About

Any object moving along a curved path experiences centripetal force - the net inward force that keeps it from flying off tangentially. Getting this value wrong in engineering contexts leads to structural failure in banked roads, derailed trains, or snapped cables in centrifuges. This calculator solves F = mv²r for any unknown variable given the other three. It accepts multiple unit systems and applies exact SI conversion factors internally.

The formula assumes uniform circular motion - constant speed along a circular path with negligible friction losses. At very high velocities (above 0.1c), relativistic corrections apply and this Newtonian model loses accuracy. For most terrestrial engineering, automotive, and laboratory applications the approximation is precise to the limits of your input measurements. Pro tip: remember that centripetal force is not a separate force - it is the resultant of real forces (tension, gravity, normal force) directed toward the center.

Formulas

The centripetal force acting on an object in uniform circular motion is given by Newton's second law applied radially inward:

F_c = m ⋅ v²r

Rearranging to solve for each variable:

m = F_c ⋅ rv²

v = √F_c ⋅ rm

r = m ⋅ v²F_c

Where F_c is the centripetal force in N (Newtons), m is the mass in kg, v is the tangential velocity in m/s, and r is the radius of the circular path in m. The centripetal acceleration component is a_c = v²r, so equivalently F_c = m ⋅ a_c. The angular velocity relation is v = ω ⋅ r, which yields the alternate form F_c = m ⋅ ω² ⋅ r.

Reference Data

Scenario	Mass	Velocity	Radius	Centripetal Force
Car on highway curve	1500 kg	25 m/s	200 m	4687.5 N
Electron in magnetic field	9.109×10⁻³¹ kg	2.2×10⁶ m/s	5.3×10⁻¹¹ m	8.23×10⁻⁸ N
Satellite in LEO	500 kg	7670 m/s	6.731×10⁶ m	4370 N
Ball on string (horizontal)	0.5 kg	3 m/s	0.8 m	5.625 N
Washing machine drum	7 kg	10 m/s	0.25 m	2800 N
Roller coaster loop top	80 kg	14 m/s	10 m	1568 N
Centrifuge (lab)	0.01 kg	314 m/s	0.1 m	986,960 N
Moon orbiting Earth	7.342×10²² kg	1022 m/s	3.844×10⁸ m	1.98×10²⁰ N
Hammer throw (athletics)	7.26 kg	28 m/s	1.8 m	3163 N
Figure skater spin	55 kg	4.7 m/s	0.3 m	4050 N
Earth orbiting Sun	5.972×10²⁴ kg	29,780 m/s	1.496×10¹¹ m	3.54×10²² N
Race car (F1 corner)	800 kg	60 m/s	100 m	28,800 N
Bicycle turn	85 kg	8 m/s	5 m	1088 N
ISS in orbit	420,000 kg	7660 m/s	6.781×10⁶ m	3.63×10⁶ N
Conical pendulum	0.2 kg	1.5 m/s	0.4 m	1.125 N

Frequently Asked Questions

Centripetal force is not a distinct force type like gravity or tension. It describes the net radial component of real forces (gravity, friction, tension, normal force) that must point toward the center to maintain circular motion. If the net inward force is insufficient - for example, friction < mv²/r on a wet road - the object leaves the circular path tangentially.

Substituting v = ω ⋅ r into F_c = mv²/r yields F_c = mω²r. Here ω is in rad/s. This form is more convenient when RPM or angular frequency is known directly, as in centrifuges or rotating machinery.

As r → 0, the required centripetal force F_c → ∞ for any nonzero velocity. Physically, no material can supply infinite force, so the object cannot follow an infinitely tight curve. This is why sharp turns at high speed cause skidding or structural failure - the available friction or tension cannot meet the force requirement.

No. The formula gives the total net inward force needed. On a flat road, friction alone provides this force, limited by μ_smg. On a banked road at angle θ, the normal force contributes a horizontal component Nsin(θ) that supplements friction. To find the maximum safe speed, set the sum of friction and banking components equal to mv²/r.

The Newtonian formula is accurate for speeds well below the speed of light (c ≈ 3×10⁸ m/s). Relativistic corrections become significant above roughly 0.1c (3×10⁷ m/s), where the Lorentz factor γ exceeds 1.005. For particle accelerators or astrophysical jets, use the relativistic form F = γmv²/r.

Multiply RPM by 2πr/60. For example, a centrifuge spinning at 3000 RPM with radius 0.1 m has tangential velocity v = 3000 × 2π × 0.1 ÷ 60 ≈ 31.42 m/s. Enter that value into the velocity field.