A uniform cylindrical rod of length L and cross-sectional area A having Young's modulus Y is acted upon by forces as shown in figure. [CG PMT 2015]

The elongation produced in the rod is

Elongation produced, if the  $\frac{2}{3}$ part of length L of rod is stretched by a force of 3F, is given by

$\mathrm{\Delta }{L}_1=\frac{F{L}_1}{AY}=\frac{3F\times \frac{2}{3}L}{AY}=2\frac{FL}{AY}$

Elongation produced, if the remaining (1/3) part of length L of rod is stretched by a force of 2F, is given by

$\mathrm{\Delta }{L}_2=\frac{F_2{L}_2}{AY}=\frac{2F\times \frac{1}{3}L}{AY}=\frac{2}{3}\frac{FL}{AY}$

Total elongation produced in the rod,

$\mathrm{\Delta }L=\mathrm{\Delta }{L}_1+\mathrm{\Delta }{L}_2=2\frac{FL}{AY}+\frac{2}{3}\frac{FL}{AY}$

$\therefore \mathrm{\Delta }L=\frac{8FL}{3AY}$