Question

Show that work done by a stretching force to produce a certain extension in the wire is given by W = 1/2 stretching force x extension.

Answer 1

Consider a wire of length L and area of cross-section A. Let a force F be applied to stretch the wire. If l be the length through which the wire is stretched, then

Longitudinal strain = l/L

and longitudinal stress = F/A

Young's modulus of elasticity,

Y = Stress/Strain = {F/A}/{l/L} = FL/Al

or, F = YAl/L

If the wire is stretched through a length dl, then work done is given by

dW = F dl = YAl/L dl

Total work done to stretch the wire through length l is given by

or,

Hence, work done = 1/2 x Stretching force x extension

This work done is stored as the potential energy of the stretched force

Work done per unit volume = 

P.E. per unit volume = 1/2 Y x Strain²

= 1/2 x Stress/Strain x Strain²

= 1/2 x (Stress x Strain)

In case of Bulk modulus of elasticity and shear modulus of elasticity equation (ii) holds good.