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Newton's Laws of Motion

Perhaps the most heavily used equation in physics is Newton's second law of motion:

$\displaystyle \zbox {\mbox{\emph{Force = Mass $\times$\ Acceleration}}}

That is, when a force is applied to a mass, the mass experiences an acceleration proportional to the applied force. Denoting the mass by $ m$ , force at time $ t$ by $ f(t)$ , and acceleration by

$\displaystyle a(t)\isdefs {\ddot x}(t) \isdefs \frac{d^2 x(t)}{dt^2},

we have

$\displaystyle \zbox {f(t) = m\,a(t) = m\,{\ddot x}(t).} \protect$ (B.1)

In this formulation, the applied force $ f(t)$ is considered positive in the direction of positive mass-position $ x(t)$ . The force $ f(t)$ and acceleration $ a(t)$ are, in general, vectors in three-dimensional space $ x\in\mathbb{R}^3$ . In other words, force and acceleration are generally vector-valued functions of time $ t$ . The mass $ m$ is a scalar quantity, and can be considered a measure of the inertia of the physical system (see §B.1.1 below).

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-05-16 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University