Notes : Modulus of Elasticity: Young's, Bulk , Shear Modulus | Formulas, Derivations, Units , Dimensions

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Modulus of Elasticity: Young’s Modulus, Bulk Modulus, Shear Modulus & Compressibility | Class 11 Physics Notes chapter 8 Mechanical Properties of Solids - Physicskund

The modulus of elasticity is defined as the ratio of stress to the corresponding strain produced within the elastic limit of a material. It is a measure of the stiffness or rigidity of a material.

$ \text{Modulus of Elasticity} = \frac{\text{Stress}}{\text{Strain}} $

Types of Modulus of Elasticity

There are three types of modulus of elasticity:

  • Young's Modulus (Y)
  • Bulk Modulus (K)
  • Shear Modulus or Modulus of Rigidity (G)

1. Young's Modulus (Y)

Definition

Young's modulus is defined as the ratio of longitudinal stress to longitudinal strain.

$ Y = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} $

Derivation

Consider a wire of length $L$, radius $r$, and cross-sectional area $A$. When a force $F$ is applied along its length producing an extension $\Delta L$:

$ \text{Longitudinal Stress} = \frac{F}{A} $

$ \text{Longitudinal Strain} = \frac{\Delta L}{L} $

$ Y = \frac{F/A}{\Delta L/L} $

$ Y = \frac{FL}{A\Delta L} $

Since,

$ A = \pi r^2 $

and

$ F = mg $

Therefore,

$ Y = \frac{mgL}{\pi r^2 \Delta L} $

SI Unit

$ N\,m^{-2} $ or Pascal (Pa)

Dimensional Formula

$ [ML^{-1}T^{-2}] $

Applicable To

Solids only.

2. Bulk Modulus (K)

Definition

Bulk modulus is defined as the ratio of bulk stress to volume strain.

$ K = \frac{\text{Bulk Stress}}{\text{Volume Strain}} $

Derivation

Bulk stress is equal to pressure.

$ \text{Bulk Stress} = P $

Volume strain:

$ \text{Volume Strain} = \frac{-\Delta V}{V} $

Therefore,

$ K = \frac{P}{-\Delta V/V} $

$ K = -\frac{PV}{\Delta V} $

The negative sign indicates that volume decreases when pressure increases.

SI Unit

$ N\,m^{-2} $ or Pascal (Pa)

Dimensional Formula

$ [ML^{-1}T^{-2}] $

Applicable To

Solids, liquids and gases.

Compressibility (B)

Definition

The reciprocal of bulk modulus is called compressibility.

$ B = \frac{1}{K} $

or

$ B = -\frac{\Delta V}{PV} $

SI Unit

$ Pa^{-1} $

Dimensional Formula

$ [M^{-1}LT^2] $

3. Shear Modulus or Modulus of Rigidity (G)

Definition

Shear modulus is defined as the ratio of tangential stress to shear strain.

$ G = \frac{\text{Tangential Stress}}{\text{Shear Strain}} $

Derivation

Tangential stress:

$ \frac{F}{A} $

For a cube of side $L$, if the upper face shifts by $\Delta x$:

$ \tan\theta = \frac{\Delta x}{L} $

For very small angles:

$ \tan\theta \approx \theta $

Therefore,

$ \theta = \frac{\Delta x}{L} $

Hence,

$ G = \frac{F/A}{\Delta x/L} $

$ G = \frac{FL}{A\Delta x} $

SI Unit

$ N\,m^{-2} $ or Pascal (Pa)

Dimensional Formula

$ [ML^{-1}T^{-2}] $

Applicable To

Solids only.

