Centripetal force is the inward pull that keeps an object travelling in a circle instead of flying off in a straight line. Its size is F = mv²/r — mass times speed squared, divided by the radius — and it always points toward the centre. The matching acceleration is a = v²/r. The standout feature is the square on the speed: go twice as fast around the same circle and you need four times the force, which is why fast cornering is so demanding and why orbits, rotors and banked tracks are all designed around this one relationship.
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: the uniform circular motion relation F = mv²/r.
The centripetal force equations
The force and acceleration both point toward the centre of the circle. Speed enters as a square, so it dominates the result; radius is in the denominator, so tighter circles need more force. To solve for speed, take the square root of F·r/m. Remember the centripetal force is whatever real inward force is acting — friction, tension or gravity — and the formula tells you how large that must be to maintain the circle.
Worked example — a ball on a string
Scenario: A 2 kg ball is whirled at 4 m/s in a horizontal circle of radius 2 m. What centripetal force does the string supply?
The string must pull inward with 16 N, giving the ball an acceleration of 8 m/s² toward the centre. Speed it up to 6 m/s and the force jumps to 36 N — the square law at work, multiplying by (6/4)² = 2.25. Tighten the circle to 1 m radius at the original 4 m/s and the force doubles to 32 N. If the string can't supply the required tension, it snaps and the ball flies off tangentially.
Frequently Asked Questions
F = m·v²/r. e.g. 2 kg × (4 m/s)² ÷ 2 m = 16 N, pointing toward the centre.
a = v²/r toward the centre. It changes direction, not speed. Here 16/2 = 8 m/s².
A real inward force: friction (cars), gravity (orbits), tension (string), normal force (banked track).
Speed is squared in F = mv²/r. v → 2v makes v² → 4v², so the force quadruples.
Inversely — tighter circle (smaller r) needs more force. Halving r doubles F.