>[!info] [Open Stax Page on this topic](https://openstax.org/books/university-physics-volume-1/pages/15-1-simple-harmonic-motion) Simple Harmonic Motion is a [[../Math/Subset|subset]] of [[Harmonic Motion]] where an object's [[Acceleration]] is proportional to its distance from some point. $\huge \begin{align} m\deriv{^2 \vec x}{t^2} &= -k\vec x\\ \end{align} $ Where $m$ is the mass of the object, and $k$ is some propotionality or [[Spring Constant]]. Two solutions to this [[../Math/Differential Equations|Differential Equation]] are versions of $\sin$ and $\cos$: $\huge \begin{align} \vec x(t) &= \vec A \sin(\omega t + \phi) + \vec B \\ \vec x(t) &= \vec A \cos (\omega t + \phi) + \vec B \end{align} $ Where $\vec A$ is the [[Amplitude]] of the wave, $\omega$ is the [[Angular Frequency]], $\phi$ is the [[Phase]], and $\vec B$ determines the starting condition (or the [[Position]] offset). >[!tip] Relaxed Proof >![[../../06 Mind Dump/202509151415|Proof]] For some [[Simple Harmonic Motion|simple harmonic oscillator]], the [[Angular Frequency]] can be calculated as the root of the force coefficient ([[Spring Constant]]) divided by the [[Mass]]. $\huge \omega = \sqrt{ \frac{k}{m} } $