When a rubber-band is stretched by a distance $x$, it exerts a restoring force of magnitude $F=ax+b{x}^2,$ Where $a$ and $b$ are constants. Determine the work done in stretching the outstretched rubber band by length is,

A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses $0.36 \mathrm{kg}$ and $0.72 \mathrm{kg}.$ Taking $g=10$ $\mathrm{m} {\mathrm{s}}^{-2}$, find the work done (in joules) by the string on the block of mass $0.36 \mathrm{kg}$ during the first second after the system is released from rest.

A block of mass $0.18 \mathrm{kg}$ is attached to a spring of force constant $2 \mathrm{N} {\mathrm{m}}^{-1}$. The coefficient of friction between the block and the floor is $0.1$. Initially the block is at rest and the spring is unstretched. An impulse is given to the block as shown in the figure. The block slides a distance of $0.06 \mathrm{m}$ and comes to rest for the first time. The initial velocity of the block in $\mathrm{m} {\mathrm{s}}^{-1}$ is $v=\frac{N}{10}$. Then $N$ is

Consider an elliptically shaped rail $PQ$ in the vertical plane with $OP=3 \mathrm{m}$ and $OQ=4 \mathrm{m}.$ A block of mass $1 \mathrm{kg}$ is pulled along the rail from $P$ to $Q$ with a force of $18 \mathrm{N},$ which is always parallel to line $PQ$ (see the figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches $Q$ is $(n\times 10) \mathrm{joules}$ . The value of $\mathrm{n}$ is (take acceleration due to gravity $=10{ \mathrm{ms}}^{-2}$)