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 ______.

A uniform solid sphere of mass $M$ is at rest on a rough horizontal surface. The coefficient of friction between ground and sphere is $\mu$. Force $F$ is applied horizontally such that its line of action passes through the centre of the sphere. If maximum value of $F$, so that the sphere will not slip, is equal to $\frac{N}{4}\mu \mathrm{Mg}$, then value of $N$ is _____.

A disc of radius $R$ rotating with an angular velocity ${\omega }_0$ is placed on a rough horizontal surface. If initial velocity of centre of disc is zero, then velocity of centre of disc when it ceases to slip, comes out to be $\frac{{\omega }_{0}R}{k}$, value of $k$ is _____.