If a solid sphere is rolling the ratio of its rotational energy to the total kinetic energy is given by

$A$ non uniform rod $OA$ of linear mass density $\lambda = \lambda_0x$ $(\lambda_0 =$ const.) is suspended from ceiling with hinge joint $O$ & light string as shown in figure. Find the angular acceleration of rod just after the string is cut.

A disc of mass $M$ and radius $R$ rolls on a horizontal surface and then rolls up an inclined plane as shown in the figure. If the velocity of the disc is $v,$ the height to which the disc will rise will be

A tube of length $L$ is filled completely with an incompressible liquid of mass $M$ and closed at both ends. The tube is then rotated in a horizontal plane about one of its end with a uniform angular velocity $\omega $ . Then the force exerted by the liquid at this other end is

A uniform bar of length $'6l'$ and mass $'8m'$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ moving in the same horizontal plane with speed $2v$ and $v$ respectively, strike the bar (as shown in the fig.) and stick to the bar after collision. Total energy (about the centre of mass, $c$ ) will be

A disc initially at rest, is rotated about its axis with a uniform angular acceleration. In the first $2$ $s$ , it rotates an angle $\theta$. In the next $2\, s$, the disc will rotate through an angle