What is the centripetal force?

How it works

A mass m moving at speed v on a circle of radius r needs a net force toward the centre:

• Force: F = m v² ÷ r, or F = m r ω² with the angular velocity ω.
• Acceleration: a = v² ÷ r = rω², toward the centre; F = m × a.
• Speed: v = √(F × r ÷ m), or ω = √(F ÷ (m × r)) with a rotation rate.
• Radius: r = m v² ÷ F, or r = F ÷ (m ω²).
• Mass: m = F × r ÷ v², or m = F ÷ (r ω²).

The motion is either the speed v along the circle or a rotation rate ω (rpm, rev/s, °/s or rad/s), with v = ω × r. The page shows F in N and lbf, a in m/s² and in g (a ÷ 9.80665), v in m/s and mph, ω in rad/s and rpm (ω × 60 ÷ 2π), the time for one lap T = 2π ÷ ω, r, m, and the bank angle θ = arctan(a ÷ g) = arctan(v² ÷ (rg)), in degrees: the road tilt that holds the car with no friction. On a flat road, the friction coefficient needed is a ÷ g, the acceleration in g.

Units (exact): 1 g = 0.001 kg; 1 t = 1,000 kg; 1 oz = 0.028349523125 kg; 1 lb = 0.45359237 kg; 1 short ton = 907.18474 kg; 1 mm = 0.001 m; 1 cm = 0.01 m; 1 km = 1,000 m; 1 in = 0.0254 m; 1 ft = 0.3048 m; 1 mi = 1,609.344 m; 1 km/h = 5 ÷ 18 m/s; 1 mph = 0.44704 m/s; 1 ft/s = 0.3048 m/s; 1 kN = 1,000 N; 1 lbf = 4.4482216152605 N; 1 kgf = 9.80665 N; 1 rpm = 2π ÷ 60 rad/s; 1 rev/s = 2π rad/s; 1°/s = π ÷ 180 rad/s; g = 9.80665 m/s².

Rules

• Mass from 10⁻⁶ kg to 10¹² kg; radius from 10⁻⁶ m to 10⁹ m; speed from 10⁻⁹ m/s to 10⁹ m/s; force from 10⁻¹² N to 10¹⁵ N. Limits are in SI units and include their ends; a value typed in another unit is converted first.
• A rotation rate is a number from 10⁻⁹ to 10⁹ in the unit chosen beside it.
• A result too large or too small to show as a number gives no answer. Within these limits it does not happen.

Output format

The answer line reads F = a N (b lbf), v = a m/s (b mph), ω = a rad/s (b rpm), r = a m (b ft) or m = a kg (b lb) for the value found. It and every number show to 6 significant figures, rounded half up, with trailing zeros dropped and thousands commas, and below 10⁻⁶ or from 10¹⁵ up as a × 10ⁿ. Typed values are read back as the exact decimals you typed, and every unit size is exact (a rotation rate in rpm, rev/s or °/s is an exact fraction times π), so F, a, r and m are exact fractions until shown, times a power of π with such a rate. A speed or rotation rate that is found is the square root of an exact fraction, rounded once. The bank angle is a 64-bit float.

Worked examples by hand

A car on a curve. OpenStax Example 6.15: m = 900 kg, v = 25 m/s, r = 500 m. F = 900 × 625 ÷ 500 = 1,125 N = 252.910 lbf; a = 625 ÷ 500 = 1.25 m/s² = 0.127465 g, so a flat road needs μ ≥ 0.127 (OpenStax: 0.13 with g = 9.80 m/s²). The no-friction bank angle is arctan 0.127465 = 7.26401°.

The speed for a force. F = 1,125 N, m = 900 kg, r = 500 m: v = √(1,125 × 500 ÷ 900) = √625 = 25 m/s = 55.9234 mph.

From rpm. m = 1 kg on a 1 m string at 60 rpm: ω = 60 × 2π ÷ 60 = 2π = 6.28319 rad/s, F = 1 × 1 × (2π)² = 39.4784 N, and one lap takes 2π ÷ 2π = 1 s.

The radius or the mass. With F = 1,125 N and v = 25 m/s: r = 900 × 625 ÷ 1,125 = 500 m for a 900 kg car, and m = 1,125 × 500 ÷ 625 = 900 kg on a 500 m curve.

A banked curve. OpenStax Example 6.16 banks a 100.0 m curve at 31.0° for 24.3 m/s. Back the other way: θ = arctan(24.3² ÷ (100 × 9.80665)) = arctan 0.602132 = 31.0535°.