Video Transcript
The objects shown in the diagram
both rotate around the same axis and have the same moment of inertia. The radius 𝑟 of both objects is
the same. What is the ratio of the mass of
the disk to the mass of the hoop?
In our diagram, we see first this
solid disk of radius 𝑟 rotating about an axis through its center. And then above it, we see this hoop
also with radius 𝑟 rotating about that same axis. Because each of these objects has a
mass and is also rotating about some axis, each one has a moment of inertia. And we’re told that those moments
are actually the same. So if we call the moment of inertia
of the hoop 𝐼 sub h and the moment of inertia of the disk 𝐼 sub d, then we know
that these two quantities are equal.
Our question goes on to ask, what
is the ratio of the mass of the disk to the mass of the hoop? To answer this question, we’ll need
to know what is the moment of inertia of the hoop and of the disk in terms of each
object’s mass and size. Often, for a given shape rotating
about a certain axis, we’re able to look up that object’s moment of inertia in a
table. And if we do this for a hoop
rotating about an axis through its center, we find that its moment of inertia is
equal to its mass times its radius squared. In a similar way, if we look up the
moment of inertia for a disk which notice has the same shape as a cylinder, we find
that moment of inertia is equal to one-half the disk’s mass times its radius
squared.
Now, one important thing about
these two equations is that we don’t know offhand whether this 𝑟 is the same as
this 𝑟 and whether this 𝑚 is the same as this 𝑚. For each equation, we’re speaking
specifically about that object’s mass and that object’s radius. In general then, we can’t assume
that these 𝑟’s are the same or that these 𝑚’s are the same. In our specific scenario, though,
we are told that the radius of both objects is the same. So we’ll refer to both of those
radii with the same symbol; we’ll just use lowercase 𝑟, while the masses of these
objects, which we are not told are the same, we’ll represent using 𝑚 sub h for the
mass of the hoop and 𝑚 sub d for the mass of our disk.
All this means that we can write
the moment of inertia of our hoop as 𝑚 sub h times 𝑟 squared. And we can write the moment of
inertia of our disk as one-half 𝑚 sub d times 𝑟 squared. And as we’ve seen, these two
quantities are equal. Now, we want to find the ratio of
the mass of the disk to the mass of the hoop. In other words, we want to
calculate 𝑚 sub d divided by 𝑚 sub h. That’s the ratio of interest. And we can solve for that ratio by
rearranging this equation.
As a first step, we can notice that
the same radius, 𝑟, multiplied by itself, 𝑟 squared, appears on both sides. So that factor can cancel out of
this expression. And now, knowing that we want to
get the fraction 𝑚 sub d divided by 𝑚 sub h in our equation, we can divide both
sides by 𝑚 sub h, which means that our left-hand side simplifies to one. And then lastly, if we multiply
both sides of this expression by two, we find that the ratio 𝑚 sub d to 𝑚 sub h is
equal to two. This then is our answer. The mass of the disk is twice as
large as the mass of the hoop.
