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,
$](http://www.dsprelated.com/josimages_new/pasp/img2679.png)
![$\displaystyle m{\ddot x}(t) + k\,x(t) = 0 \quad \forall t
$](http://www.dsprelated.com/josimages_new/pasp/img2680.png)
![$ {\dot x}(t)=v(t)$](http://www.dsprelated.com/josimages_new/pasp/img2681.png)
![\begin{eqnarray*}
0
&=& m{\ddot x}(t){\dot x}(t) + k\,x(t){\dot x}(t)\\
&=& m\...
...{d}{dt} \left[ E_m(t) + E_k(t) \right]\\
&=& \frac{d}{dt} E(t).
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img2682.png)
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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