Hooke's Law (Spring Force)

Multiply the spring constant k by the displacement x from the rest position: F = k·x. Insert k in newtons per metre and x in metres, then you get the force in newtons. Example: a spring with k = 200 N/m stretched by x = 0.05 m exerts F = 200·0.05 = 10 N. The spring constant describes the stiffness; a large value means a hard spring that needs a greater force for the same stretch. Make sure to measure x as the displacement from the relaxed position, not as the total length of the spring. The stored elastic energy is E = ½·k·x².

The spring constant k is a measure of the stiffness of a spring and states how much force is needed to displace it by a certain distance. It has the unit newtons per metre. A large k means a hard, stiff spring that is hard to stretch; a small k a soft, yielding spring. Graphically k is the slope of the line in the force-displacement diagram: plotting F against x gives, in the elastic range, a line through the origin with slope k. From two measured values of F and x, k can be determined as the ratio F/x. The spring constant depends on the material, shape and length of the spring.

Hooke law applies only in the linear elastic range, where the force is proportional to the displacement and the spring returns to its original shape after unloading. If you stretch too far, you exceed the limit of proportionality, after which the force no longer rises linearly with displacement. Under even greater load plastic deformation sets in: the material stays permanently deformed and no longer returns. At the very end comes fracture. A common mistake is to apply the linear law even at large strains. The limits depend on the material; steel stays linear for a long time, rubber shows nonlinear behaviour early on.

The energy stored in a stretched spring is the work done against the spring force while stretching it. Because the force grows linearly with the displacement, the average force is just half of the final force. From this follows the elastic energy E = ½·k·x². It rises quadratically with the displacement: at double stretch the stored energy is four times as large. Example: a spring with k = 200 N/m, stretched by 0.05 m, stores E = ½·200·0.0025 = 0.25 J. Graphically this energy corresponds to the triangular area under the line in the force-displacement diagram. On relaxing, the spring gives back exactly this energy.

When you combine springs, the effective spring constant changes depending on the arrangement. In a parallel arrangement, where the springs side by side share the same displacement, the spring constants add: k_total = k₁ + k₂. The combination becomes stiffer. In a series arrangement, where the springs hang one behind another and carry the same force, the reciprocals add: 1/k_total = 1/k₁ + 1/k₂. The combination becomes softer than the softest single spring. This is analogous to resistors in electrical engineering, only with the roles swapped. Two equal springs in parallel double the stiffness, in series they halve it.