Unit 7 Oscillations and Simple Harmonic Motion — AP Physics C Test
Practice AP Physics C: Mechanics Unit 7 with SHM differential equation derivations, period analysis, phase relationships, and energy in simple harmonic motion.
SHM from Differential Equations to Energy Analysis
Unit 7 of AP Physics C: Mechanics treats simple harmonic motion rigorously through differential equations, connecting the force law F = -kx to the full mathematical solution x(t) = A cos(ωt + φ). This calculus-based treatment distinguishes AP Physics C: Mechanics from all algebra-based AP Physics courses and is a frequent focus of the FRQ section.
Core Concepts in Unit 7
Deriving SHM from the Differential Equation
For a spring-mass system, Newton's second law gives m(d²x/dt²) = -kx, a second-order linear ODE. The general solution is x(t) = A cos(ωt + φ), where ω = √(k/m). Students must demonstrate that this function satisfies the ODE by differentiating twice and substituting — a key FRQ task. The angular frequency ω, amplitude A, and phase constant φ each carry physical meaning that students are expected to interpret.
Period of Springs and Pendulums
For a spring-mass system, the period is T = 2π√(m/k). For a simple pendulum undergoing small-angle oscillations, the linearised equation of motion gives T = 2π√(L/g). The small-angle approximation (sin θ ≈ θ in radians) is derived explicitly in AP Physics C: Mechanics using the differential equation approach, not simply stated as a given.
Phase Relationships Between x, v, and a
Since v(t) = dx/dt = -Aω sin(ωt + φ) and a(t) = dv/dt = -Aω² cos(ωt + φ), the velocity leads the displacement by 90° and the acceleration is exactly opposite in phase to the displacement. These derivative relationships are central to AP Physics C: Mechanics questions that ask about the relative phases of x, v, and a at specific points in the oscillation.
Energy in SHM
The total mechanical energy in SHM is constant: E = (1/2)kA². The kinetic and potential energies oscillate out of phase — when x is maximum, v = 0 and all energy is potential; when x = 0, all energy is kinetic. AP Physics C: Mechanics problems on SHM energy ask students to derive the speed at an arbitrary position using energy conservation and to sketch or interpret energy-position graphs.
Key AP Skills for Unit 7
- Deriving the general SHM solution by verifying it satisfies the differential equation.
- Determining ω, T, A, and φ from given initial conditions.
- Finding velocity and acceleration functions by differentiating x(t).
- Applying energy conservation in SHM to find speed at arbitrary positions.
- Setting up the differential equation for a physical pendulum or other SHM system.
- Always verify your x(t) solution by substituting back into the ODE — examiners reward this step explicitly.
- Phase constant φ is determined by initial conditions: x(0) = A cos(φ) and v(0) = -Aω sin(φ).
- Do not assume φ = 0 unless the object starts at maximum displacement.