SPRING
A circle’s shadow — the phasor shows why everything oscillates in cosines.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A block on a frictionless table, tied to a wall by a spring. Pull it out a distance A and let go. The spring’s complaint is proportional to the offense: F = −kx.
Force
The minus sign is the whole story — the force always points home. Push the block right, the spring pulls left; the farther out, the harder the pull. Newton turns that into a = −(k/m)x: acceleration proportional to displacement, and opposite.
Solve
That equation has one native motion: the cosine. Its clock rate is ω = √(k/m) — stiffer springs tick faster, heavier masses slower — and the phasor shows why: SHM is a circle’s shadow, uniform rotation seen edge-on.
Oscillate
Two full cycles. Watch the phase parade: velocity peaks where x crosses zero, acceleration mirrors x exactly. The dot on the circle never speeds up or slows down — only its shadow does.
Energy
The energy ledger closes to the last digit: ½mv² + ½kx² = ½kA² at every instant, kinetic and potential trading twice per cycle. After 2T the block is back at A, at rest, books balanced — ready to repeat forever.