Centripetal Force Calculator

About the Centripetal Force Calculator

Anything moving in a circle is accelerating towards the centre, even at a steady speed, because its direction keeps changing. The net inward force that does it is the centripetal force: F = m v² ÷ r = m ω² r, supplied by whatever holds the object on its path — a string’s tension, a road’s friction, a planet’s gravity. This calculator solves it for the force, the mass, the radius or the motion, with the motion given as a speed, a rotation rate in rpm or rad/s, or a period, and gives the centripetal acceleration (also as a multiple of g), the angular velocity, the period and the frequency.

For a car on a curve it gives, on a flat road, the least tyre friction coefficient a speed needs, the top speed for a given grip or the tightest radius; on a banked road, the ideal angle at which no friction is needed; and on a banked road with friction, the slowest and fastest speeds at which the car does not slide.

Without a pass, the free preview shows the value you solve for in full, with the first step of the working; the other figures, the whole working and Copy result need a Pro pass, or unlocking this one result.

Common uses

• Physics homework on uniform circular motion, centripetal force and acceleration, with the working shown.
• Centrifuges, spin dryers and rotors: the g-force at a radius and rpm.
• Cars, bikes and trains on curves: the grip needed, the safe speed or the banking angle — as physics estimates.
• Orbits, swings, fairground rides and objects on strings.

The formulas

• Centripetal force: F = m v² ÷ r = m ω² r, with v = ω r.
• Centripetal acceleration: a = v² ÷ r = ω² r, in g when divided by 9.80665 m/s².
• Rotation: ω = 2π ÷ T = 2π f, and 1 rpm = 2π/60 rad/s. Doubling the speed (or the rpm) needs four times the force.
• Rearranged: m = F r ÷ v², r = m v² ÷ F, v = √(F r ÷ m); with ω, r = F ÷ (m ω²) and ω = √(F ÷ (m r)).

A car on a curve

• Flat road: friction supplies the inward force, so the least static friction coefficient is μs = v² ÷ (r g), the top speed √(μs r g) and the tightest radius v² ÷ (μs g).
• Banked road, no friction: the slope of the road supplies it at one speed, where tan θ = v² ÷ (r g) (OpenStax Eq. 6.4).
• Banked road with friction: faster than that, friction acts down the slope; slower, up it. The limits are v_max = √(r g (sin θ + μ cos θ) ÷ (cos θ − μ sin θ)) (no limit once μ ≥ 1 ÷ tan θ) and v_min = √(r g (sin θ − μ cos θ) ÷ (cos θ + μ sin θ)) (zero once tan θ ≤ μ: the car can stand still).
• The mass cancels in all of these; it is needed only for the forces.

Friction coefficients

The list uses the approximate static coefficients of OpenStax Table 6.1: rubber on dry concrete 1.0, rubber on wet concrete 0.5–0.7, shoes on ice 0.1. Real tyre grip depends on the tyre, its temperature and wear, the surface, water, speed and load, so treat results as estimates and leave a margin.

Sources

• OpenStax, University Physics Volume 1: §6.3 Centripetal Force (Eq. 6.3 and 6.4, Examples 6.15 and 6.16) and §6.2 Friction (Table 6.1).
• Standard gravity 9.80665 m/s²: NIST, SP 811 Appendix B.8.

Limitations

• Uniform circular motion: a constant speed on a circle. Speeding up or slowing down adds a forward or backward acceleration.
• The car is treated as a point on the road: weight transfer, rollover, aerodynamic downforce and tyre slip angles are not modelled.
• Friction is the simple maximum μN; real tyres behave less neatly near their limit.
• Physics estimates only, not road or vehicle design: real roads are designed to standards with safety margins.

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