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)

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 $(2)$:

$A$ uniform rod of length $L$ and mass $M$ is placed on a smooth horizontal surface. $A$ particle of mass $m$ moving with velocity $v$ strikes the rod at one end perpendicular to the rod. After the collision,the particle comes to rest. What is the angular velocity of the rod about its centre of mass after the collision?

State whether the following statements are True or False:
$(1)$ Angular position $\theta$ is a scalar,while angular displacement $\Delta \theta$ is a vector.
$(2)$ The relation between linear velocity $\vec{v}$ and angular velocity $\vec{\omega}$ for a particle in rotational motion is given by $\vec{v} = \vec{r} \times \vec{\omega}$.
$(3)$ The moment of inertia of a rigid body is constant.
$(4)$ The moment of momentum is called angular momentum.

$A$ uniform cube of mass $m$ and side $a$ is placed on a frictionless horizontal surface. $A$ vertical force $F$ is applied to the edge as shown in the figure. Match the following (most appropriate choice):

$(a)$ $\frac{mg}{4} < F < \frac{mg}{2}$	$(i)$ Cube will move up
$(b)$ $F > \frac{mg}{2}$	$(ii)$ Cube will not exhibit motion
$(c)$ $F > mg$	$(iii)$ Cube will begin to rotate about $A$
$(d)$ $F = \frac{mg}{4}$	$(iv)$ Normal reaction effectively at $a/3$ from $A$,no motion

Two point-like objects of masses $20 \text{ g}$ and $30 \text{ g}$ are fixed at the two ends of a rigid massless rod of length $10 \text{ cm}$. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass,as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2 \times 10^{-8} \text{ N m rad}^{-1}$. The angular frequency of the oscillations is $n \times 10^{-3} \text{ rad s}^{-1}$. The value of $n$ is