A steel cylindrical rod of length `l` and radius `r` is suspended by its end from the ceiling.
(a) Find the elastic deformation energy U of the rod.
(b) Define U in terms of tensile strain `Deltal//l` of the rod.

Explanation

When a rod is deformed by its own weight the stress increases as one moves up, the stretching force being the weight of the portion below the element considered.
The stress on the element `dx` is
`rhopir^2(l-x)g//pir^2=rhog(l-x)`
The extension of the element is
`Deltadx=dDeltax=rhog(l-x)dx//E`
Integrating `Deltal=1/2rhogl^2//E` is the extension of the whole rod. The elastic energy of the element is
`1/2rhog(l-x)(rhog(l-x))/(E)pir^2dx`
Integrating
`DeltaU=1/6pir^2rho^2g^2l^3//E=2/3pir^2lE((Deltal)/(l))^2`