$A$ particle performs rotational motion with an angular momentum $L$. If the frequency of rotation is doubled and its kinetic energy becomes one-fourth,the new angular momentum becomes:

$A$ solid cylinder of mass $3 \ kg$ is rolling on a horizontal surface with velocity $4 \ m/s$. It collides with a horizontal spring whose one end is fixed to a rigid support. The force constant of the spring is $200 \ N/m$. The maximum compression produced in the spring will be (assume the collision between the cylinder and the spring is elastic). (in $m$)

State whether the following statements are True or False:
$(1)$ The angular acceleration of an object rotating with a constant angular velocity is always zero.
$(2)$ An object can have a moment of inertia without energy.
$(3)$ The radius of gyration of an object is a constant quantity.
$(4)$ $A$ figure skater spins faster when they pull their arms in because their moment of inertia decreases.

$A$ uniform bar of length $12 \text{ cm}$ and mass $20m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ are moving in opposite directions with the same speed $v$ and in the same plane as the bar. These masses strike the bar simultaneously and get stuck to it. After the collision,the entire system is rotating with an angular frequency $\omega$. The ratio of $v$ and $\omega$ is:

For the pivoted slender rod of length $l$ as shown in the figure,the angular velocity as the bar reaches the vertical position after being released from the horizontal position is:

$A$ rod $AB$ is free to rotate in a vertical plane about a horizontal axis through $A$ as shown in the figure. It is slightly disturbed from rest in its position of unstable equilibrium and when it is next vertical,the end $B$ collides with a fixed peg and rebounds. If the rod comes to instantaneous rest when $AB$ is horizontal (as shown in the figure),then: