$A$ disc having velocity $v_0$ and angular speed $\omega_0$ in the anticlockwise direction is placed on a rough plank. Initially,the plank is at rest. (Assume the length of the plank is very large). Choose the $INCORRECT$ option.

$A$ solid cylinder of mass $2 \ kg$ and radius $0.2 \ m$ is rotating with an angular velocity of $3 \ rad/s$. $A$ particle of mass $0.5 \ kg$ moving with a velocity of $5 \ m/s$ strikes its periphery and sticks to it. The loss in kinetic energy due to the collision is ....... $J$.

$A$ sphere rotates with an angular velocity $\omega_0$ at a temperature of $0^{\circ}C$. What will be its new angular velocity when the temperature is increased to $100^{\circ}C$? (Given: $\alpha = 2.0 \times 10^{-5} \, ^{\circ}C^{-1}$)

This question has Statement $1$ and Statement $2$. Of the four choices given after the Statements,choose the one that best describes the two Statements.
Statement $1$ : When the moment of inertia $I$ of a body rotating about an axis with angular speed $\omega$ increases,its angular momentum $L$ remains unchanged,but the kinetic energy $K$ decreases if no external torque is applied.
Statement $2$ : $L = I\omega$ and the rotational kinetic energy $K = \frac{1}{2}I\omega^2 = \frac{L^2}{2I}$.

$A$ 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 $t$?

One twirls a circular ring (of mass $M$ and radius $R$) near the tip of one's finger as shown in Figure $1$. In the process,the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone,shown by the dotted line. The radius of the path traced out by the point where the ring and the finger are in contact is $r$. The finger rotates with an angular velocity $\omega_0$. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger are in contact (Figure $2$). The coefficient of friction between the ring and the finger is $\mu$ and the acceleration due to gravity is $g$.
$(1)$ The total kinetic energy of the ring is
$[A]$ $M \omega_0^2 R^2$ $[B]$ $\frac{1}{2} M \omega_0^2(R-r)^2$ $[C]$ $M \omega_0^2(R-r)^2$ $[D]$ $\frac{3}{2} M \omega_0^2(R-r)^2$
$(2)$ The minimum value of $\omega_0$ below which the ring will drop down is
$[A]$ $\sqrt{\frac{g}{\mu(R-r)}}$ $[B]$ $\sqrt{\frac{2 g}{\mu(R-r)}}$ $[C]$ $\sqrt{\frac{3 g}{2 \mu(R-r)}}$ $[D]$ $\sqrt{\frac{g}{2 \mu(R-r)}}$
Given the answers to questions $(1)$ and $(2)$: