$A$ disc of radius $r$ is rotating about its centre with an angular speed $\omega_0$. It is gently placed on a rough horizontal surface. After what time will it be in pure rolling?

$A$ solid homogeneous sphere of mass $M$ and radius $r$ is moving on a rough horizontal surface,partly rolling and partly sliding. During this kind of motion of this sphere,

$A$ uniform solid cylinder of mass $m$ and radius $R$ is set in rotation about its axis with an angular velocity $\omega_0$,then lowered with its lateral surface onto a horizontal plane and released. The coefficient of friction between the cylinder and plane is equal to $\mu$. The time after which the cylinder starts rolling without slipping is

$A$ uniform solid sphere of mass $m$ and radius $R$ is being pulled on a horizontal surface with a force $F$ parallel to the surface,applied at its topmost point. If the acceleration of the center of mass of the sphere is $a$ and it is rolling without slipping on the surface,the value of $F$ is:

$A$ 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,
$(A)$ $\vec{V}_C-\vec{V}_A=2(\vec{V}_B-\vec{V}_C)$
$(B)$ $\vec{V}_C-\vec{V}_B=\vec{V}_B-\vec{V}_A$
$(C)$ $|\vec{V}_C-\vec{V}_A|=2|\vec{V}_B-\vec{V}_C|$
$(D)$ $|\vec{V}_C-\vec{V}_A|=4|\vec{V}_B|$

$A$ solid sphere is in rolling motion. In rolling motion,a body possesses translational kinetic energy $(K_t)$ as well as rotational kinetic energy $(K_r)$ simultaneously. The ratio $K_t : (K_t + K_r)$ for the sphere is