Part of JWAVE-01 — Simple Harmonic Motion

The SHM Equation and Its Three Parameters

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Every SHM is fully described by three parameters: amplitude AA, angular frequency ω\omega, and initial phase ϕ\phi. The displacement equation x=Asin(ωt+ϕ)x = A\sin(\omega t + \phi) contains all information. From it, velocity v=dx/dt=Aωcos(ωt+ϕ)v = dx/dt = A\omega\cos(\omega t + \phi) and acceleration a=d2x/dt2=Aω2sin(ωt+ϕ)a = d^2x/dt^2 = -A\omega^2\sin(\omega t + \phi). The defining condition is a=ω2xa = -\omega^2 x, meaning acceleration is always directed toward the mean position and proportional to displacement. Any system satisfying this condition executes SHM, regardless of whether the restoring force comes from a spring, gravity, or any other source.

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