Question

The stress along the length of a rod (with rectangular cross section) is \(1 \%\) of the Young's modulus of its material. What is the approximate percentage of change of its volume? (Poisson's ratio of the material of the rod is \(0.3\) )
(A) \(3 \%\)
(B) \(1 \%\)
(C) \(0.7 \%\)
(D) \(0.4 \%\)

Answer 1

Ans: (D)
Hint \(: \frac{F}{A}=\frac{1}{100} \times Y, Y=\frac{F / A}{\Delta \ell / L} \therefore \frac{\Delta L}{L}=\frac{F / A}{Y}=\frac{1}{100}\),
$$
\begin{array}{l|l}
\frac{\Delta \mathrm{V}}{\mathrm{V}}=\frac{2 \Delta \mathrm{r}}{\mathrm{r}}+\frac{\Delta \mathrm{L}}{\mathrm{L}} & \sigma=-\frac{\Delta \mathrm{r} / \mathrm{r}}{\Delta \mathrm{L} / \mathrm{L}} \\
& 0.3=\frac{-\Delta \mathrm{r} / \mathrm{r}}{1 / 100} \\
= & 2 \times\left(\frac{-0.3}{100}\right)+\frac{1}{100} \quad \frac{0.3}{100}=-\Delta \mathrm{r} / \mathrm{r} \\
= & +\frac{0.4}{100}=0.4 \%
\end{array}
$$