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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 $ x(t)$ from the ceiling is fixed.

Figure B.1: Mass hung by a spring from a rigid ceiling.
\includegraphics{eps/springmass-gravity}

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

$\displaystyle f_m = g m,
$

where

$\displaystyle g = G\frac{M}{r^2}
$

is called the acceleration due to gravity. Changes in $ r$ due to the motion of the mass are assumed negligible relative to the radius of the earth (about $ 4000$ miles), and so $ g$ is treated as a constant for most practical purposes near the earth's surface. We see that if we double the mass $ m$ , we double the force $ f_m$ pulling on the spring. It is an experimental fact that typical springs exhibit a displacement $ x_m$ that is approximately proportional to the applied force $ f_m$ for a wide range of applied forces. This is Hooke's law for ideal springs:

$\displaystyle \zbox {f(t) = k\,x_m(t),}$   (Hooke's Law)$\displaystyle \protect$ (B.3)

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

$\displaystyle x(t) = x_k + x_m(t),
$

where $ x_k$ 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 $ k$ 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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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