### Video Transcript

In this video, our topic is
calculating density. Density has to do with how much
mass is contained within a given amount of space. The more mass there is, the more
dense the object. Every material has some density and
we’re able to use this property to compare materials. For example, say that we have a
pool of water. And into this pool, we toss a block
of solid iron and a block of solid wood. When these materials land in the
water, the iron sinks to the bottom while the wood floats on the surface. Now, we may look at this outcome of
the wood floating and the iron sinking and think that we could change it by using
different sizes of materials.

We might imagine that if we used a
much bigger block of wood and a much smaller piece of iron, then maybe now the wood
will sink and the iron will float. But instead, we find the same thing
happening as before; even the very large block of wood floats and even the very
small chunk of iron sinks. What we’re discovering is that the
difference between these three materials, the wood, the water, and the iron, doesn’t
have to do with their volume, that is, how much space they take up. But it has to do with some other
material property. That property is called the
material’s density. Density involves both an object’s
mass as well as its volume. Density is a ratio that describes
how much matter a given material has in a certain amount of volume.

If we consider our two wood blocks,
the large one and the smaller one, the large wood block has more mass than the
smaller one. But we can see it also has more
volume. That is, it takes up more
space. And because density is a ratio of
an object’s mass to its volume, even though the masses as well as the volumes of
these two wood blocks are different, their densities are exactly the same. That is, if we take the mass of
this wood block and divide it by the volume of that block, then that ratio is
exactly equal to the mass of this larger wood block divided by its volume.

And this is the reason that both of
these blocks, even though they have very different sizes, behave the same way, that
they both float. It’s because compared to the water
they’re floating in, they both have the same density. The same thing holds true for our
two bits of iron. Even though these pieces have
different sizes and different masses, if we took the mass of this little bit of iron
and divided it by the volume of this bit, then that ratio would be exactly equal to
the mass of this larger piece divided by its volume.

So the density of an object, and
all objects have some density, depends on more than just its volume and more than
just its mass. It depends on both mass and
volume. And specifically, density is the
ratio of mass to volume. We can write this as an
equation. We can say that density, which
we’ll represent using the Greek letter 𝜌, is equal to the ratio of an object’s mass
to that same object’s volume. With this equation, we now have a
reliable recipe for calculating the density of some object. If we know or are able to find out
its mass and the same thing with its volume, then we can calculate its mass per unit
volume, its density.

As we mentioned earlier, density is
a property of materials. Any size object made purely out of
a certain material has the same density, no matter how big or how small the object
is. That’s because as the volume of an
object made of a certain material increases, the object’s mass increases by the same
proportion. These four blocks are all made of
the same material. And let’s say that if the volume of
this very small block is capital 𝑉, then the volume of this block here is twice as
big as that. It’s two 𝑉. But the density of these blocks, as
well as the other two, are all the same because they’re made of the same material,
which must mean that if the mass of this smallest block is 𝑚, then the mass of the
next smallest block must be two 𝑚.

Then, when we calculate the density
of the first block, we’ll call that 𝜌 sub one, it’s equal to the block’s mass
divided by its volume 𝑚 over 𝑉. And then if we calculate the
density of the second block, we’ll call it 𝜌 sub two, that’s equal to two 𝑚
divided by two 𝑉 because this block has twice as much mass and twice as much
volume. So then, the two in the numerator
and the two in the denominator cancel one another out. And we’re left with a density for
our second block, which is equal to the density of the first block. And this is what we did expect
because these blocks are made of the same material. Therefore, they must have the same
density.

Because the density of a given
material is always the same regardless of the size of the object made of that
material, density is helpful in letting us compare materials. For example, say that these two
blocks here are made of two different materials; we’ll call them A and B. Say that we wanted to figure out which of these materials is more dense. We can tell by eye that the volume
of material B is bigger than the volume of material A. But this doesn’t mean it will
be more or less dense.

To figure that out, we would need
to know the densities of these two materials. Let’s say that each of these
objects is in the shape of a cube and that the side length of the object made of
material A is two meters, while the side length of the larger object is four
meters. And let’s say further that the mass
of our smaller object is 16 kilograms, while the mass of our larger one is four
times that, 64 kilograms. To compare the densities of these
two cubes, let’s first calculate those values. We’ll let 𝜌 sub A be the density
of material A and 𝜌 sub B be the density of material B.

We can see that based on our
equation for density, it’s equal to the mass of a given object divided by its
volume. Now, we know what the masses of the
smaller and the larger objects are. And with those substituted in, our
next task is to solve for the volumes of these two objects. Because both of our objects are
cubic in shape, that means that their total volume will be equal to the length of
one of their sides cubed. Another way to say this is that if
we have a cube with side length capital 𝐿, then the volume of that cube is 𝐿 times
𝐿 times 𝐿 or 𝐿 cubed.

So then, when we consider the
volume of our two cubes made of materials A and B, the volume of the smaller object
made of material A is equal to its side length, two meters quantity cubed, and the
volume of the larger object is equal to its side length, four meters cubed. Now, when we perform this
operation, when we cube these distances, it’s important to apply this cube both to
the units as well as to the numbers. That is, we’ll cube the meters to
get meters cubed and we’ll cube the number in front of that unit. So then, the volume of our smaller
object is two times two times two cubic meters. That’s eight meters cubed. And our larger object has a volume
of four times four times four cubic meters or 64 meters cubed.

Now, before we calculate these
densities, notice the units that are involved. We have the SI base unit of mass,
the kilogram, divided by the SI base unit of distance, the meter cubed, which is the
SI base unit of volume. This set of units, kilograms per
cubic meter, is the standard way of expressing material densities. So knowing that, what are these
densities, 𝜌 sub A and 𝜌 sub B? We see that 𝜌 sub A is 16 divided
by eight or two kilograms per cubic meter, while 𝜌 sub B is 64 divided by 64 or one
kilogram per cubic meter. So based on our calculation,
material B is less dense than material A, which means, for example, that if we were
to toss these two objects into some liquid, even though the object made of material
B is larger, it will be more likely to float in that liquid.

And that’s because compared to
material A, its density is less. Now, even though these two objects
were cubic in shape, and so we could solve for their volume using this relationship,
we know that that won’t always be the case for objects whose density we want to
calculate. For example, we could have an
object in the shape of a sphere with a radius 𝑟 or we might have a more irregularly
shaped object with a given length and width and height. If this happens, we can recall
different formulas for calculating the volume of our shape. The volume of a sphere is
four-thirds times 𝜋 times the radius of the sphere cubed. And an object like this here, which
is called a cuboid, has a volume equal to its length times its width times its
height.

These formulas can come in useful
when we’re calculating volume in order to determine density. Perhaps the best way to get
familiar with density is to work through some examples. Let’s try one of those now.

Two spheres have the same mass, but
the second sphere has a volume half as big as the first. How much greater is the density of
the second sphere than that of the first sphere?

All right, in this exercise we have
these two different spheres. We’ll call them sphere one and
sphere two. When it comes to the masses of
these spheres, we’re told that those are the same. We could write that this way. We could say that 𝑚 one, the mass
of the first sphere, is equal to the mass of the second sphere 𝑚 two. But then our problem statement goes
on to say that the volume of the second sphere is half as big as the volume of the
first. Another way to say this is that the
volume of the first sphere, we can call it 𝑉 one, is two times as big as the volume
of the second sphere, 𝑉 two.

This means the same thing as saying
that the second sphere has a volume half as big as the first. So now that we know how the mass
and volumes of these two spheres compare, we want to know about the density of the
second sphere compared to the first. In particular, we want to know how
much greater the density of this second sphere is compared to the density of the
first. To figure this out, we can recall
that in general the density 𝜌 of an object is equal to the object’s mass divided by
its volume.

