The potential energy of a long spring when stretched by $2\,cm$ is $U$. If the spring is stretched by $8\,cm$, potential energy stored in it will be $…….\,U$

slowing down of neutrons: In a nuclear reactor a neutron of high speed (typically $10^{7}\; m s ^{-1}$ ) must be slowed to $10^{3}\; m s ^{-1}$ so that it can have a high probability of interacting with isotope $^{235} _{92} U$ and causing it to fission. Show that a neutron can lose most of its kinetic energy In an elastic collision with a light nuclel like deuterlum or carbon which has a mass of only a few times the neutron mass. The material making up the light nuclel, usually heavy water $\left( D _{2} O \right)$ or graphite, is called a moderator.

Two bodies $A$ and $B$ of masses $m$ and $2m$ respectively are placed on a smooth floor. They are connected by a spring. A third body $C$ of mass $m$ moves with velocity $V_0$ along the line joining $A$ and $B$ and collides elastically with $A$ as shown in fig. At a certain instant of time $t_0$ after collision, it is found that instantaneous velocities of $A$ and $B$ are the same. Further at this instant the compression of the spring is found to be $x_0$. Determine the spring constant

A ball of mass $100 \,g$ is dropped from a height $h =$ $10\, cm$ on a platform fixed at the top of vertical spring (as shown in figure). The ball stays on the platform and the platform is depressed by a distance $\frac{ h }{2}$. The spring constant is………. $Nm^{-1}$ . (Use $g=10\, ms ^{-2}$ )

Inside a lift, a spring (Force constant $k = 1000\ N/m$) and block ($mass = 1\  kg$) are both in a state of rest. Now the lift suddenly starts moving upwards with acceleration $a = g$. Find the maximum total compression in the spring in centimeter. ($g =10\ m/s^2$) :-