Stretch a spring and it pulls back. Hooke’s law says that pull is proportional to how far you stretched it, at least while the deformation stays modest.
k is the spring constant in newtons per metre — a stiff spring has a large k. The minus sign says the force opposes the displacement: pull the spring right and it pulls left. When you are only after magnitudes the sign can be dropped, but it is what makes the spring a restoring force and therefore what makes oscillation possible.
Energy stored
Because the force grows as you stretch, the work done is not simply force times distance. Integrating F dx from 0 to x gives:
The square matters. Stretching twice as far stores four times the energy. This is why a slightly over-drawn bow or an over-compressed spring releases far more energy than intuition suggests.
Springs in series and parallel
The arithmetic is the reverse of resistors, which catches people out. Springs side by side share the load, so their constants add: k = k₁ + k₂. Springs end to end each stretch under the full force, so they combine reciprocally: 1/k = 1/k₁ + 1/k₂. Two identical springs in series make a system half as stiff and twice as extensible.
Oscillation
Hang a mass on a spring and release it, and it oscillates with period:
The amplitude does not appear — a large swing and a small swing take the same time. This is simple harmonic motion, and it emerges from any restoring force proportional to displacement, which is why the same mathematics describes pendulums, LC circuits and vibrating molecules.
The elastic limit
Hooke’s law is an approximation valid over a limited range. Stretch a spring past its elastic limit and it deforms permanently — released, it no longer returns to its original length, and k is no longer meaningful. Push further and it reaches the breaking point.
Rubber is a useful counterexample: it stores and returns energy but is markedly non-linear, with a force curve that stiffens sharply at high extension. Treating a rubber band as a Hookean spring gives poor answers outside a narrow range.