Derive an expression for potential energy of a spring and show that spring force is a conservation force.

Explanation

The type of force acts on a spring is a conservation force and as well as a variable force. The force law also called as Hooke's law with respect to the force on the spring `f_(a)=-kx`. Where `k-` is known as the spring constant in `Nm^(-1) and x-` is elongation. For a compression the displacement will be `'-x'` so that `F_s=+kx`.
For extension `W_s=underset(0)overset(x_m)(int)F_s dx=-underset(x_m)overset(0)(int)kxdx=(kx_m^2)/(2)`
The work done by the spring force for displacement `x_i` to `x_f` is given by
`W_s=-underset(x_i)overset(x_f)(int) kxdx=-(k)/(2)(x_i-x_f)`
It is clear that if the spring returns to initial state `W=0`. Since there is no loss in the force is retrieved, the force is conservative.
using `(1)/(2) mv_m^2=(1)/(2)kx_m^2`
we get `v_m=sqrt((k)/(m)).x_m`