How to Calculate Centripetal Force (F = mv²/r): Formula & Examples

Calculate centripetal force with F = mv²/r, convert km/h safely, and rearrange for speed or radius. Verify every answer with a free centripetal force calculator.

What Centripetal Force Means

Centripetal force is the net inward force that keeps an object moving in a circle. Even when speed is constant, direction keeps changing, so there is a continuous inward acceleration a = v²/r. Newton’s second law then gives F = m a = m v² / r.

The word “centripetal” describes the required direction — toward the centre. Tension in a string, friction on a flat turn, gravity in orbit, or a normal force on a banked curve can supply that inward net force. The formula tells you how large it must be.

If you already have mass, speed, and radius and only need a checked number, open the free centripetal force calculator first, then return here for unit traps and worked examples.

The Formula F = mv²/r

For uniform circular motion:

F = m v² / r — centripetal force in newtons (N) with m in kg, v in m/s, r in m.

v = √(F r / m) — rearrange when force, radius, and mass are known.

r = m v² / F — rearrange when force and speed are known.

If the problem gives angular speed ω in rad/s, use F = m ω² r (and remember v = ω r). Stay with mv²/r when linear speed is given.

Units That Keep Marks Safe

Mass m → kilograms (kg).

Speed v → metres per second (m/s). From km/h, divide by 3.6.

Radius r → metres (m). Use radius, not diameter.

Force F → newtons (N), directed toward the centre.

Trap: 72 km/h is 20 m/s. Leaving 72 in F = mv²/r inflates force by more than an order of magnitude because speed is squared.

Worked Examples

Example 1: 2.0 kg at 4.0 m/s, r = 2.0 m

F = (2.0)(4.0)² / 2.0 = 32 / 2.0 = 16 N toward the centre. Check with a = v²/r = 16/2 = 8 m/s², then F = ma = 16 N.

Example 2: find speed from 45 N

A 5.0 kg mass needs 45 N on a 5.0 m radius. v = √(45 × 5.0 / 5.0) = √45 ≈ 6.71 m/s. Check: m v² / r ≈ 45 N.

Example 3: km/h conversion

A 0.50 kg ball on a 1.0 m string moves at 36 km/h. First v = 36 / 3.6 = 10 m/s. F = (0.50)(10)² / 1.0 = 50 N. Using 36 without converting wrongly suggests hundreds of newtons.

Centripetal vs Centrifugal

In the usual school (inertial) frame, draw real forces and require their net radial component to equal mv²/r inward. “Centrifugal force” is a fictitious outward force used in rotating frames. Do not add both in the same free-body diagram.

For straight-line F = ma problems without a circle, use the force calculator.

Common Mistakes

Leaving speed in km/h while mass and radius are SI.

Putting diameter into r instead of radius.

Drawing the force tangent to the path instead of toward the centre.

Inventing an extra centrifugal push alongside real inward forces.

Sanity check: for a few kilograms at everyday walking speeds on a metre-scale radius, F is often tens of newtons — not kilonewtons.

Check With the Free Calculator

The centripetal force calculator applies F = mv²/r with visible steps so you can verify force, speed, or radius.

Centripetal force calculator — F, v, or r.

Force calculator — F = ma.

Velocity calculator — find speed for circular problems.

Friction calculator — when friction supplies the turn.

Learn F = mv²/r with clean SI units, then confirm every answer on the calculator.

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