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Let's say we have
a cup of water.

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Let me draw the cup.

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This is one side of the cup,
this is the bottom of the cup,

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and this is the other
side of the cup.

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Let me say that it's
some liquid.

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It doesn't have to be water,
but some arbitrary liquid.

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It could be water.

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That's the surface of it.

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We've already learned that the
pressure at any point within

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this liquid is dependent
on how deep

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we go into the liquid.

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One point I want to make before
we move on, and I

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touched on this a little bit
before, is that the pressure

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at some point isn't just acting
downwards, or it isn't

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just acting in one direction.

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It's acting in all directions
on that point.

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So although how far we go down
determines how much pressure

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there is, the pressure is
actually acting in all

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directions, including up.

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The reason why that makes sense
is because I'm assuming

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that this is a static system,
or that the fluids in this

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liquid are stationary, or you
even could imagine an object

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down here, and it's
stationary.

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The fact that it's stationary
tells us that the pressure in

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every direction must be equal.

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Let's think about a
molecule of water.

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A molecule of water, let's say
it's roughly a sphere.

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If the pressure were different
in one direction or if the

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pressure down were greater than
the pressure up, then the

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object would start accelerating
downwards,

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because its surface area
pointing upwards is the same

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as the surface area pointing
downwards, so the force

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upwards would be more.

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It would start accelerating
downwards.

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Even though the pressure is a
function of how far down we

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go, at that point,
the pressure is

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acting in every direction.

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Let's remember that, and now
let's keep that in mind to

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learn a little bit about
Archimedes' principle.

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Let's say I submerge a cube into
this liquid, and let's

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say this cube has dimensions
d, so every side is d.

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What I want to do is I want to
figure out if there's any

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force or what is the net
force acting on this

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cube due to the water?

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Let's think about what the
pressure on this cube is at

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different points.

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At the depths along the side of
the cube, we know that the

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pressures are equal, because
we know at this depth right

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here, the pressure is going to
be the same as at that depth,

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and they're going to offset each
other, and so these are

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going to be the same.

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But one thing we do know, just
based on the fact that

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pressure is a function of depth,
is that at this point

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the pressure is going to be
higher-- I don't know how much

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higher-- than at this point,
because this point is deeper

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into the water.

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Let's call this P1.

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Let's call that pressure on top,
PT, and let's call this

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point down here PD.

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No, pressure on the
bottom, PB.

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What's going to be the net
force on this cube?

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The net force-- let's call that
F sub N-- is going to be

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equal to the force acting
upwards on this object.

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What's the force acting
upwards on the object?

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It's going to be this pressure
at the bottom of the object

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times the surface area at the
bottom of the object.

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What's the surface area at
the bottom of the object?

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That's just d squared.

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Any surface of a cube is d
squared, so the bottom is

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going to be d squared minus--
I'm doing this because I

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actually know that the pressure
down here is higher

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than the pressure here, so this
is going to be a larger

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quantity, and that the net force
is actually going to be

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upwards, so that's why I can
do the minus confidently up

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here-- the pressure
at the top.

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What's the force at the top?

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The force at the top is going to
be the pressure on the top

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times the surface area of
the top of the cube,

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right, times d squared.

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We can even separate out the d
squared already at that point,

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so the net force is equal to
the pressure of the bottom

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minus the pressure of the top,
or the difference in pressure

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times the surface area of either
the top or the bottom

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or really any of the
sides of the cube.

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Let's see if we can figure
what these are.

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Let's say the cube is submerged
h units or h meters

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into the water.

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So what's the pressure
at the top?

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The pressure at the top is
going to be equal to the

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density of the liquid-- I keep
saying water, but it could be

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any liquid-- times how
far down we are.

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So we're h units down, or maybe
h meters, times gravity.

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And what's the pressure
the bottom?

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The pressure at the bottom
similarly would be the density

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of the liquid times the depth,
so what's the depth?

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It would be this h and then
we're another d down.

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It's h plus d-- that's our total
depth-- times gravity.

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Let's just substitute both of
those back into our net force.

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Let me switch colors to keep
from getting monotonous.

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I get the net force is equal to
the pressure at the bottom,

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which is this.

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Let's just multiply it out, so
we get p times h times g plus

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d times p times g.

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I just distributed this out,
multiplied this out.

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That's the pressure at the
bottom, then minus the

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pressure at the top, minus phg,
and then we learned it's

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all of that times d squared.

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Immediately, we see something
cancels out.

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phg, phg subtract.

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It cancels out, so we're
just left with--

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what's the net force?

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The net force is equal to dpg
times d squared, or that

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equals d cubed times
the density of the

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liquid times gravity.

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Let me ask you a question:
What is d cubed?

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d cubed is the volume
of this cube.

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And what else is it?

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It's also the volume of
the water displaced.

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If I stick this cube into the
water, and the cube isn't

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shrinking or anything-- you
can even imagine it being

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empty, but it doesn't have to be
empty-- but that amount of

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water has to be moved out
of the way in order for

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that cube to go in.

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This is the volume of
the water displaced.

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It's also the volume
of the cube.

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This is the density-- I keep
saying water, but it could be

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any liquid-- of the liquid.

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This is the gravity.

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So what is this?

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Volume times density is the mass
of the liquid displaced,

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so the net force is
also equal to the

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mass of liquid displaced.

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Let's just say mass times
gravity, or we could say that

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the net force acting on this
object is-- what's the mass of

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the liquid displaced
times gravity?

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That's just the weight
of liquid displaced.

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That's a pretty interesting
thing.

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If I submerge anything, the net
force acting upwards on

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it, or the amount that I'm
lighter by, is equal to the

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weight of the water
being displaced.

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That's actually called
Archimedes' principle.

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That net upward force due to
the fact that there's more

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pressure on the bottom than
there is on the top, that's

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called the buoyant force.

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That's what makes
things float.

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I'll leave you there to just to
ponder that, and we'll use

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this concept in the next couple
of videos to actually

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solve some problems.
I'll see you soon.

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