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.
