We're probably all familiar with the demonstration of gyroscopic precession where a man sits on a swivel chair holding a bicycle wheel (like this: https://www.youtube.com/watch?v=1fWl2LNpcak).
And the rate of precession is given by
Suppose that we don't neglect the mass of the man on the chair, and instead, he had a relatively large mass (but still consider a friction-less swivel). What impact would this have on:
- the final rate of precession
- the rate of acceleration of the mass of the man.
What impact does the mass of the man have on the torque required to turn the wheel?
I expect that the angle the wheel is turned would have maximum effect (and therefore require maximum torque) at $90$ degrees to the vertical - beyond this the effect (and torque) would be reduced.
The above formula suggests that increasing the torque ($\tau$), or decreasing the angular momentum ($L_s$) will increase the rate of precession. Except that I expect there are limits (critical points) at which maximum torque will have no effect, or decreasing angular momentum (reduced spin velocity) will have no effect. - How do we consider these critical points mathematically?