Three solid spheres are made to move on a rough horizontal surface. Sphere $P$ is given a spin and released. Sphere $Q$ is given a forward linear velocity. Sphere $R$ is given linear and rotational motions as shown in the figure. Directions of the friction force on spheres $P, Q, R$ are respectively

$A$ 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.

$A$ sphere is rolling without slipping on a fixed horizontal plane surface. In the figure,$A$ is the point of contact,$B$ is the centre of the sphere and $C$ is its topmost point. Then,
$(A)$ $\vec{V}_C-\vec{V}_A=2(\vec{V}_B-\vec{V}_C)$
$(B)$ $\vec{V}_C-\vec{V}_B=\vec{V}_B-\vec{V}_A$
$(C)$ $|\vec{V}_C-\vec{V}_A|=2|\vec{V}_B-\vec{V}_C|$
$(D)$ $|\vec{V}_C-\vec{V}_A|=4|\vec{V}_B|$

$A$ disc is rolling, the velocity of its centre of mass is $v_{cm}$. Which one will be correct?

$A$ force $F$ is applied at the centre of a disc of mass $M$. The minimum value of the coefficient of friction of the surface for pure rolling is:

$A$ wheel is rolling straight on ground without slipping. If the axis of the wheel has speed $v$,the instantaneous velocity of a point $P$ on the rim,defined by angle $\theta$ relative to the vertical,relative to the ground will be: