Spinning Wheel and Rotating Stool
Q: In DR I sat on a stool and held the spinning bicycle wheel. When I flipped over the wheel I started to spin. What was going on??
A: The key point here is that angular momentum (just like linear momentum) is conserved. That means that, if there are no external torques, the angular momentum is constant. No matter when we measure it we will get the same value. We have to be careful to always take all of the angular momentum into account to find that the total angular momentum is conserved.
So, first we had the wheel spinning with some angular momentum. Let's call that L1. With the right hand rule we find that the angular momentum of the wheel is up. You are sitting on the stool, but since you're not spinning you have no angular velocity and thus no angular momentum. So, at the beginning the total angular momentum is just that angular momentum of the wheel, L1 (up).
Then you flipped over the wheel. Since the wheel still has the same angular speed and the same moment of inertia the magnitude of the angular momentum is the same as before, L1. However, since you flipped it over the right hand rule will tell you that the angular momentum points down. Now you are spinning as well, so we will say that your angular momentum is L2. With the right hand rule we find that your angular momentum is up. So, the total angular momentum at the end is L1 (down) + L2 (up)
The conservation of angular momentum says that the total angular momentum can not change (since there are no torques from an external source acting on you or the wheel). So, that means that the angular momentum before and after you flipped the wheel should be the same. That is:
L1(up) = L1(down) + L2(up)
Since down and up are opposite directions we can switch the direction just by adding a negative sign. We rewrite L1 (down) as -L1(up) to get:
L1(up) = -L1(up) + L2(up)
For this equation to be true, it must be the case that:
L2 (up)=2 * L1(up)
Well isn't that interesting! We've just discovered that our angular momentum due to the spinning stool is twice the angular momentum of the wheel.
To explain a bit more conceptually, when you flipped over the wheel you changed its angular momentum. Since the total angular momentum of the system has to be conserved this caused you to spin in the opposite direction with an angular momentum twice as large as the wheel’s (since you have to both zero out the downward angular momentum of the wheel and add more upward angular momentum to equal the original upward angular momentum)