A small block slides down from the top of hemisphere of radius $R=3 \mathrm{m}$ as shown in the figure. The height $h$ at which the block will lose contact with the surface of the sphere is $m$. (Assume there is no friction between the block and the hemisphere)

A large block of wood of mass $M=5.99 \mathrm{kg}$ is hanging from two long massless cords. A bullet of mass $m=10 \mathrm{g}$ is fired into the block and gets embedded in it. The (block $+$ bullet) then swing upwards, their center of mass rising a vertical distance $h=9.8 \mathrm{cm}$ before the (block $+$ bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before the collision is: (Take $g=9.8{ \mathrm{m} \mathrm{s}}^{-2}$)

Two identical blocks $A$ and $B$ each of mass $m$ resting on the smooth horizontal floor are connected by a light spring of natural length $L$ and spring constant $K$. A third block $C$ of mass $m$ moving with a speed $v$ along the line joining $A$ and $B$ collides with $A$.The maximum compression in the spring is

As shown in the figure, a particle of mass $10 \mathrm{kg}$ is placed at a point $A.$ When the particle is slightly displaced to its right, it starts moving and reaches the point $B.$ The speed of the particle at $B$ is $x \mathrm{m} {\mathrm{s}}^{-1}$. (Take $\left.g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$ The value of $x$ to the nearest integer is