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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}

h4. Description and Assumptions

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_.

h4.Problem Cues

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.


h4. Learning Objectives

Students will be assumed to understand this model who can:

* Explain why an object moving in a circle at constant speed must be [accelerating|acceleration], and why that acceleration will be [centripetal|centripetal acceleration].
* Give the relationship between the speed of the circular motion, the radius of the circle and the [magnitude] of the [centripetal acceleration].
* Define the [period] of circular motion in terms of the speed and the radius.
* Describe the relationship of the [centripetal acceleration] to the [forces|force] applied to the object executing circular motion.


h2. Model

h4. Compatible Systems

A single [point particle|point particle].

h4. Relevant Interactions

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.

h4. Relevant Definitions

h6. Phase
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{latex}\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{R}\right) = \sin^{-1}\left(\frac{y_{i}}{R}\right) \]\end{large}{latex}

h4. Laws of Change

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h6. Position
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{latex}\begin{large}\[ x(t) = R\cos\left(\frac{2\pi Rt}{v} + \phi\right)\]\end{large}{latex}
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{latex}\begin{large}\[ y(t) = R\sin\left(\frac{2\pi Rt}{v} + \phi\right)\]\end{large}{latex}
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h6. Centripetal Acceleration
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{latex}\begin{large}\[ \vec{a}_{\rm c} = -\frac{v^{2}}{R} \hat{r}\]\end{large}{latex}
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h4. Diagrammatic Representations

* [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.

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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!gravitron.jpg!
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Photos courtesy:

* [Wikimedia Commons|http://commons.wikimedia.org] by [David Burton|http://www.ride-extravaganza.com/intermediate/gravitron/]
* [NASA Johnson Space Center - [Earth Sciences and Image Analysis|http://eol.jsc.nasa.gov/]
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