with such large distances in any case, we don’t have to worry that much about the distance between them versus the distance between their centers of mass.0170

Don’t worry about it too much, but keep in mind there is a slight difference there.0182

G is the universal gravitational constant, which is the thing that makes this formula run.0185

The idea is mass times mass divided by the square of the distance.0191

That’s what affects it, but we need to have a specific thing that’s going to let us generate actual numbers and this is scaling factor that actually lets us get numbers by multiplying these things and dividing.0196

6.67x10^-11. Newton’s times metered squared divided by kilograms squared.0207

Because remember we want in the end to get Newton’s out of this.0214

We want to get four sides of this, so if we got masses up top then we’re going to have kilograms squared up top.0218

That will cancel out there. If we got on the bottom; meters. Then we got to cancel it out up top.0226

We got meters squared times kilograms squared, so it will cancel out the two masses and dividing by a distance squared.0232

That leaves us with Newton's. That's why the law gives us a unit that seems so bizarre.0241

Now notice that G is a really tiny number. G is just incredibly small.0249

The reason that we don't feel a pull from buildings around us is because G is so small.0257

We're comparatively way closer to a building than the center of the Earth.0263

That building has so much little mass compared to the Earth as a whole and we'll talk about the mass of the Earth later on.0269

It's a big number, somewhere on the scale of 10 to the 24th.0277

It is big. It's really big. That building, it just doesn't have the mass to compete with how tiny that number G is.0281

Unless you are an absolutely giant thing. Unless you're basically a stellar body, you're just not going to have the ability to have effective powers in gravity.0287

Finally, force gravity is a vector. You have to remember it points between the two objects.0297

Object 1, Object 2. Object 1 gets pulled towards Object 2, just like Object 2 gets pulled towards Object 1.0303

Equal and opposite reactions; Newton's third law still applies. So the two objects pulled towards each other.0312

Gravity is not just a number, it's a vector. You have to have a direction to go with that size.0319

To go with that amount of force. Remember that it's always pulling towards the other object.0325

Normally we'll be able to treat this as if it's single dimensional, but if you needed it would be actually vector quantity.0331

So previously, we simply thought of gravity as a general acceleration.0339

We knew G was equal to 9.8 meters per second per second.0344

Now we're talking about universal gravitation. So what does that mean?0349

What does that make our old conception of 9.8 per second per second into?0351

Such an acceleration, we call a gravitation field.0356

We know that this is still valid and useful and worthwhile because we can actually model lots of real things with 9.8 meters per second per second.0359

It works, we've probably by now done a few labs or at the very least we've done so many examples that make intuitive sense that we see that 9.8 meters per second per second is actually is pretty reasonable thing.0367

The world pretty much runs on that.0378

How do we make these two come together?0383

A gravitational field is a way of saying at a certain distance, you're going to experience a certain acceleration.0385

How can we find gravitational fields in general?0393

A gravitational field imposes a constant acceleration on anything inside of it.0397

Remember before, we had force of gravity equal to the mass times the acceleration of gravity.0401

Force of gravity equals the mass times the acceleration of gravity.0406

For now, any object. This will work on Earth, but it will also work on anything.0410

We saw this before with the force of gravity on Earth, but we can do this on Mars if we knew what the things involved were there.0414

We could do it on the surface of the Sun, we could do with any object that we felt like.0421

Connect that formula with the law for universal gravitation. We're going to have that force of gravity is going to equal the mass times the acceleration of gravity on one side and gravity times M1 times M2 over R squared on the other side.0426

For example let’s talk about me. I will consider myself to be one of the masses.0438

I'm Mass 1. I'm M1. I'm M1 times acceleration of gravity, is the force of gravity currently pulling on me.0449

From universal gravitation, we also know that the force of gravity currently pulling me is G times M1, my mass, times M2, the earths mass, divided by the distance between my center of mass and the earths center of mass.0455

The distance between here and the center of the Earth.0467

Mass times acceleration of gravity equals G times M1M2 over R squared.0471

That means my M1 and the M1 of the universe of gravitation cancel out and we're left with the acceleration of gravity is equal to the mass of the object we're looking for the gravitational field of times G divided by R squared.0476

However far up we're putting our gravitational field.0489

So in case of the Earth, for me standing up here talking to you. The distance I'm going to get, whether it's here or I climb a mountain, or I dive into the sea.0492

I'm going to change my distance by a kilometer, two kilometers. The size of the Earth is so much larger than that, that my change in R is a drop in the bucket compared to it.0505

While the exact, the precise amount of gravity that is affecting me will change slightly.0517

Its going to change a negligible amount. Which means that gravitational fields will work when we've got a very large, very massive object.0522

The distance we're going to get from that object center point is very little compared to the distance of the whole thing.0528

Our change in distance is going to be so small compared to the full mass of the distance that we can basically treat it as a constant acceleration as opposed to having to re-calculate the force at all times.0537

That's why G equals 9.8 meters per second per second worked, because no matter where I'm going to go on the surface of the Earth, I'm really not going to get very far from the surface of the Earth.0547

Unless I'm getting in a space ship. We can treat it as if I got a constant acceleration because R just is not going to change that much and everything else is going to remain constant.0556

In orbit. Orbit is one body rotating around another.0567

From our work in uniform circular motion, we already know to be in a circle, the acceleration has to be equal to the speed squared divided by the radius of the circle and that immediately gives us that the force to cause that to happen.0570

The force is equal to the mass times speed squared over the radius.0583

So what is the centripetal force that keeps a celestial body rotating? That keeps celestial bodies rotating each other.0586

What would that force be? Gravity.0596

If the objects have no other forces acting on them, which makes sense if we're in deep space or we're fairly out in space and we don't have to worry about other things pulling.0601

We're moving in a circle and then we get force of gravity is equal to force centripetal, which we can expand into the gravity times M1 times M2 over R squared equals M times speed squared divided by R.0607

One thing to point out, this isn't just M. It's M1 or M2, depending on which one we want to make it.0620

The object that's moving around, the M1's are going to cancel out on either side.0626

The other thing to note is that I want to point out that in real life, orbits are almost never circular.0632

Orbits can be close to circular but normally orbits are actually elliptical.0638

A circle is something that has a constant radius. An ellipse is something that is able to squish out.0643

An egg is kind of an ellipse. Things that get squish.0648

An ellipse is something that, we can have an object that can go around in an ellipse or it can go around in a circle.0663

We've been dealing with circles because they're much more sensible, much easier to work with, but in real life orbits are actually ellipses.0670

Also, in real life, when the Earth is going around the Sun, there is something else working on it.0676

All these other planets around us. Now comparatively the Sun is Big Pop in our universe.0682

The Sun, it's got the most mass by far. It's able to have the most effect on our orbit.0687

There is a whole but of other planets out there. One of the important planets that also has a really big mass, Jupiter.0692

Jupiter has a really large mass compared to the mass of Earth.0699

It's able to also have some effect on our orbit. Very little compared to the effect of the Sun.0702

If this real life, if we want to as correct as possible, we're actually dealing with an ellipse, we not technically dealing with a circle.0709

We're actually having to deal with other stuff, we're not having to just put this in a vacuum of force of gravity, one force of gravity is equal to the centripetal force.0715

There's more stuff happening here. At the same time, we don't have to necessarily worry about it to be able to get pretty good answers.0722

Just like when we were like 'technically there is air resistance, technically there is the other things when dealing with objects falling' at the same time, we can normally still forget air resistance and be able to get lots useful answers.0730

Except in really egregious cases where it's moving really fast, we have to clearly care about it.0742

In this case, it's one of these things were not it's a really egregious case. The mass of Jupiter is comparatively little to the Sun.0747

We don't have to worry about the fact that we're not going to calculate with it if we wanted to figure out something between the Sun and the Earth.0755

At the same, if wanted to be really rigorous, we would have deal other calculations and make it a whole lot harder.0761

So like air resistance, we kind of put in on the table, left it for a later physics course.0766

We're going to wind up doing the same thing with weird orbits that are not circular and other forces of gravity operating, but it's important to remember that there are other things out there.0770

One really cool idea before we get started in our examples. A famous thought experiment that Newton put forward. Isaac Newton, gives us another way to think about gravity and orbits.0780

Imagine, and before I get too far, I would like to apologize for the bad drawing of this Earth, I am terrible at drawing.0789

If you live in Morocco or Tangiers or anywhere in the north of Africa or England. I'm sorry, I have basically ruined your place on the Earth. It's just kind of not there.0797

This guy is supposed to be Greenland, so if you're in England, my apologizes, if you're in Morocco or Tangiers or any of the other many places that I ruined with my poor artistic ability. I'm sorry.0808

Now moving on. Imagine a very tall mountain on Earth. So tall as to be above the atmosphere.0821

There will be no air resistance, so we don't have to worry about friction slowing down the object.0827

Great. On top of this mountain, we'll put a cannon and we'll fire cannon balls out of it with greater and greater velocities.0831

What's going to happen as those velocities increase?0837

Let's start doing it, we'll play around with it. Here is the center of the Earth, we shoot something out, stuff is going to get pulled towards the center right?0840

That's how gravity works. Let's say we put the cannon in and we practically don't shoot at all, we just let the cannon ball roll out.0847

The ball comes out and boom, falls right into the Earth. Well what if we put it would with a slight amount of force.0854

Its going to shoot out, then it’s going to fall into the Earth.0859

What's going to happen if we put more force? It's going to shoot out...and then boom, it's going to fall out, because it's getting sucked into the center of the circle, remember?0864

At every point on the circle, it's getting pulled in. Well if we shoot it harder, it's going to shoot out...0873

It'll get pulled in and then it lands eventually. But, if we shoot it really, really hard. Let's say if we shoot it at super extreme, it'll get shot out, it'll get pulled slightly by that and then it'll just fly off into space.0882

It'll just go off forever. If it goes off forever, we've lost it, there is nothing there.0896

There's not nothing there, it's gone into an escape velocity. It's managed to get pulled far enough away from the Earth that it'll manage to escape the gravity of the Earth.0903

If we shoot at the right speed, instead of falling into the Earth or falling out or away from the Earth.0912

It's going to get pulled in and it's going to fall into the Earth forever and ever and ever.0921

It's just going to keep spinning around the Earth because it's getting pulled in at all moments. So it just keeps going.0932

The best one, the one that will be in orbit is a permanent fall. So the permanent fall is way to think of gravity.0941

Gravity isn't just pulling a thing directly in, it's a way of thinking of a fall. An orbit isn't something that's not falling, it's something that's falling at just the right rate.0950

It's falling in such a way as to constantly miss the ground. Flying isn't necessarily not falling, it's missing the ground as you fall.0962

The important thing is that it's still being effected by gravity, but instead of being pulled into the ground, it's getting pulled towards the ground but it's moving fast enough forward that it just keeps going around and around and around.0970

This a thought experiment. Thought experiments are a class ideas where you can, instead of having to actually do a physics problem, because clearly you're not going to be able to go up and build a mountain so high up and put a cannon on top of it and shoot it so fast, that's not really plausible.0983

We can think about it from all the ideas that we know we can trust at this point. All the things that we've learned so far, we can test that on an idea and come up with all sorts of things.0999

That's what a thought experiment does and lots of modern physics and other stuff previously is developed by that, and that's basically how we all do puzzles, we know what we know and we work around it and we're able to come up with all sorts of things.1007

That's exactly what this is, it's a thought experiment that lets us understand a cool thing about the way the world works.1018

Onto the examples. Two objects have masses of 4.7 times 10 to the 7th kilograms and 2.0 times 10 to the 9th kilograms.1028

If the centers of mass are 850 kilometers away from one another, what's the force of gravity between them?1036

This is just a really blunt use of the force of gravity formula.1041

Universal gravitation, we know that the force of gravity, the magnitude of the force of gravity is equal to G times M1 times M2 over R squared.1045

Throw in all the numbers we have. We have 6.67 times 10 to the -11th.1055

I'll tell you right now, you just have to memorize that. You're just going to have to write it down and keep it on a card with you or you're just going to have to keep it in your brain.1060

Like we need to keep 9.8 meters per second, if you got a lot of gravity problems, you're just going to have to know it. It's something that's just an important number to remember.1068

It's one of the basic fundamental concepts of the universe, so it matters.1076

6.67 times 10 to the -11th times 4.7 times 10 to the 7th kilograms times 2.0 times 10 to the 9th kilograms.1081

So mass object 1, mass object 2 divided by 850 kilometers, wait a second, standard units.1092

What's the standard units here? Is kilometers standard? No.1100

When we dealt with G, G required M squared. Remember it was M squared over kilometers squared times Newton’s.1104

That M squared, we're going to have to be working in meters.1111

Remember if somebody gives you a unit and it's not in SI units, it's not in normal metric units. Change it, change it to a normal metric unit.1114

Otherwise things can go so very wrong. Sometimes it will work out, some of the easier problems will work out just fine and you'll be able to keep it that but if you want to be able to really trust what you're doing and be sure that it will work out, change it into SI units.1122

Do the problem in normal metric units and then at the very end convert back to the unit that they gave you.1135

That's the best way you want to be sure of it. If you get really used to doing lots of things, you'll start to catch more stuff, but really you want to get used to using metric units.1140

If you want to be a scientist, if you're going to do a lot of physics, if you're curious about living anywhere else outside of America, you're going to have to do that.1149

To all of my viewers outside of America, you're probably not going to have to worry about other units.1157

If you live in America, you might want to consider getting more used to metric units and just get a feel for what they're like.1162

Anyway. Back to the problem. 6.67 times 10 to the -11th times, etc., divided by 850 kilometers, so what's that in meters.1168

So 850 kilometers. 850 times 10 to the 3rd, because it's kilometers; kilo 1,000. So 10 to the 3rd is a 1,000.1175

Then we have to remember it is R squared. You pop all that into your calculator and what do we get? Some big giant number?1186

No, no, no. We get this here, which is tiny. Then we also got this here which also going to make it really small, we get 0.00009 Newtons.1193

That's right, 900,000ths. 900,000ths of a Newton is how much they manage to pull on one another.1207

These are fairly massive objects that are at distance that we wouldn't think is that huge.1220

Keep in mind that's why this stuff is so, that's why we don't really experience gravity other than the gravity of Earth, because most of the other stuff just doesn’t have that much effect on us.1224

Example 2. If the Earth has a radius of 6.378 x 10^6 meters and the mass of 5.974 x 10^24 kilometers.1236

What is the gravitational field on the surface of Earth?1245

Remember, the gravitational field way for us to go from knowing what the force of gravity was to something telling us about the acceleration of the gravity at a certain distance away from a place.1248

Force of gravity is equal to, we want to be something, mass times acceleration of gravity.1258

We know force of gravity, we can change this into our general one. G M1 M2 over R squared equals mass.1265

Lets make it mass 1 times acceleration of gravity.1273

If we have an object of mass 1 on the surface of the Earth, then it's going to wind up having a force of gravity that's G times M1 M2 over R squared.1277

But we also want it to have this acceleration of gravity, something else that allows us to come up with a gravitational field for it.1286

If that's the case, we got M1's cancel and we get G M2 over R squared equals the acceleration of gravity.1293

We plug in all those numbers we know, we get 6.67 x 10^-11 times what's the mass of the Earth, 5.974 x 10^24 kilograms divided by 6.378 x 10^6 meters, remember we're in meters.1302

We've got to remember we need to square it when we replace it. So we punch all that into a calculator and what do we get out of it?1334

Eventually it simplifies to 9.795 meters per second per second. Hey, that makes a lot of sense.1340

What do we normal use? 9.8 meters per second per second.1352

So this turns out to be something that works out well.1356

Now there's a couple of simplifications that we made doing this problem.1359

The Earth does not actually have the radius of this, because the Earth is not actually a circle.1362

The Earth is slightly oblong, it's not quite a perfect circle.1366

So when you're dealing with it, we don't actually don't get the chance to deal with it as a perfect circle, so we made this problem a little bit easier on ourselves.1371

Also we don't necessarily have that the center of mass for the Earth is precisely in the center of the Earth.1378

That might be the case, but we haven't been guarantee yet, we need to find out more about the composition of the Earth.1386

So there's more things to keep in mind here.1391

On to the next problem. So assume that Earth's orbit is circular. Once again it's one of those things we said assume we can disregard air resistance.1394

If the Sun has the mass of 1.9898 x 10^30 then that's also a big number.1403

1.496 x 10^11 meters from the Earth, what velocity does Earth orbit the Sun at?1409

How long does it take for the Earth to complete one orbit?1415

We know that the force of gravity, because it's moving in a circle. Is there any other forces operating on it?1419

No, we know that it's just centripetal force pulling. Just gravity pulling, so that must be the entirety of our centripetal force.1424

We have to force of gravity equal to the centripetal force. Once again, there are other things in the solar system but Sun is big poppa.1432

G times M1 M2 over R squared equals, what's the centripetal force, M1 V squared over R.1440

M1 in this case is the object moving around. It can be canceled.1450

We've got that G M2 divided by, let's multiply both sides by R, equals V squared.1455

Now we'll take the square root and we'll get G M2 over R equals V.1464

So we can toss all these things in. We get the square root of 6.67 x 10^-11 times the mass of the Sun, 1.989 x 10^30.1469

All divided by the distance between the Earth and the Sun square, sorry not squared, because we managed to cancel out those R's.1487

1.496 x 10^11th.1495

Now we take the square root of that whole thing and after a whole bunch of calculating, we get 29,779.3 meters per second.1504

The Earth is really whizzing through the solar system.1516

Now if we want to find out how long it takes for the Earth to complete one orbit, how far a path does it have to travel?1519

Well the circumference, the path it has to follow is 2πR right?1525

So the orbit time is going to be, how far a distance it has to go, 2πR divided by the speed that it's moving at, V.1532

We start substituting in the numbers that we know. 2 times π times the radius, the distance from the Sun to the Earth, 1.496 x 10^11 divided by the speed that it's traveling at, 29,779.3 meters per second.1544

Punch that into a calculator and what do we get? We get that it manages to make an orbit around the Sun in 3.156 x 10^7 seconds.1566

What does that mean? I don't know how much that is really, I'm not very good at knowing how many seconds is meaningful after a 100.1582

Let's figure out what that is for ourselves.1593

So how many seconds, what does that number of seconds mean in terms of minutes, in terms of hours, in terms of days?1596

Days would be good, we already know that the answer should be pretty close to 365 otherwise something has gone wrong, right?1600

We put that in and we have 3.156 x 10^7.1607

How many seconds are in a minute? 60. How many minutes are in an hour? 60.1612

How many hours are in a day? 24. Punch that into a calculator and we'll get 365.33 days.1619

Which is really good considering that the actual orbit of the Sun is a little bit less than 365 and quarter days.1629

Now you'll probably think that the orbit, that the Sun, that the Earth manages to get around the Sun every 365 days, that's not quite true.1638

365 days is the closest round numbers of day, but you know how we have leap years every four years?1645

The leap year every four years is to catch up with the fact that the Earth takes just a longer than a year to make it around the Sun.1653

So just because we talk a little bit longer than a year to make it around the Sun. We take one quarter of a day more.1661

We have to have 365 days, 365 days, 365 days, 366 days, 365 days and then in reality there is even more things that have to be corrected when you start to expand it.1666

It's actually 365 and a quarter minus just a little bit. So the fact that we have got 365.33 days when we simply this, we dealt with it as a circular orbit, which it's not perfectly.1677

We dealt with it as if the only force involved was the force of the Sun. The Sun's gravity on the Earth, which is not the case.1687

There is actually a bunch other things going on. We got a really, really good answer.1693

So just like with air resistance, you can manager to pretend it's not there sometimes and still get really good answers.1699

It's only when it's a really egregious thing to keep it out. When it's really bad, it's really important that it not be forgotten.1703

Like say, dropping a piece of paper, flat side down, that we're going to have to worry about the fact that we're getting rid of it.1709

In this case though, we really able to get a really close estimation.1716

Example 4, final example. A neutron star is a very dense type of star that rotates extremely quickly.1720

If a neutron star has a radius of 15 kilometers, which is the same thing as 1.5 x 10^4 meters, because we got to have things in standard meters.1727

It spins at a rate of 1 revolution per second, we can figure out what its minimum mass must be based on the fact that it doesn't fling itself apart.1734

What is that minimum mass?1740

We've got this thing spinning around very, very quickly. Say we consider some chuck of its surface, there's not really things on top of a neutron star.1743

The force of gravity so strong that it's going to just a pulp.1751

There is something, some chunk of it on the surface. Let's say that chunk has mass 1.1755

For it to continue to spin in a circle, because we're saying the neutron star is circular.1761

For it to be able to spin in a circle, it's got to have a centripetal force on it.1765

Force centripetal, what's that have to equal? Always has to be pointing in the center and equal the mass of the chunk times V squared over R.1769

V squared being actually the speed squared. Being a little bit lazy, sorry.1780

What forces are keeping it down? There is nothing holding it down, it's not tensioned to the surface.1787

The only thing holding it down is raw gravity. So we know that the force of gravity has to be at least equal to the force of centripetal.1792

We could have more than that right? Because there is pressure inside, so it could be larger gravity than that because just like you could have more gravity and still be attached to the Earth, you that the force of gravity has to be at least enough to hold up the centripetal force.1799

Then there could be a normal force to cancel out that extra gravity, but we know that the force of gravity...1814

has to be at least greater than or equal to the necessary centripetal force for that object to stay on the surface otherwise if the force of gravity is less than the centripetal force...pleh the entire neutron star will just explode out every which way and we won't have a neutron star anymore.1821

What will that minimum mass be? So let's look at the minimum case, which is going to be when the force of gravity is equal to the centripetal force.1837

If the force of gravity is equal to the centripetal force, we're going to get G M1 M2 over R squared equals that mass, that chunk on the surface, times V squared over R.1845

In this case, let's make that chunk on the surface M1 V squared over R.1859

M1's cancel out and we get G M2 over R squared equals V squared over R.1864

What are we looking for? Where looking for what the mass of the neutron star is, what the minimum mass is.1872

Remember we know force of gravity must be greater than or equal to, so the minimum mass is going to be when the force of gravity is equal to the centripetal force.1879

We want to solve for M2. M2 is going to equal to V squared.1886

We multiply both sides by R squared and that will leave us with an R up top and divided both sides by G, so we'll have G on the bottom.1894

What’s V? Well, how fast is it moving around? If it makes one revolution per second, then that means how much distance it covers in that second.1901

So V is going to be equal to distance over time. Which is going to be equal to the circumference of the object divided by that one second because it manages to make one revolution in a second, right?1911

Circumference of the object is 2πR divided by one second so we're left with just...1925

2πR meters per second. So that's what the velocity is. We sub that in.1933

We get 2πR squared times R over G or 4π squared distance the surface cubed divided by G.1938

We plug in a bunch of numbers, all those numbers that we have.1956

4π squared. What's R? The radius is 15 kilometers, so 1.5 x 10^4.1960

Now, it's not just squared now, it's not to the 1, it's cubed.1972

We divided this whole thing by G or 6.67 x 10^11. Sorry, not 10^11, 10^-11.1977

What do we get when we punch this all into the calculator? We're going to have to get some pretty large number right?1988

We've got 10^4 cubed up here times these other numbers and then divided by 10^11.1992

Since it’s a negative exponent on the bottom it's going to wind up adding 10^11 on the top.1999

We punch that all through and the number that we wind up getting is 1.998 x 10^24 kilograms.2003

It has to be at least more than a third of the mass of the Earth otherwise it'll explode out in all directions.2015

In reality neutron stars turn out to be way more than that but we figured out what the minimum is based on the simple thing we've got right here.2022

The fact that it's rotating very quickly and it doesn't want to fling itself apart, it's got to have something holding itself in and we're assuming it's going to have to be holding itself in by gravity.2029

Because it's not a solid object, it's under that kind of gravity, it's kind of a soupy mass of things.2041

So to be able to keep itself from flinging itself apart it's got to have enough gravity to hold itself together.2046

Any object, any piece of itself, any mass of itself, must have enough force of gravity to overcome the necessary centripetal force.2053

Welcome back to Educator.com Today we’re going to be talking about an introduction to waves.0000

We’re going to finally get an understanding of how waves work and they have an incredibly large presence in our daily lives.0005

Waves are way for one object or location to have energy transferred to another object or location.0011

That seems like a really large definition and it is. It’s not the only way for energy to be transferred but it a motion of energy from location to location.0018

It also has a lot of other effects and we’ll be investigating some of those.0028

There are many different kinds of waves and you’re constantly around them.0032

Some basic examples that you’re being currently being exposed to: sound and light.0036

Also any vibrating strings, waves in water, and the list goes way on.0041

There’s all sorts of other things, seismic waves in the ground, vibration along a steel pipe.0046

There’s quantum motion in waves, though we won’t be getting into that.0052

Waves make up a fundamental part of the universe and nature around us.0055

To begin with, let’s imagine the idea of a pulse. Imagine you’ve tied one end of a string to a wall, so it’s tied over here and you pull the string taunt.0061

Then you whip it, really suddenly, once. You whip it hard, what’s going to happen?0072

Well you’re going to create a pulse of energy that’s going to be sent down the string.0077

As the energy might be here, we’ll have a sort of raised up area and then as time goes on, not long time probably, it’s going to move to here and then it’ll move to here then it’ll move to here until eventually it hits the wall.0081

So that pulse of energy will be sent down the string and it’ll keep moving down the string towards the wall.0095

We whip it once and we move the energy, the energy that we put into our whip gets transmitted down the string.0099

Some of it goes into heat and other things, we won’t actually be investigating the energy, we will just be investigating the motion of waves.0105

There is definitely a change of energy from a location to another location.0111

You transmit that motion down the line into the wall.0119

Do the same situation but this time you start whipping the string up and down regularly, creating multiple pulses at even time intervals.0122

Some of these pulses are pointing up and some of these pulses are pointing down. What’s going to happen then? You’re creating a wave.0128

Instead we’ve got one pulse here and one pulse here but they form together to make a continuous wave shape. The whole thing makes a wave.0133

Once again, as time moves on, the things going to slide over and the whole thing will move farther along the line and get closer to the wall.0141

As time goes on, the thing shifts forward an amount. It has a constant velocity moving towards wherever it’s moving towards.0150

We’ve created a wave, there is a number of characteristics about the wave. Some of these drawings aren’t quite perfect but the important things is that waves have regularity.0159

It’s the fact that we can always trust it to get to the same height or pressure differential as we’ll talk about much later in sound.0168

We can always trust it to get to the same levels every amount that it goes.0173

Once it goes through one oscillation, one period of itself, it manages to repeat all the same effects.0178

So looking at one snapshot, the right chunk snapshot, it can look just like another chunk snapshot, can look just like another chunk snapshot.0184

That’s when the important ideas of waves, the fact that we’ve got regularity, something we can trust is going to be regular coming down the line.0191

There are three main categories of waves.0199

Mechanical waves are the waves that you probably have the most familiarity with at this point.0202

These are the waves that travel through any sort of material medium.0206

Common examples, the most common one that we’re all used too, sound waves.0209

Every time you hear something, if you hear a big bass drum get kicked at a concert and you feel those vibrations in your chest, in your body, that’s actually another form of sound waves.0212

They are just a very low frequency, so they’re able to vibrate your entire body.0223

Another type would be water waves. Waves on the top of waters is another form of waves.0228

Seismic waves, waves in the motion of the Earth. Things that cause earthquakes, these are all different forms of waves and different propagations of energy.0234

Electrometric waves, EM waves, don’t require material medium to exist.0244

We’re also exposed to these constantly, but you might have less of an intuitive understanding of them.0248

Examples of this is radio waves, the visible light that you’re currently seeing from the screen, X-rays.0254

Anything that has, that is, is light, is a part of the electrometric spectrum, but we’ll talk about that more.0258

That’s a little bit difficult to understand, because light has a whole bunch of special properties that really affect the nature of the universe in incredibly strong ways.0267

That’s going to be its on special section. We’ll very lightly touch the surface of that.0277

But for now, we can just think of it as light.0281

Finally, matter or quantum mechanical waves. These are waves that describe the motion of elemental particles, like the electron, the proton, or even smaller, the quark or the photon on the atomic level.0284

They’re very important to modern physics. They have a whole big impact and what we understand now and where our research is going currently.0296

We’re not going to be investigating them in this course. They behave a little bit differently than the classical waves we’re going to be studying.0302

They’re going to be more challenging to understand, so that’s something we’re going to save a future course.0307

Alright, in addition to the previous categories, the mechanical waves, EM waves, and mater waves.0313

We won’t be talking about mater waves. Mechanical waves are the ones we most understand.0320

They’re the ones that have motion medium. EM waves, they sort of come with their own medium.0325

In addition to the previous categories, we can also classify waves based on how they move.0329

Transverse waves. The particles of the waves move perpendicular to the motion of the waves.0334

If we got something that would normally be a flat line like the surface of the sea.0339

Then we’ve got a wave, an undulating curve on top of it that causes the surface of that sea to move up and down depending on the location of the wave.0344

That’s going to be a transverse wave. It’s happening transverse to the motion, it’s happening perpendicular to the motion of the wave.0353

Examples include a vibrating string, EM waves, any waves in water; not any waves, sorry you can also have sound waves in water.0361

Waves that we’re used to seeing in water. There is a huge varieties of things like this.0369

If you were to knock on a pipe, you’d also once again…if you were to shake a pipe, you’d get waves. If you were to knock on a pipe, you’d get sound waves.0375

Longitude in the waves. The particles of the wave move parallel to the motion of the waves.0387

So this a little bit harder to see and we’ll get the chance to get a better visual understand of this once we get to sound.0390

The particles of the wave move parallel to the motion of the wave. This done through compression and rarefaction, expansion.0395

So if we got two particles like this and remember they’re charged particles so they don’t like getting to close to each other normally.0401

The wave is going to be transmitted by pressure changes. If one particle pushes towards the other one for a moment, a brief moment, they’ll be together and push back away from one another.0407

That pressure wave will wind up getting transmitted all the way through.0418

A pressure front and once they push away from each other, we’ll have a rarefaction of area and they’ll wind up getting pushed back together again by the next pressure front coming through.0423

Pressure changes will transmit the information, will transmit the energy. The motion is being caused by pressure changes.0431

Most common examples is sound. But it also exists in certain kinds of seismic waves.0439

Alright, let’s investigate the wave more deeply. Here’s an okay drawing of how a wave looks.0446

This is a general sinusoidal wave and we can treat most waves as sinusoidal.0451

Meaning it comes from sin, as in the sin function that looks like just like this when you graph it out.0456

First off, amplitude. How tall the wave is at its maximum height.0464

In this case, the amplitude would show up right here. We’d get A there.0469

Somewhere else we’d see it here, we’d see it also right here. Anytime we see the maximum height.0472

What about down here? Well down here, it’s also going to be a height of A.0477

Instead if we were to look at it from an absolute point of view it’s going to show up at –A, although the length will be A.0482

Amplitude is the absolute value of you location from the medium location.0489

Its how far up or down you go, but it’s always going to wind up being a positive number.0495

How tall the wave is at its maximum height is amplitude. A.0499

Wave length, the distance between repeating points on the wave, such as top to top.0504

Remember, one of the important qualities of the wave is that it repeats itself after a certain amount of time.0508

If we go from here to here, notice that this entire area here looks just like this entire area here.0513

Just like this entire area to the front of it and this entire area to the front of it looks the same.0523

And if fact, if we were to go all the way through, we’d wind up getting all the same information.0528

It’s going to look like the exact same thing from the wave length point of view.0532

When you get a one wave length, the distance that it takes to do an oscillation in the wave, you’re going to see repeating points.0537

Everything will wind up repeating. If we go from top to top, we’ll have a wave length.0545

The guy that we use to call out wave length is lambda. Lambda is another Greek letter, he gets thrown around now in waves, he becomes very important.0553

He’s pronounced lambda and spelled in case you’re curious, lambda. Lambda. Lambda is the guy.0561

What some other repeating points on this? What about from here to here?0569

Well that isn’t going to wind up being lambda because notice how this section right here is doing something different than this section right here.0573

If we wanted to say, look from this point, we’d have to go all the way to here because know we’ve got just to the right of it.0582

It’s going up a bit and just to the right of it over here it’s going up a bit.0588

So from here to here would also be another example of lambda.0592

We could do to this from anything as long as we wound up being at the same height and we went through the right amount to wind up having it experience the same affects inside the wave.0597

We’re able to get one wave length. The important part about a wave length is that after you go that distance everything repeats in the wave.0607

More characteristics. Waves speed, how fast the wave is moving.0615

Remember this whole thing is moving forward at some speed. We’ve got some velocity that it’s got.0619

After one second it will be that many meters farther ahead.0625

For velocities and meters per second, we take that and one second later it will be that much velocity meters ahead.0630

Remember the wave is moving along. And as this moves forward we’re going to wind up seeing more of it coming out of it.0638

There is always more wave that’s going to be put in. Either it’s too the left outside of where we’re looking or it gets whipped into place by the motion of whatever is creating the wave.0645

The wave continues out on either side, and the whole thing moves forward.0656

Period. The period is the time it takes to go through a full oscillation.0664

Remember if we got some speed V, that this whole thing has, we’re going to be able to get from here to here and we’ll have a repartition.0668

Now one way is we could change the location we’re looking, but we could also fix the location that we’re looking here.0678

We’d be able to see that T later. We start here, say we start here, but T later because of the velocity will wind up showing up there.0684

If you notice, the velocity times the period is always going to wind up equaling the wave length.0698

Because the amount of time that it takes, if the velocity goes this way and T later is how long it takes to go through a single oscillation, the period is how long it takes from wave peak to get to another wave peak.0708

Or one point on the wave to get to its corresponding twin point on the wave.0721

That’s going to wind up having to be the wave length right? That’s what the definition of the wave length is.0727

We can think of it as a movement, the time that we’ve allowed it to move, and has given us a repetition, a period.0732

Or we could think of it as how far different we’ve seen in the distance that we’re looking in the wave, has given us a repetition on the wave.0740

So we’ve got the velocity times the time that it takes for the period is going to equal that wave length because they’ve got to be equivalent.0747

Frequency, the number of oscillation’s that occur per second.0754

Say we’ve got some period. T is equal to the period right? So in here…it’s gets from here to here if we’re looking from some single place.0759

If we fix our gaze here, so fix our gaze on this blue line, this blue dash line and we look up, we’re going to see this point here.0775

T time later will see the red dot above us. T time later the red dot will have moved to where the blue dot is.0783

If that’s the case, we can ask, alright so, it takes T many seconds and T could a larger number, greater than one or T can be a small number that’s well less than one.0791

We could ask, if it takes us T many seconds to get a repetition in the wave, how many repetitions are we going to get every second?0805

