Centripetal force calculator

What is centripetal force?

The net force that keeps an object moving in a circle. It points toward the center of the circle and equals F = m·v² ÷ r, where m is the mass, v the speed and r the radius (OpenStax University Physics).

Example: a car on a curve

A 900 kg car at 25 m/s (about 56 mph) on a curve with a 500 m radius needs 900 × 25² ÷ 500 = 1,125 N of inward force (an OpenStax example). Its centripetal acceleration is 1.25 m/s², about 0.13 g.

Is centripetal force a separate kind of force?

No — it is the role played by some real force. On a flat road it is friction between the tires and the road, for a ball on a string it is the tension, and for an orbiting moon it is gravity.

How do speed and radius change the force?

Force grows with the square of speed: twice as fast needs four times the force. For the same speed, half the radius needs twice the force.

What about centrifugal force?

It is not a real force acting on the object. It appears only when motion is described from a rotating frame of reference; seen from outside, there is just the inward centripetal force.

How do I use revolutions per minute instead of speed?

Convert to the speed along the circle: v = 2π r × rpm ÷ 60. Equivalently, use F = m ω² r with ω in radians per second, ω = rpm × 2π ÷ 60. A 0.15 kg object turning at 60 rpm on a 1.2 m radius has ω = 6.28 rad/s and F = 0.15 × 6.28² × 1.2 = 7.1 N.

What happens to the force if the string breaks?

The inward force disappears and the object moves off in a straight line along the direction it was moving at that instant, tangent to the circle, not outward away from the center.

How do I convert km/h or mph to meters per second?

Divide km/h by 3.6 to get m/s, and multiply mph by 0.44704. For example 90 km/h is 25 m/s, and 55 mph is 24.6 m/s.

How the calculation works

For motion at constant speed around a circle, the acceleration points to the center and has size a = v² ÷ r. Newton's second law then gives the net inward force F = m a = m v² ÷ r. The calculator also reports the angular velocity ω = v ÷ r and the time for one lap, 2π r ÷ v. It does not name the force: on a real system some actual force, such as tension, friction, a normal force or gravity, supplies the centripetal force.

Worked example

A 0.15 kg ball is swung on a string in a circle of radius 1.2 m at 6 m/s. The acceleration is 6² ÷ 1.2 = 30 m/s², which is 3.1 g. The force in the string is 0.15 × 30 = 4.5 N. The angular velocity is 5 rad/s and one lap takes 1.26 s. If the speed doubles to 12 m/s, the force goes up to 18 N, four times as much. If the radius is halved at the original speed, the force doubles to 9 N.

For a 1,200 kg car at 90 km/h (25 m/s) on a curve of radius 150 m, the friction needed is 1,200 × 25² ÷ 150 = 5,000 N, or 0.42 g of acceleration.

Common mistakes

• Using km/h or mph directly. The formula in newtons needs the speed in meters per second.
• Using the diameter as the radius. r is the distance from the center of the circle to the object.
• Adding centripetal force as another force on the object. It is the sum of the real forces pointing to the center, not an extra one. Draw the real forces first.

Sources and related calculators

Formula and examples: OpenStax University Physics Volume 1, section 6.3 (Centripetal Force) and its treatment of uniform circular motion. Related: angular velocity calculator, buoyancy calculator and acceleration calculator.