UNIT 2: Elasticity and Fluid Mechanics

Elasticity

Overview
  • Lecturer: Mr. MA Malape
  • Module: MBHA010
  • Year: 2026
  • Office: 2051 Old Q-Block
Bungee Jumping Example
  • Involves a long elastic strap that stretches proportionally to the weight of the jumper.
  • Elasticity: The property determining the amplitude of vibrations resulting from the elastic strap.
  • Elastic Limit: The point beyond which the strap will break if exceeded.
Definition of Elasticity
  • Elasticity is a branch of physics studying the properties of elastic materials.
Module Objectives

After completing this module, you should be able to:

  • Demonstrate understanding of:
    • Elasticity
    • Elastic Limit
    • Stress
    • Strain
    • Ultimate Strength
  • Write and apply formulas for:
    • Young’s Modulus
    • Shear Modulus
    • Bulk Modulus
  • Solve problems with these parameters.
Elastic Properties of Matter

  • Elastic Body: Returns to its original shape after deformation (e.g., golf ball, soccer ball, rubber band).

    • Illustrated by undamped spring-mass systems.

  • Inelastic Body: Does not return to its original shape after deformation (e.g., dough, clay).

Elastic vs. Inelastic Deformation
  • Elastic Deformation: Energy is fully restored after deformation.
  • Inelastic Deformation: Energy is lost, and the deformation can be permanent.
  • Collision Types:
    • Elastic Collision: No energy loss; deformation fully restored.
    • Inelastic Collision: Energy loss occurs; deformation may be permanent.
Hooke’s Law
  • Definition: Describes the restoring force in springs, given by the formula: F=−kxF = -kx
    • Where:
    • F: Restoring force
    • k: Spring constant
    • x: Displacement from equilibrium
  • Stiffness of a Spring: Measured by the spring constant, kk. Higher kk values indicate stiffer springs.
Energy of an Elastic Material
  • In a frictionless spring-mass system, the total mechanical energy is constant: Em=K+U=constantE_m = K + U = constant
    • Energy transfers continuously between elastic potential energy and kinetic energy.
Stress and Strain
  • Stress (σ): The cause of deformation, defined as: extStress=FAext{Stress} = \frac{F}{A}
    • Where F is the applied force and A is the area.
  • Strain (ε): The effect of deformation, defined as:
    extStrain=extChangeinLengthextOriginalLengthext{Strain} = \frac{ ext{Change in Length}}{ ext{Original Length}}
Types of Stress and Strain
  • Tensile Stress: Forces acting away from each other.
  • Compressive Stress: Forces acting towards each other.
  • Shear Stress: Forces acting parallel to the surface.
  • Bulk Stress: Change in volume due to external pressures.
Modulus of Elasticity
  • The relationship between stress and strain for elastic materials is given by the Modulus of Elasticity:
    extModulusofElasticity=extStressextStrainext{Modulus of Elasticity} = \frac{ ext{Stress}}{ ext{Strain}}
Young’s Modulus
  • Represents the longitudinal modulus of elasticity:
    Y=extLongitudinalStressextLongitudinalStrain=F/A∆LL0Y = \frac{ ext{Longitudinal Stress}}{ ext{Longitudinal Strain}} = \frac{F/A}{\frac{∆L}{L_0}}
The Elastic Limit
  • Maximum stress before permanent deformation occurs.
  • Beyond this limit, the material fails to return to its original dimensions.
The Ultimate Strength
  • The maximum stress a material can withstand before rupture.
Bulk Modulus
  • Related to volume stress and strain when materials compress, given by:
    B=−extVolumeStressextVolumeStrainB = -\frac{ ext{Volume Stress}}{ ext{Volume Strain}}
Shear Modulus
  • Deals with forces that cause shear stress:
    S=extShearStressextShearStrainS = \frac{ ext{Shear Stress}}{ ext{Shear Strain}}
Examples of Stress and Strain Calculations
  • 15 cm long tendon stretches 3.7 mm under 13.4 N:
    • Calculate:
      a) Strain
      b) Stress
      c) Young’s modulus
      d) Stretch with radius tripled.
Conclusion
  • Elasticity defines how materials deform under stress and recover, establishing a foundational understanding of material properties.

Fluid Mechanics

Phases of Matter
  • Common Phases: Solid, Liquid, Gas.
    • Solid: Definite shape and size.
    • Liquid: Fixed volume, no definite shape.
    • Gas: No fixed shape or volume, easily compressed.
Fluid Types
  • Fluid Statics: Study of fluids at rest.
  • Fluid Dynamics: Study of fluids in motion.
Density and Specific Gravity
  • Density (ρ): Mass per unit volume (kg/m³).
    • Water density at 4°C: 1 g/cm³ = 1000 kg/m³.
  • Specific Gravity: Ratio of density of a substance to the density of water (at 4°C).
Pressure in Fluids
  • Definition: Force per unit area:
    extPressure=FAext{Pressure} = \frac{F}{A}
  • SI unit: Pascal (Pa) where 1 Pa = 1 N/m².
  • Pressure varies with depth in fluids.
Atmospheric Pressure
  • At sea level: 1 atmosphere (atm) ≈ 1.013 x 10⁵ N/m².
  • Absolute pressure includes atmospheric pressure plus gauge pressure.
Pascal’s Principle
  • Pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid.
  • Used in hydraulic systems like car lifts and brakes.
Buoyancy and Archimedes’ Principle
  • Archimedes’ Principle: The buoyant force on an object in fluid equals the weight of the fluid displaced.
  • Floating condition: An object will float if its density is less than that of the fluid (pobject < pfluid).
Bernoulli’s Equation
  • Relates pressure, velocity, and height in flowing fluids:
    P+12ρv2+ρgh=constantP + \frac{1}{2}ρv² + ρgh = constant
  • This principle explains that where fluid velocity is high, pressure is low.
Example Scenarios
  • Blood Flow: Between major arteries and tiny capillaries; applies Bernoulli's principles to understand flow dynamics.
  • Fluid Flow from a Tank: Governed by differences in height and resulting pressure changes, exemplified by Torricelli’s theorem.