A wire of length l and cross-sectional are A is suspended at one of its ends from a ceiling. What will be its strain energy due to its own weight, if the density and Young's modulus of the material of the wire be d and Y?

Explanation

Consider an elemental length of the wire of length dx, at a distance x from the lower end. This length is acted upon by the external force equal to the weight of the portion of wire below it = xdg. In equilibrium, the restoring force F is xAdg.
Thus, `"stress"=(F)/(A)=xdg`
Now elastic potential energy stored in the elemental length will be
`dU=(1)/(2)"stress"xx"strain"xx"volume"`
`=(1)/(2)(("stress")/(Y))"stress"xx"volume"`
`=(1)/(2)((xdg)^(2))/(Y).Adx`
`=(1)/(2)(Ad^(2)g^(2)x^(2))/(Y).dx`
`therefore" Total elastic potential energy"=int_(0)^(L)dU`
`=int_(0)^(L)(1d^(2)g^(2)A)/(Y)x^(2)dx`
`=(d^(2)g^(2)AL^(3))/(6Y)`