A sleeve A can slide freely along a smooth rod bent in the shape of a half-circle of radius R(figure). The system is set in rotation with a constant angular velocity `omega` about a vertical axis `OO^'`. Find the angle `theta` corresponding to the steady position of the sleeve.

Explanation

There are two forces on the sleeve, the weight `F_1` and the centrifugal force `F_2`. We resolve both forces into tangential and normal component then the net downward tangential force on the sleeve is,
`mgsin theta(1-(omega^2R)/(g)cos theta)`
This vanishes for `theta=0` and for
`theta=theta_0=cos^-1((g)/(omega^2R))`, which is real if
`omega^2Rgtg`. If `omega^2R lt g`, then `1-(omega^2R)/(g)cos theta`
is always positve for small values of `theta` and hence the net tangential force near `theta=0` opposes any displacement away from it.
`theta=0` is then stable.
If `omega^2Rgtg`, `1-(omega^2R)/(g)cos theta` is negative for small
`theta` near `theta=0` and `theta=0` is then unstable.
However `theta=theta_0` is stable becuase the force tends to bring the sleeve near the equilibrium position `theta=theta_0`.
If `omega^2R=g`, the two positions coincide and becomes a stable equilibrium point.