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Question: A material has Poisson’s ratio 0.3. If a uniform rod of it suffers a longitudinal strain of \[3 \tim...

A material has Poisson’s ratio 0.3. If a uniform rod of it suffers a longitudinal strain of 3×10−33 \times {10^{ - 3}}, what will be the percentage increase in volume?

Explanation

Solution

In this question we have to find the value of increase in the volume of the material. The Poisson’s ratio and longitudinal strain of the material are given. To find the increase in volume we will use the formula for Poisson’s ratio,
Poisson’s ratio(σ)=−(ΔR/RΔL/L){\text{Poisson's ratio(}}\sigma {\text{)}} = - \left( {\dfrac{{\Delta R/R}}{{\Delta L/L}}} \right)

Complete step by step solution:
Given,
⇒ΔLL=3×10−3\Rightarrow \dfrac{{\Delta L}}{L} = 3 \times {10^{ - 3}}
⇒Poisson’s ratio(σ)=−(ΔR/RΔL/L)\Rightarrow {\text{Poisson's ratio(}}\sigma {\text{)}} = - \left( {\dfrac{{\Delta R/R}}{{\Delta L/L}}} \right)
⇒0.3=−(ΔR/R3×10−3)\Rightarrow 0.3 = - \left( {\dfrac{{\Delta R/R}}{{3 \times {{10}^{ - 3}}}}} \right)
⇒ΔRR=−0.3×3×10−3\Rightarrow \dfrac{{\Delta R}}{R} = - 0.3 \times 3 \times {10^{ - 3}}
⇒ΔRR=−0.9×10−3\Rightarrow \dfrac{{\Delta R}}{R} = - 0.9 \times {10^{ - 3}}
Volume of rod is V=πR2LV = \pi {R^2}L
To find an increase in volume we will convert it in this form.
⇒ΔVV=2ΔRR+ΔLL\Rightarrow \dfrac{{\Delta V}}{V} = 2\dfrac{{\Delta R}}{R} + \dfrac{{\Delta L}}{L}
⇒ΔVV=(−2×0.9×10−3+3×10−3)\Rightarrow \dfrac{{\Delta V}}{V} = \left( { - 2 \times 0.9 \times {{10}^{ - 3}} + 3 \times {{10}^{ - 3}}} \right)
⇒ΔVV=1.2×10−3\Rightarrow \dfrac{{\Delta V}}{V} = 1.2 \times {10^{ - 3}}
Now, we will find the percentage increase in volume
⇒ΔVV×100=1.2×10−3×100\Rightarrow \dfrac{{\Delta V}}{V} \times 100 = 1.2 \times {10^{ - 3}} \times 100
⇒ΔVV×100=0.12%\Rightarrow \dfrac{{\Delta V}}{V} \times 100 = 0.12\%
Hence, from the above calculation we have found the value of percentage increase in volume and it comes out to be, ΔVV×100=0.12%\dfrac{{\Delta V}}{V} \times 100 = 0.12\% .

Additional Information:
Strain is a measure of how much a body has been deformed or stretched. Strain in a body occurs when a force is applied on it. This is a unit less quantity. Strain in a body is given by following formula;
strain=extensionlengthstrain = \dfrac{{extension}}{{length}}
If the extension in length is ΔL\Delta Land the total length of the body is LL, then strain is given by following formula;
strain=ΔLLstrain = \dfrac{{\Delta L}}{L}
There are three types of stain-
Longitudinal strain ΔLL\dfrac{{\Delta L}}{L}
Shearing strain ΔLL\dfrac{{\Delta L}}{L}
Volumetric strain ΔVV\dfrac{{\Delta V}}{V}

Note: Poisson’s ratio is a measure of deformation of a material in different directions perpendicular to the direction of the force applied. In other words, a Poisson’s ratio is the ratio of the transverse strain to the longitudinal strain. It is represented by σ\sigma . Since, it is a ratio so it does not have any dimension. It is a scalar quantity.
Poisson’s ratio(σ)=−(Transverse strainLongitudinal strain){\text{Poisson's ratio(}}\sigma {\text{)}} = - \left( {\dfrac{{Transverse{\text{ strain}}}}{{Longitudinal{\text{ strain}}}}} \right)
The deformation in the material in different directions perpendicular to the direction of force applied on the material is also known as Poisson’s effect.