Fluids (part 3) Pressure in a fluid at depth
Fluids (part 3)
- 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.
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