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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)\isdef {\ddot x}(t) \isdef \frac{d^2 x(t)}{dt^2},
$

we have

$\displaystyle \zbox {f(t) = m\,a(t) = m\,{\ddot x}(t).} \protect$ (F.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{\bf 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 §F.1.2).



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written by 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.


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