How many oscillations are we going to have per second? How many wave peaks are we going to have per second?0814

Well if that’s the case we’re going to need something that measures per second and that’s where the hertz comes in.0820

That’s one hertz equals one per second. One over one second. So that means the frequency is equal to one second divided by the period.0825

The period, so if we have something is the period of T=0.2 seconds then that means the frequency is going to be 5 because in one second we managed to have 0.2 seconds occur five times.0836

Frequency is the inverse of the period. And the period is equal to the inverse of the frequency.0854

Anytime we want one for the other, we just flip it and we’re able to get the answer.0859

One divided by the other and we take reciprocal and we’ll get it.0863

That’s because the fact we’re looking for long it takes to get from one like point on the wave to another point.0866

How long it takes for a single oscillation to occur is going to also be if we divide one second by how long it takes we’ll get the number of times we get an oscillation.0873

If we have the number of times oscillation happens in a second and we divide one second by those numbers of oscillations, we’ll get how long it takes each oscillation to occur.0882

Finally because speed, frequency, and wave length are all deeply related, we’re going to have the fact that velocity is equal to lambda times frequency.0892

Which is equivalent to velocity times time equals lambda. Remember we know that velocity that we’re moving, times the time it takes to from peak to peak has to be equal to lambda.0900

We’ve got the same thing going on here since V=λf. Well if we divide by F on both sides, we get v/f=λ. Which 1/f=t, so we get vt=λ.0914

So these are equivalent expressions. Why is V=λf make sense? Well if we’ve got lambda distance here, this is lambda…distance.0926

And we know that time, in one second we’re going to see the frequency occur. So if we see ten oscillations in one second, then how much distance have we covered?0939

We’ve covered ten oscillations to each oscillation as one wave length, then ten times the wave length.0949

So the frequency, the number of oscillations we occur, that occur in one second, we have in one second, times how long each one of those oscillations is, is going to be the velocity since we can just divide by one second.0955

Velocity is equal to lambda, the distance, times frequency, which comes in one of our seconds. So we’ve got meters per second.0967

Great. Wave equation, finally we can talk about if want to know the height of a point on a wave.0975

We’ll need to know two inputs. The horizontal location x and the time we’re looking at the wave.0982

Here’s just a quick sketch of a wave. If we got some wave like this, we could either look at this location x or maybe we want to look at this location x.0987

We’re going to get totally different results for that. However, what happens if then we also say, one of them is T=1?0999

What happens when we look at T=2? Well then the entire wave is going to have slid over some amount.1006

We’re going to have the same wave, but it’s going to have slid over and we’re going to get totally different answers for each one of those x’s.1013

That means, the time that we’re looking and also the location on the wave that we’re looking matters.1021

We’ve got a two variable function, we’ve got an equation that’s going to be based off of two variables that give us that dependent variable, that why, what output value.1027

Notice that one other thing about this, if we only care about the point where the wave originates, the very beginning point, x=0, we can simplify this because we can just knock out that x term.1037

We can make it simple at A times sin of omega T. Now, at this point you’re probably wondering what k and omega is, that’s a great question, we’re about to answer it.1046

Our equation is Y of x,t equals a, amplitude, times sin of kx minus omega t and quantity.1056

So before we can use our equation we’re going to define what k and omega mean.1065

K is the angler wave number and it causes our function to give the same value for every wave length lambda.1070

Now remember, in case you don’t remember some of the stuff from trig, every time sin of 2π times any number, n.1075

Where n equals -2, -1, 0, 1, 2… on either side. Any integer number.1085

Sin of 2π repeats, the whole thing repeats, no matter what we’ve got added to that inside of it, we’re going to wind up getting the same answer.1096

Because what we’ve done, whenever we do 2π, we’ve lapped the circle. And we’ve winding up starting here and we lap the circle, then we’re still going to get the same answer.1103

If we lap the circle in the other direction twice it doesn’t matter, we’re going to the same answer as long as we land at the same point.1113

As long as we change by things of 2π we’re going to have the same thing show up.1119

Which makes a lot of sense because on our wave, as long as we change by some wave length distance, we’re going to have all the same stuff show up.1124

If we change by one wave length and we’re using sin, we’re going see inside of it, a factor of 2π show up.1132

So for every wave length we change, we show up by 2π, that’s why we get k here.1140

Because if k is equal to 2π/λ and then we move over by 2λ amount, well the lambdas cancel out and we get 4π, which is equal to 2π times some n.1146

So we’re going to see the exact same thing as if we hadn’t shifted over, which is exactly what makes sense.1159

When we shift over by some wave length amount, it shouldn’t have an effect on what we would have seen otherwise.1164

Shifting over by a wave length is no different from the point of view of what value we’re going to see for the height or the pressure differentials that we’ll talk about later in sound.1168

Same basic ideas going for omega. Omega is the angler frequency and it causes our function to give the same value for every period time. So every period, big T.1178

Omega, not sure if I said, I don’t think I said, this guy right here, is once again another Greek letter and its pronounced omega. O M E G A.1187

You’re probably seen his bigger brother at some point. Capital Omega is that guy but this is lowercase omega.1201

Back to the thing at hand. Omega is the angler frequency and it cause our function to give the same value for every period.1209

Remember once again, every time we pass one period of time, as long as we’re looking at the same location, we should experience no difference right?1216

Every time you go through one period you’ve gone through one entire oscillation, so the wave just looks the same to you from that point of view.1224

As long as you wait 1T to look, everything is going to be the same.1230

So if we plug in omega for 2π over T. And then we hit it with some time that is factor of a period, 3t. Those t’s cancel out and we get something that is just going to get canceled out by sin.1234

We’re going to care about the other information inside of there. So as long as we wind up shifting some amount by a period, it’s not going to have an effect.1246

Let’s look at a slightly more complex example. We’re not going to see the time be just a factor of the period.1256

What if we look at t=1 and t=1+t? So if we look at that then we can say, let’s make it easy on ourselves, we’ll look at just generally…we’ll look at the same location x.1260

X is going to be fixed, otherwise we wouldn’t wind up seeing the periodic nature of the oscillations reoccur.1276

So our two values is going to be, our function is going to be y=a(sin)kx. And what’s k? We’ve got as 2π/λx - ω2π/. x t. The time.1282

If plug in t=1, we’ll plug in t=1 over here so t=1, we’re going to get y is equal to a(sin)2π/λ whatever x we’d chosen minus 2π/tx1, so there we are. That’s what we’re going to wind up seeing.1305

What happens when we plug in 1 plus one more period of time? Well we’re going to hope that we’re going to see the exact same value.1328

We’ll plug that one over here, we’ve got y=a times sin of 2πλ times x minus, now what…times one plus t, we’ll get 2π over t plus just 2π right?1334

So if that’s the case then we’ve got y=a sin 2π λ x - 2π/t - 2π.1358

This part right here winds up being the exact same as this part right here.1371

The only part that’s different is this here, but remember since we’re dealing with sin, since we’ve moved just one more 2π, we’re some location.1379

Then we just spin it around and boom we wind up being at the exact same location because we’ve moved by -2π.1389

So this location here, this stuff all here, is our real location. So if wind up shifting things by one whole period, even if we start at some time that isn’t a whole thing of a period, the way we’ve got this set up.1396

Since we’re using sin, a regular periodic function, we’ve got the fact we’re able to make those oscillations occur in our mathematics, our algebra, is able to support our visual oscillations.1409

We move a period temporally in algebra, we wind up moving that same full length time, so we see the exact thing.1422

The exact same would happen if we used k, if we used…if we moved x around and this sort of thing, but this is just to illustrate that the math here is really working and why it is.1430

We’ve got to make sure that 2π over t is able to handle a motion of a period as causing no effect to the values we’ll get out. Same thing with the motions of a wave length.1439

That’s why that 2π comes in, is because sins, sins natural period is 2π right? Its period before it repeats is 2π.1448

So we have to have a way to have those two periods, the period of the wave we’re working with and the period of the natural function we’re working with to be able to communicate with each other and that’s what this is all about.1456

Then finally, if we have 2π/t since over t is the same thing as times frequency, we get omega is also equal to 2π times the frequency, as simple as that.1465

Finally we’re ready to work on some examples. If we have a CPU on some device that has a frequency of 1 gigahertz, then we’ve got a frequency equals 1 gigahertz.1476

How many hertz is that? Well it goes, kila, mega, giga. So 10^3, 10^6, 10^9.1487

So 1 gigahertz becomes 1x10^9 hertz.1494

What’s the time that it would take to complete a single cycle? So if we want to know what the period is, then we remember the fact that frequency equals 1 over t.1501

T equals 1 over f. If want to know the period, t equals 1 over 1 gigahertz. 1x10^9.1512

Which becomes 10^-9, which is the same thing as mila, micro, nano. So we get one nanosecond is how long it takes.1523

There are 10^9 nanoseconds in a single second, which makes sense because 10^9, there has to be 10^9 nanoseconds because each of those cycles has to be complete for it to be able a whole frequency of 10^9 cycles completing.1524

So CPU, we’ve got the same, it’s not directly a wave in the same way, but we’ve got the fact that it’s having repetitive things happen.1539

Its going through cycles, it’s cycling through data. We’re able to talk about it in terms of frequency and in terms of periods just like we do with waves.1566

If the speed of light is velocity 3x10^8 meters per second, mighty fast. And you receive a wave with a wave length of 500 nanometers. Very small wave length. What is its frequency?1573

Speed of light is v=3x10^8 meters per second and you receive a wave with a wave length of 500 nanometers. We already know what v is, v is here.1588

So 500 nanometers is the wave length, right? The wave length equals 500 nanometers, which is the same thing as nana, 10^-9, 500x10^-9 meters.1598

If we want to know what its frequency is, we remember that velocity is equal to the frequency times lambda.1611

The frequency times lambda has to give us the velocity because that’s how much distance has been covered.1620

Wave length is a chuck of distance and frequency is how many times you have those chunks distance in one second to we cover. Frequency times wave length.1625

We cover fxλ and that gives us our v.1634

v=fxλ, so v/λ=f, so if we want to know what f is. F is going to be equal to 3x10^8/500x10^-9.1637

We plug that into a calculator and we get 6x10^14 hertz. That’s a giant frequency compared to the stuff that we’re used to seeing in sound.1655

For light though that’s pretty reasonable and you’re probably seeing a color pretty close to that.1666

6x10^14 hertz is a receivable for the human eye, it’s something in the visual spectrum and I think, don’t hold me to this, it’s probably pretty close to either yellow, green, or blue, somewhere in that sort of range.1671

Probably a little bit closer to the blue, green or blue. Anyway, that’s something that your eye is actually really able to see and so as opposed to seeing a numerical frequency data when we look at something.1687

We don’t say “Oh, that frequency is that”. We see a color. We’ve got these other ways of interpreting the information that the universe is sending to us.1697

It is a real thing, we are getting real information here just like when we go to some height, we’re really at a height but we can measure that height and periodically we’re able to measure the frequency we see in periodically.1704

If you’re driving a car going 30 meters per second and the car beings to run over a rumble strip, rumble strips are these evenly spaced grooves used to alert drivers, so little down grooves in the road like this.1716

So when a tire rolls over it, the tire falls in the up and down motion, winds up jostling the driver and they notice something, or they’re accidentally swerving off to the side, they notice that they’re swerving off to the side or if they’re coming up on some toll, they notice that they’re coming up to some toll.1733

It’s just something to alert drivers. If the strip vibrates your car at 98 hertz, what’s the spacing of the grooves?1747

First off, I’d like to point out that this isn’t technically a wave. Just like in #1 we weren’t technically working with a wave, but we can still apply many of the concepts.1754

In this case, the wave speed, we know the velocity of the car is equal to 30 meters per second.1762

While the wave might not be moving, the way it’s experiencing the wave is moving. In another way we could consider the tire is still, and the wave is the thing moving.1771

From your point of view you can’t really tell who’s moving when you’re inside of the car, although we look at our scenery and we can easily tell.1783

But you know, we can’t be sure who’s the thing that’s moving. It’s possible the road is the thing moving. So you can take it as the wave moving underneath you.1789

Once again, we’re not getting full repartition quite the same, maybe. But we’re not necessarily thinking it purely in terms of wave lengths, but that isn’t the issue.1796

We can expand a lot of these ideas to more things.1806

So we’re running over some rumble strips and it’s moving by us at 30 meters per second, either because the cars moving, or because it’s moving relative to us.1810

If the strip vibrates your car or bounces you up and down 98 hertz or 98 times per second, what the spacing of the grooves got to be?1818

It’s the exact same thing, if it manages to bump us up and down 98 times in a second and the wave length is the spacing of the strips. The spacing of the grooves is going to be the wave length.1827

How far they spread apart, then the 30 meters per second that we experience has got to be that number of times that the wave length times how many times they show up in a second.1838

Velocity is equal to the frequency times lambda. So we plug in our numbers and if we want to know what lambda, is we’re going to have the velocity divided by the frequency.1849

We plug in our numbers 30 meters per second divided by a frequency of 98 hertz and we get 0.306 meters.1860

There we are, the spacing of those grooves is about a third of a meter which makes a lot of sense if you’ve ever looked at them on the street.1872

Example four. Use the following diagram to give an equation describing the wave in the diagram.1882

Now to begin with, lets point out, we just want to remember y equals a, the amplitude, times sin, the function that allows us to have periodicity, allows us to have oscillations occur algebraically.1887

kx - ω x t. So x, the location we are in the wave. T the time that we’re looking at the wave. Omega is the factor that allows us to handle the fact that periods cause repetitions.1901

K is the fact that allows us to handle that wave lengths cause repetitions. To begin with, we know that the period is equal to 0.02 seconds.1915

We know that amplitude is equal 0.5 meters. Okay, great. What is this here?1927

Well that is not equal to the wave length. Remember, if we look just to the right of this point, and look just to the right of this point. We should see the exact same thing.1935

We don’t see the exact same thing, it’s going down over. If we go over here though, we will see it.1944

And because its sin wave, it’s evenly spaced out throughout, we know that 3.5 meters isn’t going to be the wave length, but it’s going to be half the wave length because it’s in one of the dips.1950

So the wave length over 2, so that means that our wave length is equal to 7 meters.1961

That’s all the data that we wind up needing. Now we want to solve for what k has to be.1968

K is equal to 2π/λ, so k is equal to 2π/7. Omega is equal 2π over the period.1972

By the way, k, we’re throwing around k a lot. I never mentioned this but k is not the same k as when we’re dealing with springs. It’s a different k, we’re using it to mean a totally different thing at this point.1985

The k that we’re using in this stuff is different than the k we used before. Just like how little t and big t don’t necessarily mean the same thing. We can even wind up using the same letter for different things, and we have know contextually what we’re talking about.1999

Sorry, I made that assumption, but that’s actually a really important thing. You don’t want to get confused and think that springs and waves necessarily have to do with each other every time. It’s not that same spring constant we were talking about before.2011

Totally different use of k. Anyway back to the problem. Omega is equal to 2π divided by the period.2024

If we know the period is, 2π/0.02 seconds and that will wind up giving us 100π.2030

Simple as this at this point, we just plug in all of our numbers. Y is equal to that amplitude, 0.5 meters times sin, of the numbers 2π/7x-ω100π times the time that we’re looking.2039

That right here is our answer. Simple as that. So we just want to be able to analyze the diagram.2065

One of the most important things to pay attention to the fact that wave length has to be not just some distance where you get the same point, but some distance where you get a point that means the exact same thing.2071

That if you look just a little bit further on and a little bit further behind you’re going to see a full repetition.2083

It’s not just the same point, because the same point occurs at any horizontal thing, except for the very tops and bottoms.2089

These pairs of points are not enough to determine a wave length. What you need is to reach is to reach a little bit farther and look here.2097

The very top to top because tops themselves only occur once every wave length.2104

Whereas middles occur twice. Same with bottoms, you can go from the bottom to the bottom and the top to the top.2111

That’s normally the easiest thing to measure, but if you’re measuring from the middle to the middle, this isn’t enough.2117

You need to also go to here. Okay, great. I hope you enjoyed that, I hope that made sense, waves are a whole bunch of big ideas, but we’re going to…2123

Hello and welcome back to educator.com. Today we’re going to be continuing with more waves. Last time we talked about our introduction to waves, we got our feet under us.0000

Now we’re going to start talking about some more complex ideas.0007

So we’ve got the basics of waves under our belts and we’re ready for the tough stuff.0011

First thing we’re going to discuss is how waves interact with each other, called interference.0014

It’s an idea that’s going to have a lot of ramifications and it’s really important.0019

Then we’re going to also consider when a wave emitter is moving in relation to the receiver.0022

So when, if it were sound waves, when the listener and the speaker are moving at possible rates.0027

They might be moving away, one might be moving in one direction while the other one is following, all sorts of possibilities.0033

Alright, first we’re going to talk about how waves interact with each other.0039

When two waves are in the same medium at the same time they interact.0042

The result though is remarkably simple, it’s actually really easy.0046

Their individual effects simply add together. Visually we superpose, so if one wave is here and another wave is here, we… and our base line is here.0050

We’re going to add this little bit on top of this and the point there would be that.0059

We do that point wise for every little point and we can draw a new wave that gives us what the total is of the two waves put together.0064

We’ll have a lot more visual examples coming up soon.0072

Algebraically we just add our equations together, if we got one wave equation, y1, that describes the first wave by itself.0075

y2, that describes the second wave on its own, then we put the two together to get the result in wave of what happens in the medium when they’re both going at the same time.0082

We call this phenomenon interference because the two waves are interfering with each other.0092

If the waves interfere in such a way so as they actually create a larger result, then they’re working together it’s called constructive interference because they’re building something.0097

If the result is smaller, it’s called destructive interference because they’re actually lowering, they’re making things smaller, they’re destroying the result.0106

Constructive interference, increasing. Destructive interference, decreasing.0113

Furthermore, while the interfering waves do cause a different result in the medium, it’s important to point out that they have no effect on either’s movement.0118

The waves travel unimpeded as if the other one wasn’t there at all.0124

There is in real life, in real situations, there is some slight changes in the way they move but for the most part this is actually true in real life.0128

You know, just like in some cases we made some slight things where we didn’t account for everything that could have.0134

This is actually closer than neglecting air resistance, which we did a lot. It’s not quite perfect, but for our purposes it’s pretty darn realistic to actually say that the waves travel fully unimpeded by another waves presence.0140

Interference, the idea is much easier to understand with some visual examples so let’s consider two positive pulses.0154

We’ve got the red pulse and the blue pulse. As they come together, they’re going to make a third, a combined pulse that is going to be the purple pulse.0160

The purple pulse will be the resulted…what will happen in the medium. In real life, if we were seeing the two things come together, we wouldn’t see the red and the blue pulse anymore.0168

We would just see the purple pulse. Remember, we’re going to see the added together thing, we’re going to see the final result.0176

If we know what the two pieces are coming together we can know what the third is. But just seeing just the third, we don’t know if it’s a wave on its own or if it’s two waves put together until we study it more closely.0182

We’ll see these start to move together and we’ll be able to talk about it as it goes.0191

First one, we see…it’s a little rough as a sketch. We see the distance from here to here. Winds up getting added here to here.0196

We get that point and so on and so forth throughout the whole thing. Here to here, it’s added.0208

Here to here, and so on and that’s how we get this whole curve.0212

It’s just everything added pointed wise and we go through the whole thing and we can see the purple wave sort of just grows on top and once they separate it just winds up being each wave on its own.0217

As the thing moves, the purple wave. The purple wave that’s the result of the thing, what you’d see if you were actually looking at it is a combination of the two waves that’s making it up.0232

It rides on top of the other ones. Alright, that’s with two positive pulses.0240

What’s going to happen when we have it with the waves, not the pulses?0246

So if red and blue are perfectly opposite, they’re just out of phase with one another, they’re synced so they have opposite oscillations.0248

There out of phase by π. If we look at example 3 in the…actually I’m sorry, we’re going to have that, we’re going to have that in this one, example 3, we’ll talk about that.0257

Sorry, not example 3. 2, example 2. Anyway, with waves instead of pulses, if the red and the blue, if they have…0266

This is length A and this also length A and they have the same frequency and whatnot, then they’re going to cancel each other out perfectly.0275

This height here is canceled out by this height here, because remember it’s a negative height. This height, and really they’re not quite perfect, sorry my drawing is not exactly accurate.0281

But this height should be as tall as this height, this height as tall as this height, and so on and so forth.0292

When you add them always together, one’s positive and one’s negative, and they’re the same size. Boom, they cancel to zero.0296

If we had it with instead, them being synced together we’d actually wind up getting larger positives and larger negatives.0304

The purple would sync together with the two of them and we’d get an even bigger result, we’d get a larger wave that’s the combination of the twos waves putting it together.0312

So we’d get an even bigger thing in the end. Those are both in sync in one way or another.0322

Either they were out of phase, where they opposed each other in the first example.0330

They’re in phase, where they were constructively working. First one being destructive interference, the second one being constructive interference.0333

Now we’re looking in the third example that is not just one or the other. Here the two add together and we get massive constructive interference.0340

But here, we get destructive interference. So we show up, we drop to zero, so this thing actually gets shifted around in more ways than just having things…just think of it as growing taller or getting smaller.0350

It’s actually been to the side and it gets more complicated. The only way to figure out what this would be is to actually know what the blue one is, what the red one is and add their wave equations to figure out what our purple wave equation would have to be.0363

We can also figure it out point wise and look at things by knowing what this value is and knowing what this value is and then just adding those two values and wind up getting the third value for the purple one.0375

With all of these cases, the result has to do with the phase of the cycle. One wave is compared to the other.0386

In the first example, they were out of phase because they were destructively interfering. They were out of phase by a phase angle of π.0392

Because they are…we’ll get a little bit more of that now, so we’ll not worry about it right now.0399

The phase angle, there is some amount that they’re off. They’re out of phase by some amount, I think that’s pretty clear at this point.0404

When they’re in phase they’re always working constructively together. The second example where we saw it grow larger throughout.0411

In that case, they’re going to be in phase together because they’re always working together.0417

Then, there is also a sort of middle thing where they’re neither fully in phase nor out of phase and they’re only partially...they’re between the two.0422

So, you’ll get sometimes constructive interference and sometimes destructive interference.0429

So a wave, if you have two waves interacting with one another it’s not just going to be frequency and wave length and what time we’re looking and what location we’re looking at.0433

It’s also going to have to be…you could start one wave a little bit after you start another wave and even if they’re the exact same looking wave, they’d be shifted by a certain amount, right?0442

That’s where the idea of phase shifting comes in, phase angle.0452

So we need some algebraic ways to talk about this slight shifting of phase from one wave to another phase.0455

To do this, we introduce a new variable to our wave equation. Phase shift.0461

Once again, this guy right here, yet another new Greek letter, he’s pronounced either fee or fi, fee fi fo fum.0466

But seriously, fi or fi, kind of depends on who you are. I don’t think there is actually a standard pronunciation. Me, I’m probably a fy guy…no I’m a fi guy. I’m a fi guy.0471

If somebody tells you they think it should be pronounce another way, pronounce it that way in front of them but, you know, check a pronunciation dictionary if you really want to know.0485

Anyway, this guy right here is spelled like that. Spelled like that, so that right there, fy or fi.0492

Y…so our wave equation is a, amplitude, times sin of kx. K being what we talked about before, 2 π over λ minus ωt.0501

Where ω is once again where we talked about before 2 π over period plus the new guy fi, fi.0512

Plus the new guy. So the new guy winds up, winds up giving us the ability to shift, right?0519

So kx-ωt, these things are determined here, but by this, we could have the same thing. We could have this wave get described both in x and t.0526

If we wanted to we could shift it forward by some little amount here as opposed to having to say ‘Look at a different time’, it allows us to have a time factor for the whole thing.0537

That allows us to have one that’s the original equation without the fi and another one that’s a new equation with the fi. I guess I’m a fi guy, not a fy guy.0544

Whatever we put up, the beginning one would be without this, it would be zero. Fi would be zero for this one.0553

The second one would have some fi and would could calculate what that is.0560

Now there is no given equation to determine what the value of fi is going to have to be.0564

We have to figure it out by solving for it from known data points.0567

You’re going have to compare the values you know and figure out what it’s got to be from what you know.0570

One equation has to be and what the other equation, what something’s it has to involve is and then solve for.0575

Well we know the resultant wave comes out to be this at x=1, t=2 and it comes out to be this at x=1, t=3 and from that we’re going to be able figure out.0583

It takes some effort, but by basically doing a lot of math and solving for it, you’ll be able to figure it out eventually.0593

Alright, on to standing waves. When a wave hits a rigid surface, it reflects and is inverted.0600

This means if we have a continuously generated wave being bounced off a surface, the incident, original and reflected wave will interfere with each other.0606

You shoot a wave at a hard surface and it comes in and it bounces off and it gets flipped…sorry, it’s not going to just be a direct flip of the other one.0614

It’s going to be a flip like this…and that will require…if that was up, I think this one would actually wind up being up but then down, so it’d be out of phase.0627

Its…sorry, little bit hard to be sure because it has to do with the fact that it’s being reflected. So it’s going backwards but it’s also being inverted at every moment.0636

You have to know what going in at each time is being spat out going the other direction.0645

This idea means that we got in one medium, if we’re bouncing off a hard surface, off a rigid surface that means that we’ve got the first wave that’s going in.0653

But now we’ve also got the reflected wave coming out, so if we got two waves they’re suddenly going to start interacting with each other.0662

Its certain wave lengths, a standing wave is set up. A standing wave is a special phenomenon where certain points on the wave, called nodes, remain stationary.0668

We’re going to explain more why it’s called a standing wave and what that means as we go on.0677

This is due to the interaction between the incident and the reflected waves. The original and the new wave going in and out of phase.0680

Let’s look at a visual example. This one, we’re once again using the purple-pink to refer to the color that comes out when they’re added together.0688

Then red and blue are the two original waves. So if we have two waves going forward…sorry. We have the incident wave, the red wave, and it’s moving to the right like this.0697

And it’s hitting the wall and being bounced off the wall and it’s being reflected like this. The red one slides forward, but then if we slide if forward a bit we’ll wind up getting a point where the reflected wave is going wind up being reflected in the other direction.0705

It’s going to be reflected and it’s going to be the opposite. So the top one, we’ve got a larger wave but in the bottom one the reflection is now going the other way because how they come in and come out, you’re going to wind up canceling at a certain point.0723

When you cancel at that certain point, the resultant wave, since they’re now perfectly out of phase, you’re going to cancel the whole thing and be a flat line.0738

When you’re going like this and you’re in phase, you’re going to make a larger one. So it’s going to be a larger wave than the original but then it’s going to flatten out and it’s going to be a larger wave and flatten out, then a larger wave and flatten out.0746

The points that stay the same, like that one right there, are called nodes. There is a node here and also on this graph, there would be a node here and a node here.0759

So this node here and also one here and one here because they’re staying at zero the whole time they way we’re working this.0769

Might be having the case that we’re having the wave look differently so that wouldn’t be a node, but if we had another surface here then we would have nodes there.0775

Alright, so this produces a wave that stands still even when it’s moving. What does it mean to stand still?0783

Well it means that the nodes stay fixed as the rest of it moves up and down. For this example we’ve got time going from red to green here.0788

The curve starts off red but then it begins to turn to orange then drops to yellow and turns to light green and then a more vibrant green. Just like this.0798

One possibly easy way to see it would be if you were to imagine a jump rope. If you were looking at a jump rope straight on.0811

There’s a person here and there’s a person here holding the two ends of the jump rope, right?0819

The jump rope goes like that but it’s going to also as you look at it, it’s going to wind up going through a number of different phases if they’re spinning it around.0825

As it spins around, it’s going to go up and down, and up and down. That’s a standing wave, with a node here and a node here. There’s no node in the middle for that because it stays the same.0832

So there’s no nodes in the middle because it moves around.0843

In this case though, the standing wave that we’ve got right here, we’ve also got a node in the middle.0848

So there’s a node in the middle here and a node here and a node here.0853

If we got this, which is how we’ve designed it. So we’ve got…a wave could have the front start like this and then go like that in a different version.0856

The way that the wave behaves from where its originating has to do with how it’s created, which is why I’m sort of hand waving so much here.0871

In our case, if we had it between two fixed hard objects then pluck the string, both the end points where it’s held down would be held in place.0880

If on the other hand, if we had a tube that had a hard end on one end, but then we were blowing into the other end to create sound, like say on the flute.0887

The waves would be moving up and down and it wouldn’t have a node at that end, we’d actually have an anti-node because it’s able to do its full size.0895

Standing waves is a whole thing that can be explored much more in depth than we’re going to.0902

We’re going to at least get our feet wet and get an idea of what’s going on here.0907

A node is a stationary point of the wave, so these three points on this, there’d be three nodes on this one, here they’d be two.0911

Anti-nodes, the exact opposite to a node, it’s the point of largest possible amplitude.0917

In this case, where both the end points are fixed, the number of nodes will be one more than the number of anti-nodes.0923

There’s an anti-node here, anti-node here on this line.0928

Those lines would be anti-nodes.0934

Standing waves can’t just occur at every wave length.0938

To determine what wave lengths work to create a standing wave, let’s look a little more closely at how our standing waves are being built.0941

The distance between the starting and the reflecting point is l.0947

If we’ve got two walls set up with an l distance between the two walls, we might have a standing wave that looks like this.0951

We’ve got our node in the middle, a node here, and a node here.0956

In this case we managed to repeat one whole wave, right?0959

One whole oscillation, so we’ve got λ as the distance. So in this case, l is equal to precisely λ.0963

We could also have standing waves like this, where it goes from here, so a node here and a node here.0968

Well it doesn’t manage to make an entire wave, it manages to make only half the wave.0975

This one here would be λ over 2. This one manages to make four nodes, so it manages to do one whole wave length here, 1 λ, but then here it manages only to do half a λ.0980

So it’s λ over 2. The way it’s going to work out, is that we’re going to have to have λ over 2 be the portions that it comes in, otherwise it won’t be able to fit.0996

Its not going to be able to have a fixed node at the end. It’s going to have to have a fixed node at the beginning and at the end.1005

We’re only going to be able to have a whole number of nodes in between, which means it’s going to have to come in λ over 2 chunks, right?1011

The wave length is going to determine whether or not it’s going to fit in as a standing wave.1019

For this one we’d have the length of the distance is λ over 2 or the length is equal to 3 λ over 2.1025

For the same length, we known seen at least three different standing wave patterns.1031

So if the standing wave has its end point fixed as nodes, we can only have a whole number of half wave lengths between the end points.1035

Some whole number, n times half wave lengths, λ over 2 is equal to the distance between them, l.1042

Equivalently, we want to know just the wave length from this, which is probably what we want to know.1049

The wave length, λ equals 2 times the length, over n. It can be any n from one on up to any arbitrary n you choose.1053

One marching on forever to infinity. Whatever n you plug in there, it will work because that’s just going to mean how many times you have to go up and down between the walls. Right?1062

You could have it that you go up and down, not up and down, actually yeah, that’s the number of up and downs you have.1072

I was going to say that the number of nodes is one more than the n we’re choosing, but the number of anti-nodes, in this case, there are other forms of standing waves like the flute example.1081

But for the string that’s held down on two ends and then plucked in the middle, the waves that get set up in it when it pulls off a standing wave are going to have to come from that length and we are going to have anti-nodes that are equal to whatever n we choose here.1090

It’s not like all wave lengths necessarily work at the same time. It will change depending on it, but this is the things that could work with it.1106

This is what standing waves would be allowed with in that, because it will be bouncing off of both walls simultaneously right?1113

One bounces this way and then sets up with another standing wave but then it bounces off the other wall.1119

We’ve got them all stacked on top of one another and the only way that they’re not going to wind up canceling each other out is if they’re in this precise thing that we can have that were it goes in phase and out of phase in such a way so as to sync up properly and cause it to just keep flipping between the two extremes.1125

Back to this precisely, we know that the length has to be equal to 2l/n for any n.1142

That’s just going to determine what number of anti-nodes, what number of nodes, how many bumps and valleys there are for our wave.1149

Furthermore, since the speed of waves in a medium is almost always going to be fixed, a given wave length implies a certain frequency.1159

If we know what the wave length is and we know what the medium speed is, then we know that we’ve got some frequency that’s going to be setting up that standing wave.1167

Any frequency that causes a standing wave is called a resonant frequency because it cause the medium to resonate.1175

Resonate, it picks up the motion and it amplifies it itself, so resonance is a really important property.1181

Huge amounts of study, if you’re heard about catastrophic failure of bridges in some cases, that’s possibly because of resonance.1188

In many cases it’s because some motion gets picked up and amplified and amplified and amplified till eventually it breaks.1196

If you’ve ever seen or heard of an opera singer being able to sing at a high enough pitch to break a glass.1201

That’s because the opera singer is able to hit the resonance frequency in the glass, so it’s able to set up vibrations in the glass, able to put energy into it and so the glass ultimately breaks because glass doesn’t like going through a lot of motion.1207

Its very brittle. So frequency that causes a standing wave is called a resonance frequency, this is a huge field of study but we’ve at least dipped our toes in it and got some idea of what it means.1220

It means something that’s able to set up a standing wave, that’s able to have something that just lives inside of it bouncing around.1230

Onto the Doppler Effect. So we finished with our idea of interference and we’re ready to talk about something new.1238

Interference, adding them together, and standing waves exist because the way interference allows things to interact.1244

What happens in the Doppler Effect, what happens when the emitter or the receiver is moving?1250

To figure that out, let’s first consider the much more simpler case, when they’re both still, right?1256

For this picture, we’re going to have the lines means the tops of waves, wave peaks.1259

The number of wave peaks you receive is the frequency right?1268

If you receive three wave peaks every second then that means you’ve got a frequency of 3 hertz right?1272

Because taking in five wave peaks every second is the same thing as taking in five whole waves, right?1278

There’s only one wave peak per wave. So, I mean, there’s of course one wave valley, a peak in the negative where we consider the peaking being just the top.1287

The number of wave peaks is the counts and that’s why it much more easy to visually represent it.1293

We can have a thing emitting the whole way and each one of those lines mean a wave peak coming out.1297

If you’re standing over here and its emitting these waves at regular intervals. These are the same distance for each one.1302

It’s putting out some frequency and its spitting out it that it has some regular frequency, so regular intervals one the waves because we have the mediums…the mediums speed is going to be constant.1312

Regular frequency means regular distance between the wave peaks.1327

The receiver here, is going to wind up hearing however many wave peaks hit him.1332

In this case, we’ve got these many coming towards, we don’t’ know how many are going to hit him in one second because we don’t know what the speed of this medium is.1338

We’re able to at least get a visual representation as it goes off in both sides.1345

