AP Physics C: Mechanics Unit 7: Oscillations 2027
Dedicated Unit 7 chapter hub for 5 current topics. Use it to move through the lesson sequence, connect formulas and labs, and hand off cleanly to the previous or next unit without mixing topic and unit search intent.
Unit 7 map: restoring dynamics, phase, energy, and pendulums
Oscillations are a synthesis unit. Start with dynamics to determine whether the restoring force or torque is linear near equilibrium; then use the resulting differential equation to connect displacement, velocity, acceleration, period, phase, and energy. A repeating motion is not automatically SHM, and a pendulum is only approximately SHM in the small-angle regime.
All five Unit 7 topics
Defining Simple Harmonic Motion
Identify stable equilibrium and test whether the net restoring force is proportional to displacement and directed toward equilibrium.
Current lesson · 7.1 →Frequency and Period of SHM
Derive and compare period relationships for mass-spring and pendulum models, including proportional reasoning.
Open 7.2 →Representing and Analyzing SHM
Move among x(t), v(t), a(t), phase, graphs, extrema, derivatives, and differential-equation representations.
Open 7.3 →Energy of Simple Harmonic Oscillators
Track K and U through the cycle and use energy to relate speed, amplitude, displacement, and turning points.
Open 7.4 →Simple and Physical Pendulums
Use small-angle torque models for simple, physical, and torsional oscillators and keep the pivot-axis inertia explicit.
Open 7.5 →Choose the right Unit 7 model
| Question feature | Best starting model | Key condition | Common mistake |
|---|---|---|---|
| Need to decide whether motion is SHM | Find equilibrium, then test F_net≈−k_eff x or τ≈−κ_effθ | Linear restoring behavior near a stable equilibrium. | Calling every periodic motion SHM. |
| Ideal spring period | T=2π√(m/k_eff) | Linear spring model and effective moving mass identified. | Using the geometric length of the spring as the period variable. |
| Simple pendulum period | T≈2π√(ℓ/g) | Small angular amplitude and point-mass bob model. | Using the formula at large angles as exact. |
| Physical pendulum | T≈2π√(I_p/(mgd)) | I_p is about the actual pivot. | Using I_cm without the parallel-axis shift. |
| Speed at a displacement | Energy: ½kA²=½kx²+½mv² | Ideal conservative spring oscillator. | Forgetting that x is measured from equilibrium. |
Differential-equation and phase map
Dynamics → SHM equation
For a spring, m x¨=−kx gives x¨+(k/m)x=0. Comparing with x¨+ω²x=0 identifies ω=√(k/m). The differential equation is the model; the sinusoid is its solution.
Phase derivatives
If x=A cos(ωt+φ), then v=−Aω sin(ωt+φ) and a=−Aω² cos(ωt+φ)=−ω²x. The graphs are phase shifted because differentiation changes the trigonometric form.
Energy phase
At |x|=A, speed and kinetic energy are zero while potential energy is maximum. At equilibrium x=0, speed and kinetic energy are maximum.
Pendulum linearization
For a physical pendulum, I_p θ¨=−mgd sinθ. With sinθ≈θ, the equation becomes SHM and gives ω=√(mgd/I_p).
Three Unit 7 synthesis problems
Vertical spring: equilibrium shifts but period does not
For a hanging mass, gravity changes the equilibrium position. Measure x from that equilibrium and the net restoring force becomes −kx, so ω=√(k/m). The period is the same as for a horizontal ideal spring even though the equilibrium location differs.
Physical pendulum from rotational dynamics
A rigid body pivots a distance d from its center of mass. Use τ=−mgd sinθ and I_pθ¨=τ. Under the small-angle approximation, derive T=2π√(I_p/(mgd)). Then use I_p=I_cm+md² when the known inertia is about the center of mass.
Infer oscillator parameters from graphs
From an x(t) graph, read A and T; calculate ω=2π/T. Use the initial slope to determine the sign of v(0), then choose a compatible phase φ. Confirm the result against an a(t) graph using a=−ω²x rather than treating the graphs independently.
Laboratory and data-analysis tasks
Spring-period scaling
Measure T for several masses or effective spring constants. Plot T² versus m for fixed k or T² versus 1/k for fixed m and interpret slope.
Motion-sensor phase test
Record x(t) and v(t) for a spring-mass system. Differentiate or compare phase timing to verify the quarter-cycle offset and a∝−x.
