Hooke's Law

Consider an ideal spring suspending a mass from a rigid ceiling, as depicted in Fig.B.1. Assume the mass is at rest, and that its distance from the ceiling is fixed.

If denotes the mass of the earth, and is the distance of mass 's center from the earth's center of mass, then the downward force on the mass is given by Eq.(B.2) as

where

is called the acceleration due to gravity. Changes in due to the motion of the mass are assumed negligible relative to the radius of the earth (about miles), and so is treated as a constant for most practical purposes near the earth's surface. We see that if we double the mass , we double the force pulling on the spring. It is an experimental fact that typical springs exhibit a displacement that is approximately proportional to the applied force for a wide range of applied forces. This is Hooke's law for ideal springs:

where is the displacement of the spring from its natural length. We call the spring constant, or stiffness of the spring. In terms of our previous notation, we have

where is the length of the spring with no mass attached.

Note that the force on the spring in Fig.B.1 is gravitational force. Equal and opposite to the force of gravity is the spring force exerted upward by the spring on the mass (which is not moving). We know that the spring force is equal and opposite to the gravitational force because the mass would otherwise be accelerated by the net force.^B.4 Therefore, like gravity, a displaced spring can be regarded as a definition of an applied force. That is, whenever you have to think of an applied force, you can always consider it as being delivered by the end of some ideal spring attached to some external physical system.

Note, by the way, that normal interaction forces when objects touch arise from the Coulomb force (electrostatic force, or repulsion of like charges) between electron orbitals. This electrostatic force obeys an ``inverse square law'' like gravity, and therefore also behaves like an ideal spring for small displacements.^B.5

The specific value of depends on the physical units adopted as well as the ``stiffness'' of the spring. What is most important in this definition of force is that a doubling of spring displacement doubles the force. That is, the spring force is a linear function of spring displacement (compression or stretching).

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.