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Simple Harmonic Motion and Waves

Subject: Physics
Topic: 5
Cambridge Code: 0625

Simple Harmonic Motion​

SHM - Oscillation with restoring force proportional to displacement

Defining SHM​

a=−ω2xa = -ω^2x

Where:

  • a = acceleration
  • ω = angular frequency
  • x = displacement from equilibrium
  • Negative sign indicates opposite direction

Equations of Motion​

Displacement: x=Acos⁡(ωt)x = A\cos(ωt) or x=Asin⁡(ωt)x = A\sin(ωt)

Velocity: v=−Aωsin⁡(ωt)v = -Aω\sin(ωt) or v=Aωcos⁡(ωt)v = Aω\cos(ωt)

Acceleration: a=−Aω2cos⁡(ωt)=−ω2xa = -Aω^2\cos(ωt) = -ω^2x

Where:

  • A = amplitude
  • ω = angular frequency = 2πf = 2π/T
  • f = frequency (Hz)
  • T = period (s)

Energy in SHM​

Total mechanical energy (constant): Etotal=12kA2E_{\text{total}} = \frac{1}{2}kA^2

Where k is spring constant

Kinetic energy: Ek=12mv2=12mω2(A2−x2)E_k = \frac{1}{2}mv^2 = \frac{1}{2}mω^2(A^2 - x^2)

Potential energy: Ep=12kx2=12mω2x2E_p = \frac{1}{2}kx^2 = \frac{1}{2}mω^2x^2

Energy oscillates between KE and PE

Maximum Values​

Maximum velocity: vmax=Aωv_{\text{max}} = Aω

  • Occurs at equilibrium (x = 0)

Maximum acceleration: amax=Aω2a_{\text{max}} = Aω^2

  • Occurs at maximum displacement (x = ±A)

Examples of SHM​

Mass-Spring System​

Spring force: F=−kxF = -kx a=−kmxa = -\frac{k}{m}x

Frequency: f=12πkmf = \frac{1}{2π}\sqrt{\frac{k}{m}}

Period: T=2πmkT = 2π\sqrt{\frac{m}{k}}

Simple Pendulum​

For small angles: a≈−gLxa ≈ -\frac{g}{L}x

Period: T=2πLgT = 2π\sqrt{\frac{L}{g}}

  • Independent of mass (!)
  • Independent of amplitude (for small angles)
  • Depends on length and g

Damped Oscillations​

Damping - Energy loss due to friction, air resistance

Types:

  1. Light damping: Oscillates with decreasing amplitude
  2. Critical damping: Returns to equilibrium without oscillating
  3. Heavy damping: Slow return to equilibrium

Amplitude decreases: A(t)=A0e−t/τA(t) = A_0e^{-t/τ}

Forced Oscillations and Resonance​

Forced oscillation - External driving force causes vibration

Resonance - Maximum amplitude when driving frequency = natural frequency

fdrive=fnaturalf_{\text{drive}} = f_{\text{natural}}

Characteristics:

  • Dramatic amplitude increase
  • Occurs at natural frequency
  • Damping affects amplitude and sharpness
  • Practical examples: bridges, buildings, speakers

Wave Properties​

Wave - Disturbance propagating through medium

Wave Types​

Transverse: Oscillation perpendicular to propagation

  • Light, water ripples, S-waves (earthquakes)

Longitudinal: Oscillation parallel to propagation

  • Sound, P-waves (earthquakes)

Wavelength and Frequency​

Wavelength (λ) - Distance between successive wavefronts

Wave speed: v=fλv = fλ

Where:

  • v = wave speed
  • f = frequency
  • λ = wavelength

Electromagnetic Spectrum​

Arranged by frequency/wavelength:

Radio → Microwave → Infrared → Visible → UV → X-ray → Gamma

  • Increasing frequency (left to right)
  • Decreasing wavelength (left to right)
  • All travel at speed of light (c = 3 × 10⁸ m/s) in vacuum

Sound Waves​

Speed of sound:

  • Air (20°C): 343 m/s
  • Water: 1480 m/s
  • Solids: 3000-6000 m/s
  • Increases with temperature

Frequency and Pitch​

Frequency determines pitch:

  • Higher frequency → higher pitch
  • Human hearing: 20 Hz to 20,000 Hz
  • Ultrasound: > 20 kHz

Intensity and Loudness​

Intensity: Power per unit area (W/m²)

Decibel scale: L=10log⁡10II0L = 10\log_{10}\frac{I}{I_0}

Where I0=10−12I_0 = 10^{-12} W/m² (threshold of hearing)

Superposition and Interference​

Superposition - Waves combine by adding displacements

Constructive Interference​

Waves in phase:

  • Path difference = nλ (n = 0, 1, 2, ...)
  • Amplitudes add
  • Maximum intensity

Destructive Interference​

Waves out of phase:

  • Path difference = (n + ½)λ
  • Amplitudes cancel
  • Minimum intensity

Standing Waves​

Formation: Interference of waves traveling in opposite directions

Nodes: Points of zero displacement Antinodes: Points of maximum displacement

Diffraction​

Diffraction - Bending of waves around obstacles

Single-Slit Diffraction​

First minimum: asin⁡θ=λa\sin θ = λ

Where a = slit width

Narrower slit → wider diffraction pattern

Diffraction Through Door​

Audible if wavelength > door width

  • Low frequency (longer λ): Easy to hear around corner
  • High frequency (shorter λ): Difficult to hear

Doppler Effect​

Frequency change when source moves:

f′=fv±vobserverv∓vsourcef' = f\frac{v ± v_{\text{observer}}}{v ∓ v_{\text{source}}}

Source approaching: f' > f (frequency increases, sounds higher) Source receding: f' < f (frequency decreases, sounds lower)

Applications: Radar speed guns, astronomy

Key Points​

  1. SHM: a = -ω²x
  2. Energy in SHM oscillates between KE and PE
  3. Period T = 2π/ω
  4. Pendulum period independent of mass
  5. Wave speed = frequency × wavelength
  6. Constructive interference: path difference = nλ
  7. Destructive interference: path difference = (n+½)λ
  8. Resonance at natural frequency
  9. Doppler effect: frequency changes with relative motion
  10. Standing waves have nodes and antinodes

Practice Questions​

  1. Solve SHM equations
  2. Calculate periods
  3. Analyze energy in SHM
  4. Calculate wave properties
  5. Apply wave equation
  6. Solve interference problems
  7. Analyze diffraction patterns
  8. Calculate Doppler shifts
  9. Predict standing wave patterns
  10. Solve resonance problems

Revision Tips​

  • Understand SHM definition
  • Know equations for x, v, a
  • Understand energy interchange
  • Know wave properties
  • Practice interference calculations
  • Understand constructive/destructive
  • Know Doppler effect application
  • Draw wave diagrams
  • Understand resonance concept