PlanetPhysics/Simple Harmonic Oscillator

A simple harmonic oscillator is a mechanical [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] which consists of a [[../Particle/|particle]] under the influence of a [[../HookesLaw/|Hooke's law]] [[../Thrust/|force]]. The equation of [[../CosmologicalConstant/|motion]] of such a system is

$m\ddot{x}+kx=0$

It is typical to define the quantity $\omega =\sqrt{k/m}$ and write this equation as

$\ddot{x}+{\omega }^{2}x=0$

Note that this equation is linear. Among other consequences, this means that the period of oscilltions does not depend on amplitude. It is rather simple to solve this equation in [[../ImageReconstructionByDoubleFT/|TEMs]] of trigonometric [[../Bijective/|functions]] to obtain a general solution. This solution is typically written in one of two forms.

$x=v_0\sin (\omega t)+x_0\cos (\omega t)$

$x=A\sin (\omega t+\varphi )$

Either of these solutions shows that the frequency of oscillation is $\omega$ (independent of the amplitude). The [[../Bijective/|relation]] between the two solutions is provided by the angle addition law for the sine. One finds that the constants appearing in the two solutions are related in the following way:

$v_0=A\cos \varphi$

$x_0=A\sin \varphi$

$A=\sqrt{v_0^2+x_0^2}$

$\varphi =\arctan (x_0/v_0)$

These constants have the following interpretation: $A$ is the amplitude of the oscillation. $\varphi$ is the phase of the oscillation. $v_0$ is the [[../Velocity/|velocity]] at time $t=0$. $x_0$ is the [[../Position/|position]] at time $t=0$.