One condition, a = −ω²x, captures every simple harmonic oscillation. From it follow the sine solution, the velocity equations, and the way the three graphs line up.
Simple harmonic motion is any oscillation in which the acceleration is proportional to the displacement and directed toward equilibrium: a = −ω²x. The solution is x = x₀ sin ωt, with v = ±ω√(x₀² − x²).
Set an oscillation going and three curves emerge together. The velocity runs a quarter of a cycle ahead of the displacement, and the acceleration is its exact mirror image. That mirror relationship, a opposite to x, is the signature of simple harmonic motion.
With angular frequency ω = 2π/T = 2πf, the displacement is x = x₀ sin ωt (taking x = 0 at t = 0). The velocity is v = v₀ cos ωt, with maximum speed v₀ = ωx₀ at the centre, and more generally v = ±ω√(x₀² − x²). The acceleration follows the defining condition a = −ω²x, greatest at the extremes.
Four quick checks on the defining condition, ω, and the velocity equations. Each correct answer earns XP and lights this skill on your star map.
Simple harmonic motion is defined by the condition:
The angular frequency ω of an oscillation of period T is:
An object in SHM has its maximum speed when its displacement is:
Using v = ±ω√(x₀² − x²), the speed at the amplitude (x = x₀) is:
The minus sign in a = −ω²x is the whole point: the acceleration is a restoring one, opposite to the displacement. Maximum speed ωx₀ occurs at the centre, maximum acceleration ω²x₀ at the extremes; the velocity leads the displacement by a quarter cycle while the acceleration is exactly antiphase with it.
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