$A$ uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly,what happens to its speed of rotation?

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 $(2)$:

If the earth suddenly stops revolving and all its rotational $KE$ is used up in raising its temperature and if $s$ is taken to be the specific heat of the earth's material,the rise of temperature of the earth will be: ($R =$ radius of the earth and $\omega =$ its angular velocity,$J =$ Joule's constant)

Two cylindrical hollow drums of radii $R$ and $2R$ and of a common height $h$ are rotating with angular velocities $\omega$ (anti-clockwise) and $\omega$ (clockwise) respectively. Their axes,which are fixed,are parallel and in a horizontal plane separated by $3R$. They are now brought into contact.
$(a)$ Show the frictional forces just after contact.
$(b)$ Identify forces and torques external to the system just after contact.
$(c)$ What would be the ratio of final angular velocities when friction ceases?

$A$ thin uniform bar of length $L$ and mass $8m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ move in the same horizontal plane from opposite sides of the bar with speeds $2v$ and $v$ respectively. The masses stick to the bar after collision at distances $L/3$ and $L/6$ respectively from the center of the bar. If the bar starts rotating about its center of mass as a result of the collision,the angular speed of the bar will be

$A$ uniform rod $AB$ of length $1 \ m$ and mass $4 \ kg$ is sliding along two mutually perpendicular frictionless walls $OX$ and $OY$. The velocity of the two ends of the rod $A$ and $B$ are $3 \ m/s$ and $4 \ m/s$ respectively,as shown in the figure. Which of the following statement$(s)$ is/are correct?