class 11 notes

Class 11 Physics Notes Chapter 3 (Chapter 3) – Physics Part-II Book

Philoid

Philoid

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Physics Part-II
Detailed Notes with MCQs of Chapter 3 from your Physics Part-II book, which deals with the 'Mechanical Properties of Solids'. This is a crucial chapter, not just for your Class 11 understanding, but also because concepts related to elasticity, stress, and strain frequently appear in various government exams. Pay close attention to the definitions, formulas, and the stress-strain curve.

Chapter 3: Mechanical Properties of Solids - Detailed Notes

1. Introduction

  • Solids: Have definite shape and size due to strong interatomic/intermolecular forces.
  • Deforming Force: An external force that changes the size or 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. (Ideal concept, Quartz fibre is very 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, plasticine.
    • Perfectly Plastic Body: Does not show any tendency to regain its original configuration. (Ideal concept).
  • Restoring Force: When a deforming force is applied, internal forces develop within the body that oppose the deformation 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

  • Definition: The internal restoring force developed per unit area of cross-section of the deformed body.
    • Mathematically: Stress (σ) = Restoring Force (F) / Area (A)
  • Units: N/m² (SI unit, also called Pascal, Pa), dyne/cm² (CGS unit).
  • Dimensions: [ML⁻¹T⁻²]
  • Types of Stress:
    • Longitudinal (or Normal) Stress: Restoring force acts perpendicular (normal) to the area of cross-section.
      • Tensile Stress: If there is an increase in length (body is stretched).
      • Compressive Stress: If there is a decrease in length (body is compressed).
    • Tangential (or Shearing) Stress: Restoring force acts parallel (tangential) to the surface area. It causes a change in the shape of the body without changing its volume.
      • Formula: Shearing Stress = Tangential Force (Ft) / Area (A)
    • Hydraulic (or Volume/Bulk) Stress: When a body is subjected to a uniform force from all sides (like immersion in a fluid), the force per unit area is hydraulic stress. It's usually equal to the fluid pressure (P). It causes a change in volume without changing the shape.

3. Strain

  • Definition: The ratio of the change in configuration (dimension like length, shape, or volume) to the original configuration of the body.
  • Units: It's a ratio of similar quantities, hence it is dimensionless and has no units.
  • Types of Strain:
    • Longitudinal Strain: Ratio of change in length (ΔL) to the original length (L).
      • Formula: Longitudinal Strain = ΔL / L
    • Shearing Strain: The angle (θ, in radians) through which a plane perpendicular to the fixed surface of the body is turned under the effect of tangential stress. For small angles, it's the ratio of relative displacement (Δx) between opposite faces to the distance (L) between the faces.
      • Formula: Shearing Strain (θ) ≈ tan θ = Δx / L
    • Volume Strain: Ratio of change in volume (ΔV) to the original volume (V).
      • Formula: Volume Strain = ΔV / V (Note: ΔV is negative if volume decreases under pressure).

4. Hooke's Law

  • Statement: Within the elastic limit, the stress developed in a body is directly proportional to the strain produced in it.
    • Mathematically: Stress ∝ Strain
    • Stress = E × Strain
  • E: Constant of proportionality called the Modulus of Elasticity of the material. Its value depends on the nature of the material and the type of stress/strain.
  • Elastic Limit: The maximum stress (or corresponding strain) up to which the body exhibits elasticity. If the deforming force exceeds the elastic limit, the body acquires a permanent set (deformation).

