For more information, please see full course syllabus of High School Physics

For more information, please see full course syllabus of High School Physics

## Discussion

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## Download Lecture Slides

## Table of Contents

## Transcription

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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/(m^{2})]). - In a fluid, the pressure is determined by the density of the fluid and the depth (compared to the highest point the fluid reaches):

where pp = p _{0}+ ρh g,_{0}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.| →F_{b}| = m _{f}g = ρ_{f}Vg.

### 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
- Fluid?
- Density
- Pressure
- Consider Two Equal Height Cylinders of Water with Different Areas
- Definition and Formula for Pressure: p = F/A
- Pressure at Depth
- Pressure at Depth Overview
- Free Body Diagram for Pressure in a Container of Fluid
- Equations for Pressure at Depth
- Absolute Pressure vs. Gauge Pressure
- Depth, Not Shape or Direction
- Depth = Height
- Buoyancy
- Archimedes' Principle
- Wait! What About Pressure?
- Example 1: Rock & Fluid
- Example 2: Pressure of Water at the Top of the Reservoir
- Example 3: Wood & Fluid
- Example 4: Force of Air Inside a Cylinder

- 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

### High School Physics Online Course

I. Motion | ||
---|---|---|

Math Review | 16:49 | |

One Dimensional Kinematics | 26:02 | |

Multi-Dimensional Kinematics | 29:59 | |

Frames of Reference | 18:36 | |

Uniform Circular Motion | 16:34 | |

II. Force | ||

Newton's 1st Law | 12:37 | |

Newton's 2nd Law: Introduction | 27:05 | |

Newton's 2nd Law: Multiple Dimensions | 27:47 | |

Newton's 2nd Law: Advanced Examples | 42:05 | |

Newton's Third Law | 16:47 | |

Friction | 50:11 | |

Force & Uniform Circular Motion | 26:45 | |

III. Energy | ||

Work | 28:34 | |

Energy: Kinetic | 39:07 | |

Energy: Gravitational Potential | 28:10 | |

Energy: Elastic Potential | 44:16 | |

Power & Simple Machines | 28:54 | |

IV. Momentum | ||

Center of Mass | 36:55 | |

Linear Momentum | 22:50 | |

Collisions & Linear Momentum | 40:55 | |

V. Gravity | ||

Gravity & Orbits | 34:53 | |

VI. Waves | ||

Intro to Waves | 35:35 | |

Waves, Cont. | 52:57 | |

Sound | 36:24 | |

Light | 19:38 | |

VII. Thermodynamics | ||

Fluids | 42:52 | |

Intro to Temperature & Heat | 34:06 | |

Change Due to Heat | 44:03 | |

Thermodynamics | 27:30 | |

VIII. Electricity | ||

Electric Force & Charge | 41:35 | |

Electric Fields & Potential | 34:44 | |

Electric Current | 29:12 | |

Electric Circuits | 52:02 | |

IX. Magnetism | ||

Magnetism | 25:47 |

### 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 deformation*2258

*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

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