A particle of mass $m$ is hanging with the help of an elastic string of unstretched length $a$ and force constant $\frac{mg}{a}$. The other end is fixed to a peg on vertical wall. String is given an additional extension of $2a$ in vertical downward direction by pulling the mass and released from rest. Find the maximum height reached by it during its subsequent motion above point of release. (Neglect interaction with peg if any)

A $0.5 \mathrm{kg}$ block slides from the point $A$ (see figure) on a horizontal track with an initial speed of $3 \mathrm{m} {\mathrm{s}}^{-1}$ towards a weightless horizontal spring of length $1 \mathrm{m}$ and force constant $2 \mathrm{N} {\mathrm{m}}^{-1}$. The part $AB$ of the track is frictionless and the part $BC$ has the coefficients of static and kinetic friction as $0.22$ and $0.2$ respectively.  If the distance $AB$ and $BD$ are $2 \mathrm{m}$ and  $2.14 \mathrm{m}$  respectively then the total distance through which the block moves before it comes to rest completely is?

A small bead of mass $m$ is free to slide on a fixed smooth vertical wire, as indicated in the diagram. One end of a light elastic string, of unstretched length $a$ and force constant $\frac{2mg}{a}$ is attached to $B$. The string passes through a smooth fixed ring $R$ and the other end of the string is attached to the fixed point $A$, $AR$ being horizontal. The point $O$ on the wire is at same horizontal level as $R$, and $AR=RO=a$.

(i) In the equilibrium position, find $OB$.

(ii) The bead $B$ is raised to a point $C$ of the wire above $O$, where $OC=a$, and is released from rest. Find the speed of the bead as it passes $O$, and find the greatest depth below $O$ of the bead in the subsequent motion. 

A ring of mass $m$ slides on a smooth vertical rod.  A light string is attached to the ring and is passing over a smooth peg distant a from the rod, and at the other end of the string is a mass $M\left(M>m\right)$. The ring is held on a level with the peg and released. Show that it first comes to rest after falling a distance $\frac{2mMa}{M^2-m^2}$.

A string, with one end fixed on a rigid wall, passing over a fixed frictionless pulley at a distance of $2 \mathrm{m}$ from the wall, has a point mass $M=2 \mathrm{kg}$ attached to it at a distance of $1 \mathrm{m}$ from the wall. A mass $m=0.5 \mathrm{kg}$ attached at the free end is held at rest so that the string is horizontal between the wall and the pulley and vertical beyond the pulley. What will be the speed with which the mass $M$ will hit the wall when the mass $m$ is released?

'$A$' block of mass $\mathrm{m}$ is held at rest on a smooth horizontal floor. A light frictionless, small pulley is fixed at a height of $6 \mathrm{m}$ from the floor. A light inextensible string of length $16 \mathrm{m}$, connected with $A$ passes over the pulley and another identical block $B$ is hung from the string. Initial height of $B$ is $5\mathrm{m}$ from the floor as shown in figure. When the system is released from rest, $B$ starts to move vertically downwards and $A$ slides on the floor towards right. (i) If at an instant string makes an angle $\theta$ with horizontal, calculate relation between velocity $u$ of $A$ and $v$ of B (ii) Calculate $v$ when $B$ strikes the floor.

In the figure shown the pulley is smooth. The spring and the string are light. The block '$B$' slides down from the along the fixed rough wedge of inclination $\theta$. Assuming that the block reaches the end of the wedge. Find the speed of the block at the end. Take the coefficient of friction between the block and the wedge to be $\mu$  and the spring was relaxed when the block was released from the top of the wedge.

As shown in the figure a person is pulling a mass '$m$' from ground on a fixed rough hemispherical surface upto the top of the hemisphere with the help of a light inextensible string. Find the work done by tension in the string if radius of hemisphere is $R$ and friction co-efficient is $\mu$. Assume that the block is pulled with negligible velocity.