In the figure,$E$ and $v_{cm}$ represent the total energy and speed of the centre of mass of an object of mass $1 \ kg$ in pure rolling. The object 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:

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.

$A$ 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 $cm$)

$A$ wheel is rolling along the ground with a speed of $2\ m/s$. The magnitude of the velocity of the points at the extremities of the horizontal diameter of the wheel is equal to:

Consider a cylinder of mass $M$ resting on a rough horizontal rug that is pulled out from under it with acceleration $a$ perpendicular to the axis of the cylinder. What is the friction force $F_{friction}$ at point $P$? It is assumed that the cylinder does not slip.