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Fluids

  • A fluid is a material that has no set shape: instead, it takes the shape of the container it is placed in. Thus, both liquids and gases are fluids.
  • If we have a homogeneous (evenly composed) material/object, we can connect its mass and its volume through density (ρ):
    ρ = m

    V
    .
  • Pressure (p) is a measure of how much force is applied per unit of area. The unit for pressure is pascal (Pa = [N/(m2)]).
  • In a fluid, the pressure is determined by the density of the fluid and the depth (compared to the highest point the fluid reaches):
    p = p0 + ρh g,
    where p0 is the ambient pressure (pressure that is not caused by the fluid).
  • We can separate pressure into absolute pressure, the total pressure at depth (p in the above equation), and gauge pressure, the difference between the absolute pressure and the ambient pressure (ρh g).
  • Pressure is determined by density, depth, and gravity only. Shape and direction have no effect on pressure, even though that might seem surprising at first.
  • Fluids exert an upward buoyant force on any object submerged in them. This force is equal to the mass of the fluid displaced by the object.
    |

    Fb
     
    | = mfg = ρfVg.

Fluids

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
  • Fluid? 0:48
    • What Does It Mean to be a Fluid?
  • Density 1:46
    • What is Density?
    • Formula for Density: ρ = m/V
  • Pressure 3:40
    • Consider Two Equal Height Cylinders of Water with Different Areas
    • Definition and Formula for Pressure: p = F/A
  • Pressure at Depth 7:02
    • Pressure at Depth Overview
    • Free Body Diagram for Pressure in a Container of Fluid
    • Equations for Pressure at Depth
  • Absolute Pressure vs. Gauge Pressure 12:31
    • Absolute Pressure vs. Gauge Pressure
    • Why Does Gauge Pressure Matter?
  • Depth, Not Shape or Direction 15:22
    • Depth, Not Shape or Direction
  • Depth = Height 18:27
    • Depth = Height
  • Buoyancy 19:44
    • Buoyancy and the Buoyant Force
  • Archimedes' Principle 21:09
    • Archimedes' Principle
  • Wait! What About Pressure? 22:30
    • Wait! What About Pressure?
  • Example 1: Rock & Fluid 23:47
  • Example 2: Pressure of Water at the Top of the Reservoir 28:01
  • Example 3: Wood & Fluid 31:47
  • Example 4: Force of Air Inside a Cylinder 36:20

Transcription: Fluids

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

With this idea, we’re ready to talk about pressure.0223

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

Ready for some examples.1429

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