Question

drive the expression of the potential energy of the elastic stretched spring

Answer 1

In order to find the formula for elastic potential energy of a spring we need to look at Hooke's law, which states that the force needed to stretch a spring is proportional to the displacement of the spring. Mathematically this is expressed as F = kx, where x is the displacement and k is the spring constant.

Answer 2

Hooke's Law, F = -kx, where F is the force and x is the elongation.

The work done = energy stored in stretched string =F.dx

The energy stored can be found from integrating by substituting for force,

and we find,

The energy stored = kx2/2, where x is the final elongation.

The energy density = energy/volume

= (kx2/2)/(AL)

=1/2(kx/A)(x/L) = 1/2(F/A)(x/L) = 1/2(stress)(strain)