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Hooke's Law Calculator

Free Hooke's law calculator — F = k·x for springs; solve for force, spring constant, or displacement, with elastic energy and an animated spring.

F = k·x — force is proportional to displacement from the natural length. A negative displacement is compression.

Spring constant k (N/m)Displacement x (m)Mass on the spring (kg) — optional

The shaded triangle fills as the spring stretches — its area is the energy stored.

10.00 N

force

Energy stored

250.00 mJ

½kx²

At twice the stretch

1.00 J

four times as much

Natural frequency

—

needs a mass

Static sag under that mass

—

F = k·x = 200 N/m × 0.05 = 10.00 N

Energy, as ½Fx250.00 mJ

Two of these in series100 N/m — softer

Two of these in parallel400 N/m — stiffer

Directionstretched — the spring pulls back

Instant insight

10.00 N stretches it by 50.00 mm
F = k·x, with k = 200 N/m. The spring constant is the force per metre of deflection: a stiffer spring needs more force for the same movement, and the relationship is a straight line through the origin — which is the entire content of the law.
It is storing 250.00 mJ — the triangle under the line
E = ½kx² = ½ × 200 N/m × 0.05². Because force rises linearly from zero, the average force over the stretch is half the final one, so the work done is ½Fx — the area of a triangle, which is what the shaded region in the plot is. It is not an illustration of the energy; it is the energy.
Stretch it twice as far and it stores 1.00 J — four times as much
Energy goes as x², not x. Doubling the draw on a bow quadruples the energy delivered to the arrow, and the same quadratic surprise is why a spring taken a little past its usual range can fail much more violently than expected.
Two in series are softer; two in parallel are stiffer
Two of these end to end give 100 N/m — each takes the full force and stretches, so the deflections add. Side by side they give 400 N/m, because each takes half the load. This is the one people reverse, and the way to keep it straight is to ask what each spring actually experiences.
Hooke's law is a description, not a law of nature
It holds only up to the elastic limit. Past that a spring deforms permanently and never returns to its original length, and past the yield point it fails. Nothing in F = kx warns you where that is — the constant stays obligingly constant in the arithmetic long after the real spring has stopped obeying it.
For anything load-bearing, the elastic limit comes from the material data sheet, not from this equation.

Worked out in your browser — nothing is sent anywhere.

Compare results

Save this result, change your inputs, and recalculate to compare scenarios side by side.

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Everyday Uses

Choosing a spring

Find the constant needed for a given force at a given deflection.

Rearranging F = kx

Solve F = kx for force, constant or extension, and find stored energy.

Spring scales

Understand how a spring balance converts weight into a readable displacement.

Suspension design

Compare stiffness when springs are combined in series or parallel.

Resistance bands

Over their working range bands behave like springs, so doubling the stretch roughly doubles the resistance — useful for matching a band to a rehab or training load.

Weighing a load by how far it sags

Measure how much a vehicle's suspension or a shelf drops when loaded, and the spring constant converts that sag straight back into weight.

Frequently Asked Questions

What is Hooke's law?

The force a spring exerts is proportional to how far it is stretched or compressed: F = kx, where k is the spring constant in newtons per metre. A spring with k = 200 N/m pulls back with 20 N when stretched 10 cm. The minus sign often written in front reflects that the force opposes the displacement — a stretched spring pulls back, a compressed one pushes out.

What does the spring constant actually mean?

Stiffness. A high k means a stiff spring needing large force for small movement; a low k means a soft one. Car suspension springs are in the tens of thousands of N/m; a ballpoint pen spring is a few hundred. The constant depends on the material, wire thickness, coil diameter and number of turns — thicker wire and fewer coils both make a spring stiffer.

When does Hooke's law stop working?

Beyond the elastic limit. Up to that point a material returns to its original shape and the relationship stays linear. Past it, the material yields — it deforms permanently, and the graph of force against extension curves away from the straight line. Push further and it reaches ultimate tensile strength and fractures. Every real spring has this limit, which is why over-stretching one ruins it.

How do springs combine in series and parallel?

The opposite way round from resistors, which catches people out. Springs in parallel add their constants: two 100 N/m springs side by side give 200 N/m, stiffer. In series the reciprocals add: the same two end to end give 50 N/m, softer, because each stretches under the full load. Longer springs are softer for exactly this reason.

How is energy stored in a spring calculated?

Elastic potential energy is ½kx², the area under the force-extension graph. Because it depends on displacement squared, stretching twice as far stores four times the energy. A spring at 10 cm with k = 200 N/m stores 1 joule; at 20 cm it stores 4. This is what makes heavily compressed springs genuinely dangerous — the stored energy releases all at once.

Does Hooke's law apply beyond springs?

Yes, to any elastic deformation within limits — stretching a wire, bending a beam, compressing rubber. In materials science it is expressed as stress proportional to strain, with Young's modulus playing the role of k normalised for the material's dimensions. That generalisation is why the same idea underpins structural engineering, not just spring design.