Find the displacement of the point of contact with the ground when a ring of radius $R$ completes a half-rotation.

$A$ wheel is rolling on a plane surface. The speed of a particle on the highest point of the rim is $8 \ m/s$. The speed of the particle on the rim of the wheel at the same level as the centre of the wheel will be

$A$ solid sphere of mass $5 \,kg$ rolls on a plane surface. Find its kinetic energy at an instant when its centre moves with speed $4 \,m/s$. (in $\,J$)

$A$ symmetrical body of mass $M$ and radius $R$ is rolling without slipping on a horizontal surface with linear speed $v$. Then its angular speed 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.

At time $t=0$,a disk of radius $1 \text{ m}$ starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha = \frac{2}{3} \text{ rad s}^{-2}$. $A$ small stone is stuck to the disk. At $t=0$,it is at the contact point of the disk and the plane. Later,at time $t=\sqrt{\pi} \text{ s}$,the stone detaches itself and flies off tangentially from the disk. The maximum height (in $\text{m}$) reached by the stone measured from the plane is $\frac{1}{2} + \frac{x}{10}$. The value of $x$ is. . . . . . . [Take $g=10 \text{ m s}^{-2}$.]