A load of 4 kg is suspended from a ceiling through a steel wire of length 2 m and radius 2 mm. It is found that, the length of the wire increases by 0.031 mm as equilibrium is achieved. What would be the Young's modulus of steel? (Take, g = 3.l$\pi$ ms^-2) 

Given, mass of load = 4 kg
Length of wire, l = 2 m
Radius of wire (cross-section) = 2 mm = $2\times {10}^{-3}\mathrm{m}$

and change in length,  $\mathrm{\Delta }l=0.031\mathrm{mm}=0.031\times {10}^{-3}\mathrm{m}$

Young's modulus,

$Y=\frac{\text{ Stress }}{\text{ Strain }}=\frac{\frac{mg}{A}}{\left(\frac{\mathrm{\Delta }l}{l}\right)}=\frac{\frac{4\times 3.1\pi }{\pi {\left(2\times {10}^{-3}\right)}^2}}{(\mathrm{\Delta }l/l)}$

$\Rightarrow Y=\frac{4\times 3.1\pi }{\pi \times 4\times {10}^{-6}}\times \frac{2}{0.031\times {10}^{-3}}$

$=\frac{3.1\times 2}{0.031}\times {10}^9=2\times {10}^{11}{\mathrm{Nm}}^{-2}$