A force is applied to a steel wire ' $A$ ', rigidly clamped at one end. As a result elongation in the wire is $0.2\,mm$. If same force is applied to another steel wire ' $B$ ' of double the length and a diameter $2.4$ times that of the wire ' $A$ ', the elongation in the wire ' $B$ ' will be $............\times 10^{-2}\,mm$ (wires having uniform circular cross sections)
JEE MAIN 2023, Medium
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$Y =\frac{ F / A }{\frac{\Delta \ell}{\ell}}$
$\Rightarrow F =\frac{ YA }{\ell} \Delta \ell$
$\left(\frac{ A \Delta \ell}{\ell}\right)_1=\left(\frac{ A \Delta \ell}{\ell}\right)_2$ $\Rightarrow \frac{\Delta \ell_2}{\Delta \ell_1}=\frac{ A _1}{ A _2} \times \frac{\ell_2}{\ell_1}$
$\Rightarrow \frac{\Delta \ell_2}{0.2}=\frac{1}{2.4 \times 2.4} \times \frac{2}{1}$ $\Rightarrow \Delta \ell_2=6.9 \times 10^{-2}\,mm$
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