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A uniform wire (Young's modulus $2 \times 10^{11} \mathrm{Nm}^{-2}$ ) is subjected to longitudinal tensile stress of $5 \times 10^7 \mathrm{Nm}^{-2}$. If the overall volume change in the wire is $0.02 \%$, the fractional decrease in the radius of the wire is close to:
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Verified Answer
The correct answer is:
$0.25 \times 10^{-4}$
$0.25 \times 10^{-4}$
Given, $y=2 \times 10^{11} \mathrm{Nm}^{-2}$
$$
\begin{aligned}
& \text { Stress }\left(\frac{\mathrm{F}}{\mathrm{A}}\right)=5 \times 10^7 \mathrm{Nm}^{-2} \\
& \Delta \mathrm{V}=0.02 \%=2 \times 10^{-4} \mathrm{~m}^3 \\
& \frac{\Delta \mathrm{r}}{\mathrm{r}}=?
\end{aligned}
$$
$$
\begin{aligned}
& \gamma=\frac{\text { stress }}{\text { strain }} \Rightarrow \operatorname{strain}\left(\frac{\Delta \ell}{\ell_0}\right)=\frac{\gamma}{\text { stress }} \\
& \Delta \mathrm{V}=2 \pi \mathrm{r} \ell_0 \Delta \mathrm{r}-\pi \mathrm{r}^2 \Delta \ell
\end{aligned}
$$
From eqns (i) and (ii) putting the value of $\Delta \ell, \ell_0$ and $\Delta \mathrm{V}$ and solving we get
$$
\frac{\Delta \mathrm{r}}{\mathrm{r}}=0.25 \times 10^{-4}
$$
$$
\begin{aligned}
& \text { Stress }\left(\frac{\mathrm{F}}{\mathrm{A}}\right)=5 \times 10^7 \mathrm{Nm}^{-2} \\
& \Delta \mathrm{V}=0.02 \%=2 \times 10^{-4} \mathrm{~m}^3 \\
& \frac{\Delta \mathrm{r}}{\mathrm{r}}=?
\end{aligned}
$$
$$
\begin{aligned}
& \gamma=\frac{\text { stress }}{\text { strain }} \Rightarrow \operatorname{strain}\left(\frac{\Delta \ell}{\ell_0}\right)=\frac{\gamma}{\text { stress }} \\
& \Delta \mathrm{V}=2 \pi \mathrm{r} \ell_0 \Delta \mathrm{r}-\pi \mathrm{r}^2 \Delta \ell
\end{aligned}
$$
From eqns (i) and (ii) putting the value of $\Delta \ell, \ell_0$ and $\Delta \mathrm{V}$ and solving we get
$$
\frac{\Delta \mathrm{r}}{\mathrm{r}}=0.25 \times 10^{-4}
$$
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