Question Video: Angular Velocity of Three Disks Rotating around an Axle Physics

Video Transcript

The diagram shows three disks, which can all rotate around an axle. Disks one and three have the same moment of inertia as each other. Disk two has a moment of inertia four times that of disk one. If the angular velocity of disk two increases by two radians per second, by how much must the angular velocity of disks one and three change in order to counterbalance the change in the angular momentum of disk two? Assume that disks one and three must have the same change in angular velocity as each other.

In our diagram, we see these three disks — one, two, and three — set up on this axle so that they can rotate around it. Our problem statement tells us that disks one and three, which we see in our diagram look identical, have the same moment of inertia as each other. We can write that like this. 𝐼 one, the moment of inertia of disk one, is equal to 𝐼 three, that of disk three. Disk two, our statement goes on to tell us, has a moment of inertia four times that of disk one. We can put that this way. 𝐼 two is equal to four times 𝐼 one.

We then imagine that the angular velocity, that is, the rate at which disk two spins around this axis, increases by two radians per second. The question then becomes, by how much must the angular velocity of disks one and three change in order to counterbalance this change in angular momentum of disk two? And then we’re to assume that disks one and three experience the same change in angular velocity. To begin working this out, let’s clear some space on screen. And having done that, let’s now write out what we know has happened to the angular velocity of disk two. If we describe that using 𝜔 sub two, we know that, whatever its value, eventually the angular speed of disk two became that value plus two radians per second. In other words, the angular speed of disk two increased by two radians per second.

Now, if we recall that the angular momentum of some system is equal to its moment of inertia times its angular speed, then we can see that in the case of disk two, with its angular speed increasing and its moment of inertia staying the same, that means its angular momentum must go up overall. Our question, though, asks us to think about the angular momentum of our entire system, all three disks together. In particular, it asks us to consider the condition that the change in angular momentum for this whole system is zero. That is, its overall angular momentum before the angular speed of disk two increased is the same as its angular momentum after.

If we allow that the moment of inertia of disk one and disk three do not change through this process, and that’s a reasonable assumption to make, then the only way that this condition can be satisfied, that the change in angular momentum of our system overall is zero, is for the angular speeds of disks one and three to somehow change in order to counterbalance the change brought about by disk two’s angular speed change.

So here’s what we’re saying. In this process overall, the system’s change in angular momentum is zero and that the moment of inertia of these three disks also doesn’t change, but that the angular speeds of these disks, that is, 𝜔 one, 𝜔 two, and 𝜔 three, do change. Based on this, we can write that Δ𝐿, the change in angular momentum of the system, is equal to 𝐼 one times the change in angular speed of disk one plus 𝐼 two times two radians per second, that’s the change in disk two’s angular speed, plus 𝐼 three times Δ𝜔 three, the change in disk three’s angular speed. And then we saw in the line above that Δ𝐿 is zero. So then, this is the equation we want to solve. And in it we’re told that this change, Δ𝜔 one, and this one, Δ𝜔 three, are the same.

To recognize that fact mathematically, let’s change this symbol so it’s now Δ𝜔 one three to represent that this change in angular speed applies to both disks. It’s Δ𝜔 one three that we want to solve for. And to do it, we can first subtract two radians per second times 𝐼 two from both sides of the equation. And then next, we factor out Δ𝜔 sub one three from both terms on the right. Lastly, if we divide both sides by 𝐼 one plus 𝐼 three, we get this equation. To solve for Δ𝜔 one three, all we need to do now is figure out what this ratio is. Looking down to the bottom right of our screen, we see that 𝐼 two is equal to four times 𝐼 one and also that 𝐼 one is equal to 𝐼 three. Therefore, we could write this entire fraction in terms of 𝐼 one, four 𝐼 one divided by 𝐼 one plus 𝐼 one or four 𝐼 one over two 𝐼 one, which is equal simply to two.

When we replace the fraction in our equation with this number, we find that Δ𝜔 one three is equal to negative two radians per second times two or equally negative four radians per second. This is how much the angular velocity of disks one and three would need to change so that the change in system angular momentum caused by the change in angular speed of disk two is counterbalanced.