Question

A metal cylinder of length $L$ is subjected to a uniform compressive force $F$ as shown in the figure. The material of the cylinder has Young?s modulus $Y$ and Poisson?s ratio $\sigma$. The change in volume of the cylinder is

• A. $\frac{\sigma FL}{Y}$
• B. $\frac{\left(1-\sigma\right)FL}{Y}$
• C. $\frac{\left(1+2\sigma\right)FL}{Y}$
• D. $\frac{\left(1-2\sigma\right)FL}{Y}$

Answer 1

Answer: (D)

$\frac{\left(1-2\sigma\right)FL}{Y}$

Answer 2

Volume of the cylinder, $V=\pi r^{2}L$ Volumetric strain $=\frac{\Delta V}{V}$ $=\frac{\Delta\left(\pi r^{2} L\right)}{\pi r^{2} L}$ $\frac{\Delta V}{V}$ $=\frac{\pi r^{2}\Delta L+2\pi r L\Delta r}{\pi r^{2}L}$ $=\frac{\Delta L}{L}$ $+\frac{2 \Delta r}{r}$ $\quad$ $\ldots\left(i\right)$ Poisson?s ratio, $\sigma$ $=-\frac{\left(\Delta r/ r\right)}{\left(\Delta L/ L\right)}$ or $\quad$ $\frac{\Delta r}{r}$ $=-\frac{\sigma\Delta L}{L}$ On substituting this value of $\frac{\Delta r}{r}$ in E $\left(i\right)$, we get $\frac{\Delta V}{V}$ $=\frac{\Delta L}{L}\left(1-2\sigma\right)$ $\quad\ldots\left(ii\right)$ Young?s modulus, $Y=\frac{\left(F /\pi r^{2}\right)}{\Delta L /L}$or $\frac{\Delta L}{L}=\frac{F}{\pi r ^{2}Y}$ On substituting this value of $\frac{\Delta L}{L}$ in E $\left(ii\right)$, we get $\frac{\Delta V}{V}$ $=\frac{F}{\pi r^{2} Y}$ $\left(1-2\sigma\right)$ $\frac{\Delta V}{\pi r^{2} L}$ $=\frac{F}{\pi r^{2} Y}$ $\left(1-2\sigma\right)$ $\Delta V$ $=\frac{FL}{Y}$ $\left(1-2\sigma\right)$