Lecture Notes: Standing Wave on a String

Introduction

• Concept: Standing wave on a string excited by a mechanical oscillating device
• Setup:
  • Speaker-like device with a metal rod
  • String length: 1.8 meters
  • Frequency: 30 Hz
  • String tension: Mass of 350 grams over a pulley

Part A: Fundamental Frequency

• Objective: Find the fundamental frequency of the string
• Resonant Frequencies: Multiples of the fundamental frequency (n=1)
  • Fundamental mode: One antinode in the center
  • Higher harmonics: Multiple antinodes (n=2, 3, 4, etc.)
• Current Setup: Fourth harmonic with four antinodes
  • Frequency of fourth harmonic: 30 Hz
• Calculation:
  • Fundamental frequency (n=1) = Frequency of fourth harmonic / 4
  • Result: 7.5 Hz
  • Interpretation: Tuning oscillator to 7.5 Hz will show one antinode

Part B: Linear Density of the String

• Objective: Compute the linear density of the string
• Wave Speed Relations:
  • Wave speed (v) = Frequency (f) x Wavelength (λ)
  • Wave speed related to tension (T) and linear density (µ) by ( v = \sqrt{T/µ} )
• Linear Density (µ): Measured in kilograms per meter (kg/m)
  • Describes weight of string per meter
• Wave Speed Calculation:
  • Wavelength (λ) for fourth harmonic: 0.9 meters
  • Frequency: 30 Hz
  • Wave speed: 30 Hz x 0.9 m = 27 m/s
• Tension Calculation:
  • Tension (T) = Mass (m) x Gravity (g)
  • Mass: 0.35 kg
  • Gravity: 9.8 m/s²
  • Tension: 0.35 kg x 9.8 m/s² = 3.43 N
• Solving for Linear Density:
  • Formula: ( µ = T / v^2 )
  • Calculation: µ = 3.43 N / (27 m/s)² = 0.0047 kg/m
  • Conversion: 4.71 grams/m
  • Comparison: Reasonable for a guitar string

Conclusion

• Fundamental frequency and linear density calculations explained
• Encouragement to subscribe for more physics content
• Acknowledgement of video production schedule

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