$A$ solid sphere is rolling without slipping on a horizontal plane. The ratio of the linear kinetic energy of the centre of mass of the sphere and rotational kinetic energy is $:$

Explain why friction is necessary to make the disc in the figure roll in the direction indicated.
$(a)$ Give the direction of frictional force at $B$,and the sense of frictional torque,before perfect rolling begins.
$(b)$ What is the force of friction after perfect rolling begins?

$A$ solid sphere rolls without slipping on a rough surface and the centre of mass has a constant speed $v_0$. If the mass of the sphere is $m$ and its radius is $R$,then find the angular momentum of the sphere about the point of contact $P$.

$A$ circular ring of mass $10 \text{ kg}$ rolls along a horizontal floor. The center of mass of the ring has a speed of $1.5 \text{ m/s}$. The work required to stop the ring is: (in $\text{ J}$)

If a solid sphere of mass $1\, kg$ and radius $0.1\, m$ rolls without slipping at a uniform velocity of $1\, m/s$ along a straight line on a horizontal floor,the kinetic energy is (in $, J$)

$A$ 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 $...........\,rad/s$.