Applying Newton's Laws of Motion

As a simple example, consider a mass driven along a frictionless surface by an ideal spring , as shown in Fig.B.2. Assume that the mass position corresponds to the spring at rest, i.e., not stretched or compressed. The force necessary to compress the spring by a distance is given by Hooke's law (§B.1.3):

This force is balanced at all times by the inertial force of the mass , i.e. , yielding^B.6

where we have defined as the initial displacement of the mass along . This is a differential equation whose solution gives the equation of motion of the mass-spring junction for all time:^B.7

where denotes the frequency of oscillation in radians per second. More generally, the complete space of solutions to Eq.(B.4), corresponding to all possible initial displacements and initial velocities , is the set of all sinusoidal oscillations at frequency :

The amplitude of oscillation and phase offset are determined by the initial conditions, i.e., the initial position and initial velocity of the mass (its initial state) when we ``let it go'' or ``push it off'' at time .

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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.