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h1. Simple Harmonic Motion

{excerpt:hidden=true}{*}System:* One [point particle] constrained to move in one dimension. --- *Interactions:* The acceleration must be a [sinusoidal function] of time. {excerpt}

h4. Description and Assumptions

This model applies to a single [point particle] constrained to move in one dimension whose position is a sinusoidal function of time. Simple harmonic motion is sometimes abbreviated SHM, or referred to as "Simple Harmonic Oscillation" (SHO).

h4. Problem Cues

Any object that experiences a _linear_ restoring force or torque so that the equation of motion takes the form
{latex}\begin{large}\[ a = \frac{d^{2}x}{dt^{2}} = - \omega_{0}^{2}x \]\end{large}{latex}
or
{latex}\begin{large}\[ \alpha = \frac{d^{2}\theta}{dt^{2}} = -\omega_{0}^{2}\theta\] \end{large}{latex}
will experience simple harmonic motion with [natural|natural frequency] [angular frequency] ω{~}0{~}. The most common systems whose equations of motion take this form are a [mass on a spring] or a [pendulum] (in the small-angle approximation).  Any problem requesting or giving a time for one of these systems will likely require the use of the Simple Harmonic Motion model.

Another cue that Simple Harmonic Motion is occurring is if the position, the velocity, or the acceleration are sinusoidal in time.  


h4. Learning Objectives

Students will be assumed to understand this model who can:

* Define the terms [equilibrium position] and [restoring force].
* Define the [amplitude], [period], [angular frequency] and [phase] of oscillatory motion.
* Give a formula for the [angular frequency] of the oscillation of a [pendulum] or [mass on a spring].
* Write mathematical expressions for the [position], [velocity] and [acceleration] of Simple Harmonic Motion as functions of time for the special cases that the initial velocity is zero or the initial position is equilibrium.
* Graphically represent the position, velocity and acceleration of Simple Harmonic Motion.
* Use the laws of [dynamics] to determine the [angular frequency] of a [system] in the limit of very small displacements from equilibrium.
* Describe the consequences of [conservation|conserved] of [mechanical energy] for Simple Harmonic Motion (assuming no dissipation).


h1. Models


h4. Compatible Systems

A single [point particle|point particle] (or, for the angular version of SHM, a single [rigid body]).

h4. Relevant Interactions

The system must be subject to a one-dimensional restoring force (or torque) that varies linearly with the displacement (or angular displacement) from an equilibrium position.

h4. Relevant Definitions

{section}{column}

{panel:title=Amplitude|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF}
{center}{latex}\begin{large}\[ A = \sqrt{x_{i}^{2} + \left(\frac{v_{i}}{\omega_{0}}\right)^{2}} \]\end{large}{latex}{center}{panel}

{column}{column}
{panel:title=Phase|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF}{center}
{latex}\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{A}\right) = \sin^{-1}\left(\frac{v_{i}}{\omega_{0} A}\right)\]\end{large}{latex}
{center}{panel}

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h4. Laws of Change

{deck:id=loc}
{card:label=Using Initial Time}

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{panel:title=Position|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF}{center}
{latex}\begin{large}\[ x(t) = x_{i}\cos(\omega_{0} (t-t_{i})) + \frac{v_{i}}{\omega_{0}}\sin(\omega_{0} (t-t_{i}))\]\end{large}{latex}
{center}{panel}

{column}{column:width=300}

{panel:title=Velocity|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF}{center}
{latex}\begin{large}\[ v(t) = -\omega x_{i}\sin(\omega_{0} (t-t_{i})) + v_{i}\cos(\omega_{0} (t-t_{i}))\]\end{large}{latex}
{center}{panel}

{column}{section}

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{panel:title=Acceleration|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF}{center}
{latex}\begin{large}\[ a(t) = -\omega_{0}^{2} x_{i}\cos(\omega_{0} (t-t_{i})) - \omega_{0} v_{i} \sin(\omega_{0} (t-t_{i})) = -\omega_{0}^{2} x(t) \]\end{large}{latex}
{center}{panel}
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{card}
{card:label=Using Phase}

{section}{column}

{panel:title=Position|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF}{center}
{latex}\begin{large}\[ x(t) = A\cos(\omega_{0} t + \phi)\]\end{large}{latex}
{center}{panel}

{column}{column}

{panel:title=Velocity|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF}{center}
{latex}\begin{large}\[ v(t) = -A\omega_{0}\sin(\omega_{0} t + \phi)\]\end{large}{latex}
{center}{panel}

{column}{section}

{panel:title=Acceleration|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF}{center}
{latex}\begin{large}\[ a(t) = -\omega_{0}^{2}A\cos(\omega_{0} t+\phi) = -\omega_{0}^{2} x(t)\]\end{large}{latex}
{center}{panel}
{card}
{deck}


h4. Diagrammatical Representations

* [acceleration versus time graph]
* [velocity versus time graph]
* [position versus time graph]

h1. Relevant Examples

{toggle-cloak:id=RelEx} All Examples involving this Model

{cloak:id=RelEx}
* [Lissajous Figures and the Bowditch Pendulum]
* [Big Ben]
* [Mass Between Two Springs]

{cloak} 



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