Hooke's law

Statement

F = −k·x — the elastic force exerted on a body attached to a spring is proportional to the displacement x and opposite to it. The spring constant k (N/m) measures its stiffness.

Why is it true?

Within the elastic limit, x is small so the restoring force is linear — which is why every small oscillation around equilibrium is approximately simple harmonic: springs, small-angle pendulums, vibrating molecules, crystal lattices.

Beyond the elastic limit (plastic flow or over-extension), F(x) is no longer linear and the physics turns nonlinear.

Applications

Spring scales, vehicle suspension, clock hairsprings and balance wheels, MEMS force sensors, the harmonic-oscillator model in solid-state physics.

Topics that use this theorem

Step-by-step proofs