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Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Work = Force times Distance = Energy
          Energy Conservation by Newton's Second Law

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Energy Conservation by Newton's Second Law

Recall that Newton's second law applied to a mass-spring system, as in §F.1.5, yields

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

which leads 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\... ...c{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.F.2.

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Next: Momentum
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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