A uniform frictionless ring of mass M and radius R, stands vertically on the ground. A wall touches the ring on the left and another wall of height R touches the ring on right (see figure).There is a small bead of mass m positioned at the top of the ring. The bead is given a gentle push and it being to slide down the ring as shown. All surfaces are frictionless.

(a) As the bead slides, up to what value of angle `theta` the force applied by the ground on the ring is larger than Mg?
(b) Write the torque of force applied by the bead on the ring about point A as function of `theta`.
(c) What is the maximum possible value of torque calculated in (b)? Using this result tell what is the largest value of `(m)/(M)` for which the ring never rises off the ground ?

Answer

(a) `theta = cos^(-1) ((2)/(3))`
(b) `mgR (2 cos theta - 3 cos^(2) theta)`
(c) `tau_("max") = (mgR)/(3); ((m)/(M))_("max") = 3`