Hooke's Law
The mathematics of elasticity
Summary
The more you stretch a spring or an elastic object, the harder it pulls back, in proportion to the stretch: F = k·x. Here k is the stiffness of the spring and x is how far it is stretched. The same law works everywhere, from clock springs to car suspensions, and from mattresses to the bonds between atoms.
In depth
In the late 1660s, while developing his own spring-driven clocks, Robert Hooke discovered a remarkable behaviour of elastic objects: for small stretches, the applied force and the amount of stretch are directly proportional. The modern formulation is very simple: F = k·x. Here k (the spring constant, N/m) measures the stiffness of the material: a large k means a stiff spring (it gives little under a force), a small k a soft one. Writing it with a minus sign (F = −k·x) emphasises that the spring exerts a restoring force: whichever way you pull it, it pulls the opposite way. This simple linearity leads naturally to simple harmonic motion (oscillation with period T = 2π√(m/k)) and to the energy stored in a spring (E = ½·k·x²). Hooke's law only holds in the elastic region: if you strain a material too much it undergoes plastic deformation and does not spring back.
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