A large stone of mass $\frac{M_{\mathrm{e}}}{2}$ is released when centre of mass of the stone is at a height $h$ $(h\ll R_{\mathrm{e}})$. Speed of stone when it is at a height of $\frac{h}{2}$ is $\mathrm{m} {\mathrm{s}}^{-1}$.

($M_{\mathrm{e}}$ and $R_{\mathrm{e}}$ are mass and radius of earth. Given $h=\frac{3}{20} \mathrm{m}$)

A small ring $A$ of mass $m=1/2 \mathrm{kg}$ is attached at an end of a light string the other end of which is tied to a block $B$ of mass $2m.$ The ring is free to move on a fixed smooth horizontal rod as shown. Find tension in the string when it becomes vertical.

A symmetric block of mass $m_1=1 \mathrm{kg}$ with a groove of hemispherical shape of radius $r=5 \mathrm{m}$rests on a smooth horizontal surface in contact with the wall as shown in the figure. A small block of mass $m_2=1 \mathrm{kg}$ slides without friction from the initial position. Find the maximum velocity of the block $m_1.$

A body of mass $5\mathrm{kg}$ moves along the $\mathrm{x}-\mathrm{axis}$ with a velocity $2\mathrm{m}/\mathrm{s}.$ A second body of mass $10\mathrm{kg}$ moves along the y-axis with a velocity of $\sqrt{3 }\mathrm{m}/\mathrm{s}.$They collide at the origin and stick together.

If the amount of heat liberated in the collision is $\frac{\mathrm{\lambda }}{3} \mathrm{J}$, then find $\mathrm{\lambda }$?

Two blocks initially at rest having masses ${\mathrm{m}}_1 \mathrm{and} {\mathrm{m}}_2$ are connected by spring of spring constant $\mathrm{k} = \frac{2}{3} \mathrm{N}/\mathrm{m}$ (as shown in the figure). The block of mass ${\mathrm{m}}_1$ is pulled by a constant force ${\mathrm{F}}_1 = 4\mathrm{N}$ and the other block is pulled by a constant force ${\mathrm{F}}_2 = 2\mathrm{N}$Find the maximum elongation of the spring (the spring is initially relaxed) Assuming${\mathrm{m}}_2 = 2{\mathrm{m}}_1$