Fluid pressure | AP Physics | Khan Academy

- [Instructor] We talk about
fluid pressure all the time,

for example, blood pressure
or pressure inside our tires,

but what exactly are these numbers?

What does 120 mean over here?

Or what does 40 mean over here?

So the big question over here
is what exactly is pressure?

That's what we're gonna
find out in this video,

so let's begin.

To make sense of this,

let's hang a perfectly cubicle wooden box

and think about all the forces.

We know that there's gravitational
force acting downwards,

which is perfectly balanced
by the tension force,

but there is another
set of force over here.

Remember that there are
air molecules in our room,

and so these air molecules
are constantly bumping

and exerting tiny forces.

But how do we model these forces?

There are billions of air molecules around

and they're constantly bumping,
so how do we model this?

Well, we can model them to be continuous,

that simplifies things,
but more importantly,

if you think about the surface area,

since this is the perfect cube,

and the surface areas
are exactly the same,

that means the number of
molecules bumping per second

on each surface is pretty much the same,

and therefore, we could model these forces

to be pretty much the same
from all the directions,

and that's why these
forces together cancel out

and they do not accelerate
the box or anything like that.

But these forces, look,
are pressing on the box

from all the direction,

so it's these forces that
are related to pressure.

But how exactly?

Well, to answer that question,

let's think of a bigger box.

Let's imagine that the
surface area over here

was twice as much as the
surface area over here.

Then what would be the amount of force

that the air molecules
would be putting here

compared to over here?

Pause the video and think about this.

Okay, since we have twice the area,

if you take any surface,

if you consider, for
example, the top surface,

the number of molecules bumping per second

would be twice as much
compared to over here,

because you have twice the area,

and there are air molecules everywhere.

So can you see just from that logic,

the amount of force over here

must be twice as much as over here.

If the surface area was
three times as much,

the amount of force
must be thrice as much.

In other words, you can see

the force that the air molecules
are exerting on the box

is proportional to the surface area.

Or in other words, the force
per area is a constant.

That is the key characteristics
of the force exerted by air

or, in general, any fluid.

And that ratio, force per area,

is what we call, in general, stress, okay?

But what exactly is pressure?

Well, in our example, notice that

the forces are not in
some random direction.

All the forces are perpendicular
to the surface area,

but how can we be so sure, you ask?

I mean, these are random
molecules bumping into it, right?

Well, if you zoom in, what do you notice?

You notice that when the molecules bump

and they sort of collide,

and they reflect off of the
walls of our cubicle box,

what you notice is the acceleration

is always perpendicular to the box,

and therefore, the force that
the box is exerting on them

is also perpendicular to the box.

And therefore, from Newton's Third Law,

the force that the molecules
are exerting on the box

will also be equal and opposite
perpendicular to the box.

And that's why the forces over here

are always perpendicular.

And so look, there's a special
kind of stress over here.

It's not just any force,

the force there are always perpendicular,

and that particular special kind of stress

where the forces are perpendicular,
that is called pressure.

So you can think of pressure
as a special kind of stress

where the forces are
perpendicular to the area.

But wait, this raises another question.

Is it possible for fluids to exert forces

which are paddle to the area as well?

Yes.

Consider the air molecules
again, they're all moving,

but they're all moving in
random direction, isn't it?

But now, imagine that
there was some kind of wind

that could happen if the
block itself is moving down,

or there is wind blowing
upwards, whatever that is,

there's some kind of a relative motion.

Now, if there is a wind,

let's say the air molecules
are moving upwards

along with the random motion

that they're doing along with that,

then because these molecules also interact

with the molecules of the box,

they will also exert force on them,

pulling them upwards.

This force is called the viscous force,

and at the heart of it comes from the fact

that molecules can
interact with each other

and as a result, look,

this total viscous force is parallel

to the surface area,

and this force tends to shear our cube.

It's kind of like this deck of cards.

If you press it
perpendicular to the surface,

then the stress is just the pressure.

But if you press at an angle,
then there's a pedal component

which shears the deck of cards.

And shearing stresses
can be quite complicated,

but we don't have to worry about it

because in our model, we
are considering ideal fluids

and ideal fluids do
not have any viscosity.

And we're also going to
assume that we are dealing

with static fluids, which
means no viscus forces,

no relative motion, and therefore,

we can completely ignore shearing stress.

And therefore, in our model,
we only have pressure.

Forces will always be only perpendicular

to the surface area.

All right, so let's try
to understand pressure

a little bit better.

What about its units?

Well, because it's force per area,

the unit of force is Newtons

and the unit of area is meters squared,

so the unit of pressure,

at least a standard unit of pressure,

becomes Newtons per meters squared,

which we also call pascals.

