Unit 6: Oscillations

📚Study Guide: Oscillations

Unit 6: Oscillations

Oscillations in AP Physics C: Mechanics are analyzed through differential equations, providing a deeper and more general understanding of simple harmonic motion (SHM), physical pendulums, and damped systems. The defining equation of SHM is d²x/dt² + (k/m)x = 0, whose general solution is x(t) = A cos(ωt + φ), where ω = √(k/m). The amplitude A and phase constant φ are determined by initial conditions: x(0) = x₀ and v(0) = v₀. You must be able to derive the angular frequency for any system where the restoring force is linear, not just springs. A physical pendulum—any rigid body swinging about a pivot—has period T = 2π√(I/mgd), where I is the moment of inertia about the pivot and d is the distance from pivot to center of mass. This replaces the simple pendulum formula and is valid for small angles. The unit also introduces damped harmonic oscillators, where a resistive force proportional to velocity (−bv) is included. The equation of motion becomes d²x/dt² + 2β dx/dt + ω₀²x = 0, where β = b/(2m). Depending on whether β < ω₀ (underdamped), β = ω₀ (critically damped), or β > ω₀ (overdamped), the system exhibits different behaviors: oscillatory decay, fastest return to equilibrium without oscillation, or slow exponential return. Driven oscillations and resonance occur when an external periodic force is applied at the natural frequency, causing amplitude to peak. While the full driven oscillator solution is beyond the AP scope, you must understand qualitatively that amplitude is maximized when driving frequency equals natural frequency. On the AP Exam, you will be asked to set up and solve the SHM differential equation, apply initial conditions, analyze energy transformations, and explain the effects of damping qualitatively. Mastery of calculus is essential because the entire theory of oscillations is built on solving second-order linear differential equations.

Key Concepts

• Differential Equation of SHM: d²x/dt² + ω²x = 0, where ω² = k/m. The solution is sinusoidal with angular frequency ω.
• Amplitude and Phase from Initial Conditions: A = √(x₀² + (v₀/ω)²) and tanφ = −v₀/(ωx₀). These are derived by evaluating x(t) and v(t) at t = 0.
• Physical Pendulum: T = 2π√(I/mgd). Requires knowing I about the pivot and d to the center of mass. Valid for small oscillations.
• Damped Harmonic Oscillator: Equation: d²x/dt² + 2β dx/dt + ω₀²x = 0. Underdamped (β < ω₀) oscillates with decaying amplitude; critically damped (β = ω₀) returns fastest without oscillating; overdamped (β > ω₀) returns slowly.
• Driven Oscillations and Resonance: An external driving force at frequency ω_d causes amplitude to peak when ω_d ≈ ω₀.
• Energy in Damped Systems: Total mechanical energy decreases exponentially due to work done against the damping force. E(t) ≈ E₀ e^(−2βt) for light damping.

Vocabulary

• Simple Harmonic Oscillator: A system obeying the differential equation d²x/dt² + ω²x = 0.
• Angular Frequency (ω): The rate of oscillation in radians per second. ω = 2πf = 2π/T.
• Phase Constant (φ): A constant in the SHM solution determined by initial position and velocity.
• Physical Pendulum: Any extended body oscillating about a fixed pivot under gravity.
• Damping Coefficient (β): A parameter characterizing the strength of the resistive force in a damped oscillator.
• Driven Oscillator: An oscillator subjected to an external periodic force.
• Resonance: The condition where the driving frequency matches the natural frequency, producing maximum amplitude.
• Transient: The portion of the solution that decays over time, leaving only the steady-state response in driven systems.

Essential Formulas

• d²x/dt² + (k/m)*x = 0
• x = A*cos(ω*t + φ)
• ω = sqrt(k/m); T = 2π/ω
• E = ½*k*A²
• T_physical = 2π*sqrt(I / (m*g*d))
• d²x/dt² + 2β*dx/dt + ω0²*x = 0 (damped)

Common Mistakes

• Wrong Initial Conditions for Phase: Do not assume φ = 0. Calculate it from x₀ and v₀. Starting from rest at x = A gives φ = 0; starting at equilibrium with v > 0 gives φ = −π/2.
• Forgetting Physical Pendulum Uses I and d: The simple pendulum T = 2π√(L/g) is a special case. For any extended body, use T = 2π√(I/mgd).
• Ignoring Damping in Real Systems: All real oscillators lose energy. The amplitude decreases over time unless energy is supplied by a driving force.
• Assuming Amplitude Stays Constant Without Driving Force: In damped systems, amplitude decays. Only undriven, undamped oscillators maintain constant amplitude forever.

AP Exam Strategies

• Write the Differential Equation First: Apply Newton's Second Law or τ = Iα to obtain an equation of motion. Identify ω² from the coefficient of x or θ.
• Determine ω from Coefficients: Once you have d²x/dt² + Cx = 0, immediately write ω = √C.
• Use Energy for Amplitude If Starting from Rest at x = A: E = ½kA² gives A directly if you know total energy.
• Sketch Amplitude vs. Driving Frequency for Resonance: A sharp peak at ω_d = ω₀ indicates resonance. Damping broadens and lowers the peak.

Real-World Applications

• Seismometers: Instruments that detect ground motion using damped harmonic oscillators tuned to specific earthquake frequencies.
• Tuned Mass Dampers: Mass-spring-dashpot systems installed in skyscrapers (like Taipei 101) to counteract swaying from wind and earthquakes.
• Vehicle Suspensions: Shock absorbers provide damping to prevent cars from oscillating excessively after hitting bumps.