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 thetathe 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?

A uniform frictionless ring of mass M and radius R, stands vertically

1.	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 thetathe 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» 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`