Talk:PlanetPhysics/Simple Harmonic Oscillator
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[edit source]%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: simple harmonic oscillator %%% Primary Category Code: 40. %%% Filename: SimpleHarmonicOscillator.tex %%% Version: 3 %%% Owner: rspuzio %%% Author(s): rspuzio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}
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A \emph{simple harmonic oscillator} is a mechanical \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} which consists of a \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html} under the influence of a \htmladdnormallink{Hooke's law}{http://planetphysics.us/encyclopedia/HookesLaw.html} \htmladdnormallink{force}{http://planetphysics.us/encyclopedia/Thrust.html}. The equation of \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} of such a system is
\[ m {\ddot x} + k x = 0 \] 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 \htmladdnormallink{TEMs}{http://planetphysics.us/encyclopedia/ImageReconstructionByDoubleFT.html} of trigonometric \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} to obtain a general solution. This solution is typically written in one of two forms. \[ x = v_0 \sin (\omega t) + x_0 \cos (\omega t) \] \[ x = A \sin (\omega t + \phi) \] Either of these solutions shows that the frequency of oscillation is $\omega$ (independent of the amplitude). The \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} 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: \[ v_0 = A \cos \phi \] \[ x_0 = A \sin \phi \] \[ A = \sqrt{v_0^2 + x_0^2} \] \[ \phi = \arctan (x_0 / v_0) \] These constants have the following interpretation: $A$ is the amplitude of the oscillation. $\phi$ is the phase of the oscillation. $v_0$ is the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} at time $t = 0$. $x_0$ is the \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} at time $t = 0$.
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