Two bodies have their moments of inertia $I$ and $2 I$ respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular momentum will be in the ratio

A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is $\pi: 22$ then, the value of its angular speed will be $...........\,rad / s$.

$A$ man, sitting firmly over a rotating stool has his arms streched. If he folds his arms, the work done by the man is

A rod of length $50\,cm$ is pivoted at one end. It is raised such that if makes an angle of $30^o$ fro the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in $rad\,s^{-1}$ ) will be $(g = 10\,ms^{-2})$

A thin hollow cylinder open at both ends:

$(i)$ Slides without rotating

$(ii)$ Rolls without slipping, with the same speed

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