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

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

or

{latex}\begin{large}\[ \alpha = \frac{d^{2}\theta}{dt^{2}} = -\omega^{2}x\] \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 $\sqrt{\dfrac{k}{m}}$.
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h2. Prerequisite Knowledge

h4. Prior Models

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

h4. Vocabulary and Procedures

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

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h2. System

h4. Constituents

A single [point particle|point particle] (or a system treated as a point particle with position specified by the center of mass).

h4. State Variables

Time (_t_), position (_x_) , and velocity (_v_).

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h2. Interactions

h4. Relevant Types

Some time-varying external influence that is confined to one dimension.

h4. Interaction Variables

Acceleration (_a_(_t_)).

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h2. Model

h4. Laws of Change

Differential Forms:
\\
\\
{latex}\begin{large}\[ \frac{dv}{dt} = a\]\end{large}{latex}\\
\\
{latex}\begin{large}\[ \frac{dx}{dt} = v\]\end{large}{latex}\\
\\
Integral Forms:
\\
{latex}\begin{large}\[ v(t) = v(t_{0})+\int_{t_{0}}^{t} a\;dt\]\end{large}{latex}\\
\\
{latex}\begin{large}\[ x(t) = x(t_{0})+\int_{t_{0}}^{t} v\;dt\]\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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