If the Earth were to suddenly contract to $\frac{1}{n}$ th of its present radius without any change in its mass,the duration of the new day will be nearly

$A$ thin circular ring of mass $M$ and radius $r$ is rotating about its axis with an angular speed $\omega$. Two particles each of mass $m$ are now attached at diametrically opposite points. The angular speed of the ring will become

$A$ solid sphere with uniform density and radius $R$ is rotating initially with constant angular velocity $\omega_1$ about its diameter. After some time,it starts losing mass at a uniform rate,with no change in its shape. The angular velocity of the sphere when its radius becomes $R/2$ is $x\omega_1$. The value of $x$ is . . . . . . .

$A$ 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?

$A$ boy stands at the center of a horizontal platform with $2 \ kg$ masses in each hand,held close to his body. The platform rotates about a vertical axis passing through its center at a speed of $2 \ rev/s$. The moment of inertia of the system is $1 \ kg \cdot m^2$. When the boy extends his arms fully with the masses,the moment of inertia becomes $2 \ kg \cdot m^2$. In the second state,the kinetic energy of the system compared to the first state will be:

$A$ thin rod of mass $0.9\, kg$ and length $1\, m$ is suspended,at rest,from one end so that it can freely oscillate in the vertical plane. $A$ particle of mass $0.1\, kg$ moving in a straight line with velocity $80\, m/s$ hits the rod at its bottom-most point and sticks to it. The angular speed (in $rad/s$) of the rod immediately after the collision will be