Question

A uniform metallic wire is elongated by $0.04m$ 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

Answer: 2

$F=Y.A.\frac{\Delta \ell }{\ell }$
$\Delta \ell =\frac{F}{Y.A}\cdot \ell$
$\Delta \ell =\frac{F.\ell }{Y.\pi r^2}$
$\Delta \ell \propto \frac{\ell }{r^2}$
$\frac{\Delta {\ell }_2}{\Delta {\ell }_1}=\left(\frac{{\ell }_2}{{\ell }_1}\right){\left(\frac{r_1}{r_2}\right)}^2$
$=(2){\left(\frac{1}{2}\right)}^2$
$\frac{\Delta {\ell }_2}{\Delta {\ell }_1}=\frac{1}{2}$
$\Delta {\ell }_2=\frac{\Delta {\ell }_1}{2}$
$=\frac{0.04}{2}$
$=0.02m$
$\Delta {\ell }_2=2$ cm