Oscillations

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.

Lessons in this unit

• The SHM Differential Equation14 min · 3 objectivesRecognize simple harmonic motion from the equation d²x/dt² = −ω²x · Verify that x(t) = A cos(ωt + φ) satisfies the SHM equation · Identify the amplitude, angular frequency, period, and phase of an oscillation
• Mass-Spring Systems13 min · 3 objectivesApply Hooke's law and derive the period of a mass-spring oscillator · Compute the angular frequency, period, and frequency for a spring system · Explain why the period is independent of amplitude
• Pendulums13 min · 3 objectivesDerive the simple pendulum's SHM using the small-angle approximation · Compute the period of a simple pendulum · Apply the physical-pendulum period formula T = 2π√(I/mgd)
• Energy in SHM13 min · 3 objectivesDescribe the continuous exchange between kinetic and potential energy during SHM · Apply the total-energy relation E = ½kA² for a spring oscillator · Compute the maximum speed and the speed at a given displacement

Formulas in Unit 7

Every term in Unit 7

All 14 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. The simple pendulum is the special case I = md².

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.

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.

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.

Questions about this unit

All 7 units of AP Physics C: Mechanics

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations