$A$ hollow sphere of mass $m$ and radius $R$ is rolling downward on a rough inclined plane of inclination $\theta$. If the coefficient of friction between the hollow sphere and incline is $\mu$,then ........

$A$ disc of radius $20 \, cm$ and mass $0.5 \, kg$ is rolling on an inclined plane with an angle of inclination $45^{\circ}$. Find the friction force required for the disc to perform pure rolling.

$A$ disc rolls without slipping down an inclined plane of length $L$ and inclination $\theta$. What will be its velocity at the bottom?

Prove that the velocity $v$ of translation of a rolling body (like a ring,disc,cylinder,or sphere) at the bottom of an inclined plane of height $h$ is given by $v^{2} = \frac{2gh}{1 + k^{2}/R^{2}}$ using dynamical considerations (i.e.,by considering forces and torques). Note: $k$ is the radius of gyration of the body about its symmetry axis,and $R$ is the radius of the body. The body starts from rest at the top of the plane.

The rotational kinetic energy of a solid sphere of mass $3 \; kg$ and radius $0.2 \; m$ rolling down an inclined plane of height $7 \; m$ is (in $; J$)

$A$ hollow cylinder of mass $m$ and radius $R$ is spinned to a clockwise angular velocity $\omega_0$ and then gently placed on an inclined plane for which the coefficient of friction is $\mu = \tan \theta$,where $\theta$ is the angle of the inclined plane with the horizontal. The centre of mass of the cylinder will remain stationary for time: