$A$ is a fixed point a height $H$ above a perfectly inelastic smooth horizontal plane. A light inextensible string of length $L\left(>H\right)$ has one end attached to $A$ and other to a heavy particle. The particle is held at the level of $A$ with string just taut and released from rest. Find the height of the particle above the plane when it is next instantaneously at rest.

A heavy ball of mass $2M$ moving with a velocity $v_0$ collides elastically head-on with a cradle of three identical balls each of mass $M$ as shown in figure. Determine the velocity of each ball after collision.

In figure, there are $n$ identical spheres of mass $\mathrm{m}$ suspended with wires of equal length. The spheres are almost in contact with each other. Sphere $1$ is pulled aside and released. If sphere $1$ strikes sphere $2$ with velocity $u$ find an expression for velocity $v_n$ of the $n^{th }$ sphere immediately after being struck by the one adjacent to it. The coefficient of restitution for all the impacts is $e.$

​​​​​​​

Two identical balls $A$ and $B$ lie on a smooth horizontal surface, which gradually merges into a curve to a height $3.2 \mathrm{m}.$ Ball $A$ is given a velocity of $10 \mathrm{m}/\mathrm{s},$ to collide head-on with ball $B,$ which then takes up the curved path. What is the minimum coefficient of restitution, $e,$ for the collision between $A$ and $B$ in order that ball $B$ reaches the highest point $C$ of the curve.

A smooth ball of mass $m$ is suspended from a light string of length $1 \mathrm{m}.$ Another ball of mass $2 \mathrm{m}$ strikes the ball of mass $m$ horizontally with a speed of $u=\sqrt{35} \mathrm{m} {\mathrm{s}}^{-1}.$. The coefficient of restitution for the collision is $e.$ The string becomes loose, when it makes an angle of $30°$ with the horizontal; find the value of $e.$ Use $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.