Two equal masses each of mass $M$ are joined by a massless rod of length $L$. Now an impulse $J = MV$ is given to one of the masses,making an angle of $30^{\circ}$ with the length of the rod. The angular velocity of the rod just after imparting the impulse is:

$A$ block of mass $M$ has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially,the right edge of the block is at $x=0$,in a coordinate system fixed to the table. $A$ point mass $m$ is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block,its position is $x$ and the velocity is $v$. At that instant,which of the following options is/are correct?
$[A]$ The $x$ component of displacement of the center of mass of the block $M$ is: $-\frac{m R}{M+m}$.
$[B]$ The position of the point mass is: $x=-\sqrt{2} \frac{mR}{M+m}$.
$[C]$ The velocity of the point mass $m$ is: $v=\sqrt{\frac{2 g R}{1+\frac{m}{M}}}$.
$[D]$ The velocity of the block $M$ is: $V=-\frac{m}{M} \sqrt{2 g R}$.

The position vector $\vec{r}$ of a particle of mass $m$ is given by the following equation:
$\vec{r}(t) = \alpha t^3 \hat{i} + \beta t^2 \hat{j}$
where $\alpha = 10/3 \ m \ s^{-3}$,$\beta = 5 \ m \ s^{-2}$ and $m = 0.1 \ kg$. At $t = 1 \ s$,which of the following statement$(s)$ is(are) true about the particle?
$(A)$ The velocity $\vec{v}$ is given by $\vec{v} = (10 \hat{i} + 10 \hat{j}) \ m \ s^{-1}$
$(B)$ The angular momentum $\vec{L}$ with respect to the origin is given by $\vec{L} = -(5/3) \hat{k} \ N \ m \ s$
$(C)$ The force $\vec{F}$ is given by $\vec{F} = (2 \hat{i} + 1 \hat{j}) \ N$
$(D)$ The torque $\vec{\tau}$ with respect to the origin is given by $\vec{\tau} = -(20/3) \hat{k} \ N \ m$

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)

Four spheres,each of mass $M$ and diameter $2a$,are placed at the corners of a square of side $b$. The moment of inertia of this system about an axis along one of the sides of the square is:

The angular momentum of a rotating body is $L$. When the frequency of the rotating body is tripled and its kinetic energy is made one-third, the new angular momentum becomes: