$A$ particle of mass $m$ moves with a velocity $v$ along the rim of a disc of radius $R$ in the counter-clockwise direction. The disc has a moment of inertia $I$ and is rotating in the clockwise direction with an angular velocity $\omega$. If the particle stops moving,what will be the angular velocity of the disc?

$A$ boy stands with his hands folded at the center of a platform rotating about its central axis. The kinetic energy of the system is $K$. The boy now stretches his hands out, which doubles the moment of inertia of the system. The kinetic energy of the system will become

The angular momentum of a system of particles is not conserved:

$A$ man stands on a rotating platform with his arms stretched horizontally,holding a $5\; kg$ weight in each hand. The angular speed of the platform is $30$ revolutions per minute. The man then brings his arms close to his body,with the distance of each weight from the axis changing from $90\; cm$ to $20\; cm$. The moment of inertia of the man together with the platform may be taken to be constant and equal to $7.6\; kg\; m^2$.
$(a)$ What is his new angular speed? (Neglect friction.)
$(b)$ Is kinetic energy conserved in the process? If not,from where does the change come about?

$A$ thin and circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about its geometrical axis with an angular velocity $\omega$. If another disc of same dimensions but of mass $M/4$ is placed gently on the first disc coaxially,then the new angular velocity of the system is:

If the volume of the Earth suddenly shrinks to $1/64$ of its present volume without any change in its mass,then the duration of the day on Earth will be ..... hours.