$A$ ring and a disc have the same mass and the same radius. The ratio of the moment of inertia of a ring about a tangent in its plane to that of the disc about its diameter is: (in $: 1$)

$A$ pendulum consists of a bob of mass $m=0.1 \ kg$ and a massless inextensible string of length $L=1.0 \ m$. It is suspended from a fixed point at height $H=0.9 \ m$ above a frictionless horizontal floor. Initially,the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. $A$ horizontal impulse $P=0.2 \ kg \cdot m/s$ is imparted to the bob at some instant. After the bob slides for some distance,the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is $J \ kg \cdot m^2/s$. The kinetic energy of the pendulum just after the lift-off is $K$ Joules. $(1)$ The value of $J$ is. . . . . . $(2)$ The value of $K$ is. . . . . Give the answers of the questions $(1)$ and $(2)$.

The total kinetic energy of a body of mass $10 \ kg$ and radius $0.5 \ m$ moving with a velocity of $2 \ m/s$ without slipping is $32.8 \ J$. The radius of gyration of the body is .......... $m$.

$A$ flywheel rotates about an axis. Due to friction at the axis,it experiences an angular retardation proportional to its angular velocity. If its angular velocity falls to half while it makes $n$ rotations,how many more rotations will it make before coming to rest?

Consider a disc rotating in the horizontal plane with a constant angular speed $\omega$ about its centre $O$. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown,two pebbles $P$ and $Q$ are simultaneously projected at an angle towards $R$. The velocity of projection is in the $y-z$ plane and is same for both pebbles with respect to the disc. Assume that $(i)$ they land back on the disc before the disc completes $\frac{1}{8}$ rotation,$(ii)$ their range is less than half disc radius,and $(iii)$ $\omega$ remains constant throughout. Then

The temperature of a thin uniform circular disc of $1 \ m$ diameter is increased by $10^{\circ} C$. The percentage increase in the moment of inertia of the disc about an axis passing through its centre and perpendicular to the circular face is: (linear coefficient of expansion $\alpha = 11 \times 10^{-6} /{ }^{\circ} C$)