The equation that turns every wobble into the same wobble
A pendulum, a spring, a bobbing buoy — they look nothing alike, but they are all the same motion wearing different costumes. The disguise runs deep until you write down the acceleration, and then the same equation appears every time. Drag the slider to change the displacement and watch the restoring force respond.
Simple Harmonic Motion
Equilibrium
Displacement x determines acceleration ẍ
Restoring force = 0
Displacement (x)
−A0+A
Acceleration
0
Speed
MAX
The equation ẍ = −n²(x − c) is not a description of simple harmonic motion, it IS the definition. Proving that something moves with SHM means showing that its acceleration takes exactly this form: proportional to displacement from centre c, and always pointing back towards it. The minus sign is what carries the physics — it makes F(x) = mẍ a restoring force, always aimed at the centre, never away from it. That is what makes the motion repeat instead of run away. Every spring, every pendulum, every bobbing object in stable oscillation obeys this single line, and everything else you know about SHM — the sine solution, the period, the energy exchange — follows from pushing this equation through calculus.
Know This
The equation IS the definition: proving motion is simple harmonic means producing the form ẍ = −n²(x − c), and the minus sign carries the physics — acceleration always points back towards centre c, which is what makes the motion repeat instead of run away.