# What is the angular momentum?

Finds angular momentum L = I × ω for a spinning body, or L = r × m × v × sin θ for a point mass, in kg·m²/s, with the rotational kinetic energy or the linear momentum.

- Page: https://www.acalculator.org/physics/angular-momentum-calculator
- JSON spec: https://www.acalculator.org/physics/angular-momentum-calculator.json
- Version: 0426d26fbe15

## Default answer

Example with the default inputs (For Spinning body (I and ω), Moment of inertia (I) 2, Inertia unit kg·m², Angular velocity (ω) 60, Turn rate unit rpm (rev/min)): The angular momentum is L = 12.5664 kg·m²/s.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| by | For | A body spinning about an axis (I and ω), or a point mass moving past a point (m, v and r). |
| I | Moment of inertia (I) | The body’s moment of inertia about the axis, in the unit chosen below. |
| Iu | Inertia unit | The unit of the moment of inertia. |
| w | Angular velocity (ω) | How fast it turns, in the unit chosen below. |
| wu | Turn rate unit | The unit of the angular velocity. |
| m | Mass (m) | The mass of the moving point or small object. |
| v | Speed (v) | How fast the mass moves. |
| r | Distance from the point (r) | The distance from the point or axis to the mass. |
| th | Angle between r and v (θ), degrees | The angle between the line from the point to the mass and the direction it moves: 90° in a circle. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| L | Angular momentum (L), kg·m²/s | L = I × ω, or r × m × v × sin θ, in kg·m²/s. |
| omegaRad | Angular velocity, rad/s | ω in radians per second. |
| rpm | Angular velocity, rpm | ω in turns per minute: ω × 60 ÷ 2π. |
| energy | Rotational kinetic energy, J | K = ½ × I × ω², in joules. |
| momentum | Linear momentum (p), kg·m/s | p = m × v. |
| arm | Lever arm (r sin θ), m | The distance from the point to the line the mass moves along. |

## Method

L = I × ω for a spinning body (rad/s), with K = ½Iω²; L = r × m × v × sin θ for a point mass, with p = m × v and lever arm r × sin θ.

## Assumptions

- A spinning body turns about a fixed axis, and I is its moment of inertia about that axis.
- For a point mass, θ is the angle between the line from the point to the mass and the direction it moves; L is the size of r × p.
- Typed values are read as the exact decimals you typed; each result is rounded once to show.

## Worked examples

1. by = rigid, I = 1.67, Iu = kgm2, w = 3, wu = rpm gives L = 0.524646, omegaRad = 0.314159, rpm = 3. Source: OpenStax, University Physics Volume 1, §11.2 Angular Momentum (Eq. 11.5 l = r × p, l = rp sin θ; Eq. 11.9 L = Iω; Example 11.6: a robot arm of I = 1.67 kg·m² turning at 0.1π rad/s has L = 0.17π kg·m²/s, and with a rock, I = 3.17 kg·m² and L = 0.32π kg·m²/s), https://openstax.org/books/university-physics-volume-1/pages/11-2-angular-momentum (retrieved 2026-10-19); OpenStax, University Physics Volume 1, §10.1 Rotational Variables (one revolution is 2π rad), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19).
2. by = rigid, I = 3.17, Iu = kgm2, w = 3, wu = rpm gives L = 0.995885. Source: OpenStax, University Physics Volume 1, §11.2 Angular Momentum (Eq. 11.5 l = r × p, l = rp sin θ; Eq. 11.9 L = Iω; Example 11.6: a robot arm of I = 1.67 kg·m² turning at 0.1π rad/s has L = 0.17π kg·m²/s, and with a rock, I = 3.17 kg·m² and L = 0.32π kg·m²/s), https://openstax.org/books/university-physics-volume-1/pages/11-2-angular-momentum (retrieved 2026-10-19).
3. by = rigid, I = 2, Iu = kgm2, w = 10, wu = rads gives L = 20, energy = 100, rpm = 95.493. Source: OpenStax, University Physics Volume 1, §11.2 Angular Momentum (Eq. 11.5 l = r × p, l = rp sin θ; Eq. 11.9 L = Iω; Example 11.6: a robot arm of I = 1.67 kg·m² turning at 0.1π rad/s has L = 0.17π kg·m²/s, and with a rock, I = 3.17 kg·m² and L = 0.32π kg·m²/s), https://openstax.org/books/university-physics-volume-1/pages/11-2-angular-momentum (retrieved 2026-10-19); OpenStax, University Physics Volume 1, §10.4 Moment of Inertia and Rotational Kinetic Energy (K = ½Iω², Eq. 10.17), https://openstax.org/books/university-physics-volume-1/pages/10-4-moment-of-inertia-and-rotational-kinetic-energy (retrieved 2026-10-19).
4. by = particle, m = 2, v = 3, r = 4, th = 90 gives L = 24, momentum = 6, arm = 4. Source: OpenStax, University Physics Volume 1, §11.2 Angular Momentum (Eq. 11.5 l = r × p, l = rp sin θ; Eq. 11.9 L = Iω; Example 11.6: a robot arm of I = 1.67 kg·m² turning at 0.1π rad/s has L = 0.17π kg·m²/s, and with a rock, I = 3.17 kg·m² and L = 0.32π kg·m²/s), https://openstax.org/books/university-physics-volume-1/pages/11-2-angular-momentum (retrieved 2026-10-19).
5. by = particle, m = 2, v = 3, r = 4, th = 30 gives L = 12, arm = 2. Source: OpenStax, University Physics Volume 1, §11.2 Angular Momentum (Eq. 11.5 l = r × p, l = rp sin θ; Eq. 11.9 L = Iω; Example 11.6: a robot arm of I = 1.67 kg·m² turning at 0.1π rad/s has L = 0.17π kg·m²/s, and with a rock, I = 3.17 kg·m² and L = 0.32π kg·m²/s), https://openstax.org/books/university-physics-volume-1/pages/11-2-angular-momentum (retrieved 2026-10-19).
6. by = rigid, I = 1, Iu = lbft2, w = 1, wu = rads gives L = 0.0421401. Source: OpenStax, University Physics Volume 1, §11.2 Angular Momentum (Eq. 11.5 l = r × p, l = rp sin θ; Eq. 11.9 L = Iω; Example 11.6: a robot arm of I = 1.67 kg·m² turning at 0.1π rad/s has L = 0.17π kg·m²/s, and with a rock, I = 3.17 kg·m² and L = 0.32π kg·m²/s), https://openstax.org/books/university-physics-volume-1/pages/11-2-angular-momentum (retrieved 2026-10-19); NIST Special Publication 811, Appendix B.8 (1 lb = 0.45359237 kg, 1 ft² = 0.09290304 m²), https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8 (retrieved 2026-10-19).

