$A$ man spinning in free space changes the shape of his body,e.g.,by spreading his arms or curling up. By doing this,he can change his

Two discs $A$ and $B$ are mounted coaxially on a vertical axle. The discs have moments of inertia $I$ and $2I$ respectively about the common axis. Disc $A$ is imparted an initial angular velocity $2\omega$ using the entire potential energy of a spring compressed by a distance $x_1$. Disc $B$ is imparted an angular velocity $\omega$ by a spring having the same spring constant and compressed by a distance $x_2$. Both the discs rotate in the clockwise direction.
$1.$ The ratio of $x_1/x_2$ is
$(A)$ $2$ $(B)$ $1/2$ $(C)$ $\sqrt{2}$ $(D)$ $1/\sqrt{2}$
$2.$ When disc $B$ is brought in contact with disc $A$,they acquire a common angular velocity in time $t$. The average frictional torque on one disc by the other during this period is
$(A)$ $\frac{2I\omega}{3t}$ $(B)$ $\frac{9I\omega}{2t}$ $(C)$ $\frac{9I\omega}{4t}$ $(D)$ $\frac{3I\omega}{2t}$
$3.$ The loss of kinetic energy during the above process is
$(A)$ $\frac{I\omega^2}{2}$ $(B)$ $\frac{I\omega^2}{3}$ $(C)$ $\frac{I\omega^2}{4}$ $(D)$ $\frac{I\omega^2}{6}$

$A$ thin uniform rod of length $L$ and mass $M$ is swinging freely along a horizontal axis passing through its centre. Its maximum angular speed is $\omega$. Its centre of mass rises to a maximum height of [where $g$ is gravitational acceleration]:

$A$ uniform cylinder of radius $1 \,m$, mass $1 \,kg$ spins about its axis with an angular velocity $20 \,rad/s$. At a certain moment, the cylinder is placed into a corner as shown in the figure. The coefficient of friction between the horizontal wall and the cylinder is $\mu$, whereas the vertical wall is frictionless. If the number of rounds made by the cylinder is $5$ before it stops, then the value of $\mu$ is (acceleration due to gravity, $g=10 \,m/s^2$)

$A$ uniform sphere of radius $R$ is placed on a rough horizontal surface and given a linear velocity $v_0$ and angular velocity $\omega_0$ as shown. The sphere comes to rest after moving some distance to the right. It follows that:

$A$ stick of length $l$ and mass $M$ lies on a frictionless horizontal surface on which it is free to move in any way. $A$ ball of mass $m$ moving with speed $v$ collides elastically with the stick at one of its ends as shown in the figure. If after the collision the ball comes to rest,what should be the mass of the ball?