So over an area of one meter squared,

if there's one Newton of force
acting perpendicular to it,

then we would say that the
pressure is one pascal.

And that is a very tiny pressure.

One Newton exerted over one meter squared

is incredibly tiny.

So the big question is,
what is the pressure

over here in a room?

What is the atmospheric pressure?

Pressure that the air
molecules are pushing

on the sides of this cube with?

Well, turns out that pressure

is about 10 to the power of five pascals.

In other words, that is a 100,000 pascals.

That is insanely huge.

A 100,000 Newtons of force is exerted

by the air molecules per square meter.

That is insanely high. I
wouldn't have expected.

That is the amount of
pressure we are all feeling

just sitting in our room

due to the air molecules over there.

But that was an important question.

Why don't things get
crushed under that pressure?

I mean, sure, this cube
is not getting crushed

because the internal forces
are able to balance that out.

But what if I take a plastic bag

and there's nothing inside it,

then because there's so much air pressure

over from the outside,

shouldn't the plastic bag just get crushed

due to the air pressure?

Well, the reason it doesn't get crushed

is because there's air inside as well.

And that air also has
the exact same pressure,

which means it is able to
balance out the pressure

from the outside.

But what if you could
somehow suck that air out?

Ooh, then the balance will be lost

because the pressure drops

and then we would see the
plastic bag collapsing,

getting crushed under the
pressure from the outside.

That all makes sense, right?

Okay, before moving forward,
let's also quickly talk

about a couple of other units of pressure

that we usually use in our daily life.

For example, when it
comes to tire pressure,

we usually talk in terms
of pounds per square inch.

And just to give some feeling for numbers,

10 to the power of five
pascals happens to be close

to 14.7 pounds per square inch.

So that is the atmospheric pressure

in pounds per square inch.

Another unit is millimeters of mercury,

and 10 to the power of five
pascals happens to be close

to 760 millimeters of mercury.

In other words, the atmospheric
pressure can pull up

the mercury up to 760
millimeters in a column.

Anyways, let's focus on Pascals for now.

And the big question
now, is pressure a scalar

or a vector quantity, what do you think?

Well, my intuition says it's vector

because there's force in one over here,

but let's think a little
bit more about it.

In fact, think about
pressure at a specific point.

How do we do that?

Well, one way to do that
is you take this box

and shrink it down.

Let's say we shrink the size of the box.

Well, now, the area has
become, let's say half,

but the number of air molecules
will also become half,

and therefore, the
force also becomes half,

making sure force per area stays the same.

So the pressure stays the same.

We'll keep shrinking it, keep shrinking,

and keep shrinking it.

Now shrink it all the way
to an extremely tiny point.

We call that as an infinitesimal.

Now, the areas are incredibly tiny,

the forces are incredibly tiny,

yet the force per area stays the same.

That is the pressure
at a particular point.

And now the question is,
should we assign a direction

to this number, to this pressure?

Well, not really, because
all I need is a number,

because what this number is
saying is that if you zoom in

and if you have any area,

then there will always be
forces perpendicular to the area

that is exerted by the fluid,
and that force per area

will be 10 to the power of five pascals.

And that is valid from any direction.

It doesn't matter how
your area is oriented,

that will always be the case.

Which means, look, all I need is a number.

I don't need a direction

to communicate the idea of pressure,

and therefore, pressure
is a scalar quantity.

And because the number of air molecules

bumping per square meter is
pretty much the same everywhere

and their speeds are pretty
much the same anywhere you take,

therefore, the pressure
now is the same everywhere.

But if we zoom out and look
at the entire atmosphere,

for example, that's not the case.

In fact, that 10 to the
power of five we said

is the pressure close to the sea level,

but if you were to go
a little bit above it,

say at 10 kilometers, which
is the cruising altitude

of commercial airplanes,
you would now find

that the pressure is about one fourth

is what we'd find over here.

Why is the pressure different over here

compared to over here?

Well, that's because
the molecules over here

are carrying the weight
of the entire atmosphere

on top of it.

That entire weight is pushing down,

pressing the molecules over here.

However, if you consider
the molecules at this level,

they're not carrying the entire weight,

they're carrying the
weight only on top of them.

They're not carrying the
weight of this amount

of air molecules.

And that's why the pressure
over here is slightly lower,

which means the pressure depends

on the height if you zoom out.

And so now, the next
obvious question would be,

is there a relationship between
the pressure and the height?

Well, there is, and it's
harder to think about that

for air molecules because air molecules

are very compressible, so
it's a little hard over there.

But let's consider
non-compressible fluids,

like water, for example.

So here's a specific question.

