Video: Angular Momentum of a System of Three Identical Disks

The diagram shows three disks that are all the same size and made of the same material, and they can rotate around an axle. If the angular momenta of disks 1 and 3 increase by 20 kg.m²/s each, by how much must the angular momentum of disk 2 change in order to counterbalance the increase in angular momentum of the other two?

02:25

Video Transcript

The diagram shows three disks that are all the same size and made of the same material, and they can rotate around an axle. If the angular momenta of disks one and three increase by 20 kilograms meter squared per second each, by how much must the angular momentum of disk two change in order to counterbalance the increase in angular momentum of the other two?

In our diagram, we see these three identical disks — one, two, and three — all set up on the same axle. Our question tells us that the angular momenta of disks one and three increases by this specific amount, 20 kilograms meter squared per second. If we recall that the angular momentum of an object is equal to its moment of inertia multiplied by its angular speed, then if we assume that for these two disks their moment of inertia stays constant, that must mean that for their angular momentum to increase their angular speed must be doing that.

Given this change in the angular momentum of disks one and three, the question is, by how much must the angular momentum of disk two change so that it counterbalances the increase in angular momentum of disks one and three? Thinking about the total change in angular momentum of this system of three disks, if we call that change Δ𝐿, we can write that it’s equal to the change in the angular momentum of disk one plus the change in the angular momentum of disk two plus the change in the angular momentum of disk three. And very importantly, we want to enforce the constraint that this overall change is zero. That’s what it means for the change in the angular momentum of disk two to counterbalance that in disks one and three.

In this equation then, we want to solve for Δ𝐿 sub two, the change in the angular momentum of disk two. And to do it, we’ll focus on this equality. The sum of all these changes is equal to zero. And our problem statement tells us what Δ𝐿 sub one and Δ𝐿 sub three are. Since the angular momentum of each of those disks increases by 20 kilograms meter squared per second, then we can substitute that value in for Δ𝐿 sub one and Δ𝐿 sub three. And now to solve for Δ𝐿 sub two, all we need to do is subtract two times 20 kilograms meter squared per second from both sides. When we do this, we find that Δ𝐿 sub two is negative 40 kilograms meter squared per second. This is how much the angular momentum of disk two needs to change in order to counterbalance the increase in angular momentum of disks one and three.

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