Simple Harmonic Motion

Simple harmonic motion (SHM) — oscillation in which the acceleration is directly proportional to the displacement from equilibrium, and always directed towards equilibrium.

a = −ω² x

The minus sign is the physics — it says the acceleration always points back towards equilibrium, opposite to the displacement. Without it the equation would describe something accelerating away, which is not an oscillation at all.

The two conditions — both must hold for motion to count as simple harmonic:

acceleration proportional to displacement, and

acceleration directed towards the equilibrium position.

Restoring force — since F = ma, the condition is equivalent to F = −kx: a restoring force proportional to displacement. Hooke's law has exactly this form, which is why a mass on a spring oscillates with SHM.

Equilibrium position — where the resultant force is zero. The oscillator passes through it at maximum speed, not at rest: displacement is zero there, but velocity is greatest.

Amplitude (A) — the maximum displacement from equilibrium.

Angular frequency (ω) —

ω = 2π / T = 2πf

The same symbol and definition as in circular motion, and not a coincidence: SHM is the projection of uniform circular motion onto a diameter, which is where the equations come from.

Isochronous oscillations — the period of SHM does not depend on the amplitude. A pendulum swinging through a small arc and one swinging through a larger arc take the same time per swing.

Example: this is what made pendulum clocks possible. As the swing gradually dies away, the period stays the same, so the clock keeps time even as the amplitude falls — a result that would be useless if the period depended on how far it swung.