JEE Main Analysis 2025 - Subject-wise Paper Review, Difficulty Level

# Relation Between Volumetric Strain, Lateral Strain And Poisson’s Ratio - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• 17 Questions around this concept.

## Solve by difficulty

A bottle has an opening of radius a and length b.  A cork of length b and radius (a + a) where (a<<a) is compressed to fit into the opening completely (See figure). If the bulk modulus of cork is B and frictional coefficient between the bottle and cork is µ then the force needed to push the cork into  the bottle is :

## Concepts Covered - 1

Relation Between Volumetric Strain, Lateral Strain and Poisson’s Ratio

Let us long rod have a length L and radius 'r', then volume of this rod = $\dpi{100} \pi r^2L . . . . . . . . (1)$

Now, Differentiating both the sides of $\dpi{100} (1)$, we get

$d V=\pi r^{2} d L+\pi 2 r L d r$

Now, dividing both the sides by volume of rod , i.e., $\dpi{100} \pi r^2L$, we get ,

$\dpi{100} \frac{d V}{V}=\frac{\pi r^{2} d L}{\pi r^{2} L}+\frac{\pi 2 r L d r}{\pi r^{2} L}=\frac{d L}{L}+2 \frac{d r}{r} . . . . (2)$

So we can say that,

Volumetric strain = Longitudinal strain + 2(Lateral strain)

Also, $\dpi{100} equation(2)$ can be written as,

$\dpi{100} \Rightarrow \frac{d V}{V}=\frac{d L}{L}-2 \sigma \frac{d L}{L}=(1-2 \sigma) \frac{d L}{L}$

This is because , $\dpi{100} \left[\mathrm \ \sigma=\frac{-d r / r}{d L / L} \Rightarrow \frac{d r}{r}=- \sigma \frac{d L}{L}\right]$

Special case -

•  When $\dpi{100} \sigma = 0.5$, then $\dpi{100} dV = 0$. It means that the substance is incompressible, so there is no change in volume.

• If a material having $\dpi{100} \sigma$ = 0, it means lateral strain is zero. So, when a substance is stretched its length increases without any decrease in diameter. For example - cork has $\dpi{100} \sigma = 0$. Also, in this case change in volume is maximum.

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## Books

### Reference Books

#### Relation Between Volumetric Strain, Lateral Strain and Poisson’s Ratio

Physics Part II Textbook for Class XI

Page No. : 238

Line : 4