### Video Transcript

In this video, we’re going to learn
about density and pressure. We’ll learn the definition of these
two terms, how to calculate them, and how to work with them practically.

To start out, imagine that as an
amateur magician you are getting practice learning your next trick, lying on a bed
of nails. But as you really look at the bed
of nails, you start to wonder, what if when you lie down, the nails actually do go
through your skin. Suddenly though, an idea comes on
how to make lying on this bed of nails safer.

You fit the bed inside a very large
glass aquarium tank and then you fill the tank up to the top with water. Then, strapping on a snorkel, you
decide you’re ready to start practicing. To understand more about why you
might not want to lie on a bed of nails and why your solution might help, we need to
know a bit about density and pressure.

When we talk about pressure, we’re
talking about a force over a unit area, or a force spread over an area 𝐴. One way to understand pressure is
to imagine having a board made of wood and a flat piece of metal, almost like a
sheet of metal. If we put the metal sheet over the
board and then try to hammer it into the board, well, you can imagine how well that
would work. It really wouldn’t work. No matter how hard we hammered, we
couldn’t drive the sheet of metal into the board.

However, if we took that same
sheet, heated it up, melted it down, and made a nail out of it, then when we put
that nail to the board and started to hammer in, it would drive in well. The nail goes into the board, while
the sheet doesn’t because with the nail we’re able to apply greater pressure. Our force through the hammer is the
same in both cases. But the area of the nail is so much
less than the area of the sheet.

Because pressure is equal to a
force divided by an area, its units in terms of SI base units, are newtons per meter
squared. But there is a special name for a
newton per meter squared. And that’s a pascal, abbreviated
simply pa, a unit which is named after the French polymath Blaise Pascal. Given a particular area 𝐴, over
which a force acts, there are two different ways that force can be applied.

In the first case, it can be
applied constantly at each point over that area 𝐴. In that case, the pressure exerted
over that area simply equals that constant force over the area 𝐴. Now, the force we apply over this
area varies so that, at each infinitesimal area element, we might have a different
value for a force. That changes our expression for
pressure on this area 𝐴.

Now, that our force varies
depending on where on the area 𝐴 it falls, we integrate that force over the entire
area and then divide that complete, or net, force over the area 𝐴. Once we solve for that over our
force though, our equation reduces to what it was before. When we consider pressure related
to fluids, that’s when we start to connect these two terms of pressure and
density.

If you’ve ever swum down in a deep
pool, you know that the farther down you go, the greater the pressure you feel from
the water. The pressure a height ℎ below the
surface of a fluid as we go down into the fluid is equal to that height ℎ multiplied
by 𝑔 times the Greek letter 𝜌, which stands for the fluid density. The density of an object is equal
to its mass divided by the volume, or the space it takes up. We often use the Greek letter 𝜌 to
symbolize density. And its units are kilograms per
meter cubed in SI base units. With these two expressions one for
density and one for pressure, let’s get some practice calculating these terms in an
example.

What is the density of a rock of
mass 356 grams that displaces 93.0 cubic centimeter of water?

We can call the rock density
𝜌. And we can recall that 𝜌 is equal
to an object’s mass divided by the volume it takes up. In our case, the mass of our
object, the rock, is 356 grams and the volume it takes up is 93.0 cubic
centimeters. When we convert this mass to units
of kilograms and this volume to units of cubic meters and calculate the fraction, we
find that to three significant figures the rock’s density is 3.83 times 10 to the
third kilograms per cubic meter. That’s the ratio of the rock’s mass
to its volume.

Now, let’s look at an example
involving calculating pressure.

The tip of a nail exerts tremendous
pressure when hit by a hammer because it exerts a large force over a small area. What magnitude force must be
exerted on a nail that has a circular tip with a diameter of 2.50 millimeters in
order to produce a pressure with a magnitude of 2.00 times 10 to the ninth newtons
per meter squared?

In this exercise, we want to solve
for a force magnitude we can call capital 𝐹. When we recall the pressure
relationship, pressure equals force divided by area, we see this force is equal to
the pressure applied by the tip of the nail multiplied by the area of that tip. We’re given the pressure of 2.00
times 10 to the ninth newtons per meter squared.

And since the tip of the nail is
circular, we know it will be equal to 𝜋 times the diameter of the nail in units of
meters squared divided by four. When we enter these values on our
calculator, we find a force of 9.82 times 10 to the third newtons. That’s the force that must be
applied to the head of the nail, perhaps by way of a hammer, in order to produce
this pressure.

Now, let’s summarize what we’ve
learnt about density and pressure. We’ve seen that the density of an
object is often represented by its mass per unit volume, density being represented
by the Greek letter 𝜌. And pressure is given by force over
area. That is an applied force divided by
the area over which that force acts, 𝑃 equals 𝐹 over 𝐴. And finally, pressure and density
are related through fluids. The pressure applied by a fluid of
density 𝜌 is equal to that density multiplied by 𝑔 the acceleration due to gravity
times the depth of the fluid ℎ.