What is the angular momentum?

How it works

Choose what turns:

• Spinning body (I and ω): a rigid body turning about a fixed axis has angular momentum L = I × ω (OpenStax Eq. 11.9), with I its moment of inertia about the axis in kg·m² and ω its angular velocity in rad/s. The page also shows ω in rad/s and in rpm (ω × 60 ÷ 2π), and the rotational kinetic energy K = ½ × I × ω² in J.
• Point mass (m, v, r): a mass m moving at speed v, at distance r from the point, has angular momentum l = r × p about that point (Eq. 11.5), of size L = r × m × v × sin θ, where θ is the angle between the line from the point to the mass and the direction it moves (90° for motion around a circle). The page also shows the linear momentum p = m × v in kg·m/s and the lever arm r × sin θ in m.

L is in kg·m²/s. The page gives its size; its direction is along the axis (right-hand rule).

Units (exact): moment of inertia 1 g·cm² = 10⁻⁷ kg·m²; 1 lb·ft² = 0.45359237 × 0.09290304 kg·m²; 1 lb·in² = 0.45359237 × 0.00064516 kg·m². Turn rates: 1 rpm = 2π ÷ 60 = π/30 rad/s; 1 rev/s = 2π rad/s; 1 °/s = π/180 rad/s. 1 g = 0.001 kg; 1 lb = 0.45359237 kg; 1 oz = 0.028349523125 kg; 1 km/h = 5/18 m/s; 1 mph = 0.44704 m/s; 1 ft/s = 0.3048 m/s; 1 mm = 0.001 m; 1 cm = 0.01 m; 1 in = 0.0254 m; 1 ft = 0.3048 m.

Rules

• Moment of inertia from 10⁻¹² to 10⁴⁰ and angular velocity from 10⁻¹² to 10¹², in the units chosen beside them.
• Mass from 10⁻¹² kg to 10³⁰ kg; speed from 10⁻¹² m/s to 3 × 10⁸ m/s; distance from 10⁻¹² m to 10¹⁵ m; θ from 0° to 180°. These limits are in SI units and include their ends; a value typed in another unit is converted first.
• At θ = 0° or 180° the mass moves straight toward or away from the point, so L = 0 and the lever arm is 0.
• A result too large or too small to show as a number gives no answer.

Output format

Every number shows to 6 significant figures, rounded half up, with trailing zeros dropped, thousands commas, and from 10¹⁵ up or below 10⁻⁶ as a × 10ⁿ. Typed values are read back as the exact decimals you typed, and every unit size is an exact fraction times a power of π, so each result is exact until one power of π is applied (a turn rate in rpm, rev/s or °/s brings π in). sin θ is exact where it is rational (0°, 30°, 90°, 150°, 180°); at other angles it is a 64-bit float. A result that is exactly 0 shows as 0.

The answer sentence reads The angular momentum is L = L kg·m²/s., with L written as its output is.

Worked examples by hand

A robot arm (OpenStax Example 11.6). I = 1.67 kg·m², turning at 3 rpm = 3 × π/30 = 0.1π rad/s = 0.314159 rad/s: L = 1.67 × 0.1π = 0.167π = 0.524646 kg·m²/s, the 0.17π of the example. Holding the rock, I = 3.17 kg·m² and L = 0.317π = 0.995885 kg·m²/s (0.32π).

A wheel. I = 2 kg·m² at 10 rad/s: L = 2 × 10 = 20 kg·m²/s, K = ½ × 2 × 10² = 100 J, and ω = 10 × 60 ÷ 2π = 95.493 rpm.

A point mass. m = 2 kg at v = 3 m/s, r = 4 m from the point, moving at θ = 90°: p = 2 × 3 = 6 kg·m/s, lever arm 4 × sin 90° = 4 m, L = 24 kg·m²/s. At θ = 30°, sin 30° = ½, so the lever arm is 2 m and L = 12 kg·m²/s.

US units. I = 1 lb·ft² at 1 rad/s: L = 0.45359237 × 0.09290304 × 1 = 0.0421401 kg·m²/s.