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?
So that question is a bit of a tongue-twister to read out. But let’s start by underlining all the important bits of the question. So we know we’ve got two spheres. They both have the same mass. But the second sphere has a volume that’s half as big as the first one. What we need to do is to find out how much greater the density is of the second sphere.
So we can draw a neat little diagram that show our two spheres. We can label the mass of the first sphere as lowercase 𝑚. But we know that the second sphere has the same mass as the first. So we can label its mass as lowercase 𝑚 as well.
Secondly, we can label the volume of the first sphere. Let’s call it 𝑉. We know that the second sphere has a volume half as big. So its volume will be 𝑉 over two. We need to compare their densities together. We’ll call this 𝜌 one for the density of the first sphere and 𝜌 two for the density of the second. But what exactly is density in the first place?
Well, density is defined as the mass per unit volume of an object. In other words, it’s a measure of how much mass there is in a given volume. So using this definition of density, we can work out the densities of the two spheres.
For sphere one, we have 𝜌 one is equal to the mass 𝑚 divided by the volume 𝑉. For sphere two, we have 𝜌 two is equal to 𝑚 divided by 𝑉 over two because the volume this time is 𝑉 by two. And this simplifies down a bit because dividing by 𝑉 by two is the same as multiplying by two over 𝑉.
So we know that the first density, 𝜌 one, is equal to 𝑚 over 𝑉. And we know that the second density, 𝜌 two, is equal to two 𝑚 over 𝑉. Now the 𝑚 over 𝑉 bit is identical to 𝜌 one. Therefore, we can replace 𝑚 over 𝑉 with 𝜌 one. And this gives us 𝜌 two is equal to two 𝜌 one. In other words, 𝜌 two is twice as large as 𝜌 one.
Therefore, we have the answer to our question. The density of the second sphere is twice as large as the density of the first sphere. This makes sense. Both of the spheres have the same mass. But in the second sphere, that mass squeezed into half the volume. Therefore, its density must be twice as large.