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Pressure at a depth in a fluid

Video transcript
In the last video, we showed that any external pressure on a liquid in a container is distributed evenly through the liquid. But that only applied to-- and that was called Pascal's principle-- external pressure. Let's think a little bit about what the internal pressure is within a liquid. We're all familiar, I think, with the notion of the deeper you go into a fluid or the deeper you dive into the ocean, the higher the pressure is on you. Let's see if we can think about that a little bit more analytically, and get a framework for what the pressure is at any depth under the water, or really in any fluid. Here I've drawn a cylinder, and in that cylinder I have some fluid-- let's not assume that it's water, but some fluid, and that's the blue stuff. I'm also assuming that I'm doing this on a planet that has the same mass as Earth, but it has no atmosphere, so there's a vacuum up here-- there's no air. We'll see later that the atmosphere actually adds pressure on top of this. Let's assume that there's no air, but it's on a planet of the same mass, so the gravity is the same. There is gravity, so the liquid will fill this container on the bottom part of it. Also, the gravitational constant would be the same as Earth, so we can imagine this is a horrible situation where Earth has lost its magnetic field and the solar winds have gotten rid of Earth's atmosphere. That's very negative, so we won't think about that, but anyway-- let's go back to the problem. Let's say within this cylinder, I have a thin piece of foil or something that takes up the entire cross-sectional area of the cylinder. I did that just because I want that to be an indicator of whether the fluid is moving up or down or not. Let's say I have that in the fluid at some depth, h, and since the fluid is completely static-- nothing's moving-- that object that's floating right at that level, at a depth of h, will also be static. In order for something to be static, where it's not moving-- what do we know about it? We know that the net forces on it must be zero-- in fact, that tells that it's not accelerating. Obviously, if something's not moving, it has a velocity of zero, and that's a constant velocity-- it's not accelerating in any direction, and so its net forces must be zero. This force down must be equal to the force up. So what is the force down acting on this cylinder? It's going to be the weight of the water above it, because we're in a gravitational environment, and so this water has some mass. Whatever that mass is, times the gravitational constant, will equal the force down. Let's figure out what that is. The force down, which is the same thing is the force up, is going to equal the mass of this water, times the gravitational constant. Actually, I shouldn't say water-- let me change this, because I said that this is going to be some random liquid, and the mass is a liquid. The force down is going to be equal to the mass of the liquid times gravity. What is that mass of the liquid? Well, now I'll introduce you to a concept called density, and I think you understand what density is-- it's how much there is of something in a given amount of volume, or how much mass per volume. That's the definition of density. The letter people use for density is rho-- let me do that in a different color down here. rho, which looks like a p to me, equals mass per volume, and that's the density. The units are kilograms per meter cubed-- that is density. I think you might have an intuition that if I have a cubic meter of lead-- lead is more dense than marshmallows. Because of that, if I have a cubic meter of lead, it will have a lot more mass, and in a gravitational field, weigh a lot more than a cubic meter of marshmallows. Of course, there's always that trick people say, what weighs more-- a pound of feathers, or a pound of lead? Those, obviously, weigh the same-- the key is the volume. A cubic meter of lead is going to weigh a lot more than a cubic meter of feathers. Making sure that we now know what the density is, let's go back to what we were doing before. We said that the downward force is equal to the mass of the liquid times the gravitational force, and so what is the mass of the liquid? We could use this formula right here-- density is equal to mass times volume, so we could also say that mass is equal to density times volume. I just multiply both sides of this equation times volume. In this situation, force down is equal to-- let's substitute this with this. The mass of the liquid is equal to the density of the liquid times the volume of the liquid-- I could get rid of these l's-- times gravity. What's the volume of the liquid? The volume of the liquid is going to be the cross-sectional area of the cylinder times the height. So let's call this cross-sectional area A. A for area-- that's the area of the cylinder or the foil that's floating within the water. We could write down that the downward force is equal to the density of the fluid-- I'll stop writing the l or f, or whatever I was doing there-- times the volume of the liquid. The volume of the liquid is just the height times the area of the liquid. So that is just times the height times the area and then times gravity. We've now figured out if we knew the density, this height, the cross-sectional area, and the gravitational constant, we would know the force coming down. That's kind of vaguely interesting, but let's try to figure out what the pressure is, because that's what started this whole discussion. What is the pressure when you go to deep parts of the ocean? This is the force-- what is the pressure on this foil that I have floating? It's the force divided by the area of pressure on this foil. So I would take the force and divide it by the area, which is the same thing as A, so let's do that. Let's divide both sides of this equation by area, so the pressure coming down-- so that's P sub d. The downward pressure at that point is going to be equal to-- keep in mind, that's going to be the same thing as the upward pressure, because the upward force is the same. The area of whether you're going upwards or downwards is going to be the same thing. The downward pressure is going to be equal to the downward force divided by area, which is going to be equal to this expression divided by area. Essentially, we can just get rid of the area here, so it equals PhAg divided by A-- we get rid of the A's in both situations-- so the downward pressure is equal to the density of the fluid, times the depth of the fluid, or the height of the fluid above it, times the gravitational constant Phg. As I said, the downward pressure is equal to the upward pressure-- how do we know that? Because we knew that the upward force is the same as the downward force. If the upward force were less, this little piece of foil would actually accelerate downwards. The fact that it's static-- it's in one place-- lets us know that the upward force is equal to the downward force, so the upward pressure is equal to the downward pressure. Let's use that in an example. If I were on the same planet, and this is water, and so the density of water-- and this is something good to memorize-- is 1,000 kilograms per meter cubed. Let's say that we have no atmosphere, but I were to go 10 meters under the water-- roughly 30 feet under the water. What would be the pressure on me? My pressure would be the density of water, which is 1,000 kilograms per meter cubed-- make sure your units are right, and I'm running out of space, so I don't have the units-- times the height, 10 meters, times the gravitational acceleration, 9.8 meters per second squared. It's a good exercise for you to make sure the units work out. It's 10,000 times 9.8, so the pressure is going to be equal to 98,000 pascals. This actually isn't that much-- it just sounds like a lot. We'll actually see that this is almost one atmosphere, which is the pressure at sea level in France, I think. Anyway, I'll see you in the next video.