During a test of a jet engine,its compressor rotates according to the graph shown. The total number of revolutions completed by the compressor during the test is:

$A$ solid sphere is in purely rotational motion about its diameter. The ratio of its angular momentum $(L)$ and kinetic energy $(K)$ is $\frac{\pi}{22}$. Find the angular velocity $(\omega)$ of the sphere. (Take $\pi = \frac{22}{7}$) (in $rad/s$)

$A$ 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 $I$. Which graph represents the variation of $I$ with respect to $R$,keeping $M$ constant?

$A$ solid cube of wood of side $2a$ and mass $M$ is resting on a horizontal surface as shown below. The cube is free to rotate about the fixed axis $AB$. $A$ bullet of mass $m (< M)$ and speed $v$ is shot horizontally at the face opposite to $ABCD$ at a height $h$ above the surface to impart the cube an angular speed $\omega_{c}$,so that the cube just topples over. Then,$\omega_{c}$ is (Note: the moment of inertia of the cube about an axis passing through the centre of mass and parallel to the edge is $2Ma^{2}/3$)

The moment of inertia of a ring about an axis passing through the centre and perpendicular to its plane is $I$. It is rotating with angular velocity $\omega$. Another identical ring is gently placed on it so that their centres coincide. If both the rings are rotating about the same axis,then the loss in kinetic energy is