Physics: How Pressure Varies with Depth

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We have some fluid here, liquid, and let me ask you a question. Is the density any different at the top and at the bottom? You have to know what this is, it's just colored water. And then the answer is no; there's negligible difference in density. It's 1,000 kilograms per cubic meter, no matter where you are in this little jar of water: water is an incompressible fluid.
How about the pressure? Is the pressure the same at the top and at the bottom? And the answer to that one is very different. Let me remind you: what's pressure? Pressure is force-per-area. When you're inside of a fluid, there is pressure. It's just a number; it's not a vector, it's just a number associated with any point in the fluid and it represents the force per area that the fluid is applying to either real or imaginary surfaces in any direction. So, when you have a fluid there's pressure, which is for instance, pushing against the walls.
Let's consider this same fluid and let me draw a little imaginary surface on the inside. So, this isn't a cube in the middle of the water, this is just a cube of water, and I've just decided to call it my object. I'm talking about statics here, nobody's moving. So this little cube of water must be in equilibrium. There's pressure acting on it, because it's under water. And so, which way is the pressure. Well, it's always perpendicular to the surfaces. The pressure's just a number. The force arising from that pressure is perpendicular to the surfaces. So there's a force this way, this way, in and out. And the thing's in equilibrium.
So, the magnitude of those forces is just pressure times area (that's our definition of pressure: it's force per area), so, let's just think about Newton's Law acting on this little cube and we're going to learn something about how the pressure varies with depth. That's what I'm really after, what's the difference in pressure in different spots in the fluid.
Let's look first left and right. That one's easy. There's pressure on this side and pressure on that side, which means there's force on this side equal to pressure times area, and force on that side, and since the cube isn't moving anywhere, then two forces must be equal, since it's a cube and the areas are equal, the pressures are equal. So, pressure is the same at these two points-and this is a little imaginary cube; I can make it small or big-the pressure is the same everywhere in this fluid as you go from left to right; as you go at the same level.
How about from top to bottom? Now the argument is a little bit different because there's one more force in the story. There's certainly an up force from this pressure down here; and there's certainly a down force from this pressure up here. There's also a force of gravity. You never get away from gravity; even when you're under water, you weight the same: mG; that's the force of gravity on you. Let me label the top to be one, and the bottom to be labeled point number two. And I've argued that the pressure is uniform at this level and uniform at this level, but I haven't yet argued, and it's not true that P[1] is equal P[2], and they won't be.
So, let's just use Newton's Second Law with this as my object. So, I draw a little force diagram (here is that cube represented as a point), and what are the various forces? There's an upward force arising from the water pressure below (force is pressure times area). There's a downward force, and then there's another downward force. So, this is arising from P[2] area P[1 ]area, and finally gravity. mG. So, let me summarize those.
P[2]A up P[1]A down mG down-I've decided to use a little m, so let me just be consistent-That's the mass of that cube. What is the mass of the cube? It's the density of the cube times the volume of the cube. That's the definition of density, is mass per volume times G and I could write the volume of the cube if I like, as - it's a cube - so it's area of the top times the height. And those are the symbols h and A, which I defined in my picture, area and height.
So, let me write down Newton's Second Law now. The sum of the forces in the Y direction has to add up to zero. Let's do it on a fresh slate. F[net] equals to zero in the y direction says up minus down, minus down had better equal to zero. Divide through by area and you get the following simple and very important result. In fact this is the central result which I was after. How does pressure vary in a fluid? If you go from point one, up high, down to a point P[2], down lower, a distance h lower, this formula tells you how much more pressure there is.
If you think about it, it's telling you that the pressure in increasing as you go down. P[2] is bigger by rho GH. And that makes sense: if you go diving under water, the pressure is increasing, the density rho is constant, but the pressure is increasing. And the farther you go, the more the pressure is. Don't confuse h with the height of where you're at. It's not the height above the bottom, it's how far point two is below point number one.
This formula tells me that pressure is increasing with depth, let me go back to this picture to ask you to think about the pressure up at the very top. What's that? If I wanted to know the increase in pressure starting from the top, I need to know where I started, and you might say well it's zero, there's no fluid above me. But that's wrong. There's lots of fluid above this glass of water. Air is a fluid. We are sitting at the bottom of a deep sea of fluid. And that fluid has a density and it's got a height.
So, according to this formula, the pressure right here must be much larger that the pressure at the top of the atmosphere which really is zero once you've left the atmosphere. And you can't this formula because the density of air, one kilogram per cubic meter here, at sea level and regular temperature, room temperature, it's getting less dense as you go up high and so this formula does not quite work for gases, it works for incompressible fluids. But still, the idea is correct. There's a lot of pressure here.
There's so much pressure, we've called it atmospheric pressure and the amount of pressure is approximately 100,000 Newtons per square meter. That's an approximate value of atmospheric pressure. And 100,000 Newtons per square meter, 10^5, is a lot. Look, if you think of my chest as having a surface area of, I don't know, about one-third of a meter squared, and you take pressure times area to get the force: there is a force arising from the atmosphere. And, yes, it's this direction, I'm immersed in a fluid, and remember fluids always exert force perpendicular to any surface that's in them. So it's not just the down force, it's also this one. And it's-multiply out the numbers, it's huge-it's 10,000 Newtons: right here. 10,000 Newtons, that's a force, in British units, of about a ton. Can that be true? It really is.
You have tons of force acting on your body, and of course, you are a big bag of fluid, in a certain sense. There's water in you. And so, there's pressure on the inside. There's pressure on the outside and it's all balancing. And there is lots of tension and stress and it's part of the design of the human body to withstand all that. These are big numbers. Atmospheric pressure can be very important.
On the other hand, we're used to it. We're built for it. So, you really don't care. You walk around, you do your daily business, and there's this, 100,000 Newtons per square meter acting everywhere on you. That doesn't bother you. What bothers you is when the pressure increases. If you go diving into a lake and you go down 20 meters, you know the density of water is a big number, much bigger than the density of air. And you multiply rho of water times g times h of 20 meters, and what you care about is not the absolute pressure, atmospheric plus gh, which is really the absolute pressure on your body. What you care about is P[2] minus P[1]. You care about the increase in the pressure from what you're used to, which is just gh.
And so, because that's what you often care about in many physics situations, we give it a name, we call it the gauge pressure; it's defined to be the pressure minus the atmospheric pressure. So, you're taking away this constant. And why is it the gauge pressure? If you go to the gas station and you've a tire and it's floppy; it's flat. There could still be air in there. And that air is at atmospheric pressure. So the true pressure inside minus atmospheric pressure is zero. And that's what the gauge reads. It says you have a flat tire, no pressure. And then you put pressure in there, what you're really doing is putting extra pressure above and beyond the atmospheric pressure, which is what you need to make the tire stiff. So, that's why it's called the gauge pressure or maybe that's why they call those things pressure gauges. I'm not sure which came first.
If you plug in the numbers I was talking about when you go swimming, the number comes out to be fairly large. If you're 20 meters under, it's 200,000 Newtons per square meter extra on your body. Your body can handle that, although there comes a point when you're deep under water when your body can't handle it any more. You can't dive way, way deep without some kind of pressurized suits or something, some device. You have to be in a submarine or some trick to withstand these high pressures.
And, you know, certain parts of your body are more sensitive to pressure and force than others. You're ear, for instance, the opening into your ear drum is quite sensitive. It's a small area, the force on your ear drum when you're 20 meters under, just multiply that pressure times the area, I'm not sure what that is for a big person, maybe one square centimeter. If it was one square centimeter, that would be 20 Newtons of force. How much is 20 Newtons? That's what these two kilograms weigh together. And so imagine taking these two kilograms and compressing them so that they were just the size of your ear and then trying to hold them up with your ear. That's what it feels like when you're 20 meters under if the pressure inside of you hasn't somehow adjusted.
So, this formula is very useful, very important and anytime you're, you know, building plumbing devices, or anytime you're worrying about any kind of particular applications with fluids, you certainly have to bear it in mind. There's nothing new here. It just came from Newton's Second Law. It's been written in a way that's useful and particle for problems with fluids. Talking about pressures and densities, instead of masses and velocity and stuff, and forces like we had been before.
Fluids
Fluid Statics
How Pressure Varies with Depth Page [3 of 3]