{
  "id": "spring-constant",
  "version": "0b5bb88df0c9",
  "status": "published",
  "name": "Spring Constant Calculator",
  "question": "What is the spring constant?",
  "summary": "Finds a spring’s constant k by Hooke’s law, k = F ÷ x, from a force or a hanging mass and the stretch, or from the bounce period of a mass, k = 4π²m ÷ T², with the stored energy.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/spring-constant-calculator",
  "markdown": "https://www.acalculator.org/physics/spring-constant-calculator.md",
  "kind": "function",
  "method": "k = F ÷ x (Hooke’s law), with F = m × g for a hanging mass; k = 4π² × m ÷ T² from the period of a bouncing mass. U = ½ × F × x.",
  "assumptions": [
    "The spring obeys Hooke’s law: the force grows in proportion to the stretch, within its elastic range.",
    "The spring’s own mass is small next to the mass on it (for the bounce time).",
    "Typed values are read as the exact decimals you typed; each result is rounded once to show."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "by": {
        "title": "I know",
        "description": "What you measured: a force and the stretch, a hanging mass and the stretch, or a mass and its bounce time.",
        "type": "string",
        "enum": [
          "force",
          "mass",
          "period"
        ]
      },
      "F": {
        "title": "Force (F)",
        "description": "The force that stretches or compresses the spring.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "force",
        "minimum": 1e-9,
        "maximum": 1000000000000
      },
      "m": {
        "title": "Mass (m)",
        "description": "The mass hanging from the spring, or the mass bouncing on it.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 1e-9,
        "maximum": 1000000000000
      },
      "g": {
        "title": "Gravity (g), m/s²",
        "description": "The acceleration of gravity: 9.80665 m/s² is standard gravity on Earth.",
        "type": "number",
        "minimum": 0.000001,
        "maximum": 1000000
      },
      "x": {
        "title": "Stretch or compression (x)",
        "description": "How far the spring stretches or compresses from its rest length.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-9,
        "maximum": 1000000
      },
      "T": {
        "title": "Time of one bounce (T)",
        "description": "The period: the time for one full up-and-down bounce of the mass.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "time",
        "minimum": 1e-9,
        "maximum": 1000000000
      }
    }
  },
  "outputs": {
    "k": {
      "label": "Spring constant (k), N/m",
      "description": "k in newtons per metre.",
      "format": "number"
    },
    "kNmm": {
      "label": "Spring constant, N/mm",
      "description": "k in newtons per millimetre: k ÷ 1,000.",
      "format": "number"
    },
    "kLbfIn": {
      "label": "Spring constant, lbf/in",
      "description": "k in pounds-force per inch, the spring rate unit used in the US.",
      "format": "number"
    },
    "force": {
      "label": "Force on the spring (F), N",
      "description": "The force in newtons: as typed, or the hanging weight m × g.",
      "format": "number"
    },
    "energy": {
      "label": "Energy stored (U), J",
      "description": "The elastic potential energy at this stretch: U = ½kx² = ½Fx.",
      "format": "number"
    },
    "omega": {
      "label": "Angular frequency (ω), rad/s",
      "description": "ω = 2π ÷ T = √(k ÷ m).",
      "format": "number"
    },
    "freq": {
      "label": "Frequency (f), Hz",
      "description": "Bounces per second: f = 1 ÷ T.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "by": "force",
      "F": "10 N",
      "m": "2 kg",
      "g": 9.80665,
      "x": "2 cm",
      "T": "1 s"
    },
    "outputs": {
      "k": 500,
      "kNmm": 0.5,
      "kLbfIn": 2.855073577366323,
      "force": 10,
      "energy": 0.1
    },
    "text": "The spring constant is 500 N/m (2.85507 lbf/in)."
