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Applying Newton's Laws of Motion

Figure B.2: Mass-spring system.

As a simple example, consider a mass $ m$ driven along a frictionless surface by an ideal spring $ k$ , as shown in Fig.B.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 lawB.1.3):

$\displaystyle f_k(t) = -k\,x(t)

This force is balanced at all times by the inertial force $ f_m(x)=-m{\ddot x}$ of the mass $ m$ , i.e. $ f_k+f_m=0$ , yieldingB.6

$\displaystyle m{\ddot x}(t) + k\,x(t) = 0\, \quad \forall t\ge 0, \quad x(0)=A, \quad {\dot x}(0)=0, \protect$ (B.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:B.7

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

where $ \omega_0\isdeftext \sqrt{k/m}$ 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 $ 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\mathbb{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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University