A mathematical approximation to the restoring behavior of springs and other elastic solids under small deformations.
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Motivation for Concept
Elastic objects are objects which rebound to their original shape from a temporary deformation. All solids are basically elastic under small deformations, but "small" is a relative term. A metal ball, for instance, is only elastic for deformations that are so small as to be invisible to the naked eye. If you press the ball hard enough to deform it noticeably, it will retain a dent. A rubber ball is elastic under a much wider range of deformations, and can have its shape noticeably changed temporarily without causing a permanent dent. Springs are objects which are designed "spread out" large deformations over a series of coils, so that the complete object can change shape dramatically while each portion of the coil deforms only a relatively small amount.
The elastic properties of objects is vital to understanding the engineering of all structures, from airplanes to skyscrapers. As such, Hooke's description of the restoring force produced by an object undergoing elastic deformation is an extremely useful piece of mathematics, and has acquired the title "Hooke's Law", even though it is not a universal Law in the same sense as, e.g. [Newton's Law of Universal Gravitation]. Hooke's Law is really a parameterization, so that every situation requires a different value for the elastic force constant (contrast this with Newton's Law of Gravity, with its universal G). This fact does not detract from its enormous utility. Objects experiencing elastic deformation are often said to "obey Hooke's Law".
Elastic Potential Energy
Assuming an object attached to a spring that obeys Hooke's Law with the motion confined to the x direction, it is customary to choose the coordinates such that x = 0 when the object is in a position such that the spring is at its natural length. The force on the object from the spring is then:
\begin
[ \vec
= - kx \hat
]\end
It is also customary to make the assignment:
\begin
[ U(0) \equiv 0]\end
Thus, the potential can be defined:
\begin
[ U
= U(0) - \int_
^
(-kx)\:dx = \frac
kx^
]\end
For an object moving under the influence of a spring only, the associated potential energy curve would then be:
POTENTIAL ENERGY CURVE
The graph indicates the presence of one stable equilibrium point at x = 0.