Simple harmonic motion

The defining condition is a linear restoring force, written as F = −kx. The negative sign means the force always points back toward equilibrium, and the proportionality to displacement means the force grows the farther the object moves away. Because acceleration is proportional to displacement, the resulting motion is sinusoidal in time: x = A cos(ωt + φ), where A is the amplitude and ω is the angular frequency.

For a mass on a spring, the period is T = 2π√(m/k) — it depends on mass and spring constant but not on amplitude. For a simple pendulum, T = 2π√(L/g), which depends on length and gravitational acceleration but neither on mass nor, for small angles, on amplitude. The pendulum is only approximately harmonic; the small-angle approximation sin θ ≈ θ is what makes the restoring force linear. This amplitude independence, called isochronism, is what made pendulums useful as clocks.

Energy analysis makes the motion easy to reason about. Total mechanical energy stays constant at E = ½kA², shifting entirely to potential energy at the turning points where speed is zero and entirely to kinetic energy at equilibrium where speed is maximum. Maximum speed is v = Aω and maximum acceleration is a = Aω², both occurring at opposite ends of the cycle.

AP Physics 1 devotes an oscillations unit to simple harmonic motion, covering the spring and pendulum period formulas and energy in SHM. The MCAT covers the same material more briefly within translational motion, forces, work, energy, and equilibrium, where SHM connects to periodic motion and wave characteristics.