Question

The Poisson's ratio of a material is 0.4. If a force is applied to a wire of this material, there is a decrease of 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 $2.5$%

Poisson's 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