Two wires of length l , radius r and length 2l , radius 2r respectively having same Young’s modulus are hung with a weight mg . Net elongation is

Young's modulus

 $$\begin{gathered} \mathrm{Y}=\frac{\mathrm{mgl}}{{\mathrm{a}}_1{\mathrm{l}}_1}\therefore {\mathrm{l}}_1=\frac{\mathrm{mgl}}{{\mathrm{Y\pi r}}^2}\mathrm{ }\dots \dots ..\left(1\right) \mathrm{and} \\ \mathrm{Y}=\frac{\mathrm{mg}\left(2\mathrm{l}\right)}{{\mathrm{a}}_2{\mathrm{l}}_2}=\frac{\mathrm{mg}\left(2\mathrm{l}\right)}{\mathrm{\pi }(2\mathrm{r}{)}^2{\mathrm{l}}_2} \mathrm{or} {\mathrm{l}}_2=\frac{\mathrm{mgl}}{2{\mathrm{Y\pi r}}^2}\dots \dots ..\left(2\right) \end{gathered}$$

From Eqs. (1) and (2), we have 

$\therefore {\mathrm{l}}_1+{\mathrm{l}}_2=\frac{\mathrm{mgl}}{{\mathrm{Y\pi r}}^2}+\frac{\mathrm{mgl}}{2{\mathrm{Y\pi r}}^2}=\frac{3}{2}\frac{\mathrm{mgl}}{{\mathrm{Y\pi r}}^2}$