Hooke's Law Calculator
Find the force, spring constant or extension of a spring from F = kx, combine springs in series or parallel, and see when you pass the elastic limit.
🌀 What is Hooke's Law?
Hooke's law describes how an ideal spring responds when you stretch or compress it. The force needed is directly proportional to how far the spring is moved from its relaxed length: F = k x. The constant k is the spring constant, or stiffness, measured in newtons per metre. A stiff spring has a large k and needs a lot of force for a small extension, and a soft spring has a small k. The spring pulls back with an equal restoring force pointing the other way, which is why the law is often written F = -k x. It is named after the 17th century scientist Robert Hooke and is the starting point for almost every spring problem in physics and engineering.
This calculator solves the law for whichever quantity you do not know. Give it k and the extension and it returns the force. Give it a force and an extension, for example from a weighing test, and it returns the spring constant. Give it a force and k and it returns how far the spring will stretch. You can work in N/m, N/cm, kN/m or lbf/in for stiffness, in metres, centimetres, millimetres or inches for length, and in newtons, kilonewtons or pound-force for load. A second mode combines up to four springs in series or in parallel and shows how the applied force and the extension are shared between them. Typical uses are spring scales, suspension and mattress springs, retractable pens, door closers, trampolines and lab experiments that measure k from a hanging mass.
Hooke's law is a model, not a law of nature that always holds. It works for small deformations. Every real spring has an elastic limit, the extension beyond which it stays permanently deformed and the straight-line relationship breaks down. A coil spring also stops being linear when its coils touch. Rubber bands and biological tissue are non-linear almost from the start. Enter an elastic limit in the optional box and the calculator warns you when the extension passes it and shades that region on the chart. The model also describes only the spring, not the object attached to it: how that object moves is a separate question, and the elastic potential energy calculator covers the energy and launch speed side, while the extension spring calculator and compression spring calculator design real coil springs.
A common mix-up is between the extension and the length. The x in Hooke's law is the change from the relaxed length, not the total stretched length. Another is thinking that the spring constant depends on how far you stretch the spring. It does not, it depends on the wire, the coil diameter and the number of coils. Finally, for a hanging mass at rest the spring force equals the weight, which is the same balance of forces handled by the net force calculator.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Force on a Spring
A spring with k = 200 N/m is stretched 5 cm
Example 2 — Spring Constant in Imperial Units
A 10 lbf load stretches a spring 2 in
Example 3 — Extension Past the Elastic Limit
A 15 N force on a spring with k = 300 N/m and an elastic limit of 4 cm
Example 4 — Three Springs in Series
Springs of 200, 300 and 600 N/m in series under a 20 N load
Example 5 — Two Springs in Parallel
Springs of 150 and 250 N/m side by side under a 40 N load
❓ Frequently Asked Questions
🔗 Related Calculators
What is Hooke's law and what is the formula?
Hooke's law says the force needed to stretch or compress a spring is proportional to the distance it is moved from its relaxed length: F = k x. The spring pulls back with the restoring force F = -k x. For k = 200 N/m and x = 0.05 m the force is 10 N.
How do you calculate the spring constant?
Divide the force by the extension: k = F / x. A 10 lbf load that stretches a spring 2 in gives k = 5 lbf/in, which is 875.63 N/m. The spring constant is always positive and is measured in newtons per metre in SI units.
How do you find the extension of a spring?
Divide the force by the spring constant: x = F / k. A 15 N force on a spring with k = 300 N/m stretches it 0.05 m, which is 5 cm. Double the force and the extension doubles, as long as the spring stays inside its elastic limit.
What are the units of the spring constant?
The SI unit is newtons per metre (N/m). Catalogues also use N/mm, N/cm, kN/m and lbf/in. One lbf/in equals 175.127 N/m, so a 5 lbf/in spring is 875.63 N/m.
How do you combine springs in series?
For springs in series the reciprocals add: 1/k_eq = 1/k1 + 1/k2 + 1/k3. Every spring carries the full force and the extensions add. Springs of 200, 300 and 600 N/m in series give 100 N/m, softer than any of the three.
How do you combine springs in parallel?
For springs in parallel the constants add: k_eq = k1 + k2. All springs share the same extension and the force splits in proportion to stiffness. Springs of 150 and 250 N/m give 400 N/m, and a 40 N load stretches both 0.1 m.
What is the elastic limit of a spring?
The elastic limit is the largest extension from which a spring still returns to its original length. Beyond it the spring deforms permanently and the force is no longer proportional to the extension, so Hooke's law stops being valid.
Is Hooke's law always true?
No. It is a linear approximation that works for small deformations of springs and many elastic solids. Rubber, soft tissue and any spring pushed near its limit show non-linear behaviour, and a coil spring that closes up to solid height stops behaving linearly.
How much energy is stored in a stretched spring?
The elastic potential energy is U = 1/2 k x squared. With k = 200 N/m and x = 0.05 m it is 0.25 J. It is the area under the force-extension line, which is why it grows with the square of the extension.
Why is there a minus sign in F = -kx?
The minus sign shows that the spring's force is a restoring force that points opposite to the displacement. Stretch the spring to the right and it pulls left. The magnitude is still k times the distance, which is what this calculator reports.