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Energy Conservation in the Mass-Spring System

Recall that Newton's second law applied to a mass-spring system, as in §B.1.4, yields

$\displaystyle f_m(t) + f_k(t) = 0, \quad \forall t,
$

which led to the differential equation obeyed by the mass-spring system:

$\displaystyle m{\ddot x}(t) + k\,x(t) = 0 \quad \forall t
$

Multiplying through by $ {\dot x}(t)=v(t)$ gives

\begin{eqnarray*}
0
&=& m{\ddot x}(t){\dot x}(t) + k\,x(t){\dot x}(t)\\
&=& m\dot{v}(t)v(t) + k\,x(t){\dot x}(t)\\
&=& \frac{d}{dt} \left[\frac{1}{2}m v^2(t)\right]
+ \frac{d}{dt} \left[\frac{1}{2}k x^2(t)\right]\\
&=& \frac{d}{dt} \left[ E_m(t) + E_k(t) \right]\\
&=& \frac{d}{dt} E(t).
\end{eqnarray*}

Thus, Newton's second law and Hooke's law imply conservation of energy in the mass-spring system of Fig.B.2.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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