Question

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.

Answer 1

$F = Y.A. \frac{\Delta \ell}{\ell}$
$\Delta \ell=\frac{ F }{ Y.A } \cdot \ell$
$\Delta \ell=\frac{ F . \ell}{ Y . \pi r ^{2}}$
$\Delta \ell \propto \frac{\ell}{ r ^{2}}$
$\frac{\Delta \ell_{2}}{\Delta \ell_{1}} =\left(\frac{\ell_{2}}{\ell_{1}}\right)\left(\frac{ r _{1}}{ r _{2}}\right)^{2}$
$=(2)\left(\frac{1}{2}\right)^{2}$
$\frac{\Delta \ell_{2}}{\Delta \ell_{1}}=\frac{1}{2}$
$\Delta \ell_{2} =\frac{\Delta \ell_{1}}{2}$
$=\frac{0.04}{2}$
$=0.02 m$
$\Delta \ell_{2} =2$ cm