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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 for elastic interactions]:
{latex}\begin{large}\[ a = -\frac{kx}{m} \]\end{large}{latex}
giving simple harmonic motion with angular frequency
{latex}$\sqrt{\dfrac{k}{m}}${latex}

In the real world, most situations in which there is a stable position with a restoring force can be treated (for small oscillations at least) as if the restoring force is linear. Therefore, even systems with non-ideal springs and the oscillations of a pendulum can be treated as cases of Simple Harmonic Motion as long as the motion is small.

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

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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)]
* [Mechanical Energy and Non-Conservative Work]

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h4. {toggle-cloak:id=vocab} 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. Frequency

The frequency of oscillation of a simple oscillator is represent by a small Greek letter omega. For a simple mass *m* on a spring of constant *k* the frequency is given by

{latex}\begin{large}\[ \omega = \sqrt{\frac{k}{m}} \]\end{large}{latex}

For a simple pendulum of length *L* making small oscillations under the influence of gravity (with the acceleration due to gravity denoted by *g*) the frequency is

{latex}\begin{large}\[ \omega = \sqrt{\frac{g}{L}} \]\end{large}{latex}

  

h5. Initial Conditions

h6. Acceleration
\\
{latex}\begin{large}\[a_{i} = a(t = t_{i}) \]\end{large}{latex}

h6. Velocity
\\
{latex}\begin{large}\[v_{i} = v(t = t_{i}) \]\end{large}{latex}

h6. Position
\\
{latex}\begin{large}\[x_{i} = x(t = t_{i}) = -\frac{a_{i}}{\omega^{2}} \]\end{large}{latex}


h5. Amplitude of Motion

\\
{latex}\begin{large}\[ A = \sqrt{x_{i}^{2} + \left(\frac{v_{i}}{\omega}\right)^{2}} \]\end{large}{latex}


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

h4. {toggle-cloak:id=laws} {color:red}Laws of Change{color}

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h5. Position

{latex}\begin{large}\[ x(t) = x_{i}\cos(\omega t) + \frac{v_{i}}{\omega}\sin(\omega t)\]\end{large}{latex}
\\
or, equivalently
\\
{latex}\begin{large}\[ x(t) = A\cos(\omega t + \phi)\]\end{large}{latex}

h5. Velocity

{latex}\begin{large}\[ v(t) = -\omega x_{i}\sin(\omega t) + v_{i}\cos(\omega t)\]\end{large}{latex}
\\
or, equivalently:
\\
{latex}\begin{large}\[ v(t) = -A\omega\sin(\omega t + \phi)\]\end{large}{latex}

h5. Acceleration

{latex}\begin{large}\[ a(t) = -\omega^{2} x_{i}\cos(\omega t) - \omega v_{i} \sin(\omega t) = -\omega^{2} x(t) \]\end{large}{latex}
\\
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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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. {toggle-cloak:id=RelEx}Relevant Examples
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* [Lissajous Figures and the Bowditch Pendulum]
* [Big Ben]
* [Mass Between Two Springs]

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