Solution to Equation of Motion

The solution to

$\displaystyle m\frac{d^2x}{dt^2} = -Kx,$

is a function proportional to its second derivative.

This condition is met by the sinusoid:

$\displaystyle x$	$\displaystyle =$	$\displaystyle A\cos(\omega_0t + \phi)$
$\displaystyle \frac{dx}{dt}$	$\displaystyle =$	$\displaystyle -\omega_0A\sin(\omega_0t + \phi)$
$\displaystyle \frac{d^2x}{dt^2}$	$\displaystyle =$	$\displaystyle -\omega_0^2A\cos(\omega_0t + \phi)$

Substituting into the equation of motion yields:

$\displaystyle m\frac{d^2x}{dt^2}$	$\displaystyle =$	$\displaystyle -Kx,$
$\displaystyle -\omega_0^2\cancel{A\cos(\omega_0 t + \phi)}$	$\displaystyle =$	$\displaystyle -\frac{K}{m}\cancel{A\cos(\omega_0 t + \phi)},$

showing that the natural frequency of vibration is

$\displaystyle \omega_0 = \sqrt{\frac{K}{m}}.$

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