Question

A material has Poisson's ratio $0.50$. If a uniform rod of it suffers a longitudinal strain of $2\times {{10}^{-3}}$, then the percentage change in volume is
$\begin{align}
  & \left( A \right)0.6 \\
 & \left( B \right)0.4 \\
 & \left( C \right)0.2 \\
 & \left( D \right)0 \\
\end{align}$

Answer 1

Hint: The volumetric strain is defined as that the unit change in volume, i.e. the change in volume divided by the first or initial volume. The Poisson effect, that describes the expansion or contraction of an object or material in directions perpendicular to the direction of loading. the worth of Poisson's ratio is the negative of the ratio of transverse strain to axial strain.
Formula used: Poisson’s Ratio:
$\sigma =-\dfrac{\left( \dfrac{dr}{r} \right)}{\left( \dfrac{dL}{L} \right)}$
Where:
$r$- radius of rod
$L$- length of rod

Complete step-by-step solution:
As the longitudinal strain is given in question
$\dfrac{dL}{L}=2\times {{10}^{-3}}$
Poisson's ratio, ​​ $\sigma =-\dfrac{\left( \dfrac{dr}{r} \right)}{\left( \dfrac{dL}{L} \right)}$
\[\begin{align}
  & \Rightarrow ~~0.5=\dfrac{-\left( \dfrac{dr}{r} \right)}{2\times {{10}^{-3}}}\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~ \\
 & \text{ }~\Rightarrow \dfrac{dr}{r}=-{{10}^{-3}} \\
\end{align}\]
Volume of the rod, $V=\pi {{r}^{2}}L$
By Differentiating, we get
\[dV=\pi \left( {{r}^{2}}dL+2Lrdr \right)\]

$\begin{align}
  & \dfrac{dV}{V}\times 100=\dfrac{\pi \left( {{r}^{2}}dL+2rLdr \right)}{\pi {{r}^{2}}L}\times 100 \\
 & \Rightarrow \left( \dfrac{dL}{L}+2\dfrac{dr}{r} \right)\times 100 \\
 & \Rightarrow \dfrac{dV}{V}\times 100=\left[ 2\times {{10}^{-3}}+2\left( -{{10}^{-3}} \right) \right]\times 100=0 \\
\end{align}$
So, the correct option for this is option (D).
Additional Information:
Poisson's ratio may be a measure of the Poisson effect, the phenomenon during which a cloth tends to expand in directions perpendicular to the direction of compression. Conversely, if the fabric is stretched instead of compressed, it always tends to accept the directions transverse to the direction of stretching. it's a standard observation that when an elastic band is stretched, it becomes noticeably thinner. Again, the Poisson ratio is going to be the ratio of relative contraction to relative expansion and can have an equivalent value as above.

Note: If the deforming force produces a change in volume along the strain produced within the body is named Volumetric Strain, the ratio of the change in radius or diameter to the first radius or diameter is named Lateral Strain, the ratio of lateral strain to longitudinal strain is named Poisson's Ratio.