Question

A vertically held wire is stretched by suspending is weight from its lower free end.If α is the longitudinal strain and Y is the Young's Modulus of the material of the wire , show that elastic potential energy per unit volume in the stretched wire is given by ½Yα².

Answer 1

$$\begin{gathered} strain=\alpha =∆l/l \\ Stress=Y∆l/l=Y\alpha \\ \mathrm{elastic}\mathrm{potential}\mathrm{energy}\mathrm{per}\mathrm{unit}volume=1/2*stress*strain \\ =1/2*Y\alpha *\alpha =\mathbf{1}\mathbf{/}\mathbf{2}\mathbf{}\mathit{Y}{\mathit{\alpha }}^{\mathbf{2}} \end{gathered}$$