Question

A uniform wire (Young?s modulus $2 \times 10^{11}\,Nm^{-2}$) is subjected to longitudinal tensile stress of $5 \times 10^{7}\,Nm^{-2}$. If the overall volume change in the wire is $0.02\%$, the fractional decrease in the radius of the wire is close to:

• A. $1.0 \times 10^{4}$
• B. $1.5 \times 10^{4}$
• C. $0.25 \times 10^{4}$
• D. $5 \times 10^{4}$

Answer 1

The Correct Option is C

Given, $y=2\times10^{11}\,Nm^{-2}$
Stress $\left(\frac{F}{A}\right)=5\times 10^{7}\,Nm^{-2}$
$\Delta V=0.02\%=2\times 10^{-4}\,m^{3}$
$\frac{\Delta r}{r}=?$
$\gamma=\frac{stress}{strain} \Rightarrow strain \left(\frac{\Delta\ell}{\ell_{0}}\right)=\frac{\gamma}{stress} ...\left(i\right)$
$\Delta V=2\pi r\ell_{0} \Delta r-\pi r^{2}\,\Delta\ell ...\left(ii\right)$
From eqns (i) and (ii) putting the value of
$\Delta\ell, \ell_{0}$ and $\Delta V$ and solving we get
$\frac{\Delta r}{r}=0.25\times10^{-4}$