Question

A compressive force is applied to a uniform rod of rectangular cross-section so that its length decreases by 1%. If the Poisson's ratio for the material of the rod is $0.2$, then the volume approximately decreases by (in percentage).

Answer 1

Volume of rod, $V=Al=abl$
where $l$ is the length and $a$ and $b$ are dimensions of cross-section.

If area of cross section ($ab$) remains same,

$\frac{\Delta a}{a}=-\frac{\Delta b}{b}$

Poisson's ratio is defined as
$\sigma =\frac{\frac{-\Delta a}{a}}{\frac{\Delta l}{l}}=\frac{\frac{\Delta b}{b}}{\frac{\Delta l}{l}}$

$V=abl$
$\Rightarrow \frac{\Delta V}{V}=\left|\frac{\Delta a}{a}\right|+\left|\frac{\Delta b}{b}\right|+\left|\frac{\Delta l}{l}\right|=2\frac{\Delta a}{a}+\frac{\Delta l}{l}$
$=-2\sigma \frac{\Delta l}{l}+\frac{\Delta l}{l}$
$\Rightarrow \frac{\Delta V}{V}=\frac{\Delta l}{l}(1-2\sigma )=-1\times (1-2\times 0.2)$
$=-1(1-0.4)=-0.6$
$\therefore$ The volume approximately decreases by 0.6%.