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Mass Kinetic Energy from Virtual Work

From Newton's second law, $ f=ma=m{\ddot x}$ (introduced in Eq.(B.1)), we can use d'Alembert's idea of virtual work to derive the formula for the kinetic energy of a mass given its speed $ v={\dot x}$ . Let $ d x$ denote a small (infinitesimal) displacement of the mass in the $ x$ direction. Then we have, using the calculus of differentials,

f(t) &=& m\, {\ddot x}(t)\\
\,\,\Rightarrow\,\,\quad d W\isdefs f\,d x &=& m\, {\ddot x}\,d x
\eqsp m\, d\left(\frac{1}{2}{\dot x}^2\right)\\
&=& d\left(\frac{1}{2}m\,v^2\right).

Thus, by Newton's second law, a differential of work $ dW$ applied to a mass $ m$ by force $ f$ through distance $ d x$ boosts the kinetic energy of the mass by $ d(m\,v^2/2)$ . The kinetic energy of a mass moving at speed $ v$ is then given by the integral of all such differential boosts from 0 to $ v$ :

$\displaystyle E_m(v) = \int_0^v dW = \int_0^v d\left(\frac{1}{2}m \nu^2\right)
= \frac{1}{2}m v^2 = \frac{1}{2}m\,{\dot x}^2,

where $ E_m(v)$ denotes the kinetic energy of mass $ m$ traveling at speed $ v$ .

The quantity $ dW=f\,dx$ is classically called the virtual work associated with force $ f$ , and $ d x$ a virtual displacement [548].

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University