# What is the spring constant?

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.

- Page: https://www.acalculator.org/physics/spring-constant-calculator
- JSON spec: https://www.acalculator.org/physics/spring-constant-calculator.json
- Version: 0b5bb88df0c9

## Default answer

Example with the default inputs (I know Force and stretch, Force (F) 2.24808943099711 lbf, Stretch or compression (x) 0.78740157480315 in): The spring constant is 500 N/m (2.85507 lbf/in).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| by | I know | What you measured: a force and the stretch, a hanging mass and the stretch, or a mass and its bounce time. |
| F | Force (F) | The force that stretches or compresses the spring. |
| m | Mass (m) | The mass hanging from the spring, or the mass bouncing on it. |
| g | Gravity (g), m/s² | The acceleration of gravity: 9.80665 m/s² is standard gravity on Earth. |
| x | Stretch or compression (x) | How far the spring stretches or compresses from its rest length. |
| T | Time of one bounce (T) | The period: the time for one full up-and-down bounce of the mass. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| k | Spring constant (k), N/m | k in newtons per metre. |
| kNmm | Spring constant, N/mm | k in newtons per millimetre: k ÷ 1,000. |
| kLbfIn | Spring constant, lbf/in | k in pounds-force per inch, the spring rate unit used in the US. |
| force | Force on the spring (F), N | The force in newtons: as typed, or the hanging weight m × g. |
| energy | Energy stored (U), J | The elastic potential energy at this stretch: U = ½kx² = ½Fx. |
| omega | Angular frequency (ω), rad/s | ω = 2π ÷ T = √(k ÷ m). |
| freq | Frequency (f), Hz | Bounces per second: f = 1 ÷ T. |

## 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.

## Worked examples

1. by = mass, m = 80, g = 9.8, x = 0.012 gives F = 784, k = 65,333.3, 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).
2. by = force, F = 7.5, x = 0.15 gives 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).
3. by = period, m = 2, T = 1.57 gives k = 32.0325, freq = 0.636943, omega = 4.00203. 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).
4. by = force, F = 44.4822, x = 0.0254 gives kLbfIn = 10, k = 1,751.27, F = 44.4822. 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).
5. by = mass, m = 1, g = 9.80665, x = 0.1 gives F = 9.80665, k = 98.0665, energy = 0.490333. 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).

## FAQ

### How do you find the spring constant?

Divide the force by the stretch it causes: k = F ÷ x (Hooke's law). A 7.5 N force that compresses a spring 0.15 m gives k = 7.5 ÷ 0.15 = 50 N/m.

### How do I find the spring constant from a hanging mass?

The hanging weight is the force: F = m × g. An 80.0 kg person who settles a car 1.20 cm gives F = 80 × 9.8 = 784 N and k = 784 ÷ 0.012 = 65,333 N/m, the 6.53 × 10⁴ N/m of OpenStax Example 16.1.

### How do I find the spring constant from the period of oscillation?

A mass m bouncing on a spring has period T = 2π√(m ÷ k), so k = 4π² × m ÷ T². A 2.00 kg mass that bounces once every 1.57 s gives k = 4π² × 2 ÷ 1.57² = 32.03 N/m.

### What are the units of the spring constant?

Force per length: newtons per metre (N/m) in SI. Spring makers also quote N/mm (1 N/mm = 1,000 N/m) and, in the US, pounds-force per inch (1 lbf/in = 175.127 N/m).

### How much energy does a spring store?

U = ½kx², or ½ × F × x at the same stretch. A 50 N/m spring compressed 0.15 m stores ½ × 50 × 0.15² = 0.5625 J, the 0.563 J of OpenStax Example 16.2.

### Why is there a minus sign in Hooke’s law?

F = −kx gives the force the spring pushes back with: it points opposite to the stretch. The spring constant itself is positive, so the page works with sizes: k = F ÷ x.

## 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