Oscillations (Copy)
A2 Level Physics – Section 17: Simple Harmonic Motion and Resonance (Detailed Notes)
17.1 Simple Harmonic Oscillations
1. Key Terms and Definitions
| Quantity | Symbol | Definition |
|---|---|---|
| Displacement | x | Distance from the equilibrium position |
| Amplitude | xâ‚€ | Maximum displacement from equilibrium |
| Period | T | Time for one complete oscillation (s) |
| Frequency | f | Number of oscillations per second (Hz), f = 1 / T |
| Angular frequency | ω | ω = 2πf = 2π / T (rad/s) |
| Phase difference | — | Angular difference in motion between two oscillations (rad or °) |
2. Conditions for SHM
- An object performs simple harmonic motion (SHM) if:
- Its acceleration is proportional to displacement
- And directed towards the equilibrium position
Mathematical form:
a = –ω²x
- Negative sign: acceleration always directed opposite to displacement (restoring force)
3. SHM Equation of Motion
Displacement as a function of time:
x(t) = x₀·sin(ωt) (or x₀·cos(ωt), depending on starting phase)
Velocity:
v(t) = v₀·cos(ωt), where v₀ = ω·x₀
Or from energy considerations:
v = ±ω·√(x₀² – x²)
- Velocity is maximum at equilibrium, zero at amplitude
Acceleration:
a(t) = –ω²·x
- Acceleration is maximum at amplitude, zero at equilibrium
4. Graphical Representations
- Displacement–time graph: sine/cosine curve
- Velocity–time graph: cosine/sine curve (90° out of phase with displacement)
- Acceleration–time graph: negative sine/cosine (180° out of phase with displacement)
- All graphs are periodic with the same period T
17.2 Energy in Simple Harmonic Motion
1. Energy Exchange
- Total mechanical energy in SHM is constant (if no damping)
- Interchange between:
- Kinetic energy (KE): max at equilibrium
- Potential energy (PE): max at amplitude
Total energy (E):
E = (1/2)·m·ω²·x₀²
- At any point:
- KE = E – PE
- PE = (1/2)·m·ω²·x²
- KE = (1/2)·m·ω²·(x₀² – x²)
- Energy vs time and Energy vs displacement graphs are sinusoidal/parabolic
17.3 Damped and Forced Oscillations, Resonance
1. Damping
- Damping: occurs when resistive forces (like friction or air resistance) remove energy from an oscillating system
Types of damping:
| Type | Description |
|---|---|
| Light damping | Amplitude gradually decreases; oscillations continue |
| Critical damping | System returns to equilibrium in shortest time without oscillating |
| Heavy damping | System returns slowly to equilibrium; no oscillation |
Displacement–time graphs differ in rate and presence of oscillation
2. Forced Oscillations and Resonance
- Forced oscillation: External periodic force applied to system
- Natural frequency: Frequency at which system oscillates freely
- Resonance: Occurs when driving frequency = natural frequency
At resonance:
- Amplitude is maximum
- Energy transferred most efficiently from driver to system
- Occurs in many physical systems: bridges, buildings, musical instruments
Damping affects resonance:
- Light damping: sharp, high amplitude resonance peak
- Heavy damping: broader, lower resonance peak