Centripetal Force Calculator (F = mv² ÷ r)
Find the inward force that keeps an object moving in a circle.
SI units: newtons, kilograms, m/s, metres.
Double the speed and the force quadruples — the curve is a parabola in v.
Centripetal force is F = m × v² ÷ r: the inward force that keeps an object moving in a circle equals mass times speed squared, divided by the radius. A 2 kg object moving at 3 m/s on a 1.5 m radius needs F = 2 × 3² ÷ 1.5 = 12 N directed toward the centre.
What centripetal force is
Anything moving in a circle is constantly changing direction, so it is accelerating toward the centre even at constant speed. The force that produces this inward (centripetal) acceleration is the centripetal force, F = mv² ÷ r. It grows with the square of the speed and shrinks as the circle gets bigger. Once you know the force, you can compare it against whatever supplies it — the tension, gravity, or friction acting on the object.
F = force (N), m = mass (kg), v = speed (m/s), r = radius (m)
Worked example
A 2 kg ball is whirled on a string in a circle of radius 1.5 m at a steady 3 m/s. What inward force must the string provide?
- 1 Write the centripetal force formula. F = m × v² ÷ r, with mass in kilograms, speed in m/s, and radius in metres.
- 2 Substitute the known values. F = 2 kg × (3 m/s)² ÷ 1.5 m = 2 × 9 ÷ 1.5.
- 3 Compute the force. F = 18 ÷ 1.5 = 12 N, directed toward the centre of the circle.
How speed and radius change the force
Starting from 12 N (m = 2 kg, v = 3 m/s, r = 1.5 m); speed dominates because it is squared.
| Change | New value | Resulting force |
|---|---|---|
| Baseline | v = 3 m/s, r = 1.5 m | 12 N |
| Double the speed | v = 6 m/s | 48 N (×4) |
| Double the radius | r = 3 m | 6 N (÷2) |
| Double both | v = 6 m/s, r = 3 m | 24 N (×2) |
It always points to the centre
Centripetal force is not a new kind of force. It is the name for whatever net force happens to point toward the centre and bends the path into a circle — the tension in a string, gravity on an orbiting satellite, friction between tyres and road, or the normal force on a banked turn. Identify that real force and set it equal to mv² ÷ r.
Speed matters most. Because v is squared, doubling the speed quadruples the required force, while doubling the radius only halves it. That is why fast cornering throws so much demand onto the tyres.
Do not confuse it with centrifugal force. The outward push you feel in a turn is your inertia resisting the inward pull, not a real force acting on the object. The genuine force here is centripetal and points inward.