{
  "id": "centripetal-force",
  "version": "962c6f020ccc",
  "status": "published",
  "name": "Centripetal Force Calculator",
  "question": "What is the centripetal force?",
  "summary": "Centripetal force calculator: F = m v² ÷ r from mass, speed and radius, or with rpm (F = m r ω²), or the speed, radius or mass for a force, with the acceleration and bank angle.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/centripetal-force-calculator",
  "markdown": "https://www.acalculator.org/physics/centripetal-force-calculator.md",
  "kind": "function",
  "method": "F = m v² ÷ r = m r ω²; a = v² ÷ r = rω²; v = √(F r ÷ m); ω = √(F ÷ (m r)); r = m v² ÷ F; m = F r ÷ v²; bank angle θ = arctan(v² ÷ (rg)).",
  "assumptions": [
    "Uniform circular motion: a steady speed on a circle of fixed radius; F is the net force toward the centre.",
    "The bank angle is for a frictionless road, with g = 9.80665 m/s²; on a flat road the friction coefficient needed is a ÷ g."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "find": {
        "title": "Find",
        "description": "Which value to work out from F = m v² ÷ r.",
        "type": "string",
        "enum": [
          "force",
          "motion",
          "radius",
          "mass"
        ]
      },
      "by": {
        "title": "Motion as",
        "description": "Give the motion as the speed along the circle or as a rotation rate such as rpm.",
        "type": "string",
        "enum": [
          "speed",
          "rate"
        ]
      },
      "m": {
        "title": "Mass (m)",
        "description": "The mass moving in the circle.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "r": {
        "title": "Radius (r)",
        "description": "The radius of the circle (of the curve, for a car).",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.000001,
        "maximum": 1000000000
      },
      "v": {
        "title": "Speed (v)",
        "description": "The speed along the circle.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "speed",
        "minimum": 1e-9,
        "maximum": 1000000000
      },
      "n": {
        "title": "Rotation rate (ω)",
        "description": "How fast it goes around, in the unit chosen below.",
        "type": "number",
        "minimum": 1e-9,
        "maximum": 1000000000
      },
      "nu": {
        "title": "Rate unit",
        "description": "The unit of the rotation rate.",
        "type": "string",
        "enum": [
          "rpm",
          "revs",
          "degs",
          "rads"
        ]
      },
      "F": {
        "title": "Centripetal force (F)",
        "description": "The net force toward the centre of the circle.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "force",
        "minimum": 1e-12,
        "maximum": 1000000000000000
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The value worked out, to 6 significant figures, in SI and US units.",
      "format": "text"
    },
    "force": {
      "label": "Centripetal force (N)",
      "description": "F = m v² ÷ r, in newtons.",
      "format": "number"
    },
    "lbf": {
      "label": "Centripetal force (lbf)",
      "description": "F in pounds-force.",
      "format": "number"
    },
    "accel": {
      "label": "Centripetal acceleration (m/s²)",
      "description": "a = v² ÷ r = rω², toward the centre.",
      "format": "number"
    },
    "gs": {
      "label": "Acceleration (g)",
      "description": "a ÷ 9.80665 m/s²; on a flat road, also the friction coefficient needed.",
      "format": "number"
    },
    "speed": {
      "label": "Speed (m/s)",
      "description": "v along the circle, in metres per second.",
      "format": "number"
    },
    "mph": {
      "label": "Speed (mph)",
      "description": "v in miles per hour.",
      "format": "number"
    },
    "omega": {
      "label": "Angular velocity (rad/s)",
      "description": "ω = v ÷ r.",
      "format": "number"
    },
    "rpm": {
      "label": "Rotation rate (rpm)",
      "description": "Turns per minute: ω × 60 ÷ 2π.",
      "format": "number"
    },
    "period": {
      "label": "Time for one lap (s)",
      "description": "T = 2πr ÷ v.",
      "format": "number"
    },
    "radius": {
      "label": "Radius (m)",
      "description": "r in metres.",
      "format": "number"
    },
    "mass": {
      "label": "Mass (kg)",
      "description": "m in kilograms.",
      "format": "number"
    },
    "bank": {
      "label": "Bank angle with no friction (°)",
      "description": "θ = arctan(v² ÷ (rg)): the road tilt that needs no friction.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "find": "force",
      "by": "speed",
      "m": "900 kg",
      "r": "500 m",
      "v": "25 m/s",
      "n": 60,
      "nu": "rpm",
      "F": "1125 N"
    },
    "outputs": {
      "result": "F = 1,125 N (252.91 lbf)",
      "force": 1125,
      "lbf": 252.9100609871743,
      "accel": 1.25,
      "gs": 0.12746452662224103,
      "speed": 25,
      "mph": 55.92340730136006,
      "omega": 0.05,
      "rpm": 0.477464829275686,
      "period": 125.66370614359172,
      "radius": 500,
      "mass": 900,
      "bank": 7.26400847516269
    },
    "text": "F = 1,125 N (252.91 lbf), with a centripetal acceleration of 1.25 m/s²."