  },
  "examples": [
    {
      "given": {
        "by": "mass",
        "m": 80,
        "g": 9.8,
        "x": 0.012
      },
      "expect": {
        "F": 784,
        "k": 65333.333333333336,
        "energy": 4.704
      },
      "source": "OpenStax, College Physics 2e, §16.1 Hooke’s Law: Stress and Strain Revisited (Eq. 16.1 F = −kx; Eq. 16.4 PE = ½kx²; Example 16.1: an 80.0 kg person settles a car 1.20 cm, k = 6.53 × 10⁴ N/m; Example 16.2: k = 50.0 N/m compressed 0.150 m stores 0.563 J), https://openstax.org/books/college-physics-2e/pages/16-1-hookes-law-stress-and-strain-revisited (retrieved 2026-10-19); hand calculation in content.mdx: F = 80 × 9.8 = 784 N, k = 784 ÷ 0.012 = 65,333.3 N/m (6.53 × 10⁴)"
    },
    {
      "given": {
        "by": "force",
        "F": 7.5,
        "x": 0.15
      },
      "expect": {
        "k": 50,
        "energy": 0.5625,
        "kNmm": 0.05
      },
      "source": "OpenStax, College Physics 2e, §16.1 Hooke’s Law: Stress and Strain Revisited (Eq. 16.1 F = −kx; Eq. 16.4 PE = ½kx²; Example 16.1: an 80.0 kg person settles a car 1.20 cm, k = 6.53 × 10⁴ N/m; Example 16.2: k = 50.0 N/m compressed 0.150 m stores 0.563 J), https://openstax.org/books/college-physics-2e/pages/16-1-hookes-law-stress-and-strain-revisited (retrieved 2026-10-19); hand calculation in content.mdx: k = 7.5 ÷ 0.15 = 50 N/m, U = ½ × 7.5 × 0.15 = 0.5625 J (0.563 J)"
    },
    {
      "given": {
        "by": "period",
        "m": 2,
        "T": 1.57
      },
      "expect": {
        "k": 32.0324699617489,
        "freq": 0.6369426751592356,
        "omega": 4.002028858076169
      },
      "source": "OpenStax, University Physics Volume 1, §15.1 Simple Harmonic Motion (Eqs. 15.9 to 15.11, T = 2π√(m/k); Example 15.2: 2.00 kg on k = 32.00 N/m has T = 1.57 s), https://openstax.org/books/university-physics-volume-1/pages/15-1-simple-harmonic-motion (retrieved 2026-10-19); hand calculation in content.mdx: k = 4π² × 2 ÷ 1.57² = 32.0325 N/m",
      "tolerance": 1e-14
    },
    {
      "given": {
        "by": "force",
        "F": 44.482216152605,
        "x": 0.0254
      },
      "expect": {
        "kLbfIn": 10,
        "k": 1751.2683524647637,
        "F": 44.482216152605
      },
      "source": "NIST Special Publication 811, Appendix B.8 (1 lbf = 4.4482216152605 N, 1 in = 0.0254 m, standard gravity 9.80665 m/s²), https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8 (retrieved 2026-10-19); hand calculation in content.mdx: k = 44.482216152605 N ÷ 0.0254 m = 1,751.27 N/m = 10 lbf/in"
    },
    {
      "given": {
        "by": "mass",
        "m": 1,
        "g": 9.80665,
        "x": 0.1
      },
      "expect": {
        "F": 9.80665,
        "k": 98.0665,
        "energy": 0.4903325
      },
      "source": "NIST Special Publication 811, Appendix B.8 (1 lbf = 4.4482216152605 N, 1 in = 0.0254 m, standard gravity 9.80665 m/s²), https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8 (retrieved 2026-10-19); OpenStax, College Physics 2e, §16.1 Hooke’s Law: Stress and Strain Revisited (Eq. 16.1 F = −kx; Eq. 16.4 PE = ½kx²; Example 16.1: an 80.0 kg person settles a car 1.20 cm, k = 6.53 × 10⁴ N/m; Example 16.2: k = 50.0 N/m compressed 0.150 m stores 0.563 J), https://openstax.org/books/college-physics-2e/pages/16-1-hookes-law-stress-and-strain-revisited (retrieved 2026-10-19); hand calculation in content.mdx: k = 1 × 9.80665 ÷ 0.1 = 98.0665 N/m"
    }
  ],
  "sources": [
    "OpenStax, College Physics 2e, §16.1 Hooke's Law: Stress and Strain Revisited: Eq. 16.1 (F = −kx), Eq. 16.4 (PE = ½kx²), Example 16.1 (k = 6.53 × 10⁴ N/m for a car that settles 1.20 cm under 80.0 kg), Example 16.2 (k = 50.0 N/m compressed 0.150 m stores 0.563 J), retrieved 2026-10-19. https://openstax.org/books/college-physics-2e/pages/16-1-hookes-law-stress-and-strain-revisited",
    "OpenStax, University Physics Volume 1, §15.1 Simple Harmonic Motion: Eqs. 15.9 to 15.11 (ω = √(k/m), T = 2π√(m/k), f = 1/T), Example 15.2 (2.00 kg on a 32.00 N/m spring, T = 1.57 s), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/15-1-simple-harmonic-motion",
    "NIST Special Publication 811, Appendix B.8: 1 lbf = 4.4482216152605 N, 1 in = 0.0254 m, 1 lb = 0.45359237 kg, standard gravity 9.80665 m/s², retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8"
  ],
  "related": [
    "force",
    "potential-energy",
    "youngs-modulus",
    "frequency"
  ],
  "changelog": []
}