$A$ metal sphere of radius $r$ and specific heat $S$ is rotated about an axis passing through its centre at a speed of $f$ rotations per second. It is suddenly stopped,and $50\%$ of its rotational kinetic energy is used in increasing its temperature. The rise in temperature of the sphere is

$A$ cube of side $a$ is moving with velocity $v$ on a smooth horizontal surface. It hits a linear raised obstacle $O$ on the horizontal surface (as shown in the figure). The angular speed of the block after hitting $O$ will be

$A$ yo-yo is resting on a rough horizontal table. Forces $F_1, F_2$ and $F_3$ are applied separately as shown. The correct statement is

$A$ thin circular coin of mass $5 \text{ g}$ and radius $4/3 \text{ cm}$ is initially in a horizontal $xy$-plane. The coin is tossed vertically up ($+z$ direction) by applying an impulse of $J = \sqrt{\frac{\pi}{2}} \times 10^{-2} \text{ N-s}$ at a distance $r = 2/3 \text{ cm}$ from its center. The coin spins about its diameter and moves along the $+z$ direction. By the time the coin reaches back to its initial position,it completes $n$ rotations. The value of $n$ is. . . . . [Given: The acceleration due to gravity $g = 10 \text{ m/s}^2$]

The moment of inertia of a sphere of mass $M$ and radius $R$ is $I$. If the mass is kept constant,what will be the nature of the graph between $I$ and $R$?

Two flywheels are connected by a non-slipping belt as shown in the figure. $I_1 = 4 \ kg \ m^2$,$r_1 = 20 \ cm$,$I_2 = 20 \ kg \ m^2$,and $r_2 = 30 \ cm$. $A$ torque of $10 \ Nm$ is applied on the smaller wheel. Match the entries of column $I$ with appropriate entries of column $II$.

Quantities	Their numerical values (in $SI$ units)
a. Angular acceleration of smaller wheel	$1$. $5/3$
b. Torque on the larger wheel	$2$. $100/3$
c. Angular acceleration of larger wheel	$3$. $5/2$