$A$ body weighs $6$ g when placed in one pan and $24$ g when placed on the other pan of a false balance. If the beam is horizontal when both the pans are empty,the true weight of the body is ....... g.

$A$ thin string is wrapped around the circumference of a wheel of radius $r$. The wheel has a horizontal axle and a moment of inertia $I$ about it. $A$ weight $mg$ is attached to the end of the string,which falls from rest. After falling through a distance $h$,the angular velocity of the wheel will be:

$A$ uniform thin rod of length $2L$ and mass $m$ lies on a horizontal table. $A$ horizontal impulse $J$ is given to the rod at one end. There is no friction. The total kinetic energy of the rod just after the impulse will be

Statement-$1$: $A$ body is rotating about an axis with angular velocity $\omega$ and moment of inertia $I$. Its angular momentum $L$ remains constant,but its rotational kinetic energy $K$ decreases,provided no external torque is applied.
Statement-$2$: $L = I\omega$ and $K = \frac{L^2}{2I} = \frac{1}{2} I\omega^2$.

$A$ thin ring of mass $2 \ kg$ has a radius of $0.5 \ m$. It is rolling without slipping on a horizontal plane with a velocity of $1 \ m/s$. $A$ small ball of mass $0.1 \ kg$ moving in the opposite direction with a velocity of $20 \ m/s$ hits the ring at a height of $0.75 \ m$ and moves vertically upward with a velocity of $10 \ m/s$ after the collision. Immediately after the collision:

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?