$A$ uniform rod is fixed to a rotating turntable so that its lower end is on the axis of the turntable and it makes an angle of $20^o$ to the vertical. (The rod is thus rotating with uniform angular velocity about a vertical axis passing through one end.) If the turntable is rotating clockwise as seen from above,what is the direction of the rod's angular momentum vector (calculated about its lower end)?

The temperature of a thin uniform circular disc of $1 \ m$ diameter is increased by $10^{\circ} C$. The percentage increase in the moment of inertia of the disc about an axis passing through its centre and perpendicular to the circular face is: (linear coefficient of expansion $\alpha = 11 \times 10^{-6} /{ }^{\circ} C$)

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
$(A)$ $\frac{I\omega^2}{2}$ $(B)$ $\frac{I\omega^2}{3}$ $(C)$ $\frac{I\omega^2}{4}$ $(D)$ $\frac{I\omega^2}{6}$

An annular disk of mass $M$,inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$,as shown in the figure. At some time,an impulse $J_0 \hat{x}$ is applied at a height $h$ above the center of the disk. If $h=h_m$ then the disk rolls without slipping along the $x$-axis. Which of the following statement$(s)$ is(are) correct?
$(A)$ For $\mu \neq 0$ and $a \rightarrow 0, h_m=b / 2$
$(B)$ For $\mu \neq 0$ and $a \rightarrow b, h_m=b$
$(C)$ For $h=h_m$,the initial angular velocity does not depend on the inner radius $a$.
$(D)$ For $\mu=0$ and $h=0$,the wheel always slides without rolling.

$A$ ring,a solid sphere,and a thin disc of different masses rotate with the same kinetic energy. Equal torques are applied to stop them. Which will make the least number of rotations before coming to rest?

$STATEMENT-1$ If there is no external torque on a body about its center of mass,then the velocity of the center of mass remains constant. because
$STATEMENT-2$ The linear momentum of an isolated system remains constant.