A solid cylinder of mass m and radius R is suspended by a light inextensible string wrapped around it. The other end of the string is fixed to a point O on the ceiling. The cylinder is released from rest and falls vertically, unwinding the string without slipping. At an instant when the downward speed of the center of mass of the cylinder is v , the magnitude of the angular momentum of the cylinder about the point O

Step-by-step solution

The cylinder undergoes both translational and rotational motion. The velocity of the center of mass is v downwards. Since the string is fixed at O and unwinds without slipping, the angular velocity of the cylinder is = v R . The angular momentum of the cylinder about the point O is the sum of its translational and rotational angular momenta: L_O = L_ trans + L_ rot The translational angular momentum about O is: L_ trans = m v r_ = m v R The rotational angular momentum about the center of mass is: L_ rot = I_ cm = (