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.
