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Applying Newton's Laws of Motion

Figure E.2: Mass-spring system.
\begin{figure}\input fig/springmass-phy.pstex_t

As a simple example, consider a mass $ m$ driven along a frictionless surface by an ideal spring $ k$, as shown in Fig. E.2. Assume that the mass position $ x=0$ corresponds to the spring at rest, i.e., not stretched or compressed. The force necessary to compress the spring by a distance $ x$ is given by Hooke's lawE.1.4):

$\displaystyle f(t) = k\,x(t)

This force is balanced at all times by the inertial force $ m{\ddot x}$ of the mass $ m$, yieldingE.7

$\displaystyle m{\ddot x}(t) + k\,x(t) = 0\, \quad \forall t\ge 0, \quad x(0)=A, \quad {\dot x}(0)=0. \protect$ (E.4)

where we have defined $ A$ as the initial displacement of the mass along $ x$. This is a differential equation whose solution gives the equation of motion of the mass-spring junction for all time:E.8

$\displaystyle x(t) = A\cos(\omega_0 t), \quad \forall t\ge 0, \protect$ (E.5)

where $ \omega_0\isdef \sqrt{k/m}$ denotes the frequency of oscillation in radians per second. More generally, the complete space of solutions to Eq. (E.4), corresponding to all possible initial displacements $ x(0)$ and initial velocities $ {\dot x}(0)$, is the set of all sinusoidal oscillations at frequency $ \omega_0$:

$\displaystyle x(t) = A\cos(\omega_0 t + \phi), \quad \forall A,\phi\in{\bf R}

The amplitude of oscillation $ A$ and phase offset $ \phi$ are determined by the initial conditions, i.e., the initial position $ x(0)$ and initial velocity $ {\dot x}(0)$ of the mass (its initial state) when we ``let it go'' or ``push it off'' at time $ t=0$.

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[How to cite and copy this work] 
``Physical Audio Signal Processing for Virtual Musical Instruments and Digital Audio Effects'', by Julius O. Smith III, (December 2005 Edition).
Copyright © 2006-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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