A uniform solid sphere is given velocity $v_0$ and angular velocity ${\omega }_0=\frac{v_0}{2\mathrm{R}}$ and placed on a rough horizontal surface as shown. The velocity of centre of mass when pure rolling starts, is

 

A hollow sphere is rolling (without slipping) on horizontal surface with velocity $v_0$. It then moves up a curved track. If sliding does not occur, then maximum height moved-up by sphere is

A block of mass $m$ is attached at one end of string and other end of string is wrapped over a solid cylinder of mass $2\mathrm{m}$ and radius $R$ as shown. When released, the tension in the string is

A solid sphere of mass $M$ and radius $R$ is pulled by a force $F$ as shown in figure. If the sphere does not slip over the surface, then frictional force acting on the sphere is

A hollow sphere is released from rest on rough inclined surface of inclination $\theta$. If coefficient of friction is $\mu$, then maximum value of $\theta$ so that the sphere rolls without slipping is

Three identical rings of mass $M$ each and radius $R$ are joined as shown in figure. If moment of inertia about an axis passing through point $O$ and perpendicular to the plane of rings is equal to ($O$ is geometrical centre of system) $kM{R}^2$, then value of $k$ is _____.

A solid cylinder of mass $3\mathrm{kg}$ and radius $R$ is arranged as shown in figure. A string is wrapped over the cylinder. The system is released from rest. The tension in the string is ______ newton. $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

A block of mass $M$ is released from rest on top of a smooth inclined plane of inclination $\theta$. The block is connected to a light string which is wound over a disc shaped pulley of mass $M$ and radius $R$ as shown. If length of inclined surface is $l$, then time (in second) taken by the block to reach the bottom (Given $\frac{l}{g\sin \theta }=3$, in S.I. unit)

A force $F$ acts at the end $Q$ of a uniform rod placed at rest on a smooth horizontal plane. If ratio of magnitude of initial acceleration of the point $P$ and $Q$, is $1:k$, then value of $k$ is ______.