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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, F s = 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.
Da Vinci lacked Hooke's law and calculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made. [3] The Bernoulli beam is named after Jacob Bernoulli, who made the significant discoveries. Leonhard Euler and Daniel Bernoulli were the first to put together a useful theory circa 1750. [4]
Another physical setting for derivation of the wave equation in one space dimension uses Hooke's law. In the theory of elasticity, Hooke's law is an approximation for certain materials, stating that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress).
Simple harmonic motion can serve as a mathematical model for a variety of motions, but is typified by the oscillation of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's law. The motion is sinusoidal in time and demonstrates a single resonant frequency.
For a stretched spring fixed at one end obeying Hooke's law, the elastic potential energy is = where r 2 and r 1 are collinear coordinates of the free end of the spring, in the direction of the extension/compression, and k is the spring constant.
The first constitutive equation (constitutive law) was developed by Robert Hooke and is known as Hooke's law.It deals with the case of linear elastic materials.Following this discovery, this type of equation, often called a "stress-strain relation" in this example, but also called a "constitutive assumption" or an "equation of state" was commonly used.
[note 1] This relationship can be interpreted as a generalization of Hooke's law to a 3D continuum. A general fourth-rank tensor in 3D has 3 4 = 81 independent components , but the elasticity tensor has at most 21 independent components. [3]
It can be mathematically modeled with Hooke's Law to show the elastic characteristics while using Lamé constants. [13] Generally the ice has its linear elasticity constants averaged over one dimension of space to simplify the equations while still maintaining accuracy. [13]