{
  "id": "angular-momentum",
  "version": "0426d26fbe15",
  "status": "published",
  "name": "Angular Momentum Calculator",
  "question": "What is the angular momentum?",
  "summary": "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.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/angular-momentum-calculator",
  "markdown": "https://www.acalculator.org/physics/angular-momentum-calculator.md",
  "kind": "function",
  "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."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "by": {
        "title": "For",
        "description": "A body spinning about an axis (I and ω), or a point mass moving past a point (m, v and r).",
        "type": "string",
        "enum": [
          "rigid",
          "particle"
        ]
      },
      "I": {
        "title": "Moment of inertia (I)",
        "description": "The body’s moment of inertia about the axis, in the unit chosen below.",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 1e+40
      },
      "Iu": {
        "title": "Inertia unit",
        "description": "The unit of the moment of inertia.",
        "type": "string",
        "enum": [
          "kgm2",
          "gcm2",
          "lbft2",
          "lbin2"
        ]
      },
      "w": {
        "title": "Angular velocity (ω)",
        "description": "How fast it turns, in the unit chosen below.",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 1000000000000
      },
      "wu": {
        "title": "Turn rate unit",
        "description": "The unit of the angular velocity.",
        "type": "string",
        "enum": [
          "rads",
          "rpm",
          "revs",
          "degs"
        ]
      },
      "m": {
        "title": "Mass (m)",
        "description": "The mass of the moving point or small object.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 1e-12,
        "maximum": 1e+30
      },
      "v": {
        "title": "Speed (v)",
        "description": "How fast the mass moves.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "speed",
        "minimum": 1e-12,
        "maximum": 300000000
      },
      "r": {
        "title": "Distance from the point (r)",
        "description": "The distance from the point or axis to the mass.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-12,
        "maximum": 1000000000000000
      },
      "th": {
        "title": "Angle between r and v (θ), degrees",
        "description": "The angle between the line from the point to the mass and the direction it moves: 90° in a circle.",
        "type": "number",
        "minimum": 0,
        "maximum": 180
      }
    }
  },
  "outputs": {
    "L": {
      "label": "Angular momentum (L), kg·m²/s",
      "description": "L = I × ω, or r × m × v × sin θ, in kg·m²/s.",
      "format": "number"
    },
    "omegaRad": {
      "label": "Angular velocity, rad/s",
      "description": "ω in radians per second.",
      "format": "number"
    },
    "rpm": {
      "label": "Angular velocity, rpm",
      "description": "ω in turns per minute: ω × 60 ÷ 2π.",
      "format": "number"
    },
    "energy": {
      "label": "Rotational kinetic energy, J",
      "description": "K = ½ × I × ω², in joules.",
      "format": "number"
    },
    "momentum": {
      "label": "Linear momentum (p), kg·m/s",
      "description": "p = m × v.",
      "format": "number"
    },
    "arm": {
      "label": "Lever arm (r sin θ), m",
      "description": "The distance from the point to the line the mass moves along.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "by": "rigid",
      "I": 2,
      "Iu": "kgm2",
      "w": 60,
      "wu": "rpm",
      "m": "2 kg",
      "v": "3 m/s",
      "r": "4 m",
      "th": 90
    },
    "outputs": {
      "L": 12.566370614359172,
      "omegaRad": 6.283185307179586,
      "rpm": 60,
      "energy": 39.47841760435743
    },
    "text": "The angular momentum is L = 12.5664 kg·m²/s."
  },
  "examples": [
    {
      "given": {
        "by": "rigid",
        "I": 1.67,
        "Iu": "kgm2",
        "w": 3,
        "wu": "rpm"
      },
      "expect": {
        "L": 0.5246459731494955,
        "omegaRad": 0.3141592653589793,
        "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); hand calculation in content.mdx: L = 1.67 × 0.1π = 0.167π = 0.524646 kg·m²/s (0.17π)",
      "tolerance": 1e-15
    },
    {
      "given": {
        "by": "rigid",
        "I": 3.17,
        "Iu": "kgm2",
        "w": 3,
        "wu": "rpm"
      },
      "expect": {
        "L": 0.9958848711879644
      },
      "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); hand calculation in content.mdx: L = 3.17 × 0.1π = 0.317π = 0.995885 kg·m²/s (0.32π)",
      "tolerance": 1e-15
    },
    {
      "given": {
        "by": "rigid",
        "I": 2,
        "Iu": "kgm2",
        "w": 10,
        "wu": "rads"
      },
      "expect": {
        "L": 20,
        "energy": 100,
        "rpm": 95.4929658551372
      },
      "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); hand calculation in content.mdx: L = 2 × 10 = 20 kg·m²/s, K = ½ × 2 × 10² = 100 J",
      "tolerance": 1e-15
    },
    {
      "given": {
        "by": "particle",
        "m": 2,
        "v": 3,
        "r": 4,
        "th": 90
      },
      "expect": {
        "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); hand calculation in content.mdx: L = 4 × 2 × 3 × sin 90° = 24 kg·m²/s"
    },
    {
      "given": {
        "by": "particle",
        "m": 2,
        "v": 3,
        "r": 4,
        "th": 30
      },
      "expect": {
        "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); hand calculation in content.mdx: sin 30° = ½ exactly, so L = 24 × ½ = 12 kg·m²/s"
    },
    {
      "given": {
        "by": "rigid",
        "I": 1,
        "Iu": "lbft2",
        "w": 1,
        "wu": "rads"
      },
      "expect": {
        "L": 0.0421401100938048
      },
      "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)"
    }
  ],
  "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"
  ],
  "related": [
    "moment-of-inertia",
    "angular-velocity",
    "momentum",
    "torque"
  ],
  "changelog": []
}