Vibration and Waves: Periodic Motion

Key Concepts

• Periodic Motion: Motion that is repeating and regular.
  • Repeating: The motion occurs over and over again (cycle after cycle).
  • Regular: The motion occurs in the same amount of time for each cycle.

Period, Frequency, and Amplitude

• Period (T):
  • The time taken to complete one full cycle of vibration.
  • For any object in periodic motion, the period is constant.
• Frequency (f):
  • The number of complete cycles per unit of time.
  • Formulas:
    • Period: ( T = \frac{\text{time in seconds}}{\text{number of complete cycles}} )
    • Frequency: ( f = \frac{\text{number of cycles}}{\text{time}} )
    • These two are reciprocals of each other.
• Amplitude:
  • The maximum displacement from the resting position.
  • Diminishes over time due to damping.

Examples of Periodic Motion

• Mass on a spring: Vibrates back and forth.
• Pendulum: Back and forth motion of a bob hanging from a string.
• Earth's rotation: Takes 24 hours for one full cycle.
• Earth's orbit: Takes 365.25 days to orbit the sun.

Graphical Representation

• Position-Time Graph:
  • Shape resembles a sine wave.
  • Example: Mass vibrating from 20 cm to 120 cm about a resting position of 70 cm.
• Velocity-Time Graph:
  • Sinusoidal relationship; shows speed with direction.
  • Example behaviors: Mass slows down, changes direction, speeds up repeatedly.

Damping

• Energy dissipates due to interaction with surroundings.
• Results in decreasing amplitude over time while the period remains constant.
• Important: "Slowing down" refers to speed, but for periodic motion, the period is constant.

Example Calculation

• If an object completes 60 cycles in 10 seconds:
  • Period: ( \frac{10 \text{ seconds}}{60 \text{ cycles}} = 0.25 \text{ seconds/cycle} )
  • Frequency: ( \frac{60 \text{ cycles}}{10 \text{ seconds}} = 6.0 \text{ cycles/second} )

Action Plan

1. Explore the simulation page on a vibrating mass on a spring.
2. Review the tutorial page on periodic motion.
3. Practice with the calculator pad problem set on frequency and period.