Question

A material has Poisson’s ratio 0.5. If a uniform rod of it suffers a longitudinal strain of \[3\times {{10}^{-3}}\], what will be the percentage increase in volume?
\[\begin{align}
& \text{A) 2 }\!\!\%\!\!\text{ } \\
& \text{B) 3 }\!\!\%\!\!\text{ } \\
& \text{C) 5 }\!\!\%\!\!\text{ } \\
& \text{D) 0 }\!\!\%\!\!\text{ } \\
\end{align}\]

Answer 1

Hint: We need to understand the relation between the Poisson’s ratio, longitudinal strain or the Young’s modulus and the Volume strain or Bulk modulus experienced by a uniform rod of a material to solve this problem to get the required solution.

Complete answer:
We are given the longitudinal strain experienced by a uniform rod which is made of a material which has a Poisson’s ratio of 0.5. We know that the Poisson’s ratio is a relation between the bulk or the volume modulus and the elastic or the Young’s modulus. It is given as –
\[B=(1-2\eta )Y\]
where \[\eta \] is the Poisson’s ratio for a material, B is the bulk modulus and Y is the Young’s modulus.
Now, let us substitute the given values to find the required percentage in volume. We know that the stress applied in the system will be same regardless of the type of strain considers, so we can rewrite the relation in terms of the strains as –
\[\begin{align}
& B=(1-2\eta )Y \\
& \Rightarrow \dfrac{S}{\dfrac{\Delta V}{V}}=(1-2\eta )\dfrac{S}{\dfrac{\Delta l}{l}} \\
& \therefore \dfrac{\Delta l}{l}=(1-2\eta )\dfrac{\Delta V}{V} \\
\end{align}\]
Now, we are given the longitudinal strain as \[3\times {{10}^{-3}}\]. But we know that the change is accompanied by the change in the radius of the cross section of the rod. As a result of which the volume doesn’t change.
\[\therefore \dfrac{\Delta V}{V}=0\]
From this, we can find the percentage change in volume to be equal to zero percentage.

The correct answer is option D.

Note:
The Poisson’s ratio is a constant for a given material. It is a characteristic property of the material which relates the two major stress-strain ratios – the bulk modulus and the Young’s modulus. Also, the volume of material will not when it is deformed.