{
  "id": "angular-acceleration",
  "version": "080a32476f8f",
  "status": "published",
  "name": "Angular Acceleration Calculator",
  "question": "What is the angular acceleration?",
  "summary": "Angular acceleration calculator: α = Δω ÷ Δt from two angular speeds and a time, or from the angle turned, torque and moment of inertia (α = τ ÷ I), or tangential acceleration (α = a ÷ r).",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/angular-acceleration-calculator",
  "markdown": "https://www.acalculator.org/physics/angular-acceleration-calculator.md",
  "kind": "function",
  "method": "α = (ω − ω₀) ÷ Δt; α = (ω² − ω₀²) ÷ (2Δθ) with Δt = 2Δθ ÷ (ω₀ + ω); α = τ ÷ I; α = a_t ÷ r; 1 rpm = 2π ÷ 60 rad/s.",
  "assumptions": [
    "The angular acceleration is constant over the time or angle (the average for a change that is not steady).",
    "A rigid body turning about a fixed axis; the torque and the moment of inertia are about that same axis."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "by": {
        "title": "From",
        "description": "What you know about the spin.",
        "type": "string",
        "enum": [
          "speeds",
          "angle",
          "torque",
          "tangential"
        ]
      },
      "w0": {
        "title": "Starting angular speed (ω₀)",
        "description": "How fast it turns at the start; 0 from rest.",
        "type": "number",
        "minimum": -1000000000,
        "maximum": 1000000000
      },
      "w": {
        "title": "Final angular speed (ω)",
        "description": "How fast it turns at the end.",
        "type": "number",
        "minimum": -1000000000,
        "maximum": 1000000000
      },
      "wu": {
        "title": "Angular speed unit",
        "description": "The unit of both angular speeds.",
        "type": "string",
        "enum": [
          "rads",
          "rpm",
          "degs",
          "revs"
        ]
      },
      "t": {
        "title": "Time (Δt)",
        "description": "How long the change in speed takes.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "time",
        "minimum": 1e-9,
        "maximum": 1000000000
      },
      "th": {
        "title": "Angle turned (Δθ)",
        "description": "How far it turns while the speed changes, in the unit chosen below.",
        "type": "number",
        "minimum": 1e-9,
        "maximum": 1000000000000
      },
      "thu": {
        "title": "Angle unit",
        "description": "The unit of the angle turned.",
        "type": "string",
        "enum": [
          "rad",
          "rev",
          "deg"
        ]
      },
      "tq": {
        "title": "Net torque (τ)",
        "description": "The sum of the torques on the body about its axis; negative slows a forward spin.",
        "type": "number",
        "minimum": -1000000000000,
        "maximum": 1000000000000
      },
      "tqu": {
        "title": "Torque unit",
        "description": "The unit of the torque.",
        "type": "string",
        "enum": [
          "Nm",
          "lbft",
          "lbin"
        ]
      },
      "I": {
        "title": "Moment of inertia (I)",
        "description": "The rotational inertia of the body about the same axis.",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 1000000000000000
      },
      "Iu": {
        "title": "Inertia unit",
        "description": "The unit of the moment of inertia.",
        "type": "string",
        "enum": [
          "kgm2",
          "gcm2",
          "lbft2",
          "lbin2"
        ]
      },
      "at": {
        "title": "Tangential acceleration (a_t)",
        "description": "How fast a point on the rim speeds up along its path.",
        "type": "number",
        "minimum": -1000000000,
        "maximum": 1000000000
      },
      "atu": {
        "title": "Acceleration unit",
        "description": "The unit of the tangential acceleration.",
        "type": "string",
        "enum": [
          "ms2",
          "cms2",
          "fts2",
          "g"
        ]
      },
      "r": {
        "title": "Radius (r)",
        "description": "The distance from the axis to the point.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-9,
        "maximum": 1000000000
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The angular acceleration to 6 significant figures, in rad/s² and °/s².",
      "format": "text"
    },
    "alpha": {
      "label": "Angular acceleration (rad/s²)",
      "description": "α in radians per second squared; negative means slowing a forward spin.",
      "format": "number"
    },
    "degS2": {
      "label": "Angular acceleration (°/s²)",
      "description": "α in degrees per second squared.",
      "format": "number"
    },
    "revS2": {
      "label": "Angular acceleration (rev/s²)",
      "description": "α in turns per second squared.",
      "format": "number"
    },
    "rpmS": {
      "label": "Angular acceleration (rpm/s)",
      "description": "How many rpm the speed changes each second.",
      "format": "number"
    },
    "turns": {
      "label": "Angle turned (rev)",
      "description": "Δθ = (ω₀ + ω) ÷ 2 × Δt, in turns.",
      "format": "number"
    },
    "radians": {
      "label": "Angle turned (rad)",
      "description": "The same angle in radians.",
      "format": "number"
    },
    "seconds": {
      "label": "Time taken (s)",
      "description": "Δt = 2Δθ ÷ (ω₀ + ω), in seconds.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "by": "speeds",
      "w0": 0,
      "w": 250,
      "wu": "rpm",
      "t": "5 s",
      "th": 10,
      "thu": "rev",
      "tq": 375,
      "tqu": "Nm",
      "I": 56.25,
      "Iu": "kgm2",
      "at": 4.95,
      "atu": "ms2",
      "r": "4.5 cm"
    },
    "outputs": {
      "result": "α = 5.23599 rad/s² (300°/s²)",
      "alpha": 5.235987755982989,
      "degS2": 300,
      "revS2": 0.8333333333333334,
      "rpmS": 50,
      "turns": 10.416666666666666,
      "radians": 65.44984694978736
    },
    "text": "The angular acceleration is 5.23599 rad/s²."