5. Stress-Strain Curve

  • A graph plotted between stress (usually on y-axis) and strain (usually on x-axis) for a material under load.
  • 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 / Yield Point): Stress is not proportional to strain, but the material is still elastic (returns to original shape if load is removed). Point B is the yield point (also called elastic limit). The stress at this point is the Yield Strength (Sy).
    • B to D (Plastic Region): If the load is increased beyond B, the strain increases rapidly even for a small change in stress. The material undergoes plastic deformation. If the load is removed at some point C between B and D, the body does not regain its original configuration (permanent set, represented by O' on the strain axis).
    • D (Ultimate Tensile Strength, Su): The maximum stress the material can withstand before starting to fracture. Beyond this point, necking (thinning of the wire) starts.
    • E (Fracture Point): The point where the material breaks.
  • Ductile Materials: Materials that show large plastic deformation before fracture (e.g., copper, aluminium, mild steel). They have a large difference between ultimate strength and fracture point.
  • Brittle Materials: Materials that fracture soon after the elastic limit is crossed, showing very little plastic deformation (e.g., glass, ceramics, cast iron). Ultimate strength and fracture points are close.
  • Elastomers: Materials that can be stretched to cause large strains (e.g., rubber, aorta tissue). They do not obey Hooke's law over most of the region, and the stress-strain curve is not linear. They have no plastic range.

6. Moduli of Elasticity

  • a) Young's Modulus (Y):
    • Ratio of longitudinal stress to longitudinal strain (within the elastic limit).
    • Formula: Y = Longitudinal Stress / Longitudinal Strain = (F/A) / (ΔL/L) = (F L) / (A ΔL)
    • Measures resistance to change in length.
    • Units: N/m² or Pa. Dimensions: [ML⁻¹T⁻²]
    • Higher Y means more elastic (stiffer). Steel has a higher Y than copper or aluminium.
  • b) Shear Modulus (G) or Modulus of Rigidity:
    • Ratio of shearing stress to shearing strain (within the elastic limit).
    • Formula: G = Shearing Stress / Shearing Strain = (Ft/A) / θ = (Ft/A) / (Δx/L)
    • Measures resistance to change in shape.
    • Units: N/m² or Pa. Dimensions: [ML⁻¹T⁻²]
    • Liquids and gases have zero shear modulus.
  • c) Bulk Modulus (B):
    • Ratio of hydraulic stress (change in pressure, ΔP) to the corresponding volume strain (ΔV/V) (within the elastic limit).
    • Formula: B = Hydraulic Stress / Volume Strain = - ΔP / (ΔV/V) = - (ΔP V) / ΔV
    • The negative sign indicates that an increase in pressure (ΔP positive) causes a decrease in volume (ΔV negative), making B positive.
    • Measures resistance to change in volume.
    • Units: N/m² or Pa. Dimensions: [ML⁻¹T⁻²]
    • Higher B means less compressible. Solids > Liquids > Gases in terms of Bulk Modulus.
    • Compressibility (k): Reciprocal of Bulk Modulus. k = 1/B. Unit: Pa⁻¹ or m²/N.

7. Poisson's Ratio (σ)

  • When a wire is stretched (longitudinal strain), its diameter decreases (lateral strain).
  • Definition: Within the elastic limit, the ratio of lateral strain to the longitudinal strain.
    • Lateral Strain = Change in diameter (ΔD) / Original diameter (D)
    • Longitudinal Strain = Change in length (ΔL) / Original length (L)
    • Formula: σ = - (Lateral Strain) / (Longitudinal Strain) = - (ΔD/D) / (ΔL/L)
    • The negative sign ensures σ is positive (as ΔL is positive, ΔD is negative, and vice versa).
  • Units: Dimensionless ratio.
  • Theoretical Limits: -1 < σ < 0.5
  • Practical Limits (for most materials): 0 < σ < 0.5 (e.g., Steel ≈ 0.3, Rubber ≈ 0.5)

8. Elastic Potential Energy in a Stretched Wire

  • Work done in stretching a wire is stored as elastic potential energy (U) within it.
  • Work Done (dW) = Average Force × Extension = (0 + F)/2 × ΔL = ½ F ΔL
  • Using Stress = F/A and Strain = ΔL/L, F = Stress × A and ΔL = Strain × L
  • U = ½ (Stress × A) × (Strain × L) = ½ × Stress × Strain × (A × L)
  • Since A × L = Volume (V) of the wire,
    • U = ½ × Stress × Strain × Volume
  • Also, U = ½ × (Y × Strain) × Strain × Volume = ½ × Y × (Strain)² × Volume
  • And U = ½ × Stress × (Stress / Y) × Volume = ½ × (Stress)² / Y × Volume
  • Elastic Potential Energy Density (u): Potential energy stored per unit volume.
    • u = U / Volume = ½ × Stress × Strain
    • u = ½ × Y × (Strain)²
    • u = ½ × (Stress)² / Y
  • Units of U: Joule (J). Units of u: J/m³.