## FAQ

### How do you calculate angular momentum?

For a body spinning about a fixed axis, multiply the moment of inertia by the angular velocity in rad/s: L = I × ω. A 2 kg·m² wheel at 10 rad/s has L = 20 kg·m²/s. For a point mass, L = r × m × v × sin θ.

### What is the angular momentum formula for a particle?

l = r × p, a cross product, with size l = r × p × sin θ, where p = m × v and θ is the angle between r and the motion (OpenStax Eq. 11.5). A 2 kg mass at 3 m/s, 4 m from the point and moving at 90°, has l = 4 × 6 × 1 = 24 kg·m²/s.

### What are the units of angular momentum?

Kilogram metres squared per second, kg·m²/s, which is the same as a joule second (J·s): a moment of inertia (kg·m²) times an angular velocity (1/s).

### How do I use rpm in the angular momentum formula?

Turn rpm into rad/s first: one turn is 2π rad, so 1 rpm = 2π ÷ 60 = π/30 rad/s. The robot arm of OpenStax Example 11.6 turns at 3 rpm (0.1π rad/s) with I = 1.67 kg·m², so L = 0.167π = 0.525 kg·m²/s.

### Is angular momentum conserved?

Yes, when no net external torque acts. A spinning skater who pulls in her arms lowers I, so ω rises to keep L = Iω the same.

### What is the difference between angular momentum and rotational kinetic energy?

L = Iω grows with ω, while K = ½Iω² grows with ω². The page shows both for a spinning body: 2 kg·m² at 10 rad/s has L = 20 kg·m²/s and K = 100 J.

## Sources

- OpenStax, University Physics Volume 1, §11.2 Angular Momentum: Eq. 11.5 (l = r × p, size rp sin θ), Eq. 11.9 (L = Iω), Example 11.6 (a robot arm of I = 1.67 kg·m² at 0.1π rad/s has L = 0.17π kg·m²/s; with a 1.5 kg rock, I = 3.17 kg·m² and L = 0.32π kg·m²/s), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/11-2-angular-momentum
- OpenStax, University Physics Volume 1, §10.1 Rotational Variables: one revolution is 2π rad, retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables
- OpenStax, University Physics Volume 1, §10.4 Moment of Inertia and Rotational Kinetic Energy: K = ½Iω² (Eq. 10.17), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/10-4-moment-of-inertia-and-rotational-kinetic-energy
- NIST Special Publication 811, Appendix B.8: 1 lb = 0.45359237 kg, 1 ft = 0.3048 m, 1 in = 0.0254 m, 1 mph = 0.44704 m/s, retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8