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Lecture Comments (9)

2 answers

Last reply by: Marcos Castillo
Sun Apr 13, 2014 12:57 PM

Post by Marcos Castillo on April 13, 2014

Hi, related to Newton's thought experiment, for the case we shoot the canon ball at the right velocity, in orden to describe an orbit, can we say that in this case gravity becomes centripetal acceleration?.

0 answers

Post by Ali barkhurdar on November 11, 2012

thanks

4 answers

Last reply by: Professor Selhorst-Jones
Tue May 28, 2013 4:22 PM

Post by ahmed raza on September 28, 2012

Is the formula F=G*M*m/r^2 related to newton's law of gravity which follows as "any two point masses attract each other with the force that is directly proportional to the product of their masses and inversely proportional to the square of their separation ", right?

Gravity & Orbits

  • The force of gravity is based off of the mass of the objects involved (m1,  m2), the distance between the objects (r), and the universal gravitational constant (G).
    |

    F
     

    g 
    | = G · m1 ·m2

    r2
    .
  • The universal gravitational constant is
    G = 6.67 ·10−11  N · m2

    kg2
    .
  • The force of gravity acts on each object equally, and the direction of the force is towards the center of the other object.
  • A gravitational field is an area we can treat as having a constant acceleration. Like the surface of the Earth, in some places the force will only change a negligible amount in the area around the object's location.
  • We denote a gravitational field with ag. Thus, for an object of mass m in the field, Fg = m ·ag.
  • If we want to find the gravitational field for a given object with mass M at distance r, it is
    ag = G · M

    r2
    .
  • If something is in orbit, it must have a centripetal force to keep it in the orbit. Gravity provides this centripetal force. For a simple circular orbit where one object is much more massive than the second object, it follows the relation
    G · m1 ·m2

    r2
    = m2
    |

    v
     
    |2

    r
    .

Gravity & Orbits

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Law of Universal Gravitation 1:39
    • Law of Universal Gravitation
    • Force of Gravity Equation
  • Gravitational Field 5:38
    • Gravitational Field Overview
    • Gravitational Field Equation
  • Orbits 9:25
    • Orbits
  • The 'Falling' Moon 12:58
    • The 'Falling' Moon
  • Example 1: Force of Gravity 17:05
  • Example 2: Gravitational Field on the Surface of Earth 20:35
  • Example 3: Orbits 23:15
  • Example 4: Neutron Star 28:38

Transcription: Gravity & Orbits

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