A stick of length $L$ and mass $M$ lies on a frictionless horizontal surface on which it is free to move in any ways. A ball of mass $m$ moving with speed $v$ collides elastically with the stick as shown in the figure. If after the collision the ball comes to rest, then what should be the mass of the ball ?

A solid sphere rolls without slipping, first horizontally and then up to a point $X$ at height $h$ on an inclined plane before rolling down, as shown in the figure below. The initial horizontal speed of the sphere is

$A$ rod hinged at one end is released from the horizontal position as shown in the figure. When it becomes vertical its lower half separates without exerting any reaction at the breaking point. Then the maximum angle $‘\theta ’$ made by the hinged upper half with the vertical is ......... $^o$.

The $M.I.$ of a body about the given axis is $1.2\,kg \times m^2$ and initially the body is at rest. In order to produce a rotational kinetic energy of $1500\,joule$ an angular acceleration of $25\,rad/sec^2$ must be applied about that axis for a duration of ........ $\sec$.

A solid cylinder $P$ rolls without slipping from rest down an inclined plane attaining a speed $v_p$ at the bottom. Another smooth solid cylinder $Q$ of same mass and dimensions slides without friction from rest down the inclined plane attaining a speed $v_q$ at the bottom. The ratio of the speeds $\frac{v_q}{v_p}$ is

Two coaxial discs, having moments of inertia $I_1$ and $\frac{I_1}{2}$ are a rotating with respectively angular velocities $\omega_1$ and $\frac{\omega_1}{2}$, about their common axes. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If $E_f$ and $E_i$ are the final and initial total energies, then $(E_f -E_i)$ is