Which of the following remains constant in a central force field?

$A$ circular disc of mass $M$ has an initial angular velocity $\omega_1$. Two small spheres,each of mass $m$,are attached to the disc at opposite points on its diameter. What will be the final angular velocity of the disc?

$A$ round disc of moment of inertia $I_{2}$ about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia $I_{1}$ rotating with an angular velocity $\omega$ about the same axis. The final angular velocity of the combination of discs is

$(a)$ $A$ child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of $40\; rev/min$. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to $2/5$ times the initial value? Assume that the turntable rotates without friction.
$(b)$ Show that the child's new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?

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:

$A$ uniform rod $AB$ of length $L$ and mass $M$ is pivoted at its center $O$ such that it can rotate freely in a vertical plane. The rod is initially in a horizontal position. $A$ particle $S$ of the same mass $M$ falls vertically with velocity $v$ and strikes the rod at point $C$. $C$ is the midpoint between $O$ and $B$. Find the angular velocity of the rod immediately after the particle strikes it.