Notes on Vibration and Waves: Periodic Motion

Introduction

• Topic: Periodic Motion
• Key Traits: Repeating and Regular motion

Definition of Periodic Motion

• Periodic Motion: Motion that is repeating and occurs in a regular manner.
  • Period: Time taken to complete one full cycle of vibration.
  • Example: A mass on a spring takes the same time (e.g., 2.5 seconds) for each cycle.

Examples of Periodic Motion

• Mass on a spring
• Pendulum swinging back and forth
• Earth’s rotation (24 hours for one complete rotation)
• Earth’s orbit around the Sun (365.25 days for one orbit)

Position-Time Graphs

• Motion can be represented as a sine wave.
• Example: Mass vibrates from:
  • Nearest position: 20 cm
  • Furthest position: 120 cm
  • Resting position: 70 cm (fixed position)

Sinusoidal Relationship

• Position varies sinusoidally as a function of time.
• The repeating shape on the graph confirms the periodic nature of the motion.

Period of Motion

• Period is constant across cycles; for example:
  • First cycle: 0 to 6.3 seconds
  • Second cycle: 6.3 to 12.6 seconds
  • Period remains 6.3 seconds.

Damping in Periodic Motion

• Damping: Energy dissipation due to friction and resistance.
• Effect: Amplitude decreases over time while period remains constant.
• Amplitude: Displacement from the resting position.

Velocity-Time Relationship

• Velocity has both speed and direction; represented as a graph.
• Changes in velocity over time include:
  • Slowing down, changing direction, and speeding up.
  • This sequence repeats periodically.

Summary of Key Terms

• Period (T): Time per cycle.
• Frequency (f): Number of cycles per unit time.

Formulas

• Period: T = Time (seconds) / Number of Cycles
• Frequency: f = Number of Cycles / Time (seconds)

Example Calculation

• 60 cycles in 10 seconds:
  • Period = 10 seconds / 60 cycles = 0.25 seconds per cycle
  • Frequency = 60 cycles / 10 seconds = 6 cycles per second
• Interpretation:
  • Period (0.25 seconds): Time taken for one cycle.
  • Frequency (6 cycles per second): How often cycles occur.

Action Plan for Further Learning

• Resources available on the website:
  1. Simulation page on vibrating mass on a spring.
  2. Tutorial page on periodic motion.
  3. Calculator problem set on frequency and period.