Question Video: Calculating the Ratio between the Masses of Two Objects Rotating around the Same Axis | Nagwa Question Video: Calculating the Ratio between the Masses of Two Objects Rotating around the Same Axis | Nagwa

Question Video: Calculating the Ratio between the Masses of Two Objects Rotating around the Same Axis Physics

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?

03:05

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.

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