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{td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]*
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h2. Description and Assumptions

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

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.{excerpt}


h2. Problem Cues

Any object that experiences a _linear_ restoring force or torque so that the equation of motion takes the form 

{latex}\begin{large}\[ a = \frac{d^{2}x}{dt^{2}} = - \omega^{2}x \]\end{large}{latex}

or

{latex}\begin{large}\[ \alpha = \frac{d^{2}\theta}{dt^{2}} = -\omega^{2}\theta\] \end{large}{latex}

will experience simple harmonic motion with angular frequency ω.  The prototypical example is an object of mass _m_ attached to a spring with force constant _k_, in which case, by [Hooke's Law]:

{latex}\begin{large}\[ a = -\frac{kx}{m} \]\end{large}{latex}

giving simple harmonic motion with angular frequency {latex}$\sqrt{\dfrac{k}{m}}${latex}.
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h2. Prerequisite Knowledge

h4. Prior Models

* [1-D Motion (Constant Velocity)]
* [1-D Motion (Constant Acceleration)]

h4. Vocabulary and Procedures

* [restoring force]
* [periodic motion]
* [angular frequency]
* [phase]

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h2. System

h4. Constituents

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

h4. State Variables

Time (_t_), position (_x_) , velocity (_v_) and acceleration (_a_) or their angular equivalents.

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h2. Interactions

h4. Relevant Types

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. Interaction Variables

Force (_F_) or the angular equivalent.

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h2. Model

h4. Relevant Definitions

h5. Initial Conditions
\\
{latex}\begin{large}\[ x_{0} = x(t=0) = -\frac{a(t=0)}{\omega^{2}}\]
\[ v_{0} = v(t=0)\]\end{large}{latex}

h5. Amplitude of motion:
\\
{latex}\begin{large}\[ A = \sqrt{x_{0}^{2} + \left(\frac{v_{0}}{\omega}\right)^{2}}\]\end{large}{latex}

h5. Phase:
\\
{latex}\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{0}}{A}\right) = \sin^{-1}\left(\frac{v_{0}}{\omega A}\right)\]\end{large}{latex}

h4. Laws of Change

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h5. Position:
\\
{latex}\begin{large}\[ x(t) = x_{0}\cos(\omega t) + \frac{v_{0}}{\omega}\sin(\omega t)\]\end{large}{latex}
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or, equivalently
\\
{latex}\begin{large}\[ x(t) = A\cos(\omega t + \phi) \]\end{large}{latex}
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h5. Velocity:
\\
{latex}\begin{large}\[ v(t) = -\omega x_{0}\sin(\omega t) + v_{0}\cos(\omega t)\]\end{large}{latex}
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or, equivalently:
\\
{latex}\begin{large}\[ v(t) = -A\omega\sin(\omega t + \phi)\]\end{large}{latex}
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h5. Acceleration:
\\
{latex}\begin{large}\[ a(t) = -\omega^{2} x_{0}\cos(\omega t) - \omega v_{0} \sin(\omega t) = -\omega^{2} x(t) \]\end{large}{latex}
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or, equivalently:
\\
{latex}\begin{large}\[ a(t) = -\omega^{2}A\cos(\omega t+\phi) = -\omega^{2} x(t)\]\end{large}{latex}

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h2. Diagrammatical Representations

* Acceleration versus time graph.
* Velocity versus time graph.
* Position versus time graph.

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h2. Relevant Examples

None yet.
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