When torque applied on a system is zero,which of the following will be constant?

$A$ uniform rod of length $6L$ and mass $8m$ is pivoted at its centre $C$. Two masses $m$ and $2m$ with speeds $2v$ and $v$ respectively,as shown in the figure,strike the rod and stick to it. Initially,the rod is at rest. If it rotates with angular velocity $\omega$ due to the impact,then $\omega$ will be:

Consider a disc of mass $5 \,kg$ and radius $2 \,m$, rotating with an angular velocity of $10 \,rad/s$ about an axis perpendicular to the plane of rotation and passing through its center. An identical disc is placed gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is . . . . . . $J$.

$A$ thin smooth rod of length $L$ and mass $M$ is rotating freely with angular speed $\omega_0$ about an axis perpendicular to the rod and passing through its centre. Two beads of mass $m$ and negligible size are at the centre of the rod initially. The beads are free to slide along the rod. The angular speed of the system,when the beads reach the opposite ends of the rod,will be:

$A$ disc of mass $M$ and radius $R$ is rolling with angular speed $\omega$ on a horizontal plane as shown. The magnitude of angular momentum of the disc about the origin $O$ is

$A$ particle of mass $m$ moves with a velocity $v_0$ in a circle of radius $R_0$ on a smooth horizontal plane. The mass is tied to a string passing through a hole in the smooth plane as shown in the figure. The tension in the string is gradually increased,and finally,the mass $m$ moves in a circle of radius $\frac{R_0}{2}$. What is the ratio of the final kinetic energy to the initial kinetic energy?