So then, let’s do this. Let’s let 𝜌 sub one represent the
density of the first sphere, and 𝜌 sub two will represent the density of the
second. We know from our equation for
density that the density of sphere one is equal to 𝑚 one divided by 𝑉 one. And then the density of sphere two
is 𝑚 two divided by 𝑉 two. We want to make a comparison
between these two densities, 𝜌 one and 𝜌 two. And to do it, we’ll express the
density of sphere two entirely in terms of variables having to do with sphere
one. Here’s what we mean by that.

First off, we’ll use the fact that
𝑚 two, the mass of the second sphere, is equal to the mass of the first sphere. That means we can replace 𝑚 two
with 𝑚 one. They’re the same. Then, considering sphere volumes,
we have this equation that 𝑉 one is equal to two times 𝑉 two. If we divide both sides of this
equation by two, then we see that two cancel out on the right. And we find that 𝑉 two is equal to
𝑉 one divided by two. So then we can take 𝑉 one divided
by two and substitute it in for 𝑉 two in our equation for 𝜌 two. When we do that, we now have this
fraction 𝑚 one divided by 𝑉 one divided by two.

If we multiply this fraction by two
divided by two, we’re not changing the number at all because technically we’re
multiplying by one. But we see that in the denominator,
the twos cancel one another out. And we wind up with a result of two
times 𝑚 one over 𝑉 one. Now, the density of our second
sphere is expressed entirely in terms of the mass and volume of the first
sphere. Since 𝜌 one, the first sphere’s
density, is equal to 𝑚 one over 𝑉 one, we can replace 𝑚 one over 𝑉 one here with
𝜌 one. And we find that 𝜌 two, the
density of the second sphere, is twice 𝜌 one, the density of the first. Our question asked, “How much
greater is the density of the second sphere than that of the first sphere?” And this is our answer. It’s two times greater.

Let’s look now at a second example
exercise.

A small cube of iron has sides with
length 0.15 meters. If the mass of the cube is 26.6
kilograms, what is its density? Give your answer to three
significant figures.

Okay, so in this example, we have a
cube made of iron. And the sides are all the same
length, 0.15 meters. Along with this, we’re told the
mass of the cube — we can refer to it as 𝑚. And that’s given as 26.6
kilograms. Based on this, we want to calculate
the cube’s density. To do that, let’s recall the
mathematical relationship between density, mass, and volume. The density 𝜌 of a given object is
equal to its mass divided by its volume. So in our case, the density of our
cube, we can call it 𝜌 sub c, is equal to the cube’s mass, 26.6 kilograms, divided
by its volume.

To solve for its volume, we can
recall that since we are working with a cube, the volume of our cube is equal to its
side length cubed. In our case, that side length is
0.15 meters. Therefore, our volume is 0.15
meters quantity cubed. Those parentheses are important
because they tell us that we’ll apply this cube both to the unit of meters as well
as to the number, 0.15. So the volume of our cube is 0.15
cubed cubic meters. When we calculate this density, we
find it’s equal to 7881.48 and so on and so forth kilograms per cubic meter.

But our statement tells us to give
our answer to three significant figures. So let’s start at the front of our
answer and count off three. Here’s one significant figure,
there’s two, and there’s the third one. Now to figure out whether this
second eight here, our third significant figure, will round up or stay the same,
we’ll look at the next digit in our answer. That digit is a one, which is less
than five. So this eight will not round up to
a nine. It will stay as it is. To three significant figures then,
our density is 7880 kilograms per cubic meter. That’s the density of this iron
cube.

Now, let’s summarize what we’ve
learned about calculating density. Starting off, we learned that
density is a material property that depends on mass and volume. For a given material, its density
is always the same, regardless of how large or small of a piece of that material we
have. Written as an equation, an object’s
density, symbolized using the Greek letter 𝜌, is equal to its mass divided by its
volume.

And lastly, we saw that an object’s
volume depends on its shape. Where the volume of a cube is equal
to its side length cubed, the volume of a sphere is equal to four-thirds times 𝜋
times its radius cubed. And the volume of a shape called a
cuboid is equal to its length times its width times its height. This is a summary of calculating
density.