simple harmonic motion
/ SHM /
Imagine a frictionless world: a mass on a spring, pulled aside and released, would swing back and forth forever, tracing a perfect, never-fading sine wave. That smooth, endlessly repeating, lossless wobble is simple harmonic motion. It is the purest vibration there is — the shape a tuning fork's tip would trace if it never ran out of energy.
Drop the damping (c = 0) and the force (F = 0) from the spring equation and you are left with m x'' + k x = 0, or x'' = -(k/m) x. In words: acceleration is always pointed back toward the centre and is proportional to how far you have strayed. Every solution is x(t) = A cos(omega t - phi), where omega = sqrt(k/m). The motion is a single cosine of fixed amplitude A, fixed frequency omega, and a starting phase phi — it neither grows nor decays.
Simple harmonic motion is the idealized backbone everything else is measured against. Add a little damping and the sine slowly shrinks; add forcing and you ask how the system answers being pushed. The reason it appears everywhere — pendulums for small swings, atoms in a crystal, the bob of a clock — is that near any stable balance point, the restoring force is approximately proportional to displacement, so nature is full of approximate simple harmonic motion.
For m = 1 and k = 9, the equation x'' + 9 x = 0 with x(0) = 2, x'(0) = 0 gives x(t) = 2 cos(3t): amplitude 2, angular frequency 3, forever.
No damping means the amplitude 2 never changes — a perfect, undying oscillation.
True simple harmonic motion is a frictionless idealization that no real system achieves — every real vibration loses a little energy and decays. It is still the right first picture, because for short times and light damping the motion is very nearly a clean sine.