A student measures the volume of
some sand by pouring it into a measuring cylinder. The student also measures the
volume of some large pebbles using the same method. Explain why this method for
measuring the volumes of these materials yields a more accurate value for the volume
of the sand than for the volume of the pebbles.
Looking at Figure one, we see the
sand as well as the large pebbles in the measuring cylinders. We can see that the sand fully
occupies the volume of this measuring cylinder below its maximum height, while on
the other hand, we can see lots of large air gaps in the measuring cylinder filled
with the large pebbles.
We want to explain just why this
method of measurement is more accurate for finding the volume of the sand than for
the large pebbles. We could think of it this way: say
that we have a measuring cup and we decide to fill this measuring cup with
oranges. We’ll put as many oranges as we can
fit into it. Well, oranges are pretty big
relative to a measuring cup. So we find we’re only able to fit
two. We could say the measuring cup is
full of oranges, but we still see that there are large gaps of air.
So let’s try something smaller than
oranges. Let’s try filling the cup with
eggs. Working carefully not to break any
of the eggs, we see that they do a better job than the oranges at filling up the
measuring cup. But still, we see the measuring cup
isn’t completely full. There are still air gaps we’re able
Let’s go even smaller. Let’s fill the cup not with eggs,
but with coffee beans. In this case, we see that these
coffee beans being smaller than the eggs and oranges do a much better job filling
the measuring cup. Nonetheless, if we’re going to be
very particular, we can still find some small spaces in the cup, which are not
occupied by its solid contents.
What we’re seeing as we fill our
measuring cup with progressively smaller materials is a trend. We can say that the smaller the
object, the better it fills the cup, where by “better,” we just mean more
completely. This brings us back to our volume
measurements of sand and pebbles.
We now have a better understanding
of just why this method is better for sand than it is for the large pebbles. It comes down to the size and shape
of what we’re measuring. We can start out by saying that the
pebbles are much larger than the grains of sand. That’s certainly true. Each pebble must be hundreds or
even thousands of times bigger than one of the grains of sand.
Therefore, we can say that the
shape of the pebbles has a more significant effect on how will the pebbles can fit
into the same measuring cylinder as the sand. And if we combine these two pieces
of information, we can say that they imply that a greater proportion of the volume
of the cylinder is taken up by empty space.
And as we look at the cylinder with
the pebbles in it, we see that’s true. If we measure the maximum height of
the pebbles in the cylinder and use that to find their volume, we’d actually be
counting lots of empty space as pebble volume. All this then tells us just why
this measurement method is better for the sand than for the pebbles.
Now, let’s consider a method for
accurately measuring the volume of both the sand and the pebbles.
Describe a better method for
finding the volumes of the sand and the pebbles.
To do this, let’s go back to our
measuring cup. We saw earlier with this measuring
cup that the smaller the object we put in the cup, the better it was filled up. In this case, we like to consider
what is the limit of filling up the cup. In other words, what can we put in
it such that there will be no air gaps left over?
Well, what if we went from coffee
beans to granulated sugar. The sugar would definitely have
fewer air gaps, but there would still be some. But then, what if we went from
sugar down to flour — something that had been ground into a powder? Even in that case, there would
still be air gaps, so small that we couldn’t see them left in the cup.
So is there anything then that we
could put in the cup that would completely fill it? There is and it’s a substance we
use all the time. If we fill the cup with water, that
would do a great job of completely expelling all the air from the measuring cup. And we can use this fact to help us
measure the volume of objects that don’t fill the cup perfectly.
Imagine that we had this cup half
filled with water and say we want to use this water to help us measure the volume of
an egg, to do that, we carefully put the egg in the water. And when we do and the egg is fully
under water, we find that the level of water in the cup has moved up. Now that the egg is in the cup, all
of the water which is above the original level of the water before the egg was there
represents the volume of the egg.
This method for measuring the
volume of a large solid object is part of what is known as Archimedes’
principle. This principle tells us that when
an object is completely submerged in a liquid, the object displaces an amount of
that liquid, which is equal to the volume of the object itself.
Knowing that, let’s return to our
sand and pebble volume measurement. Borrowing from Archimedes’
principle, we could take two empty measuring cylinders. We could then fill the cylinders
with a known volume of water and then pour our sand into one of the cylinders and
drop the pebbles into the other. And then once the sand and pebbles
had reached the bottom of their respective cylinders, they would raise the level of
the water in each one.
The new water level would be an
indication of the volume of the contents of the cylinder in total, the water plus
the sand or the pebbles. And so to get the volume of the
sand and the pebbles, respectively, we would take the difference in those two
volumes between the initial and the final heights.
Putting that in words, we can say
that the sand and the pebbles could be added to measuring cylinders containing a
known volume of liquid. Then, as we saw, the amount by
which the volume in the measuring cylinder increases is the volume of each of the
sand and the pebbles. This is a method we could use to
accurately measure the volume of both sand and pebbles.