In physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660. It is the fundamental principle behind the spring scale, the manometer, the galvanometer, and the balance wheel of the mechanical clock. The equation holds in many situations where an elastic body is deformed. An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean. Hooke's law is a first-order linear approximation to the real response of springs and other elastic bodies to applied forces. It fails once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those elastic limits are reached.
Definition The modern theory of elasticity generalizes Hooke's law to say that the strain (deformation) of an elastic object or material is proportional to the stress applied to it. However, since general stresses and strains may have multiple independent components, the "proportionality factor" may no longer be just a single real number, but rather a linear map (a tensor) that can be represented by a matrix of real numbers. In this general form, Hooke's law makes it possible to deduce the relation between strain and stress for complex objects in terms of intrinsic properties of the materials they are made of. For example, one can deduce that a homogeneous rod with uniform cross section will behave like a simple spring when stretched, with a stiffness k directly proportional to its cross-section area and inversely proportional to its length.
Linear springs
Consider a simple helical spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is Fs. Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. Let x be the amount by which the free end of the spring was displaced from its "relaxed" position (when it is not being stretched). Hooke's law states that F s = k x {\displaystyle F_{s}=kx} or, equivalently, x = F s k {\displaystyle x={\frac {F_{s}}{k}}}
where k is a positive real number, characteristic of the spring. A spring with spaces between the coils can be compressed, and the same formula holds for compression, with Fs and x both negative in that case.
According to this formula, the graph of the applied force Fs as a function of the displacement x will be a straight line passing through the origin, whose slope is k. Hooke's law for a spring is also stated under the convention that Fs is the restoring force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes F s = − k x {\displaystyle F_{s}=-kx} since the direction of the restoring force is opposite to that of the displacement.
Torsional springs The torsional analog of Hooke's law applies to torsional springs. It states that the torque (τ) required to rotate an object is directly proportional to the angular displacement (θ) from the equilibrium position. It describes the relationship between the torque applied to an object and the resulting angular deformation due to torsion. Mathematically, it can be expressed as:
τ = − k θ {\displaystyle \tau =-k\theta }
Where:
τ is the torque measured in Newton-meters or N·m. k is the torsional constant (measured in N·m/radian), which characterizes the stiffness of the torsional spring or the resistance to angular displacement. θ is the angular displacement (measured in radians) from the equilibrium position. Just as in the linear case, this law shows that the torque is proportional to the angular displacement, and the negative sign indicates that the torque acts in a direction opposite to the angular displacement, providing a restoring force to bring the system back to equilibrium.
General "scalar" springs Hooke's spring law usually applies to any elastic object, of arbitrary complexity, as long as both the deformation and the stress can be expressed by a single number that can be both positive and negative. For example, when a block of rubber attached to two parallel plates is deformed by shearing, rather than stretching or compression, the shearing force Fs and the sideways displacement of the plates x obey Hooke's law (for small enough deformations). Hooke's law also applies when a straight steel bar or concrete beam (like the one used in buildings), supported at both ends, is bent by a weight F placed at some intermediate point. The displacement x in this case is the deviation of the beam, measured in the transversal direction, relative to its unloaded shape.
Vector formulation In the case of a helical spring that is stretched or compressed along its axis, the applied (or restoring) force and the resulting elongation or compression have the same direction (which is the direction of said axis). Therefore, if Fs and x are defined as vectors, Hooke's equation still holds and says that the force vector is the elongation vector multiplied by a fixed scalar.
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