Pendulum small-angle boundary
Measure period for increasing amplitudes. Determine the amplitude range in which the small-angle period is supported within experimental uncertainty.
Oscillator energy audit
Use measured x and v to calculate U and K through the cycle. Test whether E=K+U is approximately constant and identify dissipative trends.
Unit 7 traps that reveal model weakness
Periodic ≠ SHM
A repeating motion can be nonlinear. SHM requires a restoring acceleration proportional to −x near equilibrium.
Wrong origin
For spring SHM, displacement is measured from equilibrium, not automatically from the unstretched spring position.
Amplitude-dependent ideal period
For the ideal spring and small-angle pendulum models, period is amplitude independent. A strong amplitude dependence signals model breakdown or nonideal effects.
Phase-sign mistakes
x, v, and a are not all in phase. Use derivatives or a=−ω²x instead of guessing graph shifts.
Pendulum inertia axis
A physical pendulum uses I about the pivot, not about the center of mass unless the pivot is there.
Large-angle pendulum
sinθ≈θ requires small angles in radians; it is an approximation, not an identity.
Where Unit 7 connects across Physics C
Unit 2 · Dynamics
Hooke’s law plus Newton’s second law creates the spring SHM differential equation.
Unit 3 · Energy
Potential-energy curvature around a stable minimum explains why many systems behave approximately like SHM for small displacements.
Units 5–6 · Rotation
Physical and torsion pendulums use torque, rotational inertia, and rotational energy from the rotation units.
E&M Unit 13 · LC circuits
Charge in an ideal LC circuit obeys the same mathematical SHM form, d²q/dt²=−q/(LC), creating a direct Mechanics-to-E&M differential-equation bridge.
Unit 7 mastery checklist
Unit 7 lesson sequence
This dedicated unit hub owns the chapter intent. Every lesson keeps its own canonical topic URL and returns here for unit context.
Topic 7.1 · Defining Simple Harmonic Motion (SHM)
Simple harmonic motion is a special kind of periodic motion in which the net restoring force is proportional to displacement from equilibrium and points toward equilibrium. In one dimension, F_net=-k_eff x, so a=-(k_eff/m)x=-ω²x. The word net matters: individual forces do not each have to be proportional to x as long as their sum produces the linear restoring relation about a stable equilibrium. Periodic motion by itself is not enough.
Open Topic 7.1 →Topic 7.2 · Frequency and Period of SHM
The period T is the time for one complete oscillation, frequency f is cycles per second, and angular frequency ω is the phase rate in radians per second: f=1/T and ω=2πf=2π/T. For an ideal mass-spring oscillator, T=2π√(m/k_eff). For a simple pendulum at small angle, T=2π√(L/g). These periods come from the system’s restoring-force-to-inertia ratio, not from amplitude.
Open Topic 7.2 →Topic 7.3 · Representing and Analyzing SHM
Once a system has been identified as SHM, its displacement can be written x(t)=A cos(ωt+φ) or an equivalent sine form. Differentiation gives v(t)=-Aω sin(ωt+φ) and a(t)=-Aω² cos(ωt+φ)=-ω²x. The extrema and zero crossings of x, v, and a are therefore locked by phase: speed is greatest at equilibrium, acceleration magnitude is greatest at the turning points, and changing amplitude does not change the period of an ideal SHM system. This topic also includes natural frequency and resonance under a sinusoidal drive.
Open Topic 7.3 →Topic 7.4 · Energy of Simple Harmonic Oscillators
In ideal SHM the total mechanical energy is constant while kinetic and potential energy exchange continuously. For a spring–mass oscillator measured from equilibrium, U=½kx² and E=½kA², so K=E-U=½k(A²-x²). At equilibrium U is minimum and K is maximum; at the turning points x=±A, K=0 and U=E. Because E∝A², changing amplitude changes the oscillator’s total energy even though the ideal period remains unchanged.
Open Topic 7.4 →Topic 7.5 · Simple and Physical Pendulums
A physical pendulum is a rigid body oscillating about a fixed pivot. Gravity acts at the center of mass a distance d from the pivot, giving restoring torque τ=-mgd sinθ. For small amplitude, sinθ≈θ and I_p θ¨=-mgdθ, so ω=√(mgd/I_p) and T=2π√(I_p/(mgd)). A simple pendulum is the special point-mass case I_p=mℓ² and d=ℓ, giving T=2π√(ℓ/g). A torsion pendulum instead uses τ=-κθ and T=2π√(I/κ).
Open Topic 7.5 →Unit 7 resource network
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