Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and directed toward the equilibrium position: F = -kx. This relationship produces sinusoidal oscillations in time, characterized by amplitude, period, frequency, and phase. SHM is one of the most important models in physics because it describes a vast range of phenomena — from vibrating guitar strings to oscillating atoms in a crystal lattice to the behavior of AC electrical circuits.

The two canonical SHM systems are the mass-spring system and the simple pendulum (for small angles). For a mass on a spring, the period depends only on the mass and the spring constant: T = 2*pi*sqrt(m/k). For a simple pendulum, the period depends on the length and gravitational acceleration: T = 2*pi*sqrt(L/g). In both cases, the period is independent of the amplitude — a property called isochronism. The position, velocity, and acceleration of an SHM oscillator can be described by sinusoidal functions, and the energy continuously converts between kinetic and potential forms while the total mechanical energy remains constant.

AP Physics 1 requires students to analyze SHM quantitatively and qualitatively. Students must derive and apply period equations, describe the energy transformations during oscillation, interpret position-velocity-acceleration graphs, and connect SHM to circular motion. Extensions include damped oscillations (where energy is gradually lost to friction), driven oscillations (where an external periodic force sustains motion), and resonance (where the driving frequency matches the natural frequency, producing maximum amplitude). Understanding resonance is critical for engineering applications from bridge design to radio tuning.

Practice a little. See where you stand.