A small sphere of radius $R$ is held against the inner surface of a smooth spherical shell of radius $6R$ as shown in the figure. The masses of the shell and the small sphere are $4M$ and $M$, respectively.

 

This arrangement is placed on a smooth horizontal table. The small sphere is now released. The $x$-coordinate of the center of the shell when the smaller sphere reaches the other extreme position is,

Two blocks $A$ and $B$ are connected by a massless string (shown in the figure). A force of $30 \mathrm{N}$ is applied on block $B$. The distance travelled by centre of mass in $2 \mathrm{s}$ starting from rest is:

A circular loop of rope of length $L$ rotates with uniform angular velocity $\omega$ 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 both are in opposite directions, then the velocity of the pulse with respect to the ground will be:

A rectangular plate of length$L$, width$B$ and mass $M$ rotates with angular velocity$\omega$ about an axis in$\mathrm{X}-\mathrm{Y} \text{plane}$. The axis of rotation passes through centre of mass and cuts the length of the plate at an acute angle $\theta$. Find the angular momentum about$\mathrm{Z}-\text{axis}$.