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In the last video, we showed
that any external pressure on

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a liquid in a container
is distributed

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evenly through the liquid.

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But that only applied to-- and
that was called Pascal's

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principle-- external pressure.

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Let's think a little bit about
what the internal pressure is

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within a liquid.

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We're all familiar, I think,
with the notion of the deeper

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you go into a fluid or the
deeper you dive into the

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ocean, the higher the
pressure is on you.

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Let's see if we can think about
that a little bit more

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analytically, and get a
framework for what the

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pressure is at any depth
under the water, or

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really in any fluid.

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Here I've drawn a cylinder, and
in that cylinder I have

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some fluid-- let's not assume
that it's water, but some

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fluid, and that's
the blue stuff.

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I'm also assuming that I'm doing
this on a planet that

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has the same mass as Earth, but
it has no atmosphere, so

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there's a vacuum up here--
there's no air.

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We'll see later that the
atmosphere actually adds

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pressure on top of this.

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Let's assume that there's no
air, but it's on a planet of

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the same mass, so the
gravity is the same.

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There is gravity, so the
liquid will fill this

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container on the bottom
part of it.

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Also, the gravitational constant
would be the same as

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Earth, so we can imagine this is
a horrible situation where

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Earth has lost its magnetic
field and the solar winds have

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gotten rid of Earth's
atmosphere.

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That's very negative, so we
won't think about that, but

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anyway-- let's go back
to the problem.

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Let's say within this cylinder,
I have a thin piece

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of foil or something that
takes up the entire

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cross-sectional area
of the cylinder.

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I did that just because I want
that to be an indicator of

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whether the fluid is moving
up or down or not.

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Let's say I have that in the
fluid at some depth, h, and

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since the fluid is completely
static-- nothing's moving--

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that object that's floating
right at that level, at a

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depth of h, will
also be static.

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In order for something to be
static, where it's not

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moving-- what do we
know about it?

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We know that the net forces on
it must be zero-- in fact,

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that tells that it's
not accelerating.

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Obviously, if something's not
moving, it has a velocity of

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zero, and that's a constant
velocity-- it's not

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accelerating in any direction,
and so its net

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forces must be zero.

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This force down must be
equal to the force up.

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So what is the force down
acting on this cylinder?

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It's going to be the weight of
the water above it, because

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we're in a gravitational
environment, and so this water

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has some mass.

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Whatever that mass is, times
the gravitational constant,

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will equal the force down.

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Let's figure out what that is.

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The force down, which is the
same thing is the force up, is

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going to equal the mass of
this water, times the

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gravitational constant.

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Actually, I shouldn't say
water-- let me change this,

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because I said that this is
going to be some random

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liquid, and the mass
is a liquid.

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The force down is going to be
equal to the mass of the

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liquid times gravity.

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What is that mass
of the liquid?

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Well, now I'll introduce you to
a concept called density,

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and I think you understand what
density is-- it's how

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much there is of something in
a given amount of volume, or

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how much mass per volume.

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That's the definition
of density.

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The letter people use for
density is rho-- let me do

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that in a different
color down here.

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rho, which looks like a p to
me, equals mass per volume,

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and that's the density.

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The units are kilograms per
meter cubed-- that is density.

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I think you might have an
intuition that if I have a

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cubic meter of lead-- lead is
more dense than marshmallows.

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Because of that, if I have a
cubic meter of lead, it will

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have a lot more mass, and in a
gravitational field, weigh a

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lot more than a cubic meter
of marshmallows.

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Of course, there's always that
trick people say, what weighs

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more-- a pound of feathers,
or a pound of lead?

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Those, obviously, weigh the
same-- the key is the volume.

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A cubic meter of lead is going
to weigh a lot more than a

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cubic meter of feathers.

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Making sure that we now know
what the density is, let's go

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back to what we were
doing before.

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We said that the downward force
is equal to the mass of

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the liquid times the
gravitational force, and so

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what is the mass
of the liquid?

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We could use this formula right
here-- density is equal

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to mass times volume, so we
could also say that mass is

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equal to density times volume.

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I just multiply both sides of
this equation times volume.

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In this situation, force down is
equal to-- let's substitute

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this with this.

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The mass of the liquid is equal
to the density of the

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liquid times the volume of the
liquid-- I could get rid of

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these l's-- times gravity.

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What's the volume
of the liquid?

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The volume of the liquid
is going to be the

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cross-sectional area of the
cylinder times the height.

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So let's call this
cross-sectional area A.

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A for area-- that's the area
of the cylinder or the foil

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that's floating within
the water.

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We could write down that the
downward force is equal to the

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density of the fluid-- I'll stop
writing the l or f, or

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whatever I was doing
there-- times the

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volume of the liquid.

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The volume of the liquid is just
the height times the area

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of the liquid.

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So that is just times the height
times the area and then

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times gravity.

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We've now figured out if we knew
the density, this height,

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the cross-sectional area, and
the gravitational constant, we

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would know the force
coming down.

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That's kind of vaguely
interesting, but let's try to

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figure out what the pressure
is, because that's what

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started this whole discussion.

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What is the pressure when you go
to deep parts of the ocean?

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This is the force-- what is the
pressure on this foil that

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I have floating?

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It's the force divided by the
area of pressure on this foil.

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So I would take the force and
divide it by the area, which

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is the same thing as A,
so let's do that.

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Let's divide both sides of this
equation by area, so the

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pressure coming down--
so that's P sub d.

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The downward pressure at that
point is going to be equal

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to-- keep in mind, that's going
to be the same thing as

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the upward pressure, because the
upward force is the same.

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The area of whether you're going
upwards or downwards is

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going to be the same thing.

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The downward pressure is going
to be equal to the downward

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force divided by area, which is
going to be equal to this

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expression divided by area.

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Essentially, we can just get
rid of the area here, so it

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equals PhAg divided by A-- we
get rid of the A's in both

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situations-- so the downward
pressure is equal to the

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density of the fluid, times the
depth of the fluid, or the

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height of the fluid above it,
times the gravitational

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constant Phg.

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As I said, the downward pressure
is equal to the

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upward pressure-- how
do we know that?

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Because we knew that the upward
force is the same as

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the downward force.

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If the upward force were less,
this little piece of foil

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would actually accelerate
downwards.

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The fact that it's static-- it's
in one place-- lets us

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know that the upward force is
equal to the downward force,

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so the upward pressure
is equal to

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the downward pressure.

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Let's use that in an example.

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If I were on the same planet,
and this is water, and so the

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density of water-- and this is
something good to memorize--

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is 1,000 kilograms
per meter cubed.

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Let's say that we have no
atmosphere, but I were to go

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10 meters under the
water-- roughly 30

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feet under the water.

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What would be the
pressure on me?

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My pressure would be the density
of water, which is

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1,000 kilograms per meter
cubed-- make sure your units

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are right, and I'm running out
of space, so I don't have the

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units-- times the height,
10 meters, times the

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gravitational acceleration, 9.8
meters per second squared.

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It's a good exercise for
you to make sure

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the units work out.

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It's 10,000 times 9.8, so the
pressure is going to be equal

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to 98,000 pascals.

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This actually isn't that
much-- it just

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sounds like a lot.

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We'll actually see that this
is almost one atmosphere,

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which is the pressure at sea
level in France, I think.

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Anyway, I'll see you
in the next video.

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