Circular Motion Calculator (centripetal force)

Understanding the circular motion calculator — concept and principle

An object moving in a circle at constant speed is still accelerating, because its direction keeps changing. The acceleration points toward the centre, and a matching centripetal force must supply it, through tyre friction, a string's tension, gravity or a track. This calculator gives the centripetal acceleration and force, the acceleration in g, the time for one lap and the angular velocity. It is used for road curves, rides, centrifuges, orbits and physics problems.

The force points to the centre, not along the motion; without it the car would go straight.

How it is calculated

With m in kg, v in m/s and radius r in m:

Centripetal acceleration: a = v² ÷ r (m/s²).
Centripetal force: F = m v² ÷ r (N).
In g: a ÷ 9.80665.
Period: T = 2π r ÷ v (s), the time for one full circle.
Angular velocity: ω = v ÷ r (rad/s).

The principle

In a short time Δt the object turns through an angle ωΔt, and its velocity vector, of length v, turns by the same angle. The change in velocity therefore has size about v × ωΔt and points toward the centre, so the acceleration is vω = v² ÷ r. By Newton's second law, something must provide a net inward force mv² ÷ r. Nothing pushes the object outward; the "centrifugal" feeling is your body's inertia trying to continue in a straight line while the seat or door pushes you inward.

Worked example

Inputs: m = 1,000 kg, v = 20 m/s (72 km/h), r = 50 m.
Acceleration: 20² ÷ 50 = 400 ÷ 50 = 8 m/s².
Force: 1,000 × 8 = 8,000 N.
In g: 8 ÷ 9.80665 ≈ 0.8158 g.
Period: 2π × 50 ÷ 20 ≈ 15.708 s; angular velocity 20 ÷ 50 = 0.4 rad/s.

Each point is the calculator re-run with only “Speed (m/s)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Speed (m/s)	Centripetal acceleration (m/s²)
5	0.5
10	2
15	4.5
20	8
25	12.5
30	18
40	32

How to read the result

On a flat road the needed sideways friction coefficient is at least a ÷ g, here about 0.82. That is near the grip limit of good tyres on dry asphalt and well beyond what a wet or icy road offers, so this curve at 72 km/h would be risky. Because a grows with v², a 10% higher speed needs 21% more grip, while doubling the radius halves the required force. Passengers typically find sideways accelerations above a few tenths of a g uncomfortable, which is why road design keeps them much lower.

Common mistakes

Entering diameter instead of radius.
Using km/h without dividing by 3.6.
Adding a separate centrifugal force in an inertial (ground) frame.
Forgetting that gravity or a bank angle can supply part of the force.

Keep in mind: The model is uniform motion on a perfect circle with constant speed. It ignores tyre slip, load transfer and road camber, and is not a substitute for road, ride or vehicle safety design by qualified professionals.

Frequently asked questions

What is the difference between centripetal and centrifugal force?

Centripetal force is the real inward net force; centrifugal force is an apparent outward force that appears only when you describe motion from the rotating frame.

How do I convert rpm to angular velocity?

ω = rpm × 2π ÷ 60. For example, 60 rpm is about 6.28 rad/s.

What keeps a satellite in a circular orbit?

Gravity supplies the centripetal force, so the orbital speed satisfies GM ÷ r² = v² ÷ r.

Sources: Halliday, Resnick and Walker, Fundamentals of Physics, Wiley · Isaac Newton, Philosophiæ Naturalis Principia Mathematica, 1687

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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