Period Derivation for the Simple Harmonic OscillatorPeriodDerivationForTheSimpleHarmonicOscillator2026-10-10 18:30:302026-10-10 18:30:30Derivationsimple harmonic oscillatoradded the tikz source for the figurePeriodicictyPeriodSineSimple Harmonic Oscillator% this is the default PhysicsLibrary preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics).
% If you use \includegraphics{figure} with no file extension, PhysicsLibrary
% will prefer figure.png before falling back to figure.eps.
\usepackage{graphicx}
\DeclareGraphicsExtensions{.png,.jpg,.jpeg,.pdf,.eps}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here\section*{Period Derivation for the Simple Harmonic Oscillator}
A solution for the simple harmonic oscillator is
\begin{equation}
x = A \sin (\omega t + \phi)
\end{equation}
The sine function as a function of $\omega t$ is shown in figure 1.
\begin{center}
\includegraphics{sine_function.png}
{\small Figure1. Sine function.}
\end{center}
One can see that the sine function repeats every $2\pi$. Periodicicty was defined in Oscillations as a function that is equal to itself at $t+T$,
\begin{equation}
f(t+T) =f(t)
\end{equation}
Therefore, the arguments to the sine function must be equal to itself again at $2\pi$ (plug in $t+T$ in for $t$).
\begin{equation}
\omega (t+T) + \phi = \omega t + \phi + 2\pi
\end{equation}
expand paranthesis and subtract each side by $\phi$
\[ \omega t + \omega T = \omega t + 2\pi \]
$\omega t$ cancels to leave
\[ \omega T = 2\pi \]
Finally, rearrange to get the eqaution for period
\begin{equation}
T= \frac{ 2\pi }{\omega}
\end{equation}
\begin{thebibliography}{9}
\bibitem{French}
A. P. French,
\emph{Vibrations and Waves},
W. W. Norton, 1971.
\bibitem{Loftin}
PhysicsLibrary,
\emph{Oscillation}.
\end{thebibliography}