I've wondered about this for quite a while (perhaps fifteen years but forgot about it periodically) and have not been able to find the answer. I can't find the answer on Google, either.
My question is this: why does a spinning flywheel, supported by reasonably sensible bearings* that is allowed to coast down to rest, rock backwards slightly before finally stopping completely? I assume that it doesn't come to rest after the first "rock" or reversal of direction, rather it decays.
*I mention this because I haven't observed it when the bearings of whatever it is are particularly tight or knackered, or there is a lot of parasitic drag, such as when the flywheel has an engine attached. As you'd probably expect.
I have a rubbish video: https://www.youtube.com/watch?v=khXpTaNs9Fw
That's a Technics 1210 with the magnet removed. I removed it for the video to make sure that the rocking at the end of the coasting wasn't caused by the back EMF induced in the motor.
I tried to illustrate that it does it in both directions.
I've been wondering about the zero crossing point, or the point in time between (arbitrarily) clockwise rotation and anticlockwise rotation. In my mind, if the reason it does this is simply that the friction is of a higher magnitude than the inertia (it has to be, otherwise it would not decelerate if it were of an equal magnitude and would accelerate if it were of a lower magnitude) then how can it be of a magnitude significant to overcome the reverse static friction, which is higher than the dynamic friction seen during the deceleration?
Best Answer
When a flywheel decelerates from its initial angular velocity $\omega_0$ to standstill, that is caused by torque $\tau$, acc. Newton:
$$\tau=I\alpha,$$
where $I$ is the inertia moment of the flywheel and $\alpha$ the angular deceleration:
$$\alpha=-\frac{\mathrm{d}\omega}{\mathrm{d}t}$$
(The minus sign represents deceleration)
If for simplicity's sake we assume $\tau$ to be constant, then:
$$\alpha=-\frac{I}{\tau}$$
And the angular velocity $\omega(t)$, as a function of time:
$$\omega(t)=\omega_0-\frac{I}{\tau}\Delta t$$
If torque persists then $\omega \leq 0$ when:
$$\Delta t \geq \frac{\omega_0 \tau}{I}$$
So this is the moment in time the flywheel's sense of rotation would reverse.
Now, if we consider a freewheeling flywheel with no external, intentional braking torque imposed on it, then the only phenomenon that can reduce the angular velocity is friction. Even the best constructed flywheel experiences some friction in the bearings, as well as some air drag. These forces do provide a torque, in the opposite sense the of the motion and this works to reduce the angular velocity, as described above.
However, friction forces always act in the opposite sense of motion and decay completely when motion decays. Mathematically:
$$\omega=0 \implies \tau_{\mathrm{friction}}=0$$
This means that friction force (torque, to be precise) can never be responsible for any reversal of the sense of rotation.
But if we look at the case of the turntable (YouTube in the OP's comment), the required reversing torque is quite easily explained.
The 'secret' lies in the turntable's drive belt. The drive belt is not perfectly unstretchable (ineleastic, usually they're made of rubber, sometimes fabric reinforced).
When the turntable comes to a halt, the drive belt becomes slightly stretched, making it act like a stretched spring. Like a spring it now provides a small restoring torque, acting in the opposite sense of the original sense of rotation. As above, this torque now causes angular acceleration in the opposite sense of the original sense of rotation, causing the brief reversal.
A very related way of looking at it is that the drive belt stores a small amount of potential energy $U$. during the final stages of braking. This is then converted to rotational kinetic energy $K$:
$$U=K=\frac12 I\omega_1^2,$$
where $\omega_1$ is the angular velocity at the end of the reversal.
'On paper' the turntable should enter an oscillation (you can kind of see it in the video) but friction causes that motion to cease quickly because friction expends work (energy).