A wedge of mass $M=4m$ lies on a frictionless plane. A particle of mass $m$ approaches the wedge with speed $v.$ There is no friction between the particle and the plane or between the particle and the wedge. The maximum height climbed by the particle on the wedge is given by:

A rod of length $50 cm$ is pivoted at one end. It is raised such that it makes an angle of ${30}^o$ from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in $rad s^{-1}$ ) will be $\left(g=10 m{s}^{-2}\right)$

Four particles $\mathrm{A},\mathrm{ }\mathrm{B},\mathrm{ }\mathrm{C}$ and $\mathrm{D}$ with masses ${\mathrm{m}}_{\mathrm{A}}=\mathrm{m},{\mathrm{m}}_{\mathrm{B}}=2\mathrm{m},{\mathrm{m}}_{\mathrm{C}}=3\mathrm{m}$  and ${\mathrm{m}}_{\mathrm{D}}=4\mathrm{m}$ are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is:

A uniform rectangular thin sheet $ABCD$ of mass $M$ has length $a$ and breadth $b$, as shown in the figure. If the shaded portion $HBGO$ is cut-off, the coordinates of the centre of mass of the remaining portion will be:

A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights $h_{sph}$ and $h_{cyl}$ on the incline. The ratio $\frac{h_{sph}}{h_{cyl}}$ is given by: