Class 11 Physics Chapter 8: Mechanical Properties of Solids Notes & Formulas

Class 11 Physics Chapter 8: Mechanical Properties of Solids

Mechanical Properties of Solids is an important chapter of Class 11 Physics. It explains how solid objects change their shape, size and volume when external forces act on them. This chapter helps us understand elasticity, deformation, stress, strain and the mechanical behaviour of different materials.

In our daily life, we observe that a rubber band stretches when pulled, a spring becomes longer under a load, and a metal rod may bend when a force is applied. These changes in the shape and dimensions of objects are explained through the mechanical properties of solids.

Chapter Focus: Elasticity, plasticity, stress, strain, Hooke's law, stress-strain curve, Young's modulus, bulk modulus, shear modulus, Poisson's ratio, elastic potential energy and applications of elasticity.

In This Chapter:

• Introduction to Mechanical Properties of Solids
• Elasticity and Plasticity
• Deformation of Solids
• Stress
• Strain
• Hooke's Law
• Stress-Strain Curve
• Young's Modulus
• Shear Modulus
• Bulk Modulus and Compressibility
• Poisson's Ratio
• Elastic Potential Energy
• Applications of Elasticity
• Quick Revision and Important Formulas
• Important Questions
• Important MCQs
• Solved Numerical Problems
• Frequently Asked Questions

1. Introduction to Mechanical Properties of Solids

A solid is a state of matter which generally has a definite shape and volume. The particles of a solid are closely packed and experience strong intermolecular forces.

When an external force is applied to a solid, its shape, size or volume may change. This change is called deformation.

For example, when a force is applied to a metal wire, its length may increase slightly. Similarly, a rubber ball changes its shape when pressed.

The study of how solids respond to external forces is known as the mechanical properties of solids.

Important Points:

• Solids have a definite shape and volume.
• External forces can change their dimensions.
• Some solids regain their original shape after removing the force.
• The behaviour of a solid depends on its material and the magnitude of the applied force.

2. Elasticity and Plasticity

Elasticity

The property of a material by which it regains its original shape and size after the removal of an external deforming force is called elasticity.

Example: A steel spring returns to its original length when the stretching force is removed, provided the force does not exceed its elastic limit.

Plasticity

The property of a material by which it does not completely regain its original shape and size after the removal of an external deforming force is called plasticity.

Example: Clay or putty retains its changed shape after being pressed.

Difference Between Elasticity and Plasticity

Elasticity	Plasticity
Body regains its original shape after removing the force, within the elastic limit.	Body retains some permanent deformation.
Deformation is reversible.	Deformation is partly or completely irreversible.
Example: Steel spring.	Example: Clay.

Important: Elasticity and plasticity are properties of materials. A material may show elastic behaviour under a small force and plastic behaviour under a sufficiently large force.

3. Deformation of Solids

Deformation refers to the change in the shape, length or volume of a body due to an external force.

There are three important types of deformation in solids:

1. Longitudinal Deformation

When a force acts along the length of a body and changes its length, the deformation is called longitudinal deformation.

It may be of two types:

• Tensile deformation: The length of the body increases.
• Compressive deformation: The length of the body decreases.

2. Volume Deformation

When an external force changes the volume of a body, the deformation is called volume deformation.

Example: A solid object compressed uniformly from all sides experiences a change in volume.

3. Shearing Deformation

When a tangential force changes the shape of a body without necessarily changing its volume significantly, the deformation is called shearing deformation.

Example: The upper surface of a rectangular block shifts sideways while its lower surface remains fixed.

4. Stress

When an external deforming force acts on a body, internal restoring forces develop within the material. The restoring force acting per unit area is called stress.

Formula:

Stress = Restoring Force / Area

σ = F/A

Where:

• σ = Stress
• F = Magnitude of restoring force
• A = Area of cross-section

SI Unit of Stress

The SI unit of stress is Pascal (Pa).

1 Pa = 1 N/m²

Dimensional Formula:

[ML^-1T^-2]

Types of Stress

1. Longitudinal Stress

The restoring force per unit area acting perpendicular to the cross-section of a body and producing a change in its length is called longitudinal stress.

It is classified into:

• Tensile Stress: Produced when a body is stretched.
• Compressive Stress: Produced when a body is compressed.

