Questions in rotational-motion

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The moment of inertia of a solid cylinder about its axis is I. It is allowed to roll down an incline without slipping. If its angular velocity at the bottom be $\omega $, then K.E. of the cylinder will be
A solid ball of mass m and radius r rolls without slipping along the track shown in the fig. The radius of the circular part of the track is R. The ball starts rolling down the track from rest from a height of 8R from the ground level. When the ball reaches the point P then its velocity will be
A ring takes time ${{t}_{1}}$ in slipping down an inclined plane of length L and takes time ${{t}_{2}}$ in rolling down the same plane. The ratio $\frac{{{t}_{1}}}{{{t}_{2}}}$ is
A ring of radius 4a is rigidly fixed in vertical position on a table. A small disc of mass m and radius a is released as shown in the fig. When the disc rolls down, without slipping, to the lowest point of the ring, then its speed will be
A disc of mass M and radius R rolls in a horizontal surface and then rolls up an inclined plane as shown in the fig. If the velocity of the disc is v, the height to which the disc will rise will be
Two uniform similar discs roll down two inclined planes of length S and 2S respectively as shown is the fig. The velocities of two discs at the points A and B of the inclined planes are related as
A mass M is supported by a mass less string would wound a uniform cylinder of mass M and radius R. On releasing the mass from rest, it will fall with acceleration
A uniform disc of radius R and mass M can rotate on a smooth axis passing through its centre and perpendicular to its plane. A force F is applied on its rim. See fig. What is the tangential acceleration
A massless string is wrapped round a disc of mass M and radius R. Another end is tied to a mass m which is initially at height h from ground level as shown in the fig. If the mass is released then its velocity while touching the ground level will be
A cylinder of mass M and radius r is mounted on a frictionless axle over a well. A rope of negligible mass is wrapped around the cylinder and a bucket of mass m is suspended from the rope. The linear acceleration of the bucket will be

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