9. Applications of Elasticity

  • Material Selection: Knowledge of moduli helps choose materials for specific purposes (e.g., steel for bridges and heavy machinery due to high Y, rubber for shock absorbers).
  • Structural Design:
    • Bridges: Designed to handle load without exceeding the elastic limit to avoid permanent bending. Beam shapes (like I-shape) are used to maximize strength and minimize buckling/weight.
    • Cranes: Ropes (usually steel) are chosen based on their elastic limit and required load capacity, ensuring a sufficient safety factor (usually 5 to 10 times the expected load). Thickness is determined by F = Sy × A / (Safety Factor).
  • Maximum Height of a Mountain: Limited by the elastic properties (specifically, the yield strength for compression) of the rock at the base. The pressure due to the weight of the mountain (hρg) must be less than the critical shear stress or compressive strength of the rock.

Multiple Choice Questions (MCQs)

  1. The property of a material by virtue of which it regains its original shape and size after the removal of deforming force is called:
    (a) Plasticity
    (b) Elasticity
    (c) Ductility
    (d) Brittleness

  2. The SI unit of Stress is the same as that of:
    (a) Force
    (b) Pressure
    (c) Strain
    (d) Work

  3. According to Hooke's law of elasticity, if stress is increased, the ratio of stress to strain:
    (a) Increases
    (b) Decreases
    (c) Becomes zero
    (d) Remains constant

  4. A wire of length L and cross-sectional area A is made of a material of Young's modulus Y. If the wire is stretched by an amount x, the work done is:
    (a) YAx / L
    (b) ½ (YAx / L) x
    (c) YAx² / L
    (d) ½ (Y A / L) x²

  5. Which of the following materials is the most elastic?
    (a) Rubber
    (b) Glass
    (c) Steel
    (d) Copper

  6. The Bulk modulus for an incompressible liquid is:
    (a) Zero
    (b) Unity
    (c) Infinite
    (d) Between 0 and 1

  7. Poisson's ratio is defined as the ratio of:
    (a) Longitudinal strain to lateral strain
    (b) Lateral strain to longitudinal strain
    (c) Longitudinal stress to lateral stress
    (d) Lateral stress to longitudinal stress

  8. In the stress-strain curve for a metal wire, the point where the material undergoes plastic deformation even on removing the load is known as:
    (a) Proportional limit
    (b) Fracture point
    (c) Elastic limit / Yield point
    (d) Ultimate tensile strength point

  9. Shearing strain is given by:
    (a) Change in length / Original length
    (a) Change in volume / Original volume
    (c) The angle of twist or relative displacement between faces divided by the distance between faces
    (d) Change in pressure / Original pressure

  10. The energy stored per unit volume in a stretched wire is:
    (a) ½ × Stress × Strain
    (b) Stress × Strain
    (c) ½ × Load × Extension
    (d) Y × (Strain)²

Answer Key for MCQs:

  1. (b)
  2. (b)
  3. (d)
  4. (d) [Work Done = ½ F x = ½ (Stress × A) x = ½ (Y × Strain × A) x = ½ (Y × (x/L) × A) x = ½ (YA/L) x²]
  5. (c) [Elasticity here refers to the ability to resist deformation, indicated by a high Young's Modulus. Steel has a higher Y than rubber, glass, or copper.]
  6. (c) [Incompressible means ΔV = 0 for any ΔP. B = -ΔP / (ΔV/V) -> ∞ as ΔV -> 0]
  7. (b) [Remember the negative sign in the formula, but the definition is the ratio itself.]
  8. (c) [Beyond the elastic limit/yield point, permanent deformation occurs.]
  9. (c)
  10. (a)

Study these notes thoroughly, focusing on the definitions, formulas, units, dimensions, and the concepts behind the stress-strain curve. Good luck with your preparation!

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