Class 11 Physics Notes Chapter 9 (Chapter 9) – Examplar Problems (English) Book

Alright class, let's delve into Chapter 9: Mechanical Properties of Solids from the NCERT Exemplar. This chapter is crucial for understanding how solid materials behave under external forces, a fundamental concept frequently tested in various government exams. Pay close attention to the definitions, laws, and graphical representations.

Chapter 9: Mechanical Properties of Solids - Detailed Notes

1. Introduction

• Solid: A state of matter characterized by a definite shape and volume. The constituent atoms or molecules are held in fixed equilibrium positions by strong interatomic/intermolecular forces.
• Deforming Force: An external force that changes or tends to change the size, shape, or both, of a body.
• Elasticity: The property of a body by virtue of which it tends to regain its original size and shape after the removal of the deforming force. Examples: Steel, Rubber (to some extent), Quartz.
  • Perfectly Elastic Body: Regains its original configuration completely and immediately after the removal of the deforming force. (Quartz fibre is close).
• Plasticity: The property of a body by virtue of which it does not regain its original size and shape even after the removal of the deforming force. It undergoes permanent deformation. Examples: Putty, Mud, Wax.
  • Perfectly Plastic Body: Does not show any tendency to regain its original configuration.
• Restoring Force: When a deforming force is applied, internal forces are set up within the body that oppose the deforming force and try to restore the body to its original state. In equilibrium, the restoring force is equal in magnitude and opposite in direction to the applied deforming force.

2. Stress

• It is defined as the internal restoring force set up per unit area of cross-section of the deformed body.

• Mathematically, Stress (σ) = Restoring Force (F) / Area (A)

• Under equilibrium conditions, Restoring Force = Applied Deforming Force. So, σ = F_applied / A.

• SI Unit: N/m² or Pascal (Pa)

• Dimensional Formula: [ML⁻¹T⁻²]

• Stress is a tensor quantity, but for simple cases, it's often treated based on its effect (tensile, compressive, shearing).

• Types of Stress:

  • Normal Stress: When the deforming force acts perpendicular (normal) to the surface area.
    • Tensile Stress: If there is an increase in length or extension of the body. (Produced by stretching forces).
    • Compressive Stress: If there is a decrease in length or compression of the body. (Produced by compressing forces).
    • Longitudinal Stress: General term for tensile or compressive stress. σ_longitudinal = F_normal / A.
  • Hydraulic Stress (or Volume Stress): When a body is subjected to a uniform force from all sides (like immersion in a fluid), the force is perpendicular to the entire surface area. It leads to a change in volume without a change in shape. It is equivalent to pressure (P). σ_hydraulic = F_normal / A = ΔP.
  • Tangential or Shearing Stress: When the deforming force acts parallel (tangential) to the surface area. It tends to change the shape of the body without changing its volume. σ_shear = F_tangential / A.

3. Strain

• It is defined as the ratio of the change in configuration (dimension like length, shape, or volume) to the original configuration of the body.

• Strain = Change in Dimension / Original Dimension

• It is a dimensionless and unitless quantity.

• Types of Strain:

  • Longitudinal Strain: The ratio of change in length (ΔL) to the original length (L).
    • Longitudinal Strain = ΔL / L
    • Tensile Strain: If length increases.
    • Compressive Strain: If length decreases.
  • Volume Strain: The ratio of change in volume (ΔV) to the original volume (V).
    • Volume Strain = ΔV / V
    • Usually associated with hydraulic stress. A negative sign is often used (-ΔV/V) because an increase in pressure (stress) usually causes a decrease in volume.
  • Shearing Strain: When a tangential stress is applied, layers of the body are displaced relative to each other. If 'x' is the relative displacement between two layers separated by a distance 'L', then shearing strain is defined as the angle 'θ' (in radians) through which a line originally perpendicular to the fixed surface gets turned.
    • Shearing Strain = tan θ ≈ θ = x / L (for small angles).

4. Hooke's Law

• This fundamental law of elasticity states that, within the elastic limit, the stress developed in a body is directly proportional to the strain produced in it.
• Stress ∝ Strain
• Stress = E × Strain
• Where 'E' is the constant of proportionality called the Modulus of Elasticity or Coefficient of Elasticity of the material.
• The value of E depends on the nature of the material of the body and the type of stress applied and strain produced.
• Elastic Limit: The maximum stress (or corresponding strain) up to which a material behaves elastically. Beyond this limit, the material does not fully regain its original state, and permanent deformation occurs.

5. Modulus of Elasticity

• It measures the resistance of a material to elastic deformation under stress. A higher modulus indicates a stiffer material.

