A block of mass $m$ is attached to a pulley disc of equal mass $m$ and radius $r$ by means of a slack string as shown. The pulley is hinged about its centre on a horizontal table and the block is projected with an initial velocity of $5 \mathrm{m}/\mathrm{s}$. Its velocity when the string becomes taut will be

A uniform solid sphere of radius $r$ is rolling on a smooth horizontal surface with velocity $V$ and angular velocity $\omega =(V=\omega R)$. The sphere collides with a sharp edge on the wall as shown in figure. The coefficient of friction between the sphere and the edge $\mu =1/5$. Just after the collision the angular velocity of the sphere becomes equal to zero. The linear velocity of the sphere just after the collision is equal to

A conical pendulum consists of a simple pendulum moving in a horizontal circle as shown. $C$ is the pivot, $O$ the centre of the circle in which the pendulum bob moves and $\omega$ the constant angular velocity of the bob. If $\overrightarrow{L}$ is the angular momentum about point $C$, then

A solid sphere of mass $m$ is lying at rest on a rough horizontal surface. The coefficient of friction between the ground and sphere is $\mu$. The maximum value of $F$, so that the sphere will not slip, is equal to