{
  "id": "rpm",
  "version": "bbf3f026f4da",
  "status": "published",
  "name": "RPM Calculator",
  "question": "What is the RPM?",
  "summary": "Finds revolutions per minute (rpm) from the speed at the edge of a wheel, tire or pulley and its diameter, or the speed or diameter from the rpm, with the angular velocity and frequency.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/rpm-calculator",
  "markdown": "https://www.acalculator.org/physics/rpm-calculator.md",
  "kind": "formulas",
  "method": "rpm = 60 × v ÷ (π × d): one turn moves the rim π × d, and there are 60 seconds in a minute. ω = 2π × rpm ÷ 60 = v ÷ r.",
  "assumptions": [
    "The speed is the speed of the rim (tangential speed); for a wheel rolling without slipping it equals the vehicle speed.",
    "The diameter is the outside diameter where the speed is measured; tire wear and squash are not modelled.",
    "Values run in metres and seconds, in double precision."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "v": {
        "title": "Edge speed",
        "description": "The speed of a point on the rim; for a rolling wheel, the vehicle’s speed.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "speed",
        "exclusiveMinimum": 0
      },
      "d": {
        "title": "Diameter",
        "description": "The distance across the wheel, tire or pulley through its centre.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "exclusiveMinimum": 0
      },
      "rpm": {
        "title": "Revolutions per minute",
        "description": "How many full turns the wheel makes each minute.",
        "type": "number",
        "exclusiveMinimum": 0
      }
    }
  },
  "solve": {
    "fillAny": 2,
    "variables": [
      "v",
      "d",
      "rpm"
    ],
    "inputsOnly": []
  },
  "outputs": {
    "speed": {
      "label": "Edge speed",
      "description": "The speed of a point on the rim; for a rolling wheel, the vehicle’s speed.",
      "format": "quantity"
    },
    "diameter": {
      "label": "Diameter",
      "description": "The distance across the wheel, tire or pulley through its centre.",
      "format": "quantity"
    },
    "rpm": {
      "label": "RPM",
      "description": "Full turns per minute.",
      "format": "number"
    },
    "omega": {
      "label": "Angular velocity (rad/s)",
      "description": "ω = 2π × rpm ÷ 60, in radians per second.",
      "format": "number"
    },
    "hz": {
      "label": "Turns per second (Hz)",
      "description": "rpm ÷ 60: the frequency of rotation.",
      "format": "number"
    },
    "period": {
      "label": "Time for one turn (s)",
      "description": "60 ÷ rpm seconds.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "v": "10 mph",
      "d": "26 in"
    },
    "outputs": {
      "rpm": 129.28278454233958,
      "omega": 13.538461538461537,
      "hz": 2.1547130757056596,
      "period": 0.46409891473485587
    },
    "text": "It turns at 129.28 rpm, or 13.5385 rad/s."
  },
  "examples": [
    {
      "given": {
        "rpm": 250,
        "d": 0.7
      },
      "expect": {
        "omega": 26.17993877991494,
        "hz": 4.166666666666667,
        "v": 9.162978572970228
      },
      "source": "OpenStax, University Physics Volume 1, §10.1 Rotational Variables (v_t = rω; ω in rad/s; 250 rev/min × 2π rad/rev × 1 min/60 s = 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-02); hand calculation in content.mdx: v = 250 × π × 0.7 ÷ 60; Python 3: 250*math.pi*0.7/60"
    },
    {
      "given": {
        "v": 4.4704,
        "d": 0.6604
      },
      "expect": {
        "rpm": 129.28278454233958
      },
      "source": "hand calculation in content.mdx: 4.4704 m/s × 60 ÷ (π × 0.6604 m); Python 3: 60*(1609.344/3600*10)/(math.pi*26*0.0254); OpenStax, University Physics Volume 1, §10.1 Rotational Variables (v_t = rω; ω in rad/s; 250 rev/min × 2π rad/rev × 1 min/60 s = 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-02)"
    },
    {
      "given": {
        "v": 3,
        "rpm": 60
      },
      "expect": {
        "d": 0.9549296585513721,
        "period": 1
      },
      "source": "hand calculation in content.mdx: d = 60 × 3 ÷ (π × 60) = 3 ÷ π m; OpenStax, University Physics Volume 1, §10.1 Rotational Variables (v_t = rω; ω in rad/s; 250 rev/min × 2π rad/rev × 1 min/60 s = 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-02)"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §10.1 Rotational Variables (tangential speed v_t = rω; angular velocity in rad/s; 250 rev/min × 2π rad/rev × 1 min/60 s = 26.2 rad/s). https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-02)"
  ],
  "related": [
    "gear-ratio",
    "speed",
    "tire-size"
  ],
  "changelog": []
}