If we know the pressure
at some level over here,

what is the pressure at
some depth, say h, below?

So let's say the pressure at the top is PT

and the pressure at the bottom is PB.

Well, we know that the pressure
at the bottom is higher

than the pressure at the top.

So we could say pressure at the bottom

equals pressure at the top,
plus some additional pressure

due to this weight of the water.

But how do we figure that out?

Well, for that, let's just
draw a cuboidal surface,

having the surface area A.

Now, the additional pressure
that we are getting over here

is due to the weight of
this cuboidal column.

To be precise, it's going
to be the weight per area.

So this term over here is
gonna be weight per area.

But what exactly is the
weight of this cubital column?

Well, weight is just the force of gravity,

so it's gonna be MG, where
M is the mass of the water

in this column divided
by the area, which is A,

but how do we figure out what is the mass

of the column of this water?

Well, we know that density
is mass per volume.

So mass can written as
density times volume,

and that's because we know
the density of a fluid.

So we can write this as
density of the fluid,

density of water, times
the volume of the column

times GD divided by A.

But the final question is what
is the volume of this column?

Hey, we know the volume of the column.

Volume of this cuboidal column

is just going to be area times the height.

And so we can plug that in over here,

and if we cancel out the areas,

we finally get our expression.

The pressure at the bottom will equal

the pressure at the top plus
this additional pressure

due to the weight of this column.

And this equation will work anywhere

as long as you're dealing
with a non-compressible fluid

because we are assuming
the density to be the same.

If you consider a
compressible fluid, like air,

the density varies and this
calculation becomes harder,

and so you'll get a considerably
different expression.

But what I find really
surprising about this

is that for a given
non-compressible fluid,

which means it has a specific density,

the pressure difference between two points

only depends on their
height, nothing else.

In other words, this means
the pressure difference

between two points, say
10 centimeter apart,

whether you consider that in an ocean

or a tiny test tube, it's the same.

It doesn't matter how much
water you're dealing with,

it's just the height that matters.

Anyways, now we can introduce
two kinds of pressure.

The pressure that we have
over here, these two,

they're called absolute pressures.

For example, if this was
the atmospheric pressure,

then that is the absolute pressure.

The absolute pressure of the atmosphere

close to the sea level

is about 10 to the power of five pascals.

But now, look at this term.

What does that term represent?

That represents the extra pressure

that you have at this point

over and above the atmospheric pressure.

That extra pressure is what
we call the gauge pressure.

And most of the time when
we're talking about pressure

in our day-today life,

we are not talking about
the absolute pressure,

we're talking about the gauge pressure.

So for example, when we talk
about the blood pressure,

we say it's 120 millimeters of mercury.

What does that even mean?

Well, remember that the
atmospheric pressure

is 760 millimeters of mercury.

This is the pressure over and above that.

So the pressure in the arteries, or veins,

during a cysto for example,

is 760 millimeters plus
120 millimeters of mercury.

That's what it really means.

So this is the additional pressure

above the atmospheric pressure,

so this is the gauge pressure.

The same as the case with our tires.

For example, if you look at
the pressure inside the tire,

you can see it's about 40 PSI,

but that is a gauge pressure,

meaning it's over and above
the atmospheric pressure.

Remember, the atmospheric
pressure is 14.7 PSI.

So the pressure in the tire is 40 PSI

above the atmospheric pressure.

So most of the time, we're
dealing with gauge pressures.

Okay, finally. coming back over here,

suppose we were to draw a
graph of the gauge pressure

versus the depth.

Okay, what do you think the
graph would look like for,

say, a lake and for the ocean?

Why don't you pause it and
have a think about this?

Okay, let's consider the lake.

Right at the surface, the
gauge pressure is zero

because the pressure
over there is the same

as the atmospheric pressure,
so we start from zero,

and then you can see that the
gauge pressure is proportional

to the height, it's proportional to it,

so we get a straight line.

And so we'd expect the
pressure to increase linearly.

That's what we would get for
the lake. What about the ocean?

Well, ocean is also water,
so it has the same density,

or does it?

Remember, ocean has salt water,

so the density is slightly higher.

So for the ocean, we expect
the line to be steeper,

having slightly higher slope.

Finally, before wrapping up the video,

if you were to submerge
a cube inside water,

earlier, we said that the
pressure is gonna be the same

from all directions,

but now we know that the
pressure on the bottom

is slightly higher than
the pressure on the top,

which means the forces on the bottom

would be slightly higher
than the force on the top

because the area is the same.

So wouldn't that produce
a net upward force?

Yes, it would, and that's
called the buoyant force,

which is responsible for
making certain things float.

And that's something we'll
talk about in a future video.