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 over all volume change in the wire is $0.02\%,$ the fractional decrease in the radius of the wire is close to
JEE MAIN 2013, Medium
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$Given,\,y = 2 \times {10^{11}}N{m^{ - 2}}$
$Stress\left( {\frac{F}{A}} \right) = 5 \times {10^7}N{m^{ - 2}}$
$\Delta V = 0.02\% = 2 \times {10^{ - 4}}{m^3}$
$\frac{{\Delta r}}{r} = ?$
$\gamma = \frac{{stress}}{{strain}} \Rightarrow strain\left( {\frac{{\Delta \ell }}{{{\ell _0}}}} \right) = \frac{\gamma }{{stress}}\,\,...\left( i \right)$
$\Delta V = 2\pi {\ell _0}\Delta r - \pi {r^2}\Delta \ell $ $...\left( {ii} \right)$
From eqns $(i)$ and $(ii)$ putting the value of
$\Delta \ell ,{\ell _0}\,and\,\Delta V\,and\,solving\,we\,get$
$\frac{{\Delta r}}{r} = 0.25 \times {10^{ - 4}}$
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