Notes on Simple Harmonic Motion

Simple Harmonic Motion (SHM)

  • Periodic Motion: Motion that repeats with a set frequency.

  • Simple Harmonic Motion: Type of periodic motion where the restoring force is proportional to the displacement and acts in the opposite direction.

  • Restoring Force: Force that returns the system toward equilibrium after displacement.

Types of Simple Harmonic Motion

  • Vibrating Spring/Mass System

  • Oscillating Pendulum

Oscillating Spring/Mass Systems

  • Can oscillate horizontally or vertically.

  • Maximum Displacement: When the spring is at maximum stretch/compression.

  • Equilibrium Position: Where the net force is zero (spring force = 0).

  • Potential Energy (PE) and Kinetic Energy (KE) vary:

    • At maximum displacement, KE = 0, PE is maximum.

    • At equilibrium, PE = 0, KE is maximum.

Damping

  • Ideal mass-spring systems would oscillate indefinitely.

  • Damping: Caused by friction, slows motion, and eventually brings the system to rest.

  • Over short timeframes, damping can be minimal, treating the system as ideal.

Simple Pendulum

  • Consists of a mass (bob) attached to a fixed string.

  • At maximum displacement, maximum KE converts to PE.

  • Total mechanical energy remains constant throughout the motion.

Key Terms in Simple Harmonic Motion

  • Equilibrium: Position where net force (F net) is zero.

  • Amplitude: Maximum displacement from equilibrium.

  • Period (T): Time to complete one full cycle of motion (measured in seconds).

  • Frequency (f): Number of cycles per second (units: s⁻¹ or Hertz).

    • Frequency is the reciprocal of the period: f=1Tf = \frac{1}{T} or T=1fT = \frac{1}{f}

Sample Problems

  1. If a string vibrates at a frequency of 20 Hz, what is its period?

  2. A swing completes one cycle in 2 seconds; what are its period and frequency?

Damping is the reduction in the amplitude of oscillation of a system due to external forces like friction, which ultimately brings the system to rest after a period of time.