Longitudinal Stress = F/A

2. Hydraulic or Bulk Stress

When a body is subjected to uniform pressure from all directions, the stress developed is called hydraulic stress or bulk stress.

It generally produces a change in the volume of the body.

3. Shearing Stress

The tangential restoring force acting per unit area of a body is called shearing stress.

Shearing Stress = F/A

Remember: Stress describes the internal force intensity developed in a material, not simply the external force acting on the whole body.

5. Strain

The fractional change in the dimensions of a body due to an external deforming force is called strain.

Strain = Change in Dimension / Original Dimension

Strain is a dimensionless physical quantity because it is the ratio of two quantities having the same unit.

Types of Strain

1. Longitudinal Strain

The ratio of change in length to the original length of a body is called longitudinal strain.

Longitudinal Strain = ΔL/L

Where:

• ΔL = Change in length
• L = Original length

It can be tensile or compressive depending on whether the length increases or decreases.

2. Volumetric Strain

The ratio of change in volume to the original volume is called volumetric strain.

Volumetric Strain = ΔV/V

Where:

• ΔV = Change in volume
• V = Original volume

3. Shearing Strain

The angular deformation produced in a body due to shearing stress is called shearing strain.

For a small angular deformation:

Shearing Strain = tan θ ≈ θ

Where θ is the angular displacement measured in radians.

Difference Between Stress and Strain

Stress	Strain
Restoring force per unit area.	Fractional change in dimension.
Formula: F/A	Formula: ΔL/L
SI unit: Pascal.	No SI unit.
Dimensional formula: [ML-1T-2]	Dimensionless.

6. Hooke's Law

Hooke's law is an important principle of elasticity. According to this law, within the proportional limit, stress is directly proportional to strain.

Mathematical Expression:

Stress ∝ Strain

Stress = k × Strain

Where k is the modulus of elasticity of the material for the given type of deformation.

For longitudinal deformation:

F/A = Y(ΔL/L)

Therefore,

Y = FL/AΔL

Where Y represents Young's modulus.

Conditions for Hooke's Law

• The deformation should be small.
• The material should remain within its proportional limit.
• Stress must be directly proportional to strain.

Important Note: Hooke's law is not valid for all materials under every condition. It is applicable only in the linear elastic region for materials that exhibit Hookean behaviour.

7. Stress-Strain Curve

A stress-strain curve is a graph showing the relationship between stress and strain for a material when it is subjected to an external deforming force.

It helps us understand the elastic and plastic behaviour of materials.

In a typical tensile test of a ductile metal, the curve contains the following important regions and points:

1. Proportional Limit

It is the maximum stress up to which stress is directly proportional to strain. Hooke's law is valid in this region.

2. Elastic Limit

It is the maximum stress up to which a material can return to its original dimensions after removing the deforming force.

3. Yield Point

The point beyond which a material begins to undergo considerable plastic deformation is called the yield point.

In this region, a small increase in stress may produce a large increase in strain.

4. Ultimate Tensile Strength

It is the maximum tensile stress that a material can withstand during a tensile test.

5. Breaking Point

The point at which the material finally fractures or breaks is called the breaking point.

Important: The exact shape of the stress-strain curve depends on the material. A ductile material and a brittle material do not show identical curves.

8. Young's Modulus

Young's modulus is defined as the ratio of longitudinal stress to longitudinal strain, provided the material is within its elastic region.

Formula:

Y = Longitudinal Stress / Longitudinal Strain

Y = (F/A)/(ΔL/L)

Therefore,

Y = FL/(AΔL)

Where:

• Y = Young's modulus
• F = Applied force
• L = Original length
• A = Cross-sectional area
• ΔL = Change in length

SI Unit: Pascal (Pa)

Dimensional Formula: [ML^-1T^-2]

Physical Meaning of Young's Modulus

Young's modulus measures the resistance of a material to longitudinal deformation. A material with a higher Young's modulus generally undergoes a smaller change in length for the same stress.

Example: Steel has a much higher Young's modulus than rubber. Therefore, steel is much more resistant to stretching under the same stress.

9. Shear Modulus or Modulus of Rigidity

The ratio of shearing stress to shearing strain is called shear modulus or modulus of rigidity.

It is represented by the symbol G or η (depending on the textbook notation).

Formula:

Shear Modulus = Shearing Stress / Shearing Strain

G = (F/A)/θ

Therefore,

G = F/(Aθ)

Where:

• G = Shear modulus
• F = Tangential force
• A = Area of the surface
• θ = Small angular deformation in radians

SI Unit: Pascal (Pa)

Dimensional Formula: [ML^-1T^-2]

Important: Shear modulus measures the resistance of a solid to change in shape due to tangential forces.

10. Bulk Modulus and Compressibility

Bulk Modulus

Bulk modulus is defined as the ratio of hydraulic stress to volumetric strain, with a negative sign to indicate that volume decreases when pressure increases.

Formula:

K = -ΔP/(ΔV/V)

Therefore,

K = -V(ΔP/ΔV)

Where:

• K = Bulk modulus
• ΔP = Change in pressure
• V = Original volume
• ΔV = Change in volume

SI Unit: Pascal (Pa)

Dimensional Formula: [ML^-1T^-2]

A higher bulk modulus indicates that a material is more resistant to compression.

Compressibility

Compressibility is the reciprocal of bulk modulus. It measures how easily a material undergoes a change in volume under applied pressure.

Formula:

β = 1/K

SI Unit: Pa^-1

Important Points:

• Bulk modulus is applicable to solids, liquids and gases.
• Compressibility is high for gases compared with most liquids and solids.
• Water has a relatively high bulk modulus and is difficult to compress.

11. Poisson's Ratio

When a wire is stretched along its length, its length increases while its lateral dimensions generally decrease. The ratio of lateral strain to longitudinal strain, taken with a negative sign, is called Poisson's ratio.

Formula:

ν = - Lateral Strain / Longitudinal Strain

Where ν (nu) represents Poisson's ratio.

For a wire stretched along its length:

ν = -(Δd/d)/(ΔL/L)

Where:

• Δd = Change in diameter
• d = Original diameter
• ΔL = Change in length
• L = Original length

Poisson's ratio is dimensionless because it is a ratio of two strains.

Example: When a rubber band is stretched, its length increases and its thickness generally decreases.

12. Elastic Potential Energy

When an elastic body is stretched or compressed within its elastic limit, work is done against its restoring force. This work is stored in the body as elastic potential energy.

Consider a wire of original length L and cross-sectional area A. If it is stretched by a small extension ΔL, the restoring force increases from zero to F.

Average force = F/2

Work done = Average Force × Extension

U = (1/2)FΔL

This work is stored as elastic potential energy.

Using Young's modulus:

Y = FL/(AΔL)

Therefore,

U = YA(ΔL)²/(2L)

Elastic energy per unit volume is:

u = (1/2) × Stress × Strain

For a stretched wire:

u = (1/2)Y(ΔL/L)²

SI Unit of Elastic Potential Energy: Joule (J).

Important: The above expressions apply to linear elastic deformation where force increases linearly with extension.

13. Applications of Elasticity

The principles of elasticity are widely used in engineering, construction and everyday life.

1. Construction of Buildings and Bridges

Engineers select materials with suitable elastic properties to ensure that buildings and bridges can withstand loads safely without excessive deformation.

2. Railway Tracks

Railway tracks are designed to withstand the weight and repeated forces produced by moving trains.

3. Springs

Elastic materials are used in springs, shock absorbers and measuring instruments because they can return to their original shape within the elastic limit.

4. Cranes and Lifting Equipment

Elastic properties help engineers select suitable cables and supporting materials for lifting heavy loads.

5. Sports Equipment

Elastic materials are used in sports equipment such as balls, rackets and protective items to provide suitable flexibility and strength.

6. Safety and Structural Design

Engineers consider stress, strain, elastic limits and material strength while designing machines, vehicles and structures.

14. Quick Revision – Important Formulas

Physical Quantity	Formula
Stress	σ = F/A
Longitudinal Strain	ε = ΔL/L
Volumetric Strain	ΔV/V
Shearing Strain	θ (for small angles)
Hooke's Law	Stress = k × Strain
Young's Modulus	Y = FL/(AΔL)
Shear Modulus	G = (F/A)/θ
Bulk Modulus	K = -ΔP/(ΔV/V)
Compressibility	β = 1/K
Poisson's Ratio	ν = -Lateral Strain/Longitudinal Strain
Elastic Potential Energy	U = ½ FΔL
Elastic Energy Density	u = ½ × Stress × Strain

Important SI Units

Quantity	SI Unit
Stress	Pascal (Pa)
Strain	Dimensionless
Young's Modulus	Pascal (Pa)
Shear Modulus	Pascal (Pa)
Bulk Modulus	Pascal (Pa)
Compressibility	Pa-1
Poisson's Ratio	Dimensionless
Elastic Potential Energy	Joule (J)

15. Important Questions – Mechanical Properties of Solids

Very Short Answer Questions

1. What is elasticity?
2. Define plasticity.
3. What is deformation?
4. Define stress.
5. What is the SI unit of stress?
6. Define strain.
7. Why is strain dimensionless?
8. State Hooke's law.
9. What is Young's modulus?
10. Define bulk modulus.
11. What is shear modulus?
12. Define Poisson's ratio.
13. What is the elastic limit?
14. Define compressibility.
15. What is elastic potential energy?

Short Answer Questions

1. Differentiate between elasticity and plasticity with examples.
2. Explain the different types of stress.
3. Explain the different types of strain.
4. Differentiate between stress and strain.
5. Explain Hooke's law with its mathematical expression.
6. What is a stress-strain curve? Explain its importance.
7. Differentiate between Young's modulus and bulk modulus.
8. Why is steel considered more elastic than rubber in physics?
9. What is the physical significance of Young's modulus?
10. Explain Poisson's ratio with an example.

Long Answer and Derivation Questions

1. Explain the mechanical properties of solids.
2. Derive the expression for Young's modulus of a wire.
3. Derive the expression for the bulk modulus of a material.
4. Explain the stress-strain curve of a ductile material.
5. Derive the expression for elastic potential energy stored in a stretched wire.
6. Explain shear modulus and its physical significance.
7. Discuss the applications of elasticity in everyday life.

16. Important MCQs – Class 11 Physics Chapter 8

1. The SI unit of stress is:

• (a) Newton
• (b) Pascal
• (c) Joule
• (d) Watt

Answer: (b) Pascal

2. Strain is a:

• (a) Vector quantity
• (b) Dimensionless quantity
• (c) Quantity having unit metre
• (d) Quantity having unit Newton

Answer: (b) Dimensionless quantity

3. Hooke's law states that within the proportional limit:

• (a) Stress is inversely proportional to strain
• (b) Stress is directly proportional to strain
• (c) Stress is equal to mass
• (d) Strain is always zero

Answer: (b) Stress is directly proportional to strain

4. Young's modulus is the ratio of:

• (a) Stress to longitudinal strain
• (b) Force to mass
• (c) Energy to volume
• (d) Pressure to density

Answer: (a) Stress to longitudinal strain

5. Which of the following is dimensionless?

• (a) Stress
• (b) Young's modulus
• (c) Strain
• (d) Pressure

Answer: (c) Strain

6. The reciprocal of bulk modulus is called:

• (a) Young's modulus
• (b) Compressibility
• (c) Stress
• (d) Rigidity

Answer: (b) Compressibility

7. The property of regaining original shape after removing the deforming force is:

• (a) Plasticity
• (b) Elasticity
• (c) Viscosity
• (d) Density

Answer: (b) Elasticity

8. The SI unit of Young's modulus is:

• (a) N
• (b) J
• (c) Pa
• (d) m/s

Answer: (c) Pa

9. Shear modulus measures resistance to:

• (a) Change in shape
• (b) Change in mass
• (c) Change in temperature
• (d) Change in density only

Answer: (a) Change in shape

10. Elastic potential energy stored in a stretched wire is:

• (a) FΔL
• (b) ½FΔL
• (c) 2FΔL
• (d) F/ΔL

Answer: (b) ½FΔL

17. Solved Numerical Problems

Numerical 1: Calculation of Stress

Question: A force of 200 N acts normally on a cross-sectional area of 0.01 m². Calculate the stress produced.

Given:

F = 200 N

A = 0.01 m²

Formula:

Stress = F/A

Solution:

Stress = 200/0.01

Stress = 20,000 Pa

Answer: 2 × 10⁴ Pa

Numerical 2: Calculation of Longitudinal Strain

Question: A wire of length 2 m increases its length by 0.002 m when stretched. Calculate the longitudinal strain.

Given:

L = 2 m

ΔL = 0.002 m

Formula:

Strain = ΔL/L

Solution:

Strain = 0.002/2

Strain = 0.001

Answer: 1 × 10^-3

Numerical 3: Calculation of Young's Modulus

Question: A wire of length 2 m and cross-sectional area 2 × 10^-6 m² is stretched by a force of 100 N. If its extension is 0.001 m, calculate Young's modulus.

Given:

L = 2 m

A = 2 × 10^-6 m²

F = 100 N

ΔL = 0.001 m

Formula:

Y = FL/(AΔL)

Solution:

Y = (100 × 2)/(2 × 10^-6 × 0.001)

Y = 200/(2 × 10^-9)

Y = 1 × 10¹¹ Pa

Answer: 1 × 10¹¹ Pa

Numerical 4: Calculation of Bulk Modulus

Question: The pressure on a liquid is increased by 2 × 10⁵ Pa. Its volume decreases by 0.01% of its original volume. Calculate its bulk modulus.

Given:

ΔP = 2 × 10⁵ Pa

Volumetric strain = -0.01/100 = -10^-4

Formula:

K = -ΔP/(ΔV/V)

Solution:

K = -(2 × 10⁵)/(-10^-4)

K = 2 × 10⁹ Pa

Answer: 2 × 10⁹ Pa

Numerical 5: Elastic Potential Energy

Question: A spring is stretched by 0.05 m when a force of 20 N is applied. Calculate the elastic potential energy stored in it, assuming the spring obeys Hooke's law.

Given:

F = 20 N

ΔL = 0.05 m

Formula:

U = ½FΔL

Solution:

U = ½ × 20 × 0.05

U = 10 × 0.05

U = 0.5 J

Answer: 0.5 J

18. Important Points to Remember for Exams

• Stress is the restoring force per unit area.
• Strain is the fractional change in dimension.
• Stress has the SI unit Pascal.
• Strain is dimensionless.
• Hooke's law is valid within the proportional limit.
• Young's modulus measures resistance to longitudinal deformation.
• Shear modulus measures resistance to change in shape.
• Bulk modulus measures resistance to change in volume.
• Compressibility is the reciprocal of bulk modulus.
• Poisson's ratio relates lateral strain to longitudinal strain.
• Elastic potential energy is stored when an elastic body is deformed within its elastic limit.
• Young's modulus and shear modulus are properties relevant to solids.
• Bulk modulus is applicable to solids, liquids and gases.

19. Frequently Asked Questions (FAQs)

Q1. What is meant by mechanical properties of solids?

Mechanical properties of solids describe how solid materials respond to external forces, including their deformation, elasticity and resistance to stress.

Q2. What is the difference between stress and strain?

Stress is the restoring force per unit area, whereas strain is the fractional change in the dimensions of a body.

Q3. Why is strain dimensionless?

Strain is the ratio of change in dimension to the original dimension. Both quantities have the same unit, so their units cancel.

Q4. What is Hooke's law?

Hooke's law states that stress is directly proportional to strain within the proportional limit of a material.

Q5. Which material is more elastic, steel or rubber?

Steel is considered more elastic than rubber in physics because steel has a higher Young's modulus and undergoes less strain for the same stress.

Q6. What is Young's modulus?

Young's modulus is the ratio of longitudinal stress to longitudinal strain within the elastic region.

Q7. What is the SI unit of bulk modulus?

The SI unit of bulk modulus is Pascal (Pa).

Q8. What is the physical meaning of compressibility?

Compressibility indicates how easily the volume of a material changes when pressure is applied.

Q9. What is Poisson's ratio?

Poisson's ratio is the negative ratio of lateral strain to longitudinal strain.

Q10. What is elastic potential energy?

Elastic potential energy is the energy stored in an elastic body when it is stretched or compressed within its elastic limit.

20. Conclusion

Mechanical Properties of Solids is an important chapter of Class 11 Physics. It explains the behaviour of solid materials under the influence of external forces. Concepts such as stress, strain, elasticity, Hooke's law and elastic moduli help us understand how different materials respond to deformation.

Students should pay special attention to the definitions, mathematical expressions, SI units, stress-strain curve and numerical problems. A clear understanding of these concepts will help in school examinations, CBSE examinations and future studies in Physics.

Keep Learning, Keep Practising!

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