What happens when its starts to move?1347

If the emitter is still producing the same frequency but now it’s moving towards you.1352

We’ve got those same distances, same circles but now as it moves along, it emits it, but then it moves inside of the circle.1356

It pops out the circle that’s growing around it but then before it makes it, before it would have been pop out a circle, stay in the same place, right?1365

But now we’ve switched instead to pop out a circle and before the next time you’re about to pop out a circle. Instead of staying in the same place in the middle, you’re now over on the right side, right?1372

You pop out a new circle and now you’re going to wind up having two circles around you.1382

You’re going to be building them out closer to one side right? Because as you move, you’re putting out these pulses out of you.1389

Your frequency, we can consider the wave peaks as just being pulses, right, it makes it a little bit easier to think about.1397

You put out these pulses around you but you’re moving, so while they’re spreading out evenly, you’re getting ahead of the point that they’d have to be to all be equal distance from that same point.1402

You’re moving forward, they’re wind up bunching up in front of you. If it’s moving you, I switched to being the emitter.1414

If the receiver is standing still and the emitter is moving towards the receiver, it’s going to bunch up those peaks in front of it, it’s going to make basically this wall of peaks in front of it that’s going to hit the receiver with really quickly.1420

On this side, the receiver is going suddenly receive a lot more peaks coming in at once because the emitter’s motion has effetely caused those peaks to get bunched up.1435

When they come towards the receiver, it’s going to get a whole bunch more of them at once because they’ve been bunched up by the emitter’s motion.1446

Similarly, if the emitter’s moving away from you, it’s going to look like this. You experience a lower frequency because there extra space between the wave peaks.1454

It’s bunched up one side, but in exchange it’s caused more space, it’s caused a widening of gaps between the wave peaks.1461

Its going to sound like the frequency has gone down, sound…I haven’t necessarily made this just about sound, it’s about any wave in fact.1469

You’re going to wind up experiencing fewer of them getting to you. You’re going to receive fewer wave peaks per second because it’s moving away from you.1477

We can also make these same ideas for the person if the emitter was still and the person was moving.1486

It’s going to wind up being the case that this person is going to be able to sort of out run these peaks to some extent and it’s going to be harder for this peak to catch up with the person.1495

Its going to increase the space effectively. Similarly, with the receiver running towards the emitter, it is it’s going to run through the wave fronts.1506

You could think of it as a very slowly moving bunch of bushes, I not quite sure what the best thing to do here is, like bubbles.1515

If you had a constantly strung together stream of bubbles right? Between every bubble there was one meter, right?1521

They’re moving along, they’re floating through the space, you could either run away from the bubbles and they could pop on you much more slowly because it takes that much longer for each one to get up to you.1530

Because not only does it need to cross the distance between the bubble in front of it and itself, it has to cross that plus you running away from it for a while.1538

If you ran towards the bubbles you’d wind up popping them on you much faster because it doesn’t have to just cross that distance, it now only has to cross to you, right?1546

The motion of the receiver is going to effect the experienced frequency.1553

Finally we’re ready for a formula, but I want you to keep in mind the reasons why this…what the expectation for how the frequency should shift, right?1559

If the emitter is moving towards the receiver, frequency should go up, because we’ve got them bunched together.1569

If the emitter is moving away from the receiver, frequency should go down because there’s now space between the wave peaks.1573

Same thing with the receiver moving towards…well similar ideas is the receiver moving towards the emitter is going to mean bunched up effectively is going to mean that they pop sooner.1579

It’s going to mean that the frequency goes up if the receiver is moving towards the emitter. If the receiver is moving away from the emitter, the frequency is going to go down.1589

You have to be able to imagine this because the formula is going to require you to understand what to expect before you’re able to use it and we’ll see why in just a moment.1597

We’re going to need a bunch of factors. V, speed of waves in the medium, so if we don’t know how fast the medium is moving, if those bubbles are moving at 1 meter per second, it’s a big difference if those bubbles are flying towards you at a 100,000 meters per second.1606

If they’re zipping along you don’t have any chance at out running them right?1620

If they’re moving along at only 1 meter per second, you’ve got a really good chance at being able to out run any bubbles.1623

You might be able to lower your rate to zero frequency.1627

The E, the speed of the emitter. How fast it’s moving is going to have an effect.1632

Vr, the speed of the receiver. How the fast the receiver is moving is going to have an effect.1635

FE, the emitted frequency, what everyone would hear if everyone was sitting still. What the person on the emitter would hear if they were doing it.1638

Hear is probably the wrong thing, I keep using sound here because we’re used thinking the Doppler Effect in sound, but it actually works with any wave.1646

Once again, works with light waves, that’s how we’re able to tell how far planets…not necessarily how far planets, what the motion of planets is.1654

It is either red shifting or blue shifting in the electromagnetic spectrum. It’s a little beyond us right now to talk about to in depth, but it is one of the uses for the Doppler Effect.1663

Anytime that you know there is going to be motion, you can compare what the emitted frequency is, what the expected emitted frequency is to the actually received and to be able to figure out some things about the motion going on.1673

If we have all these different ideas, we’re finally ready to have a formula.1683

Put together, we have the receive frequency, f. So the receive frequency f is equal to the speed of the waves in the medium, plus or minus the speed of the receiver, divided by the speed of waves in the medium, plus or minus the speed of the emitter, times the frequency of the emitter.1689

Now this isn’t like the quadratic formula where it’s plus or minus and it means both of those at once and you’re supposed to get two different answers.1705

For the Doppler Effect, you have to choose which one you’re going to use and the choice is important and it’s up to you.1712

You have to know what the expected is. You assign plus’ or minus’ by thinking through those above examples about the emitter and the motion.1718

If the wave fronts are going to get bunched up or bounced…or gets contacted by the receiver faster, you’re going to want to use the appropriate sign.1725

For example if the receiver was moving towards the emitter, we’d want a plus sign because we’d want a bigger number on top.1733

If the emitter was away from the receiver, we’d want a plus sign on the bottom, because we’d want a bigger number on the bottom.1745

You want to think in terms of bigger number on the bottom means lower frequency. Smaller number on the bottom means larger frequency.1754

Bigger number on top means larger frequency. Smaller number on top means smaller frequency.1760

You have to think about which you’re going to want here, so there’s some flipping, it’s really takes real thought here, so don’t just rush into it, don’t just throw things into this formula.1765

You have to understand what you’re expected output is and be able to think about that and that’s one of the really important to learn in physics.1774

Is being able to know what you’re sort of expecting and being able to guide your use of the formulas. Guide your use of information based on how you’re doing it.1781

Math is an important set of machinery, but you have to build the scaffold of understanding to know how to use that machinery.1788

This is just one more example of when it’s up to you to pay attention to what you’re doing.1796

Ready for some examples. If two waves have the equations y1 equals this and y2 equals this.1800

Now keep in mind, there are no units with these so this is really just something totally in a vacuum.1809

This is not actually applicable to real life things, because we don’t know if this in meters, we don’t know if this is in centimeters, we don’t know if this is in like the king’s feet.1814

It’s something, but we can talk about it mathematically and we can get a good idea.1820

We can use it as a practice problem but it’s not good physics, it’s just a reasonable practice problem for in a class, which is what we’re doing.1827

Its okay here, but you want to make sure that you know what units you’re using. You want to mark units down for the real thing, when you’re actually doing a lab for example.1834

If the waves are superimposed on one another, we place the waves together. What’s the value for y?1845

X equals 2. T equals 22.1849

The new value, the y that we’re going to make is just y1 + y2. We just add the two together.1852

It’s as simple as that. 3 sin of 1.2x + 600t + 4sin of 0.9x + 625t + 0.3.1858

That’s what the general formula is. If we want to plug in specific values to be able to get what the location is going to be like.1880

What the height is going to be at x=2. At 2 far down and 22 time later in the future. We have to plug them in.1888

X equals 2. T equals 22. That means y is going to be equal to 3 sin 1.2 x 2 + 600 x 22 + 4 sin 0.9 x 2 + 625 x 22 + 0.3.1897

We simplify these things out a little and we get y is equal to 3 sin of this big number, 13,202.4 + 4 sin of 13,751.4.1930

Now before we actually figure out what this is, I have some very important thing to talk about.1950

When you’re working with waves, you work are working with radians. Let me repeat that again, when you are working with waves, you are working with radians. Right?1956

Remember radians from trig, when things that go in terms of 2 π. 2 π is your one whole way around the circle.1970

Whenever we’re working with waves, we work with radians. Those phase angles we talked about, they aren’t phase angles in terms of 45 degrees or 127 degrees, or a 147 degrees.1977

They’re in radians, so they’re going to be some number.1987

Now remember you can convert from degrees to radians but we’re working in radians.1992

That’s why in terms of 2 π divided by λ, or 2 π divided by the period.1995

Because we’ve got 2 π is really fundamental to the way we’re using waves. It’s really fundamental.2000

Radians is the currency of physics. Now keep in mind, we do still occasionally use when we’re surveying things.2006

We use degrees, right? We’ve had plenty of problems where we say something goes off at a 50 degree angle or something is below something else by 15 degrees.2016

In this case, we’re now switching over to radians wholeheartedly. Radians and waves go together like, I don’t know, chocolate and peanut butter.2025

I’m actually not a big fan of chocolate and peanut butter, sorry if you are. Anyway, bad example, don’t need to tell you about my personal life.2035

W equals 3 sin, so we’ve got this, right? So we have to plug in it into radians.2042

If you are using a calculator and you probably are, you need to make sure your calculator is in radians when you’re working with waves.2047

In some way there is some mode function in your calculator, some way to get from degrees to radians.2054

Make sure you’re going to switch it over before you’re going to do stuff with this.2059

That’s where you’re going to have to start thinking in terms of. Now what value might we expect this to output when we plug this into a calculator?2061

We probably expect something pretty big, right? We’ve got sin acting on some giant number, 13,000 and sin acting on some other giant number, even more 13,000, so we’d expect a big number.2069

Remember, sin just repeats itself every 2 π. If sin repeats itself every 2 π, sin really only gives a value between -1 and +1.2078

The biggest and the smallest. The lowest and the tallest value that the unit circles going to give.2086

In reality, this number can never get very big. It’s going to repeat and do a lot of bouncing around.2092

It’s going to be kind of complicated because these don’t really interact that well as two equations.2098

They’re not clearly in phase, they’re clearly out of phase.2101

It is never going to actually turn out to be a very large thing. It’s just determined by those amplitudes.2106

Remember, it’s not like when you get to a really far number and your wave equation, it just off the amplitude. The amplitude is always maximum height it achieves.2110

We plug in these numbers and we wind up getting the kind of lack luster, 0.552.2119

You just plug in x and plug in t. If you’ve got two equations working together, you just put them together and you plug in separately.2127

Then you just add everything together and remember even more important when you’re working with waves, you’re working with radians.2134

We haven’t talked about that before but it’s a really important thing to notice, is that everything is in terms of radians.2139

If you switch to using degrees, it can screw up a whole mess of things. So double check on your calculator, make sure you’re switching in radians.2146

Try taking the sin of 90 and see what comes out. If the sin of 90 gives you 1, you’re not in radians.2152

Example two. We’ve got a wave that’s a wave equation of y1 equals a, amplitude, times sin of kx minus ωt.2158

Another wave, y2 is added to it. Superposed and they completely cancel each other out.2166

Fully destructive interference. If we’ve got that…so let’s say this is the baseline, alright.2172

We want to now figure out what the equation of y2 is going to be.2180

Well graphically, we know that y1 is going to look something like this, right?2182

I don’t know what’s it’s precisely going to be, but it’s going to look something like this.2190

Now, for y2 to come along and cancel out, it’s going to have to be the perfect mirror opposite of this.2193

It’s able to come up with a resultant thing here that just turns it to zero.2203

Y2 has to be its negative at all points, so one thing we can do is say, ‘Y1, well –y1 equals y2, right? Because y2 is you’re negative”.2208

True, but we can’t have a negative in front because amplitude doesn’t come negative.2217

Amplitude always positive so we need to be able to shift this around by a phase shift.2221

We do know, clearly they must have the same frequency.2226

Y1 frequency is equal to the y2 frequency.2230

Which also means that y1 t equals y2 t.2235

I’m not saying y1 x t, I’m just saying the guy that belongs to it.2240

Similarly we also know that y1 amplitude equals y2 amplitude because they’ve got to be the same height for them to cancel each other out.2244

At this point, we know that the equation for y2 is going to have to have the same amplitude, going to be using that same function sin.2251

Its going to have to have the same k, right? Because it’s got the same frequency, same wave length, all that sort of thing…frequency.2259

Since we’re assuming that velocity of the medium is the same, speed of the medium is the same.2268

Frequency, the fact they’ve got the same frequency implies. Same that they’ve got the same period implies. The must have the same wave length.2272

Kx minus ωt, but that’s still just the exact same thing.2278

So one last thing we have to do, add in φ. We add in φ and that gives us our phase angle change.2282

So we’ve got some phase shift. How do we figure out what angle φ has to be?2292

Well let’s look at the units circle. The units circle is a perfect circle.2295

If we start here at 0 and go up, we’ve got positive right?2302

This guy effectively has +0. He’s got an angle φ of 0, so he’s really easy going.2307

If we want to make the exact opposite, what’s the perfect opposite to the circle at this point?2312

The perfect opposite of the circle is to start here and go this way, you’re negative.2318

You’re just the negative mirror of what’s going on if you go positive.2322

If you’re the negative mirror, what’s going on if you go positive?2325

That means over here you start at 0, here you start at π.2328

If we add in π, we get a negative, we’re able to flip the whole thing.2331

Normally we’re looking starting from the positive side of the circle, now we’re going to start from the negative side of the circle.2336

That means φ equals π for this. So we’ve got y2 equals a times sin kx minus ωt plus π.2340

We add that in and we’re set. It’s as simple as that.2354

We are able to figure out where it has to be located based on the fact that we know it’s going to be completely in opposition the whole time.2361

So it has to have a phase shift that will put it out of phase all the time and the thing that’s going to make it out of phase, full destructive is if its phase shifted π over.2368

That’s where you get the opposite part of the sin wave.2377

We know that everything else has to be the same in order to figure this out and this is an easy example to some extent because we know that we’ve got this perfect phase shifting.2382

Now if it were going to have to be a phase shifting of three quarters π, it might be harder to see if coming when we probably have to have a couple of values given to us and more things filled out.2390

But it’s the same basic idea, we’ll plug in y1 and we’ll plug in: a sin k – ωt equals that.2400

So we know that y = y1 + y2. You plug in all your values and you’ll eventually be able to get enough information to solve for what φ has to be.2406

You’ll have to use of the information you know from trig, but at that point, it’s just solving for what φ has to be from that.2416

That’s how you can do it. In this case we can do it graphically because we can see what has to happen.2422

This will hopefully give you a slightly deeper intuitive understanding of how this stuff is working, what it means to have a phase angle.2427

Where are you starting from on the circle effectively? How are you different? What is your different way of looking at the same wave?2432

The same wave but you’re starting in a slightly different place so you just a different vantage point from the beginning.2440

String length of 2 meters is put under tension so that waves travel along it of v equals 450 meters per second.2446

We’ve got some string, it’s tensioned between two things. Two meters and we know that the velocity of this medium is 450 meters per second.2452

What frequencies would produce standing waves on this string? What frequencies would produce standing waves?2466

Well, we know that λ is equal to 2l over n, for any n contained in the natural numbers; 1, 2, 3, and up.2473

We can figure out what λ is. We know what the speed is, if we can know what the λ is, we can figure out what the frequencies have to be.2488

We know that for it to have a standing wave it’s got to have a wave length that’s equal to 2 times the length divided by n.2497

That’s going to give a whole infinite array of possibilities, but it will give us what the possibilities are.2504

We know that λ has to equal 2 times 2 meters over n. Anything that’s 4 divided by n and we’ll have the wave length.2510

Anything that’s 4 over n. If we want to know its frequency is, we know that velocity equals frequency times λ.2523

That’s going to give us that frequency is equal to the velocity over λ. Which is equal to 450 meters per second divided by 4 over n.2531

Which gives us n times 450 over 4. Or 112.5n.2543

You plug in any n and multiply it by 112.5 and that gives you a possible frequency, a possible resonate frequency.2553

If gives you one frequency that would produce a standing wave.2562

Some examples frequencies would be f = 112.5, the next one up; 225, the next one up; 337.5 and so on and so forth.2565

These are all in hertz. Both this and this are in hertz.2578

If you’re curious, the λ was in meters, right? Because we’ve got some length, 2 meters divided by n, and n doesn’t have any units, it’s just a number.2585

So 4 over n, we’ve still got 4 meters. Which makes sense because wave length has to be in length.2593

If we want to produce standing waves we just have to determine what wave length possibilities are and use that to figure out the frequency.2599

Example four. If you’re standing just off the side of the road and a police car is flying down the road at a very fast 40 meters per second, with the siren going at a frequency of 800 hertz.2607

If the speed of sound is 340 meters per second, what frequency will you hear as the car is approaching?2623

Once it passes you, what are the sounds you’re going to hear?2629

Clearly we’re going to need to use the Doppler Effect because the formula for that gives us the change in frequency due to the motion of the emitter, motion of the receiver.2632

In general, it was f = v, speed of the medium plus or minus v, speed of the receiver divided by v, speed of medium plus or minus v, emitter, times the frequency emitted.2641

Now remember it’s up to us to figure out plus’ and minus’ and all those sorts of things.2655

Just to mark everything off from the beginning. Velocity here, speed of our medium, 340 meters per second.2660

That’s the speed sounds here. Car is flying down the road at 40 meters per second, so the emitter is moving at 40 meters per second.2665

The receiver, is the receiver moving? No, you’re just standing there, so you’ve got 0 meters per second.2677

Finally, frequency emitted is equal to 800 hertz. In the case where the cop car is going by you, you’re going have your frequency that you’re going to hear is going to be equal to v.2683

Now what do we want to use? Do we want to use plus or do we want to use minus.2697

Well if its moving toward you, if you were moving towards it, you could look at it from either point of view right?2700

The two things are moving towards one another then you’ve got…you’re going to get more, right?2705

It’s going to go up because the two objects are moving towards one another.2713

It’s going to be v plus the receiver because you’re going to be increasing it as you move towards those wave fronts.2717

These two objects moving towards one another divided by v, plus or minus, once again, the emitter is coming towards you so you’ve got more wave fronts coming at you so you’re frequency is going to go up.2727

Because it’s packing those wave fronts at its front. So are you going to use plus or are you going to use minus.2738

Remember, smaller denominator means bigger overall number so you actually use minus v e times the frequency emitted.2743

We’ve got v, so 340, you’re speed is 0 divided by 340. Its speed is 40 times 800. 340 over 300 times 800.2752

We’ve got 906.7 hertz. So that what’s you’ll hear as it move towards you.2767

Now just after it passes you, once it passes you, it’s going to be going the opposite right? It’s going to be going very differently than it just did.2777

Change up the color. So, you’re standing here but now the gulf between you two is widening.2785

You can effectively think of you’re moving away from it, it’s moving away from you. Now we’re going to have to actually use a slightly different version of this formula.2795

All of our constants just stayed the same, but it’s the way that formula changes with those plus’ and minus’ that will give us a different experience.2800

When it’s coming towards you, you’re going to hear a higher frequency than it naturally emits.2809

When it’s going away from you, you’re going to hear a lower frequency than it naturally emits.2812

If frequency is equal to v, plus or minus vr. Which does it become?2817

Well if you’re moving away from an object, if you’re moving away from an object, you’re going to cause yourself to experience less wave peaks because you’re going to be running away from the wave peaks.2821

If you’re running away from the wave peaks, we need a smaller number up top, so v minus vr divided by, now the emitter.2831

Is the emitter going to be bunching up or is it spreading out then from the point of view of the receiver?2841

We’ll it’s spreading them out because it goes farther away before dropping the next peak.2847

That means that we’re going to have to have a smaller frequency. It’s going to have to contribute to making a smaller frequency, so it’s v plus ve.2852

Because it’s making a larger denominator times the frequency emitted.2861

We get 340 once again, minus 0 divided 340 plus 40 times 800 equals 340 over 380 times 800.2866

That gives us the value 715.8 hertz. When it’s moving towards you, you get 906.7 hertz of frequency.2878

When it’s moving away from you, you get 715.8 hertz. And if you were just standing still while it was sitting still next to you, if you were both still next to one another, you’d hear 800 hertz.2890

Alternately, if you were the cop in the car because you’re moving with it, you’re moving at the same speed so to you it’s as if it’s effectively still, it’s going to be 800 hertz for the cop in the car the whole time.2900

If you’re curious what these sounds like because frequency actually what is going into our ears.2912

If you wanted to know what this was, keep in mind, a siren isn’t going to sound like a pure tone.2917

We made this a little bit easier but staying it was just giving out a single hertz and we’ll talk a little bit more about how sound is actually working out after this example.2921

Let’s actually listen to it. I’ve got an example sound here, so here’s an example sound. I’ll be playing it in just a few moments.2927

The first sound you’re going to hear is the 800 hertz sound. Just to give you a sense of what 800 hertz sounds like tonally.2935

Two, you’re going to hear the 906.7 hertz, what it would be as it approaches you.2942

Then three, you’re going to hear 715.8 hertz. Then finally four, you’re going to hear simulation of what it might sound like to have it drive past you.2949

Remember it’s going to actually at some point, swap between these two and if it were driving directly at you, it wouldn’t’ ever swap until it went through you.2958

But because you don’t get hit by the car hopefully, you’re actually going to have this period of time where it’s going to slide between the two.2967

It’s going to slide between the high frequency to the low frequency. Or from the low frequency to the high frequency as it passes you or you pass it depending on the situation.2975

It’s going to slide frequencies because you were actually not going to be passing directly through its point.2985

This works reasonable well when you’re close to the path that it’s taking, but since you can’t ever actually on the path and not get run over.2991

You’re going to actually hear a change when its passing…say when it gets here, the wave changes, is going to about on the distance.2999

In the extreme, we can effectively treat it as a straight line, but as it gets close to you there’s going to be a change in the way the velocities are working.3008

We aren’t dealing with that because that’s much more complicated but that’s why it slides when you hear it in real life.3015

I’m sure you’ve heard a cop’s or car’s siren or something go off that has moved by you quickly while it’s been emitting noise and you’ve heard this consistent slide sound and that’s what we’re going to hear here.3021

Finally, we’ll hear a slide from 906.7. It will start at 906.7 hertz, it’ll go on for just a little bit and then it will slide down to 715.8 hertz.3033

This is gives you some idea of what you might actually hear. Alright, ready for that example sound? Go.3043

– 5055 [Pitch sounds varying]3048

Alright, sounds pretty good. There you go, there’s an idea of what it sounds like.3055

Keep in mind a couple things, real sounds in real life are not just pure tones.3059

Those that you just heard now was a pure tone. It was a pure tone of 800 hertz.3065

It was a pure tone of 906.7 hertz. In real life, you are not going to actually hear pure tones because there is many, many tones.3069

If you hear an actually cop car siren, it goes way between all these different things. It’s trying to catch your attention by sliding through a huge variety of tones.3079

In real life, we hear many, many tones at once compacted together, interacting with each other.3088

Which is actually how we experience light too.3094

The way we experience is different than breaking it up in single values that we get here.3099

This still gives you a good idea. It’s much easier to work these single values, it’d be hard to describe the whole range of possibilities.3106

Which is why when you get a very high level of physic, when you are really trying to describe it as opposed to just understanding what’s going fundamentally.3113

Which this does, it gives you the chance to understand it fundamentally. It gets hard to deal with all the things happening at once.3120

In real life there’s many frequencies going on. There is many things happening. The real sound is compounded by a number of things.3127

It doesn’t just directly between these two, it slide by you.3135

The angle that you are from the car, I mean the distance that you are from the path of the car is taking as it goes by you.3139

If maybe it turns around you as it goes by, all these different things will change the sounds that we’ll hear but this is one possibility that gives you a reasonable approximation of what it’s really like.3144

I hope that made sense, I hope that waves are beginning to come together.3154

It’s a really, it’s a huge can of worms that we’ve opened and we only a little bit of time to experience some of it.3157

There is so much more stuff in waves that we can talk about and go in depth but we’re just sort of scratching the surface so we can at least get an idea of what’s going… on here before moving to the next thing.3163

Alright, hope it made sense and I’ll see you in sound which will be a great use of this stuff that we’ve been talking about.3171

Hello and welcome back to educator.com. Today we’re going to be talking about sound.0000

Sound is one of the waves you encounter the most often in life.0005

Such as right now unless you’ve got me on mute, but probably you don’t.0007

So sound waves are longitudinal pressure differentials.0013

If you had just a normal column of air, all of the air molecules would be equally spaced.0017

If you were to push on it suddenly, you’d get this effect where you’d have a bunch of them get bunched up.0025

Then that bunching would translate into them being really far spaced out and then they’d get bunched up again and then they’d be far spaced out, and then bunched up again and far spaced out.0030

That bunching and spacing is the equivalent of a wave going up and down.0040

Suddenly we’re seeing the pressure deferential either positive pressure deferential or negative pressure deferential.0046

That’s how the wave information is being transmitted. That’s how energy is being transmitted.0052

Is through these compressions and rarefactions moving through it, the compression being where it’s bunched and rarefaction being where there is very few air molecules at that moment.0056

We’re able to have longitudinal pressure differentials and that’s what’s happening as I’m speaking right now.0065

My vocal folds, my vocal cords are vibrating the air and those vibrations are being transmitted to the mike0070

Which is then being moved, turned into data, which is then transmitted to your speaker, and then the speakers vibrate that same amount of vibration so that you hear my voice reproduced.0076

The speed of sound varies greatly from medium to medium, even within the same types of mediums; the speed of sound can vary based on other factors.0088

Here’s some example mediums, so when see air at 0 degrees centigrade and one atmosphere of pressure, the speed of sound is 331 meters per second.0095

On a hot day like 30 degrees centigrade suddenly it’s moved up 349 meters per second.0104

This is because now the air has more energy in it, it moves faster so it’s going to bounce around so those pressure differentials are going to get moved faster.0110

Water at 0 degrees centigrade, one atmosphere is 1482 meters per second, whoops that shouldn’t be 0 degrees centigrade, that should be 20 degrees centigrade right there.0117

Water at 20 degrees centigrade, once again more energy, it moves around faster so it’s 1482 meters per second.0128

We increase the density by adding salt, turning it into sea water. We get an even higher one, 1520 meters per second.0134

Steel. Very uncompressible, so those pressure differentials get moved very quickly as its very hard for it to move.0140

That stiffness causes the information, the energy to be transmitted very quickly from one location to the next location along our steel column, bar, what have you.0146

So we have a much faster speed of sound. Depending on the material, depending on what we’re dealing with, speed of sound is going to change.0157

That’s something to keep in mind while we’re working with this.0163

Pitch. The human ear is capable of hearing a huge array of frequencies.0166

We’re able to hear the buzzing of a TV set, that sort of background hum of electronics.0171

We’re also able to hear the low base note of a kick drum.0178

That huge variety of frequencies that we’re able to hear is what we consider pitch.0183

A high pitch sound is one that is a large high frequency, like a violin or a piccolo or the right end of a piano.0189

A low pitch on the other hand is one that has a small frequency, like a double bass, a tuba, or the left end of a piano.0196

High frequency gives us that high sound like ‘Ah.’ That’s a higher frequency than when I talk like this in a low frequency voice’.0204

We experience this…experientially, we experience it qualitatively, as a feeling but it is also connected to a qualitative change in the way the thing is measured.0211

A young person with good hearing can normally hear sounds anywhere between 20 hertz and 20,000 hertz.0226

This is a really wide range of possible frequencies we can pick up.0232

As we get older, we start to lose some of that range, especially the higher end.0236

Certain diseases and injuries can also damage your ability to hear.0239

Pitches below human hearing a called infrasonic and pitches above human hearing are called ultrasonic.0243

Infra being Latin for below. Ultra being beyond and sonic being sound.0250

Certain animals can actually hear sounds that we can’t hear.0255

A dog whistle puts out a frequency that above human hearing in the ultrasonic range.0258

The dog’s ears are better calibrated to be able to hear those higher sound notes and so they’re able to pick it up.0263

Bats use ultrasonic sound to be able to locate their prey, locate the area around them, to be able to see, quote unquote ‘see’ the area around them.0268

Ultrasonic things have certain uses and so they’re able to send out an ultrasonic chirp and hear it back.0278

The way they hear it back, they’re able to pick up knowledge about the area around them.0286

We’re not able to hear those chirps because they’re ultrasonic, they’re above our hearing.0289

Intensity. Clearly the frequency or frequencies that we hear is part of the experience of sound but that’s only part of it.0296

The other part is how loud it is, the intensity of the sound.0302

A really loud sound is experienced differently than a really quiet sound.0306

That dynamic change is part of our experience of sound.0310

This is actually a qualitatively measure as well. It is how much power is being put out.0314

Remember waves are the transfer of energy through a medium.0320

An emitter must be putting out some constant supply of energy or a varying supply of energy but it makes it a little bit easier to talk about as if it’s constant first.0323

It has some power output, p, right?0330

If our emitter is putting out energy, putting out information to the world around it, it’s putting it out by putting out power.0333

Of course, if we’re far from an emitter, the power is more spread out and thus less loud.0340

If we walk really far away from an emitter it’s going to seem quieter to us. That’s because its power may remain constant but our experience of that power is now spread out over a wider distance.0344

It has to travel farther and so the sound waves might start off like this and they spread out a little bit but if we’re over here, they’re going to spread out and they’re going to be way wider out.0356

We’re going to get less information for the same area, so we’re going to have the intensity of the sound is going to be determined by how much area the power has to spread over.0366

A high intensity is either going to be a small area or a very large power.0378

As we either make the power smaller or the area wider we’re going to experience less loudness, less intensity in the sound.0385

What area does our power spread over? This is actually a really difficult question because sound often behaves in very complex ways in real life.0394

For example if I was sitting in the shower, probably standing, but whatever.0402

If I was standing in a shower, say it’s just a square around me.0407

My ear is over here and I shoot out sound waves.0413

It is not just the sounds waves that manage to shoot backwards towards my ear and get picked up.0417

Some of these sounds waves are going to bounce off the container and bounce back to my ear.0422

They’re going to get spread out and some of the energy will wind up being absorbed by the container, but some of it’s going to get bounced back towards me.0427

This is actually, can be a really complicated question, that’s why your voice can sound and have such different dynamics depending on the room you’re in.0434

If you’re in a very large cathedral chamber and you’ve got all this wood around you, you’ve got this big echoey open space.0440

The wood picks it up, changes the vibrations slightly. If on the other hand you’re in a tiny, tiled bathroom, you’re going to get a different sound.0447

Once again, those very hard tiles vibrate; reflect almost all the energy, almost all that information.0455

It’s a small space so it all bounces back and gets picked up by your ear again.0462

Different spaces, different perception of the sound because it becomes reflected differently.0465

Really complicated question, so we’re going to do a reasonable approximation.0471

We’re going to treat sound as if it spreads in all directions equally and simultaneously.0475

An outwardly radiating sphere, so that makes sense.0479

If we’re on a clear plain, we’re out somewhere in a grass land and we shout.0482

Its going to expand everywhere at once. We’re going to have some point source and it’s going to burst outwards just in a continuing radiating sphere.0487

Remember, this is actually a sphere that it’s bursting out in.0494

The point source and it blows out in all positions. Some of it may wind up being reflected but it’s a reasonable approximation.0500

Not perfect, but it gives us something to hold on to and have an understanding of how sound works.0505

If that’s the case, what’s the surface area of a sphere?0510

Well the area of a sphere is equal 4 times pi times the radius of the sphere squared. 4 pi r squared.0513

If the center of an imaginary sphere is put on our emitter, so we have some imaginary sphere that we’re considering.0519

We put it so that the center of that sphere is around the emitter. We are standing r away.0526

The intensity of the sound is going to be that area of the sphere dividing the power of our emitter.0531

The intensity will equal the power divided by 4 pi r squared.0536

The area divides the power. If we’ve got that simplified version of the expanding sphere then we’re able to use this nice little formula.0541

Intensity is a great way to measure sound as a quantitative thing, but it’s not a very good way to talk about it.0551

The range of possibilities for human hearing is really, really large. 1 x 10^-12 watts per meters squared.0558

Sorry I should have mentioned that before. If intensity is equal to power over area.0567

Remember we talk about power as watts. Area, well we use for our unit of length, meters.0571

Area is going to be meter time meter or meters squared.0578

Intensity is equal to the watts divided by meters squared.0582

Human threshold for hearing is around 1 x 10^-12 watts per meters squared.0585

That’s a really, really small amount that we’re able to pick up.0590

The pain threshold is massively larger, 10^13 times more.0594

Pain threshold when we start experience real physical pain from the sounds going off around us is 10 watts per meters squared.0598

That’s a really huge array of different possible values that we’re able to pick up.0606

We might be able to talk about it intensity like this thing but our experience of sound isn’t with each of these little tiny amounts different or these very large possible amounts different.0610

Our experience is actually based on a different scale. It’s based on a log rhythmic one.0620

We experience sounds multiplicatively, so that’s where the idea of decibels is going to come in.0624

To deal with this, we’re going to introduce the idea of the decibel.0630

DB which uses a log rhythm to help us manage this vast range of possibilities.0633

If you don’t remember what a log is. You’re going to want to go back and you’re going to want to check either pre-calculus or calculus and get a real quick understanding of it.0638

The basic idea is log base a of x equals y. Means the same thing as x to the y equals, sorry, not x to the y. Screwed that up.0645

Log base A of x to equal to y, gives us a to the y equals the quantity x.0662

For example, log base 10, which is what we’ll be using for the log rhythm for decibels.0671

Log base 10 of 10 equals 1 because 10^1 equals 10.0676

But log base 10 of 100 is equal to only 2 because 10 squared equals 100.0684

Similarly, 1,000, log 10 of 1,000 would be 3.0690

For simplicity we’re just going to start saying log even though we’re going to be talking about log base 10.0694

You could have the base of anything but we’re going to be talking about log base 10.0699

For ease we’ll just say log from here on out.0703

This is the important thing, if you don’t know, we’re going to wind up needing to use some of the properties that log rhythms have later on in the examples.0705

You might want to go back and refresh some of your memory on this.0711

On with decibels. We’re going to define the sound level beta, and this guy right here is called beta.0714

B E T A. He’s another one of our Greek letter friends.0721

The sound level beta, by referring to the ‘I knot’, the intensity, the basic intensity, the lower threshold of our hearing.0725

We’re going to utilize log. So beta, the sound level is equal to 10 decibels and that gives us a nice chunky number that we can hang on it.0732

