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{composition-setup}{composition-setup}

{excerpt:hidden=true}{*}System:* One [point particle] constrained to move in one dimension. --- *Interactions:* The particle must experience a force (or torque) that attempts to [restore|restoring force] it to equilibrium and is directly proportional to its displacement from that equilibrium.{excerpt}
h4. Description and Assumptions

This model applies to position of a single [point particle], or to the angle of a rigid body, constrained to move in one dimension that experiences a linear restoring force toward its equilibrium.  Consequently, its position, velocity, and acceleration will each be 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 position or velocity vs. 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 (in which case they all will be sinusoidal).


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], [natural|natural frequency] [angular frequency] and [phase] of oscillatory motion.
* Give a formula for the [natural|natural frequency] [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 [natural|natural frequency] [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 a stable equilibrium position.

h4. Relevant Definitions

{section}{column}

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

{column}{column}
{panel:title=Phase|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF}
{panel}{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:Phase}
{panel}

{column}{section}
h4. Laws of Change

{panel:title=Using Initial Time|borderWidth=1|borderStyle=solid}
{panel}
{section}{column}
{panel:title=Position|bgColor=#FFFFFF}
{panel}
{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:Position}
{panel}

{column}{column}

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


{column}{section}
{section}{column}
\\
{panel:title=Acceleration|bgColor=#FFFFFF}
{panel}
{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:Acceleration}
{panel}
{column}{section}
{panel:Using}
{panel}
{panel:title=Using Phase|borderWidth=1|borderStyle=solid|bgColor=#F0F0F0}
{panel}
{section}{column}

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

{column}{column}

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

{column}{column}

{panel:title=Acceleration|bgColor=#FFFFFF}
{panel}
{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:Acceleration}
{panel}

{column}{section}
{panel:Using Phase}
{panel}
h4. Diagrammatic Representations

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


|[!images^MathematicaPlayer.png!|^SHM-Phase.nbp]|[Click here for a _Mathematica Player_ application illustrating these representations.|^SHM-Phase.nbp]|


|[!images^download_now.gif!|http://www.wolfram.com/products/player/download.cgi]|[Click here|http://www.woldfram.com/products/player/download.cgi] to download the (free) _Mathematica Player_ \\ from [Wolfram Research|http://www.wolfram.com]|


h1. Relevant Examples

h4. {toggle-cloak:id=Pend} Examples involving Pendulums

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{contentbylabel:example_problem,SHM,pendulum|operator=AND|maxResults=50|showSpace=false|excerpt=true}
{cloak:Pend}
h4. {toggle-cloak:id=Spr} Examples involving Springs

{cloak:id=Spr}
{contentbylabel:example_problem,SHM,spring|operator=AND|maxResults=50|showSpace=false|excerpt=true}
{cloak:Spr}
h4. {toggle-cloak:id=RelEx} All Examples involving this Model

{cloak:id=RelEx}
{contentbylabel:example_problem,SHM|operator=AND|maxResults=50|showSpace=false|excerpt=true}
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