If the earth suddenly shrinks to $\frac{1}{64}$th of its original volume with its mass remaining the same,the period of rotation of earth becomes $\frac{24}{x} \text{ h}$. The value of $x$ is $.......$

$A$ circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two particles of mass $M$ are gently attached to the ring at the ends of a diameter. The ring will now rotate with an angular velocity $\omega'$ equal to:

$Assertion$ : For a system of particles under a central force field,the total angular momentum is conserved.
$Reason$ : The torque acting on such a system is zero.

$A$ circular disc with moment of inertia $I_t$ is rotating in a horizontal plane about its symmetry axis with a constant angular speed $\omega_i$. Another disc with moment of inertia $I_b$ is placed coaxially onto the rotating disc. Initially,the second disc has zero angular speed. Finally,both discs rotate with a constant angular speed $\omega_f$. What is the energy lost due to friction by the rotating disc initially?

$A$ ring of mass $m$ and radius $r$ is placed on a rough surface with an initial angular velocity $\omega_0$. The initial velocity of the center of the ring is zero. What will be the velocity of its center when the ring stops slipping?

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are $0.1 \; kg \cdot m^{2}$ and $10 \; rad \cdot s^{-1}$ respectively,while those for the second one are $0.2 \; kg \cdot m^{2}$ and $5 \; rad \cdot s^{-1}$ respectively. At some instant,they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is ........... $J$.