10 decibels times log of the intensity of the sound that we’re listening to divided by that base threshold I knot.0740

Log of I over I knot times 10 decibels gives us our sound level.0750

Notice, a rise in 10 decibels means 10 times the intensity because we’re dealing with log 10.0755

If we go from 10 decibels to 20 decibels, that means the intensity has gone up by a power, not by a power of 10, has gone up multiplicatively by 10.0761

Because of log has gone from 1 to 2, it has gone up one number.0770

Going up one number we multiply by 10 decibels, so we’re going to have a change in the power of 10 times that we started off with.0775

If we go from 100 decibels to a 110 decibels, the exact same thing going on, because it’s a difference of 10 decibels.0782

That means our log creates a difference of 1, so it’s times 10.0788

If we had a difference of 20 decibels, it’d be a difference of a 100.0792

Sorry not a difference, a multiplicative 100.0796

100 times the power. If we had a difference of 30 decibels, it’d be 1,000 times the power, etc., etc.0799

Remember, we’re dealing with a log so it’s going to be connected to how exponents work.0806

You’re going to have to remember how logs work. If you remember that, sorry to harp on it so long, but if you don’t remember it right now, go back, relearn it really quickly.0810

It will make this understanding how decibels business work really, really easy if you compared to what you’re standing with right now if you don’t remember how logs work.0816

Here’s some loudness examples. A bunch of examples.0825

The example sound and the sound level that it gives you.0829

The auditory threshold for human hearing is at 0 decibels, because remember I over I knots.0833

If that’s equal to 1, log of 1 equals 0.0838

Light rustling leaves, very light, vague slight sound. 10 decibels.0841

A whisper, 20 decibels. Normal conversation gives us anywhere between 40 and 60 decibels depending on the loudness of the conversation.0846

A washing machine might come in around 50 decibels although it’s going to depend on the washing machine.0853

Car at 10 meters, once again, depends on the type of car.0858

Hearing damage from long term exposure, so this just means that you’re being exposed to something constantly and you’re not using any sort of hearing protection.0862

Notice how many things are going to be above that. A busy highway at 10 meters. A chainsaw at 1 meter.0868

A rock concert. So if you’re going to rock concerts regularly. If you’re constantly using loud motor driven things.0874

Either working around large powerful engines, going to like a construction yard or doing any sort of serious construction work.0881

You’re going to definitely want wear hearing protection because you’ll start to experience hearing damage.0890

If you’re shooting at a range, you’re definitely going to want to have at least one form of hearing protection on if not two forms of hearing protection.0894

Look at how incredibly high that is. 150-160 decibels, that means that we’re a 1,000 times potentially or even more depending on the kind of gun being shot.0900

A 1,000 times the power of a pain threshold. Guns really, really powerful amount of noise coming out of them.0910

Any sort of explosion, really powerful amount of noise coming out it, you’re defiantly want to wear some sort of hearing protection if you’re around that sort of noise.0918

Keep in mind; anything that’s particularly loud can cause hearing damage over long term exposure or even short term exposure if it’s very loud.0926

You’re going to want to keep that in mind because when you get older it’s going to be a real disappointment when you start to lose your hearing.0934

Beats. So from earlier work we know that waves interfere with own another.0942

If the waves have the same frequency, the amount of constructive or destructive interference comes from how in or out of phase they are.0947

As they sync up, they’ll manage to make themselves louder but as they come out of sync they’re going to wind up becoming quieter as they become more and more destructive on one another.0953

What if they’re different frequencies? From before it was just a question of how out of phase that they are and that determined how much destructive or constructive interference we had.0964

What if they’re different frequencies? That means we’re going to have the constructive interference, the amount that they’re syncing up is going to change as they cycle through their different frequencies.0972

They have a 1 hertz difference in their frequencies, every second they’re going to cycle from fully in phase to fully out of phase.0981

We’re going to experience the sense of them working together and then canceling one another out.0990

Working together and then canceling one another out. That’s going to be called the beat frequency.0994

The beat frequency is equal to the difference between the two numbers. It doesn’t matter which one is the larger one and which one is the smaller once because it’s the absolute value.0998

That’s how we’re going to experience it. It’s going to just be the difference between the two numbers.1007

They waves will come in and out of phase based on the difference in their frequencies.1011

It doesn’t matter if its 10 hertz and 11 hertz or a 1,001 hertz and a 1,000 hertz.1015

That’s still going to be the same beat frequency from our experience.1021

Visualizing this and visualizing beats, drawing them out, really difficult potentially.1025

They’re really easy to hear, so that’s how we’ll demonstrate this one here.1030

In a few moments we’re going to play an audio clip. The order of the clip will go like this; a 440 hertz pure tone just by itself.1033

Then a 445 hertz pure tune by itself. Now notice, these sound pretty darn close, these sound really alike.1040

If you’re not good at hearing differences in small frequencies you might not even be able to notice it.1049

It’s just barely noticeable that these two things are different tones, but when we play them simultaneously right after that, 440 hertz and 445 hertz.1054

We’ll be able to really notice the fact that they’re not the same thing anymore. We’re going to notice a beat frequency of 5 hertz as they pull in and out of phase.1066

We’re going to hear that sound. The next one we’ll play after that is 440 hertz compared to 441 hertz.1073

You’re going to hear a beat frequency of 1 hertz as they come in and out of phase.1079

You’ll hear the sound louder and then quiet down to nothing and then louder and then quiet down to nothing.1082

Then finally, we’ll have the first frequency stay at 440 hertz and just remain there while the second one will slide from 441 hertz to 450 hertz over ten seconds.1087

We’re going to hear those beat frequencies go from 0 to 10 hertz.1098

We’ll hear that beat frequency of getting louder, getting quieter. It’s going to speed up over those 10 seconds as the beat frequencies become larger and larger and larger.1101

Okay, so we’re going to listen to it and there we go.1109

Okay, now we have some idea of how those beat frequencies work.1132

The thing that’s really interesting to notice is how those two tones that don’t really seem that different suddenly produce this really noticeable phenomenon in this beat frequency.1136

At 440 hertz and 441 hertz, heard on their own we wouldn’t be able to notice the difference, but put right next to one another, put right on top of one another, not next to each.1143

Put right on top of another, suddenly that beat frequency becomes really, really noticeable.1153

We’re definitely able to hear those things. As the slide happens we start to hear those sounds change, we start to hear those beat frequencies change more and more.1158

In fact, you might have noticed at the very end, as the tail end of it. It starts to sound like the beat frequency is in some way its own sound.1165

That’s kind of true. The beat frequency becomes perceived as its own sound as the two sort of fight one another and we get this extra sound that is the two working against one another or together.1170

They’re creating this extra sound. Sound is a very complex phenomenon because once again we’re hearing many, many different waves.1182

It’s easy to talk about as a single wave, but in real life we’re hearing many, many different waves working together, working against one another all simultaneously.1188

Because we’ve had a lifetime experience listening to sound, we’re able to have an idea of how to turn all that auditory information into something we can actually process and operate on.1195

Once again, we’ve had a lifetime experience; we know what these sounds mean.1205

But really, when we you get right down to it, it’s a lot of very complicated information.1209

It’s because we’re so good at understanding and analyzing. We’ve got brains and they’re developed to do this.1213

They’re able to make sense, to get some meaningful information.1218

Finally, sonic boom. What happens when an object exceeds the speed of sound?1222

It moves faster than sound, producing a conical shockwave in its wake.1227

When you go past the speed of sound suddenly a lot of things are going to change.1232

Normally if you’re moving slower than sound, the sound wave, the fact that you’re pushing on the air in front of you tells the sound in front of it because it’s basically translating that information through the speed of sound.1236

Tells it to get out the way, there’s this really fast guy coming. So it’ll be able to create a moving pressure wave and you’ll be able to sort of push the sound out of your way before you have to actually run into it.1248

Suddenly once you’re moving faster than sound, you’re movement is faster than the sound can propagate the information of your moment coming.1260

So you’re actually slamming, you have to…your aircraft or spaceship has to be able to slam the air out of the way.1267

You’re actually slamming all of those molecules, so suddenly the fact that you’re slamming all of these molecules, slamming all of these atoms out of the way.1275

There is going to be way more friction, way more drag, way more heat produced on your thing.1283

Whole bunch of complicated ideas to talk about here, but we’re going to just talk about real quick, real simple idea.1287

We’re going to see the fact that every time you emit a sound or just the fact that you’re going faster than sound is able to handle.1293

That information is emitted spherically out in a sphere from you, or a circle if we’re looking two dimensionally.1300

The next time you emit it, you actually manage to already pass the edge of that expanding circle.1306

See over here, you emit here, but by the next time you emit, you’ve already passed the edge of that expanding thing.1312

It’s a little harder to see over here because we’re looking so far behind in time.1319

Over here where we’re looking closer to the instant that this picture is taken, we’re able to see the fact that the emitting of information happens after you passed the front of the information that you’re coming.1323

We’re going to get this depending on the speed that we’re moving, we’re going to see this conical wave front out of you.1337

This sonic boom appearing and that’s what people on the ground or other people in the air would wind up hearing is that slamming of air getting transmitted to them.1343

That slamming of air can only move at the speed of sound, so you’re able to actually get passed the sound of your own coming because you’re going faster than the speed of sound.1352

Also if the speed of sound is vs and the object is traveling at v, we give it a Mach number.1365

V over vs. So Mach 1 is just the speed of sound, you’re traveling at sound.1370

Mach 2 is two times the speed of sound because you have to be traveling double the speed of sound etc., etc.1375

Mach 5 would mean that you’re managing to move at five times the speed of sound.1381

Really interesting ideas here but we really have quite enough understanding or time at this moment to tackle all of them.1384

But we’re getting the chance to dip our toes in what’s going on here.1391

We’re ready for some examples. On a warm summer evening, the speed of sound in the air is 340 meters per second. If you see a firework explode in the air, so boom, firework explodes, and we’re standing over here.1395

We see the firework explode almost instantly because light travels so incredibly fast as we’ll talk about in the next section.1408

We see…we can effectively pretend that it’s instantaneous.1416

We see it explode the instance it explodes, but it takes some time for the sound to reach us.1420

It explodes and there is some distance between us and it. It’s going to take 4 seconds.1424

The speed of sound is 340 meters per second and the time that it takes is 4 seconds, we just figure out the distance is equal to the velocity times the time.1433

So 340 times 4 seconds and we see the fact that it must be 1,360 meters away from us because it takes time for that much distance to be crossed by those pressure waves coming from the explosion.1445

However light 1,360 meters is practically nothing to light. Light moves super, super fast as we’re going to talk about in the next section.1461

We’re able to effectively treat it as moving over that distance as instantaneously.1472

If we’re really, really far distance, light wouldn’t necessarily at something that we can treat as instantaneous but the distance for the sound would be so incredibly far at that point that we pretty much wouldn’t be able to hear it at all.1478

Example 2. If you double the intensity of a sound, what increase does that cause in decibels?1489

We start off with some sound at intensity I, what increase are we going to see in the decibels?1495

Say our old sound came in at b, so old intensity is as same as beta.1500

Then our new intensity is going to make beta new.1509

What is beta new? Beta new, remember, I new, intensity new is equal to 2 times I old, which for e, we’ll just say I old equals i.1515

Because that’s what we had it as before. If we double the intensity, I, of a sound, what increases does that cause in decibels?1530

So I starts off being the same as beta and I new is going to be equal to beta new and I new equals 2 times i.1537

Beta new is equal to our formula for decibels is 10 decibels times the log base 10 of 2 I, because that’s the intensity of our sound, divided by I knot.1544

10 decibels times log base 10 of 2 times I over I knot, you can’t stop me from separating that.1562

Still the same thing; either multiply it on the numerator, just multiplying the whole fraction, same thing.1575

Now, because the nature of log rhythms, we can separate those two. Log of xy is equal to log x plus log y.1580

At this point we’ve now got log 10 of 2 plus, and we should multiply because remember that 10db has to go over the entire thing.1590

Plus log 10 of I over I knot. We spread that out, we distribute, we get 10db times log 10 of 2 plus 10db log 10 of I over I knot.1602

Or remember, log 10 of I over I knot times 10db, that’s just…that’s the general expression for what something has as a sound level, what something is in decibels.1628

If that’s what we’ve got and we started off originally with I, then that means this whole thing on the right is just equal to our original beta. Beta old, right?1639

Our original beta is over here on the right and now we just have to figure out 10db times, well log base 10 of 2 is approximately equal to 0.301, plus that old beta.1648

That means 3.01 decibels plus the old number of decibels. The increase that we have is how much we’ve added to the old number of decibels.1666

So our increase in sound is that 3.01 decibels. If you double the intensity, you don’t experience a doubling in the number of decibels.1679

Not by a long shot. You’re just going to wind up adding on 3.01 decibels to the amount that you had originally.1687

Doubling the intensity does not mean doubling the decibels. Anything between intensity and decibels has to go through a log first.1694

We have to calculate this stuff out otherwise we’re going to wind up tripping over stuff.1701

Now we’re going to see something again about the importance of how much logs are going to play with this stuff by a slightly more complicated example.1705

Assume we can treat sound as an expanding evenly in all directions. So that same spherical idea we talked about before.1712

It’s not perfect but it’s a pretty good approximation and just like when before when we got rid of air resistance.1719

Not perfect but often a good approximation for a basic physics idea.1724

If we’re currently 1 meter from an emitter and we hear a sound at x decibels, what distance away from the emitter will lower the sound by 10 decibels, i.e. bring it to x-10 decibels?1728

To begin with, let’s note the fact that the emitter, no matter what distance, the emitter is still going to be putting out p equals the power.1739

The intensity is equal to the power over the area and since we’re dealing with a sphere, we’ve got power over 4 pi r squared.1752

Now we can start using our decibels connection, so we know that for the new one that we want to create, we want to see the distance r that we have to be at.1764

What distance r will create x minus 10 decibels. Our formula, 10 decibels and for ease, I’m just going to drop the base but we still know I’m talking about log base 10.1774

Of the power, divided by 4 pi r squared. Now power doesn’t change because power is constant for this emitter, is going to be equal to x minus 10 decibels.1785

Whatever we started with, lowered by 10.1796

If we want to know what this is, we know what x is, right?1798

X was the original amount. So let’s write everything in again so we can keep our equation proper.1802

X, x was the original number of decibels. So the original number of decibels was what the original r was.1813

Our old r was 1, right? We used to be 1 meter away from the sound source, so it’s going to be pi times 1 square minus, then we have to keep going up from what was above, 10 decibels.1818

Now at this point, that means we’ve got 10 decibels showing up everywhere.1835

This becomes…we divide by 10 decibels, these cancel out here and here and this becomes 1 right here.1840

At this point we can now do something else. Once again, if you don’t remember too much about logs, you’re going to want to double check, I mean you’re going to want go back, relearn this really quickly.1848

We can cancel out a log by raising it to whatever the power of the base is.1858

In this case, since we know those two sides are equal, 10 to each of those sides would still be equal.1863

So we raise both sides with a power underneath of 10. So 10 to the log p over 4 pi r squared is equal to 10 to the quantity log power over 4 pi, because 1 squared is just 1, minus 1.1867

Now 10 to the log, that cancels out and we get what’s inside, p over 4 pi r squared equals p over 4 pi.1886

One thing to notice is that we’ve got 10 to the…we can separate it, 10 to the xy is equal to 10 to the x times 10 to the y.1901

It just gets…Sorry, 10 to the x plus y. Misspoke there.1909

10 to the x plus y is equal to 10 to the x times 10 to the y because we just add things together.1912

X squared times x becomes x to the 2 plus 1 x cubed.1918

We have to remember that and we’ve got still over here, 10^-1 because we’re separating those two different things.1923

At this point we can now cancel p over 4 pi from both sides and what we’ve got now is 1 over r squared equals 1 over 10.1929

At that point, we’ve got r squared equals 10, so r is equal to the square root of 10, which is approximately equal to 3.16 meters.1943

So notice that because the way we did this, this whole thing occurred on the fact that we were dealing with r squared over here, here right.1963

If instead of starting at 1, we’d start it at r old, that r old would still have shown up over here and we would have had that.1973

What it is; is it’s not just going 3.16 meters away. It’s not that at all, its 3.16 meters times whatever the original distance was.1981

It’s our original distance times 3.16. So in our case, we looked at a slightly easier one because we dealt with an initial distance of 1 meter, but we can also expand it to a slightly more complicated thing.1990

It’s actually route 10 times the original distance that we started at if we want to get a 10 decibel lower.2001

Lowering it by 10 decibels means multiplying our original distance by route 10, which is why we get going from 1 meter to 3.16 meters.2008

Say we started off at 5 meters and we wanted to lower it by 20 decibels, then we’d have to take that 5 meters originally, multiply it by route 10 twice.2017

Route 10 twice, becoming 10, so we’d have to go from 5 to 50 meters if we wanted to be able to get a 20 decibel lower in the sound.2026

It’s important to note, once again, dealing this stuff can be a little confusing at first because we’re dealing with logs.2036

You’re definitely going to want to work on calculating this thing out by hand because you might be surprised by how some of the results are going to work out.2042

Think about this, the fact that we’re going from 1 meter to 3.16 meters to get a lower of 10 decibels might not be inherently oblivious at first because we’re working with the way log rhythm works.2047

Log rhythm can be a little, a little new, a little odd at the first time you’re dealing with it because we’re normally can…we’re normally…the way we understand the world first is through additive understanding.2058

Adding things together. It is a little bit more difficult to think in terms of multiplication through exponents and that’s how logs look at the world.2070

Definitely the sort of thing, I definitely want to caution you if you have to do a problem with decibels, make sure you’re working through the formula of how we get decibels first.2078

Otherwise it’s really, really easy to wind up making mistakes.2085

Final example, a nice easy example to finish things off with.2088

Say we have a 256 hertz tuning fork and we put it to a violin and we hear a beat frequency of 3 hertz.2092

What are the possible frequencies that the violin is emitting?2101

Remember, f beat is equal to the difference in those two things.2103

If we’ve got a 250 hertz tuning fork and we’re comparing it to the frequency of the violin and we’ve got 3 hertz beats coming out, then the two possibilities for the violin are the two things that are 3 away from 256.2110

We’ve got the frequency of the violin, it must currently be emitting. The violin can emit many things but we’re going to treat it as if you’re just bowing on a single thing, at a single tension.2127

At this point, we’re going to see the two possibilities are 253 hertz, the lower one or 259 hertz.2136

The two things that wind up being 3 hertz away from what we’re comparing at.2146

That’s how beats frequencies work if you need to solve anything with beat frequencies.2152

Once again, if you’re not used to logs, go back, definitely going to behoove you to take 20 minutes just refreshing yourself on how logs work.2155

Otherwise some of this is going to be really, really complicated to understand how decibels work to solve any problems with it.2162

If you’re working with decibels, definitely want to work through it, otherwise you can easily trip yourself up.2166

You want to remember everything is based on multiplicative ideas.2172

That’s how you want to be approaching it through log rhythmic scale.2175

Alright, hope it made sense. Hope you have a better understanding of how sounds work and next time we’ll wind up talking about light waves. Thanks.2178

Hi, welcome back to educator.com. Today we’re going to be talking about light.0000

Light is a complex phenomenon that is extremely important to a huge variety of aspects in physics and we’re not going to have enough time to fully explore it in this course.0005

Instead we’re going to be able to limit our discussion of light to ideas we can easily pull from understanding waves.0014

That’s still going to give us a much better understanding of what’s going on with light.0019

Keep in mind there is a lot of stuff to talk about in light.0023

We’ll talk about a little bit of that at the end of this. We’re still going to get some idea of what’s going on here.0027

First off, light is fast, really, really fast. In a vacuum, light travels at a rate of c equals 299,792,458 meters per second.0033

Which is approximately to equal to 3x10^8 meters per second and for the most part, all the problems that we’re going to be working on and all the problems you’ll ever have to work on, unless you get into serious, serious theoretical physics.0048

You’re going to be enough with 3x10^8 meter per second, that’s pretty much what all physicists wind up memorizing.0060

So 3x10^8 meters per second equals c, the speed of light. This value is so important it’s given its own constant, once again that’s c.0067

Speed of light, 3x10^8 meters per second and that is fast.0074

Light also has this unique property that stationary and moving observers all measure the same speed for light.0081

That’s not true for all other types of waves. Consider if we had a wave on the ocean moving at 3 meters per second this way.0088

If we were in a boat that was going at 2 meters per second, we’d wind up seeing that wave at only moving at 1 meters per second, right?0097

From our point of view, if we can’t see any stationary objects around us, we’re moving because we can’t see that fact that we’re moving if we aren’t experiencing acceleration.0105

We don’t have any reference points. So if we’re in the middle of the sea and we’re moving at a certain speed and we look down at the waves in the water underneath us.0115

It’s going to seem like its moving depending on our speed. But if on the other hand, we were going in the opposite direction.0123

We’d wind up seeing it moving at 5 meters per second, right? So the experience would be 1 meter per seconds if we were going with it, 5 meters per second if we were going against it.0132

If we’re sitting still in the water, it’d be 3 meters per second.0141

For light, it has the phenomenon that for every direction you’re moving, you record the exact same speed.0144

That is fantastically interesting. That’s so different from all other types of waves, it’s just…it’s really interesting, really important.0151

Also what we’re going to wind up…what we’re not where going to wind up looking at, but one of the cornerstones of relativity is this fact.0160

This doesn’t mean that light travels through all materials at equals rates though.0170

In a vacuum, light travels at c and it travels very close to sea through air, like we talked about before, c for a vacuum.0174

In water, it travels at 0.75 c. Through glass, 0.67 c. Through diamond at 0.4 c.0180

Still really, really fast but it does mean when its moving through another material, it winds up being slowed down.0188

It’s able to move at its full speed in a perfect vacuum, but as it winds up having to encounter other objects to pass through, it’s not able to keep up its speed quite as much.0195

Light speed. Since light moves so fast and it’s so important to the nature of the universe, occasionally we talk distance using the speed of light.0205

If we’re talking about astronomical stuff, it’s a really great thing to be able to talk about, because we’re talking about really, really large distances in space.0212

So being able to talk about how far light manages to travel in year, we’re able to talk the speed of light times the time of one year.0219

Multiply it by the time of one year, we’re able to talk about a light year.0226

A light year is a measurement of distance because it’s how far light could travel in on year.0229

For large, large distances in space being able to talk about the distance in terms of light years is a really handy thing to be able to talk about.0236

It also tells us how long it would take for information from that galaxy to make it to us, because of its sun, its sun for example goes supernova and explodes but its 50 light years away from us.0243

We’re not going to be able to get the information of it exploding, the light of its explosion isn’t going to get to us until 50 years after it explodes.0255

Some of the information we’re getting, some of the stars we see in the sky could actually wind up being dead stars, we just don’t know they’re dead yet because we haven’t seen the information of their explosion come to us yet or their nullification in some way.0261

Medium. Normally a wave needs a medium to propagate through. If we have waves and water, we need the water to be able to move up and down.0275

If we have a wave in a string, we need the string to whip up and down.0283

Me speaking needs this air for it to be able to bounce against and have those differentials.0286

Light on the other hand needs no such medium. Light is able to move through a vacuum without issue.0290

Nobody else can do that. If there’s nothing there, other waves can’t transmit themselves because they don’t have anything to vibrate against, they don’t have anything to move.0296

Light is in itself its own motion. It’s its own medium. We’re starting to get into a complicated thing here about the dual particle wave nature of light, so we’re not going to talk too much about this.0306

But light suffices to say doesn’t need a medium, it’s able to just go on its own.0319

Unlike all the other kinds of waves that we learned about. Now notice the speed can still be affected by what it travels through.0324

Remember if you’re traveling through diamond, you go slower. You travel through water, you go slower.0329

It’s not the same as that material being what’s transmitting the wave.0333

It’s traveling through the water doesn’t mean that the water’s motion is what moves the wave along.0337

The wave is able to go through the water but then also hop out to going through just pure raw vacuum.0342

It’s a really big difference.0346

Electromagnetic spectrum. Light is kind of a misnomer and me using the word light; we want to expand that to more than just the light.0350

Just the light that you’re seeing me with. There is way more quote on quote light than just the light that you’re seeing me with.0358

The light that we see, visual light is only one of many forms of light.0364

Light is really just part of the electromagnetic spectrum.0369

The EM’s spectrum is a really wide set of possibilities. It’s going to go over a huge amount and we’ll talk about some of those.0373

Visible light, the light that we see, is just a small fraction of the EM spectrum.0380

There is many more possibilities and it’s those other possibilities that allow us to transmit other kinds of information.0385

We’ll talk about that in a little bit. Any electromagnetic wave, they share many similarities for every electromagnetic wave.0390

Such as their speed and their ability to move without a medium. Different frequencies and wave lengths give some different properties though, such as the amount of energy that the wave carries.0397

A higher frequency has more energy because that means that it’s vibrating more.0404

If we had a string for example and I whipped it up and down only once a second, I’d have way more energy in that string if I was whipping it up and down a hundred times per second right?0408

There’d be more energy being put into it and a similar idea is happening with waves.0416

More energy is in the higher frequencies because they’re vibrating more.0420

The different classifications for electromagnetic waves.0427

So the long radio wave has a frequency of anywhere from 10^0, so just 1 hertz to 10^5 hertz.0430

Radio waves, the kind of stuff we use to send television or cellphones or…cellphones actually start to verge into microwaves depending…well anyway, there’s a lot of different possibilities in different sections of the spectrum.0439

Radio waves, we use to send radio as you might guess. We used to send, I believe radar, I’m not actually sure about that, don’t hold me to it.0455

We definitely use to send television. A lot of information gets transmitted from place to place.0464

Because by the way that we move the wave by taking slight variations off that frequency we can send that light, quote on quote light, we can send those electronic, those electromagnetic waves through a space.0469

Say from the top of a mountain with a transmitter to a city that has a radio in it.0482

We’re able to put out a certain kind of electromagnetic spectrum and by vibrating slightly different than the expected frequency, that radio on the other end is able to pick up those slightly different vibrations and turn that into some sort of information.0486

Like say sound information that it then puts out through a speaker.0502

Those slight variations off of the starting base level put out information, right?0505

Microwave, the same sort of thing that you in a microwave is 10^9 to 10^11 hertz.0510

The reason why that works even though they’re lower energy waves than the visual spectrum.0517

We’re able to have it putting out energy because it’s able to sync to the appropriate frequency that water and sugars and many kinds of fats, that they’re on so that it’s able to vibrate them and cause them to pick it up and have a resonate frequency as we talked about before.0521

And able to increase the vibrations in those atoms, thus increasing the energy, the heat in it.0537

Infrared, 10^12, 10^14. Visual spectrum, see how small it is, it’s just from 4x10^14 to 7.9x10^14 hertz.0542

ROYGBIV. Red, orange, yellow, green, blue, indigo, violet.0551

Haven’t put down the specific times, if you want to know more, easy to look up.0557

As it cycles up, we go from red up to violet as we go through higher and higher frequencies.0560

That’s why it’s called infrared, below red and for the next energy above it, ultraviolet, above violet.0567

More than that we get ultraviolet 10^15, 10^16.0574

X-rays, even more and finally gamma rays covers the really extreme high, high, high frequency. Really high energy things.0579

Now the reason that ultraviolet rays, x-rays, gamma rays, though you’re really unlikely to be exposed to those.0585

The reason why these can be potentially damaging to us is there is such high energy things and they have such a small wave length.0594

They’re able to pierce through our body and they’re able to hit a cell and they’ve got enough energy in them to be able to potentially cut off a piece of DNA.0598

Cut DNA in a certain way. Potentially, if you’re really unlikely, that DNA will be cut in just the wrong way, and this happening to you lots of times.0607

You get exposed to a lot of ultraviolet light, suddenly you’re exposed to more possibility of skin cancer.0615

You get exposed x-rays, suddenly you’re exposed to the possibility of maybe getting some kind of cancer inside of yourself.0620

There is a lot of other things that cause other carcinogens out there, but this one possible way to get cancer is because DNA winds up getting split.0625

So suddenly the code that tells the cell how to behave, how to multiply, goes haywire and most of the time the cell fails to work at all and just dies.0633

Sometimes it manages to get cut in the quote on quote right way, not right at all, very bad from our point of view.0643

Cut in just the wrong way and suddenly it goes haywire, goes out of control and it starts to produce many, many of itself and we get a cancer.0649

It does something bad to the body. The reason why that’s possible is because we’ve got such high energy in them that they’re able to actually effect ourselves and they’re able to potentially do things to our DNA.0656

Now there’s huge amounts left to explore. We could talk more, I mean that idea that I was just talking about; x-rays and ultraviolet and gamma rays being able to cause cancer damage to cells.0668

So much we could talk about just in that tiny little idea, but there is way more things here.0679

We’ve just begun to talk about the tip of the iceberg in terms of light.0682

There is so much more. We could fill many college physics’ courses and then more onto a lifetime of research after this.0686

Huge amount of things. Some of the topics you might one day encounter if you’re interested in light and there is lots to be interested in about.0693

Just go ahead, search for yourself. You find out all sorts cool ideas or go ahead and take more courses.0698

This is…I mean there’s all sorts of cool things in physics and this is one of them.0703

First off, optics, how light behaves with various materials.0705

The way light is going to move through them, pass through them. Change, be reflected.0709

The energy in light, we started to talk about there is more energy in light and less energy in different things but we didn’t get into any specific numbers.0713

Way more to be talked about there. The fact that energy in light, not energy in light.0720

Light can both be treated simultaneous as wave and as partially.0725

You’ve probably heard of photons, that’s a single packet of light.0728

Light behaves, light has some of the effects of a wave. It has those interference effects that we’ve talked about previously with waves.0732

The same time it also can be broken down into discretized quantities of single chunks.0739

It’s got a really strange thing going on there. We don’t normally thing of a wave as something that can broken up into a single piece.0743

Light able to do both at once. Once again, really unique phenomenon.0749

Finally, all of relativity. Everything in relativity is based on this fact that light is this unique thing and that light is top speed limit of the universe.0753

And why that is a whole kettle of fish to get into. There is huge amounts of stuff here.0765

Light is interesting. Still at least we’ve managed to crack up an amazing new vista to be interested in and we’ve dipped our toes.0769

We’ve got some new ideas and it helped us understand our universe just a little bit more and it’s a really cool thing.0777

I’d really encourage you, go ahead, just do some research. Get yourself exposed to a whole bunch of ideas.0783

You’ll get the chance if you want to, to start taking more courses or just do some personal reading and you can learn a lot about what’s going on in the world and the universe.0788

I mean everything at once. Alright, ready for some examples.0794

How much distance is in a light year?0798

We start off, we know the speed of light, right? C is equal to 3x10^8 meters per second.0800

We’ve got that down. If we want to know what distance is covered, we need to know how much time is in a year.0805

If time is equal to 1 year. Well 1 year is equal to 365.25 days. We’ve got that leap year every four years.0811

We’ve got a quarter of a year in there, so 365.25 days.0820

If we want to know how many hours are in there, 365.25 times 24 hours in a day.0826

We get that we’ve got 8,766 hours in a year.0839

Which means that we can multiply that by 60 and we’ll get how many minutes are in a year, 525,960 minutes.0846

Which we could turn into seconds. We can multiply that number by another 60 seconds.0863

We’re going to get 3.156 x 10^7 seconds.0872

If we want to know what the distance that it covers, we just put the two together.0879

Distance equals velocity times time. So the velocity of our thing is 3 x 10^8 and we multiply that by the time that it has to travel, 3.156 x 10^7 seconds.0882

We get a distance of 9.467 x 10^15 meters. That is a huge, huge distance.0894

10^6 means that we’re dealing…so 10^3 means we’re dealing with thousands.0908

10^6 means we’re dealing with millions. 10^9 means that we’re dealing with billions.0913

10^12 means that we’re dealing with trillions. 10^16 means that we’re dealing with quadrillions.0918

That’s almost 1 quadrillion meters. That’s a massive amount of distance, this is just a huge amount of distance that we’re able to cover.0925

Now keep in mind that the closest star system to us, Alpha Centauri is approximately 4.3 light years away.0933

We’re dealing with an absolutely massive amount of distance between us and that other system.0942

If we want to get there in any reasonable amount of time, we’re going to have to figure out some way to get a close approximation to the speed of light.0948

Once again to relativity, you start to realize that getting into reasonable amounts of speed like that, really, really difficult and there is even more complex stuff going on there.0954

Just even moving at 1/10th the speed of light, think about how much energy you’d have to put into that.0964

Moving at 1/10th the speed of light, 3 x 10^7. If we’re dealing classical mechanics, that would be 3 x 10^7, the whole thing squared times ½ times the mass of the object.0970

½ mv squared. Huge, huge, huge amount of energy.0983

To be able to get any sort of space ship to any other solar system is going to require some incredible feat of engineering or some incredible feat of scientific process for us to be able to cover these huge distances.0989

Right now we’re basically in the solar system for at least a while longer.1000

Being able to actually touch the other stars is going to take something really, really cool and really, really smart from humanity.1004

Example two. If we perceive an electromagnet wave at the color green it’s going to have something…so we perceive a 525 nanometer wave if it’s moving through the air at the color green.1011

What frequency is that? Remember electromagnetic waves very near to sea and air.1023

Remember, we know that the speed of a wave to equal to its frequency times its wave length.1027

If we’ve got the speed of the wave at 3 x 10^8 and we want to know what the frequency is.1035

Well we know the wave length, 525 nanometers. So 525 x 10^-9 meters.1043

Now we can easily just solve for frequency. So frequency is equal to 3 x 10^8 / 525 x 10^-9 which is equal to 5.71 x 10^14 hertz.1050

Smack dap in the middle of that ROYGBIV spectrum, right in where G is going to be.1065

That’s what the frequency that we’d wind up getting out of that wave.1071

Now say you want to tune into my favorite radio station, KSBC at 88.7 megahertz.1076

If KSBC is at 88.7 megahertz, which it is, what wave length does that mean you’d have to be scanning for?1081

If you’re scanning for KSBC at 88.7 megahertz what wave length would that mean that we’re looking to be able to pick up?1089

Well once again we use the exact same thing just slightly different.1097

Frequency times wave length. Well the velocity we’re dealing with electromagnetic waves is 3 x 10^8 equals whatever frequency it is, so in this case 88.7 megahertz.1100

88.7 x 10^6 hertz because it’s mega. Times the wave length, so the wave length is going to be equal to 88.7…oh whoops, sorry, put that the wrong way on.1111

3 x 10^8 / 88.7 x 10^6. Which means that we’re going to be looking for a wave length that’s 3.38 meters.1125

That’s pretty big. It’s really interesting to compare how much difference there was between the wave length of that green light.1138

Tiny, tiny thing. 525 nanometers to 3.38 meters. That’s practically two of me standing on my shoulders.1145

That’s a really tall wave length. That means if you’re going to want to pick it up, you’re got to have some way of being able to see all that information in that really long wave length passing by you.1152

Which has to do with the way waves work, but we once again aren’t going to quite get into that.1160

3.38 meters, really long wave length because it’s got a really small frequency compared to some of the other ones.1165

Alright, hopefully that gives you some idea of how light works and possibly spark your interest in some of the many, many interesting…1170

Hi and welcome back to educator.com. Today we’re going to be talking about fluids.0000

It certain makes things easier to concern ourselves just with rigid blocks and point masses.0005

Clearly the world is made up of up more than just solid objects. At this point we’ve only talked about fixed things and point masses and things that we could push any part of it and have the entire thing move.0010

What happens when we’re dealing with a liquid, where you push on the liquid and you…you’re hand just goes right into it.0020

What about when you’re walking and a gas and you walk through it and there is no issue?0026

You try to walk through a solid of block of metal and you just bounce off it.0029

You walk through a gas and it easily lets you pass through.0032

There is clearly a lot of different stuff going on when we’re dealing with fluids.0035

What do we do if want to talk about a lake of water or an atmosphere full of gas or a flow of lava coming down the side of a mountain?0038

This is where the idea of fluids come into play.0046

First off, what does it mean to be a fluid?0049

It just means that you flow, that you’re able to have a malleable shape.0051

Unlike solids, fluids don’t a set shape. Instead they take the shape of whatever container they’re placed in.0056

If you put in a liquid into it, gravity will hold it against the walls of the container.0062

If you put a gas into a container, it’s the same thing, it’ll just go out the top.0065

So if you put a closed container, it will take the shape of the closed container if you place a gas into a closed container.0070

A liquid doesn’t have to a closed one because it’s not able to float out of the top of it, but a gas does because it can potentially float out of the top of it if it’s light enough.0076

Liquid and gasses are fluids, while there are differences between liquids and gasses.0086

The ideas that we’ll discuss in this section are applicable to both of them.0092

We will be able to talk about both of those ideas, a lot of the ideas we’re going to talk about are going to be just as usable at whether we’re talking about water, whether we’re talking about the air we’re breathing, whatever we want to talk about as long it’s a fluid.0095

First thing we’re going to want to talk about is the idea of density. Density is something that we can even apply to solid objects.0108

As long as it’s a homogenous, which means it’s even distributed, that there are not like one really heavy chunk and one really light chunk next to one another.0113

For example, most milk, milk is homogenized to make it a nice homogenize texture so it doesn’t eventually separate into milk fat and the milk liquid.0121

Homogenous just means it’s been mixed together and it’s evenly mixed together, evenly composed.0132

No big chunks of one type compared to big chunks of another type. They’re all about the same size, evenly distributed.0139

This means that instead of having to talk about mass and volume separately, if we’ve got this homogeneous mixture, we can suddenly relate them.0145

We can relate them through the idea of density. If we have a homogeneous substance, than the density, rho, this guy right here is called rho.0152

R H O. Another Greek friend that we’ll be using now that we’ve in the need for another Greek letter.0162

The density rho of a homogeneous substance or object is the mass divided by the volume.0170

So whatever the mass of our thing is divided however much volume we have.0177

That way we have a relationship between the amount of volume and the amount of mass.0180

If you have one liter of water and you compare that to ten liters of water, it’s perfectly reasonable in your mind to be able to expect that mass difference is just going to be 10x more.0184

When you jump up to 10x the quantity. If for the most part, most of the things we already think about have fairly homogenous structures so if we have one car and then we upgrade to ten cars.0194

We’re able to treat it in the same idea as density because we know that we’re dealing with the whole thing.0205

If we taking just the engines though, we wouldn’t be able to follow that mass.0210

In our case, we’re just going to be working with homogenous things, not actually this car example, but mass divided by volume when we’ve got something evenly distributed.0214

Consider if we had two equal heights cylinders of water but with totally different areas on the bottom.0226

If one of them had a small area and the other one had a big area and they had the same height, then this one right here would have a little force, but this one here would have a giant force.0230

The amount of pressure pushing down on the base of these two cylinders is….sorry not the amount of pressure, the amount of force pushing down on the base of these two cylinders.0243

Its going to be very, very different. They’ve got different volumes of water and so they must have forces pushing down because they’ve got to have something holding that water up in the container.0251

What if we look at how the load is spread over the base?0262

Instead of looking at the sum force, we look at what it is by chunk by chunk.0266

Then we can see how each point at the bottom of both cylinders pushes down equally.0272

The difference is the larger cylinder has more points. This small cylinder can fit into this big cylinder what…one, two, three, four, five, six, about six times.0275

So it makes perfectly good sense that the big cylinder is going to push down 6x harder just because we’ve got 6x more of it.0287

The issue isn’t that it’s more massive or heavy or anything like that. The issue is just that it’s got more of it to push out over more.0295

It’s the fact that there is more of it and what we want to do is, we want to be able to disengage just talking about more to talking about the pressure on that small point and that small area.0303

What’s that area have to tell us?0317

This is where pressure comes in. This observation leads us to create a new definition.0320

Pressure. P. Pressure and here’s some little problem I like to point out to you.0325

At this point we’re now talking about rho, which is this slantingly p, like this.0330

We’re talking about pressure that is a small p, just like we’re used to.0336

This causes some problems, some physics groups, some physics text books will wind up doing it as a capital P and that makes a lot of sense but we’ve already used capital P for power.0343

We’re kind of stuck between a rock and hard place here. Either we’re going to have to use the small letter for two different things or we’re going to have a little bit of difficulty telling the difference between density and pressure.0353

It’s up to you to be careful, really pay attention to the letter you’re reading.0363

You might think it’s a p, pressure, but it actually turns out it’s a density.0366

Be careful, pay attention to if the p looks slantingly.0370

Pressure is the magnitude of the force applied divided by the area it’s applied over.0374

With the small cylinder, we had a small force but it was applied over a small area, so that made some pressure.0381

With the big cylinder, we had a large force but it was applied over a large area.0386

So if we manage to divide those areas into the respected forces we’ll wind up getting the same pressure, because they have same height and the same liquid.0391

It makes sense that that liquid is going to push down as hard on a per area basis.0398

Because it’s just a question of many quote on quote points you have to push down with.0403

The unit for pressure is the Pascal. Pascal is the Newton, unit of force divided by the meters squared unit of areas.0410

Newton’s over meters squared is the Pascale. The Pascale is named for a famous scientist that did studies in pressure.0416

Pressure at depth. If we have a container of fluid, how do we tell how much pressure there will be in that container?0423

What do we want to do if we want to tell what it is at different heights in the container or in different locations in the container?0430

If you’ve ever dived deeply in a body of water or driven up a mountain, you’ve felt this inside of your ear.0436

That feeling when you’re at the bottom of a pool, the water pushing in on your ears is because the water is really pushing in on your ears.0442

The weight of all that water above you is actually pushing in harder.0448

If you drive up a mountain and you feel your ears pop that’s because there is less air pressure around you, so instead you’ve got more air pressure inside of your head that you built up when you were lower down.0451

Now it’s pushing out, so you pop your ears so you can regulate the air pressure between your inner ear and the outside of the air.0461

The outside atmosphere, the air around you. You’ve felt this pressure before; this is definitely a real thing.0471

Pressure is connected to your depth in the fluid. Let’s figure out what that pressure is.0479

It makes sense, if we’re farther down at the bottom of the pool we’re going to feel it more.0484

If we’ve dived into a pool, we’re certainly felt that difference as we go down between starting at the top to going down a meter to going down and touching the deep end at 3 or 4 meters.0487

You’ve definitely felt that difference in your ear. Or if you’ve ever taken an empty soda bottle and brought it down with you, you’ve seen it start to crush down more and more.0500

That’s because of the pressure increase in the water around you.0508

Imagine we have some container of fluid that is totally at rest, so it’s perfectly still just sitting there.0512

Let’s consider some portion of that fluid as just an imaginary column.0517

We box out some of the imaginary column and while this is done as a box, we want to think that this is actually a column.0522

It will make our computation just a little bit easier.0528

We’ve got this column and it’s sitting there. Now let’s examine it a little more closely.0531

If it’s sitting there then the fluid’s at rest. If the fluids at rest it means that it has to have the forces on it in static equilibrium.0536

We know that the fluid has mass, so the fluid if it has mass must have some force of gravity on it.0544

That force of gravity has to be canceled out by something else.0551

If we consider all the forces that are on this column. It’s currently sitting still and it’s got some force pushing up from the water underneath it that’s holding it up.0555

Sorry, water, I meant to speak generally although I’m imaging it as water but this is going to work for any derivation that we want to talk about fluids.0564

That’s why this is so great. So there’s something holding this fluid up.0571

We’re pushing up with some big f but at the same time we’ve got the weight of gravity that has to be overcome, mg.0574

Then we also have something else, what if we had to say a bucket of water outside in our atmosphere.0581

Our atmosphere has some air pressure pushing down so we’re going to also have to keep in mind that the fact that there is some air pressure or some outside other fluid that is pushing on our fluid already.0588

That extra pressure is going to have be kept in mind as we’re working with this.0598

If the fluid’s at rest we know that all of these things have to be in equilibrium.0604

The up pressure, the up force on this column is going to have to be canceled out by the downward force of whatever is outside and just the raw force of gravity.0608

We put this all together and we get f, the force of the fluid on the fluid from below is equal to f knot.0618

The force from above plus mg, the weight of the column.0625

With this in mind we can remember pressure is p equals the force divided by the area.0629

So pressure is force over area and density, rho, remember, and notice how close those two symbols look.0636

It’s important to keep them different in your mind.0642

Density is the mass divided by the volume. With that in mind we can set up our equilibrium equation and we can start playing with it.0645

And we can get an idea for how pressure works.0653

Here’s our equilibrium equation right here. Now at this point, we know that f is equal to the pressure times the area.0657

We substitute that in for this, so f will use p and f knot will use p knot.0666

For mass we know that density equals mass over volume.0672

If we just multiply the density by the volume, we’ll be able to get another expression for the mass.0676

We’ve got pressure times the area of the cylinder, the area being pushed up, is equal to the outside pressure, pushing down on the cylinder, times the area it’s allowed to pushed down, the one from the top.0682

Plus the weight of gravity, the weight of it is going to be the mass, rho, times volume times gravity.0695

Then remember that volume on a cylinder is just the area of the cylinder times that height of the cylinder.0702

We know that volume is area times height. At this point we’ve got area showing up on all three sides and we can cancel that out and we’re able to get that the pressure, the pressure at any point in our liquid is equal to whatever the outside pressure is, the external pressure.0708

Plus the density of our fluid times the height depth of the fluid times gravity.0722

If we have the same exact thing on Earth, it’s going to have a different density looking at the same spot on the Moon, looking at the same spot on Saturn.0728

Every difference place with a different gravity will wind up changing the experience of pressure.0737

Pressure will change based on gravity, without any gravity we don’t have anything to pull down and cause pressure.0742

Gravity is intrinsically connected to pressure.0748

So absolute pressure versus gauge pressure. Now we have an equation for pressure at a depth, so the pressure is equal to the outside pressure plus density times depth times gravity.0752

We can separate this into two different ideas. Absolute pressure, what the total pressure is at a given location, p, what we’re used to, what it was at the bottom of that cylinder.0769

We can also create another idea, gauge pressure. The difference between the pressure at that point, the absolute pressure and the ambient pressure is.0781

The ambient pressure is p knot, what the external pressure is.0791

If we want to know what is being contributed just by the fluid we’re looking at that’s going to be our gauge pressure.0795

So gauge pressure is a little bit different than absolute pressure.0801

If we had a bucket of water and we looked at some point down here, at the same time we’ve also got air pushing down.0806

So we might be curious to find out what’s the absolute pressure at this point.0814

But we also might be curious to find out what is the pressure just of the water, so that just of the water would be the gauge pressure and the amount of the water plus the air, what you’d actually experience when you were down there would be the absolute pressure.0818

Now why does gauge pressure matter? It seems like maybe it’s an interesting idea but why would really care about it?0833

Consider the following idea, if we were to fill up a tire with air, which we all have to do if we’re going to drive in a car.0838

If we want to cause the tire to inflate, it’s going to have to have more pressure inside of it than the outside external pressure.0844

Otherwise it’s not going to be able to overcome the air pressure pushing down on it.0850

We put more air pressure into that tire to beat out the air pressure pushing down on it.0854

Do we care about how much that air pressure is?0860

We only care how much the tire is inflated. So it’s not…it has to even begin inflating, it has to cancel it out.0864

So its absolute pressure is going to be whatever pressure we want to put into the tire plus the pressure that it has to have to even start at ambient air pressure.0871

What we’re really comparing, we’re really seeing how much more pressure is in our tire than our ambient pressure.0879

If we were to measure this, we’d measure it with a gauge.0886

If you’ve used a tire gauge you know what we’re talking about.0889

What you’re really measuring with a tire gauge, is you’re measuring the difference between the ambient air pressure and the tire’s air pressure.0892

That’s why it’s called gauge pressure and that’s why we care about. There is all sorts of applications where you’re going to care about what’s the difference of this thing versus the outside thing.0899

Same thing if you were to look at a balloon, you’re going to care about what’s the difference in this pressure versus the difference of the pressure inside of the balloon versus the pressure of the air outside.0907

It’s going to have more pressure inside of the balloon to be able to push out against the air pressure and actually become larger.0917

Depth. Not shape or direction, so at this point, the way that we derived it we’ve got pressure is equal to whatever that external pressure is plus the density of our fluid times the height, the depth that we are in the fluid times gravity.0925

It seems kind of counter intuitive to think if we had a big v thing, it might have more pressure here because it’s got more water than if we had the reverse, a pyramid.0942

Well that’s not the case; it’s going to actually turn out that it’s just depth.0954

The only thing that matters is depth, not shape, not the direction we’re trying to look at.0958

The force is going to be pointing in all the directions equally and the way that we can see this is through the idea of Pascal’s vases.0963

We can prove this imperatively, it’s a possibly a little bit difficult to prove theoretically, so not something that we’re going to get into.0970

This point you’ve probably lived pretty long enough to be ready to see…if you were to see something that looked like this.0976

If you were to see a three dimensional creation of this and it was just a bunch of different vases that were all connected at the bottom by a glass tube, so there is a bunch of glasses vases that were all ultimately emptied into the bottom tube.0981

And you put water into this, if you poured water into any one of these, you’d probably expect inherently at this point for it to look something like this.0994

It’s going to wind up filling up to the exact same level throughout. It doesn’t matter what the shape of each one of those individual vases are.1005

It’s going to wind up filling to the same height. Fluids always seek their own level.1013

This an idea you’re almost certainly used to by now. It’s just something that we get used to, but we don’t necessarily think about it too much.1018

This fact that they seek their own level is imperially proof that the way we measure pressure is only going to be depend on depth.1025

If this weren’t true, if it weren’t only based on depth, then say the pressure here might be stronger than the pressure here.1033

If there is more pressure over here than pressure here then that would cause water to push this way, so we’d have water flow up and ultimately this one would get a higher level and this one would get a lower level.1043

We’ve got to have the case that pressure has to be the same based on height only, otherwise it wouldn’t be able to work out and we’d wind up getting different levels in these vases depending on their different shapes.1055

At this point we’re used to seeing the fact that the shape doesn’t matter, it’s just going to be the water being poured in.1068

If you pulled up a hose and you pour in water, it’s just going to come up equally on both sides.1075

If you move the hose around the fluid is going to stay at the same height no matter what you do because the fluids always seek their own level.1078

This idea gives us that it’s not going to be based on shape, it’s not going to be based on direction, it’s just based on the height, how deep you are in the water or what’s your depth.1084

Sorry, not the water once again, the fluid, whatever fluid we’re in.1096

This works for atmosphere, this works for noble gases, this works for lava flows, it’s just any fluid this works for.1099

One more thing, depth is equal to height. So notice the depth, h, is measured to the highest point the fluid achieves.1108

Not the distance to the top, but its how high up the top is from where you’re measuring.1117

In this one, measuring from here to here is the same thing as measuring from here to here, as the same thing as measuring from here to here, as measuring the same thing from here to here.1121

It doesn’t matter if there are different distances or what the shape is or what the total length of the container is.1134

All that matters is how the high it is to the highest point. It doesn’t matter if you’re put on a slant or if you’re all curvy.1140

It doesn’t matter what the shape of the container is, all that matters is how high up, how tall is the highest level of the liquid and what is your depth compared to that liquid.1146

It doesn’t matter what the distance to get to that highest point is, all that matters is what the height difference is.1155

That’s why we have just h. So for all of these pictures, the pressure is the exact same at that dotted line.1161

Regardless of which one we’re looking at. They have the same top height for the fluid and since they all have the same top height and they’re all measured at the same depth, they all have the exact same pressure.1166

Shape has no impact on pressure, it’s just depth. How far down you are from the highest level that the fluid achieves.1180

Buoyancy. Why does a piece of styrofoam float in water? Why does a helium balloon float up in the air? Why does a rock feel lighter when we’re in the water then when you’re holding it up in the air?1186

The answer to all these questions is buoyancy. Fluids impart an upwards buoyant force, fb, on any object inside of them.1197

If we’re going to see this lets consider the container full of some liquid.1206

If we draw an imaginary boundary around any portion of the liquid, the fluid’s still so it must have some upward force of mg to cancel out that gravity.1210

The liquid, it’s got mass, so it’s currently being pulled down by gravity, by some mg.1217

If it’s being pulled down by gravity there must be something canceling that out because we know that in the end it just sits still.1223

So if it sits still there has to be something to defeat gravity.1229

If the volume of the portion, v, and the density of the fluid was rho fluid, we’d get that the mass of the gravity is equal to the mass, rho fluid times volume times gravity.1233

Density times volume times gravity would be the same weight that gravity is exerting.1245

The force of gravity is going to just be this right here, so if that’s the force of gravity we know that our buoyant force, the amount pushing up on that liquid is going to have to be the exact same amount fighting it in the opposite direction.1251

The buoyant force will push away from gravity and it’s going to have to be the same amount that the force of gravity would be on that volume of liquid.1262

If were replace our imaginary chunk of liquid with some object the buoyant force isn’t going to apply only to the imagined chunk of liquid, it’s going to apply to whatever object is taking up that space.1270

If we put something else in there, that submerged object will have a buoyant force that is equal to the amount…the volume, sorry not the volume but the weight of the fluid displaces.1281

Once again it depends on the gravity that we’re dealing with. Since we’re on Earth, we’ll just locally always use 9.8 but whatever the submerged object displaces in the weight of the liquid, it’s going to have its own buoyant force.1292

Because the liquid is pushing up on it by that same amount, whether it’s just liquid taking the space or it’s a piece of Styrofoam or a rock.1307

So the rock feels lighter in water because it’s displaced that much water. It’s displaced itself worth of water and so it’s being pushed up by the amount of mass of the water that would take up the space of that rock.1316

So that is going to help us make it…helps us lift it, make it seem lighter.1329

This gives us the exact same formula that we just derived.1334

The buoyant force is equal to the mass of the fluid that was displaced times gravity and mass of fluid is the exact same thing as density of fluid times volume.1337

Density of fluid times volume times gravity is our buoyant force.1346

Simple as that. Wait, I hear you wondering about pressure.1350

Doesn’t pressure change at our different heights?1354

Of course. You might be tempted to think that the buoyant force is going to have to change with the depth because the pressure changes with the depth.1357

That’s not the case. Consider this picture. Buoyancy is called by the pressure differential, so on this one, we’ve got three arrows pushing up.1363

That’s going to be this one and it’s going to push up with one arrow total and it’s going to move up.1373

For this one, we’re pushing down with five arrows since we’re way deeper, so there is way more pressure.1378

That same distance down is going to result in the same difference of arrows, so we’re going to be pushing up with seven arrows.1383

Whoops, sorry. We’re pushing up with two arrows here, not just one.1391

This one would be pushing down with five arrows; here we’re pushing up with seven arrows.1395

Once again we’re pushing up with the exact same amount. The difference between the pressures is the same.1398

This also explains why we don’t have any sideways buoyant force.1404

There is no sideways push because at every height there going to have the exact same pressure from the right and from the left because they’re going to be at the same height.1408

The left and right, there is no pressure because the pressure is going to be…there is no pressure differential because the pressure is always equal to the same height.1416

Since there is no pressure differential we only have to worry about buoyant force going straight up.1423

If we have a graduated cylinder that’s full to the edge, absolutely full to the edge with 1 liter of fluid and then we dip a rock into that cylinder, it’s going to cause some of the water to come out.1431

As we put the rock in some of the water will slosh out of the sides, so when we lift it out we’ve managed to find out what the volume of the rock was.1443

If we start off with 1 liter of fluid and then we pull the rock out and we’re at 700 milliliters of fluid we know now that the volume of the rock is going to be equal to 300 milliliters.1450

Now if we know what the density of the rock is and what the volume of the rock is, then we can immediately toss the two together and boom, we’ve got mass.1463

We multiply 2.5 x 300 x…wait a second, that means we’ve got a rock that’s 700 x 10^3 kilogram. What just happened?1473

We used the wrong volume. Volume…well liter is a good standard measure of volume.1481

Liter is one of the measures of volume and we’re using milliliters. Once again we want to at least catch that fact and switch it 10^-3 x 300 x liters.1487

We aren’t using liters. Up here it’s kilograms per cubic meters.1498

We have to make sure what we’re dealing with is what we’re also dealing with cubic meters.1503

One trick that you might not know but we didn’t mention in the lesson is that the milliliter is the same thing as 1 cubic centimeter.1508

1 milliliter is equal to 1 centimeter cubed. So what’s 1 centimeter cubed in terms of meters?1517

Well 1 centimeter is the same thing as 0.01 meters. If we’re at .01 meters and we cube that, then we’re going to be at 10^-6 meters cubed.1528

10^-6 cubic meters is the same thing as 1 milliliter of space.1544

Remember, liters, their way of measuring volume, their way of measuring a liquid, so the only way to measure a liquid is with a volume.1550

To know how much space it takes up. Volume, liter, they’re the same thing; they’re the same measure of an idea.1558

We’re not using that measure; it’s like when you talk about Celsius versus Fahrenheit.1565

You might be…you’re talking about the same idea, you’re talking about temperature but one of them is going to give you very different results than the other one.1569

You have to know which one you’re talking in terms of. In our case we’re talking about cubic meters.1575

If we’re taking about cubic meters, we need to change this to cubic meters.1581

So 300 milliliters is going to be 300, and each milliliter is 10^-6 cubic meters.1584

We know what the density of the rock, so since density is equal to the mass divided by the volume.1592

All we have to do is just toss that volume to the density and we’ve got that’s equal to the mass.1598

We replace those and we get volume of 300 times 10^-6 cubic meters times the density, which is 2.5 times 10^3 kilograms per cubic meter will be our mass.1606

We put those two together, we multiply them and we get 0.75 kilograms is the mass of our rock.1624

Now one thing to point out, if it’s .75 kilograms…I just like to point out that that makes perfect sense because 300 milliliters, well we all have reasonable idea of what…actually…1634

Here’s something I believe is 1 liter, 600 milliliters, so if this is…if that is going to be 600 milliliters then about 300 milliliters is about the size of my fist, maybe a little bit smaller.1644

Little bit smaller than my fist, if we had a rock about that big, .75 kilograms, that seems about right.1658

That’d be about the right mass for that rock so it makes sense.1664

We can think about the idea and it works out. That’s one of the great things about physics, is for the most part you can use some idea of your intuition and you’ll actually be able to figure out how things work because we’ve been living in this world for a long time.1667

We’ve been exposed to a lot of classical physics by now.1678

Example 2. We’ve got tank containing 5 degrees centigrade water and the important part about 5 degrees Celsius water is that that means we know what the density is.1683

Density of water changes slightly as we go through different temperatures so that’s why we have to go and get what that density is.1692

The density of water for this is 1 x 10^3 kilograms per cubic meter.1699

That’s our density and we’ve got a large squat main reservoir, so tall thin chimney connected together and one quick point, while I was talking that volume is equal…volume can be measured in liters, volume can be measured in cubic meters.1704

We’ve talked about density and so density could in theory be kilograms per liter.1719

But our equation for pressure is based off depth, is based off of meters.1725

If we use something other than cubic meters there our equation won’t work because we would have to derive it a different way.1731

Since we derive ours based on the idea that were working with meters, whenever we have a density it’s going to have to be dealing with cubic meters.1739

Otherwise our results won’t come out right. Just something to keep in mind if you come across a different one and want to find out what the pressure is at a depth.1746

If our density for our liquid is this and we’ve got this picture where we’ve got a large squat main reservoir and then we’ve got this chimney that’s connected to it and goes really, really tall this sort of bizarre thing will happen.1753

Where the pressure that we get might be way higher than what we’d expect because the pressure isn’t just going to be the fact that it’s a no height here.1767

What’s the tallest height that the liquid in this container achieve?1777

The tallest height the liquid in this container achieves is way up here, it’s at 20 meters.1780

The height, the depth that it goes, its height is actually going 18 meters. It’s going to be this giant thing.1783

That thin chimney, all of the weight, all of the pressure of the water coming down this actually manages to change the pressure of the whole reservoir.1793

If we were to change just the chimney we’d be able to massively change the pressure.1802

This amazing thing about the way pressure works, it doesn’t violate the law of conservation of energy or any of the laws that we’ve learned so far.1807

It is this really surprising, slightly unintuitive thing. You want to keep in mind it’s just about the depth of highest point our liquid achieves.1813

The highest point our liquid achieves, 18 meters higher, so that’s our h.1822

We’ve got our density, we know what gravity, boom we’re ready to figure out what the pressure is.1826

In this case, do we need to know what air pressure is? No, we’re just looking for the gauge pressure, not the absolute pressure.1830

If all we’re looking for is gauge pressure, gauge pressure is equal to the density times the height times gravity.1835

Our density is 10^3 kilograms per cubic meter. Our height is 18 meters and our gravity is 9.8 meters per second per second.1842

We toss all those in together, we multiply it through and we get that we’ve got…let me check real quick.1853

We’ve got 176,400 Pascal. So 176,400 Newton per square meter.1862

Way more pressure, a good chunk more pressure than we get…actually way, way more pressure than we get at air pressure because we’ve got 20 meters of water above us.1876

For those of us who use the English American system, that’s going to be well more than 60 feet of water above you.1886

If you’re drove down to the bottom of the deep end of a pool, you know what that pressure is like.1893

Imagine multiplying that by four or five or six times that depth.1897

Its going to be way, way more pressure on you.1901

176,400 Pascal, that’s a lot of pressure.1904

We have a 10 centimeter cube of oak and we’ve got the density of that oak and it’s held at rest completely submerged under 40 degree centigrade water.1910

Now that one is going to have a slightly different density, it’s going to have 9.92 x 10 squared.1918

Just a little bit less than the other one because the other one was 10^3.1923

If we were to release that wood, what would be the acceleration on the wood when we released it?1928

10 centimeter cube of oak, we hold it at rest, we release it under submerged, under that water.1934

We know the density, so what are we going to have?1939

Well first off let’s figure out what is…what’s the force of gravity going to be on this oak?1942

Well once again, 10 centimeter cube volume of the oak is going to be equal to 10 centimeter cubes, so .1 meters cubed.1947

We have to multiply it on each of the cube so we’ve got 10^-3 cubic meters is the volume of that oak.1958

10^-3 cubic meters, so at this point we can find out what’s the mass.1965

The mass of the oak is going to be equal to the density times the volume.1970

So 7.5 times 10 squared times 10^-3. We put those together and we get...hey look, it’s the exact same mass as our rock was earlier, .75 kilograms.1977

Then if we want to figure out what the buoyant force is on this we need to figure out what’s the weight of the water that it’s displacing.1995

We know the buoyant force is equal to the density of the water times the volume of the water that’s been displaced, so exact same amount, 10^-3 times gravity.2002

We substitute in everything, we get 9.92 x 10^2 x 10^-3 x gravity.2018

We put those altogether and we’re going to get…oh shoot, I didn’t actually calculate this number.2030

But we know that fb is going to be fighting mg, so we have fb minus mg, m of the oak times gravity is equal to mass times acceleration.2035

Some of the forces is going to mass times acceleration. So fb minus m knot g equals mass times acceleration.2051

At this point and this is also mass of the oak, that’s the thing we’re curious about the acceleration of.2057

The buoyant force, 9.92 times 10 squared times 10^-3 times 9.8 minus what’s the mass of…what’s the force of gravity going to be.2063

Well the mass of our oak is 0.75 times the force of gravity, also at 9.8 is going to equal the mass of our oak times acceleration.2078

We calculate with this whole number gives us and we get 2.37 is equal to the mass of the oak times the acceleration it has, we divide out by the mass of that oak, which was .75 and we get 3.16 meters per second per second equals our acceleration.2087

First thing to do, calculate what density you’ve got…sorry, not calculate with the density.2116

Use the density you have or you might have to find out what the density is first, but use that density to find out what the mass of our object is.2119

Find out what the volume of the object is, find out what the weight of the water displaces is, the buoyant force and then you just do a normal sum of forces equals mass times acceleration.2128

The weight of gravity on the object versus the buoyant force is acting on it.2136

If the density of our object is greater than the density of our fluid, it’s going to sink.2142

If the density of our object is less than the density of our fluid, it’s going to float.2147

That’s exactly why helium balloons, because they’ve got such a lower density, look on a periodic table, look at how low helium’s mass is, atomic mass is versus the atomic mass of most of the elements that make up our atmosphere like say nitrogen or oxygen.2153

Those are going to be way more massive so helium is going to have less density for the same pressure and it’s going to float into the air.2169

A rock is going to have more density than water and so it’s going to sink.2176

That’s why all this stuff happens.2180

Example 4. Ambient air pressure at sea level is generally about 1 x 10^5 Pascal.2183

If we have a cylinder with a radius of r equal to .035 meters and height equal to .17 meters, so that’s just about the size of a smallish soda bottle.2190

And we manage to create a pure vacuum inside of it and real quick note, actually impossible to create a pure vacuum, at least as far as science has figured out to do so far.2203

Laboratory vacuums have never managed to be perfectly pure, you can get…it gets harder and harder to suck out the last thing.2213

Just imagine if you were trying to suck all the dirt out of a carpet, it’s easy at first but those very, very last few grains get really difficult because it get kind of hard to catch that last few because they’ve got so much…2220

It’s harder to get at those very last few because there is a difference in the pressure that you’re trying pull at it with.2232

Not exactly a perfect metaphor with the dirty carpet but hopefully you understood.2240

So if we want to figure out what the force of the air pushing on this bottle will be once there is no air inside.2244

Normally we’ve got air pressure, we open a bottle, we’ve got air pressure inside of the bottle, we’ve got air pressure outside of the bottle.2252

There is no difference because they’re both being pushed on the exact same amount of air pressure, so we’ll see no deformation2258

We’ve got static equilibrium, same amount of pressure on one side as the other side so no change is going to happen.2263

If we’ve got the forces canceled out, we’ve got the pressures cancelled out, nothing happens.2270

If we managed to make it a perfect vacuum, suddenly there is going to be all that force of air pressure pushing down on it and there will be nothing to resist it with.2274

We’re going to actually see some really big changes.2280

First we have to figure out how much area does the air pressure have to push with.2283

We need to figure out what’s the surface area of that cylinder. The surface area, what are the two ends of our cylinder, each end of our cylinder is pi r squared.2288

How many n’s do we have? Well we’ve got two ns. So 2 times pi r squared plus what’s the area of the outside of the cylinder.2298

Well the length of outside of the cylinder, a cross section length is just the circumference of a circle, 2 pi r, or the diameter times pi.2308

Then if we want to figure out what the total area is, we slide that down and it slides down by height and so the swept area is going to be that circumference times the height that it sweeps to, 2 pi r times height.2318

We substitute everything in, we’ve got 2 times pi times 0.035 squared plus 2 times pi times 0.035 times the height, 0.17.2331

We toss that all in together, we put it into a calculator and we get that the total surface area our bottle has exposed or our cylinder has exposed to the air is .0451 square meters.2349

If we want to figure out what the force pushing on that was, we want to look at, what’s the pressure?2361

Well we know pressure is equal to force over area. We know what the pressure is here, we know what the area is.2367

We just toss those two together and we have that the area times the pressure is equal to the force.2372

We plug in the area, we know our area is 0.0451 meters squared times the pressure, which is 10^5 Pascal’s.2378

Multiply those two together and that’s equal to 4,510 Newton’s.2388

Which is a whole lot of force. I mean imagine how much force that is.2393

That’s enough force to lift about 460 kilograms. If you can lift about 460 kilograms, that’s enough to be able to pick a motorcycle up off the ground and lift it over your head.2398

If you have enough strength to do that, if that’s the amount of push that you’re putting on this cylinder, that you’re pushing on a soda bottle, it’s just going to absolutely crush.2411

There is huge amounts of pressure and that’s why when we take a soda bottle and we go under water with it and we look at it, it gets deformed by even just 1 meter of water.2420

It gets reasonable deformed because it crushes because there is massive amounts of pressure in there.2428

That is with air pressure already inside the bottle. If you’ve ever put a soda bottle to your mouth and sucked the air out, you use the…you’ve been able to use muscle expansion in your chest to change the pressure differential in your lungs so that some of the air in the bottle comes in to your lungs.2432

You’ve seen it squeeze down just a little bit, you changed it by like 10-20% of the internal pressure, probably way, way less actually now that I think about it.2447

You’re changing the internal amount of pressure by very, very little.2454

That change, that small change is able to cause massive crushing.2459

Imagine if you were to have a perfect vacuum, which would just smash that bottle.2463

This also is the reason why straws work. If you’ve ever taken a straw and put into a simplified drawing of a liquid.2467

If you managed to suck out some of the air pressure in here then all the air pressure of the water…sorry all of the air pressure of the air is going to push on our liquid and it’s going to push that drink up our straw.2477

However, if you’re tried…if you want to prove that its air pressure and not suction to cause it come up.2491

Which a lot people think at first that it’s suction in the straw.2497

Take a straw, suck up some amount of liquid into the straw.2500

Now take the bottom and pinch it off. So take a straw and then fold it up and hold it pinched.2505

So you’ve still got some liquid inside of it and now try to suck out of that straw.2510

If you try to suck out of that straw, you’re going to notice that the water won’t come up to your lips because you don’t have enough strength in your lungs to be able to crush that whole thing.2513

You’d have to put a huge amount of anti-pressure there. You’d have to create pretty much a vacuum inside of that.2521

You can’t create enough vacuum with just your lungs, that’s just not by a long shoot, enough mechanical power to beat the pressure of the air.2528

