$A$ wheel of radius $2 \ cm$ is at rest on a horizontal surface. $A$ point $P$ on the circumference of the wheel is in contact with the horizontal surface. When the wheel rolls without slipping on the surface,the displacement of point $P$ after half a rotation of the wheel is:

$A$ wheel of radius $1 \ m$ rolls through $180^{\circ}$ over a plane surface. The magnitude of the displacement of the point of the wheel initially in contact with the surface is:

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?

$A$ 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.

$A$ 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 $\,J$)

$A$ circular hoop of radius $50 \ cm$ and mass $1 \ kg$ rotating with an angular velocity $\omega_0$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. Let $v$ be the velocity of the centre of the hoop when it ceases to slip. The ratio $v / \omega_0$ will be (in $cm$)