This chapter covers the mechanical properties of solids (elasticity) and fluids (liquids and gases).
Overview
graph TD
A[Properties of Matter] --> B[Elasticity]
A --> C[Fluid Statics]
A --> D[Fluid Dynamics]
A --> E[Surface Tension]
B --> B1[Stress-Strain]
B --> B2[Moduli of Elasticity]
C --> C1[Pressure]
C --> C2[Pascal's Law]
C --> C3[Buoyancy]
D --> D1[Bernoulli's Theorem]
D --> D2[Viscosity]Elasticity
Stress
Force per unit area:
$$\sigma = \frac{F}{A}$$Types:
- Tensile stress
- Compressive stress
- Shear stress
Strain
Relative deformation:
$$\epsilon = \frac{\Delta L}{L}$$Hooke’s Law
Within elastic limit:
$$\text{Stress} \propto \text{Strain}$$Moduli of Elasticity
| Modulus | Formula | Type of Deformation |
|---|---|---|
| Young’s (Y) | $\frac{FL}{A\Delta L}$ | Longitudinal |
| Bulk (B) | $\frac{-PV}{\Delta V}$ | Volume |
| Rigidity (η) | $\frac{F/A}{\theta}$ | Shape |
Poisson’s Ratio:
$$\sigma = \frac{\text{Lateral strain}}{\text{Longitudinal strain}}$$Fluid Statics
Pressure in Fluids
$$P = P_0 + \rho g h$$Pressure at depth h below surface
Pascal’s Law
Pressure applied to enclosed fluid is transmitted equally in all directions.
Hydraulic Lift:
$$\frac{F_1}{A_1} = \frac{F_2}{A_2}$$Buoyancy
Archimedes’ Principle: Buoyant force = Weight of displaced fluid
$$F_B = \rho_{fluid} \cdot V_{submerged} \cdot g$$Floating Condition:
$$\frac{V_{submerged}}{V_{total}} = \frac{\rho_{object}}{\rho_{fluid}}$$Fluid Dynamics
Equation of Continuity
For incompressible flow:
$$A_1 v_1 = A_2 v_2$$Bernoulli’s Theorem
For steady, irrotational flow:
$$\boxed{P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}}$$Applications:
- Venturi meter
- Airplane lift
- Magnus effect
Torricelli’s Theorem
Speed of efflux from tank:
$$v = \sqrt{2gh}$$Viscosity
Newton’s Law of Viscosity
$$F = \eta A \frac{dv}{dx}$$where η = coefficient of viscosity
Stokes’ Law
Viscous force on sphere:
$$F = 6\pi \eta r v$$Terminal Velocity
$$v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$$where ρ = density of sphere, σ = density of fluid
Reynolds Number
$$R_e = \frac{\rho v d}{\eta}$$- Laminar flow: $R_e < 2000$
- Turbulent flow: $R_e > 3000$
Surface Tension
Definition
$$S = \frac{F}{L}$$Force per unit length at the surface
Surface Energy
$$E = S \times A$$Energy per unit area = Surface tension
Pressure Difference
Soap bubble (two surfaces):
$$\Delta P = \frac{4S}{R}$$Liquid drop (one surface):
$$\Delta P = \frac{2S}{R}$$Capillary Rise
$$h = \frac{2S\cos\theta}{\rho g r}$$Angle of Contact:
- Water-glass: 0° (wets)
- Mercury-glass: 140° (doesn’t wet)
Practice Problems
A wire of length 2 m and diameter 2 mm extends by 1 mm when a force of 100 N is applied. Find Young’s modulus.
Water flows through a pipe of varying cross-section. At one point, the radius is 10 cm and velocity is 2 m/s. Find velocity where radius is 5 cm.
A sphere of radius 1 cm falls through glycerine. Find terminal velocity. (Given: η = 0.8 Pa·s, ρ_sphere = 8000 kg/m³)
Further Reading
- Thermodynamics - Heat and thermal properties
- Oscillations and Waves - Wave motion in fluids