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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.