Two discs with moments of inertia $I_1$ and $I_2$ and angular speeds $\omega_1$ and $\omega_2$ are rotating about collinear axes passing through their centers of mass and perpendicular to their planes. If they are joined together to rotate about the same axis,the rotational kinetic energy of the system will be:

$A$ smooth uniform rod of length $L$ and mass $M$ has two identical beads of negligible size,each of mass $m$,which can slide freely along the rod. Initially,the two beads are at the center of the rod and the system is rotating with angular velocity ${\omega _0}$ about an axis perpendicular to the rod and passing through the midpoint of the rod (see figure). There are no external torques. When the beads reach the ends of the rod,the angular velocity of the system is

$A$ dancer on a smooth floor is spinning about a vertical axis at an angular velocity of $20 \ rad/s$ with her arms folded. When she stretches her arms out,the angular speed decreases to $10 \ rad/s$. If the initial moment of inertia of the dancer is $I$,what will be the new moment of inertia?

$A$ 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 uniform circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity $\omega$. Another disc of same dimension but of mass $M/4$ is placed gently on the first disc coaxially. The angular velocity of the system now is

$A$ swimmer curls their body before entering the water so that...