$A$ spherical rigid ball is released from rest and starts rolling down an inclined plane from height $h=7 \, m$,as shown in the figure. It hits a block at rest on the horizontal plane (assume elastic collision). If the mass of both the ball and the block is $m$ and the ball is rolling without sliding,then the speed of the block after collision is close to ............. $m/s$.

In the given figure,a ring of mass $m$ is kept on a horizontal surface,and a body of equal mass $m$ is attached through a string wound on the ring. When the system is released,the ring rolls without slipping. Consider the following statements and choose the correct option.
$(i)$ Acceleration of the centre of mass of the ring is $\frac{g}{3}$.
$(ii)$ Acceleration of the hanging particle is $\frac{2g}{3}$.
$(iii)$ Frictional force (on the ring) acts in the forward direction.
$(iv)$ Frictional force (on the ring) acts in the backward direction.

In the figure shown,a ring $A$ is initially rolling without sliding with a velocity $v$ on the horizontal surface of the body $B$ (of the same mass as $A$). All surfaces are smooth. $B$ has no initial velocity. What will be the maximum height reached by $A$ on $B$?

$A$ particle performs rotational motion with an angular momentum $L$. If the frequency of rotation is doubled and its kinetic energy becomes one-fourth,the new angular momentum becomes:

Two cylindrical hollow drums of radii $R$ and $2R$ and of a common height $h$ are rotating with angular velocities $\omega$ (anti-clockwise) and $\omega$ (clockwise) respectively. Their axes,which are fixed,are parallel and in a horizontal plane separated by $3R$. They are now brought into contact.
$(a)$ Show the frictional forces just after contact.
$(b)$ Identify forces and torques external to the system just after contact.
$(c)$ What would be the ratio of final angular velocities when friction ceases?

Moment of Inertia of a thin uniform rod rotating about the perpendicular axis passing through its centre is $I$. If the same rod is bent into a ring and its moment of inertia about its diameter is $I^{\prime}$,then the ratio $\frac{I}{I^{\prime}}$ is