Comparison of Elastic Moduli

Modulus Formula Applicable To
Young's Modulus (Y) $ \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} $ Solids
Bulk Modulus (K) $ \frac{\text{Bulk Stress}}{\text{Volume Strain}} $ Solids, Liquids, Gases
Shear Modulus (G) $ \frac{\text{Tangential Stress}}{\text{Shear Strain}} $ Solids

Dimension and Unit Summary

Physical Quantity Symbol SI Unit Dimensional Formula
Young's Modulus Y Pa $ [ML^{-1}T^{-2}] $
Bulk Modulus K Pa $ [ML^{-1}T^{-2}] $
Shear Modulus G Pa $ [ML^{-1}T^{-2}] $
Compressibility B $ Pa^{-1} $ $ [M^{-1}LT^2] $

Formula Summary

$ Y = \frac{FL}{A\Delta L} $

$ Y = \frac{mgL}{\pi r^2\Delta L} $

$ K = -\frac{PV}{\Delta V} $

$ B = \frac{1}{K} $

$ G = \frac{FL}{A\Delta x} $

Very Short Answer Questions

Q1. What is modulus of elasticity?

Answer: The ratio of stress to strain within the elastic limit.

Q2. What is Young's modulus?

Answer: Ratio of longitudinal stress to longitudinal strain.

Q3. What is bulk modulus?

Answer: Ratio of bulk stress to volume strain.

Q4. What is shear modulus?

Answer: Ratio of tangential stress to shear strain.

Q5. What is compressibility?

Answer: Reciprocal of bulk modulus.

Short Answer Questions

Q1. Why is bulk modulus applicable to liquids and gases?

Answer: Liquids and gases can undergo volume changes under pressure, therefore bulk modulus is applicable to them.

Q2. Why does bulk modulus contain a negative sign?

Answer: The negative sign indicates that volume decreases when pressure increases.

Q3. What does a high value of Young's modulus indicate?

Answer: It indicates greater rigidity and stiffness.

Long Answer Questions

Q1. Define Young's modulus and derive its expression.

Answer: Young's modulus is the ratio of longitudinal stress to longitudinal strain.

$ Y=\frac{F/A}{\Delta L/L}=\frac{FL}{A\Delta L} $

Q2. Define bulk modulus and derive its formula.

$ K=\frac{P}{-\Delta V/V}=-\frac{PV}{\Delta V} $

Q3. Define shear modulus and derive its formula.

$ G=\frac{F/A}{\Delta x/L}=\frac{FL}{A\Delta x} $

Fill in the Blanks

1. The ratio of stress to strain is called Modulus of Elasticity.

2. Young's modulus is denoted by Y.

3. Bulk modulus is denoted by K.

4. Shear modulus is denoted by G.

5. Compressibility is the reciprocal of Bulk Modulus.

6. SI unit of modulus of elasticity is Pascal.

7. Bulk stress is equal to Pressure.

8. Compressibility is represented by B.

True or False

1. Young's modulus is applicable to solids only. True

2. Bulk modulus is applicable to liquids and gases. True

3. Compressibility is equal to bulk modulus. False

4. Shear modulus is also called modulus of rigidity. True

5. All elastic moduli have SI unit Pascal. True

6. Higher modulus indicates greater stiffness. True

MCQ Quiz

1. Young's modulus is the ratio of:

A) Stress to Strain ✔️

B) Force to Area

C) Pressure to Volume

D) Area to Force

2. SI unit of modulus of elasticity is:

A) Newton

B) Joule

C) Pascal ✔️

D) Watt

3. Which modulus is applicable to liquids?

A) Young's Modulus

B) Shear Modulus

C) Bulk Modulus ✔️

D) None

4. Compressibility is:

A) K

B) G

C) 1/K ✔️

D) Y

5. Shear modulus is also known as:

A) Bulk Modulus

B) Modulus of Rigidity ✔️

C) Elastic Constant

D) Compressibility

Frequently Asked Questions (FAQ)

Q1. What is modulus of elasticity?

Answer: It is the ratio of stress to strain within the elastic limit.

Q2. How many types of modulus of elasticity are there?

Answer: Three—Young's Modulus, Bulk Modulus and Shear Modulus.

Q3. Which modulus is valid for liquids and gases?

Answer: Bulk modulus.

Q4. What is the SI unit of modulus of elasticity?

Answer: Pascal (Pa).

Q5. What is compressibility?

Answer: The reciprocal of bulk modulus.

$ B=\frac{1}{K} $

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