Stage 3 – Page 9
Simple Harmonic Motion – Forced Oscillations & Resonance

                                  
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1. What are Forced Oscillations?

When an external periodic force acts on an oscillator, the motion is called forced oscillation.

F = F₀ sin(ωt)

  • System oscillates with driving frequency
  • Independent of natural frequency (after transient)

2. Equation of Motion

m d²x/dt² + b dx/dt + kx = F₀ sin(ωt)

Steady-state motion is important for exams

3. Steady-State Amplitude

Amplitude depends on driving frequency:

A = F₀ / √[(k − mω²)² + (bω)²]

Maximum amplitude occurs near resonance

4. Resonance (Most Tested Concept)

Resonance occurs when amplitude becomes maximum.

ωr = √(ω₀² − (b² / 2m²))

Resonant frequency ≠ natural frequency

5. Phase Relation (IIT Favourite)

Frequency Phase Difference
Low (ω ≪ ω₀) 0° (in phase)
At resonance 90°
High (ω ≫ ω₀) 180° (out of phase)

6. Velocity Resonance

Velocity becomes maximum at:

ω = ω₀

Different resonance conditions for displacement and velocity

7. Power Absorbed by Oscillator

Average power absorbed:

Pavg = (F₀² b ω²) / [2((k − mω²)² + (bω)²)]

Power maximum occurs at ω = ω₀

8. Bandwidth & Q-Factor

Bandwidth:

Δω = b / m

Quality factor:

Q = ω₀ / Δω

High Q → sharp resonance

9. Common JEE Traps

  • Assuming resonance at ω₀ always
  • Ignoring damping in resonance problems
  • Confusing power resonance with displacement resonance
  • Wrong phase angle assumptions

Stage 3 – Page 9 Summary

✔ Forced oscillations depend on driving frequency
✔ Resonance shifts due to damping
✔ Power maximum at natural frequency

— Stage 3 | Forced Oscillations & Resonance —

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