Notes : Poisson’s Ratio: Definition, Formula, Derivation, Properties, Numericals, MCQs, FAQs - Physicskund

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Poisson’s Ratio: Definition, Formula, Derivation, Properties, Numericals, MCQs, FAQs - Physicskund

1. Introduction

When a force is applied to a material, its dimensions change. For example, when a wire is stretched:

  • Its length increases.
  • Its diameter decreases.

The strain produced along the direction of the applied force is called longitudinal strain, while the strain produced perpendicular to the applied force is called lateral strain.

French mathematician and physicist Siméon Denis Poisson observed that within the elastic limit, lateral strain is proportional to longitudinal strain. This observation led to the concept of Poisson’s Ratio.

2. Lateral Strain

The strain produced perpendicular to the direction of the applied force is called lateral strain.

Formula

If:

  • Original diameter of wire = \(d\)
  • Change (decrease) in diameter = \(\Delta d\)

\[ \text{Lateral Strain} = \frac{\Delta d}{d} \]

3. Longitudinal Strain

The strain produced in the direction of the applied force is called longitudinal strain.

Formula

If:

  • Original length = \(L\)
  • Increase in length = \(\Delta L\)

\[ \text{Longitudinal Strain} = \frac{\Delta L}{L} \]

4. Poisson’s Observation

Within the elastic limit:

\[ \text{Lateral Strain} \propto \text{Longitudinal Strain} \]

Therefore:

\[ \frac{\text{Lateral Strain}} {\text{Longitudinal Strain}} = \text{Constant} \]

This constant is called Poisson’s Ratio.

5. Definition of Poisson’s Ratio

Poisson’s ratio is defined as the ratio of lateral strain to longitudinal strain.

Formula

\[ \mu = \frac{\text{Lateral Strain}} {\text{Longitudinal Strain}} \]

Substituting the strain formulas:

\[ \mu = \frac{\frac{\Delta d}{d}} {\frac{\Delta L}{L}} \]

Multiplying by the reciprocal:

\[ \mu = \frac{\Delta d}{d} \times \frac{L}{\Delta L} \]

Hence:

\[ \boxed{ \mu = \frac{\Delta d\,L} {d\,\Delta L} } \]

6. Explanation of Symbols

Symbol Meaning
\(d\) Original diameter
\(\Delta d\) Change in diameter
\(L\) Original length
\(\Delta L\) Change in length
\(\mu\) Poisson’s Ratio

7. Properties of Poisson’s Ratio

1. Pure Number

Poisson’s ratio is the ratio of two strains.

\[ \text{Strain} = \frac{\text{Change in Dimension}} {\text{Original Dimension}} \]

Therefore:

\[ \boxed{\text{Poisson’s Ratio is a pure number}} \]

2. No Unit

Poisson’s ratio has no unit because it is a ratio of similar quantities.

\[ \boxed{\text{No Unit}} \]

3. Dimensionless Quantity

Dimensional Formula:

\[ \boxed{ [M^0L^0T^0] } \]

4. Material Property

Poisson’s ratio depends only on the nature of the material. Different materials have different values.

8. Typical Values of Poisson’s Ratio

Material Poisson’s Ratio
Steel 0.28 – 0.30
Aluminium Alloy 0.33
Copper 0.34
Brass 0.35
Rubber 0.48 – 0.50

9. Physical Significance

Small Poisson’s Ratio

A small value indicates very little lateral contraction when the material is stretched.

Large Poisson’s Ratio

A large value indicates considerable lateral contraction when the material is stretched.

10. Numerical Example

Problem

  • Original diameter \(d = 2\,mm\)
  • Decrease in diameter \(\Delta d = 0.01\,mm\)
  • Original length \(L = 100\,cm\)
  • Increase in length \(\Delta L = 1\,cm\)

Find Poisson’s ratio.

Solution

\[ \mu = \frac{\Delta d\,L} {d\,\Delta L} \]

\[ = \frac{0.01\times100} {2\times1} \]

\[ = \frac{1}{2} \]

\[ \boxed{\mu=0.5} \]

11. Key Points to Remember

  • Lateral strain acts perpendicular to the applied force.
  • Longitudinal strain acts along the applied force.
  • Poisson’s ratio is the ratio of lateral strain to longitudinal strain.
  • It is represented by \(\mu\) or \(\nu\).
  • It is dimensionless.
  • It has no unit.
  • It depends on the material.
  • For steel, Poisson’s ratio is approximately 0.29.
  • For aluminium alloy, Poisson’s ratio is approximately 0.33.

12. FAQ (Frequently Asked Questions)

Q1. What is Poisson’s Ratio?

Answer: Poisson’s ratio is the ratio of lateral strain to longitudinal strain.

Q2. Who introduced Poisson’s Ratio?

Answer: It was introduced by Siméon Denis Poisson.

Q3. What is lateral strain?

Answer: The strain produced perpendicular to the applied force is called lateral strain.

Q4. What is longitudinal strain?

Answer: The strain produced along the direction of the applied force is called longitudinal strain.

Q5. Does Poisson’s ratio have a unit?

Answer: No, it has no unit.

Q6. Is Poisson’s ratio dimensionless?

Answer: Yes, it is a dimensionless quantity.

Q7. What is the Poisson’s ratio of steel?

Answer: Approximately 0.28–0.30.

Q8. What is the Poisson’s ratio of aluminium alloy?

Answer: Approximately 0.33.

13. Quiz Section

Multiple Choice Questions (MCQs)

  1. Poisson’s ratio is defined as:
    A) Stress/Strain
    B) Longitudinal Strain/Lateral Strain
    C) Lateral Strain/Longitudinal Strain
    D) Force/Area
    Answer: C
  2. Poisson’s ratio is represented by:
    A) Y
    B) K
    C) μ
    D) η
    Answer: C
  3. The SI unit of Poisson’s ratio is:
    A) N/m²
    B) Joule
    C) Meter
    D) No unit
    Answer: D
  4. Lateral strain occurs:
    A) Along the force
    B) Perpendicular to the force
    C) Opposite to the force
    D) At 45°
    Answer: B
  5. Poisson’s ratio is:
    A) Vector quantity
    B) Dimensionless quantity
    C) Unit of stress
    D) Unit of strain
    Answer: B

True or False

  1. Poisson’s ratio is a scalar quantity. True
  2. Poisson’s ratio has dimensions. False
  3. Steel has a Poisson’s ratio of about 0.29. True
  4. Lateral strain acts parallel to the applied force. False
  5. Poisson’s ratio depends on the nature of the material. True

Fill in the Blanks

  1. Poisson’s ratio is denoted by μ.
  2. The strain perpendicular to the applied force is called lateral strain.
  3. The strain along the force is called longitudinal strain.
  4. Poisson’s ratio is a dimensionless quantity.
  5. Steel has a Poisson’s ratio of about 0.29.

14. Numerical Problems

Numerical 1

The longitudinal strain in a rod is \(0.002\) and lateral strain is \(0.0006\). Find Poisson’s ratio.

\[ \mu = \frac{0.0006}{0.002} = 0.3 \]

\[ \boxed{\mu=0.3} \]

Numerical 2

A material has a Poisson’s ratio of \(0.25\). If the longitudinal strain is \(0.004\), find the lateral strain.

\[ \text{Lateral Strain} = 0.25\times0.004 = 0.001 \]

\[ \boxed{0.001} \]

15. Long Answer Question

Explain Poisson’s Ratio and derive its expression.

When a wire is stretched, its length increases and its diameter decreases. The increase in length produces longitudinal strain, while the decrease in diameter produces lateral strain.

According to Poisson:

\[ \text{Lateral Strain} \propto \text{Longitudinal Strain} \]

Therefore:

\[ \mu = \frac{\text{Lateral Strain}} {\text{Longitudinal Strain}} \]

Using:

\[ \text{Lateral Strain} = \frac{\Delta d}{d} \]

and

\[ \text{Longitudinal Strain} = \frac{\Delta L}{L} \]

we get:

\[ \mu = \frac{\frac{\Delta d}{d}} {\frac{\Delta L}{L}} = \frac{\Delta d\,L} {d\,\Delta L} \]

Thus, Poisson’s ratio is the ratio of lateral strain to longitudinal strain. It is dimensionless, has no unit, and depends only on the nature of the material.

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