The portion $AB$ of the wedge shown in the figure is rough and $BC$ is smooth. $A$ solid cylinder rolls without slipping from $A$ to $B$. The ratio of translational kinetic energy to rotational kinetic energy,when the cylinder reaches point $C$,is:

$A$ solid sphere of mass $m$,radius $R$,having moment of inertia about an axis passing through its center of mass as $I$ is recast into a disc of thickness $t$ whose moment of inertia about an axis passing through the rim (edge) and perpendicular to its plane remains $I$. Then the radius of the disc is:

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)

$A$ 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 $g = 10 \text{ m/s}^2$]

$A$ projectile of mass $3m$ explodes at the highest point of its path. It breaks into three equal parts. One part retraces its path,the second one comes to rest. The distance of the third part from the point of projection when it finally lands on the ground is ........$m.$ (The range of the projectile was $100\,m$ if no explosion would have taken place.)

$A$ thin uniform rod,pivoted at $O$,is rotating in the horizontal plane with constant angular speed $\omega$,as shown in the figure. At time $t = 0$,a small insect of mass $m$ starts from $O$ and moves with constant speed $v$ with respect to the rod towards the other end. It reaches the end of the rod at time $t = T$ and stops. The angular speed of the system remains $\omega$ throughout. The magnitude of the torque $(|\vec{\tau}|)$ on the system about $O$,as a function of time,is best represented by which plot?