Keeping the mass of the Earth constant, if its radius is halved, then what will be the duration of the day (in $\text{ hours}$)?

$A$ thin circular ring of mass $M$ and radius $R$ is rotating in a horizontal plane with angular velocity $\omega$ about an axis passing through its center and perpendicular to the plane. If another ring of the same size but mass $\frac{M}{4}$ is placed coaxially on the first ring,then the new angular velocity of the system is:

$A$ circular stage is free to rotate about a vertical axis passing through its centre. $A$ tortoise is sitting at the corner of the stage. The stage is provided with an initial angular velocity $\omega_0$. If the tortoise starts moving along one chord at a constant speed with respect to the stage,then how does the angular velocity of the stage $\omega(t)$ vary with time $t$?

$A$ child is standing at the center of a rotating platform with hands folded. The kinetic energy of the system is $K$. The child now stretches his hands,doubling the moment of inertia. The kinetic energy of the system will now become ........

$A$ thin smooth rod of length $L$ and mass $M$ is rotating freely with angular speed $\omega_0$ about an axis perpendicular to the rod and passing through its centre. Two beads of mass $m$ and negligible size are at the centre of the rod initially. The beads are free to slide along the rod. The angular speed of the system,when the beads reach the opposite ends of the rod,will be:

$A$ solid sphere is rotating in space. If the radius of the sphere is increased while keeping its mass constant,which of the following physical quantities will not change?