Unit 7: Oscillations
1. Simple Harmonic Motion (SHM)
Simple harmonic motion occurs when a restoring force is directly proportional to displacement from equilibrium.
Basic SHM Equations:
F = -kx
ω = √(k/m)
T = 2π√(m/k)
f = 1/T = ω/(2π)
Motion Equations:
x(t) = A cos(ωt + φ)
v(t) = -Aω sin(ωt + φ)
a(t) = -Aω² cos(ωt + φ)
Key Concepts:
• Amplitude (A)
• Angular frequency (ω)
• Period (T)
• Phase constant (φ)
2. Energy in SHM
Energy constantly transforms between kinetic and potential during oscillation.
Energy Equations:
K = ½mv²
U = ½kx²
E = K + U = ½kA²
Energy Conservation:
½mv² + ½kx² = ½kA²
Example:
Mass-spring system with k = 100 N/m, A = 0.2 m:
Total Energy = ½(100)(0.2)² = 2 J
3. The Simple Pendulum
A mass suspended by a light string exhibits SHM for small angles.
Pendulum Equations:
T = 2π√(L/g)
ω = √(g/L)
Small Angle Approximation:
sin θ ≈ θ (for θ < 15°)
Angular Motion:
θ(t) = θ₀ cos(ωt + φ)
Applications:
• Timekeeping
• Seismographs
• Physical pendulum analysis
• Small oscillation studies
4. Damped Oscillations
Real oscillators experience energy loss through friction or resistance.
Damped Motion:
x(t) = Ae⁻ᵇᵗ cos(ω't + φ)
Damped Frequency:
ω' = √(ω² - b²)
Quality Factor:
Q = ω/(2b)
Example:
Damped oscillator with b = 0.5 s⁻¹:
Amplitude decay: A(t) = A₀e⁻⁰·⁵ᵗ
5. Forced Oscillations and Resonance
External periodic forces can drive oscillations and lead to resonance.
Forced Oscillation:
F = F₀ cos(ωₑt)
Resonance Frequency:
ωᵣ = √(ω² - 2b²)
Amplitude Response:
A(ωₑ) = F₀/√((k-mωₑ²)² + (bωₑ)²)
Key Parameters:
• Driving frequency
• Natural frequency
• Damping coefficient
• Resonance amplitude
6. Coupled Oscillators
Systems of connected oscillators exhibit normal modes and energy transfer.
Normal Modes:
ω₁ = √(k/m)
ω₂ = √(3k/m)
Beat Frequency:
ωᵦ = |ω₁ - ω₂|
Energy Transfer:
T = 2π/ωᵦ
Example:
Two identical masses coupled by springs:
Normal modes: in-phase and anti-phase
7. Wave Motion
Oscillations can propagate through media as waves.
Wave Equation:
∂²y/∂t² = v²(∂²y/∂x²)
Wave Speed:
v = √(tension/μ) [string]
v = √(Y/ρ) [solid]
Wave Function:
y(x,t) = A cos(kx - ωt)
Applications:
• String instruments
• Seismic waves
• Sound waves
• Electromagnetic waves
8. Angular Oscillations
Rotational systems can exhibit oscillatory motion.
Torsional Pendulum:
τ = -κθ
ω = √(κ/I)
Physical Pendulum:
T = 2π√(I/mgd)
Example:
Physical pendulum with I = 0.5 kg·m², d = 0.3 m:
T = 2π√(0.5/(9.8×0.3)) = 2.57 s
Applications:
• Anharmonic oscillators
• Double pendulum
• Nonlinear springs
• Chaos theory
Practice and Advanced Applications
To master Oscillations in AP Physics C:
- Analyze energy transformations
- Study damped systems
- Explore resonance phenomena
- Practice phase diagrams
- Review past AP exam questions
Common Challenges:
• Phase relationships
• Energy analysis
• Damping effects
• Resonance conditions
• Coupled systems