• SI Unit: N/m² or Pascal (Pa) (Same as Stress, since Strain is dimensionless).

• Dimensional Formula: [ML⁻¹T⁻²]

• Types of Modulus of Elasticity:

  • Young's Modulus (Y): It is defined as the ratio of longitudinal stress to longitudinal strain, within the elastic limit.
    • Y = Longitudinal Stress / Longitudinal Strain = (F/A) / (ΔL/L) = (F × L) / (A × ΔL)
    • It relates to the resistance of a solid to change in its length.
    • Applicable only to solids. Steel has a very high Young's Modulus compared to rubber or copper.
  • Bulk Modulus (B): It is defined as the ratio of hydraulic (volume) stress to the corresponding volume strain, within the elastic limit.
    • B = Hydraulic Stress / Volume Strain = -ΔP / (ΔV/V) = - (ΔP × V) / ΔV
    • The negative sign indicates that as pressure increases (ΔP positive), the volume decreases (ΔV negative), making B positive.
    • It relates to the resistance of a substance (solid, liquid, or gas) to change in its volume.
    • Compressibility (k): The reciprocal of the Bulk Modulus. k = 1/B. It measures how easily a substance can be compressed. Gases are highly compressible (low B, high k), while solids are least compressible (high B, low k).
  • Shear Modulus (G) or Modulus of Rigidity: It is defined as the ratio of shearing stress to the corresponding shearing strain, within the elastic limit.
    • G = Shearing Stress / Shearing Strain = (F_tangential / A) / θ
    • It relates to the resistance of a solid to change in its shape.
    • Applicable only to solids. Liquids and gases cannot sustain shearing stress, so their Shear Modulus is zero.

6. Poisson's Ratio (σ)

• When a body (like a wire) is stretched longitudinally (longitudinal strain), its length increases, but its diameter decreases (lateral strain).
• Within the elastic limit, the lateral strain (ΔD/D) is directly proportional to the longitudinal strain (ΔL/L).
• Poisson's Ratio (σ) is defined as the ratio of lateral strain to longitudinal strain.
• σ = - (Lateral Strain) / (Longitudinal Strain) = - (ΔD/D) / (ΔL/L)
• The negative sign ensures that σ is positive (since lateral and longitudinal strains usually have opposite signs).
• It is a pure number (dimensionless and unitless).
• Theoretical Limits: -1 < σ < 0.5
• Practical Limits (for most materials): 0 < σ < 0.5

7. Stress-Strain Curve

• A graph plotted between the stress (usually tensile) and the corresponding strain for a material. It provides valuable information about the material's elastic and plastic properties.

• Key Points on the Curve (for a typical ductile material like mild steel):

  • O to A (Proportional Limit): Stress is directly proportional to strain (Hooke's Law is obeyed). The graph is a straight line.
  • A to B (Elastic Limit): Strain is not strictly proportional to stress, but the material still returns to its original state when the load is removed. Point B is the Yield Point (often considered the same as the elastic limit for practical purposes).
  • B to C' (Yield Point): Beyond B, the material shows plastic behaviour. Strain increases rapidly even for a small increase in stress. C' is the lower yield point. The stress corresponding to the yield point is called Yield Strength (S_y).
  • C' to D (Strain Hardening): In this region, further elongation requires an increased stress. The material seems to regain some strength.
  • D (Ultimate Tensile Strength, S_u): The point corresponding to the maximum stress the material can withstand before starting to form a 'neck' (local thinning).
  • D to E (Necking): After reaching the ultimate strength, the cross-sectional area starts decreasing rapidly in a localized region (necking), and the stress decreases.
  • E (Fracture Point): The point where the material finally breaks.

• Classification based on the curve:

  • Ductile Materials: Exhibit a large plastic deformation range between the elastic limit and the fracture point (e.g., Copper, Aluminium, Mild Steel). They show significant necking before fracture. Can be drawn into wires.
  • Brittle Materials: Have a very small or negligible plastic range. Fracture occurs soon after the elastic limit is crossed (e.g., Glass, Ceramics, Cast Iron). They break without significant prior deformation or warning.
  • Elastomers: Materials like rubber that can be stretched to very large strains. They do not obey Hooke's law over most of the region but return to their original shape. They don't have a well-defined plastic region (e.g., Rubber, Aorta tissue).

