TITLE: Transmission/Equilibrium Question
QUESTION [0 upvotes]: Suppose there are two wheels linked by some kind of apparatus with a transmission, in a vacuum. Wheel A is spinning, wheel B is not. Is it empirically possible for the transmission to transfer the energy from wheel A to wheel B such that wheel A stops spinning and wheel B spins at a speed very close to that which wheel A was originally spinning? If so, how would such a transmission look?
If I understand correctly, this should be a question of gear ratios and how close the gear ratio can come to infinity. Please help me understand this better. Thanks!

REPLY [1 votes]: The final state you propose is admitted by conservation of angular momentum and energy (assuming that $I_B = I_A$), so we're not sunk in the first case. Now lets consider the intermediate cases so see if the transfer is possible all the way through.
Simplest case: a passive transfer with no loss to or storage in the transmission itself.
We'll take $\omega_A = f\omega_0$ with $f \in [0,1]$ and from the conservation of angular momentum we get $\omega_B = (1-f) \omega_0$. In this case we have assumed that the transmission and any superstructure are not taking up any of the angular momentum. By construction the angular momentum is conserved, so we need only check on the energy:
$$ \begin{align*}
\frac{1}{2}I_A\omega_0^2 
&\overset{?}= \frac{1}{2}I_A \left[ f \omega_0 \right]^2 + \frac{1}{2}I_B \left[ (1-f) \omega_0 \right]^2 \\
&\overset{?}= \frac{1}{2}I_A \left[ f^2+(1-f)^2 \right] \omega_0^2 \\
&\overset{?}= \frac{1}{2}I_A [1 - 2f + 2f^2] \omega_0^2 \\
&\ge \frac{1}{2}I_A [1 - 2f + 2f^2] \omega_0^2 \\
\end{align*}$$
The last line comes from the range restrictions on $f$.
The energy drops steadily until $f = 0.5$ at which point the angular kinetic energy is only half of its original value. From then on the system needs a net energy input to continue the transfer.
It is clear that we need to store some energy somewhere during the transfer.
Next step: a system that can store energy
We need to figure out how to store energy without affecting the net angular momentum. Possibly some arrangement with counter rotating flywheels (enter stage left: a differential) would do the job, but this isn't obvious and I don't know how to proceed off hand.
We've entered the realm of mechanical cleverness. I want my Lego Technic back, but it all went to charity years ago.

NB: the likely hood that a purely passive transfer between exactly two components wouldn't work should jump out at you because of the different polynomial dependence of the angular kinetic energy and the angular momentum.
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