  },
  "examples": [
    {
      "given": {
        "find": "force",
        "by": "speed",
        "m": 900,
        "r": 500,
        "v": 25
      },
      "expect": {
        "F": 1125,
        "accel": 1.25,
        "gs": 0.12746452662224103,
        "lbf": 252.9100609871743,
        "bank": 7.26400847516269,
        "result": "F = 1,125 N (252.91 lbf)"
      },
      "source": "OpenStax, University Physics Volume 1, §6.3 Centripetal Force (a_c = v² ÷ r = rω², F_c = m v² ÷ r = m r ω²; Example 6.15: 900.0 kg at 25.00 m/s on a 500.0 m curve needs 1,125 N and μ_s = 0.13; Example 6.16: tan θ = v² ÷ (rg)), https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force (retrieved 2026-10-19); hand calculation in content.mdx: F = 900 × 25² ÷ 500 = 1,125 N; μ = a ÷ g = 1.25 ÷ 9.80665 = 0.127 (OpenStax: 0.13 with g = 9.80 m/s²)"
    },
    {
      "given": {
        "find": "motion",
        "by": "speed",
        "F": 1125,
        "m": 900,
        "r": 500
      },
      "expect": {
        "v": 25,
        "result": "v = 25 m/s (55.9234 mph)"
      },
      "source": "OpenStax, University Physics Volume 1, §6.3 Centripetal Force (a_c = v² ÷ r = rω², F_c = m v² ÷ r = m r ω²; Example 6.15: 900.0 kg at 25.00 m/s on a 500.0 m curve needs 1,125 N and μ_s = 0.13; Example 6.16: tan θ = v² ÷ (rg)), https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force (retrieved 2026-10-19); hand calculation in content.mdx: v = √(1,125 × 500 ÷ 900) = √625 = 25 m/s"
    },
    {
      "given": {
        "find": "force",
        "by": "rate",
        "m": 1,
        "r": 1,
        "n": 60,
        "nu": "rpm"
      },
      "expect": {
        "F": 39.47841760435743,
        "omega": 6.283185307179586,
        "period": 1
      },
      "source": "OpenStax, University Physics Volume 1, §6.3 Centripetal Force (a_c = v² ÷ r = rω², F_c = m v² ÷ r = m r ω²; Example 6.15: 900.0 kg at 25.00 m/s on a 500.0 m curve needs 1,125 N and μ_s = 0.13; Example 6.16: tan θ = v² ÷ (rg)), https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force (retrieved 2026-10-19) (F = m r ω²); hand calculation in content.mdx: ω = 2π rad/s, F = 1 × 1 × 4π² = 39.4784 N"
    },
    {
      "given": {
        "find": "radius",
        "by": "speed",
        "F": 1125,
        "m": 900,
        "v": 25
      },
      "expect": {
        "r": 500
      },
      "source": "OpenStax, University Physics Volume 1, §6.3 Centripetal Force (a_c = v² ÷ r = rω², F_c = m v² ÷ r = m r ω²; Example 6.15: 900.0 kg at 25.00 m/s on a 500.0 m curve needs 1,125 N and μ_s = 0.13; Example 6.16: tan θ = v² ÷ (rg)), https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force (retrieved 2026-10-19); hand calculation in content.mdx: r = 900 × 25² ÷ 1,125 = 500 m"
    },
    {
      "given": {
        "find": "mass",
        "by": "speed",
        "F": 1125,
        "r": 500,
        "v": 25
      },
      "expect": {
        "m": 900
      },
      "source": "OpenStax, University Physics Volume 1, §6.3 Centripetal Force (a_c = v² ÷ r = rω², F_c = m v² ÷ r = m r ω²; Example 6.15: 900.0 kg at 25.00 m/s on a 500.0 m curve needs 1,125 N and μ_s = 0.13; Example 6.16: tan θ = v² ÷ (rg)), https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force (retrieved 2026-10-19); hand calculation in content.mdx: m = 1,125 × 500 ÷ 25² = 900 kg"
    },
    {
      "given": {
        "find": "force",
        "by": "speed",
        "m": 1000,
        "r": 100,
        "v": 24.3
      },
      "expect": {
        "bank": 31.053501143693964
      },
      "source": "OpenStax, University Physics Volume 1, §6.3 Centripetal Force (a_c = v² ÷ r = rω², F_c = m v² ÷ r = m r ω²; Example 6.15: 900.0 kg at 25.00 m/s on a 500.0 m curve needs 1,125 N and μ_s = 0.13; Example 6.16: tan θ = v² ÷ (rg)), https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force (retrieved 2026-10-19), Example 6.16 (θ = 31.0° gives 24.3 m/s with g = 9.80 m/s²); hand calculation in content.mdx: θ = arctan(24.3² ÷ (100 × 9.80665))"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §6.3 Centripetal Force: a_c = v² ÷ r = rω²; F_c = m v² ÷ r = m r ω²; Example 6.15 (900.0 kg at 25.00 m/s on a 500.0 m curve: 1,125 N, μ_s = 0.13); Example 6.16 (ideal banking, tan θ = v² ÷ (rg); 31.0° and 100.0 m give 24.3 m/s), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force",
    "OpenStax, University Physics Volume 1, §10.1 Rotational Variables: v = rω; 250 rpm = 250 × 2π ÷ 60 rad/s, retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables",
    "NIST, CODATA 2022: standard acceleration of gravity g_n = 9.80665 m/s² (exact), retrieved 2026-10-19. https://physics.nist.gov/cgi-bin/cuu/Value?gn",
    "NIST Special Publication 811 (2008), Appendix B.8: 1 lb = 0.45359237 kg, 1 lbf = 4.4482216152605 N, 1 kgf = 9.80665 N, 1 ft = 0.3048 m, 1 mi = 1,609.344 m, 1 mph = 0.44704 m/s, retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "angular-velocity",
    "force",
    "friction",
    "acceleration"
  ],
  "changelog": []
}