In absence of external forces on a rigid system, which of the following quantities must remain constant?

In the given figure a ball strikes a rod elastically and rod is smoothly hinged at point $A$. Then which of the statement(s) is/are correct for the collision?

A wheel (to be considered as a ring) of mass $m$ and radius $R$ rolls without sliding on a horizontal surface with constant velocity $v$. It encounters a step of height $\frac{R}{2}$ at which it ascends without sliding.

A hollow sphere is set into motion on a rough horizontal surface with a speed $v$ in the forward direction and a rotational speed $v/R$ in the anticlockwise direction as shown in the figure. Find the translational speed of the sphere $\left(\mathrm{a}\right)$ when it stops rotating and $\left(\mathrm{b}\right)$ when slipping finally ceases and pure rolling starts.

Two small particles $A \text{and} B$ of masses $M \text{and }2M$ respectively, are joined rigidly to the ends of a light rod of length $l$ as shown in the figure. The system translates on a smooth horizontal surface with a velocity $u$ in a direction perpendicular to the rod. A particle $P$ of mass $M$ kept at rest on the surface sticks to the particle $A$ as the particle $P$ collides with it. Then choose the correct option(s) 

A block with a square base measuring $a\times a$ and height $h$, is placed on an inclined plane. The coefficient of friction is $\mu$. The angle of inclination $\left(\theta \right)$ of the plane is gradually increased. The block will:

A uniform disc of mass $M$ and radius $R$ is lifted using a string as shown in the figure. Then,

A circular loop of rope of length $\mathrm{L}$ rotates with uniform angular velocity $\mathrm{\omega }$ about an axis through its centre on a horizontal smooth platform.Velocity of pulse (with respect to rope ) produce due to slight radial displacement is given by$\left(\frac{\omega L}{2\mathrm{\pi }}\right)$, If the motion of the pulse and rotation of the loop, both are in same direction then the velocity of the pulse w.r.t. to ground will be: