{
  "id": "tension",
  "version": "cdf118644c5e",
  "status": "published",
  "name": "Tension Calculator",
  "question": "What is the tension in the rope?",
  "summary": "Tension calculator: the tension in a rope holding a weight (T = m(g + a)), in two ropes at angles, in an Atwood machine, or pulling a block across a table, in newtons and pounds-force.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/tension-calculator",
  "markdown": "https://www.acalculator.org/physics/tension-calculator.md",
  "kind": "function",
  "method": "One rope: T = m(g + a). Two ropes: T₁ = mg cos θ₂ ÷ sin(θ₁ + θ₂), T₂ = mg cos θ₁ ÷ sin(θ₁ + θ₂). Atwood: T = 2m₁m₂g ÷ (m₁ + m₂), a = (m₂ − m₁)g ÷ (m₁ + m₂). Table: T = m₁m₂g ÷ (m₁ + m₂), a = m₂g ÷ (m₁ + m₂).",
  "assumptions": [
    "Light ropes that do not stretch, over pulleys with no mass and no friction; the table has no friction.",
    "The masses are at rest or move in a straight line along the rope; for two ropes the weight hangs still."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "setup": {
        "title": "Setup",
        "description": "How the rope holds or pulls the masses.",
        "type": "string",
        "enum": [
          "hanging",
          "two",
          "atwood",
          "table"
        ]
      },
      "m": {
        "title": "Mass (m)",
        "description": "The mass the rope or ropes hold.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "m1": {
        "title": "Mass 1 (m₁)",
        "description": "Atwood: the mass on one side. Table: the block on the table.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "m2": {
        "title": "Mass 2 (m₂)",
        "description": "The hanging mass on the other side of the pulley.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "a": {
        "title": "Acceleration (a, m/s²)",
        "description": "How fast the mass speeds up: positive up, negative down, 0 at rest or at a steady speed.",
        "type": "number",
        "minimum": -1000,
        "maximum": 1000
      },
      "th1": {
        "title": "Angle of rope 1 (θ₁, degrees)",
        "description": "The angle of the first rope above level, more than 0° and up to 90°.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 90
      },
      "th2": {
        "title": "Angle of rope 2 (θ₂, degrees)",
        "description": "The angle of the second rope above level, more than 0° and up to 90°.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 90
      },
      "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
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The tension (or both tensions) to 6 significant figures.",
      "format": "text"
    },
    "tension": {
      "label": "Tension (N)",
      "description": "T in newtons (rope 1 with two ropes).",
      "format": "number"
    },
    "lbf": {
      "label": "Tension (lbf)",
      "description": "T in pounds-force (rope 1 with two ropes).",
      "format": "number"
    },
    "tension2": {
      "label": "Tension in rope 2 (N)",
      "description": "T₂ in newtons.",
      "format": "number"
    },
    "lbf2": {
      "label": "Tension in rope 2 (lbf)",
      "description": "T₂ in pounds-force.",
      "format": "number"
    },
    "weight": {
      "label": "Weight (N)",
      "description": "m × g of the held mass, or m₂ × g of the hanging mass.",
      "format": "number"
    },
    "accel": {
      "label": "Acceleration (m/s²)",
      "description": "Atwood: m₂ moves down (negative: up). Table: the blocks speed up.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "setup": "hanging",
      "m": "70 kg",
      "m1": "2 kg",
      "m2": "4 kg",
      "a": 0,
      "th1": 30,
      "th2": 45,
      "g": 9.80665
    },
    "outputs": {
      "result": "T = 686.466 N (154.324 lbf)",
      "tension": 686.4655,
      "lbf": 154.3235835294143,
      "weight": 686.4655
    },
    "text": "T = 686.466 N (154.324 lbf)."
  },
  "examples": [
    {
      "given": {
        "setup": "hanging",
        "m": 70,
        "a": 0,
        "g": 9.80665
      },
      "expect": {
        "tension": 686.4655,
        "result": "T = 686.466 N (154.324 lbf)"
      },
      "source": "OpenStax, University Physics Volume 1, §5.6 Common Forces (a rope holding a weight: T = w = mg; Example 5.13: a 70.0 kg tightrope walker, wire at 5.0° each side, T = mg ÷ (2 sin 5.0°) = 3,930 N), https://openstax.org/books/university-physics-volume-1/pages/5-6-common-forces (retrieved 2026-10-19); hand calculation in content.mdx: T = 70 × 9.80665 = 686.4655 N"
    },
    {
      "given": {
        "setup": "hanging",
        "m": 75,
        "a": 1.2,
        "g": 9.8
      },
      "expect": {
        "tension": 825
      },
      "source": "OpenStax, University Physics Volume 1, §6.1 Solving Problems with Newton’s Laws (Example 6.1: a 15.0 kg traffic light on wires at 30° and 45°, T₁ = 108 N and T₂ = 132 N; Example 6.3: 75.0 kg at 1.20 m/s² up, 825 N; Example 6.4: a = m₂g ÷ (m₁ + m₂), T = m₁m₂g ÷ (m₁ + m₂); Example 6.5: an Atwood machine of 2.00 kg and 4.00 kg, a = 3.27 m/s², T = 26.1 N), https://openstax.org/books/university-physics-volume-1/pages/6-1-solving-problems-with-newtons-laws (retrieved 2026-10-19), Example 6.3 (the same sum for a scale in an elevator); hand calculation in content.mdx: T = 75 × (9.8 + 1.2) = 825 N"
    },
    {
      "given": {
        "setup": "two",
        "m": 15,
        "th1": 30,
        "th2": 45,
        "g": 9.8
      },
      "expect": {
        "tension": 107.61146871262497,
        "tension2": 131.79659440870387
      },
      "source": "OpenStax, University Physics Volume 1, §6.1 Solving Problems with Newton’s Laws (Example 6.1: a 15.0 kg traffic light on wires at 30° and 45°, T₁ = 108 N and T₂ = 132 N; Example 6.3: 75.0 kg at 1.20 m/s² up, 825 N; Example 6.4: a = m₂g ÷ (m₁ + m₂), T = m₁m₂g ÷ (m₁ + m₂); Example 6.5: an Atwood machine of 2.00 kg and 4.00 kg, a = 3.27 m/s², T = 26.1 N), 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: T₁ = 147 × cos 45° ÷ sin 75° = 107.6 N, T₂ = 147 × cos 30° ÷ sin 75° = 131.8 N"
    },
    {
      "given": {
        "setup": "two",
        "m": 70,
        "th1": 5,
        "th2": 5,
        "g": 9.8
      },
      "expect": {
        "tension": 3935.4836432647603,
        "tension2": 3935.4836432647603
      },
      "source": "OpenStax, University Physics Volume 1, §5.6 Common Forces (a rope holding a weight: T = w = mg; Example 5.13: a 70.0 kg tightrope walker, wire at 5.0° each side, T = mg ÷ (2 sin 5.0°) = 3,930 N), https://openstax.org/books/university-physics-volume-1/pages/5-6-common-forces (retrieved 2026-10-19); hand calculation in content.mdx: T = 686 × cos 5° ÷ sin 10° = 686 ÷ (2 sin 5°) = 3,935 N (OpenStax: 3,930 N)"
    },
    {
      "given": {
        "setup": "atwood",
        "m1": 2,
        "m2": 4,
        "g": 9.8
      },
      "expect": {
        "tension": 26.133333333333333,
        "a": 3.2666666666666666
      },
      "source": "OpenStax, University Physics Volume 1, §6.1 Solving Problems with Newton’s Laws (Example 6.1: a 15.0 kg traffic light on wires at 30° and 45°, T₁ = 108 N and T₂ = 132 N; Example 6.3: 75.0 kg at 1.20 m/s² up, 825 N; Example 6.4: a = m₂g ÷ (m₁ + m₂), T = m₁m₂g ÷ (m₁ + m₂); Example 6.5: an Atwood machine of 2.00 kg and 4.00 kg, a = 3.27 m/s², T = 26.1 N), 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: a = (4 − 2) × 9.8 ÷ 6 = 3.27 m/s², T = 2 × 2 × 4 × 9.8 ÷ 6 = 26.1 N"
    },
    {
      "given": {
        "setup": "table",
        "m1": 5,
        "m2": 3,
        "g": 9.8
      },
      "expect": {
        "tension": 18.375,
        "a": 3.675
      },
      "source": "OpenStax, University Physics Volume 1, §6.1 Solving Problems with Newton’s Laws (Example 6.1: a 15.0 kg traffic light on wires at 30° and 45°, T₁ = 108 N and T₂ = 132 N; Example 6.3: 75.0 kg at 1.20 m/s² up, 825 N; Example 6.4: a = m₂g ÷ (m₁ + m₂), T = m₁m₂g ÷ (m₁ + m₂); Example 6.5: an Atwood machine of 2.00 kg and 4.00 kg, a = 3.27 m/s², T = 26.1 N), https://openstax.org/books/university-physics-volume-1/pages/6-1-solving-problems-with-newtons-laws (retrieved 2026-10-19), Example 6.4 (a = m₂g ÷ (m₁ + m₂), T = m₁m₂g ÷ (m₁ + m₂)); hand calculation in content.mdx: T = 5 × 3 × 9.8 ÷ 8 = 18.375 N"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §5.6 Common Forces: tension in a rope holding a weight, T = w = mg; Example 5.13 (a 70.0 kg tightrope walker on a wire at 5.0° each side: T = mg ÷ (2 sin 5.0°) = 3,930 N, almost six times the 686 N weight), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/5-6-common-forces",
    "OpenStax, University Physics Volume 1, §6.1 Solving Problems with Newton’s Laws: Example 6.1 (a 15.0 kg traffic light on wires at 30° and 45°: T₁ = 108 N, T₂ = 132 N); Example 6.3 (75.0 kg at 1.20 m/s² up: 825 N); Example 6.4 (a block on a frictionless table and a hanging block: a = m₂g ÷ (m₁ + m₂), T = m₁m₂g ÷ (m₁ + m₂)); Example 6.5 (an Atwood machine of 2.00 kg and 4.00 kg: a = 3.27 m/s², T = 26.1 N), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/6-1-solving-problems-with-newtons-laws",
    "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, retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "normal-force",
    "friction",
    "pulley",
    "force"
  ],
  "changelog": []
}