Lecture Notes: Harmonic Motion

Introduction

• Harmonic Motion Overview
  • Often confused with harmonic convergence (planets lining up), which is unrelated.
  • Real harmonic motion relates to forces and their behaviors over distances.

Force and Distance

• General Principle
  • Most forces decrease with distance (e.g., gravity, electromagnetism).
  • Described by the formula: 1/R², where R is the radius or distance.
  • Example:
    • Triple the distance = 1/3² = 1/9th of the force.

Forces Increasing with Distance

• Examples
  • Rubber Bands & Springs: As they stretch, the force increases to bring them back to equilibrium.
  • Formula: Force = -kx
    • k = constant
    • x = displacement from equilibrium
    • Minus sign: Force is always opposite to displacement.

Harmonic Oscillator

• Concept
  • When a spring is displaced and released, it oscillates around the equilibrium point.
  • Speed: Maximum at the equilibrium point, zero at maximum displacement.

Role of Trigonometry

• Trigonometry in Harmonic Motion
  • Time equations include trigonometric functions (e.g., sine curves).
  • Position vs. Time: A sine curve describes the oscillation.

Physics and Calculus

• Calculus in Harmonic Motion
  • First Derivative (Slope): Represents speed.
  • Sign Curve: Shows displacement and speed variations over time.
  • Critical Points:
    • Slope (speed) is zero at maximum displacement.
    • Slope (speed) is maximum at equilibrium position.

Real-Life Examples

• Rocking Chair: Example of harmonic motion.
• Pendulum
  • Moves slowest at the extremes of its swing.
  • Gravity acts as the restoring force.
  • Takes the same time to complete a swing, regardless of amplitude.
  • Discovered by Galileo; further developed into the pendulum clock.

Historical Insight

• Galileo's Observation: Realized pendulum’s constant swing time using pulse counting.
• Development of Pendulum Clock
  • Invented by Christian Huygens, who understood pendulum characteristics.
  • Horology: Study of time and oscillations.

Applications and Conclusion

• Harmonic Motion in Everyday Life
  • Swings, springs, and other oscillating systems.
  • Restoring forces increase with displacement, leading to oscillation.

Final Thoughts

• Harmonic motion is foundational in physics and can be observed in many everyday objects and phenomena.
• Encouragement to observe and appreciate these principles in real life.