$A$ 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:

Read each statement below carefully,and state,with reasons,if it is true or false;
$(a)$ During rolling,the force of friction acts in the same direction as the direction of motion of the $CM$ of the body.
$(b)$ The instantaneous speed of the point of contact during rolling is zero.
$(c)$ The instantaneous acceleration of the point of contact during rolling is zero.
$(d)$ For perfect rolling motion,work done against friction is zero.
$(e)$ $A$ wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling) motion.

$A$ 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

$A$ sphere of mass $50 \, g$ and diameter $20 \, cm$ is rolling without slipping with a velocity of $5 \, cm/s$. Its total 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 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 $...........\,rad/s$.