{
  "id": "gravitational-force",
  "version": "466bf71be7aa",
  "status": "published",
  "name": "Gravitational Force Calculator",
  "question": "How strong is the gravitational force?",
  "summary": "Computes the gravitational force between two masses a distance apart with Newton’s law F = G m₁ m₂ ÷ r², in newtons and pounds-force, with the acceleration each mass feels.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/gravitational-force-calculator",
  "markdown": "https://www.acalculator.org/physics/gravitational-force-calculator.md",
  "kind": "function",
  "method": "F = G × m₁ × m₂ ÷ r², G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻².",
  "assumptions": [
    "The masses are points or spheres (whose mass is spread evenly in shells), and r is the distance between their centers.",
    "G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² (CODATA 2022). Exact unit sizes: 1 lb = 0.45359237 kg; 1 short ton = 907.18474 kg; 1 mi = 1,609.344 m; 1 ft = 0.3048 m."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "m1": {
        "title": "First mass",
        "description": "The mass of the first object: Earth is 5.972 × 10²⁴ kg.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 1e-30,
        "maximum": 1e+40
      },
      "m2": {
        "title": "Second mass",
        "description": "The mass of the second object: the Moon is 7.346 × 10²² kg.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 1e-30,
        "maximum": 1e+40
      },
      "r": {
        "title": "Distance between centers",
        "description": "The distance from the center of one mass to the center of the other.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-15,
        "maximum": 1e+30
      }
    }
  },
  "outputs": {
    "newtons": {
      "label": "Force (N)",
      "description": "The force of gravity between the two masses, in newtons.",
      "format": "number"
    },
    "lbf": {
      "label": "Force (lbf)",
      "description": "The same force in pounds-force (1 lbf = 4.4482216152605 N).",
      "format": "number"
    },
    "a1": {
      "label": "Acceleration of the first mass (m/s²)",
      "description": "F ÷ m₁: how fast the pull speeds up the first mass.",
      "format": "number"
    },
    "a2": {
      "label": "Acceleration of the second mass (m/s²)",
      "description": "F ÷ m₂: how fast the pull speeds up the second mass.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "m1": "5.972e24 kg",
      "m2": "7.346e22 kg",
      "r": "384400 km"
    },
    "outputs": {
      "newtons": 198157123241918700000,
      "lbf": 44547493443694820000,
      "a1": 0.000033181032023094225,
      "a2": 0.0026974833003256017
    },
    "text": "The gravitational force between the two masses is 1.98157 × 10²⁰ N (4.45475 × 10¹⁹ lbf)."
  },
  "examples": [
    {
      "given": {
        "m1": 5.972e+24,
        "m2": 7.346e+22,
        "r": 384400000
      },
      "expect": {
        "newtons": 198157123241918700000
      },
      "source": "OpenStax, University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation (F₁₂ = G m₁ m₂ ÷ r²), https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation; NIST, CODATA 2022 value: Newtonian constant of gravitation G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², https://physics.nist.gov/cgi-bin/cuu/Value?bg; hand calculation in content.mdx: 6.6743e-11 × 5.972e24 × 7.346e22 ÷ (3.844e8)² = 1.98157 × 10²⁰ N"
    },
    {
      "given": {
        "m1": 1,
        "m2": 1,
        "r": 1
      },
      "expect": {
        "newtons": 6.6743e-11,
        "a1": 6.6743e-11
      },
      "source": "OpenStax, University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation (F₁₂ = G m₁ m₂ ÷ r²), https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation; NIST, CODATA 2022 value: Newtonian constant of gravitation G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², https://physics.nist.gov/cgi-bin/cuu/Value?bg; hand calculation in content.mdx: F = G × 1 × 1 ÷ 1² = 6.6743 × 10⁻¹¹ N"
    },
    {
      "given": {
        "m1": 5.972e+24,
        "m2": 70,
        "r": 6371000
      },
      "expect": {
        "newtons": 687.398139835728,
        "a2": 9.819973426224687
      },
      "source": "OpenStax, University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation (F₁₂ = G m₁ m₂ ÷ r²), https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation; NIST, CODATA 2022 value: Newtonian constant of gravitation G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², https://physics.nist.gov/cgi-bin/cuu/Value?bg; hand calculation in content.mdx: 6.6743e-11 × 5.972e24 × 70 ÷ (6.371e6)² = 687.398 N; ÷ 70 = 9.81997 m/s²"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation (F₁₂ = G m₁ m₂ ÷ r²). https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation (retrieved 2026-10-03)",
    "NIST, CODATA 2022 recommended value: Newtonian constant of gravitation, G = 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻². https://physics.nist.gov/cgi-bin/cuu/Value?bg (retrieved 2026-10-03)",
    "NASA, Earth Fact Sheet (mass 5.9722 × 10²⁴ kg, volumetric mean radius 6,371 km) and Moon Fact Sheet (mass 0.07346 × 10²⁴ kg, mean distance 384,400 km). https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html (retrieved 2026-10-03)"
  ],
  "related": [
    "force",
    "free-fall",
    "mass"
  ],
  "changelog": []
}