- [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.