  },
  "examples": [
    {
      "given": {
        "by": "speeds",
        "w0": 0,
        "w": 250,
        "wu": "rpm",
        "t": 5
      },
      "expect": {
        "alpha": 5.235987755982989,
        "degS2": 300,
        "rpmS": 50,
        "turns": 10.416666666666666,
        "result": "α = 5.23599 rad/s² (300°/s²)"
      },
      "source": "OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.6 and 10.8; Example 10.2: 0 to 250 rpm in 5.00 s is 5.24 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19); hand calculation in content.mdx: α = 250 × 2π ÷ 60 ÷ 5 = 5π ÷ 3 rad/s²"
    },
    {
      "given": {
        "by": "speeds",
        "w0": 26.2,
        "w": 0,
        "wu": "rads",
        "t": 0.3
      },
      "expect": {
        "alpha": -87.33333333333333,
        "degS2": -5003.831410809189
      },
      "source": "OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.6 and 10.8; Example 10.2: 0 to 250 rpm in 5.00 s is 5.24 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19), Example 10.2(b) (α = −87.3 rad/s² stops 26.2 rad/s in 0.300 s); hand calculation in content.mdx: −26.2 ÷ 0.3"
    },
    {
      "given": {
        "by": "angle",
        "w0": 0,
        "w": 220,
        "wu": "rads",
        "th": 220,
        "thu": "rad"
      },
      "expect": {
        "alpha": 110,
        "seconds": 2
      },
      "source": "OpenStax, University Physics Volume 1, §10.2 Rotation with Constant Angular Acceleration (Equations 10.11 to 10.13; Example 10.4: a reel at 110 rad/s² reaches 220 rad/s after 220 rad and 2.00 s), https://openstax.org/books/university-physics-volume-1/pages/10-2-rotation-with-constant-angular-acceleration (retrieved 2026-10-19); hand calculation in content.mdx: α = 220² ÷ (2 × 220) = 110 rad/s², Δt = 2 × 220 ÷ 220 = 2 s"
    },
    {
      "given": {
        "by": "torque",
        "tq": 375,
        "tqu": "Nm",
        "I": 56.25,
        "Iu": "kgm2"
      },
      "expect": {
        "alpha": 6.666666666666667
      },
      "source": "OpenStax, University Physics Volume 1, §10.7 Newton’s Second Law for Rotation (Equation 10.25, Στ = Iα; Example 10.16: 375 N·m on 56.25 kg·m² gives 6.67 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-7-newtons-second-law-for-rotation (retrieved 2026-10-19); hand calculation in content.mdx: α = 375 ÷ 56.25 = 6.6667 rad/s²"
    },
    {
      "given": {
        "by": "tangential",
        "at": 4.95,
        "atu": "ms2",
        "r": 0.045
      },
      "expect": {
        "alpha": 110
      },
      "source": "OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.6 and 10.8; Example 10.2: 0 to 250 rpm in 5.00 s is 5.24 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19), Equation 10.8 (a_t = rα), with OpenStax, University Physics Volume 1, §10.2 Rotation with Constant Angular Acceleration (Equations 10.11 to 10.13; Example 10.4: a reel at 110 rad/s² reaches 220 rad/s after 220 rad and 2.00 s), https://openstax.org/books/university-physics-volume-1/pages/10-2-rotation-with-constant-angular-acceleration (retrieved 2026-10-19): 4.95 ÷ 0.045 = 110 rad/s²"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §10.1 Rotational Variables: Equation 10.6 (α = Δω ÷ Δt), Equation 10.8 (a_t = rα); Example 10.2 (0 to 250 rpm in 5.00 s: 26.2 rad/s and 5.24 rad/s²; braking at −87.3 rad/s² stops it in 0.300 s), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables",
    "OpenStax, University Physics Volume 1, §10.2 Rotation with Constant Angular Acceleration: Equations 10.9 to 10.13; Example 10.4 (a reel at 110 rad/s² reaches 220 rad/s in 2.00 s and 220 rad), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/10-2-rotation-with-constant-angular-acceleration",
    "OpenStax, University Physics Volume 1, §10.7 Newton’s Second Law for Rotation: Equation 10.25 (Στ = Iα); Example 10.16 (375 N·m on I = 56.25 kg·m² gives 6.67 rad/s²), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/10-7-newtons-second-law-for-rotation",
    "NIST Special Publication 811 (2008), Appendix B.8: 1 ft = 0.3048 m, 1 in = 0.0254 m, 1 lb = 0.45359237 kg, 1 lbf = 4.4482216152605 N, standard gravity 9.80665 m/s², retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "angular-velocity",
    "torque",
    "moment-of-inertia",
    "rpm"
  ],
  "changelog": []
}