### Video Transcript

In this video, we’re talking about measuring density. And here on screen, we have four objects of different density: a balloon, an indoor soccer ball, a block of wood, and a coin. Now interestingly, for this set of objects, the smaller the objects get, the greater their density is. This isn’t always true. But as we’ll see, there is more to density than meets the eye. In order to know the density of an object, say this one right here, there’re two things we’ll need to know about it. We’ll need to know the mass of the object, that is, how much material, physical material, an object is made of. And we also need to know its volume, how much space the object takes up.

The notion of density involves a combination of mass and volume. That is, if we consider the four objects we saw on the opening screen, the balloon, the soccer ball, the block of wood, and the coin, it’s not enough to say that the one with the most mass, for example, has the greatest density or the one with the most volume. The density of an object is equal to the ratio of an object’s mass to its volume. It’s this ratio, mass to volume, that we need to know.

Now, speaking of mass and volume, it’s helpful to talk a bit about the units of those terms. We can use this notation of square brackets around the term to indicate the units of that term. The SI based unit of mass is the kilogram, abbreviated kg. Sometimes though, we don’t know the mass of an object in kilograms. But rather we know it in units of grams. Now, because that k in kg stands for kilo, which is a prefix meaning thousand, we can tell that one kilogram — one kilogram is equal to one thousand grams. Knowing this relationship between kilograms and grams, this means that, now, if we were given a mass value in grams, we can convert that into a value in kilograms.

To see how this works, let’s consider a quick example. Let’s say we have an object whose mass is 25 grams. And we want to solve for its mass in kilograms. We know that one kilogram is one thousand grams. Or if we divide both sides of this equation by one thousand, then on the right-hand side, one thousand in the numerator cancels with one thousand in the denominator. This tells us that one one thousandth of a kilogram is equal to one gram. And then, if we multiply both sides of this equation by 25, we have 25 grams on the right-hand side. That’s the value we want to convert to kilograms, which we now know is equal to 25 one thousandth of a kilogram, or another way to say the same thing, 0.025 kilograms. So that’s a bit about the mass part of our equation for density. Now let’s talk about volume. And to do that, let’s change our shape from what we have here to something that looks more like a box.

What we have here is a cube. And we’ll say that the length of each side of this cube is one metre. To find the overall volume, we’ll call it 𝑉 of this cube, we’ll multiply the height of it, that’s one metre, by the depth of it, one metre, by the width of it, one metre again. One metre times one metre times one metre can be expressed a different way. We can write it like this, one metre quantity cubed. Mathematically, we know that this exponent, the three, applies to both the unit, metres, as well as the number, one. So we could say that this volume is equal to one cubed times metres cubed. But then, since one cubed is just one, we can write this overall cube volume simply as one metre cubed.

Now, this volume calculation went pretty smoothly. But partly, it was made easier by the fact that each one of our length, width, and height values was one. But what if they were at different value. For example, what if instead of using one metre, we expressed these length, width, and height values in terms of centimetres. That brings to mind the conversion between metres and centimetres. We can write that one metre is equal to 100 centimetres. Or expressed at different but equivalent way, one one hundredth of the metre is equal to one centimetre.

Knowing that, let’s recalculate the volume of our cube. This time, we want it in units of centimetres cubed rather than metres cubed. As we do this, we can recall that because 100 centimetres is equal to one metre, we can rewrite the length, width, and height of our cube in terms of these units. So the volume of this cube is equal to 100 centimetres, that’s the width, times 100 centimetres, that’s the depth, times 100 centimetres, the height. And then, just like we did with our metres calculation of volume, we can condense all this into one parenthetical term. Instead of writing all that out, we can put parentheses around 100 centimetres and then cube that.

At this point, it’s very important to recall what we saw earlier that this cube, the three, applies both to the units as well as to the number. That is, this is equal to 100 cubed times centimetres cubed. Calculating this result, we realise that 100 times 100 times 100, that is, 100 cubed, is equal to 1000000. This means that one metre cubed is not equal to 100 centimetres cubed, but rather 1000000 centimetres cubed.

This perhaps surprising result traces back to the fact that we’re cubing this distance rather than just leaving it as it is or even squaring it. So typically, we’ll see a volume expressed in units of cubic metres. But sometimes, we’ll see it in cubic centimetres. And now we know how to convert between the two. Let’s get some practice now with these ideas through a couple of examples.

The density of nickel is 8908 kilograms per cubic metre. What is this value in grams per cubic centimetre?

Okay, so we’re given the density of this metal, nickel, in a particular set of units, kilograms per cubic metre. What we want to do is reexpress this value in a different set of units, grams per cubic centimetre. Making this conversion will involve two steps. First, we’ll convert kilograms to grams. And then, we’ll convert cubic metres to cubic centimetres. As we convert kilograms into grams, let’s recall the conversion between those two units. One kilogram is equal to 1000 grams. That means, in our current units expression, kilograms per cubic metre, we can replace one kilogram with 1000 grams.

