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 be $0.2$, then volume approximately decreases by (in percentage).

Answer 1

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