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{excerpt:hidden=true}*[System|system]:* One [point particle] constrained to move in a circle at constant speed. --- *[Interactions|interaction]:* [Centripetal acceleration|centripetal acceleration].{excerpt}

h1. Uniform Circular Motion

h4. {toggle-cloak:id=desc} Description and Assumptions

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This model applies to a single [point particle] moving in a circle of fixed radius (assumed to lie in the _xy_ plane with its center at the origin) with constant speed.  It is a subclass of the [Rotational Motion] model defined by {latex}$\alpha=0${latex} and _r_ = _R_.

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

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Usually uniform circular motion will be explicitly specified if you are to assume it.  (Be especially careful of _vertical_ circles, which are generally _nonuniform_ circular motion because of the effects of gravity.  Unless you are specifically told the speed is constant in a vertical loop, you should not assume it to be.)  You can also use this model to describe the acceleration in _instantaneously_ uniform circular motion, which is motion along a curved path with the tangential acceleration instantaneously equal to zero.  This will usually apply, for example, when a particle is at the top or the bottom of a vertical loop, when gravity is not changing the _speed_ of the particle.

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

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

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

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* [tangential acceleration]
* [centripetal acceleration]
* [angular frequency]

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h2. Model

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

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A single [point particle|point particle].

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

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The system must be subject to an acceleration (and so a net force) that is directed _radially inward_ to the center of the circular path, with no tangential component.

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

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h5. Initial conditions

{latex}\begin{large}\[ x_{0} = x(t=0)\]
\[ y_{0} = y(t=0) \]
\[\theta_{0} = \theta(t=0)\]\end{large}{latex}
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h5. Centripetal acceleration
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{latex}\begin{large}\[ \vec{a}_{c} = -\frac{v^{2}}{R}\hat{r} = -\omega^{2}R\;\hat{r}\]\end{large}{latex}
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h5. Phase
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{latex}\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{0}}{R}\right) = \sin^{-1}\left(\frac{y_{0}}{R}\right) \]\end{large}{latex}
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h4. {toggle-cloak:id=laws} {color:red}Laws of Change{color}

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h5. Angular Position
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{latex}\begin{large}\[ \theta(t) = \theta_{0}+\omega t\]\end{large}{latex}
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h5. Position
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{latex}\begin{large}\[ x(t) = R\cos(\omega t + \phi)\]\end{large}{latex}
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{latex}\begin{large}\[ y(t) = R\sin(\omega t + \phi)\]\end{large}{latex}
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h5. Velocity
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{latex}\begin{large}\[ \vec{v} = -R\omega\sin(\omega t+\phi) \;\hat{x} + R\omega\cos(\omega t +\phi)\;\hat{y} = \omega R \hat{\theta}\]\end{large}{latex}
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h4. {toggle-cloak:id=diag} {color:red}Diagrammatic Representations{color}

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* [Free body diagram|free body diagram] (used to demonstrate that a net radial force is present).
* [Delta-v diagram|Delta-v diagram].
* x- and y-position versus time graphs.
* θ versus time graph.

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h2. Relevant Examples

h4. {toggle-cloak:id=uni} Examples Involving Uniform Circular Motion
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{contentbylabel:example_problem,uniform_circular_motion|maxResults=50|showSpace=false|excerpt=true|operator=AND}
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h4. {toggle-cloak:id=inst} Examples Involving Non-Uniform Circular Motion
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{contentbylabel:example_problem,circular_motion,centripetal_acceleration|maxResults=50|showSpace=false|excerpt=true|operator=AND}
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h4. {toggle-cloak:id=all} All Examples Using the Model
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{contentbylabel:example_problem,uniform_circular_motion|maxResults=50|showSpace=false|excerpt=true|operator=AND}
{contentbylabel:example_problem,circular_motion,centripetal_acceleration|maxResults=50|showSpace=false|excerpt=true|operator=AND}
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