Lecture on Hooke's Law and Related Concepts

Hooke's Law

• Definition: Within the limit of proportionality, the extension of a material is proportional to the applied force.
  • Graph Representation: A force-extension graph is a straight line from the origin (0) to the elastic limit.
  • Equation: Extension is proportional to the applied force, represented as:
    • $\text{Extension} = T \times \text{Applied Force}$, where $T$ is a constant.

Stress and Strain

• Derived Equation: From Hooke's Law, stress and strain are related.
  • Within the limits of proportionality, the strain is proportional to the stress.
  • Equation: Stress is proportional to strain:
    • $\text{Stress} = E \times \text{Strain}$, where $E$ is Young's Modulus of Elasticity.
  • Young's Modulus ($E$) is the ratio of stress to strain:
    • $E = \frac{\text{Stress}}{\text{Strain}}$

Key Concepts

• Stiffness: The ability of a material to return to its original shape after the applied force is removed.
  • Graph Interpretation: Stiffness is the gradient of the force-extension graph.
• Limit of Proportionality: The point on a material's force-extension graph beyond which extension is no longer proportional to the force applied.

Equations Derived

• Young's Modulus as related to gradient:
  • $E = \frac{F \times L}{A \times x}$, where:
    • $F$ is force, $L$ is the original length, $A$ is area, and $x$ is extension.
  • Can use stiffness (gradient of force-extension graph) to solve for Young's Modulus:
    • $E = \text{Stiffness} \times \frac{L}{A}$

Example Problem: Copper Rod

1. Given Data:
   • Diameter: 20mm = 0.02m
   • Length: 2.0m
   • Force: 5kN = 5000N
   • Young's Modulus ($E$): 96 GPa = $96 \times 10^9$ Pa
2. Find:
   • Stress: $\text{Stress} = \frac{\text{Force}}{\text{Area}}$
     • Area: $A = \frac{\pi d^2}{4}$
   • Extension: Use $E = \frac{\text{Stress}}{\text{Strain}}$

Example Problem: Mined Steel Sample

1. Given Data:
   • Diameter: 1.3mm = 0.0013m
   • Length: 8.0m
2. Find:
   • Modulus of Elasticity using graph and stiffness.
   • Stress at limit of proportionality from graph.

Graphing and Calculations

• Graphing Steps:

  • Plot load vs. extension to find stiffness.
  • Identify elastic limit and calculate stiffness as the gradient of the linear portion of the graph.

• Calculating Stiffness:

  • Use gradient formula: $\frac{F_2 - F_1}{x_2 - x_1}$
  • Stiffness helps find Young’s Modulus.

• Conversion:

  • Converting stress to Mega Pascals and modulus to Giga Pascals for final answers.

Conclusion

• Understanding of Hooke's Law and how it applies to material properties and equations.
• Application of concepts through problem-solving with real-world examples and graph interpretations.