Last updated: 2026-06-14

Centripetal Force

Any object moving along a circular path is constantly changing direction, which means it is accelerating even at constant speed. The inward force that produces this acceleration is the centripetal force: $F = \frac{m v^2}{r}$, where $m$ is the mass, $v$ is the speed, and $r$ is the radius of the path. The accompanying centripetal acceleration is $a = \frac{v^2}{r}$. Both point toward the centre of the circle — centripetal means "centre-seeking."

This calculator returns the centripetal force and acceleration for an object given its mass, speed, and the radius of its circular path.

Why circular motion needs a force

Newton's first law says an object moves in a straight line unless a force acts on it. To keep something on a circular path, a force must continually pull it away from that straight-line tendency and toward the centre. Without it, the object would fly off along a tangent. Because the required force depends on the square of the speed, going around a bend twice as fast needs four times the force — the reason tight, fast turns are so demanding.

Formulas

The acceleration depends only on speed and radius, not on mass; the mass enters only when converting that acceleration into a force.

Worked example

A 1000 kg car rounds a bend of radius 50 m at 15 m/s (54 km/h). The centripetal acceleration is:

and the force the road must supply through tyre friction is:

Entering a mass of 1000 kg, a speed of 15 m/s, and a radius of 50 m reproduces these values. If the available friction cannot provide 4500 N, the car cannot hold the curve and slides outward.

What supplies the centripetal force

Centripetal force is not a separate kind of force — it is the role played by whatever real force happens to point toward the centre. For a car it is friction between tyres and road; for the Moon orbiting Earth it is gravity; for a ball whirled on a string it is the string's tension; for an electron in an atom it is the electrostatic attraction to the nucleus. Identifying that real force is the key step in any circular-motion problem.

Centripetal versus centrifugal

The outward "centrifugal force" a passenger feels on a sharp turn is not a true force. It is the sensation of inertia: the body tends to continue in a straight line while the vehicle turns beneath it, so it seems pushed outward. In the ground-based frame there is only the inward centripetal force; the centrifugal effect appears solely in the rotating frame of the passenger.