Rotational Work-Energy Theorem and uniform precession of a gyroscope

Question

For a gyroscope in uniform precession, what would happen if we suddenly turn off gravity? Will it stop precessing because there is no torque and so there is nothing causing the direction of the angular momentum vector to change? But if that happens, then what would happen to all that rotational kinetic energy due to precession? Also, isn't the torque provided by gravity doing work on the gyroscope according to $dW = \tau d\theta$ ? If it does do work, then shouldn't the precession angular velocity increase as time goes on according to the rotational work energy theorem $\int \vec{\tau} \cdot\vec{d\theta} = \frac{1}{2}I(\omega^2 - \omega_0^2) $ ?

Answer 1

Sudden removal of the precession inducing torque has the same effect as sudden onset of the precession inducing torque, but in opposite direction. In both cases the sudden change kicks in a nutation.

Onset of precession: if the gyroscope wheel is spinning very fast then the nutation frequency is high, and the amplitude is small. In cases where the spin rate is very fast the nutation tends to go unnoticed. Also, the nutation decays fast due to friction, and that too tends to keep the nutation unnoticed/discarded.

When a gyro wheel is released it initially gives in to the torque from gravity. Once the nutation has decayed the wheel has settled in steady precessing motion. The final height of the center of mass is lower than at the outset. As Feynman put it: "it has to go down a little, in order to go around".

When the torque is suddenly removed a nutation is kicked in too. Depending on how the gyro wheel is suspended that nutation may decay quickly, it depends on how much friction there is. Without suspension (hence without friction) the motion pattern is also known as 'Feynman's wobbling plate'.

So: when the torque inducing precession is suddenly removed not only will the precession motion cease but also the center of mass will climb a little. This climb accounts both for the change of angular momentum and the change of kinetic energy.

This behavior of precession and nutation has been experimentally verified in a tabletop experiment by Svilen Kostov and Daniel Hammer. The paper that describes their verification is titled after the Feynman quote: "It has to go down a little, in order to go around"

(Incidentally, in demonstrations the torque is almost always added gingerly, which has the side effect of suppressing nutation. So there are multiple factors that tend to hide the importance of nutation. )

About doing work

Once the motion pattern of steady precessing motion is going the torque from gravity isn't doing work.

For comparison: the case of a centripetal force sustaining circular motion. The centripetal force is a necessary condition in order for circular motion to happen, but the centripetal force is not doing work. Similarly, when there is steady precessing motion the presence of the torque is necessary for the precessing motion to continue, but that torque isn't doing work

LATER EDIT
The rotational version of the work-energy theorem relates to the case of an object spinning up to a uniform angular velocity.

Onset of gyroscopic precession is a more complex case, it inherently involves three axes of rotation.

So: onset of gyroscopic precession and an object spinning up to uniform angular velocity are fundamentally distinct cases.