Last updated: 2026-06-14

Hooke's Law

Hooke's law states that the force needed to stretch or compress an elastic object is proportional to how far it is displaced from its natural length: $F = k \cdot x$. The force $F$ is measured in newtons, the displacement $x$ in metres, and the spring constant $k$ — the object's stiffness — in newtons per metre. Robert Hooke published the principle in 1678 as the Latin anagram ut tensio, sic vis, "as the extension, so the force."

This calculator solves $F = k \cdot x$ for any one of the three quantities and reports the elastic potential energy stored in the spring.

Force, stiffness, and displacement

The spring constant $k$ captures how stiff a spring is: a high value means a large force produces only a small stretch, while a low value means the spring extends easily. The displacement $x$ is always measured from the spring's relaxed, unloaded length — not from any other reference point. The spring responds with a restoring force equal in size and opposite in direction to the applied force, which is why the law is often written $F = -k \cdot x$ when describing that restoring force.

Formulas

The stored energy grows with the square of the displacement, so stretching a spring twice as far stores four times the energy.

Worked example

A spring with a stiffness of $k = 200$ N/m is stretched by $x = 0.1$ m. The force required is:

and the energy stored in the stretched spring is:

Setting the mode to "Find force" and entering 200 N/m and 0.1 m reproduces both values.

Measuring a spring constant

Hooke's law gives a simple way to measure stiffness: hang a known weight from a spring and record how far it stretches. A 0.5 kg mass exerts a force of about $0.5 \times 9.81 = 4.9$ N. If it stretches the spring by 0.025 m, the spring constant is $k = F / x = 4.9 / 0.025 \approx 196$ N/m. Repeating with several weights and plotting force against extension gives a straight line whose slope is $k$ — a standard physics laboratory exercise.

The elastic limit

Hooke's law is linear only within a material's elastic limit, the range where the object springs back to its original shape once the load is removed. Stretch a spring or wire too far and it passes its yield point, deforms permanently, and the straight-line relationship breaks down. Beyond that, the force needed grows nonlinearly until the material eventually fractures. Real springs also have a finite range of motion and obey the law only while their coils are neither fully compressed nor over-extended.