8. Elastic Potential Energy in a Stretched Wire

• When a wire is stretched by a force, work is done against the internal restoring forces. This work is stored in the wire as elastic potential energy (U).
• Consider a wire of length L and area A stretched by a force F, causing an extension ΔL.
• Work done (dW) for a small extension (dx) = F_internal × dx
• Assuming elastic limit is not crossed, F_internal = Applied Force (F') = (Y A / L) x
• Total Work Done (U) = ∫ dW = ∫₀^ΔL (Y A / L) x dx = (Y A / L) [x²/2]₀^ΔL
• U = (1/2) × (Y A / L) × (ΔL)²
• Using F = (Y A / L) ΔL (Force causing total extension ΔL), we get:
  • U = (1/2) × F × ΔL
  • U = (1/2) × (Stress × A) × (Strain × L)
  • U = (1/2) × Stress × Strain × (A × L)
  • U = (1/2) × Stress × Strain × Volume
• Elastic Energy Density (u): Potential energy stored per unit volume.
  • u = U / Volume = (1/2) × Stress × Strain
  • u = (1/2) × Y × (Strain)² (Since Stress = Y × Strain)
  • u = (1/2) × (Stress)² / Y (Since Strain = Stress / Y)

9. Applications of Elastic Behaviour of Materials

• Design of Bridges and Buildings: Beams and columns are designed such that they do not bend or buckle excessively under load. Materials with high Young's Modulus (like steel) are preferred. The dimensions are chosen to keep the stress well below the elastic limit. I-shaped girders are used to provide strength while minimizing weight and cost.
• Cranes: The ropes used in cranes to lift heavy loads are made of materials with high Ultimate Tensile Strength. The thickness is chosen such that the stress remains within safe limits (usually much lower than the ultimate strength, incorporating a safety factor).
• Shock Absorbers: Materials exhibiting elastic hysteresis (elastomers) are used. The area of the hysteresis loop (stress-strain curve for loading and unloading) represents the energy dissipated as heat during deformation, which helps dampen vibrations.
• Maximum height of a mountain depends on the elastic properties (specifically shear modulus and yield strength) of the rock at the base.

Multiple Choice Questions (MCQs)

1. The property of a material by virtue of which it permanently retains the deformation produced by an external force, even after the force is removed, is called:
   (a) Elasticity
   (b) Plasticity
   (c) Ductility
   (d) Brittleness

2. Hooke's law is valid up to the:
   (a) Proportional limit
   (b) Elastic limit
   (c) Yield point
   (d) Fracture point

3. The dimensional formula for Young's Modulus is:
   (a) [MLT⁻²]
   (b) [ML⁻¹T⁻²]
   (c) [ML⁻²T⁻²]
   (d) [ML⁻¹T⁻¹]

4. Two wires A and B are of the same material. The length of A is twice the length of B, and the diameter of A is half the diameter of B. If they are stretched by the same force, the ratio of elongation of A to elongation of B (ΔL_A / ΔL_B) will be:
   (a) 1:8
   (b) 8:1
   (c) 1:4
   (d) 4:1

5. Which of the following materials is expected to have the lowest value of Young's Modulus?
   (a) Steel
   (b) Copper
   (c) Aluminium
   (d) Rubber

6. The Bulk Modulus of a fluid is inversely proportional to its:
   (a) Density
   (b) Volume
   (c) Compressibility
   (d) Pressure

7. A material breaks just after the elastic limit is crossed. This material is:
   (a) Ductile
   (b) Brittle
   (c) Malleable
   (d) Elastic

8. The work done per unit volume in stretching a wire is equal to:
   (a) Stress × Strain
   (b) (1/2) × Stress × Strain
   (c) Stress / Strain
   (d) Strain / Stress

9. Poisson's ratio cannot have the value:
   (a) 0.1
   (b) 0.3
   (c) 0.5
   (d) 0.7

10. Shear modulus is zero for:
    (a) Solids only
    (b) Liquids only
    (c) Gases only
    (d) Both liquids and gases

Answer Key for MCQs:

1. (b)
2. (a) Note: Often, the proportional limit and elastic limit are very close, but strictly, Hooke's law (Stress ∝ Strain) holds only up to the proportional limit.
3. (b)
4. (b) Hint: ΔL = (F L) / (A Y) = (F L) / (π (D/2)² Y). ΔL ∝ L / D². ΔL_A / ΔL_B = (L_A / L_B) × (D_B / D_A)² = (2) × (2)² = 8.
5. (d)
6. (c) Compressibility k = 1/B.
7. (b)
8. (b) This is the formula for elastic energy density.
9. (d) Practical upper limit is 0.5.
10. (d) Fluids cannot sustain shear stress.

Study these notes thoroughly. Focus on understanding the definitions, the physical meaning of the moduli, the stress-strain curve interpretation, and the formulas. Practice numerical problems based on these concepts. Good luck with your preparation!