Question

The Poissons ratio of a material is $0.4$. If a force is applied to a wire of this material, there is a decrease of the cross-sectional area by $2$%. The percentage increase in its length is

• A. $3$%
• B. $2.5$%
• C. $1$%
• D. $0.5$%

Answer 1

The correct option is B $2.5$%

Poissons ratio, $0.4=\frac{\Delta d/d}{\Delta l/l}\Rightarrow \frac{\Delta l}{l}=\frac{\Delta d/d}{0.4}$-------------(1)

$A=\pi r^2=\frac{\pi d^2}{4}$
Taking log both sides:
$\log A=\log \left(\frac{\pi }{4}\right)+2\log d$
Taking relative change both the sides:
$\frac{\Delta A}{A}=2\frac{\Delta d}{d}\frac{\Delta d}{d}=\frac{1}{2}\frac{\Delta A}{A}=\frac{1}{2}\times 2\%=1\%$------------------------------(1)
From equation (1) and (2), the percentage increase length can be given as:
$\frac{\Delta l}{l}=\frac{1\%}{0.4}=2.5\%$

Option B is correct