A light thread with a body of mass `m` tied to its end is wound on a uniform solid cylinder of mass `M` and radius `R`. At a moment `t=0` the system is set in motion. Assuming the friction in the axle of the cylinder to be negligible, find the time dependence of
(a) the angular velocity of the cylinder and
(b) the kinetic energy of the whole system ( `M=2m`).

In the following figure, a body of mass m is tied at one end of a light string and this string is wrapped around the solid cylinder of mass M and radius R. At the moment t = 0 the system starts moving. If the friction is negligible, angular velocity at time t would be

In the following figure, a body of mass m is tied at one end of a light string and this string is wrapped around the solid cylinder of mass M and radius R. At the moment t = 0 the system starts moving. If the friction is negligible, angular velocity at time t would be

A rope is wound round a hollow cylinder of mass M and radius R. If the rope is pulled with a force F newton, then the angular acceleration of the cylinder will be

A rope is wound round a hollow cylinder of mass M and radius R. If the rope is pulled with a force F newton, then the angular acceleration of the cylinder will be

A uniform disc of radius R and mass M is mounted on an axis supported in fixed friction less bearing. A light cord is wrapped around the rim of the wheel and suppose we hang a body of mass m from the cord. (a) Find the angular acceleration of the disc and tangential acceleration ofpoint on the rim. (b) At a moment t= 0, the system is set in motion, find the time dependence of the angular velocity of the system and the kinetic energy ofthe whole system.

Find the moment of inertia of a solid cylinder of mass M and radius R about a line parallel to the axis of the cylinder and on the surface of the cylinder.

Find the moment of inertia of a solid cylinder of mass M and radius R about a line parallel to the axis of the cylinder and on the surface of the cylinder.

Find the moment of inertia of a solid cylinder of mass M and radius R about a line parallel to the axis of the cylinder and on the surface of the cylinder.

A bullet of mass m moving with a velocity of u just grazes the top of a solid cylinder of mass M and radius R resting on a rough horizontal surface as shown and is embedded in the cylinder after impact. Assuming that the cylinder rolls without slipping, find the angular velocity of the cylinder and the final velocity of the bullet.