Oscillations

Name: AP Physics C: Mechanics Unit 7: Oscillations
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10–15% of the exam6 lessons · 81 min36 terms

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What this unit covers

The topics below follow the published Physics C: Mech course framework for Unit 7. This unit is worth 10–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.

SHM differential equationSpringsPendulumsEnergy in SHM

Lessons in this unit

Formulas in Unit 7

The SHM equation and its solution

d²x/dt² = −ω²x x(t) = A cos(ωt + φ)

A is the amplitude, ω the angular frequency, φ the phase constant set by initial conditions. The period is T = 2π/ω.

Period, frequency, and angular frequency

T = 2π/ω f = 1/T ω = 2πf

ω is in rad/s, f in hertz, T in seconds. Together they describe how fast the oscillation repeats, independent of amplitude.

Mass-spring oscillator

ω = √(k/m) T = 2π√(m/k) f = 1/T

Angular frequency rises with stiffness and falls with mass. The period depends only on m and k — not on how far the spring is pulled.

Simple pendulum

ω = √(g/L) T = 2π√(L/g)

Valid in the small-angle limit sin θ ≈ θ. Remarkably, the period depends on length and g only — not on the bob's mass or (for small swings) the amplitude.

Physical pendulum

T = 2π√(I / (mgd))

I is the moment of inertia about the pivot; d is the pivot-to-center-of-mass distance. Reduces to the simple-pendulum result when the mass is concentrated at distance L.

Energy in SHM

E = ½kA² = ½kx² + ½mv² v_max = Aω = A√(k/m)

Total energy is fixed by the amplitude. The speed is greatest at equilibrium (v_max = Aω) and zero at the turning points.

Physical, simple and torsional pendulums

T = 2π√(I/Mgd) simple: T = 2π√(L/g) torsional: T = 2π√(I/κ)

d is the distance from the pivot to the center of mass. The simple pendulum is the special case I = ML² and d = L, which reduces the first formula to the second.

Damped oscillation

x(t) = A₀e^(−bt/2m) cos(ω′t + φ) ω′ = √(ω₀² − (b/2m)²) E ∝ A² so E decays as e^(−bt/m)

Because energy goes as amplitude squared, it decays at twice the rate of the amplitude. The damped frequency ω′ is always below the natural frequency ω₀ and falls to zero at critical damping.

Every term in Unit 7

All 36 terms we publish for Oscillations, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Physical pendulum: T = 2π√(I/mgd), where d is the distance from pivot to center of mass. A uniform rod about its end gives T = 2π√(2L/3g), shorter than a simple pendulum of the same length.
Defining condition for SHM: A restoring force proportional to displacement, F = −kx, giving d²x/dt² = −(k/m)x — a differential equation whose solution is sinusoidal.
General solution for SHM: x(t) = A cos(ωt + φ), with ω = √(k/m). Amplitude and phase are set by the initial conditions.
Angular frequency vs frequency: ω = 2πf = 2π/T. Substituting f for ω is a common and costly slip.
Velocity and acceleration in SHM: v = −Aω sin(ωt + φ) and a = −Aω²cos(ωt + φ) = −ω²x. Maximum speed is Aω and maximum acceleration Aω².
Energy in SHM: E = ½kA², constant. It divides between ½kx² and ½mv², exchanging completely twice per cycle.
Small-angle approximation: sin θ ≈ θ in radians, which is what makes the pendulum equation linear and the motion simple harmonic. It fails at large amplitude.
Damped oscillation: A resistive force proportional to velocity gives exponentially decaying amplitude with slightly reduced frequency.
Recognizing SHM from a differential equation: Any equation of the form d²x/dt² = −(constant)x describes SHM, and ω is the square root of that constant.
Deriving the pendulum period: Torque −mgL sin θ ≈ −mgLθ gives Iα = −mgLθ, so ω = √(mgL/I) — which reduces to √(g/L) for a simple pendulum.
Effect of amplitude on a real pendulum: The small-angle approximation fails at large amplitude, and the true period is slightly longer than 2π√(L/g).
Energy method for finding ω: Write total energy as a function of x and dx/dt, differentiate with respect to time and set to zero. Recovers the equation of motion without free-body diagrams.
Springs in series and parallel: Parallel springs add stiffness (k_eff = k₁ + k₂); series springs combine reciprocally, so the combination is softer than either.
Phase constant from initial conditions: φ is fixed by x(0) and v(0). Released from rest at maximum displacement gives φ = 0 with a cosine.
The condition for simple harmonic motion: A restoring force proportional to displacement, F = −kx, giving d²x/dt² = −ω²x. Any system reducing to that equation oscillates sinusoidally.
Angular frequency of a mass–spring system: ω = √(k/m), so a stiffer spring or a lighter mass oscillates faster. Note it is the square root, so quadrupling the mass only halves the frequency.
Period independent of amplitude: For true SHM, T = 2π√(m/k) contains no amplitude. This isochronism is what made pendulum clocks possible and is exactly what fails at large pendulum amplitudes.
Displacement, velocity and acceleration in SHM: x = A cos(ωt + φ), v = −Aω sin(ωt + φ), a = −Aω² cos(ωt + φ) = −ω²x. Each differentiation multiplies the amplitude by ω and advances the phase by 90°.
Phase relationships: Velocity leads displacement by a quarter cycle and acceleration is exactly out of phase with displacement. Velocity is maximum where displacement is zero and vice versa.
Maximum speed in SHM: v_max = Aω, occurring at the equilibrium position where all the energy is kinetic.
Maximum acceleration in SHM: a_max = Aω², occurring at the extremes of the motion where the restoring force is largest and the speed is zero.
Total energy in SHM: E = ½kA² = ½mv_max², constant and proportional to the SQUARE of the amplitude. Doubling the amplitude quadruples the energy.
Where kinetic and potential energy peak: K is maximum at equilibrium and zero at the extremes; U is the reverse. Each is at half its maximum when x = A/√2.
Average energy over a cycle: Averaged over a full cycle, kinetic and potential energy are each exactly half the total. This equipartition is a general feature of harmonic systems.
Vertical mass–spring equilibrium: The hanging equilibrium sits x₀ = mg/k below the natural length. Measuring displacement from there removes gravity from the equation of motion entirely.
Period of a vertical spring: T = 2π√(m/k), identical to the horizontal case, because gravity only shifts the equilibrium and does not change the restoring constant.
Simple pendulum period: T = 2π√(L/g) for small amplitudes. It depends on length and local gravity only.
Pendulum period independent of mass: The mass appears in both the restoring force and the inertia and cancels, which is why a heavy and a light bob on equal strings keep the same time.
Physical pendulum period: T = 2π√(I/Mgd), with d the pivot-to-center-of-mass distance. It reduces to the simple pendulum when I = ML² and d = L.
Torsional oscillator: T = 2π√(I/κ) with κ the torsion constant of the wire. Gravity does not appear, so a torsional clock keeps the same time on the moon.
Validity of the small-angle approximation: sin θ ≈ θ in radians. The period error is about 0.2% at 10° and grows past 4% by 45°, so "small" means a few degrees for precision work.
Amplitude dependence of a real pendulum: Because the restoring torque goes as sin θ rather than θ, a real pendulum's period INCREASES with amplitude. Claiming amplitude independence without qualification is the trap.
Damped oscillation solution and its envelope: With F = −bv, x = A₀e^(−bt/2m) cos(ω′t + φ): sinusoidal motion inside an exponentially decaying envelope, at a frequency slightly below the natural one.
Energy decay in a damped oscillator: Since E ∝ A², energy decays at twice the exponential rate of the amplitude. When the amplitude has halved, the energy is a quarter of its original value.
Underdamped, critically damped, overdamped: Underdamped oscillates with decaying amplitude; critically damped returns to equilibrium fastest without overshoot; overdamped returns without oscillating but more slowly. More damping past critical means slower settling, not faster.
Resonance: A driven oscillator responds most strongly when the driving frequency matches its natural frequency. Light damping gives a tall narrow peak, heavy damping a low broad one, and zero damping an unbounded response.

What examiners penalize here

State the small-angle approximation explicitly on free-response: sin θ ≈ θ is what turns d²θ/dt² = −(g/L)sin θ into SHM. Remember the simple-pendulum period is independent of mass, and use the physical-pendulum formula T = 2π√(I/mgd) whenever the swinging object is an extended body.
Resonance is defined by the **driving** frequency matching the natural frequency, not by any property of the driving force's strength. A weak force applied at resonance can produce a larger response than a strong force applied far from it — which is the whole reason resonance is worth a name.

Practice Physics C: Mech

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

168 practice questionsEvery one with a written explanation FRQ trainerWritten answers scored against the rubric Physics C: Mech study guideAll units, weights and score data in one place All 7 Physics C: Mech unitsThe full course

Questions about this unit

How much of the AP Physics C: Mechanics exam is Unit 7?

Unit 7, Oscillations, is worth 10–15% of the Physics C: Mech multiple-choice section according to the published course framework. Across all 7 units that makes it a substantial share — heavier than an even split would give it.

What topics are covered in Physics C: Mech Unit 7?

Oscillations covers SHM differential equation, Springs, Pendulums and Energy in SHM. We publish 36 terms with definitions for this unit, all of them on this page.

How should I study Physics C: Mech Unit 7?

Read the 6 lessons below first — about 80 minutes — then drill the 36 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.

All 7 units of AP Physics C: Mechanics

Unit names, topics and exam weights follow the published College Board course framework for AP Physics C: Mechanics. AP® is a trademark registered by the College Board, which does not endorse this site.