Every time a car rounds a bend, a satellite circles the Earth, or a ball swings on a string, the same physics governs the motion: a net inward force — the centripetal force — continuously redirects the object’s velocity without changing its speed. This calculator applies Newton’s classical formula F = m·v²/r to any circular-motion problem. You choose which of the four variables (force, mass, velocity, or radius) is the unknown and supply the other three; the result appears instantly alongside four secondary outputs that complete the picture of the motion.
The physics behind circular motion
When an object moves in a circle of radius r at constant tangential speed v, its direction changes continuously, meaning its velocity vector changes — and a changing velocity requires a net force by Newton’s second law. That force always points toward the centre of the circle and has magnitude:
F = m · v² / r
The same relationship can be written in terms of angular velocity ω (rad/s), where ω = v/r:
F = m · ω² · r
Both forms are equivalent; the calculator uses the v form internally and derives ω from v/r.
The centripetal acceleration follows directly from Newton’s second law: aₒ = F/m = v²/r, measured in m/s². It is always directed inward. The period of one complete revolution is T = 2πr/v, and the frequency is f = 1/T. All four secondary quantities are computed and displayed automatically every time you change an input.
Worked example — car on a curved road
A 1,200 kg car travels at 20 m/s (72 km/h) around a circular bend of radius 80 m. What centripetal force must friction supply?
- F = m·v²/r = 1 200 × (20)² / 80 = 1 200 × 400 / 80 = 6,000 N
- Centripetal acceleration: aₒ = v²/r = 400/80 = 5 m/s²
- Angular velocity: ω = v/r = 20/80 = 0.25 rad/s
- Period: T = 2π × 80 / 20 = 25.13 s
The 6,000 N sideways friction force is about half the car’s weight (1 200 × 9.81 ≈ 11,772 N), so a dry-road friction coefficient of ~0.5 is just enough — which matches real-world highway design guidelines.
Solve-for-variable feature
Most physics calculators only solve F. This tool rearranges the formula algebraically so you can solve for any variable:
| Solve for | Rearrangement |
|---|---|
| Force F | F = m·v²/r |
| Mass m | m = F·r/v² |
| Velocity v | v = sqrt(F·r/m) |
| Radius r | r = m·v²/F |
Select the unknown from the dropdown. The corresponding input field disappears (because it is the unknown), and the result panel updates in real time as you change the remaining fields.
Formula note
The formula F = m·v²/r is exact for uniform circular motion in classical (Newtonian) mechanics and holds at everyday speeds. At relativistic speeds (v approaching c = 3 × 10⁸ m/s) the relativistic mass correction becomes significant, but for any terrestrial or near-Earth scenario the classical formula is accurate to many significant figures.
Physical constants used: g = 9.81 m/s² (standard gravity, for reference comparisons), h = 6.626 × 10⁻³⁴ J·s (Planck — not directly used here but available in the suite).