Hooke’s Law CalculatorSpring constant · F = kx
Hooke’s law says a spring’s force is proportional to how far you stretch or compress it: F = kx, where k is the spring constant. Enter any two of force, spring constant and extension to find the third. You can also hang a known mass to measure k. The results include the elastic energy stored (½kx²) and how fast a mass would bounce on the spring.
Formula verified against OpenStax — University Physics, Volume 1 · HyperPhysics, Georgia State University
By Manzoor Ahmad · Pharmacist & calculator developerReviewed & updated
Spring values
Extension x is the change from the spring’s natural length, not its total length.
Step-by-step working
What is hooke’s law?
Hooke’s law describes how springs and other elastic objects behave: the restoring force is proportional to the displacement from the natural length. The constant of proportionality, k, measures stiffness.
Robert Hooke published it in 1678 as the anagram “ceiiinosssttuv”, Latin for “as the extension, so the force.” It underpins spring scales, vehicle suspension, seismometers and the oscillation of atoms in solids.
Quick answers
How do you calculate the spring constant?
Divide the force by the extension: k = F ÷ x. If 15 N stretches a spring 3 cm, k = 15 ÷ 0.03 = 500 N/m. Use the hanging-mass tab if you only know the mass.
What does the spring constant tell you?
How stiff the spring is: the force needed per metre of stretch. A spring with k = 500 N/m needs 5 N to stretch it 1 cm; one with k = 50 N/m needs only 0.5 N.
How do you calculate the energy stored in a spring?
E = ½kx². A 200 N/m spring compressed 10 cm stores ½ × 200 × 0.01 = 1 J. That energy can turn into kinetic energy when released.
What happens to k when springs are combined?
In series, 1/k = 1/k₁ + 1/k₂, so the pair is softer. In parallel, k = k₁ + k₂, so it is stiffer. This is the reverse of how resistors combine.
Hooke’s law formula
The restoring force acts opposite to the displacement, which is why textbooks often write F = −kx.
| Symbol | Meaning | SI unit |
|---|---|---|
| F | Spring force | N |
| k | Spring constant (stiffness) | N/m |
| x | Extension or compression from natural length | m |
| E | Elastic potential energy | J |
| T | Period of a mass bouncing on the spring | s |
How to find the spring constant
- Hang a known mass from the spring and measure how far it stretches.
- Work out the force: F = mg.
- Divide force by extension (in metres): k = F ÷ x.
- Repeat with several masses and plot F against x. The gradient of the straight line is k.
Worked example: measuring k with a 500 g mass
A 500 g mass stretches a spring by 4.9 cm. What is the spring constant?
k = F ÷ x = 4.903 ÷ 0.049 = 100.1 N/m
Stored energy: E = ½ × 100.1 × 0.049² = 0.120 J
Pull the mass down and let go, and it bounces with a period of 2π√(0.5 ÷ 100.1) ≈ 0.444 s.
Typical spring constants
Order-of-magnitude values; real springs vary widely with wire size and coil count.
| Spring | k (N/m) | k (lbf/in) |
|---|---|---|
| Ballpoint pen | ≈ 500 | ≈ 2.9 |
| School lab spring | 10–100 | 0.06–0.6 |
| Trampoline spring | ≈ 5,000 | ≈ 29 |
| Car coil spring | 20,000–50,000 | 115–285 |
| Truck leaf spring | ≈ 200,000 | ≈ 1,140 |
Common mistakes to avoid
- Using the total length instead of the extension. x is measured from the natural, unloaded length.
- Leaving x in centimetres. 5 cm is 0.05 m. Getting this wrong makes k a hundred times too small.
- Applying Hooke’s law beyond the elastic limit. Past it the spring deforms permanently and the line on the graph curves.
Manzoor Ahmad
Pharmacist turned Web Developer & Calculator Engineer · 13+ years in web developmentI’m a pharmacist by training who turned into a full-time tech geek, and for 13+ years I’ve designed and coded web tools. My pharmacy degree included physical pharmacy and pharmaceutics — states of matter, heat, flow and reaction kinetics — so the physics behind these tools is familiar ground, not a new hobby.
I’m not a physicist, so I don’t ask you to take my word for it: every formula on this page is traced to an open university textbook or NIST reference data, and the calculator is tested against published worked examples before it goes live. Spot an error? Tell me and it will be fixed and dated on the next update.
Hooke’s Law Calculator: frequently asked questions
Hooke’s law states that the force needed to stretch or compress a spring is directly proportional to the distance: F = kx, as long as the spring stays within its elastic limit.
Newtons per metre (N/m). Engineers also use N/mm and lbf/in.
It shows the spring’s restoring force points opposite to the displacement. Stretch it down, and it pulls up.
Only roughly, and only for small stretches. Rubber is non-linear, so its force–extension curve bends.
A force proportional to displacement and opposite to it produces simple harmonic motion, with period T = 2π√(m ÷ k).
References & sources
- OpenStax — University Physics, Volume 1 — Ch. 7.1 and 15.1, springs and simple harmonic motion
- HyperPhysics, Georgia State University — elastic potential energy
- OpenStax — College Physics 2e — Ch. 5.3, Elasticity and Hooke’s Law
Disclaimer: This calculator is an educational tool built on idealised textbook models. It is not a substitute for professional engineering or safety calculations.