A cylinder of mass $M=2 \mathrm{kg }$and radius $R=12 \mathrm{cm}$ lies on a plank of the same mass as shown in the figure. The surface between plank and ground is smooth but there is friction between cylinder and plank. If the coefficient of friction between the cylinder and the plank is $\mu =0.4$, then what maximum initial compression (in $\mathrm{cm}$) can be given to the spring such that the cylinder moves without slipping with respect to the plank? [Given, $k=200 \mathrm{N} {\mathrm{m}}^{-1}$]

Two blocks of equal masses $M$ are tied with a light string which passes over a massless pulley as shown in the figure.

The magnitude of acceleration of center of mass of both the blocks is (neglect friction everywhere),

In a vertical plane inside a smooth hollow thin tube, a block of same mass as that of tube is released as shown in the figure. When it is slightly disturbed, it moves towards right. By the time the block reaches the right end of the tube, the displacement of the tube will be (where $R$ is mean radius of tube), Assume that the tube remains in

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,

The centre of mass of triangle shown in figure has coordinates

A disc of mass $M$ and radius $R$ rolls on a horizontal surface and then goes up a smooth inclined plane as shown in the figure. If the velocity of the disc is $v$, the height to which the disc will rise will be:

A thin hollow sphere of mass $m$ is completely filled with an ideal liquid of mass $m$. when the sphere rolls with a velocity $v$. Kinetic energy of the system is equal to