Lesson Overview

Unit 10 explains elasticity, stress–strain behavior, Young’s modulus, energy stored in stretched materials, viscosity, Stokes’ law, terminal velocity, surface tension, capillarity, and pressure differences across curved surfaces. These concepts help explain the mechanical behavior of solids and fluids.

1. Core Concepts (Short Notes)

10.1 Elasticity

• Describes how materials return to original shape when forces are removed.

• Tensile stress: σ = F/A.

• Tensile strain: ε = ΔL/L₀.

• Young’s modulus: E = σ/ε.

• Hooke’s Law: F ∝ ΔL (up to limit of proportionality).

10.2 Stress–Strain Curve

• Shows material behavior under tension.

• Key points:

  • Proportional limit

  • Elastic limit

  • Yield point

  • Breaking point

  • Breaking stress

• Ductile vs brittle behavior.

10.3 Energy Stored in Elastic Materials

• Elastic potential energy:U = ½ F ΔLor U = ½ kx² for springs.

10.4 Viscosity

• Internal friction of a fluid.

• Coefficient of viscosity:η = (shear stress)/(velocity gradient).

• SI unit: Pa·s.

10.5 Stokes’ Law & Terminal Velocity

• Viscous force on a sphere:F = 6πηav.

• Terminal velocity:v_t = (2/9)(r²(ρ_s – ρ_f)g / η).

10.6 Surface Tension

• Force per unit length along a liquid surface.

• Pressure difference across curved surface:ΔP = 2T/r.

• Capillary rise:h = (2T cosθ)/(ρgr).

2. Detailed Notes for Each Section

10.1 Hooke’s Law & Elasticity

Elastic Behavior

• When force removed → material returns to original shape.

• Hooke’s region: linear relationship.

Force Constant (k)

F = kx.Depends on material & dimensions.

Stress & Strain

• Stress: σ = F/A.

• Strain: ε = ΔL/L₀.

• Both can be applied to solids.

Young’s Modulus

10.2 Stress–Strain Curve

Key Regions

• Proportional limit: obeys Hooke’s law.

• Elastic limit: returns to original shape.

• Yield point: significant extension without extra force.

• Ultimate tensile stress: maximum stress.

• Breaking point: material fractures.

Types of Materials

• Ductile: large plastic region.

• Brittle: breaks soon after elastic limit.

10.3 Energy Stored in a Stretched Spring

Work–Extension Graph

• Area under graph = elastic potential energy.

Expressions

• U = ½Fx.

• U = ½kx².

• Units: joules.

10.4 Viscosity & Fluid Flow

Laminar vs Turbulent Flow

• Laminar: layers flow smoothly.

• Turbulent: chaotic flow.

Velocity Gradient

velocity gradient = (v₁ – v₂)/d.

Coefficient of Viscosity

Poiseuille’s Formula (for laminar flow through capillary):

Flow rate:

10.5 Terminal Velocity & Stokes’ Law

Forces on a Sphere

• Weight: mg.

• Upthrust: ρ_f V g.

• Viscous force: 6πηav.

Terminal Velocity Condition

Net force = 0.

Expression

10.6 Surface Tension

Definitions

• T = force/length.

• Unit: N/m.

Angle of Contact

• θ < 90°: liquid wets surface (water–glass).

• θ > 90°: liquid does not wet surface (mercury–glass).

Pressure Difference Across Curved Surface

• For spherical bubble:

  ΔP = 2T/r.

Capillary Rise

3. Formula Summary (Unit 10)

• σ = F/A

• ε = ΔL/L₀

• E = σ/ε

• F = kx

• U = ½kx²

• η = (F/A) / (velocity gradient)

• F_viscous = 6πηav

• v_t = (2/9)(a²(ρ_s – ρ_f)g/η)

• ΔP = 2T/r

• h = (2T cosθ)/(ρgr)

4. Common Mistakes to Avoid

• Confusing stress with pressure.

• Forgetting that strain has no units.

• Misreading stress–strain graph regions.

• Using diameter instead of radius in Stokes’ law.

• Using wrong angle of contact in capillarity.

5. Exam Tips

• Draw clear stress–strain graphs.

• Show all steps when deriving v_t.

• In viscosity questions, state laminar flow assumption.

• For surface tension, always indicate direction of forces.

• Use correct units (Pa·s, N/m, etc.).

6. Quick Revision Table