What this calculator does
Find the inward force needed to keep something moving in a circle, like a car on a curve. An object going around a circle keeps changing direction. So it's always accelerating toward the center, even at a steady speed. The inward force that causes this is the centripetal force: F = m × v² ÷ r. Here m is the mass, v is the speed along the circle and r is the radius. You can also write it with angular velocity, ω (omega), how fast the object turns: F = m × ω² × r.
Centripetal force isn't a new kind of force. It's whatever real force happens to point inward. That could be tension in a string, friction between tires and road, gravity holding a satellite, or the wall of a spinning drum. This calculator finds the force or solves for the mass, speed or radius. It always shows the centripetal acceleration, the acceleration in g (multiples of Earth's gravity), the time for one lap and the matching linear or angular speed.
Where it's used
Why this math matters outside the classroom:
- Road curves. A 1,000 kg car taking a 50 m radius curve at 15 m/s needs 1,000 × 15² ÷ 50 = 4,500 N of grip from its tires.
- Amusement rides. Ride engineers work out the force a spinning swing ride's chains must hold as riders circle at speed.
- Centrifuges. Lab technicians know a centrifuge pushes samples harder at higher speed or with a larger rotor, since force grows with ω²r.
- Satellite orbits. In orbit, gravity supplies the centripetal force, which is why satellites at one height must all travel at the same speed.
- Ball on a string. Students swing a 0.2 kg ball on a 1 m string at 4 m/s and find the string pulls with 0.2 × 16 ÷ 1 = 3.2 N.
Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSA.CED.A.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose what to solve for: force, mass, speed or radius.
- Choose whether you know the linear speed (such as km/h) or the angular velocity (such as rpm).
- Enter the known values with their units.
- Read the result together with the centripetal acceleration and the time for one revolution.
What the inputs mean
- Speed along the circle (v)
- The speed along the circular path. In steady circular motion, it stays the same even though the direction changes.
- Angular velocity (ω)
- Angular velocity is how many radians the object turns through per second. A radian is a unit of angle, and one revolution is 2π radians. So 1 rpm = 2π ÷ 60 ≈ 0.1047 rad/s.
- Radius of the circle (r)
- Measured from the center of the circle to the object. For a car, measure to its center of mass, its balance point.
Formula and method
F = m × v² ÷ r F = m × ω² × r a = v² ÷ r = ω² × r = F ÷ m v = ω × r T = 2π ÷ ω = 2π × r ÷ v
Here a is the centripetal acceleration and T is the period, the time for one lap. π is pi, the circle constant. The calculator first converts every input to standard SI units: kg, m/s or rad/s, and m. So the force comes out in newtons (N) and the acceleration in m/s². The two force formulas match because v = ω × r.
Speed is squared, so doubling the speed means four times the force to hold the same circle. A tighter circle at the same speed also needs more force. Halve the radius, and the force doubles.
Worked example
A 1,200 kg car drives around a flat curve of radius 50 m at 20 m/s (72 km/h). How much sideways grip does it need?
- F = m × v² ÷ r = 1,200 × 20² ÷ 50 = 9,600 N.
- a = F ÷ m = 9,600 ÷ 1,200 = 8 m/s².
- In g: 8 ÷ 9.80665 ≈ 0.8158 g.
- One full lap of the circle would take 2π × 50 ÷ 20 ≈ 15.708 s.
Result: The tires must supply 9,600 N of inward friction. That's a centripetal acceleration of 8 m/s², about 0.82 g.
Understanding the result
The force has to come from somewhere. On a flat road, it comes from tire friction. If the curve needs more force than the friction available, the car slides outward along a straighter path.
The "outward" push you feel in a turning car is your body's tendency to keep going straight. It isn't an outward force acting on you. Seen from inside the turning car, it's described as the centrifugal effect.
Assumptions and limitations
- Steady circular motion (uniform circular motion): constant speed on a circle of fixed radius.
- The object can be treated as a single point at its center of mass.
- Speeds are far below the speed of light.
- When the speed changes, there's also acceleration along the path (tangential acceleration). This calculator doesn't include it.
- Banked curves, loops and cone-shaped pendulum swings need more work. You have to split the real forces (normal force, gravity, tension) into parts to find what supplies the centripetal force.
- Oval or elliptical paths have a changing radius, so the force changes around the path.
Frequently asked questions
What is the difference between centripetal and centrifugal force?
Centripetal force is the real inward force that keeps an object on a curved path. Centrifugal force is an apparent outward force. It only shows up when you describe the motion from inside the spinning frame, such as from a seat in the car.
How do I convert rpm to rad/s?
Multiply by 2π and divide by 60. For example, 60 rpm is one revolution per second, which is 2π ≈ 6.283 rad/s.
Why does doubling the speed need four times the force?
The force depends on v². Going twice as fast means the direction changes twice as quickly. Each change is also twice as big in velocity. Together, that gives four times the acceleration.
Does centripetal force do work?
No. It always points at right angles to the motion. So it changes the direction of travel but not the speed, and the object's kinetic energy (energy of motion) stays the same.
How this calculator is checked
Method: Kinematics of uniform circular motion (a = v²/r = ω²r) with Newton's second law (F = ma). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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