Circular Motion and Centripetal Force Calculator

Uniform circular motion

When an object moves around a circle at constant speed, the motion is called uniform circular motion. The speed stays the same but the direction changes all the time. Because the direction changes there is an acceleration, and it always points toward the centre of the circle.

The basic quantities

• Period T: the time for one full turn. Frequency f = 1 / T: turns per second.
• Angular velocity: ω = 2 × π / T (rad/s)
• Speed: v = 2 × π × r / T = ω × r
• Centripetal acceleration: a = v² / r = ω² × r
• Centripetal force: F = m × v² / r

What is centripetal force?

Centripetal force is not a new kind of force. It is the name for the net force that keeps the object on the circle. For a stone whirled on a string it is the tension, for a car in a bend it is friction between tyres and road, and for a satellite it is gravity. If that force disappears the object carries on in a straight line along the tangent.

Example

A 1000 kg car takes a bend of 50 m radius at 20 m/s (72 km/h). The centripetal acceleration is 20² / 50 = 8 m/s² and the force required is 1000 × 8 = 8000 N. Tyre friction has to supply it. Doubling the speed makes the required force four times as large.

Limits

The calculation holds while the speed is constant. The tool gives only the force required: whether the road can supply it depends on the coefficient of friction, the banking and the rollover limit of the vehicle, and has to be checked separately.