### 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 โ one, two, and three โ 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. ๐ผ one, the moment of inertia of
disk one, is equal to ๐ผ 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. ๐ผ two is equal to four times ๐ผ
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 weโre to assume that disks
one and three experience the same change in angular velocity. To begin working this out, letโs
clear some space on screen. And having done that, letโs now
write out what we know has happened to the angular velocity of disk two. If we describe that using ๐ 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 thatโs 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 twoโs angular speed change.

So hereโs what weโre saying. In this process overall, the
systemโs change in angular momentum is zero and that the moment of inertia of these
three disks also doesnโt change, but that the angular speeds of these disks, that
is, ๐ one, ๐ two, and ๐ three, do change. Based on this, we can write that
ฮ๐ฟ, the change in angular momentum of the system, is equal to ๐ผ one times the
change in angular speed of disk one plus ๐ผ two times two radians per second, thatโs
the change in disk twoโs angular speed, plus ๐ผ three times ฮ๐ three, the change in
disk threeโs angular speed. And then we saw in the line above
that ฮ๐ฟ is zero. So then, this is the equation we
want to solve. And in it weโre told that this
change, ฮ๐ one, and this one, ฮ๐ three, are the same.

To recognize that fact
mathematically, letโs change this symbol so itโs now ฮ๐ one three to represent that
this change in angular speed applies to both disks. Itโs ฮ๐ one three that we want to
solve for. And to do it, we can first subtract
two radians per second times ๐ผ two from both sides of the equation. And then next, we factor out ฮ๐
sub one three from both terms on the right. Lastly, if we divide both sides by
๐ผ one plus ๐ผ three, we get this equation. To solve for ฮ๐ 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 ๐ผ two is equal to four times ๐ผ one and also that ๐ผ one is
equal to ๐ผ three. Therefore, we could write this
entire fraction in terms of ๐ผ one, four ๐ผ one divided by ๐ผ one plus ๐ผ one or
four ๐ผ one over two ๐ผ one, which is equal simply to two.

When we replace the fraction in our
equation with this number, we find that ฮ๐ 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.