An ant is sitting at the edge of a rotating disc. If the ant reaches the other end, after moving along the diameter, the angular velocity of the disc will

$A$ small bead of mass $m$ moving with velocity $v$ gets threaded on a stationary semicircular ring of mass $m$ and radius $R$ kept on a horizontal table. The ring can freely rotate about its centre. The bead comes to rest relative to the ring. What will be the final angular velocity of the system?

Consider a uniform rod of mass $M=4m$ and length $\ell$ pivoted about its centre. $A$ mass $m$ moving with velocity $v$ making an angle $\theta=\frac{\pi}{4}$ to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is

$A$ 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?

$A$ particle is moving in a circular path. Which of the following statements is true?

The Sun rotates around its centre once in $27$ days. What will be the period of revolution if the Sun were to expand to twice its present radius without any external influence (in $days$)? Assume the Sun to be a sphere of uniform density.