You’re going to have no way of going to pull that straw...pull that liquid from that straw into your mouth and so what we know, the reason why a straw is because we’ve got all this air pressure around us.2539

We’re lowering the air pressure inside of the straw and so there is now this differential in pressure and the water, the liquid, whatever our drink is, gets pushed through the straw because air pressure is pushing down on the rest of the drink.2548

That’s why it works. Pressure is absolutely an amazing thing, huge amounts of pressure, it’s a part of our daily life and we don’t even really notice it because we’ve grown up with it all of our life.2561

Hope you’ve learned some cool stuff. We’ll see you again on educator next time.2566

Hi, welcome back to educator.com. Today we’re going to be talking about temperature and heat.0000

All atoms and molecules have some vibrational motion in them. They’re shaking around just a slight amount. Even solid objects still have some of this motion.0005

Well we can’t see this motion without eyes, it is happening on an atomic level. This vibration has a huge impact on how substances interact with one another and how they behave on their own.0015

What do we call this slight shaking motion that’s so integral in the very nature that chemistry behaves?0024

We call it temperature. This is a strange thing at first but it’s that motion, that slight shaking motion that is heat.0032

That is heat actually being something that’s slightly different that we’re about to talk about.0040

That is what is warm, that is why something feels hot or something feels cold is how much of this shaking that is.0044

We call the average of the motion is in a substance, the average of this microscopic atomic level motion in a substance, temperature.0051

You denote it with the t. Note, this is the average of many different microscopic, super microscopic, not the sort of microscopic that you can see with a microscope, but something on the super micro level.0058

This is the average of many different tiny, tiny motions. On a macro level it seems like the substance is one unified temperature.0070

On a micro level one molecule might be moving slightly faster or slower than the next.0079

So on a micro level each one is moving slightly different than all of its neighbors, but when we look at the giant scale, we’re seeing so many things happening at once.0083

We just take the average because that’s what it seems like to us because they’re so many tiny particles in there each doing slightly, slightly different behavior.0091

From our point of view we can’t notice the tiny particles. It’d be talking about a beach but trying to talk about every single grain of sand.0099

We just sort of notice it as sand under our feet supporting us on the whole.0106

With this idea of temperature being based on super microscopic motion, we can see that there has to be a lower bound to that temperature.0111

When molecules completely stop moving we can’t get below the point where they aren’t moving.0118

There’s no stiller thing than just being motionless. So once their motionless we’ve gone down as far as temperature can go.0124

We call this lower limit absolute zero and it’s going to form the base of our temperature scale.0131

The base of our temperature scale will be the lowest that we can get to, the lowest amount of motion we can have as nothing as so zero is nothing.0136

What scale are we going to use? For length we use meters. For mass we use kilograms. What do we use for temperature?0145

So far we’ve talked a lot in terms of centigrade. We’ve talked about change, centigrade is actually Celsius, we’ve talked about a lot of things in terms of Celsius and what is that?0153

Well kelvin is actually what we’re going to switch over to. We’re going to have a starting point for temperature and now we can introduce temperature measurement and we’re going to use kelvin.0163

So kelvin is the exact same size, the distance on a kelvin from 0 to 1 and 1 to 2. It’s the same thing.0172

One change in a kelvin degree is the same thing as one change in a Celsius degree.0181

There are still 100 kelvin degrees to get from the frozen point of water to the boiling point of water; I mean the end of the frozen point to the beginning of the boiling point.0185

It’s not going to start at the same place so Celsius has its zero set at frozen water, a reasonable thing when you’re living in normal daily life.0195

When you want to do laboratory experiments, you’re going to want to have it set down so your kelvin is set at zero, is set at the very base of where temperature can go.0204

It’s going to matter a lot when we talk about certain other things later on.0213

The Kelvin scale is 0k is absolute zero. 0 degrees Celsius on the other hand is a freezing point of water at 1 atmosphere of pressure.0216

To convert between the two we have the temperature in Celsius is equal to the temperature in kelvin minus 273.15.0225

If you want to go from kelvin to Celsius, you just add 273.15 and you’ll get what you’re number…sorry, if you want to go from kelvin to Celsius, you subtract.0232

If you want to go from Celsius to kelvin then you’re to add 273.15 because you need to get from the fact that your 0 of frozen water is actually 273.15 above where the stopping of motion is or the end to temperature is.0240

I think it’s kind of the end. Heat versus temperature. In everyday life we often talk about heat and temperature as totally interchangeable ideas.0260

I accidentally slipped up and did it at the very beginning of this lesson in fact.0269

In physics we make a distinction, heat we denote it with a q. Heat is the transfer of thermal energy.0273

Heat is positive when we put thermal energy into a system and it’s negative when the environment takes it out.0280

So environment puts it in, you’ve got positive heat, positive q, positive thermal energy.0287

Its negative when the environment takes it out. So if the system gets cooler, it’s going to be negative heat, it’s going to be a heat flow out of it. It’s going to be negative thermal energy, negative q.0293

Notice the similarity to work in energy. When we were dealing with energy, energy was the fixed quantity that moved up and down.0304

Here temperature is the fix, once again these things aren’t really fixed, they clearly move around a lot.0311

But temperature was the thing that moved up and down based on how much work we put in.0315

Positive work when we put energy into the system, negative work meant we took energy out of the system.0318

It’s the same thing here, positive heat, positive q means that we put thermal energy into the system. Negative q means we take energy out of the system.0324

It’s the exact same thing. If we want to heat stuff, how much heat do we have to put into it?0331

If we want to heat water, if we want to say raise the temperature on a pot of water, how much heat do we have to put in to it?0337

Clearly from experience the more water in the pot the more heat we need to raise. If you have a small pot of water it’s going to boil way faster than if you have a giant, giant pot of water.0343

We’re used to this at this point. Clearly the amount of the object, the mass of the object is going to have some effect on it.0352

There’s also some other stuff coming into it. For water, one way to measure that heat is the calorie, which is shortened to Cal when we’re putting it in as a unit.0359

1 calorie is the amount of heat required to raise 1 gram, and notice that’s gram, not kilogram.0369

1 gram of room temperature water by 1 kelvin or 1 degree Celsius.0377

If you’re at room temperature and you want to increase the temperature by 1 degree Celsius or kelvin, not Fahrenheit.0382

If you want to increase it by 1 degree, you put in 1 calorie. If you have 1 gram, 1 degree gets 1 calorie.0390

Thermal energy is just energy though, so we can also use the joule, the conversion between calories and joules is 4.1868 joules to the calorie.0397

That’s a defined thing because the calorie, the amount of energy, the specific amount of energy that you’re going to have to put in to get a temperature raise of 1 degree actually varies a little bit as the temperature of the water goes up.0408

For our purposes, we’ll be perfect fine to call it one calorie, but notice there is a very slight change as we move around. As we get farther and farther away from room temperature.0418

One calorie won’t be quite enough to make the exact same change. If we want to be really precise scientist is able to use joules instead because calorie is defined to make 4.1868.0427

Having it mean 1 calorie having…for 1 calorie to heat 1 gram, 1 degree isn’t going to always be the case.0441

It’s actually more correct to base it just on ideas of energy and then we’ll have to do lots more specific measurements if we want to be really careful about this.0450

For our propose we can treat it as always being 1 gram, 1 degree. Specific heat, the amount of heat needed to raise one objects temperature is going to be different than the amount of heat needed to raise another objects temperature.0459

Water is different than steel is different than wood is different than granite is different than rubber.0473

Each one of these is going to need different amount of heat. They’re going to have a different heat and that’s based on the chemical composition and the really deep molecular atomic structure of these things.0478

That’s something we defiantly won’t be getting into. Just for our proposes, it’s enough to know that there’s going to each one’s going to need a different coefficient.0487

We define this coefficient as specific heat, c. That will give us some proportion; this proportion is going to vary based on the substance.0495

We’ll have to get it for each problem or look it up in a table if we want to find out what it is.0503

We are able to look these things up and then figure out how much heat energy we’d have to put in to an object.0508

The amount of heat needed for a given mass, m, to have a temperature change, delta t, is given by the equation q, the heat we put in, is equal c, that’s specific heat, times the mass of the object times the change in temperature.0512

This makes a lot of sense. Each object’s going to have a different type, c, different amount proportion for their heat that they need.0526

Each thing is going to care how much mass it is. A small pot of water takes a different amount of time than a large pot of water.0534

There’s also going to be a big difference if you want to raise the temperature in that water just a little bit or if you want to get it all the way from frozen ice to boiling water.0540

Totally different numbers are going to be needed in each one of these and so we take account with that with c for the proportion, m for the mass, the amount of the thing and change in t, the delta t for just how much we want to make a difference in the temperature.0548

Heat can be transferred through one of three of methods.0561

The first one is conduction. Direct contact, motion in the atoms is directly passed to adjacent atoms. If I heat one end of this pencil, pen, I’m not really sure what to call it since it writes on a tablet, but if I were to heat one end of this, over time that would heat would slowly make it way through the object, all the way through.0565

Some things are going to conduct heat at different rates. You’ve probably see this before if you use a wooden spoon to stir a pot, the heat gets transferred to the end of that spoon way slower than if you use a metal spoon.0583

They’ve got different rates of conduction. One again that’s going to be based on the chemistry involved. A really good example would be if we were to put an empty pot to heat on a stove, so if we just put an empty pot and heat it on the stove.0595

This isn’t a good idea to do at home because it probably won’t hurt you but it is going to possibly ruin a nice pot. If you were to heat an empty pot on the stove though the temperature, either the hot coils or the hot gas flame would heat the bottom of the pot and that heat would spread through the metal.0606

It would be spread directly through conduction. The next one is convection. This is fluid motion doing the work. A combination of fluid motion happening and conduction.0624

If a fluid manages to have a hot pocket and then that hot pocket gets spread through, it’ll manage to conduct way faster than if it’s trying to conduct layer through layer through layer.0636

It’s a combination of direct conduction and the fluids mixing due to pressure differentials from temperature variation.0645

Hotter water has a slightly different pressure than colder water so that hot water is going to rise which means that its’ going to spread out through the colder layers that are now above it and it’s going to spread it and it we’re going to get convention currents.0651

If fire at the bottom of a chimney, it’s going to heat the air directly above it. That hot air will rise and will now easily touch the air at the top of the chimney making it hotter.0664

Its not direct conduction, it’s not having to make it through each layer of air atoms. It’s that it’s heating the hot air and then that hot air has a different amount of pressure in it.0674

So that gust of hot air, that packet of hot air will move up the chimney and it will manage to mix with the colder air at the top and so it will heat that hot air more easily.0683

Finally, radiation, electromagnetic waves. Hot objects emit EM waves that can be received by other objects increasing their internal thermal energy.0695

Great example of this would be to go out on a sunny day. That sun, the reason that you can feel any heat from that sun is because of electromagnetic waves.0704

That sun doesn’t have anything to conduct through. It doesn’t have convention. It’s got the dead vacuum of space for a long distance between us and it.0711

The only way that heat manages to make it to us is because it’s able to directly shot it at us.0719

It uses light to shot it at us. Infrared is one of the main carries of heat energy, as one of the first thing that a hot object starts to emit.0723

Its also going to emit it through a variety of spectrums. If you’ve ever heard of something being white hot, that’s because it’s literally been heated to the point where it manages to emit white light.0731

When something is red hot it’s at less energy because it’s only emitting a lower wave length. It’s emitting red which is less than the entire spectrum.0740

White light being the entire spectrum of light seen a once. When we manage to make something really, really hot temperature, if you’ve ever seen a glowing piece of steel either in a movie or in real life that’s because it’s so hot it actually managing to broadcast light to us.0748

That’s what happening to the flame too. That thing is so hot it’s managing to broadcast light to us.0766

It’s not able to take anyone of these on its own. That fire at the bottom of chimney, it’s able to heat some of those bricks at the top of the chimney by directly shooting electromagnetic waves at them.0773

Which is then also going to be able to conduct to the air next to it. Anyone of these, they’re going to come together.0785

You can’t just say there’s just one at a time because they’re all working together. They’re all working in concert.0790

Each one works slightly differently and it’s an interesting to have an understanding of how heat is moving around.0795

We’re ready for some examples. If we had a block of steel that had a temperature of 540 kelvin what would that be in Celsius?0802

Remember, the temperature in kelvin minus 273.15 is equal to the temperature in Celsius.0808

Interesting point about kelvin is when we talk about kelvin we don’t say degrees kelvin. We just say number kelvin.0819

So 540 kelvin. When we talk about Celsius though we say degrees Celsius.0824

It’s just a thing that we do, it’s the way we describe it.0830

If a block of steel is the temperature of 540 kelvin, how do we convert that?0834

Well we’re going to need to move that down because kelvin has a higher number for this same temperature.0838

We move that down 540 subtract by 273.15, we get 266.85 degrees Celsius.0844

If on the other hand we’re going from -15 degrees Celsius, if we had a cloud of air at -15 Celsius and we needed to know what that was in kelvin.0854

We would add that same amount 273.15. So we add 273.15 and we get 258.15, no degree mark, just kelvin directly.0862

If we’re got something that’s in kelvin and we need to convert it to Celsius the number we get lower because kelvin has a lower starting base.0875

If we have something that’s in Celsius and we convert it to kelvin the number will get higher because Celsius has a higher starting base.0885

The number that we convert with is 273.15.0891

A calorie in food, notice the capital C. This is an interesting point, a calorie in food, if you look at the back of a box.0896

If you’re in the United States, I’m not quite sure about some of Europe. But if you’re in the United States and you see a calorie of food you’re going to see that it’s not actually the same thing as the calorie we were talking about.0904

We see calories on the back. Other countries though will actually stop kilocalories because what a calorie is with a capital C is it’s actually a kilocalorie.0917

It’s 1,000 of those calories that heat water that we were talking about before.0927

Some countries in fact, they don’t need calories because they could also just talk about in straight energy.0931

Other countries will use joules on the back. We’ll talk about the kilojoules that the food has that you’d be looking at.0934

It’s an interesting point of view. It’s an interesting point; we couldn’t look into this as anything as long as we’re looking in terms of the energy that one of these things can impart.0940

If we’ve got calories, as kilocalories, and a person burns 2,000 kilocalories in a day. What’s their average power output going to be?0948

Remember we had 1 calorie is equal to 4.1868 joules. SO what would one kilocalorie be?0957

It would be 1,000 times that. If we want to see the energy of the kilocalorie is then we’ve got 2,000, sorry energy of 2,000 kilocalories.0968

That’s what we want to look at, the 2,000 kilocalories from here. So energy of 2,000 kilocalories is going to be 2,000, the number of kilocalories times 10^3 because we’ve got a kilo, we’ve got a 1,000 of them, calories.0982

The conversion between calories, so at this point we’ve got 2,000 times 10^3 plain calories.0994

If we want to convert it to joules we multiply by another 4.1868 and this will tell us what the number of joules is in 2,000 kilocalories.0998

We get, multiply it out, and we get 8.374 x 10^6 joules.1007

The person goes through 8.374 x 10^6 joules in a day. If we want to know what the power is, we need to figure out how many joules you go through per second.1018

So what would be the average output over the day? Well it’s going to be power is equal to the change of energy, the amount of energy we use, this number right here.1026

Divided by the amount of time. Well the change in energy, we’ve already figured that out, we use up 8.374 x 10^6, at least for this average person using 2,000 kilocalories a day.1036

Not necessarily average the amount that you need, depends on your personal lifestyle, your personal metabolism, and how much work you had to do.1049

On a cold day, a linebacker, a heavy athletic person. On a cold day practicing can use something like 6,000 calories in a day.1058

If you’re living...if you’re a very small person, living in a warm climate, you might only need 1,500.1066

It depends on your personal life. 8.374 x 10^6 joules, we want to divide that by, well how many hours are in a day?1070

24 hours in a day. How many minutes in an hour? 60 minutes an hour. How many seconds in a minute? 60 seconds.1078

We put that all the together, we divide it out and we get 96.92 watts.1084

That’s what the person average power output is. Keep that in mind, think about that.1092

A watt, a bulb where used to using a 100 watt bulb to light our house. If you have incandescent bulb at 100 watts, that things actually pretty warm to the touch.1097

It can burn you. That’s how much heat your body is putting out pretty much. Your body is going to be putting out almost certainly at least 96.92 watts at any time.1106

That’s why a crowd of people, if you’ve ever noticed that it’s a cold day but you’re standing with a crowd of people its noticeable warmer.1116

Or if you’re in a closed classroom for a long time. It starts to get noticeably hot. Part of that is just because of the raw amount of heat that a bunch of bodies sitting around make.1123

A bunch of live people put out a lot of heat. That’s what you’re seeing, you’re actually seeing waste heat of a bunch of people just standing around.1133

Granite. Granite has a specific heat of .79 kilojoules; notice this is kilojoules, not joules.1142

If you have a 2.5 kilogram block of granite currently at temperature 280 kelvin and you want to raise the temperature by 20 kelvin, how much thermal energy needs to be put in?1149

First off, I’d like to point out; do we need to know what the starting and ending was if we know what the proportion is?1158

If we know what the specific heat is for the level we’re starting at, we don’t actually need to know the temperature.1164

We know we’ve got 280 kelvin going to 20 kelvin, so we end at 300 kelvin, but the important thing is the change in t is equal to 20.1169

That’s all that we really care about. If change in t equals 20, c is equal to .79 kilojoules per kilogram times kelvin and 2.5 kilograms is the amount of our mass, then we have q is equal to the specific heat times the mass times the change in the temperature.1177

We plug everything in, ah .79 kilojoules, so if we want to do this in joules, what we’re used to doing.1193

We’ll want to change it to kilojoules with .79 x 10^3. So it’s joules per kilogram per kelvin.1199

We multiply by the mass, 2.5 kilograms times the change we want to effect. So we want to change 20, we put it all together and we get 39,500 joules.1208

If we were curious how many kilojoules that would be, we just divide by 10^3, move that decimal over by three, 39.5 kilojoules.1221

Either way you want to look at it, same thing.1230

You use the specific heat that you’ve got, the mass that you’re working with and the temperature change. Put them all together and that tells you how much heat needs to be put in to get the change.1234

Finally, we have a perfectly insulated, this means the environment won’t effect it so we don’t have to worry about it radiating heat or heat being put in from the environment.1246

We know that we only have to care about what’s happening inside. We’ve got a perfectly insulated container that is 4 liters of water at 10 degrees centigrade.1255

You place a 3 kilogram brick of iron that is currently at 400 degrees centigrade, whoops typo right here. This should be 400 degrees centigrade, into the water.1262

Of the specific heat of iron is .47 kilojoules per kilograms times kelvin. What temperature will the water/iron brick mixture be when it comes to equilibrium?1272

First thing, do we have what the mass of water is? We don’t have it yet actually but one more thing that we haven’t mentioned before.1283

What you might have learned in pervious science classes, is that 1 milliliter of water, so 1 milliliter of water is the same thing at most pressures and most temperatures.1291

1 milliliter of water is the same thing as 1 cubic centimeter of water, which is the same thing as 1 gram of water.1301

Thus, 1 liter has got to be 1 kilogram since there is 1,000 milliliters in a liter and there is 1,000 grams in a kilogram. Then we know that 1 liter of water at normal temperature and pressure is what we’re dealing with in this problem.1315

We know that we’ve would have 4 kilograms of water for this problem. So the mass of our water is equal to 4 kilograms.1328

We know what the mass of the iron is. Do we know what the coefficient for the iron is? We know what the coefficient for the iron is.1339

Do we know what the coefficient for the water is? Well once again, we know that the coefficient for water is in general 1 calorie per gram per kelvin.1344

If we want to switch from calories to joules, we have 4.1868 joules per gram per kelvin and notice we couldn’t do this if we were in calories because we’ve got a different thing over here.1357

We’ve got a different thing for our iron, so we’d have to convert into calories there.1371

We have to make sure that we’re dealing with this same thing for both of them.1375

If we want to convert from joules per gram per kelvin into kilojoules per kilogram, well 1,000 but also divided by 1,000, so it’s going to wind up being the exact same thing since its kilo both on the top and the bottom.1378

Kilojoules per kilogram times per kelvin. At this point, we’ve got what the coefficient for water is.1391

We know what its specific heat is. We know what the specific heat of iron is.1400

One more idea, if we put iron into water it’s going to conduct its heat both through, we’re going to have conduct, we’re going to have convection, and we’re going to have electromagnetic waves bouncing around inside of that container.1405

What will be the final heat? Sorry, what will be the final temperature, not the final heat?1421

What will be the final temperature for that container? Are they going to have to agree on a temperature at the end? Yeah, when they hit equilibrium, there’s going to have to be at the same temperature otherwise they‘re going to continue transmit heat.1426

Only when something is at the same thing are they not transmitting heat back and forth.1439

What is…what’s that going to be? We know that the amount of energy we put into the water, the amount of thermal energy that gets put into the water because it’s going to have its heat, it’s going to have to take positive heat in because it’s having its temperature raised.1445

It’s going to be the amount of energy that the iron loses. If the iron, it’s going to have a negative amount because it’s putting energy out of itself. Energy is coming out of it so it’s going to have a negative heat.1459

That means that we need to change that to a positive for it to be the same thing as the water.1472

The amount…the total absolute amount of energy that the iron loses is the amount of energy that the water gains because we’ve got this perfect insulator around it.1478

Negative heat becomes the positive heat of the water. This relationship right here is how we’ll be able to solve this problem.1488

At this point we’ve finally ready to work on it. Here’s all of the things we need.1496

The mass of water is equal to 4 kilograms. The mass of the iron is equal to 3 kilograms. The temperature of the water is 10 degrees centigrade. The temperature of the iron is 400 degrees centigrade.1500

The coefficient for the water is 4.1868 kilojoules per kilogram per kelvin. The specific heat for the iron is .47 kilojoules per kilogram per kelvin.1510

What is our ending temperature? We need to know that the amount of energy that goes in for a specific heat for the amount of heat that has to go in for a temperature change is going to be the specific heat times the mass times the change in that temperature.1520

One quick question, do we have to convert from Celsius into kelvin before we can do this problem?1539

In this case we don’t actually have to do it because we’re only looking at the change, the difference between those two numbers are going to be the same difference whether we’re looking at this in kelvin or if we’re looking at this in Celsius.1545

The difference between a Celsius, 1 Celsius and 1 kelvin is the same distance temperature wise.1555

If we’re looking at the change in temperature, it’s the same thing if we do it as 400 Celsius or 673.15 kelvin.1562

Ultimately when we look at the difference it’s going to be the same difference whichever way we measure this to begin with.1571

We can actually continue to work in Celsius. We know that the change in the heat for the water is equal to the opposite of the heat for the iron because the iron heat is negative.1579

We set this two equal to one another and we’ve got that c of the water times the mass of the water times the temperature difference.1590

We want to actually use the temperature difference because we want end to show up minus the temperature of the water is equal to negative specific heat of the iron times the mass of the iron times the ending temperature.1600

Its going to be the same ending temperature on both sides minus the starting temperature for the iron.1613

Let’s start moving things around, make this a little bit easier. We’ve got –cimi x tn so we can multiply that in.1618

We distribute the right side and add it over here. We’ve got cwmw t end plus because it’s negative on the right side, plus cimi t end.1624

Then we’ll move everything that’s not t end based because we’re solving for t end to the right side.1643

We’ve got cwmw x –tw, so we add that over here, we have cwmw tw and then over here that –cimi hits the negative t end, so we get positive.1648

So we’ve got positives on both sides, cimi ti. Let’s actually take a quick pause and look at this for a moment.1662

Notice what we’ve got here. We’ve got cwmw t end plus cimi t end, so this the total inertia to speak.1669

The thermal inertia for the system as a whole, the iron and water together at the end is going to be the temperature times its specific heat times the amount of mass.1679

On the right side, we’re going to have…it’s going to become a…we’ve got what the total thermal inertia is here plus the thermal inertia here.1689

We know that the total amount of heat energy, sorry the total amount of thermal energy for the iron plus the water is going to have to be the same no matter what because we aren’t losing any thermal energy because we’ve got that perfect insulation.1697

The total amount of thermal energy at the beginning is going to be equal to the total amount of thermal energy at the end.1712

We start solving for t end. Pull out the t end, we have cwmw plus cimi equals all that same stuff over.1718

We divide by cwmw plus cimi, whoops I accidently got confused again, a lot of letters here.1734

Cwmw tw plus cimi ti all over cm, ah keep doing that. Cwmw plus cimi.1747

At this point I’d also like to point out what we’ve basically got, is we’ve got a weighted average. We’ve got a way of weighting how much the water temperatures matters because we’ve got cw times mw.1766

What sort of that thermal block is like, how much mass it has along with its specific heat, to see how hard it is to push that mass around temperature wise.1777

Along with its starting temperature plus cimi ti, what its thermal inertia is, times that starting place.1785

Then we divide it, we average it, we have to take that out because we want to just get temperature at the end.1793

If you remember back to finding the center of mass, there is a lot of similarities to center of mass here.1797

What we’re looking for in the end when we solve it out like this is we’re really looking for the weighted average between these things is.1802

We have to work this out with algebra because it’d be too easy to just go 400 plus 10 divided by 2 won’t be the correct answer.1808

Because we’ve got different amounts of water then we have iron. We’ve got difference specific heats for water and iron. We’ve got to work this thing out with math, but it is going to be a weighted average.1814

It’s how important the heat, how important the temperature in that water is versus how important the temperature in the iron is and what it becomes when we put them together.1823

We finally substitute in a huge mess of numbers, we have cw is .47, now once again, what are we working with?1833

We’re working with kilojoules. So we probably want to switch to joules because we know joules are the friendly SI unit. Just in case we’ll switch it over.1844

We’ll see in the end that wasn’t going to be too important but we’ll switch it over for now.1854

Specific heat of water, 4.1868 x 10^3 to deal with that kilojoules time the mass of the water, 4 kilograms times the starting temperature of the water, 10 degrees Celsius plus ci.1857

So .47, but once again we’ve got kilojoules so we multiply that by 10^3 times the mass of the iron, 3 kilograms times the starting temperature of the iron, 400 degrees Celsius.1873

Then we divide that by 4.1868 x 10^3 x 4 + .47 x 10^3 x 3. Now at this point notice we’ve got 10^3 showing up everywhere.1887

10^3 here, here, and here. SO every single one of our additive elements there had a 10^3 factor so we can cancel it out.1906

Ultimately in this case because we had this specific heat using the same unit on the top and the bottom and we had specific heat showing up everywhere we’re able to completely cancel out that unit.1915

So that all that matter was the coefficient in front of that unit. That’s a dangerous thing to do just like toss around so you don’t want to just use unit without understanding what you’re using.1924

It’s generally the safest thing to do is to use the SI unit, whatever it is.1935

Once again, that’s why we had to talk about why we were using Celsius instead of converting the kelvin first.1940

If we think about it, it might be the case that we can get away with using non perfect SI unit, but sometimes you’re going to completely ruin your problem by thinking you can.1944

And you won’t be able because it’s inherent in the way that the formulas we have, we have expectations.1952

We finally punch all this into a calculator and the number that we get out is 40.3 degrees Celsius.1958

40.3 degrees Celsius, notice how low that is. That iron block was almost the same mass of our water.1968

Almost the same mass as our water and it was 400 degrees Celsius when we put it into that water and yet we managed to raise the temperature of the water by 30.3 degrees.1976

Hardly anything. That’s less than a 10th of the temperature difference to the block of iron.1987

Why is that? Because the specific heat of iron is so much lower than the specific heat of the water.1993

More accurately, it’s because the specific heat of water is so freaking high. The specific heat of water is really, really, really high compared to most other things.1998

Most other things are at least below 2, if not below 1. The specific heat of water is really high number.2009

It’s actually one of the reasons that life can exist is because you can go out on a hot day and be basking in the sun and your body temperature won’t jack up because you’ve got all that water to give you this nice thermal inertia that will keep you from moving suddenly from one temperature to another.2014

It’s one of the great things that make the possibility for life here on Earth. The fact that we’ve got this wonderful stable water temperature because we’ve got this wonderful high specific heat for water.2028

Really interesting. Hope you enjoy this, hope you learned something great and we’ll meet again at educator.com later. Bye.2041

Hi. Welcome back to educator.com. Today we’re going to be talking about changes due to heat.0000

We’re already talked about how more thermal energy in a substance causes the molecules of the atoms in the substance to vibrate more and more.0007

As those molecules vibrate more against one another they’re going to be bouncing off each other more often.0014

They’re going to push against each other harder. So as they push harder against each other, they’re going to have more pressure in between them which is going to cause the substance to expand some amount.0020

If it’s a solid substance, we won’t see much expansion. If it’s a liquid substance, we’ll see of that internal pressure in the thermal energy will cause things to push out slightly, a little bit more push in our thing.0030

It’s going to grow a little bit. At very high levels of vibration, very high levels of thermal energy the material will actually change phases as the pressures inside the vibrations inside become so large that they’re able to break being in a solid substance and become able to just move on each other fluidly.0043

Like a in a liquid. Then eventually so much that they’re able to completely pull off of one another and turn into a gas.0059

First idea, linear expansion. When we heat or cool any substance we’re going to get some slight fluctuations in size.0068

The amount of that fluctuation is going to depend on the original length of the object, l. The specific properties of the substance.0074

Some substances will wind up expanding a lot more; some of them won’t expand very much.0080

The change in temperature delta t. Put together we get the equation delta l is equal to l times alpha times our change in temperature.0084

Even the ones that are large fluctuations, the large alphas are still pretty small.0093

All of these alphas are very small. It’s going to basically vary somewhere between 10^-6 and 10^-4 for most substances.0098

Even a really big alpha is still a huge amount of temperature going to have to be changed before we’re actually going to have any really noticeable difference that we’d be able to see with a naked eye.0105

For the most part we aren’t going to actually see this sort of thing. It will have effects on very large structures though, say if you were to build a bridge, you’d want to be able to have that bridge expand and contract without ripping itself apart or smashing into itself.0116

They wind up creating these slots like this so that the bridge can expand into the slots and pull back out without completely ripping itself apart.0129

If they were to build one long many hundreds of meters continuous bridge, when it got very hot it might expand so much that it actually causes itself to pull apart.0140

It’s really important to have those contractions and expansion points otherwise bridges wouldn’t be able to be built.0149

Linear expansion is great and that’s really interesting, but that’s not enough to describe lots of things out there.0156

If we wanted to talk about volume expansion, if we wanted to talk about how much liquid in a cup, what its volume changes to as it gets hotter, it will change.0162

Since its length is changing, but we don’t have length in a liquid. We can’t pour something into some funny shaped container, we can’t pour water into this funny shaped container and say ‘Oh look that has a length of…’0171

It doesn’t make sense. We can’t say…we can’t put a width and length and a depth and be able to easily come up with cubic volume.0184

Instead we just need to be able to talk about volume direct. So it’s very similar, it’s going to be that the change in v is equal to the original volume times beta, the new coefficient for that times the change in temperature.0191

Beta is related to our original alpha and beta equals 3xalpha.0203

Why is that the case? We’ll explore this actually in example two.0208

It’s going to turn out that beta isn’t precisely equal to 3alpha but it’s good enough that it’s going to work for all of our purposes and most purposes you’re going to possibly have.0212

Finally we’re ready to talk about gas expansion. Solids, liquids, they both…they push against one another they’re going to expand out just a little bit as they get higher and higher temperatures they’re going to push harder.0221

But a gas, it’s already pushing hard enough and bouncing off of itself. If you put it into a container, it fills out the whole container.0231

It’s not like liquid where it just fills up to the level you poured it in or solid where it just sits there.0237

A gas is going to expand to whatever container size it’s put inside.0242

Since gas is…are already completely filling the container, they’re not going to expand the way we talk about liquid and solids expanding.0246

Instead we’ll talk about how hard they’re pushing. How hard are they pushing against the walls of the containers and how hard they’re pushing against each other.0253

It’s their pressure. In addition we’ll also have to talk about the size of the container, v and the temperature of the gas, t, and how many molecules are in that gas.0261

If we had a box with one atom in it. One single atom in it bouncing around, it’s not going to have that many bounces.0270

If we give it a certain temperature and the same volume, it’s not going to have that many bounces because it’s just one atom.0278

If we were to do the same thing, same size box but now we put in 10 different atoms.0285

If it’s go the same temperature, meaning that each of the atoms is moving at the same rate as this one over here.0295

If we’ve got the same temperature between those two boxes, we’re going to get way more bounces going on in that second box.0301

That means that second box is hitting things more often. Means it’s pushing against stuff more often.0306

It keeps bouncing off repeatedly, that effectively turns into one continuous push when you’ve got millions upon millions upon millions upon millions of molecules doing all of these pushes one after another.0311

That’s what we’re going to feel as pressure. It’s going to really matter how many of these molecules, how many of these atoms do we have doing the bouncing.0322

That’s going to be a really important idea but we haven’t talked about a way to describe how many molecules, how many atoms we have.0329

That’s going to bring up an entirely new idea; it’s going to bring up the mol, which we shorten to m o l.0335

Now we need to talk about mols. Real quick, let’s say you worked in a nail factory and you had a crate of nails.0342

If you needed to know how many nails where in the crate you could count them but that would be a really, really slow process.0349

Say you have a big crate of nails, that’s going to take you hours if not days to be able to count through each of those nails by hand.0354

That’s not a very good way to figure out how many nails you’ve got in the crate.0362

But you might still need to know how many nails you have in the crate if you’re going to do some sort of job in construction or try to sell it somebody; you’re going to need to know how many nails are present.0365

So what you can do instead is you could measure the mass per quantity for a small amount.0372

You might be able to figure out ‘Oh, for every kilogram of nails I have, I’ve got 230 nails in that kilogram.’0379

You’d be able to come up with a conversion ratio of 230 nails per kilograms. Now if you’ve got that you can just dump the whole crate onto a scale.0386

You weigh all of those nails at once and boom; just with a little bit of math you’re able to figure out ‘Oh that’s how many nails I’ve got.’0394

Way, way easier than trying to sit there and do it by hand. Now imagine if you were trying to do something like count molecules.0401

There’s no way we could count it by hand. First off, there’s way too many molecules for us to ever have any hope of counting something by hand.0408

Two, I can’t pick up a molecule. Can you? I can pick up a bunch of molecules at once, but I certainly can’t just pick up one individually and put it over somewhere else.0413

We’ve got no hope of counting molecules but this idea of being able to change mass for quantity is going to work perfectly for us.0421

In general if you know how many objects are some unit mass, you can easily count a substance by using its weight.0429

You figure out what its weight is depending on what local gravity is. In our case we’re on Earth, you’re probably going to be using 9.8.0434

You figure out what its weight is, you then figure out what the mass is from that and using that mass you can boom…if you’ve got a conversion ratio you can easily find out how many of the thing you’ve got.0440

With that idea we can do the exact same thing with molecules. Since every atom or molecule already has an atomic or molecular mass.0449

Where molecular mass is just adding up all of the atomic masses for the number of one’s we’ve got.0459

Depending on what the molecule is. We can connect that to the masses we can measure.0463

We’ve got this idea of molecular mass. You figure out what he molecular mass for your molecule is or the atomic mass if it’s just a single atom.0468

Let little m be the molecular mass. So little m is the molecular mass, we can now also weigh the quantity we have.0476

Say we’ve got some pure amount of lead. We know it’s just lead so we go, we look up the atomic weight for lead and then weigh the quantity of lead we have.0483

We measure it in grams. It’s really important to notice it is grams that we’re using. It’s not kilograms.0493

We normally use kilograms for everything else but we get this idea from Chemistry and because we’re dealing with normally small quantities, we use grams.0499

Measure mols in grams, not in kilograms, in grams. We measure our quantity of lead and then we divide it by what that atomic mass is.0508

Whatever number we get in the end, we’ve got the number of mols.0517

n equals m over m is the number of mols. In each mol there are many, many, many molecules.0523

Specifically the number is 6.022 x 10^23 objects per mol.0530

If you’re doing it with molecules and you’ve got 1 mol of molecules. Then what you’ve got is 6.022 x 10^23 many molecules.0537

That’s a whole lot of molecules. This number is called Avogadro's number because he did pioneering work in figuring out quantities.0546

He wasn’t actually the person to figure precisely this number but he did a lot of work in figuring out how much of a substance is there and how it connects to other things.0554

Now we’re ready to get back to the idea of expanding a gas. We’re finally ready to relate gas, temperature, and the number of mols, and all these sorts of things.0563

We do this with the ideal gas law, so named because it models an ideal gas. There isn’t such a thing as an actual ideal gas.0570

It’s just a theoretical gas that we use to create a useful formula. It does allow us to really closely model real world gases.0578

It’s not absolutely perfect but for our purposes it’s going to be well within .1% or .01% of what we’re going to have if not even better.0587

It’s going to do great for our problems, but if we were trying to do something really, really careful we’d have to come up with a new way to model it.0593

We’d need more data, but for our purposes it’s really great and it’s a great way for being able to get a close estimation of what’s going to be going on.0601

Now we need to figure out how to read this. This is a lot of letters.0608

P is the pressure of the gas. Note that this absolute pressure, not gauge pressure. If we had air in a tire and we wanted to see what the pressure on that was, we wouldn’t just be able to take a measure on the tire using a gauge.0612

We’d have to figure out what’s the pressure in that tire but then also add what local air pressure is. We’d have to use the absolute pressure.0627

Next up v, is the volume of the gas. Whatever size container it is. Expanding on that tire idea, if would be the volume inside of that tire, how much space the gas is filling out.0634

N is the number of mols of the gas that we have to figure out by knowing what the mass of our quantity is. You’d have to figure out some way to measure the amount of gas you’ve got.0645

If you know how many mols you’ve got you’re ready to keep going. R is just a single number, it’s the gas constant and r is equal to 8.314 joules per mol x kelvin.0655

Interesting thing to note, that also means that the left and the right side wind up equaling joules in the end. Because we’ve got mols x kelvin, so n and t are going to cancel out the mol and kelvin.0666

So we’re only going to be left with joules, so that means that pressure times volume is actually a measure of the energy in the system.0677

The number of molecules you’ve got times the temperature times the special constant is also a measure of how much energy in the system.0685

Which makes sense, a really, really pressurized container has a lot more energy in it than a container that has almost no pressure in it.0692

If you’ve got a balloon, it’s not going to do much if it’s completely deflated and you poke a pin in it. But if you’ve got a fully inflated balloon and you poke a pin in it something’s going to happen.0700

That’s a display of the energy occurring. Not of the energy occurring, I should rephrase that as you seeing how much energy is in that. The energy is already there but you’re now causing it to change in other forms.0708

T is the temperature in kelvin of the gas. Remember we’ve got to have this kelvin otherwise it isn’t going to work.0720

We put all those things together and we’ve got this really useful equation for being able to connect how much the gas will expand and how much the gas will push in on the walls of its container with the volume and the pressure and the number of molecules there are in the special gas constant.0726

One special note on water. Water is really special. Water is maybe almost unique. It might actually be fully unique. It has many, many special properties.0742

Some of those properties are shared with other things but as far as an object that has all of those properties in it. One substance that is all of those properties, it’s really amazing. I don’t know of anything else that has as many extremely unusual properties.0753

Water has a number of these unusual properties and if you’re interested just take a quick internet search on special properties of water or waters properties.0768

You’ll find that there’s a whole bunch of these things and each one of them is incredibly important to the way our world works.0778

Life wouldn’t be able to exist without pretty much any of these.0784

First off we already its amazingly high specific heat, previously water has one of the highest specific heats out there.0787

Another thing that’s really strange about it is the fact ice floats in water. You probably haven’t thought about this too much because we’re used to it.0794

Ice has always floated for our lives so we sort of take it for granted. Everything that we’ve learned so far has been the opposite direction.0801

We’ve discussed that every time we have a hot object and you make it hotter it causes it to expand.0810

If you make it colder you cause it to become denser. So a cold object shrinks in is volume while retaining the same mass.0815

That means the colder you make an object the denser you make an object. If you make an object denser, wouldn’t that mean the solid form of it would always sink in the liquid form of it?0822

That is true for most substances, if we did this with a brick of iron and dropped it into a molten container of iron that brick would fall to the bottom before eventually melted by that heated self.0833

For water near the freezing point, at around…I think precisely 4 degrees centigrade…Celsius not centigrade. Celsius, they changed the name a while ago.0843

At 4 degrees Celsius it actually stops contracting and expands to create the crystalline structure we know as ice.0851

At a certain point it begins to line up into ice. Before that it becomes very, very dense. Once it’s at 4 degrees Celsius. But after that point, it starts to get prepared for its transition into ice.0859

Ice is a really specially arranged crystalline structure. And that crystalline structure causes it to have more space in between the molecules than the liquid form does.0871

Since it’s got that more space in its structure it’s going to be less dense than the water and it’s going to actually wind up floating.0880

This is really, really cool and it’s actually one of the things that allows life to exist on Earth.0885

Its possible life would still be able to exist without it but it certainly makes it a lot easier for complex life forms to exist.0892

If it were otherwise, if water were to get really dense when it was frozen then that would mean the coldest part of the water would be on the bottom.0897

That means if you had a lake it would wind up freezing from the bottom up. If it froze from the bottom up that would mean everything in that lake would die.0907

Because it would freeze from the bottom up to the top, you’d have a fully solid lake and fish in there, in plants in there would just wind up dying off.0917

That means that you wind up killing off a huge amount of your aquatic life. Instead the way things occur now ice rises to the top.0924

At the top it winds up actually providing an insulating layer from the cold exterior.0933

Because we’ve got that nice ice on top the fish are able to go down a lower level and they’re able to survive for the winter.0938

They’re able to survive this thing that would otherwise cause all the water to freeze up and kill them off.0945

So because of the fact that ice floats it’s one more reason we can have life on Earth. It’s really cool.0950

So far we’ve talked about expanding within a phase but we haven’t talked about how we’re going to be able to jump from being a solid to a liquid or a liquid to a gas.0960

We want to talk about…so this idea is substances are held together by inner molecule forces. We enough latent energy, enough temperature, the molecules start to bounce around more and more.0968

These inner molecular forces that hold them together can actually be overcome by these being energetic enough to just pop out.0977

If we’re in a solid, we’re in this structure where their fairly rigidly head together. Eventually we’re able to put enough energy in there for instead of being rigidly held together they’ll sort of turn into a slush and they’ll be able to slide on one another.0984

If we put in even more energy they’ll be able to jump out of the slush and fly around. They’ll be able to just fly around whatever container they’re in.0995

If it’s on Earth they’ll just be able to fly through the atmosphere. You’re able to jump out of the liquid with enough energy.1004

As we manage to switch from one phase to the next, we’re clearly going to get some different properties.1010

Solids are different than liquids are different than gases. As more and more heat is added to a substance it changes phase.1014

Starting at a solid and applying heat, it will first melt into a liquid and then it will vaporize into a gas.1021

That’s how we switch forward with more and more energy. If we do the reverse and pull heat energy out, if we cool the substance the process reverses.1028

We start at gas and then it will condense into a liquid and then it freezes into a solid. We first condense into a liquid, it condenses into a liquid and then gas will freeze into a solid. It freezes into a solid.1038

Inner molecular attraction is not the only force holding a substance together. We also have to account for pressure.1053

If we’ve got inner molecular force holding it in, pressure is doing the exact same thing from a different point of view.1059

Its just pushing down. If it’s pushing from all sides, it’s going to help keep whatever it is, it’s going to help keep it together. It’s going to pushing it together all the time.1066

If a substance is immersed in a fluid like our atmosphere the pressure of that fluid is going to keep pushing on the substance and it will back up those inner molecular forces and will help hold the substance together against rising temperatures.1074

While that temperature might be able to defeat the inner molecular force on it, it might not also be able defeat, defeat being sort of an odd word to use here, but it won’t be able to stronger than the force of the pressure and the inner molecular forces.1088

It might be strong enough to overcome one of them but not necessarily both. This means that the phase of substance is determined not by just temperature or pressure; it’s determined by the combination of them.1103

Temperature and pressure. If we want to understand how a substance is going to behave with temperature and pressure we’re going to have to look at both of those together.1112

The specific change over points will vary from substance to substance and to be able to show any given substance we make a phase diagram.1123

A phase diagram will look something like this picture right here. As we have higher and higher temperatures we become things likes gases and liquids.1129

At low temperatures and high pressure, its stays solid but if we drop the pressure, we could become a liquid at the same temperature or even a gas at the same temperature.1139

Notice how easy it is with very low pressures to be able make that switch over from solid to gas. At low pressures it’s easy for the thing to break apart.1148

At higher and higher pressures it takes more and more thermal energy.1158

It’s possible for a solid to transform directly into a gas. If we had a really low pressure we could hop over by just increasing the temperature without ever having to touch liquid.1163

This is a process called sublimation. You’ve probably seen it if you’ve ever played with dry ice or seen dry ice before.1174

It manages to jump directly from being a solid hunk of dry ice into being carbon dioxide, I’m pretty sure its carbon dioxide, but if you’re doing an experiment with it you might want to double check.1180

I should have looked this up beforehand. Anyway, dry ice is able to sublimate directly from its solid from to its gas form.1192

There’s also another special thing to talk about. Right here is the triple point. This is a special point that’s going to occur on each phase diagram at different locations.1199

That’s going to be a unique state where all three forms can exist at once. All three phases will be able to simultaneously exist. You could have solid next to liquid next to gas.1208

Which is a really amazing thing. You couldn’t really imagine something that’s able to be same temperature and have steam and ice and water all floating together happily coexisting.1218

That’s a really, really hard thing to achieve because we have to be at just the right pressure and just the right temperature but it is possible for any substance.1230

Now getting from one phase to the next isn’t just as simple stepping over a line. For a given mass to be able to change phase it has to overcome its heat of transformation.1239

What this means is that to be able to jump from being a solid to a liquid you don’t have to just hit…for water for example we don’t just get to 99 degrees and then 100 degrees and then 101 degrees and then it changes from water, water, gas.1249

Water, water, steam. It doesn’t work like that, instead we get 99 degrees centigrade, 100 degrees centigrade, and then we have to put in a whole bunch more energy before we’re able to get it to jump up to the steam level.1265

There’s a big difference between managing to get up to the line and actually crossing the line on that phase diagram.1275

This is called the heat of transformation. It’s going to vary depending on the mass of the object. More mass means it’ll take more energy to get to jump that line.1283

The amount of energy we take is heat is equal to l x m. Where l is going to be proportionately constant that will vary from substance to substance and what kind of transformation it is.1293

If it’s a solid to a liquid or a liquid gas transformation, it’s going to be different for even the same substance.1304

We’ll have to look up the specific substance we’re talking about and what kind of transformation, but whatever the l is we could figure it out or look it up in a table.1309

If a substance is moving to a higher energy phase like solid to liquid to gas it has to take in q from its environment.1318

If on the other hand it’s going to lower energy phase like gas to liquid to solid it actually releases the energy.1324

When water becomes ice it actually puts out energy into the environment around, which makes a lot of sense because that’s why we have to put it in a freezer, we’re having the environment constantly pump the heat away from that area.1331

That’s what the point of a refrigerator is. We’ve got a heat pump where it takes the heat inside of the box and it puts it somewhere else.1342

Because it’s putting all that heat somewhere else we’ve got a lower environmental temperature which allows things to give off their own temperature and become things like ice for example.1349

Now we’re ready to go on to some examples. First off being able to change the length of temperature.1360

If we’ve got a brass pipe of initial length of 20 meters and its alpha, its coefficient for transformation is 18.7 x 10^-6 per kelvin and it has a temperature rise of 80 kelvins, what would the new length be?1364

Well we’ve got that the change in the length is equal to the original length times alpha times the change in temperature. If we want to know what the change in length is we just toss each of those in.1380

We’ve got 20 meters times 18.7 x 10^-6 times 80 kelvins. We multiply those altogether and we get 0.03 meters. So it manages to gain a positive 0.03 meters.1392

If we want to know what the final length is, we add that to our original length and we’ll get 20.03 meters is our final length.1408

Notice even with the changeover of 80 kelvin, a fairly large amount of temperature from my point of view. It manages to only grow .03; this is a really small amount.1419

The amount of changeover that we’re going to get is really small compared to the temperature difference that we have to have here.1430

We’re either going to have to be talking about absolutely giant lengths or absolutely giant temperatures differences before we’re really going to notice this.1438

This is why we don’t notice it in our daily lives. Although if you wanted to have one experiment or actually a useful thing, if you ever have a stuck jar, one trick you can do is run the lid under really, really hot water.1445

Because metal has a higher coefficient of expansion than the glass so that means the metal on that lid will wind up become a little bit larger and so you can then grab it and twist off and make it a little bit easier to turn and take it off.1456

There of course be careful if you’re going to do this, you are running it under hot water and you don’t want to accidentally burn your hand when you’re taking it off.1469

It will make it slightly easier because you’ll be changing that friction because it will have less pressure holding it to the glass.1475

Let’s talk about why volume has beta equals 3 alpha. Previously I talked a little briefly about the fact that beta isn’t precisely equal to 3 alpha but it’s really good, it’s good enough for our purposes.1480

It will almost always be good enough. Let’s try to figure out how another way, how we could figure out what beta should be.1492

Another way to get to beta. The way that we’ll do this is we’ll start off the cube where all the lengths will undergo linear expansion.1500

What would the new volume of this cube be? Let’s say one of the lengths in our original thing was l. If l is what the original length is, l new, the new l would be equal to l plus the change in l.1508

Well we know what the change in l is; change in l is just the original length times alpha times the change in temperature.1524

Now it’s going to make this problem a lot easier if we set change in temperature equal to 1.1532

Because that change in temperature is just going to wind up showing up throughout the entire problem and it’s just going to be multiplicative fact that keeps stacking and so it’s actually not going to be a problem.1539

We should do this…if were being really, really rigorous we’d want to not make it simpler on ourselves, but we’ll still be able to see the same reasoning so it’s okay in this case to change it just to 1.1549

Now we’ve got at this point that it’s going to be equal. New l is equal to l plus l alpha.1558

What was the old volume of our cube? Our old volume of our cube was equal to l times l times l. Which is equal to l cubed.1566

If we want to know what the new volume is, we’ll that’s going to be v old plus whatever the change in the volume was.1577

Now we want to figure out a way to solve for change in volume. Also notice v old is just the same thing as l cubed.1586

So we’ve got l cubed plus whatever the change in volume wind up being is the way to find out what v new is.1592

If that’s the case, what’s another way to figure out what v new is? We managed to apply that heat so all of our lengths turn into l new. So v news length, the new length…sorry the new volume just has to be based off of the new lengths.1599

The way we find volume is a cube is the side cubed. So l new cubed is equal to v new. We’ve got l new cubed equals l cubed plus change in v.1613

Well l new is the same thing as l plus change in l cube equals l cubed plus change in v.1626

Start working this out, so we’ll have l plus change in l, we can do l plus change in l squared in our head so that becomes l squared...whoops.1634

L squared, what was changed in l, change in l was l alpha, that will make it easier to do. So l plus l alpha cubed, so l plus l alpha cubed. We take off one of those and we’ve now got l plus l alpha times l plus l alpha squared.1645

If we do that in our head we get l squared plus 2 l squared alpha plus l squared alpha squared equals l cubed plus the change in v.1664

We multiply that out some more and we’re going to get l times l squared l cubed plus 2 l cubed alpha plus l cubed alpha squared.1675

We also do l alpha plus l alpha times l squared becomes l cubed alpha plus 2 l cubed alpha squared plus l cubed alpha cubed equals l cubed plus change in v.1686

That this point we want to solve and figure out what is change in v in terms of those old lengths.1700

That will give us a way to connect alpha to the other stuff. Notice at this point we’ve l cubed and l cubed show up on both sides.1706

One other thing, we’ve got l cubed everywhere. L cubed is run amuck. So what is l cubed equal to? L cubed equal to the old volume.1713

Now we’re just going to say that’s just going to be v. We’ll make that v on its own.1722

So we’ve got v plus 2…sorry let’s simplify this as we go along.1726

There’s 2 l cubed alpha plus l cubed alpha so we’ve now got 3v alpha plus l cubed alpha squared. 2 l cubed alpha squared so we’ve got 3v alpha squared plus l cubed alpha cubed on its own.1732

V alpha cubed equals l cubed…oops... make that v cubed as well, plus change in v.1747

If that’s the case, not v cubed. There we are, sorry. So we’ve got a v and v on both sides, so we knock those out and what we’ve really got is 3 v alpha plus 3 v alpha squared plus v alpha cubed equals change in v.1758

Well that’s totally different than the formula we had originally. Which was change in v equals v beta change in temperature, since we’re just dealing with change in temperature is equal to 1; we’ll make that v beta.1780

Since beta we were told was equal to 3 alpha, then we’ve got 3 v alpha. Notice this is totally different than this whole long thing.1797

How is it we can get away with just using v alpha…just using alpha 3 alpha as opposed to 3 alpha plus 3 alpha squared plus 3 alpha cubed.1807

Why? Because alpha is tiny remember, alpha was somewhere between 10^-4 and 10^-6. If its 10^-4 a big alpha, 10^-4 squared is 10^-8.1816

10^-8 is almost certainly going to have no impact practically. It’s going to turn negligible.1832

This right here is going to be really negligible. And this right here is going to be super-duper negligible because of alpha cubed and alpha squared.1838

Alpha is practically negligible unless we’re dealing with really big volumes or really big temperature differences. That’s going to have almost no effect unless we’re dealing with really giant things.1846

Alpha squared and alpha cubed, they’re so small they’re going to have almost no effect entirely.1857

We’re able to drop them and use the much simpler formula of beta equals 3 alpha.1863

If we were dealing with really, really, really, really giant numbers, absolutely huge numbers we might want to wind up including the 3 v alpha squared.1869

If there absolutely fantastic ginormous we might even decide to include that one as well.1879

For the most part it’s plenty fine to just go with that. It’ll be enough information for us to be able to do a great job.1885

We’re able to set beta equal to 3 alpha and be able to have really good understanding of how it’s working.1893

Now we’re ready to try using that fancy ideal gas law. What’s our ideal gas law?1901

If we’ve got a gas comprised entirely of O2 with a mass of 1.43 grams and it’s held in a 1 liter container at this pressure and a temperature of this…okay great.1907

If we bring the temperature to a new temperature and we keep the volume fixed what will the pressure increase to?1917

What if we also doubled v while doing it? To do this we’re going to need the ideal gas law.1922

What’s the ideal gas law? It’s the pressure times the volume is equal to the number of mols times the gas constant times temperature.1927

What was the gas constant again? We go, we look it up, we get 8.314 joules per mol per kelvin.1934

To figure out what the number of mols we have is, we’ll have to know that the atomic mass for oxygen. We go we look it up on our periodic table, we get 15.9994 is the mass of oxygen.1946

Remember we’re not going to be dealing with oxygen, we’re dealing with O2. We’re dealing with molecules of oxygen.1962

If the mass for oxygen is 15.9994 the mass for O2 is going to be double that and we’re going to get 31.988 is what we’re going to get for the mass of O2.1968

If that’s the case then our number of mols is the number grams we have of our gas, 1.43 grams divided by the molecular mass, 31.988 and that gives us n is equal to 0.0447 mols.1985

We also need to talk about what’s the temperature. If ti is equal to -.5 degrees Celsius, how do we convert from Celsius to kelvin?2007

Remember we’re not allowed to use Celsius with temperature when we’re dealing with ideal gas law.2020

We can talk about it when we’re just talking about change in tempreture because Celsius and kelvin have the same change size but they have totally different absolute values.2026

If we’re talking about what the full value of it is we’re going to have to switch over to kelvin because the ideal gas law its defiantly using the full value.2036

It needs t, not change in t. So how do we convert over? Temperature in Celsius is equal to the temperature in kelvin minus 273.15.2042

If we want to know what the initial temperature is over here, temperature initial, we can switch it over by adding 273.15 and so we get 272.62 kelvin.2054

What if we want to know what t final is? T final, once again, we had 273.15 and we’re going to get 313.15 kelvin.2065

Interesting thing to note, we didn’t actually have to convert ti. In fact as far as this problems concerned we didn’t ever have to know ti because this formula right here, it doesn’t concern itself with the difference.2078

It only concerns itself with the ending value is. If we had to figure out something from the beginning we would have needed to know what that starting value was, but we didn’t.2090

Everything was given to us, so in the end we actually don’t need to know it because it’s already set what it’s going to wind up being, what it’s going to wind up ending at.2096

At this point we’ve got everything figured out so we can start plugging things in. The equation p v equals n r t.2105

We do the first part of the problem first. What’s the pressure that we’re at? We don’t know what the pressure is, we’re solving for pressure.2113

What’s the volume that we’re at? We’re in a 1 liter container…oh whoops, that’s one more thing we have to solve for.2119

What is 1 liter as a volume? Volume equals 1 liter, we’re not allowed to use liters. We don’t use liters for this, we use cubic meters.2125

One thing we…I’m not sure if you know this already, if we’ve already talked about this. 1 liter…well what’s 1 milliliter? 1 milliliter is the same thing as 1 cubic centimeter.2136

If 1 milliliter is equal to 1 cubic centimeter then 1 liter would a thousand of those. So 10 x10 x 10. So 1 liter is the same thing as .1 meter cubed.2147

If its .1 meter cubed then we’ve got 0.001 cubic meters is our volume.2163

So we plug that in and we got pressure times 0.001 cubic meters, remember we can’t use liters for this we have to do it in cubic meters because that’s how everything’s been built.2172

That’s what Pascal is based on. Then we plug in the number of mols we’ve got, 0.0447 mols times the gas constant, 8.314 times the temperature that we end at 313.15.2184

We solve for what the pressure is using a calculator and we get that the pressure wind up being 116,378 Pascals. If we want it to be something a little bit more hand able, that’s approximately same thing as 116.4 Kilopascals.2201

For the second part of this problem, we’re just going to need to also change the volume. So the new volume it’s going to change to is double its original amount, so it’s going to be pressure times 0.002 cubic meters, because it’s doubling to 2 liters now.2219

Is equal to exact same stuff for everything else; we’ve got the same number of mols, same gas constant, same ending temperature.2234

We solve that out for our pressure and get 58,189 Pascals, which is equal to something a little bit more sensible, we’ll talk about it in Kilopascals.2247

Notice changing that volume, since it’s got double the space to bounce around in, it actually winds up being half the amount of pressure.2264

Since they bounce, since they’ve got double the space to bounce around in, you’ve got half the number of bounces occurring on average. So with half the number of bounces occurring on average, you’re going to have half the pressure.2273

We’re ready to do our final example. If we’ve got 2 liters of water on a stove that’s currently at 20 degrees Celsius, how much heat energy cubed do we have to put in that water to bring it to boil?2284

How much more heat energy will we have to put in to boil off all the water?2296

First thing to notice, we don’t have liters as mass. We’ve just got 2 liters of water.2299

If we’re dealing with standard room temperature water and standard pressure, which we are reasonably thing to assume since we’re dealing with a stove.2304

We can convert that since 1 milliliter of water…sorry 1 milliliter of water, standard atmosphere and pressure is the same thing as 1 gram of water.2311

Then a liter of water is the same thing as 1 kilogram. So a liter is a kilogram of water.2323

If a liter is equal to a kilogram then 2 liters is equal to…well they’re not…liters not equal to kilogram, but liter goes to kilogram when it’s in terms of water.2329

So 2 liters goes to 2 kilograms. We’ve got a mass of 2 kilograms of water, which we’re going to need to use to be able to do any of this.2340

Also what’s the specific heat for water? Specific heat for water is equal to 1 calorie per gram per kelvin.2349

If its 1 calorie per gram per kelvin, well that’s the same thing as 4.1868 joules per gram per kelvin.2361

So 4.1868 joules per gram per kelvin, but if we want to do that we can also up convert that to kilograms, 4.1868 kilojoules per kilograms. Kilo both on the top and bottom.2369

Now we’ve got that’s what our specific heat is. That’s how much energy it takes in to be able to raise it 1 degree.2388

If we need to get from 20 degrees Celsius to 100 degrees Celsius, what’s our change have to be?2395

Our change in temperature is going to be equal to 80 kelvin. Since a kelvin and a Celsius are the same thing.2401

We’re going to start boiling once we hit 100 Celsius, but it will take a lot more energy, because we have to then push over. We have to push over all of that extra energy to be able to actually get over that line.2406

To be able to get all of our liquid over the line and into the gas phase. We’re going to need even more energy on top of just getting it to 100 Celsius.2419

Change in temperature equals 80 kelvin. Our formula is the amount of energy that we have to put in to get a given temperature increase for a given substance is the specific heat of that times the mass we’re dealing with times the change in temperature.2428

Our specific heat is 4.1868 kilojoules per kilogram, so its 4.1868 times 10^3 since we’re dealing with kilojoules, times the mass we’re dealing with is 2 kilograms, times the change in temperature, is 80 kelvin.2446

We hit that with a calculator and we get 669,888, way too many significant digits and so that becomes 670 kilojoules.2465

We need 670 kilojoules of energy to be able to move 2 liters of water from room temperature to a boiling temperature.2476

That’s not enough energy to be able to get rid of all it. From there we know how to use the heat of transformation.2485

The heat of transformation to get from liquid into gas, the heat of vaporization for water is 2,257 kilojoules per kilogram.2491

If that’s how much it is we’re going to use heat of transformation is equal to l our coefficient of transformation times the mass we’re dealing with.2502

So l is 2,257 kilojoules, so kilo x 10^3 times 2 kilograms. We get 4,514,000,000 joules which is then the same thing as 4.154 mega joules of energy.2510

4.154 mega joules of energy, that’s more than 6.67 times energy. It takes way, way more energy to get it to actually boil off than it does to get it to the point of where it can just begin to boil.2530

As we add in a little bit of energy a little bit of the water will manage to boil off, but to actually boil off all 2 liters, it takes well more than 6x, almost 7x more energy.2552

We’re not even including the fact we’re going to actually lose some of this heat energy. It doesn’t all perfectly go into heating the water.2563

As it’s a hotter temperature it’s going to be radiating more heat greatly to the area around it.2569

It’s going to take probably even more than 6.67x energy to get that water to actually boil off.2574

That’s a huge difference. That’s the reason why you can bring water to a boil, toss in your spaghetti and come back and not have all the water gone.2579

It’s because it’s actually going to take, if consuming a constant amount of heat going to that pot, it’s actually going to take more than 6x the amount of time to get it to go from starting to boil then all the water gone.2587

If it takes you 10 minutes to bring a given amount of water to boil, you know you’ve got at least an hour before all of that water will have wind up boiling off.2603

So it’s a really interesting thing to think about and that’s specific heat of vaporization is also one of the things that’s so amazing about water. That high specific heat of vaporization means that we can things like sweat happen.2611

Where the amount of heat in our skin gets transferred into the water and since it has such a high vaporization heat, we’re able to transfer lots of energy into it and have it get wicked away into the atmosphere.2621

So we’re able to conveniently, usefully cool ourselves because of this high vaporization heat for water. One of the really cool things that water gives us.2631

Alright, hope you enjoyed that, hope it made a lot sense and we’ll see you later. Bye.2639

Hi welcome to educator.com. Today we’re going to be talking about thermo dynamics.0000

At this point it should be really clear that the study of heat and temperature incredibly important to any kind of science.0005

We’ve talked about how it causes expansion, how it causes changes of phase, how there’s just all this stuff that happens based on the temperature of that substance is at.0011

So it’s going to affect chemistry, it’s going to affect biology; it’s going to affect physics. It really, really matters.0020

If you want to do electrical engineering that’s going to matter. Temperature is incredibly important.0025

It has a huge effect on how things operate and even through some clever arraignment we can start to take advantage of that.0030

We can create a thing an engine. You can use heat to do work. Through engines we’re able to cause heat to be beneficial to us in all sorts of ways.0035

In addition to the fact that we might want say a warm room, so it might be…we’ll be able to understand heat better from being able to get heat into a location or heat out of a location.0046

This study is called thermodynamics and it’s what we’ve been talking about for the past two lessons.0057

In general, thermodynamics is way to study heat and its relationship to energy and work.0061

Let’s take a look at some of the laws that make up thermo dynamics.0068

The first law of thermo dynamics is a slight expansion of conservation of energy. You’ve probably already take it for granted and probably figured this out on your own.0072

We’ve talked about it and we’ve almost certainly had this idea introduced to you for a long time. But we haven’t explicitly stated it before.0081

Let’s put it in words. The amount of heat energy we put into a system is equal to the change in that systems internal energy plus the amount of work this system does.0087

Symbolically we get that q; heat energy is equal to change in internal energy plus work.0097

Now why do we use the words internal energy instead of just temperature?0104

Well internal energy lets us talk about how much energy is in the system. Temperature is a specific measure of the average of the system.0108

If we’ve got a 50 kilogram block of steel versus a drop of water at the same temperature in both of those things it’s going to have different amounts of internal energy.0114

There’s even some other things to consider. We can instead be talking about the pressure and volume of the thing without directly relating to the temperature.0124

We can talk about internal energy as more than something that’s just a function of the temperature.0131

It’s the…how much motion is happening in internally, what’s happening, the vibration of the atoms that make up the substance that we’re talking about.0136

How can we talk about heat doing work? Let’s explore one possible way that heat could do simple work.0147

Heat does work in many ways. If we just have an open flame, it’s going to cause work to happen by making those molecules become more energetic and expand out and push out in all directions so it’s causing work by increasing their speed and moving them around.0154

That’s not a very easy way to think about of it though and its not certainly useful work. Let’s think in terms of an engine.0169

For a simplified example we’ll consider a piston, it’s a cylinder full of air where end can move but the enclosure is fully air tight.0175

What would happen if we heat the air at the bottom of it? If we heat the air down here we’re going to cause that air, remember if we had a single volume and we heated it, we caused increase in temperature, it would cause increase in pressure.0183

So in our piston, when we heat that air we’re going to cause those air molecules to start moving around more. Since they’re moving around more they’ve got more pressure.0197

Since they’ve got more pressure they’re now pushing on that piston plate more which means that they’re actually able to cancel out the force of gravity and able to push past it.0205

They’re going to push out as their volume increases, they’re going to wind up having less pressure and less temperature until more temperature gets in.0214

They’re going to be able sort of…as they become higher in temperature they’ll be able to push out the piston.0220

Pushing out the piston will have a connection to the amount of heat energy that would put in.0226

Some will go into raising the internal energy of the atoms, some of it will go into just actual doing work, actually doing work lifting the piston up.0230

We’ve got a way to be able to talk about how…we have a definite example of heat being able to both raise the internal energy but also at the same time cause work to be done external to just they system of the air and the heat.0240

We’re able to do something outside the system lifting that block.0254

The second law of thermo dynamics is similar to the first law. The second law is one that we’ve haven’t especially talked about it but it’s one that we’re definitely ready to understand.0259

One that is probably even certainly more obvious than the first one, at least the way it’s going to be stated here.0268

One way to put the second law is that without external work heat never flows from a cold object to a hot object.0275

If something's cold it isn’t going to colder to make something else even hotter, that doesn’t make sense.0283

It goes from high to low, is what we’re used to. Heat is in some way something that we think of as water.0289

If you’ve got something hot it pours out until it’s evenly spread out. Hot things go down to cold things but cold things don’t, of their own accord, go to hot things.0295

One possible way to fight this out is…to defeat this is by putting work into the system, that’s what a refrigerator does.0305

There’s a clever way to mechanically cause the heat in the box to be put into something else and have that dissipated somewhere else, but we have to put in external work.0313

Without external interference, heat always goes from hot objects to cold objects.0324

This probably seems shockingly obvious; you’d never drop an ice cube in water and expect the water to get hotter while the ice cube got colder. It’s completely intuitive to this at this point in our lives.0333

Hot things make the things around them hotter. They do not themselves get hotter from being around cold things. They make the cold things hotter and the cold things make hot things colder, simple as that.0344

But this idea has really important ramifications. One way to state this was what we just said, heat never flows too cold to hot.0357

Another way to state it is through the idea of entropy. This idea of entropy is really important.0364

We haven’t discussed entropy is yet but another way to put the second law of thermo dynamics is to say that entropy will either increase or remain the same for any process.0371

For any action for any reaction. Entropy is always going to stay the same or get larger.0380

Almost all the time, it’s going to get larger. Entropy never decreases.0385

What the heck is entropy? Entropy is a measure of disorder or chaos. Virtually any real life process will take something organized and cause it to become less organized.0390

As something becomes less organized, the organized something is the more entropy it has.0400

This idea of entropy can be put into a specific mathematical formula. We can talk about it mathematically. We’re not going to, it’s enough for our purposes right now to just explore it a little bit on the surface and get the idea that entropy means chaos, it means randomness, it means disorder.0405

A good example of this would be if you had a stick of dynamite and you lit it and it blew up.0421

A stick of dynamite starts off as this really tightly packed configuration of very complex molecules. We’ve got the very complex molecules that store all of that chemical energy inside of there.0425

We’ve got them tightly packed into this single orderly piece of dynamite that’s well wrapped up.0438

You light the fuse, so you apply a little bit of heat, the fuse runs in; it manages to apply some heat to the dynamite.0443

