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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. {toggle-cloak:id=desc} Description and Assumptions

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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.
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h4. {toggle-cloak:id=cues} Problem Cues

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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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h4. {toggle-cloak:id=pri} Prior Models

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

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h4. {toggle-cloak:id=voc} Vocabulary

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

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h2. Models

h4. {toggle-cloak:id=sys} {color:red}Compatible Systems{color}

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

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h4. {toggle-cloak:id=int} {color:red}Relevant Interactions{color}

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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.

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h4. {toggle-cloak:id=def} {color:red} Relevant Definitions{color}

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

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h5. Amplitude of Motion
{latex}\begin{large}\[ A = \sqrt{x_{0}^{2} + \left(\frac{v_{0}}{\omega}\right)^{2}}\qquad\qquad\qquad\]\end{large}{latex}      \\
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h5. Phase
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{latex}\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{0}}{A}\right) = \sin^{-1}\left(\frac{v_{0}}{\omega A}\right)\]\end{large}{latex}
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h4. {toggle-cloak:id=laws} {color:red}Laws of Change{color}

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h5. Position:
{latex}\begin{large}\[ x(t) = x_{0}\cos(\omega t) + \frac{v_{0}}{\omega}\sin(\omega t)\qquad\qquad\qquad\]\end{large}{latex}
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or, equivalently
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{latex}\begin{large}\[ x(t) = A\cos(\omega t + \phi)\]\end{large}{latex}
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h5. Velocity
{latex}\begin{large}\[ v(t) = -\omega x_{0}\sin(\omega t) + v_{0}\cos(\omega t)\qquad\qquad\qquad\]\end{large}{latex}
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or, equivalently:
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{latex}\begin{large}\[ v(t) = -A\omega\sin(\omega t + \phi)\]\end{large}{latex}
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h5. Acceleration
{latex}\begin{large}\[ a(t) = -\omega^{2} x_{0}\cos(\omega t) - \omega v_{0} \sin(\omega t) = -\omega^{2} x(t) \qquad\]\end{large}{latex}
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or, equivalently:
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{latex}\begin{large}\[ a(t) = -\omega^{2}A\cos(\omega t+\phi) = -\omega^{2} x(t)\]\end{large}{latex}
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h4. {toggle-cloak:id=diag} {color:red}Diagrammatical Representations{color}

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* 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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