Hooke's law

Hooke's law is a physics rule based on observation, describing how the force required to stretch or compress a spring changes in direct proportion to the distance it moves. This relationship is written as Fs = kx, where k is a fixed number representing the spring's stiffness, and the movement x is small relative to the spring's total possible deformation. The law is named after the 17th-century British physicist Robert Hooke, who first presented it in 1676 as a Latin anagram and published the solution in 1678: *ut tensio, sic vis* ("as the extension, so the force"). Hooke claimed he had known the law since 1660. It serves as the core principle behind the spring scale, manometer, galvanometer, and the balance wheel in mechanical clocks.

The law applies in many cases where an elastic object is deformed. A material that follows this equation is called linear-elastic or Hookean. However, Hooke's law is only a first-order linear approximation of how springs and other elastic bodies actually respond to forces. It stops working once forces exceed a certain limit, because no material can be compressed beyond a minimum size or stretched beyond a maximum size without permanent damage or a change in state. Many materials noticeably deviate from Hooke's law well before reaching those elastic limits.

In modern elasticity theory, Hooke's law is generalized to state that the strain (deformation) of an elastic object or material is proportional to the stress applied to it. Because general stresses and strains can have multiple independent components, the proportionality factor is not a single number but a linear map (a tensor), often represented by a matrix of real numbers. This general form allows the relationship between strain and stress to be deduced for complex objects based on their material properties. For instance, a homogeneous rod with a uniform cross-section behaves like a simple spring when stretched, with stiffness k directly proportional to its cross-sectional area and inversely proportional to its length.

For a simple helical spring with one end fixed and the free end pulled by a force of magnitude Fs, once the spring reaches equilibrium, the displacement x of the free end from its relaxed position follows Fs = kx, or equivalently x = Fs/k. Here, k is a positive real number unique to the spring. A spring with spaces between its coils can also be compressed, with the same formula holding when Fs and x are both negative. The graph of applied force Fs versus displacement x is a straight line through the origin with slope k. Hooke's law is also often written with Fs as the restoring force exerted by the spring, giving Fs = -kx, because the restoring force acts opposite to the displacement.

The torsional version of Hooke's law applies to torsional springs, stating that the torque τ needed to rotate an object is directly proportional to the angular displacement θ from equilibrium. This describes the relationship between applied torque and resulting angular deformation due to torsion. Mathematically, τ = -kθ, where τ is torque in Newton-meters, k is the torsional constant (in N·m/radian) characterizing the spring's stiffness, and θ is the angular displacement in radians. The negative sign indicates the torque acts opposite to the displacement, providing a restoring force.

Hooke's spring law generally applies to any elastic object, no matter how complex, as long as both the deformation and stress can be expressed by a single positive or negative number. For example, when a rubber block between two parallel plates is sheared, the shearing force Fs and sideways displacement x obey Hooke's law for small deformations. Similarly, when a straight steel bar or concrete beam supported at both ends is bent by a weight F at an intermediate point, the displacement x—the beam's deviation in the transverse direction from its unloaded shape—follows the law.