The dynamite takes that and has a chain reaction where way more heat is released. Way more heat is released; it manages to cause all of those complex molecules to transform into simpler molecules, which allows that heat to be released.0449

All of that heat and pressure causes the dynamite to blast out. We’ve got a release of heat which means our molecules are now vibrating around, they’re more random, there’s more motion going on.0462

They’re bouncing around, they’re more disordered. We’ve got all of those complex, complicated molecules that have been blow…not blow, been transformed into smaller, simpler forms of molecules.0471

Their transformation from the complex to the simple has released energy. Finally the process of exploding has caused the entire piece of dynamite to be blown over a large area randomly.0485

We’ve got all of this chaos, all of this disorder from lighting this stick of something that started off fairly ordered.0495

In general, order will tend to disorder. Over time ordered systems go to disordered system. Unless they’re perfectly isolated with no ongoing processes.0504

If nothing’s going on inside of them and they’re completely removed from everything else, they’ll be able to stay the same level of entropy.0512

But that’s not possible in the real world. We can’t perfectly insulate something from the world around it and we can’t keep everything from stopping happening inside of it.0519

Whatever we’ve what, whatever it is; it’s going to over time become more and more disordered. More and more entropic, more entropy and it’s going to have more randomness.0527

Over time order falls into disorder. A given system like a living being can increase its own order, but it causes this at a greater increase in disorder elsewhere.0537

While you might be able to lower the entropy in your body by eating a good breakfast and working out, by doing things like that your body is taking care of itself.0548

Its doing things, it’s making a very complex system. But over the course of doing that you have to eat that food, all those complex chemicals in the food are broken down into simpler chemicals, turned into waste products.0556

The air we breathe in it becomes less complex to some extent turning oxygen into carbon dioxide is an interesting question the way it’s working.0567

We’re able to take all of these things around, we produce heat, causing more motion, more randomness. The whole process of being a complex thing, being able to stay complex and not break down yourself means that you have to cause complexness in other things to break down.0575

You have to take complexity from elsewhere and break it down to keep up your own level of complexity.0592

While you’re able to have order in your own body, it causes an even greater increase in disorder elsewhere. The entropy also goes up; it isn’t something that’s conserved.0598

Entropy is just constantly marching forward. In fact we can look at entropy; entropy is a one way arrow. We can think of it as the direction that times flows.0607

Entropy is always flowing in the direction of time moving forward. Things get more chaotic with time.0614

With that idea of things getting more chaotic with time, I’d just like to point out that the poets got it right. They might not have stated it scientifically, but poets have stated this intuitively for centuries.0622

For example, so dawn goes down to day, nothing gold can stay’. Nothing complex, nothing perfect can stay. Nothing beautiful is what’s its expressed in this poem, can stay for a long time.0632

That idea of break down is right there. Robert Frost. Another one, William Butler Yates, ‘things fall apart, the center can’t fold, mere anarchy is loosed upon the world.’0644

This is a part of larger poem, which actually has some different ideas going on, but that definitely applies to the notice of entropy, the fact that things just don’t last.0654

If this sounds interesting to you, there is a lot more poetry out there. I recommend it, go check it out. There’s some really cool stuff, more than just physics.0662

If you already like poetry, I think it’s really interesting to see how much there is a connection between the things that we study and the arts that are out there.0670

There’s a lot of connection between the science and the arts and it’s really interesting.0676

Back to the science. Engines in general, an engine takes in heat and it turns some of that into useful work and puts out the rest as exhaust.0680

An example of a car, it takes in gasoline, a complex molecule. It burns that to make the wheels spin through the use of the engine and some heat comes out of the engine and the engine compartment through the exhaust pipe.0689

Understand that that’s the general way of a car working? So we can come up with a really simple diagram, we can diagram this as a hot source that goes into some engine and then energy comes out of that engine in the form of work.0700

Then the rest of the energy goes into a cold sink. We’ve got some large amount of heat going into our engine and then out of that we’ve got work being turned out of that heat.0711

At the same time, some of it manages to get just lost to that cold sink. The best kind of engine would be something that gets lots of lots of work for very, very little energy.0721

The best kind of car would be where you put in a teaspoon of gasoline and then after driving your car you’ve got a liter of gasoline.0733

You put in a tiny amount of gasoline and it actually causes you to have more. But that doesn’t make sense because that would be a destruction of the conversation of energy.0740

We can’t get more work out than the energy we put in. At best, the best kind of engine from our point of view then, knowing that the conservation of energy is defiantly something we can’t beat.0748

The best engine would be one that’s able to convert all the heat energy we put into it into useful work. Every drop of heat energy gets turned into work.0759

That would be the best kind of thing. To describe this, to be able to measure the amount of heat that gets turned into work, we need to talk about efficiency.0767

How efficient something is. The amount of energy that has to go in for the amount of energy we get out.0774

We can compare the heat in to the work out. The efficiency is the work that we get out, w divided by the amount of heat we put in, the energy we put in.0780

The energy you get out divided by the energy you put in is the efficiency. The efficiency can be anywhere between 0 to 1 but no more.0789

We can also easily turn this into a percent by multiplying by 100. It would be great if there was an engine out there that did have a 100% efficiency that would be awesome.0796

It turns out that that’s not possible. A Carnot engine is a theoretical engine that can be proven to have the maximum possible efficiency.0806

It’s not possible to get better than a Carnot engine. Sadly a Carnot engine doesn’t even allow for 100% energy conversion.0814

The most that a Carnot engine will allows for is this: the upper limit on efficiency is connected to how hot your source is, how hot your heat source is, and how cold that exhaust is.0821

The best is going to be when there’s a really, really wide thing. The absolute best would be if your cold exhaust was absolute zero but that’s not possible.0832

You can’t have something in real life that is still at absolute zero. Once again you can’t perfectly insulate something.0839

If you were to have heat flowing into it, it wouldn’t be absolute zero for long. The upper limit is going to be 1 minus the ratio of the cold to the hot.0845

1 minus temperature of the cold divided by the temperature of the hot.0856

Notice temperature cold and temperature hot have to be measured in kelvin otherwise this formula won’t work because once again, if temperature cold was below zero Celsius, we’d have a negative number.0860

Suddenly we’d be able to have an efficiency that was greater one doesn’t make any sense.0869

We have to be talking in kelvin because we have to always be above that zero. We have to be working with kelvin.0873

But this is kind of disappointing. It would be great if there could be a perfect efficient engine but we can prove that it’s not because the best kind has got to be a Carnot engine and a Carnot engine has a maximum efficiency.0878

This efficiency isn’t what you’ll get in real life because the Carnot engine is forgetting the annoying things that come with real life, like friction and heat through other sources, the fact that you can’t have perfect insulation, those sorts of things.0889

It’s like talking about sort of a perfect theoretical thing and we can’t even create a perfect theatrical thing in real life.0903

So you’ll never see something in real life that is better than this. It’s always got to be less than that.0909

Ready for some examples. First example is a nice easy one to knock out of the park.0914

If a system has 100 joules of heat energy put into it and the system does 47 joules of work on its environment, how much will the internal energy have to be raised?0918

Well the first law of thermo dynamics is the heat in is equal to the change in the internal energy plus the work out.0926

Our heat energy was 100 joules, the work out was 47 joules. So we toss those numbers in and we get that 53 joules is the amount that the internal energy has gone up by.0939

Depending on the specific heat of what we’re dealing with we’ll get different amounts.0958

If we had a really low specific heat we’d get a higher temperature raise. If we had a really high specific heat we’d get a lower temperature raise.0962

We do know that the change in the internal energy is going to be 53 joules, whatever temperature that winds up connecting to.0967

Second example, what would be the efficiency of the system from the previous question?0974

Remember we had heat of 100 joules put in, we had a work of 47 joules and we had to change in internal energy of 53 joules.0978

First thing to notice, this is actually a red herring. We don’t care what the change in internal energy is. All we care is what’s the connection between the work out and the heat in.0985

Because that’s how we figure out efficiency. Efficiency is equal to the work divided by the heat in. Work out divided by heat in.0992

47 over 100, we get 0.47 is our efficiency; we would could also talk about as 47%.1002

We’ve got a 47% efficient engine, which is actually really, really, really good.1012

We’ll talk about why that’s so great. You’ll see precisely why that’s so great when we get to the fourth example but 47 is actually a really great efficiency to get.1016

Third example. We’re going to do this one without any math, but we will talk about it.1026

If you’ve got a bunch of coins scattered on the floor and you pick them all up and you stack them into one neat ordered column. You’ve created more order, right?1030

You’ve got this disordered bunch of coins sitting on the floor and you manage to bring it into one tightly bunched thing were they’re all together, they all have a unified temperature because they’re now touching.1038

Haven’t you made there less entropy? Haven’t you lowered the entropy in those coins? Yes, you have lowered the entropy in the coins, but you’ve introduced entropy to the rest of the world.1047

The rest of the universe will now have a total of more entropy. If we look at the entropy over the whole thing, we’re going to get more entropy.1058

Where is this coming from? How are we not violating the second law of thermo dynamics with the fact that we’re stacking these coins?1065

It’s because we can be sure in the fact that we’re introducing more entropy. How are we introducing more entropy?1070

Sure, the stack has less entropy, but there's other things out there. As you move around, one of the primary things, is you move around, you’re going to be generating heat.1075

Heat is generated by your motion. As you walk around the room picking up those coins, bending over. You’re going to introduce heat.1088

You’re going to cause heat to get put into it. The motion of the air molecules is going to become more frenetic, they’re going to be bouncing around, moving around more.1099

They’re going to become more random, more chaotic, less ordered, disordered. You’ve caused disorder in the air molecules by your raw motion, walking around the room and also by the heat generated off your body.1106

As you’ve done this, sugars in your body have been broken down. So you’ve got these complex chemicals that are supplying you with the energy.1116

So you’re breaking down these complex chemicals to be able to have less complex chemicals so you can have energy. You start with a complex chemical, you break it down into something simpler, some energy is released and that’s how your body is doing its thing.1125

It’s able to eat food and break it down into things that are simpler and so you get energy out of it. But in breaking it down you’re taking a complex thing and turning it into simpler.1136

In that case, you’re also once again releasing them. Just as you go around living, parts of your body, they start to degrade. You’ve got your skin cells breaking off, turning into dust.1145

Your skin cells are decaying along with varies other cells in your body. And your body is replenishing them, it’s causing more order, but this is going to all turn into waste products.1156

You’ve got these complex systems that are by nature just breaking down over time. Well you can go around doing something where you’re creating more order in the system.1163

On the whole when we look at the large picture more entropy is introduced just by the fact that you’re moving around doing anything.1172

The only way we could keep the entropy as low as possible would be to just let the coins sit and not have any effect on them and leave the door closed to it.1178

At least in that case no more entropy would be introduced. It would stay at its already disordered state but it wouldn’t become more disordered.1185

But as we go in and start to bustle around and do things we can cause some order to show up in one spot but on the whole more disorder will be introduced than order.1192

The entropy always goes up. Entropy wins. In the long run entropy is the winner.1200

Ready to do the final example. Iron melts a little bit after 1,500 degrees Celsius. As you start to go a little bit past that iron will start to break down and melt into a liquid.1211

If we’ve got a car engine made of iron, the absolute hottest we could run that engine would 1,500 degrees Celsius right?1221

We don’t…that would probably be…well okay first of all in real life, that would be magical because the oil that’s used to lubricate the engine to make sure it runs smoothly would be burned off, completely gone by that.1228

The gaskets involved, charred. The engine would just completely stop working way before 1,500 degrees Celsius. But we can defiantly see the…but it would be hard to figure out what the precise top level would be.1238

We can defiantly see one good place to say that the absolute top value is when you’re engine turns into a molten slag of iron, right?1249

When it turns just too liquid iron it’s no good anymore. We can say that the top operating heat would be 1,500 degrees Celsius, absolute top level for an engine.1257

That’s still higher than we could possibly get in real life but the absolute highest temperature we could operate an engine at is 1,500 degrees Celsius.1268

If that car engine is running on a 20 degree Celsius day, what’s the maximum efficiency we could get out of it?1275

Remember, the maximum efficiency is equal to…by the way, this guy right here he’s called epsilon, once again yet another of our friendly Greek letters, he’s epsilon.1280

Didn’t say that earlier but all of our efficiency has been using the letter epsilon. Our maximum efficiency is 1 minus the temperature of the cold sink divided by the temperature of the hot input.1293

If we managed to run our engine at its absolute highest temperature which is 1,500 degrees Celsius and sort of a ridiculously high temperature, higher than we could ever achieve in real life.1307

The absolute top we could pull off is 1,500 degrees Celsius and that’s going to be the absolute best we could have because the bigger the denominator, the better the efficiency we can get.1316

If we can get this number equal to zero we’ve got perfect efficiency. We want to make a giant denominator and tiny cold. Well what’s the cold sink going to be?1325

The cold sink, the best we could do would be the temperature around us in the atmosphere. We’re not able to drive around with a bucket of ice attached to us.1334

Although that’s an interesting idea, but in real life we’re going to have to deal with a bunch of other things.1343

We’ve got the fact that part of the sink is going to be engine compartment around us, we’ve got the engine compartment and that’s defiantly not going to be fully room temperature.1347

We’re going to at best be able to pull off that 20 degrees Celsius cold sink. Remember we can’t use Celsius though when we’re dealing with this because it’d be possible to drop Celsius into the negatives and then this whole formula would get screwed up.1357

We have to be working in kelvin since kelvin is the SI unit. If we want to make this into kelvin, we take each of these and we add 273.15. The cold is going to be 293.15 kelvin.1368

The hot in kelvin is going to be 1773.15 kelvin. The maximum efficiency, we plug both of those in and our maximum efficiency is 1 minus the cold, 293.15 kelvin divided by the hot, 1773.15 kelvin.1382

That comes out to 0.835 which is equal to 83.5%. Keep in mind 83.5% is ridiculously high because there is no way we could get a real life engine to operate at those kinds of temperatures.1406

If we’d tried to get a real life engine to operate at those kinds of temperatures, kapoot, there’s no way we’re going to actually be able to pull it off without completely destroying, just ruining our cars engine.1423

Real life we can’t get that kind of temperatures. That kind of temperature in the engine is just ridiculous. We’re not dealing with the best temperature we could have.1433

It’d clearly be better the way this works for us to be driving around in the winter, a colder temperature than 20 degrees Celsius, is something we could defiantly pull off.1442

But we’re not going to do way better, we can’t drop more than…if we were to manage to drop to -80 Celsius, that’d be colder than any day has ever been on Earth.1449

That’d be only dropping by 100, the efficiency just not going to bump up that much more.1457

83.5, when we’re dealing with this magical super engine is the best we can do.1461

To me at least, 83.5 that doesn’t sound that great. That’s getting pretty close to a B- if this were a test.1468

Think about this, efficiency is actually really, really hard to come by in real life. 83.5% is the best this magical engine can do. It’s able to withstand these crazy internal temperatures.1474

If that’s the best we can do for our engine that’s magical and remember this not even including things like the friction, the other real life forces that are going to occur.1486

This is this theoretical maximum for an already magical engine. If we were to introduce just a little bit of real life, those numbers are start to plummet.1494

In real life for an actual car engine, the theoretical amount that could put out. The theoretical for a real car engine operating at real temperatures, so theoretical car engine is able to pull off an efficiency of around 55% or so.1503

That’s pretty great but an actual car engine once you have to start factoring in all the friction resistances, all the pressure, the various turbulence that’s starts to happen, that sort of randomness that can’t be controlled for.1520

Actual car engines manage to pull of around 25% efficiency rate. Efficiency is really, really hard to come by. Most of the energy that we wind up breaking down from our chemical bonds, actually just goes to heat.1531

We aren’t able to get most of our heat to turn into work. Which is a real shame, because if we were able to manage to just double this, we’d be able to do great things with the amount of energy we have.1547

One of the best things we could do to get more energy is to be able to increase our efficiency, but there’s this hard limit to the best efficiency we can get.1556

That’s based on our hot temperature and our cold temperature. Unless we’re able to get those really far in difference, more importantly get a really nice cold sink temperature, you just can’t get that great in efficiency because of our Carnot maximum efficiency.1565

In real life efficiency is really hard to come by, so an actual car is able to pull off around 25%. Most of the gasoline you wind up burning in your car actually just goes to heating up the air around you, kind of disappointing.1581

But certainly really interesting. That’s the nature of thermo dynamics, a lot of stuff, all the order eventually falls into disorder. It’s not exactly the happiest of endings but that’s how things are going to go with time.1595

Entropy is the winner in the long run. We can at least have long period of order in thought and intelligence and great complex beings.1607

In the really, really long end, as the universe runs to its end, things will eventually tend to disorder and just have compete fall apart as the universe goes into just hot heat death mode as it breaks apart.1615

That’s a long, long, long time from now. For our purposes we can basically forget about that, we’re going to have the chance to live long complex interesting lives in a nice cool universe that hasn’t seen the long painful end of entropy yet.1628

Things are good for a long time to come so don’t worry about it.1643

Alright, hope that was interesting, hope you learned a lot and we’ll see you at educator.com later.1645

Hi, welcome back to educator.com. Today we are going to be talking about multi-dimensional kinematics.0000

What happens when we are moving in more than just one dimension! 0005

So, previously we dealt with everything as if we were only one-dimensional, always just one dimension.0009

Otherwise, everything is a scalar, a single number on its own. 0011

But, now we are going to see some of the concepts we were dealing with before, actually are vectors in hiding. 0017

All the vectors, all the ones we were talking about, are getting from one place to another place, because a place can be many dimensions. 0022

We are used to living in a three-dimensional world. 0030

You do not just go to the store on a direct path, not everything is always going to be a straight line, if you look at two paths or more paths. 0031

So, we are going to have to talk about things using multiple dimensions, either in x-y axes or in x-y-z coordinate system, some kind of coordinate system that has more than one dimension, if we going to be talking about the real world. 0039

All of our vectors are going to be the ideas that we have a change, displacement, when we are moving from one location to another location.0045

Velocity, moving at a certain speed.0057

But, more than just speed, it is going to talk about the way we are moving. 0060

Acceleration, when we are changing the velocity.0064

Scalars, on the other hand were the things that were just raw length, if we changed it, it is really one-dimensional.0066

The distance between two points is, what it would be if you take a tape measure between those two points. 0072

It does not care what the angle is, it just cares where, how far it is from 'a' to 'b'. 0076

The displacement on the other hand, would care how did you get there, it is just not the length, there is an entire circle you could go. 0082

If you were talking about some length, that length could be pointing in any direction on a circle.0090

You need more than that if you are going to really talk about displacement. Speed is similar to distance.0095

It is just distance/time, if you are looking for the average speed. That is how fast you are traveling, but once again, it is not going to say anything about where you are actually heading.0100

Finally, time.0109

We are going to treat time as a scalar, however I do want to say that if you are to get in to more heavier Physics, you would wind up seeing that time can actually be treated as another part of the space vector, where we are located.0110

That is going to have to do with the idea of space-time, that is beyond what we are talking about right now.0123

So, we are going to be able to treat time as just a single dimensional quantity all the time.0128

First, before we get started, a quick note on vectors.0134

When we are indicating a vector, I like to use a little arrow, like this guy here.0136

Other people, they prefer to use a bold font. If you are looking in a text book do not be surprised if you are seeing bold fonts everywhere, I am writing little arrows.0140

Or, if you look in one text book, and it has bold fonts, and another has little arrows, now you know why!0149

They are both ways of indicating vectors.0154

Sometimes, when we already know we are definitely talking about vectors, it is just assumed that we write it like that.0156

Finally , when we are writing it with handwriting, I tend to write it like this, because frankly, I am a little bit lazy.0161

I could write it like that, but that takes little bit extra effort, so, instead I make this little harpooned hat that lands on top of it.0166

So, little harpoon on top makes that little arrow, and that is how I write it when we are dealing with a vector, when we are writing it up.0174

But when it is written, it has that actual arrow above it. Alright! So, position.0180

Like said before, position is just the location of an object at a given moment in time. 0185

But instead of having position be just along a single string, it is no longer just a single dimensional coordinate axis, we now would have to have some sort of grid, may be two dimensions, may be three dimensions.0189

We are going to have to talk about more than just one dimension though.0200

Here, we would be able to talk about the point (3,2), just the x axis distance, and the y axis distance. 0203

We go over 3, and then we go up 2. Simple as that.0210

Distance and displacement. So, like before, distance is a measure of length.0216

We got two points 'a' and 'b', the distance is just that line, that straight line distance from one point to the other.0220

It is just the length that you would measure if you were using a tape measure, or walking it out with your feet.0228

The displacement on the other hand is a vector that indicates the change occurring between those two points.0232

So, 'a' to 'b' would be very different vector than 'b' to 'a', and even if we are dealing with the same length over here, but pretend it is the same length, I think it is about the same length, If we had 'c' over here, 'a' to 'c' would be a completely different vector than 'a' to 'b', even though these are the same length.0238

So, the length 'ab' = the length 'ac' = the length 'ca' = the length 'ba'.0256

But, each one of those would be a totally different vector.0262

We are talking about taking a different path, We have to go there in a different way.0265

We 6ake the same number of steps, so to speak, to get there, but we are taking a totally different path to get there.0270

Because we are going to a different final location. 0276

For example, say we walk 3 km North of the house, and then 4 km East.0280

We start off at home, we go up 3 km, and then we go East another 4 km.0285

So, what would our displacement be? Our displacement will be from where we started, to where we ended.0293

So, we go like that, and that would be our displacement vector.0298

Our displacement vector would be (4,3).0301

And also notice, we are not saying this explicitly, but we almost always assume 'up' as positive, and 'right' is positive.0311

Because, that is what they are on the x-y axis on the normal Cartesian coordinate system.0321

That is what we are used to in Algebra, we tends to translate over.0325

Once in a while we might want to change which we consider to be positive, and which we consider to be negative, but for the most part we are going to treat going North as positive, going South as negative, going East, to the right as positive, going West, as negative.0329

And if we are dealing with on a flat ground, may be on a table, and had a box, if the box moved to the right, that would be positive, if it moves to the left, that would be negative.0342

If the box moved up, that would be positive, if the box moved down, that would be negative.0351

It is up to us to impose a coordinate system.0357

It is always an important thing to keep in mind.0359

It is us humans who impose a coordinate system on the world and make sense where things are, by giving them assigned values.0360

The assigned values can vary, but we have to choose how we are going to start with to make the window frame to look at the world with.0369

So, our displacement would be this vector right here, (4,3) km.0377

We traveled 4 to the East, we traveled 3 North. 0382

It does not matter which direction we have done it.0385

We could have alternatively done it, it does not matter which order we do it in. We could have alternatively done like this, we would have landed at the same spot.0387

But what is the distance between where you started and where you ended? We have got a right triangle.0394

By Pythagorean theorem, we know that 32 + 42 has to be equal to, here is the symbol that we use to show distance, when we want to show how long that vector is, we use the magnitude or the absolute value as you are used to seeing in Math. 0400

So, this would be the absolute value squared, so 9 + 16 = the distance squared, so we get 25, which is going to wind up becoming 5.0405

And we would technically get +/- 5, but when we take the square root, we know there is no such thing as a negative value in distance, so we know that we got to have a positive value.0429

We can just forget about the negative when we are doing this.0439

And finally, what distance did we travel? We noticed that there is a difference between the distance from beginning to end, a path a bird might take, versus the path that we actually took with our feet.0442

In this case, if we measure the feet path, we have to go 3 km North, to the first point, and then change to 4 km East.0453

So, we would be 3 + 4 =7 for the total km, for the total distance traveled.0464

Each one of these are being different things.0476

Displacement is the vector that says how do you get from where you started, to where you ended. Which path you have to take.0478

You have to, no matter how you do it, no matter what path you wind up taking, you could walk like this, then walk like this, then walk like this, then walk like this, and then walk like this, and get to the point, the same starting point.0485

That is going to wind up being up being the displacement vector of zero. 0496

Because you did not ultimately displace yourself. 0499

In this case, we did ultimately displace ourselves, we displaced ourselves 4 km to the East, 3 km to the North. Compare that to the distance.0502

We wound up being in a different location, we are now away from our original location by 5 km.0510

And finally, the feet steps, we had to actually walk using our feet.0517

How far we actually traveled by foot, is going to be the total distance we traveled, 3+4 = 7 km.0522

Three very different ideas, but important to remember that we can talk about each one of these things, and each one of these is going to get used, at different times.0531

Speed and Velocity.0538

Just like before, speed is how fast, it is just how fast you are going some time. 0540

So, it is that length that you have traveled, divided by the time it took you to do that travel.0544

But velocity is based on displacement. It asks how you got there, not just how fast you moved to get there, but how you got there.0549

Did you go there at this angle, did you go there at this angle, did you go there at this angle.0557

Speed, since they are all the same length, if they were all the same length, that would be the exact same speed, but traveling in three very different directions.0563

We are going to have different meeting.0573

If you are traveling 60 km/h, to the North, that is very different from if you are traveling 60 km/h to the East.0575

They would be the same speed, in the pedometer in your car, but they are going to be very different velocities, because you are traveling in a different path, you are traveling in different way, different direction.0584

Velocity is based off displacement. Velocity is a vector as well.0585

Velocity is the displacement divided by the time, so v = change in displacement/time, Δ d/t.0598

Let us go back and look at the example we just had, we started in a house, we traveled 3 km to the North, and then we travel 4 km to the east.0607

We do this in 2 hours.0619

If it is 2 hours, our displacement, is equal to (4,3) km.0623

So, what is our average velocity? Average velocity is the change in our displacement.0640

Displacement is the change in the locations, we denote with d.0648

We can talk about displacement, as d, like this, but we could also talk as the change in location.0652

Once again, we have this thing where location, displacement, sometimes they get used for the same letter, so there is little bit of confusion here.0661

But we know we need to talk about how far we traveled.0667

We traveled (4,3) km as a vector.0670

If we want to find out what the velocity is, velocity = (4,3) km / 2 hours, so velocity is going to be equal to (2,3/2) km.0673

Now, if we want to know what our average speed was, we need to see what distance did we travel.0690

Distance = 5 km .0695

So the average speed is going to be, we are going to look at the magnitude, the size of that speed vector and it is going to be 5/2 = 2.5 km/h .0701

Here is an important note. Speed from velocity. 0722

As we just saw, on the previous example, if you know something's velocity, we can easily figure out its speed.0724

You could figure out how far it traveled total and divide it by the amount of time to get there.0730

But you can even do better than that. Speed is the length of the velocity vector.0735

We are going to figure it out by looking at how long is the velocity vector.0741

If we want the speed, we are going to look at the distance that we traveled and divide it by the time, or we could look at the velocity vector, which is displacement/time.0746

So, if we just look at the length of the velocity, we already wound up dividing by the time, and it is going to work out.0756

Last time, we got that the average speed for that trip was, 2.5 km/h.0761

But velocity vector, v = (2,3/2) km/h .0770

So, if we want to make it a little bit faster, we just need to see what the magnitude of this is, and it is going to wind up being the exact same as this right here.0779

Let us double check that.0785

If we do sqrt(22 + (3/2)2), we get sqrt(4 + (9/4)), i.e. sqrt(25/4), which is equal to 5/2, exact same thing. 0789

So, it is just a question of do we wind up figuring out the distance, and then divide it by the time, or do we wind up figuring out the velocity, which already involves dividing by the time.0818

And then just figure out how long that vector is on its own. So, two ways to do it.0826

Normally it is going to be easier if we want to find out the speed of something, we know its velocity vector, we just toss it in to this.0830

(vx)2 + (vy)2, we can easily figure out what our speed is from that.0836

Acceleration just comes from velocity. Since velocity is vector, acceleration must be a vector.0850

Acceleration = change in velocity / time .0856

So, if we have two different velocities in two different times, we find out what the difference between them is, and then we divide it by the time.0860

Gravity is going to be just as the same as before, but we remember it is only going to be effective only in one axis.0868

There is no lateral gravity on the Earth.0872

If we jump in to the air, you do not get shifted to the right or to the left by gravity. 0874

So, we are not going to have any x-axis shifting and since when we normally talk about coordinate systems, this tends to be down, this tends to be up, and these are right and left.0878

Occasionally we will also look from the top-down when we talk about North and South and East and West.0890

But if we are talking about something that is lateral and moving vertically, then we are going to normally do it as up and down being the y-axis.0895

If up and down is the y-axis, then we need -9.8 m/s/s, because we are going down at 9.8 m/s/s .0901

There is not going to any real changes from the formulae before.0914

This is the exact same as it was before, except now are winding up looking at it with vectors.0919

So, everything is now working in terms of the vectors here.0924

Displacement, the location at time t = 1/2 × a t2 + vit + the initial location.0927

This one is a little bit special, in that we are going to wind up having to break it down into its components.0942

Remember, before we had, vf2 = vi2 + 2a × displacement.0948

That was what happened when we were looking in one dimension.0956

But if we are now looking in two dimensions, this equation right here applies in each dimension, so we wind up applying it over the x's and then over the y's.0959

So we just need to separate it and break it into each one.0972

We cannot talk about a vector squared, that does not mean anything.0974

We cannot just scale it, because who is going to multiply who?0977

Instead we have to break it into its components, then we easily square the components of that vector.0984

So, we just keep the x axis, the x components separate from the y components, the y axis.0991

They each do their own thing, as long as the acceleration and the distance, they are not changing at all.0996

As long as you got a steady acceleration, then we are safe, and we can talk about it this way.1001

Acceleration = change in velocity, so now we are just working in terms of vectors, and same as velocity = change in location, or the displacement.1006

Lets us look at some examples:1017

If we have a car driving North at 30 m/s, and then it turns right, so this is vi, turns right, and the final, goes East at 30 m/s.1019

If the car takes 5 s to complete the turn, what is the average acceleration that the car has to have on it for the duration of the turn?1037

The first thing to note here, is we have to be talking in multiple dimensions.1044

We are talking in multiple dimensions, because the car is moving North and then East, so we doing in two totally different directions.1047

We cannot just do this in one coordinate axis.1054

So, what we really want to do is, we want to convert these directions and lengths into actual vectors.1056

First we are going 30 m/s to the North, so that is going to be positive, so we get, (0,30) m/s is vi.1064

vf = 30 m/s to the East.1078

If we want to find out what the acceleration is, acceleration = (change in velocity vectors)/(time involved).1086

That is, (final - initial)/time = ((30,0) - (0,30)) / 5 = (30,-30) / 5 = (6,-6) m/s/s , because it is changing, for every second it goes, it winds up changing either 6 m/s faster to the East, or -6 m/s 'faster' to the North.1094

What that really means is, 6 slower.1138

-6, we mean as we are now being accelerated to the South, which means since it is already moving to the North, it is going to lose some of its speed to that southern acceleration.1143

Second example is going to be a long one, it is going to create a bunch of different ideas that we are all going to hook together to help us understand how the stuff works.1155

We have a ball being thrown out of a window from a height of 10 m, with an initial speed of 20 m/s angled 30 degrees above the horizontal, we can ask some questions.1162

What is the initial velocity vector v for the ball?1170

Then, ignoring air resistance, how long does it take the ball to hit the ground?1173

Finally, ignoring air resistance once again, what is the ball's displacement from its starting point, and what is its distance?1177

We have got some building, and a window, somebody throws a ball out of that window.1185

We know, if we were to set a horizontal straight line, this is going to be an angle of 30 degrees, and this ball comes out at 20 m/s.1194

Down here is the ground, and as time goes on, the ball is going to fly forward, and then gravity is going to take more and more of a share of its velocity, and it is going to eventually land, hit the ground somewhere.1204

We want to figure out how long is it going to be flying through the air, how long is it going to take before its y location hits zero.1218

Once we know that, we can figure out, how far is it going to make it to the right.1228

First, we are going to have to know, what is its initial velocity vector.1232

If we have got, something that is 20 long on this side, and 30 degrees here, we can figure out what the other sides have to be, just use trig.1234

So, this side, since it is the side opposite, is going to be sin(30) × 20.1247

This one, would be cos(30) × 20.1254

So, cos(30) × 20 = sqrt(3)/2 × 10 = 17.3 m/s, approximately.1259

And sine is going to be 10 m/s.1272

That is going to give us initial velocity vector of (17.3,10) m/s. (right, up).1276

Now we are ready to move on, because now we have got a vector, and we can make use of this vector, and break each component off, and work with each component separately.1289

If we want to figure out how long does it take the ball to hit the ground, we do not care what its x component is, it is the same thing if the ball hits the ground here, or here, or here.1296

All that matters is, when does it have that zero.1306

When does the ball make contact at the ground, when is the y axis location zero.1310

If we have got, d(t) = (1/2)at2 + vit + initial location, we can then just break this into looking at the y's throughout.1316

We just look at the y components throughout, and we can figure out when it is going to hit the ground.1333

What is its acceleration?1338

The acceleration is, -9.8, so (1/2) × (-9.8) × t2 + 10t + 10 .1343

We want to solve this for when does its location becomes zero.1363

We want to make that left side set to zero, and we want to see what 't' will allow that zero to appear on the left side.1366

So, -4.9t2 +10t +10.1375

How do we solve something like this, we got a couple of choices.1379

Right here, we got a polynomial.1383

If we have got a polynomial, one of the first things we can do is factor it.1386

To me, that does not look really easy to factor.1390

The easiest thing after that, is to chuck it into the quadratic formula, make that machine go through it.1392

(-b +/- sqrt(b2 - 4ac)) / 2a , and that is going to be the solutions when time is going to make it equal to zero.1402

That gives us one way to do it, one other way, you could just plug that into a computer or a powerful calculator.1414

But let us go through the quadratic formula, because we could all just work through it that way.1423

The answers, t = (-10 +/- sqrt(102 - 4 × (-4.9) × 10)) / (2× (-4.9))1427

Keep working through that, we get, (-10 +/- sqrt(100 + 196)) / (-9.8) = (-10 +/- sqrt(296))/(-9.8) = (-10 +/- 17.2) / (-9.8)1455

Now, we are going to have to ask ourselves, are we going to go with the plus, or are we going to go with the minus.1490

Because mathematically, both of those are correct answers, both of them are times when this equation is going to be fulfilled, it will be fulfilled, that equation will be fulfilled both at the plus and the minus.1493

But, we can see logically that the ball only hits the ground one side, it does not have both a forward time and a negative time.1511

So, what we want to do, is we know our answer can only be a positive time, because this equation is not true at t < 0.1520

Before this moment, we do not know where the ball was, the ball was sitting in some apartment before somebody picked it up, and decided to through it out the window.1528

At that moment, that equation right here, was not true before time = 0.1534

Once time = 0, that is the moment the ball is actually thrown, we know we can start using this equation.1540

So, the only solution that will work, is th