A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should by unbinding the strings to achieve a speed of $4\,ms ^{-1}$, is$........cm$. $\left(\right.$ take $\left.g=10\,ms ^{-2}\right)$

$A$ uniform rod of mass $M$ is hinged at its upper end. $A$ particle of mass $m$ moving horizontally strikes the rod at its mid point elastically. If the particle comes to rest after collision find the value of $M/m =?$

A solid sphere of mass $m$ and radius $R$ is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $E_{sphere}/E_{cylinder}$ will be

The angular velocity of a body is $\mathop \omega \limits^ \to   = 2\hat i + 3\hat j + 4\hat k$ and a torque $\mathop \tau \limits^ \to   = \hat i + 2\hat j + 3\hat k$ acts on it. The rotational power will be .......... $W$

Four point masses are fastened to the corners of $a$ frame of negligible mass lying in the $xy$ plane. Let $w$ be the angular speed of rotation. Then

A tangential force $F$ is applied on a disc of radius $R$, due to which it deflects through an angle $\theta $ from its initial position. The work done by this force would be