$A$ body with a moment of inertia of $3\ kg\cdot m^2$ rotating with an angular velocity of $2\ rad/s$ has the same kinetic energy as a mass of $12\ kg$ moving with a velocity of .......... $m/s$.

Consider a disc rotating in the horizontal plane with a constant angular speed $\omega$ about its centre $O$. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown,two pebbles $P$ and $Q$ are simultaneously projected at an angle towards $R$. The velocity of projection is in the $y-z$ plane and is same for both pebbles with respect to the disc. Assume that $(i)$ they land back on the disc before the disc completes $\frac{1}{8}$ rotation,$(ii)$ their range is less than half disc radius,and $(iii)$ $\omega$ remains constant throughout. Then

$A$ rotating body has angular momentum $L$. If its frequency is doubled and its kinetic energy is halved,what will be its new angular momentum?

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

$A$ circular platform is situated in a horizontal plane and can rotate about a vertical axis passing through its center. $A$ tortoise is sitting at one edge of the platform,and the platform is rotating with a constant angular velocity $\omega_0$. If the tortoise starts moving with a uniform speed along a chord of the circular platform,how will the angular velocity of the platform change with time $t$?

$A$ uniform circular disc of mass $1.5 \ kg$ and radius $0.5 \ m$ is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude $F=0.5 \ N$ are applied simultaneously along the three sides of an equilateral triangle $XYZ$ with its vertices on the perimeter of the disc (see figure). One second after applying the forces,the angular speed of the disc in $\text{rad } s^{-1}$ is: