Mechanical Properties of Solids is Chapter 8 of CBSE Class 11 Physics — the chapter that explains why a steel cable can hold a bridge but a rubber band cannot, and why a bone breaks under a certain load. It studies how solids deform under force and how they spring back, using the ideas of stress, strain, and elastic moduli. Master this chapter and you unlock a whole family of easy, formula-driven marks in boards, JEE, and NEET.
By the end of these notes you will be able to define stress and strain, state and apply Hooke’s law, read a stress-strain curve, calculate Young’s modulus, shear modulus, bulk modulus and Poisson’s ratio, and find the elastic potential energy stored in a stretched wire. This is a scoring chapter carrying roughly 3–4 marks in boards, and a reliable source of single-concept numericals in entrance exams.
Table of Contents
- Key Concepts — Elastic behaviour, stress & strain, Hooke’s law, stress-strain curve, elastic moduli, Poisson’s ratio, elastic energy
- Weightage in Board & Entrance Exams
- Important Definitions
- Solved Examples
- Important Questions for Board Exams
- Quick Revision Points
Key Concepts
1. Elastic Behaviour of Solids
A solid is made of atoms held in a regular lattice by inter-atomic bonds that act like tiny springs. When you apply a force, these “springs” stretch or compress, and the body deforms. Remove the force, and the bonds pull the atoms back — the body regains its shape.
Elasticity is the property of a body by which it regains its original shape and size after the deforming force is removed. A steel ball is highly elastic; putty or wet clay is not.
Plasticity is the opposite property — the body retains the new shape after the force is removed. A body with no tendency to return is called a perfectly plastic body.
2. Stress
Stress is the internal restoring force set up per unit area of cross-section when a body is deformed. In magnitude it equals the applied force per unit area at equilibrium.
Stress = F/A
- SI unit: N/m² or pascal (Pa)
- Dimensional formula: [ML⁻¹T⁻²]
- Stress is a scalar (more precisely a tensor), not a vector.
Types of Stress
- Tensile / Compressive (longitudinal) stress: force perpendicular to area, causing change in length.
- Shearing (tangential) stress: force parallel to the surface, causing change in shape.
- Hydraulic (volume) stress: force applied uniformly from all sides by a fluid, causing change in volume.
3. Strain
Strain is the fractional change in configuration (length, shape, or volume) of a body under stress. It is a ratio of two like quantities, so it has no units and no dimensions.
Types of Strain
- Longitudinal strain = change in length / original length = ΔL/L
- Shearing strain = relative displacement / distance between layers = Δx/L = tan θ ≈ θ (for small angles)
- Volume strain = change in volume / original volume = ΔV/V
4. Hooke’s Law
Within the elastic limit, the stress developed in a body is directly proportional to the strain produced in it.
Stress ∝ Strain, so Stress = E × Strain
- The constant of proportionality E is called the modulus of elasticity.
- Its value depends only on the material, not on the dimensions of the body.
- SI unit of modulus: N/m² or pascal (same as stress, since strain is unitless).
Key idea: Hooke’s law holds only up to the elastic limit. Beyond it, stress and strain are no longer proportional.
5. Stress-Strain Curve
If a metal wire is loaded gradually and the stress is plotted against strain, we get a characteristic curve with distinct regions.
[DIAGRAM: Stress (y-axis) vs strain (x-axis) curve — straight line O→A (proportional limit), A→B elastic limit (yield point), B→D plastic region, D the ultimate tensile strength, E the fracture point.]
- O to A — Proportional limit: stress ∝ strain; Hooke’s law obeyed; the graph is a straight line.
- A to B — Elastic limit (yield point): the wire still returns to its original length on removing the load, but the line curves.
- B to D — Plastic region: beyond the yield point, strain increases with little extra stress and a permanent (plastic) set remains.
- D — Ultimate tensile strength: the maximum stress the material can bear.
- E — Fracture point: the wire finally breaks.
Note: A large gap between the yield point and fracture point means the material is ductile (can be drawn into wires); a small gap means it is brittle (breaks soon after the elastic limit, like glass).
6. Young’s Modulus (Y)
Within the elastic limit, the ratio of longitudinal stress to longitudinal strain is a constant called Young’s modulus.
Y = Longitudinal stress / Longitudinal strain = (F/A) / (ΔL/L) = FL / (A·ΔL)
- SI unit: N/m² or pascal; dimensions: [ML⁻¹T⁻²]
- Steel has a higher Young’s modulus than rubber, so steel is more elastic and harder to stretch.
- A high Y means a small strain for a given stress — the material is stiff.
7. Shear Modulus / Modulus of Rigidity (G)
The ratio of shearing (tangential) stress to shearing strain within the elastic limit is the shear modulus or modulus of rigidity.
G = Shearing stress / Shearing strain = (F/A) / θ = F / (A·θ)
- SI unit: N/m² or pascal
- G is generally about one-third of Young’s modulus (G ≈ Y/3) for most metals.
- Only solids have rigidity; liquids and gases cannot sustain a shear stress.
8. Bulk Modulus (K)
The ratio of hydraulic (volume) stress to the volume strain within the elastic limit is the bulk modulus.
K = −p / (ΔV/V) = −pV / ΔV
- The negative sign shows that an increase in pressure decreases the volume.
- SI unit: N/m² or pascal
- The reciprocal of bulk modulus is the compressibility, k = 1/K.
- Solids have the highest K, then liquids, then gases — solids are least compressible.
9. Poisson’s Ratio (σ)
When a wire is stretched, it becomes longer but thinner — its length increases while its diameter decreases. The ratio of lateral strain to longitudinal strain is called Poisson’s ratio (σ).
σ = Lateral strain / Longitudinal strain = −(Δd/d) / (ΔL/L)
- It is a pure ratio, so it has no units and no dimensions.
- Its theoretical range is −1 to 0.5; for most materials it lies between 0.2 and 0.4.
10. Elastic Potential Energy in a Stretched Wire
Work done in stretching a wire is stored in it as elastic potential energy. Since the force builds up gradually from 0 to F, the average force is F/2.
U = ½ × Stress × Strain × Volume = ½ × F × ΔL
The energy stored per unit volume (elastic energy density) is:
u = ½ × Stress × Strain = ½ × Y × (Strain)²
Key idea: A catapult, a bow, and a spring all store elastic potential energy this way before releasing it as kinetic energy.
11. Applications of Elastic Behaviour
The choice of materials and shapes in engineering comes straight from this chapter.
- Beams and bridges: a girder is given an I-shape (I-section) so it resists bending with less material, because the depression of a loaded beam δ ∝ Wl³ / (Ybd³).
- Cranes and ropes: the rope of a crane is made thick (large area) so the stress stays safely below the elastic limit for the maximum load.
- Pillars: pillars with distributed (cross-section) ends carry more load than those with rounded ends, so building columns are designed accordingly.
- Mountains: the maximum height of a mountain on Earth (~10 km) is limited by the elastic strength of rock at its base.
Weightage in Board & Entrance Exams
| Exam | Typical Weightage | Most-Tested Areas |
|---|---|---|
| CBSE Board (Class 11) | 3–4 marks | Stress-strain definitions, Young’s modulus numericals, stress-strain curve |
| JEE Main / Advanced | 1–2 questions | Young’s modulus, elastic energy, composite wires, bulk modulus |
| NEET | 1–2 questions | Elastic moduli definitions, Poisson’s ratio, stress-strain graph reading |
[TABLE: Question-type split — VSA (1 mark): definitions of stress, strain, elastic limit, units; SA (2–3 marks): Young’s modulus derivation, stress-strain curve, Poisson’s ratio; LA (5 marks): elastic energy derivation, comparison of three moduli, applications of elasticity.]
Important Definitions
| Term | Definition |
|---|---|
| Elasticity | Property by which a body regains its original shape and size after the deforming force is removed |
| Stress | Internal restoring force per unit area: Stress = F/A; unit Pa |
| Strain | Fractional change in configuration (length, shape, or volume); dimensionless |
| Hooke’s law | Within the elastic limit, stress ∝ strain |
| Elastic limit | Maximum stress up to which a body returns to its original size on removing the load |
| Young’s modulus (Y) | Ratio of longitudinal stress to longitudinal strain: Y = FL/(A·ΔL) |
| Shear modulus (G) | Ratio of shearing stress to shearing strain: G = F/(A·θ) |
| Bulk modulus (K) | Ratio of volume stress to volume strain: K = −pV/ΔV |
| Poisson’s ratio (σ) | Ratio of lateral strain to longitudinal strain; dimensionless |
| Elastic potential energy | Energy stored in a stretched body: U = ½ × F × ΔL |
Solved Examples
Example 1
A steel wire of length 2 m and cross-sectional area 1 mm² is stretched by a force of 100 N. Find the stress in the wire.
Answer: Stress = F/A = 100 / (1 × 10⁻⁶) = 1 × 10⁸ N/m² (10⁸ Pa).
Example 2
A wire of original length 4 m stretches by 2 mm under a load. Find the longitudinal strain.
Answer: Strain = ΔL/L = (2 × 10⁻³) / 4 = 5 × 10⁻⁴ (no units).
Example 3
A force of 200 N stretches a wire of length 3 m and area 2 × 10⁻⁶ m² by 1.5 mm. Find Young’s modulus.
Answer: Y = FL / (A·ΔL) = (200 × 3) / (2 × 10⁻⁶ × 1.5 × 10⁻³) = 600 / (3 × 10⁻⁹) = 2 × 10¹¹ N/m².
Example 4
The bulk modulus of water is 2.2 × 10⁹ N/m². Find the pressure required to reduce a given volume of water by 0.1%.
Answer: ΔV/V = 0.1% = 10⁻³. p = K × (ΔV/V) = 2.2 × 10⁹ × 10⁻³ = 2.2 × 10⁶ N/m².
Example 5
A wire is stretched by 1 mm under a force of 50 N. Find the elastic potential energy stored in it.
Answer: U = ½ × F × ΔL = ½ × 50 × (1 × 10⁻³) = 0.025 J (2.5 × 10⁻² J).
Example 6
When a wire is stretched, its length increases by 0.2% and its diameter decreases by 0.05%. Find Poisson’s ratio.
Answer: σ = lateral strain / longitudinal strain = 0.05% / 0.2% = 0.0005 / 0.002 = 0.25.
Important Questions for Board Exams
1-Mark Questions (VSA)
- Define stress and give its SI unit.
- Why has strain no units or dimensions?
- Which is more elastic — steel or rubber? Justify.
- What does the negative sign in the formula for bulk modulus indicate?
- Define Poisson’s ratio and state its dimensions.
2–3-Mark Questions (SA)
- State Hooke’s law and define modulus of elasticity. Give its SI unit.
- Draw and explain the stress-strain curve for a metal wire, marking the elastic limit, yield point, and fracture point.
- Distinguish between Young’s modulus, shear modulus, and bulk modulus.
- Why are girders given an I-shape? Explain using the depression formula.
5-Mark Questions (LA)
- Define Young’s modulus and derive its expression Y = FL/(A·ΔL). Explain why solids are more elastic than liquids and gases.
- Derive an expression for the elastic potential energy stored per unit volume in a stretched wire, and show that u = ½ × stress × strain.
- Explain three applications of the elastic behaviour of materials in everyday engineering (bridges, cranes, pillars).
Quick Revision Points
- Elasticity = tendency to regain shape; plasticity = tendency to keep the new shape
- Stress = F/A (unit Pa, dimensions [ML⁻¹T⁻²]); strain = fractional change (dimensionless)
- Three stresses: tensile/compressive, shearing, hydraulic — giving length, shape, volume strain
- Hooke’s law: stress ∝ strain within the elastic limit; stress = E × strain
- Stress-strain curve: proportional limit → elastic limit (yield) → plastic region → ultimate strength → fracture
- Young’s modulus Y = FL/(A·ΔL); higher Y means stiffer material
- Shear modulus G = F/(A·θ) ≈ Y/3; only solids have rigidity
- Bulk modulus K = −pV/ΔV; compressibility = 1/K; gases most compressible
- Poisson’s ratio σ = lateral strain / longitudinal strain; range −1 to 0.5
- Elastic energy U = ½ × F × ΔL; energy density u = ½ × stress × strain = ½ Y(strain)²
- Applications: I-shaped girders, thick crane ropes, designed pillars, mountain-height limit
Next Chapter: Chapter 9 — Mechanical Properties of Fluids
Chapter Navigation
Previous: Gravitation Class 11 Notes
Next: Mechanical Properties of Fluids Class 11 Notes
Related Chapters in Class 11 Physics
Practice What You Learned
Take your mechanics further with our Class 12 Physics notes once you are board-ready.