Uniform Circular Motion Calculator
Enter a radius, speed and mass — get angular velocity, period, frequency, centripetal acceleration and force at once, with a live 3D orbit and charts to match.
Reviewed by the ToolNestr Editorial Team — July 2026
Enter radius, speed and mass — everything updates live.
Live 3D orbit
The ball circles at your radius and speed. The red arrow is the inward centripetal force; the green arrow is the tangential velocity. Drag to orbit the camera.
Circular motion graphs
How uniform circular motion is calculated
The core idea in one line: moving in a circle at constant speed still means constantly accelerating, because the direction of the velocity keeps changing — and that acceleration always points to the centre.
From the radius r and speed v, every other quantity follows:
- ω = v / r — angular velocity (rad/s), how fast the angle sweeps.
- T = 2πr / v = 2π / ω — period (s), the time for one full lap.
- f = 1 / T — frequency (Hz), laps per second.
- a = v² / r = ω²r — centripetal acceleration, directed inward.
- F = m·v² / r = m·ω²r — centripetal force, the inward net force needed.
The centripetal force is never a new force — it's whatever real force happens to point inward: the tension in a whirled string, gravity for an orbiting satellite, friction for a cornering car, or the track's push on a roller-coaster loop.
Worked example 1 — a ball on a string
Given: a 0.20 kg ball is whirled on a 0.75 m string at 4 m/s. Find the acceleration and the string tension.
Worked example 2 — a car on a bend
Given: a 1200 kg car takes a 50 m radius bend at 20 m/s (72 km/h). What friction force must the tyres provide?
Two ideas that trip students up
1. Velocity is tangent, acceleration points in
The green velocity always points along the circle (tangent); the red acceleration always points to the centre. They stay at 90° the whole way round.
2. Tighter circle, harder turn
At the same speed, the small inner circle needs a much bigger inward force than the wide outer one — because a = v²/r grows as the radius shrinks.
Centripetal acceleration at 10 m/s
Same speed, different radius — the tighter the turn, the harder the acceleration.
| Radiusm | Accel. a = v²/rm/s² | In "g"×9.81 |
|---|---|---|
| 2 | 50.0 | 5.1 g |
| 5 | 20.0 | 2.0 g |
| 10 ★ | 10.0 | 1.0 g |
| 50 | 2.0 | 0.2 g |
★ A 10 m radius turn at 10 m/s pulls about 1 g sideways. a = v²/r; "g" column divides by 9.81 m/s².
Where circular motion actually shows up
🛰️ Satellites & orbits
A satellite orbits because gravity supplies exactly the centripetal force its speed and altitude demand. Setting mg-gravity equal to mv²/r gives the orbital speed for any radius.
🏎️ Cornering & banking
Race tracks and highway bends are banked so part of the normal force points inward, supplying centripetal force without relying only on tyre friction — letting cars corner faster and safer.
🎡 Rides & centrifuges
Roller-coaster loops, rotor rides and lab centrifuges all exploit a = v²/r. A centrifuge spinning fast at small radius reaches thousands of g to separate materials by density.
Common misconceptions
"Constant speed means no acceleration."
Acceleration is any change in velocity, and velocity includes direction. In a circle the direction changes every instant, so there's always centripetal acceleration even at steady speed.
"Centrifugal force throws you outward."
There's no real outward force. You feel pushed out because your inertia wants to go straight; the seat or door pushes you inward to make you turn. "Centrifugal" is a fictitious force of the rotating frame.
"If the string breaks, the ball flies outward."
It flies off along the tangent — straight, in the direction it was heading — not radially outward. Removing the centripetal force just lets Newton's first law take over.
"Bigger radius always means faster."
Only at fixed angular velocity (v = ωr). At fixed linear speed, a bigger radius means a gentler turn and smaller acceleration — the opposite effect. Be clear which quantity is held constant.
Formula sources & further reading
Circular-motion kinematics and centripetal force are standard introductory mechanics, traceable to:
- • OpenStax, University Physics Volume 1 — §4.4 "Uniform Circular Motion" and §6.3 "Centripetal Force" (free, peer-reviewed). openstax.org
- • Halliday, Resnick & Walker, Fundamentals of Physics — Chapter 4 (Uniform Circular Motion) and Chapter 6.
- • Serway & Jewett, Physics for Scientists and Engineers — Chapter 6, Circular Motion and Other Applications of Newton's Laws.
Angular velocity is in radians per second. Results are rounded for display.
How to use this calculator
Enter r, v and m
Type the radius, linear speed and mass in SI units (m, m/s, kg).
Read five results
Angular velocity, period, frequency, centripetal acceleration and force all appear at once.
Watch the orbit
Use the radius and speed sliders to see the velocity and force arrows respond in 3D.
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Frequently asked questions
What is uniform circular motion?
Motion in a circle at constant speed. The speed is steady but the velocity is always changing direction, so the object is constantly accelerating — the acceleration points toward the centre and is called centripetal acceleration.
What is centripetal acceleration?
The centre-directed acceleration that keeps an object moving in a circle: a = v²/r = ω²r. Without it the object would fly off in a straight line (its tangential velocity), obeying Newton’s first law.
What is centripetal force?
The net inward force that produces centripetal acceleration: F = mv²/r = mω²r. It is not a new kind of force — it is provided by tension, gravity, friction or the normal force depending on the situation.
What is the difference between speed and angular velocity?
Linear speed v (m/s) is how fast the object moves along the circle; angular velocity ω (rad/s) is how fast the angle sweeps. They are linked by v = ωr, so a bigger radius means more linear speed for the same angular velocity.
Is there such a thing as centrifugal force?
Centrifugal force is a fictitious force that appears only in the rotating frame of the object. In an inertial frame there is just the real inward centripetal force; the outward "push" you feel is your inertia resisting the change in direction.
How are period and frequency related?
Period T is the time for one full revolution; frequency f is revolutions per second, and f = 1/T. Angular velocity ties them together: ω = 2π/T = 2πf.