Fluids (part 5)

Let's say we have
a cup of water.

Let me draw the cup.

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

and this is the other
side of the cup.

Let me say that it's
some liquid.

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

It could be water.

That's the surface of it.

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

this liquid is dependent
on how deep

we go into the liquid.

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

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

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

just acting in one direction.

It's acting in all directions
on that point.

So although how far we go down
determines how much pressure

there is, the pressure is
actually acting in all

directions, including up.

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

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

liquid are stationary, or you
even could imagine an object

down here, and it's
stationary.

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

every direction must be equal.

Let's think about a
molecule of water.

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

If the pressure were different
in one direction or if the

pressure down were greater than
the pressure up, then the

object would start accelerating
downwards,

because its surface area
pointing upwards is the same

as the surface area pointing
downwards, so the force

upwards would be more.

It would start accelerating
downwards.

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

go, at that point,
the pressure is

acting in every direction.

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

learn a little bit about
Archimedes' principle.

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

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

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

force or what is the net
force acting on this

cube due to the water?

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

different points.

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

pressures are equal, because
we know at this depth right

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

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

going to be the same.

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

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

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

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

into the water.

Let's call this P1.

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

point down here PD.

No, pressure on the
bottom, PB.

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

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

equal to the force acting
upwards on this object.

What's the force acting
upwards on the object?

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

times the surface area at the
bottom of the object.

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

That's just d squared.

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

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

actually know that the pressure
down here is higher

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

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

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

here-- the pressure
at the top.

What's the force at the top?

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

times the surface area of
the top of the cube,

right, times d squared.

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

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

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

times the surface area of either
the top or the bottom

or really any of the
sides of the cube.

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

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

into the water.

So what's the pressure
at the top?

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

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

any liquid-- times how
far down we are.

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

And what's the pressure
the bottom?

The pressure at the bottom
similarly would be the density

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

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

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

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

Let me switch colors to keep
from getting monotonous.

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

which is this.

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

d times p times g.

I just distributed this out,
multiplied this out.

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

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

all of that times d squared.

Immediately, we see something
cancels out.

phg, phg subtract.

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

what's the net force?

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

equals d cubed times
the density of the

liquid times gravity.

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

d cubed is the volume
of this cube.

And what else is it?

It's also the volume of
the water displaced.

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

shrinking or anything-- you
can even imagine it being

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

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

that cube to go in.

This is the volume of
the water displaced.

It's also the volume
of the cube.

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

any liquid-- of the liquid.

This is the gravity.

So what is this?

Volume times density is the mass
of the liquid displaced,

so the net force is
also equal to the

mass of liquid displaced.

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

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

the liquid displaced
times gravity?

That's just the weight
of liquid displaced.

That's a pretty interesting
thing.

If I submerge anything, the net
force acting upwards on

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

weight of the water
being displaced.

That's actually called
Archimedes' principle.

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

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

called the buoyant force.

That's what makes
things float.

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

this concept in the next couple
of videos to actually

solve some problems.
I'll see you soon.