Hooke's Law Calculator
Solve Hooke's law F = k·x for the spring force, spring constant, or displacement, and find the elastic potential energy stored in the spring.
How to use this calculator
- 1Choose whether you're solving for force, the spring constant, or displacement.
- 2Enter the two quantities you know.
- 3Read the third, computed from F = k·x.
- 4See the elastic potential energy stored in the spring.
How it works
Hooke's law
F = k × x F = spring force, k = spring constant, x = displacement k = F ÷ x, x = F ÷ k elastic potential energy = ½ k x²
Hooke's law describes how an elastic object, most simply a spring, responds to being stretched or compressed. It states that the restoring force the spring exerts is directly proportional to the displacement from its natural, unstretched length. The constant of proportionality is the spring constant k, a measure of stiffness: a large k means a stiff spring that pushes back hard for a small stretch, while a small k means a soft, easily stretched spring. Because the relationship is a simple proportion, any one of the three quantities can be found from the other two. The energy stored in the stretched or compressed spring — the elastic potential energy — is not linear but grows with the square of the displacement, given by one-half of k times the displacement squared, which is why stretching a spring twice as far stores four times the energy. The law holds only up to the elastic limit, beyond which the material deforms permanently.
Worked example
A spring with a stiffness of 200 N/m stretched by 0.1 m pulls back with a force of 200 × 0.1 = 20 newtons, and stores ½ × 200 × 0.1² = 1 joule of elastic energy. Stretch it twice as far, to 0.2 m, and the force doubles to 40 N but the stored energy quadruples to 4 J.
Hooke's Law Calculator: the complete guide
The straight-line law of springs
Hooke's law is one of the first quantitative laws a physics student meets, and its appeal is its simplicity: force is proportional to stretch. Named for the 17th-century scientist Robert Hooke, who first stated it, the law says that if you pull a spring twice as far, it pulls back twice as hard; three times as far, three times as hard. This straight-line relationship between force and displacement is what makes springs so predictable and useful, and it holds for a remarkable range of elastic materials, not just coiled metal springs.
The spring constant k is the slope of that straight line and encapsulates the spring's character in a single number. It is measured in units of force per distance — newtons per metre — telling you how many newtons of force each metre of stretch produces. Car suspension springs have a high constant to support the vehicle's weight without excessive travel; the delicate spring in a pen has a tiny one. Measuring k is often the first step in characterising any elastic system, and it is found simply by dividing a known force by the stretch it produces.
Energy stored in the stretch
When you stretch or compress a spring, the work you do is stored as elastic potential energy, ready to be released. This energy is not proportional to the displacement but to its square — half the spring constant times the displacement squared. The squared relationship has an important consequence: doubling the stretch quadruples the stored energy, and tripling it stores nine times as much. A spring pulled far back holds disproportionately more energy than one barely stretched.
This stored energy is what makes springs such versatile devices. It powers the recoil of a trampoline, the click of a pen, the drive of a wind-up toy, and the return of a valve in an engine. It cushions shocks in vehicle suspension by absorbing energy on compression and releasing it gradually. The same principle, generalised, underlies the storage of elastic energy in a drawn bow, a stretched rubber band, or a bent diving board. In every case the energy scales with the square of the deformation, which is why small increases in stretch yield large increases in stored, and then released, energy.
Where Hooke's law breaks down
Hooke's law is a superb approximation, but it is not universal, and knowing its limits matters. The law holds only within a material's elastic region — the range of deformation from which it springs fully back to its original shape. Stretch a spring or any elastic object beyond its elastic limit, and it enters plastic deformation, where it no longer returns to its natural length and the neat proportionality between force and stretch collapses. Push further still and the material eventually breaks. Beyond the elastic limit, the simple F = k·x no longer describes what is happening.
This is why the law applies to modest deformations and why engineers design springs and structures to operate well within the elastic range. It also explains why real materials show a straight force-displacement line only up to a point, after which the curve bends. Despite these limits, Hooke's law remains foundational far beyond springs: it is the basis for understanding elasticity in solids generally, for the theory of oscillations and simple harmonic motion (a mass on a spring bobs at a frequency set by k), and for the behaviour of everything from bridges flexing under load to atoms vibrating in a crystal. The humble spring equation is a gateway to a large part of physics.
Frequently asked questions
What is Hooke's law?
Hooke's law states that the force a spring exerts is proportional to its displacement from its natural length: F = k·x. Stretch it twice as far and it pulls back twice as hard. The spring constant k measures stiffness — a large k means a stiff spring. It holds within the elastic limit.
How do I find the spring constant?
Divide a known force by the displacement it produces: k = F ÷ x. If a 20 N force stretches a spring 0.1 m, then k = 20 ÷ 0.1 = 200 N/m. The units, newtons per metre, tell you how much force each metre of stretch produces.
How much energy does a stretched spring store?
The elastic potential energy is ½ k x² — half the spring constant times the displacement squared. Because it depends on the square of the stretch, doubling the displacement quadruples the stored energy. A 200 N/m spring stretched 0.1 m stores ½ × 200 × 0.01 = 1 joule.
When does Hooke's law not apply?
Beyond the elastic limit. If a spring or material is stretched too far, it deforms permanently (plastic deformation) or breaks, and force is no longer proportional to displacement. Hooke's law is a good approximation only for modest deformations within the elastic range.