{
  "id": "normal-force",
  "version": "34d7d9de5426",
  "status": "published",
  "name": "Normal Force Calculator",
  "question": "What is the normal force?",
  "summary": "Normal force calculator: N = mg on level ground, N = mg cos θ on a slope, with a push or pull at an angle (N = mg − F sin φ), or in an accelerating lift (N = m(g + a)).",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/normal-force-calculator",
  "markdown": "https://www.acalculator.org/physics/normal-force-calculator.md",
  "kind": "function",
  "method": "N = m × g on level ground; N = m × g × cos θ on a slope; N = m × g − F × sin φ with a force F at φ above level; N = m × (g + a) in a lift accelerating at a (up positive).",
  "assumptions": [
    "The surface is rigid and the object stays on it; no other forces act square to the surface.",
    "On a slope the object rests on (or slides along) the plane; a push or pull acts on level ground."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "on": {
        "title": "Situation",
        "description": "Where the object rests and what else acts on it.",
        "type": "string",
        "enum": [
          "incline",
          "level",
          "force",
          "lift"
        ]
      },
      "m": {
        "title": "Mass (m)",
        "description": "The mass of the object.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "g": {
        "title": "Gravity (g, m/s²)",
        "description": "The acceleration of gravity: 9.80665 m/s² is standard on Earth.",
        "type": "number",
        "minimum": 0.001,
        "maximum": 1000
      },
      "th": {
        "title": "Slope angle (θ, degrees)",
        "description": "The angle of the slope above level, from 0° to 90°.",
        "type": "number",
        "minimum": 0,
        "maximum": 90
      },
      "F": {
        "title": "Applied force (F)",
        "description": "A push or pull on the object, besides its weight.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "force",
        "minimum": 0,
        "maximum": 1000000000000000
      },
      "phi": {
        "title": "Force angle (φ, degrees)",
        "description": "The angle of the force above level: positive pulls up, negative pushes down into the ground.",
        "type": "number",
        "minimum": -90,
        "maximum": 90
      },
      "a": {
        "title": "Acceleration (a, m/s²)",
        "description": "The acceleration of the lift: positive up (starting up or stopping on the way down), negative down.",
        "type": "number",
        "minimum": -1000,
        "maximum": 1000
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The normal force to 6 significant figures, in newtons and pounds-force.",
      "format": "text"
    },
    "normal": {
      "label": "Normal force (N)",
      "description": "N, the push of the surface square to it, in newtons.",
      "format": "number"
    },
    "lbf": {
      "label": "Normal force (lbf)",
      "description": "N in pounds-force.",
      "format": "number"
    },
    "weight": {
      "label": "Weight (N)",
      "description": "w = m × g, in newtons.",
      "format": "number"
    },
    "share": {
      "label": "Normal force ÷ weight",
      "description": "N ÷ (m × g): 1 on level ground with nothing else acting.",
      "format": "number"
    },
    "along": {
      "label": "Weight along the slope (N)",
      "description": "m × g × sin θ: the part of the weight down the slope.",
      "format": "number"
    },
    "lift": {
      "label": "Force, up part (N)",
      "description": "F × sin φ: the part of the applied force that pulls up (negative pushes down).",
      "format": "number"
    },
    "push": {
      "label": "Force, level part (N)",
      "description": "F × cos φ: the part of the applied force along the ground.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "on": "incline",
      "m": "60 kg",
      "g": 9.80665,
      "th": 25,
      "F": "100 N",
      "phi": 30,
      "a": 1.2
    },
    "outputs": {
      "result": "N = 533.271 N (119.884 lbf)",
      "normal": 533.2705955845778,
      "lbf": 119.88399897952208,
      "weight": 588.399,
      "share": 0.9063077870366499,
      "along": 248.66816258996582
    },
    "text": "The normal force is 533.271 N (119.884 lbf), 0.906308 times the weight."
