$A$ solid sphere with a velocity (of centre of mass) $v$ and angular velocity $\omega$ is gently placed on a rough horizontal surface. The frictional force on the sphere:

$A$ wheel of radius $1 \text{ m}$ rolls forward half a revolution on a horizontal ground. The magnitude of the displacement of the point of the wheel initially in contact with the ground is

$A$ uniform non-deformable cylinder of mass $m$ and radius $R$ is rolling without slipping on a horizontal rough surface. The force of friction is

$A$ spherical shell of $1 \, kg$ mass and radius $R$ is rolling with angular speed $\omega$ on a horizontal plane (as shown in the figure). The magnitude of the angular momentum of the shell about the origin $O$ is $\frac{a}{3} R^{2} \omega$. The value of $a$ will be ..............

For a rolling spherical shell,the ratio of rotational kinetic energy and total kinetic energy is $\frac{x}{5}$. The value of $x$ is ................

In the figure,$E$ and $v_{cm}$ represent the total energy and speed of the centre of mass of an object of mass $1 \ kg$ in pure rolling. The object is