Unit 6: Simple Harmonic Motion

Executive Summary

Simple harmonic motion is oscillation driven by a restoring force proportional to displacement — springs and pendulums are the two canonical cases.

The essentials

For a mass on a spring, $F = -kx$ and the period is $T = 2\pi\sqrt{\frac{m}{k}}$ — independent of amplitude. For a pendulum (small angles), $T = 2\pi\sqrt{\frac{L}{g}}$ — independent of mass and amplitude. Frequency is $f = \frac{1}{T}$, and $\omega = 2\pi f$.

Energy in SHM

Total energy $E = \frac{1}{2}kA^2$ is constant, swapping between kinetic (max at equilibrium, where speed is max) and potential (max at the extremes). Maximum speed $v_{max} = A\omega$.

Graphs

Position, velocity, and acceleration are sinusoids: velocity leads position by a quarter cycle, acceleration is opposite in phase to position. Acceleration is always toward equilibrium and proportional to displacement.

Exam traps

Spring period does NOT depend on amplitude — doubling A changes energy, not T. Pendulum period does NOT depend on mass. The maximum speed occurs at equilibrium, maximum acceleration at the endpoints. For springs in series/parallel, only AP-1-adjacent qualitative questions appear — focus on the standard single-spring cases.