Simple Harmonic Motion Lecture Notes

Jul 28, 2024

Simple Harmonic Motion (SHM)

Key Concepts

  • Equations of Motion: Standard equations (
    • x = x0 + v0t + 1/2 at^2) work only with constant acceleration, not with variable forces like friction or springs.
    • Springs don't have constant force, leading to variable acceleration without simple kinematic equations.
  • Simple Harmonic Motion (SHM): Periodic motion of springs or pendulums where force is proportional to displacement, often modeled by sine and cosine waves.

Recognizing SHM

  • Indicators of SHM:
    • Presence of a spring or small-angle pendulum.
    • Force F = -kx leading to acceleration a = -kx/m (not constant).
  • Characteristics:
    • Periodic - returns to a starting point over time.
    • Oscillatory - moves back and forth.

Definitions and Equations

  • Amplitude (A): Maximum displacement from equilibrium.
  • Period (T): Time for one complete cycle.
  • Frequency (f): Number of cycles per second; measured in Hertz (Hz).
  • Angular Frequency (ω): ω = 2πf, measured in radians per second.

Motion Attributes

  • Amplitude Calculation: Amplitude is half the peak-to-peak distance.
  • Period Measurement: Can measure from peak to peak, trough to trough, or zero crossing with careful attention to direction.
  • Frequency Calculation: Number of complete cycles over a period of time.

Calculations and Graphs

  • Kinetic and Potential Energy:
    • Kinetic Energy: 1/2 mv^2
    • Potential Energy: 1/2 kx^2
    • Total Energy: Sum of kinetic and potential.
  • Key Formulas:
    • x(t) = A * cos(ωt)
    • v(t) = -Aω * sin(ωt)
    • a(t) = -Aω^2 * cos(ωt)
  • Graphs: Typical SHM graphs include position, velocity, and acceleration over time showing sinusoidal behavior.

Practical Notes

  • Energy Transfer: Continuous transfer between kinetic and potential energy.
  • Effect of Mass and Spring Constant:
    • Stiff springs (high k) lead to faster oscillations (high ω).
    • Larger masses (m) result in slower oscillations (small ω).

Common Misconceptions

  • Amplitude Misinterpretation: Amplitude is not the total peak-to-peak distance, but half of it.
  • Frequency Confusion: Differentiating between regular frequency (f) and angular frequency (ω).

Advanced Considerations

  • Use of Calculators: Ensure calculators are set to radian mode for trigonometric calculations in SHM.
  • Periodic Function Characteristics: Be mindful of starting points (cosine starts from max, sine starts from zero).
  • Two Frequencies and Omegas: Recognize different contexts (SHM vs rotating systems).

Conclusion

Understanding SHM requires focusing on periodic, sinusoidal motion with variable acceleration, recognizing energy transformations, and accurately applying amplitude, period, and frequency concepts. Proper calculator settings and careful interpretations ensure accurate analyses and solutions.