  },
  "examples": [
    {
      "given": {
        "on": "level",
        "m": 20,
        "g": 9.8
      },
      "expect": {
        "normal": 196,
        "share": 1,
        "result": "N = 196 N (44.0626 lbf)"
      },
      "source": "OpenStax, University Physics Volume 1, §5.6 Common Forces (w = mg; the normal force N = mg on a level surface and N = mg cos θ on an incline; Example 5.12: a 60.0 kg skier on a 25° slope), https://openstax.org/books/university-physics-volume-1/pages/5-6-common-forces (retrieved 2026-10-19); OpenStax §6.2 Example 6.10 (N = w = 196 N); hand calculation in content.mdx: 20 × 9.8 = 196 N"
    },
    {
      "given": {
        "on": "incline",
        "m": 60,
        "g": 9.8,
        "th": 25
      },
      "expect": {
        "normal": 532.9089787775501,
        "along": 248.49953790353126
      },
      "source": "OpenStax, University Physics Volume 1, §5.6 Common Forces (w = mg; the normal force N = mg on a level surface and N = mg cos θ on an incline; Example 5.12: a 60.0 kg skier on a 25° slope), https://openstax.org/books/university-physics-volume-1/pages/5-6-common-forces (retrieved 2026-10-19); hand calculation in content.mdx: N = 60 × 9.8 × cos 25° = 532.909 N; along the slope 60 × 9.8 × sin 25° = 248.500 N"
    },
    {
      "given": {
        "on": "lift",
        "m": 75,
        "g": 9.8,
        "a": 1.2
      },
      "expect": {
        "normal": 825,
        "weight": 735
      },
      "source": "OpenStax, University Physics Volume 1, §6.1 Solving Problems with Newton’s Laws (Example 6.3: a 75.0 kg person in an elevator accelerating up at 1.20 m/s² reads 825 N; 735 N at constant velocity), https://openstax.org/books/university-physics-volume-1/pages/6-1-solving-problems-with-newtons-laws (retrieved 2026-10-19); hand calculation in content.mdx: N = 75 × (9.8 + 1.2) = 825 N"
    },
    {
      "given": {
        "on": "force",
        "m": 20,
        "g": 9.8,
        "F": 100,
        "phi": 30
      },
      "expect": {
        "normal": 146,
        "lift": 50,
        "result": "N = 146 N (32.8221 lbf)"
      },
      "source": "OpenStax, University Physics Volume 1, §5.6 Common Forces (w = mg; the normal force N = mg on a level surface and N = mg cos θ on an incline; Example 5.12: a 60.0 kg skier on a 25° slope), https://openstax.org/books/university-physics-volume-1/pages/5-6-common-forces (retrieved 2026-10-19); hand calculation in content.mdx: N = 196 − 100 × sin 30° = 196 − 50 = 146 N"
    },
    {
      "given": {
        "on": "force",
        "m": 20,
        "g": 9.8,
        "F": 100,
        "phi": -30
      },
      "expect": {
        "normal": 246,
        "lift": -50
      },
      "source": "OpenStax, University Physics Volume 1, §5.6 Common Forces (w = mg; the normal force N = mg on a level surface and N = mg cos θ on an incline; Example 5.12: a 60.0 kg skier on a 25° slope), https://openstax.org/books/university-physics-volume-1/pages/5-6-common-forces (retrieved 2026-10-19); hand calculation in content.mdx: N = 196 − 100 × sin(−30°) = 196 + 50 = 246 N"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §5.6 Common Forces: weight w = mg; the normal force N = mg on a level surface and N = mg cos θ on an incline, with the weight along the slope mg sin θ; Example 5.12 (a 60.0 kg skier on a 25° slope), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/5-6-common-forces",
    "OpenStax, University Physics Volume 1, §5.3 Newton’s Second Law: Equations 5.3 to 5.5, ΣF = ma, applied to each direction (here the vertical one), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/5-3-newtons-second-law",
    "OpenStax, University Physics Volume 1, §6.1 Solving Problems with Newton’s Laws: Example 6.3 (a 75.0 kg person in an elevator: 825 N accelerating up at 1.20 m/s², 735 N at constant velocity), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/6-1-solving-problems-with-newtons-laws",
    "OpenStax, University Physics Volume 1, §6.2 Friction: Example 6.10 (a 20.0 kg crate on a level floor: N = w = 196 N), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/6-2-friction",
    "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 oz = 0.028349523125 kg, 1 lbf = 4.4482216152605 N, 1 kgf = 9.80665 N, retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "friction",
    "force",
    "tension",
    "mass"
  ],
  "changelog": []
}