{
  "id": "net-force",
  "version": "721106235f2c",
  "status": "published",
  "name": "Net Force Calculator",
  "question": "What is the net force?",
  "summary": "Adds up to 20 forces, each with a size and a direction, into the net (resultant) force: its size, direction and x and y components, and the acceleration it gives a mass (F = ma).",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/net-force-calculator",
  "markdown": "https://www.acalculator.org/physics/net-force-calculator.md",
  "kind": "function",
  "method": "Fx = Σ F cos θ, Fy = Σ F sin θ; net force = √(Fx² + Fy²); direction = atan2(Fy, Fx); acceleration = net force ÷ mass.",
  "assumptions": [
    "All forces act on one object, in one plane; directions are measured counterclockwise from the +x axis.",
    "Units convert to newtons: 1 lbf = 4.4482216152605 N, 1 kN = 1,000 N."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "forces": {
        "title": "Forces",
        "description": "Each force acting on the object, one per row: its size and its direction.",
        "type": "array",
        "items": {
          "type": "object",
          "properties": {
            "f": {
              "title": "Force",
              "description": "The size of one force.",
              "type": [
                "number",
                "string"
              ],
              "x-quantity": "force",
              "minimum": 0,
              "maximum": 1000000000000
            },
            "a": {
              "title": "Direction",
              "description": "The direction of the force, counterclockwise from the +x axis (0° right, 90° up).",
              "type": [
                "number",
                "string"
              ],
              "x-quantity": "angle",
              "minimum": -6.283185307179586,
              "maximum": 6.283185307179586
            }
          }
        }
      },
      "m": {
        "title": "Mass (optional)",
        "description": "The mass of the object, for the acceleration a = F ÷ m; leave empty to skip it.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 1e-9,
        "maximum": 1000000000000
      }
    }
  },
  "outputs": {
    "net": {
      "label": "Net force",
      "description": "The size of the sum of all the forces.",
      "format": "quantity",
      "unit": "N"
    },
    "direction": {
      "label": "Direction of the net force",
      "description": "Counterclockwise from the +x axis, from 0° up to 360°; empty when the forces cancel.",
      "format": "quantity",
      "unit": "deg"
    },
    "fx": {
      "label": "x component",
      "description": "The sum of F cos θ over the forces.",
      "format": "quantity",
      "unit": "N"
    },
    "fy": {
      "label": "y component",
      "description": "The sum of F sin θ over the forces.",
      "format": "quantity",
      "unit": "N"
    },
    "acceleration": {
      "label": "Acceleration",
      "description": "The net force ÷ the mass, in metres per second squared.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "forces": [
        {
          "f": "30 N",
          "a": "0°"
        },
        {
          "f": "40 N",
          "a": "90°"
        }
      ]
    },
    "outputs": {
      "net": 50,
      "direction": 0.9272952180016122,
      "fx": 30,
      "fy": 40
    },
    "text": "The net force is 50 N: 30 N along x and 40 N along y."
  },
  "examples": [
    {
      "given": {
        "forces": [
          {
            "f": 30,
            "a": 0
          },
          {
            "f": 40,
            "a": 1.5707963267948966
          }
        ]
      },
      "expect": {
        "net": 50,
        "direction": 0.9272952180016122,
        "fx": 30,
        "fy": 40
      },
      "source": "OpenStax, University Physics Volume 1, §5.1 Forces: 30.0 N to the right and 40.0 N up give a net force of 50.0 N at 53.1°, https://openstax.org/books/university-physics-volume-1/pages/5-1-forces; hand calculation in content.mdx: √(30² + 40²) = 50 N, atan(40/30) = 53.13°"
    },
    {
      "given": {
        "forces": [
          {
            "f": 10,
            "a": 0.5235987755982988
          },
          {
            "f": 40,
            "a": 4.71238898038469
          },
          {
            "f": 5,
            "a": 3.141592653589793
          },
          {
            "f": 2,
            "a": 1.5707963267948966
          }
        ],
        "m": 4
      },
      "expect": {
        "net": 33.202371295158365,
        "direction": 4.8228542451628975,
        "fx": 3.6602540378443873,
        "fy": -33,
        "acceleration": 8.300592823789591
      },
      "source": "OpenStax, University Physics Volume 1, §5.3 Newton’s Second Law, Example 5.7 Several Forces on a Particle, https://openstax.org/books/university-physics-volume-1/pages/5-3-newtons-second-law: a = (0.92 i − 8.3 j) m/s² at 276°; hand calculation in content.mdx (the book rounds the size to 8.4 m/s²)"
    },
    {
      "given": {
        "forces": [
          {
            "f": 10,
            "a": 0
          },
          {
            "f": 10,
            "a": 3.141592653589793
          }
        ],
        "m": 2
      },
      "expect": {
        "net": 0,
        "fx": 0,
        "fy": 0,
        "acceleration": 0
      },
      "source": "OpenStax, University Physics Volume 1, §5.1 Forces: 30.0 N to the right and 40.0 N up give a net force of 50.0 N at 53.1°, https://openstax.org/books/university-physics-volume-1/pages/5-1-forces (balanced forces give no net force); hand calculation in content.mdx"
    },
    {
      "given": {
        "forces": [
          {
            "f": 444.82216152604997,
            "a": 0
          },
          {
            "f": 444.82216152604997,
            "a": 1.5707963267948966
          }
        ]
      },
      "expect": {
        "net": 629.0735336742554,
        "direction": 0.7853981633974483
      },
      "source": "OpenStax, University Physics Volume 1, §5.1 Forces: 30.0 N to the right and 40.0 N up give a net force of 50.0 N at 53.1°, https://openstax.org/books/university-physics-volume-1/pages/5-1-forces; hand calculation in content.mdx: 100√2 = 141.42 lbf at 45°"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, section 5.1 Forces: the net external force is the vector sum of the forces; 30.0 N to the right and 40.0 N up give 50.0 N at 53.1°. CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/university-physics-volume-1/pages/5-1-forces",
    "OpenStax, University Physics Volume 1, section 5.3 Newton’s Second Law: F_net = ma; Example 5.7 Several Forces on a Particle. CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/university-physics-volume-1/pages/5-3-newtons-second-law"
  ],
  "related": [
    "acceleration",
    "power",
    "speed",
    "density"
  ],
  "changelog": []
}