$A$ 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 $t$?

$A$ smooth uniform rod of length $L$ and mass $M$ has two identical beads of negligible size,each of mass $m$,which can slide freely along the rod. Initially,the two beads are at the center of the rod and the system is rotating with angular velocity ${\omega _0}$ about an axis perpendicular to the rod and passing through the midpoint of the rod (see figure). There are no external torques. When the beads reach the ends of the rod,the angular velocity of the system is

$A$ torque is applied to a body,changing its angular velocity from $\omega_1$ to $\omega_2$. What is the ratio of the initial radius of gyration to the final radius of gyration?

$A$ 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?

$A$ person is standing at the edge of a circular plate that is rotating with a constant angular speed about an axis passing through its center and perpendicular to its plane. If the person starts walking along the radius towards the axis, the angular velocity of the system will:

$A$ bullet of mass $10 \; g$ and speed $500 \; m/s$ is fired into a door and gets embedded exactly at the centre of the door. The door is $1.0 \; m$ wide and weighs $12 \; kg$. It is hinged at one end and rotates about a vertical axis practically without friction. Find the angular speed of the door just after the bullet embeds into it.