# Oscillations
## Types of Oscillations
- **Free oscillations**: Occur in a system left to itself after being disturbed from equilibrium
- **Forced oscillations**: Occur when an external periodic force acts on the system
## Harmonic Oscillations
- Simplest type of oscillation, follows sine or cosine law
- Described by equation: x'' + ω0^2 x = 0
- Where ω0 = √(k/m) is the natural angular frequency
- Solution: x = A cos(ω0t + φ0)
- A = amplitude, φ0 = initial phase
Key parameters:
- T = period
- v = frequency = 1/T
- ω0 = angular frequency = 2πv
## Damped Oscillations
- Energy decreases over time due to friction/resistance
- Equation: x'' + 2βx' + ω0^2x = 0
- β = damping coefficient
- Solution: x = A0e^(-βt)cos(ωt + φ)
- ω = √(ω0^2 - β^2) = frequency of damped oscillations
## Forced Oscillations
- External periodic force acts on system
- Equation: x'' + 2βx' + ω0^2x = (f0/m)cosΩt
- Ω = frequency of external force
- Resonance occurs when Ω ≈ ω0
## Examples of Oscillating Systems
- Spring pendulum
- Mathematical pendulum
- Physical pendulum
- LC electrical circuit
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