$A$ solid sphere is rotating in free space. If the radius of the sphere is increased while keeping its mass constant,which of the following quantities will remain unaffected?

$(a)$ $A$ 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?

An ant is sitting on the edge of a rotating disc. If the ant walks along the diameter to reach the other end, how will the angular velocity of the disc change?

State and explain the law of conservation of angular momentum.

If the Earth shrinks to $1/8$ of its original volume, while maintaining the same mass, then the duration of the day will be (in $hrs$)

$A$ circular platform is free to rotate in a horizontal plane about a vertical axis passing through its center. $A$ tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity $\omega_0$. When the tortoise moves along a chord of the platform with a constant velocity (with respect to the platform), the angular velocity of the platform $\omega(t)$ will vary with time $t$ as: