Question

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

Answer 1

The potential energy of a spring is the energy stored by a spring due to its compression.
According to Hooke's law, the restoring force produced in a spring is given by,
F=-KX
Where, k=Constant
x=Displacement produced
Now, the potential energy would be equal to the work done on the restoring system,
Therefore, P.E=W=F.dx
$$\begin{gathered} \mathrm{On}\mathrm{integration},\mathrm{we}\mathrm{get} \\ \mathrm{P}.\mathrm{E}.=-\frac{1}{2}{\mathrm{kx}}^2 \end{gathered}$$
If the work done by a force does not depends upon the initial and final point but depends upon the path taken, it is called a non conservative force.
Spring force is non conservative force because work done depends upon the actual path.
$$\begin{gathered} \mathrm{For}\mathrm{small}\mathrm{displacement},\mathrm{spring}\mathrm{force}\mathrm{F}=-\mathrm{kx} \\ \mathrm{The}\mathrm{elastic}\mathrm{potential}\mathrm{energy}\mathrm{of}\mathrm{the}\mathrm{spring}=\mathrm{Work}\mathrm{done}\mathrm{on}\mathrm{it}\mathrm{to}\mathrm{deform}\mathrm{it}\mathrm{an}\mathrm{amount}\mathrm{x} \\ \mathrm{Thus},\mathrm{V}=-{\int }_0^{\mathrm{x}}-\mathrm{kx}\mathrm{dx}=\frac{1}{2}{\mathrm{kx}}^2 \\ \mathrm{Here}\mathrm{if}\mathrm{the}\mathrm{deformation}\mathrm{increases}\mathrm{from}{\mathrm{x}}_1\mathrm{to}{\mathrm{x}}_2\mathrm{during}\mathrm{the}\mathrm{motion},\mathrm{then}\mathrm{the}\mathrm{change}\mathrm{in}\mathrm{potential}\mathrm{energy}\mathrm{of}\mathrm{the}\mathrm{spring}\mathrm{will}\mathrm{be}\mathrm{the}\mathrm{difference}\mathrm{between}\mathrm{its}\mathrm{final}\mathrm{and}\mathrm{initial}\mathrm{values}. \\ ∆\mathrm{V}=\frac{1}{2}\mathrm{k}({{\mathrm{x}}_2}^2-{{\mathrm{x}}_1}^2) \end{gathered}$$