AP Physics C: Mechanics Unit 7: Oscillations Notes

Introduction

Simple harmonic motion emerges from a linear restoring law, producing sinusoidal motion whose phase, period, and energy follow from a second-order differential equation.

An oscillation is simple harmonic only when acceleration is proportional and opposite to displacement from equilibrium. That physical condition creates the differential equation $\ddot x+\omega^2x=0$ and its sinusoidal family of solutions. Initial conditions choose the amplitude and phase; system parameters choose the natural frequency.

Energy offers a second description of the same motion, while pendulums show where approximation matters. A gravitational pendulum is exactly nonlinear and becomes SHM only at small angle. Physical and torsion pendulums extend the same restoring-torque logic to rigid bodies.

Linear restoring law

SHM differential equation

System parameters

natural frequency and period

Initial conditions

amplitude and phase

Position in the cycle

kinetic and potential energy

Small-angle or torsion model

pendulum timing

Current exam guidance: Aligned to the current 2026–27 AP Physics C: Mechanics framework, with calculus-based modeling, derivations, experimental reasoning, and explicit assumptions.

7.1

Defining Simple Harmonic Motion (SHM)

What mathematical condition makes an oscillation simple harmonic?

Short answer

SHM requires $a=-\omega^2x$ , so acceleration is linear in displacement from equilibrium and always points back toward equilibrium.

Know firstNewton’s second law · Hooke’s law · Second-order differential equations

The restoring law defines the model

Equilibrium is the position where the net force is zero. Measure $x$ from that position, not from an arbitrary origin. For an ideal spring, $F_x=-kx$ . Newton’s second law gives $m\ddot x=-kx$ , or $\ddot x+(k/m)x=0$ . Comparing with $a=-\omega^2x$ identifies $\omega=\sqrt{k/m}$ . The negative sign makes the acceleration restorative: positive displacement produces negative acceleration and vice versa.

The differential equation admits $x(t)=A\cos(\omega t+\phi)$ because differentiating twice returns $-\omega^2x$ . The constants $A$ and $\phi$ come from the initial position and velocity. They describe which member of the solution family occurs; they do not change the system’s ideal natural frequency.

Periodicity alone is insufficient. A pendulum at large angle, a nonlinear spring, and a damped oscillator may repeat or nearly repeat without satisfying a constant proportionality between acceleration and displacement. At equilibrium the acceleration is zero, but speed is generally largest—not zero.

Physics model

Simple harmonic motion has a linear restoring acceleration

Move the oscillator through equilibrium and compare displacement with its opposite acceleration.

Displacement0.6 m

Displacement: 0.6 m
Acceleration for $\omega=2$: -2.4 m/s²

equilibriumdisplacementrestoring acceleration

a=-\omega^2x

Read this figure without the graphic

Move the oscillator through equilibrium and compare displacement with its opposite acceleration.

Current quantity	Value
Displacement	0.6 m
Acceleration for $\omega=2$	-2.4 m/s²
Model relation	$a=-\omega^2x$

Five THEMIS spacecraft probes mounted together during a vibration test. — A vibration test probes a structure’s oscillatory response
A controlled vibration test drives the spacecraft through specified motions and measures its response. Simple harmonic motion is one ideal model; real structures can have many coupled modes and damping.
Use this visual to see
Place the linear restoring-force model inside a real structural test.
What it cannot show
The image does not show displacement-time data, mode shapes, frequencies, damping, applied force, or whether the motion is simple harmonic.

Read the sinusoidal solution family

System: A mass $m$ attached to an ideal horizontal spring
Frame: Inertial frame with $x=0$ at equilibrium
Assumptions: Linear spring; negligible damping; one-dimensional motion
Givens: $m\ddot x=-kx$

Rewrite the motion law in the SHM form $\ddot x=-\omega^2x$ .
Identify $\omega^2=k/m$ from the coefficient of displacement.
Use initial position and velocity to select $A$ and $\phi$ from the known sinusoidal solution family.

Result

x=A\cos(\omega t+\phi),\qquad \omega=\sqrt{\frac{k}{m}}

Units:

x

and

A

in m,

\omega

in rad/s,

k

in N/m

Boundary cases

$A=0$ is the equilibrium solution.
A force containing significant $x^2$ or $x^3$ terms is not exact SHM.

The spring equation identifies the natural angular frequency, while the initial state selects one cosine curve from the solution family.

Classify an oscillator

Locate equilibrium and define displacement from it.
Write net force or acceleration as a function of displacement.
Check for a straight line through the origin with negative slope.
Identify the physical range over which that linear law holds.
Read $\omega^2$ from the magnitude of the acceleration–position slope.

Keep these three ideas

SHM is defined by a linear, opposite acceleration–displacement relation.
The sinusoidal solution follows from the second-order differential equation.
Amplitude and phase come from initial conditions, not system stiffness alone.

Oscillations