Oscillations: every key term you need (+ practice quiz)

More AP Physics C guides

• Kinematics
• Newton's Laws of Motion
• Work, Energy and Power
• Systems of Particles and Linear Momentum
• Rotational Motion

Simple Harmonic Motion

Periodic motion with restoring force proportional to displacement (F = -kx); sinusoidal position/velocity/acceleration.

Amplitude

Maximum displacement from equilibrium; determines energy of oscillation; remains constant (no damping).

Period

T; time for one complete oscillation; T = 1/f; independent of amplitude (isochronous).

Frequency

f = 1/T; number of oscillations per unit time; measured in hertz (Hz); f = ω/(2π).

Angular Frequency

ω = 2πf = √(k/m); rate of angle change; related to spring constant and mass.

Restoring Force

F = -kx; force toward equilibrium; proportional to displacement; opposite direction; creates SHM.

Spring Constant

k; measure of spring stiffness; F = kx; larger k → stiffer spring → higher frequency.

Kinetic Energy

KE = ½mv²; maximum at equilibrium (zero PE); oscillates between zero and maximum.

Potential Energy

PE = ½kx²; maximum at amplitude (zero KE); oscillates between zero and maximum.

Total Mechanical Energy

E = KE + PE = ½kA²; constant in SHM (conservative force); proportional to amplitude squared.

Damping

Energy dissipation reducing amplitude; caused by friction/resistance; exponential decay; quality factor Q.

Resonance

Driven oscillation at natural frequency; maximum amplitude; energy transfer most efficient; dangerous in structures.

Pendulum

Simple: T = 2π√(L/g); period independent of mass; approximately SHM for small angles (<15°).

Physical Pendulum

Rigid body rotating about point; T = 2π√(I/(mgd)); depends on moment of inertia and distance to COM.

Wave

Disturbance propagating through space; carries energy/momentum; described by wavelength, frequency, speed.

Transverse Wave

Particle motion perpendicular to wave propagation; example: electromagnetic, string vibrations.

Longitudinal Wave

Particle motion parallel to wave propagation; example: sound waves, compression waves.

Wavelength

λ; distance between adjacent crests (or troughs); v = fλ (wave speed = frequency × wavelength).

Wave Speed

v = √(T/μ) for string (T = tension, μ = linear mass density); depends on medium properties.

Superposition

Principle that when waves overlap, displacements add (constructively or destructively).

SHM Differential Equation

d²x/dt² = −ω²x; any system whose restoring force is linear in displacement obeys it with ω = √(k_eff/m_eff).

General Solution

x(t) = A cos(ωt + φ); A and φ set by initial position and velocity: A = √(x₀² + (v₀/ω)²).

Velocity and Acceleration in SHM

v = −Aω sin(ωt + φ), a = −Aω² cos(ωt + φ); v_max = Aω, a_max = Aω².

Speed at Position x

From energy: v = ω√(A² − x²); speed is maximum at equilibrium and zero at the turning points.

Vertical Spring

Gravity only shifts equilibrium by mg/k; the period 2π√(m/k) is unchanged.

Springs in Series and Parallel

Parallel: k_eff = k₁ + k₂; series: 1/k_eff = 1/k₁ + 1/k₂; series is softer, giving a longer period.

Physical Pendulum

T = 2π√(I/(Mgd)) with d the pivot-to-CM distance; reduces to a simple pendulum when I = ML², d = L.

Rod Pendulum

A uniform rod pivoted at its end: T = 2π√(2L/(3g)); it behaves like a simple pendulum of length 2L/3.

Torsion Pendulum

τ = −κθ gives ω = √(κ/I); period independent of amplitude and of g.

Energy in SHM

E = ½kA² = ½mv² + ½kx²; average KE and PE over a cycle are each E/2.

Small-Angle Approximation

sinθ ≈ θ valid for small θ (error ~1% at 14°); pendulum period grows slightly at larger amplitudes.

Effective Spring Constant from U(x)

For any potential minimum, k_eff = d²U/dx² at x₀; period of small oscillations = 2π√(m/k_eff).

Phase Constant

φ from initial conditions: x₀ = A cosφ, v₀ = −Aω sinφ; starting at rest at +A gives φ = 0.

Floating Object Oscillation

Buoyancy restoring force ρgA_cross·y gives ω = √(ρgA_cross/m), the same form as a spring.

Damped Oscillation (Qualitative)

A velocity-dependent drag force reduces amplitude over time; light damping barely changes the period.