Now that we’ve taken care of converting the units of mass from kilograms to grams, let’s convert the units of volume from cubic metres to cubic centimetres. We can recall that one metre is equal to 100 centimetres. And this means if we cube both sides of this equation, then one metre cubed, which is what we have in our denominator over here, is equal to 100 centimetres quantity cubed. We recall that this three, the cube, applies to both the units as well as the number in this parenthesis, which means that 100 centimetres quantity cubed is equal to 1000000, a one followed by six zeros, cubic centimetres. This then, 1000000 cubic centimetres, is equal to one cubic metre, which means we can substitute it in for metres cubed in our denominator.

That done, we now have 1000 grams divided by 1000000 cubic centimetres. And we see that some of the factors of zero in numerator and denominator can cancel. In particular, three zeros from the numerator and three from the denominator cancel out. What we have then is one gram per 1000 cubic centimetres. At this point, we can recall that this expression we’ve developed is equal to one kilogram per cubic metre. And the given density of this metal, nickel, is given in those units, 8908 kilograms per cubic metre.

Shifting our equation over that, here’s what this means. It means that if we multiply both sides of this equation by 8908, then on the right-hand side, we’ll have our original density of nickel. But then, on the left-hand side, we’ll have that equivalent density except in the units we’re interested in, grams per cubic centimetre. All we need to do to find our answer then is calculate this fraction, 8908 grams divided by 1000 cubic centimetres. And when we calculate that fraction, here’s the result we find, 8.908 grams per cubic centimetre. That’s the value of the density of nickel written in these units. Let’s look at a second example involving measuring density.

How many cubic centimetres are in 0.02 cubic metres?

There are a couple of different ways to answer this question. One is primarily in algebraic way. And another is mostly a geometric method. In the algebraic method, we start out by recalling the conversion between metres and centimetres. One metre is equal to 100 centimetres. Then, what we do is cube both sides of this equation, multiply each side by itself twice over. Once we’ve done this, it means we can take any given value in cubic metres, say for example, the one we’re given in this problem statement, 0.02 cubic metres. And then, we can replace the metres cubed with 100 centimetres quantity cubed. And the reason we can do that is because of the equality that we’ve just established over here. So 0.02 cubic metres is equal to 0.02 times the quantity, 100 centimetres cubed.

The next step is to cube both the units as well as the number within the parentheses. 100 cubed is 1000000, a one followed by six zeros. And centimetres cubed is cubic centimetres. All we have to do then is multiply this 1000000 cubic centimetres by 0.02. And when we do that, we find a result of 20000 cubic centimetres. This then is our answer to the question, 20000. But now, let’s look at that second method, the geometric method, for finding this answer.

Another way to do this is to draw a box and let the dimensions of this box fit certain values. What we’ll do is we’ll let the height of the box be one metre, the width of the box be one metre. But the depth of the box we’ll let be 0.02 metres. And you can see we haven’t chosen this value by accident. That value matches the prefix in 0.02 cubic metres. Now, what is the volume 𝑉 of our box? Well, it’s one metre times one metre times 0.02 metres. In other words, it’s 0.02 metres cubed. And what we want is to express this volume in a different set of units, in cubic centimetres.

To find this value, let’s return to our statement from earlier that one metre is equal to 100 centimetres. What the statement tells us is that, for each of the dimensions of our box, we can substitute in 100 centimetres for each one metre value. That means our height becomes 100 centimetres. Our width becomes 100 centimetres. And our depth becomes 0.02 times 100 centimetres.

So now, let’s write up the volume of our cube according to these values. That value is equal to 100 centimetres times 100 centimetres times 0.02 times 100 centimetres. Notice that when we multiply all these terms together, we’ll get a volume in units of cubic centimetres. And when we do that, we find the same result from earlier, 20000 centimetres cubed. So then, we’ve seen that there are 20000 cubic centimetres in 0.02 cubic metres.

Let’s summarise what we’ve learned about measuring density. We learned in this lesson that the density of an object depends on its mass as well as its volume. As an equation, we saw that density is mass divided by volume. We talked then a bit about the possible units that an object’s mass and an object’s volume might take on. Regarding mass, the standard units of mass are kilograms. And the conversion between kilograms and grams is that one kilogram is equal to 1000 grams.

We also discussed volume in a similar way. Volume, which expresses how much space an object takes up, is often written in units of cubic metres. But sometimes, it’s given in units of cubic centimetres. And we saw that because one metre is equal to 100 centimetres, then, if we cube both sides, that one cubic metre is equal to 100 centimetres quantity cubed. And we saw this turned out to mean that one cubic metre is equal to 1000000 cubic centimetres. And all this means, finally, that the expected units for an object’s density are kilograms per cubic metre.