PE and KE in the mass-spring oscillator

All vibrating systems consist of this interplay between an energy storing component and an energy carrying (``massy'') component.

The potential energy PE of the ideal mass-spring system is equal to the work done¹ stretching or compressing the spring:

$\displaystyle PE$	$\displaystyle =$	$\displaystyle \frac{1}{2} K x^{2},$
	$\displaystyle =$	$\displaystyle \frac{1}{2} K A^{2} \cos^{2}(\omega_{0}t + \phi).$

The kinetic energy KE in the system is given by the motion of mass:

$\displaystyle KE$	$\displaystyle =$	$\displaystyle \frac{1}{2} m v^{2},$
	$\displaystyle =$	$\displaystyle \frac{1}{2} \cancelto{K}{m\omega_{0}^{2}} A^{2} \sin^{2}(\omega_{0}t + \phi)$
	$\displaystyle =$	$\displaystyle \frac{1}{2} K A^{2}\sin^{2}(\omega_{0}t + \phi).$

``MUS 206: Mechanical Vibration'' by Tamara Smyth, Department of Music, University of California, San Diego (UCSD).