$A$ solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about the axis of rotation of the sphere to the total energy of the moving sphere is $\pi: 22$,then the value of its angular speed will be $...........\,rad/s$.

For a rolling body,the velocities of points $P_1$ and $P_2$ are ${\vec v_1}$ and ${\vec v_2}$,respectively. Which of the following is correct?

$A$ ball rolls without slipping. The radius of gyration of the ball about an axis passing through its centre of mass is $K$. If the radius of the ball is $R$,then the fraction of total energy associated with its rotational energy is:

$A$ disc of radius $R$ is rotating with an angular velocity $\omega_0$ about a horizontal axis. It is placed on a horizontal table. The coefficient of kinetic friction is $\mu_k$.
$(a)$ What was the velocity of its centre of mass before being brought in contact with the table?
$(b)$ What happens to the linear velocity of a point on its rim when placed in contact with the table?
$(c)$ What happens to the linear speed of the centre of mass when the disc is placed in contact with the table?
$(d)$ Which force is responsible for the effects in $(b)$ and $(c)$?
$(e)$ What condition should be satisfied for rolling to begin?
$(f)$ Calculate the time taken for the rolling to begin.

$A$ disc is rolling (without slipping) on a horizontal surface. $C$ is its centre and $Q$ and $P$ are two points on the same horizontal line passing through $C$,such that $Q$ is at a distance $r$ from $C$ and $P$ is at a distance $r$ from $C$ on the opposite side. Let $V_P, V_Q$ and $V_C$ be the magnitudes of velocities of points $P, Q$ and $C$ respectively,then:

$A$ boy is pushing a ring of mass $2 \ kg$ and radius $0.5 \ m$ with a stick as shown in the figure. The stick applies a force of $2 \ N$ on the ring and rolls it without slipping with an acceleration of $0.3 \ m/s^2$. The coefficient of friction between the ground and the ring is large enough that rolling always occurs,and the coefficient of friction between the stick and the ring is $(P/10)$. The value of $P$ is