A uniform metallic wire is elongated by $0.04\, m$ when subjected to a linear force $F$. The elongation, if its length and diameter is doubled and subjected to the same force will be ..... $cm .$
JEE MAIN 2021, Medium
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$F = Y.A \frac{\Delta \ell}{\ell}$
$\Delta \ell=\frac{ F }{ Y.A } \cdot \ell$
$\Delta \ell=\frac{ F \ell}{ Y \cdot \pi r ^{2}}$
$\Delta \ell \propto \frac{\ell}{ r ^{2}}$
$\frac{\Delta \ell_{2}}{\Delta \ell_{1}}=\left(\frac{\ell_{2}}{\ell_{1}}\right)\left(\frac{ r _{1}}{ r _{2}}\right)^{2}$
$=(2)(\frac{1}{2})^2$
$\frac{\Delta \ell_{2}}{\Delta \ell_{1}}=\frac{1}{2}$
$\Delta \ell_{2}=\frac{\Delta \ell_{1}}{2}$
$=\frac{0.04}{2}$
$=0.02\, m$
$\Delta \ell_{2}=2\, cm$
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