CIRCULAR
v⃗·a⃗ = 0, bitwise — a force that steers forever and spends nothing.
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 mass rides a circle at steady speed. The speedometer never moves — so is anything accelerating? The compass says yes: velocity is a vector, and this one never stops turning.
Velocity
The velocity arrow stays the same length and refuses to keep the same direction — always tangent, always turning. Changing direction is acceleration as surely as changing speed, and symmetry leaves that acceleration only one place to point: the center.
Solve
The geometry hands over a formula: a_c = v²/r. Double the speed and you owe four times the acceleration; tighten the circle and the bill rises again. Newton converts it to the inward force F_c = mv²/r that the string, road, or planet must supply.
Revolve
Two laps in real time. Watch the pair of arrows: velocity tangent, force inward, locked at ninety degrees the whole way around. The force steers but never pushes along the motion — it turns the velocity without ever feeding it.
Audit
The ledger: |v| constant to the last bit, v⃗·a⃗ = 0 exactly — not approximately, exactly — so the force does no work and KE never changes. A force that acts forever and spends nothing: the cheapest trick in mechanics, and the one that holds up the sky.