Circular Motion Calculator
Work out centripetal acceleration and force for anything moving in a circle, from its speed, spin rate or lap time. Also finds the grip a car needs on a flat curve and the design speed of a banked one.
How to Use
- Pick a mode: Acceleration (a and F from the speed), Speed or Radius (from the other two), Flat curve (the grip needed) or Banked curve (the design speed).
- In Acceleration mode, say how you know the speed: as a speed, as a spin rate in rpm or Hz, or as the time for one lap.
- Enter the values with a unit for each. Add a mass to get the centripetal force, and a friction coefficient on the curve modes to check whether the tyres hold.
- Read the answer in the highlighted field and the readouts; the table under the drawing gives the speed, angular speed, lap time and acceleration in several units.
- Press a preset to load an example, and follow each step in Show Work.
Worked Example
A car on a bend. A 1,200 kg car at 20 m/s (72 km/h) round a bend of radius 50 m: a = v² ÷ r = 20² ÷ 50 = 8 m/s², which is 8 ÷ 9.80665 = 0.8158 g. The tyres must supply F = m × a = 1,200 × 8 = 9,600 N towards the centre. A full lap would take 2π × 50 ÷ 20 = 15.71 s.
Is the grip enough? On a flat road the friction needed is μ = a ÷ g = 0.8158. Dry asphalt gives roughly 0.7, so at 20 m/s this bend is too tight; the most it allows is √(0.7 × 50 × 9.80665) = 18.53 m/s.
The common mistake: leaving the speed in km/h. Plugging 72 straight into v² ÷ r gives 72² ÷ 50 = 103.68, nearly 13 times the real 8 m/s². Convert first: 72 km/h ÷ 3.6 = 20 m/s. Using the diameter of the bend instead of the radius is the other classic slip; it halves the answer.
Show Work
Formulas
Huygens, Newton and the Inward Pull
Christiaan Huygens worked out the size of the force needed to hold a body on a circle, proportional to mv²/r, in his treatise De Vi Centrifuga, written in 1659 and published only in 1703, after his death. He described it from the point of view of the spinning body, as a centrifugal tendency to fly outwards.
Isaac Newton turned the picture round. In the Principia of 1687 he coined the term vis centripeta, the centre-seeking force, and showed that the same inward pull, gravity, keeps the Moon on its orbit. Nothing pushes an orbiting or cornering body outwards; it simply keeps trying to go straight, and the inward force bends its path. A satellite is circular motion with gravity as the centripetal force, which the Gravity & Orbit Calculator works out.
About This Tool
This calculator handles uniform circular motion end to end: centripetal acceleration and force from a speed, a spin rate or a lap time, the speed or radius for a given acceleration, the grip a car needs on a flat curve, and the design speed and safe speed range of a banked curve. Each value takes its own unit, including rpm, g and mph, and the working shows every conversion. The drawing shows the object going round, with its velocity along the path and its acceleration towards the centre, or the forces on a car on a banked road.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Force Calculator, Torque Calculator, and Physics Playground.
Frequently Asked Questions
What is the formula for centripetal acceleration?
a = v² ÷ r, or a = ω²r with the angular speed ω in radians per second. A car at 20 m/s round a bend of radius 50 m has a = 20² ÷ 50 = 8 m/s², 0.82 g, always pointing towards the centre of the bend.
How do I calculate centripetal force?
Multiply the acceleration by the mass: F = m × v² ÷ r. For a 1,200 kg car at 20 m/s on a 50 m bend, F = 1,200 × 8 = 9,600 N, supplied by the tyres. Because speed is squared, doubling it to 40 m/s needs four times the force, 38,400 N.
How do I convert rpm to g-force for a centrifuge?
Turn rpm into radians per second (× 2π ÷ 60), then a = ω²r. At 3,000 rpm, ω = 314.16 rad/s; at a radius of 10 cm, a = 314.16² × 0.1 = 9,870 m/s², which is 1,006 g. Labs quote this as RCF, relative centrifugal force.
How fast can a car take a flat curve?
Until the friction needed, μ = v² ÷ (r × g), reaches the grip available. On a 100 m radius with dry-road grip of about 0.7, the limit is √(0.7 × 100 × 9.80665) = 26.2 m/s, 58.61 mph. On a wet road at about 0.4 it falls to 44.3 mph.
Why are curves banked?
Tilting the road lets the normal force supply part of the centripetal force. At the design speed v = √(r × g × tan θ) no sideways friction is needed at all: a 20° bank on a 200 m radius has a design speed of 26.72 m/s, 96.19 km/h (59.77 mph).
How do I use the Circular Motion Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Road and track design
A 20° bank on a 200 m curve suits 96.19 km/h with no side friction; a flat 100 m curve at 60 mph needs a grip of 0.7336.
Lab centrifuges
3,000 rpm at a 10 cm radius gives 9,870 m/s², about 1,006 g, the relative centrifugal force most protocols ask for.
Washing machines
A 1,400 rpm spin in a drum of 25 cm radius pushes the clothes outwards at about 548 g, which is what wrings the water out.
Roller coaster loops
At the top of a 10 m radius loop a car must go at least √(g r) = 9.903 m/s (35.65 km/h) to stay on the track without help.
Driving and karting
Holding 0.8 g of sideways grip on a 30 m bend allows 55.23 km/h; at 50 mph a comfortable 0.3 g needs a radius of 169.8 m.
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