{
  "id": "angular-velocity",
  "version": "dfd1c1307583",
  "status": "published",
  "name": "Angular Velocity Calculator",
  "question": "What is the angular velocity?",
  "summary": "Angular velocity calculator: ω = Δθ ÷ Δt from an angle and time, ω = v ÷ r from a speed and radius, or from rpm or the time of one turn, in rad/s, °/s, rpm and Hz.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/angular-velocity-calculator",
  "markdown": "https://www.acalculator.org/physics/angular-velocity-calculator.md",
  "kind": "function",
  "method": "ω = Δθ ÷ Δt; ω = v ÷ r; ω = 2π × rpm ÷ 60; ω = 2π ÷ T; v = rω; a = rω².",
  "assumptions": [
    "The angular velocity is steady over the time given (an average otherwise).",
    "The radius is measured from the axis of rotation to the point that moves."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "by": {
        "title": "From",
        "description": "What you know about the turning.",
        "type": "string",
        "enum": [
          "angle",
          "speed",
          "rate",
          "period"
        ]
      },
      "th": {
        "title": "Angle turned (Δθ)",
        "description": "How far it turns, in the unit chosen below.",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 1000000000000000
      },
      "thu": {
        "title": "Angle unit",
        "description": "The unit of the angle turned.",
        "type": "string",
        "enum": [
          "rad",
          "deg",
          "rev"
        ]
      },
      "t": {
        "title": "Time (Δt)",
        "description": "How long it takes to turn that angle.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "time",
        "minimum": 1e-12,
        "maximum": 1000000000000
      },
      "v": {
        "title": "Tangential speed (v)",
        "description": "The speed of a point on the circle along its path.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "speed",
        "minimum": 1e-9,
        "maximum": 1000000000
      },
      "n": {
        "title": "Rotation rate (n)",
        "description": "How fast it turns, in the unit chosen below.",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 1000000000000
      },
      "nu": {
        "title": "Rate unit",
        "description": "The unit of the rotation rate.",
        "type": "string",
        "enum": [
          "rpm",
          "revs",
          "degs",
          "rads"
        ]
      },
      "T": {
        "title": "Time for one turn (T)",
        "description": "The period: how long one full turn takes.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "time",
        "minimum": 1e-12,
        "maximum": 1000000000000
      },
      "r": {
        "title": "Radius (r)",
        "description": "The distance from the axis to the point. Optional, except with a speed: gives the speed and acceleration there.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-9,
        "maximum": 1000000000
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The angular velocity to 6 significant figures, in rad/s and rpm.",
      "format": "text"
    },
    "omega": {
      "label": "Angular velocity (rad/s)",
      "description": "ω in radians per second.",
      "format": "number"
    },
    "degS": {
      "label": "Angular velocity (°/s)",
      "description": "ω in degrees per second.",
      "format": "number"
    },
    "rpm": {
      "label": "Revolutions per minute (rpm)",
      "description": "Full turns per minute: ω × 60 ÷ 2π.",
      "format": "number"
    },
    "hz": {
      "label": "Turns per second (Hz)",
      "description": "Full turns per second, the frequency: ω ÷ 2π.",
      "format": "number"
    },
    "period": {
      "label": "Time for one turn (s)",
      "description": "T = 2π ÷ ω, in seconds.",
      "format": "number"
    },
    "speed": {
      "label": "Tangential speed (m/s)",
      "description": "v = r × ω at the radius, in metres per second.",
      "format": "number"
    },
    "accel": {
      "label": "Centripetal acceleration (m/s²)",
      "description": "a = r × ω² at the radius, toward the axis.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "by": "angle",
      "th": 1350,
      "thu": "rad",
      "t": "30 s",
      "v": "30 m/s",
      "n": 250,
      "nu": "rpm",
      "T": "1 s",
      "r": "0.1 m"
    },
    "outputs": {
      "result": "ω = 45 rad/s (429.718 rpm)",
      "omega": 45,
      "degS": 2578.3100780887044,
      "rpm": 429.71834634811745,
      "hz": 7.16197243913529,
      "period": 0.13962634015954636,
      "speed": 4.5,
      "accel": 202.5
    },
    "text": "The angular velocity is 45 rad/s, or 429.718 rpm."
  },
  "examples": [
    {
      "given": {
        "by": "angle",
        "th": 1350,
        "thu": "rad",
        "t": 30,
        "r": 0.1
      },
      "expect": {
        "omega": 45,
        "v": 4.5,
        "accel": 202.5,
        "rpm": 429.7183463481174,
        "result": "ω = 45 rad/s (429.718 rpm)"
      },
      "source": "OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19); OpenStax, University Physics Volume 1, §6.3 Centripetal Force (a_c = v² ÷ r = rω²), https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force (retrieved 2026-10-19); hand calculation in content.mdx: ω = 1,350 ÷ 30 = 45 rad/s, v = 0.1 × 45, a = 0.1 × 45²"
    },
    {
      "given": {
        "by": "rate",
        "n": 250,
        "nu": "rpm"
      },
      "expect": {
        "omega": 26.179938779914945,
        "hz": 4.166666666666667,
        "T": 0.24
      },
      "source": "OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19); hand calculation in content.mdx: 250 × 2π ÷ 60 = 26.18 rad/s; T = 60 ÷ 250 = 0.24 s"
    },
    {
      "given": {
        "by": "speed",
        "v": 30,
        "r": 0.3
      },
      "expect": {
        "omega": 100,
        "rpm": 954.929658551372,
        "accel": 3000
      },
      "source": "OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19), Equation 10.4 (v_t = rω); hand calculation in content.mdx: ω = 30 ÷ 0.3 = 100 rad/s"
    },
    {
      "given": {
        "by": "period",
        "T": 86164.09054
      },
      "expect": {
        "omega": 0.00007292115854530769,
        "degS": 0.004178074621850468
      },
      "source": "NASA JPL Solar System Dynamics, Astrodynamic Parameters (mean sidereal day 86,164.09054 s), https://ssd.jpl.nasa.gov/astro_par.html (retrieved 2026-10-19); OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19); hand calculation in content.mdx: ω = 2π ÷ 86,164.09054 s"
    },
    {
      "given": {
        "by": "angle",
        "th": 90,
        "thu": "deg",
        "t": 2
      },
      "expect": {
        "omega": 0.7853981633974483,
        "degS": 45,
        "T": 8
      },
      "source": "OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19); hand calculation in content.mdx: 90° = π ÷ 2 rad, so ω = π ÷ 4 rad/s; T = 360 ÷ 45 = 8 s"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §10.1 Rotational Variables: Equation 10.3 (ω = Δθ ÷ Δt), Equation 10.4 (v_t = rω); Example 10.1 (a flywheel at 45.0 rad/s turns 1,350 rad in 30 s; 4.5 m/s at 0.1 m); Example 10.2 (250 rpm is 26.2 rad/s), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables",
    "OpenStax, University Physics Volume 1, §6.3 Centripetal Force: a_c = v² ÷ r = rω², retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force",
    "NASA JPL Solar System Dynamics, Astrodynamic Parameters: mean sidereal day 86,164.09054 s, retrieved 2026-10-19. https://ssd.jpl.nasa.gov/astro_par.html",
    "NIST Special Publication 811 (2008), Appendix B.8: 1 in = 0.0254 m, 1 ft = 0.3048 m, 1 mi = 1,609.344 m, 1 mph = 0.44704 m/s, 1 km/h = 1 ÷ 3.6 m/s, retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "angular-acceleration",
    "rpm",
    "centripetal-force",
    "frequency"
  ],
  "changelog": []
}