Physics Chapter 3: Elasticity

Chapter 3: Elasticity

• Elasticity is a body's ability to resist deformation and return to its original shape after the deforming force is removed.
• Solid objects deform when significant forces are applied.
• If the material is elastic, the object returns to its original shape upon force removal.
• The reason for elastic behavior varies between materials.
  • In metals, atomic lattice changes size and shape when force is applied, returning to the original lower energy state when force is removed.
  • In rubbers and polymers, stretching of polymer chains causes elasticity.

Stress and Strain

• Stress is the internal resistive force per unit area of a body.
• Stress = Force/Cross-Sectional Area
• Strain is the ratio of the change in length to the original length of a wire.
• Strain = Elongation/Original Length

Hooke's Law and Coefficient of Elasticity

• Hooke's Law states that within elastic limits, stress is proportional to strain.
• Within elastic limits, tension is proportional to extension.
• Stress ∝ Strain
• A ∝ I/L (proportional relation between area and length)

Young's Modulus

• Young's modulus (Y) is the ratio of normal stress to longitudinal strain.
• Y = (Normal Stress)/(Longitudinal Strain) = (F/A)/(ΔL/L)
• Y = (MgxL)/(πr²×L)

Bulk Modulus

• Bulk modulus (K) is the ratio of volume stress to volume strain.
• K = (Volume Stress)/(Volume Strain) = (PV)/(ΔV)

Modulus of Rigidity

• Modulus of Rigidity (η) is the ratio of tangential stress to shearing strain.
• η = (Tangential Stress)/(Shear Strain) = (F/A)/θ = T/θ

Work Done During Longitudinal Strain

• The longitudinal strain equation demonstrates the material's extension when a load or stress is applied.
• Y = Stress/Strain
• F = AY/L × (x)
• dw = Fdx
• W = Integral(AY/L x dx)
• W = 1/2 (F/A) × (ΔL/L) = 1/2 × stress × strain

Work Done During Volume Strain

• K = Pv/v
• P = Kv/V
• dw = Fdx = PA dx, dw = Pdv
• W = Integral P dv
• W = 1/2 k/v v² = 1/2 (k/v v) x (v)
• W = 1/2 Pv = 1/2 × (stress) × (volume strain)

Work Done During Shearing Strain

• Shearing strain = θ
• Modulus of rigidity = η (F/A)/ θ
• F = η ΑΘ
• F= ηL²Θ,
• W = ∫on L Idl
• W = 1/2 η LI²
• W = F/LI
• W = 1/2 Fl
• W = 1/2 × (tangential force)× (Displacement)
• W = (F/A) × (Θ)
• W = 1/2 × Shearing stress × Shearing strain

Poisson's Ratio

• Poisson's Ratio (σ) is the ratio of lateral strain to longitudinal strain.
• σ = Lateral Strain/Longitudinal Strain.
• Lateral strain = Change in diameter/Original diameter = δD/D
• Longitudinal strain = Change in length/Original length = δL/L
• σ = (δD/D)/(δL/L)

Experimental Determination of Y by Loading a Rectangular Thin Bar at The Center

• The Young's modulus (Y) of a material in a non-uniform bending bar (central loading) is given by:
• Y = (mgl³)/(48le)
  • m = Mass loaded
  • g = Acceleration due to gravity
  • l = Length between knife edges
  • b = Breadth of bar
  • d = Thickness of bar
  • e = Depression of bar

Torsional Oscillations

• A torsion pendulum involves a body suspended by a thread or wire which twists in one direction, then the opposite.
• The period of oscillation (T) for a torsion pendulum is given by:
• T = 2π√(I/C)
  • I = Moment of inertia of suspended body
  • C = Couple/unit twist
• C = (1/2)πηγ⁴/l (expression for couple per unit twist)
  • γ = rigidity modulus
  • r = radius of suspension wire
  • l = length of suspension wire
• Substituting relation for C in T, a formula for rigidity modulus is derived.