The system is released from rest when spring is at its natural length. Spring constant is $100 {\mathrm{Nm}}^{-1}$ and mass of each block is $10 \mathrm{kg}$. The velocity '$v$' of block $A$ as a function of position $x$ is given as $v^2=Ax-B{x}^2$. The value of $\frac{A}{B}$ is equal to.

A massless platform is kept on a light elastic spring, shown in Fig. When a sand particle of $0.1 \mathrm{kg}$ mass is dropped on the pan from a height $0.24 \mathrm{m}$, the particle strikes the pan, and the spring compresses by $0.01 \mathrm{m}$. From what height should the particle be dropped to cause a compression of $0.04 \mathrm{m}$ ? Give your answer to two decimal places.

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.