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Energy in the Mass-Spring Oscillator

Summarizing the previous sections, we say that a compressed spring holds a potential energy equal to the work required to compress the spring from rest to its current displacement. If a compressed spring is allowed to expand by pushing a mass, as in the system of Fig.B.2, the potential energy in the spring is converted to kinetic energy in the moving mass.

We can draw some inferences from the oscillatory motion of the mass-spring system written in Eq.(B.5):

$\displaystyle x(t) = A\cos(\omega_0 t), \quad t\ge 0

Regarding the last point, the potential energy, $ E_k(t)=k\,x^2(t)/2$ was defined as the work required to displace the spring by $ x$ meters, where work was defined in Eq.(B.6). The kinetic energy of a mass $ m$ moving at speed $ v$ was found to be $ E_m(t)=m\,v^2(t)/2=m\,{\dot x}^2/2$ . The constance of the potential plus kinetic energy at all times in the mass-spring oscillator is easily obtained from its equation of motion using the trigonometric identity $ \cos^2(\theta)+\sin^2(\theta)=1$ (see Problem 3).

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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