$A$ thin metal disc of radius $0.25\,m$ and mass $2\,kg$ starts from rest and rolls down an inclined plane. If its rotational kinetic energy is $4\,J$ at the foot of the inclined plane,then the linear velocity (in $m/s$) at the same point is:

Two discs having masses in the ratio $1:2$ and radii in the ratio $1:8$ roll down without slipping one by one from an inclined plane of height $h$. The ratio of their linear velocities on reaching the ground is ........

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.

$A$ solid sphere of mass $m$ and radius $r$ rolls down an inclined plane. What is the ratio of its rotational kinetic energy to its translational kinetic energy?

$A$ solid sphere of mass $m$ and radius $R$ is released from rest on a sufficiently rough long inclined plane of inclination $\theta$ . Consider four points $A$,$B$,$C$ and $D$ on the sphere as shown in the figure. After one revolution,which of the following statements is correct regarding the acceleration of these points?

$A$ uniform ring of radius $R$ is moving on a horizontal surface with speed $v$,then climbs up a ramp of inclination $30^{\circ}$ to a height $h$. There is no slipping in the entire motion. Then,$h$ is