Two wires of same material having radius in ratio 2 : 1 and lengths in ratio 1 : 2. If same force is applied on them, then ratio of their change in length will be

Given, ratio of radius of two wires, $\frac{r_1}{r_2}=\frac{2}{1}$

and ratio in their lengths,  $\frac{l_1}{l_2}=\frac{1}{2}$

We know that, Young's modulus,

$Y=\frac{Fl}{A\mathrm{\Delta }l}$

$\Rightarrow \mathrm{\Delta }l=\frac{Fl}{AY}$

$\mathrm{\Delta }l\propto \frac{l}{A}$

When same force is applied on them, then ratio of change in their lengths,

$\frac{\mathrm{\Delta }{l}_1}{\mathrm{\Delta }{l}_2}=\frac{l_1}{A_1}/\frac{l_2}{A_2}=\frac{l_1{A}_2}{l_2{A}_1}$

$\Rightarrow \frac{\mathrm{\Delta }{l}_1}{\mathrm{\Delta }{l}_2}=\frac{l_1}{l_2}\cdot \frac{r_2^2}{r_1^2}=\frac{1}{2}\cdot {\left(\frac{1}{2}\right)}^2=\frac{1}{8}$