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

{excerpt}This model applies to a single [point particle] moving in a circle with constant speed.{excerpt}


h2. 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.

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h2. Prerequisite Knowledge

h4. Prior Models

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

h4. Vocabulary and Procedures

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

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h2. System

h4. Constituents

A single [point particle|point particle].

h4. State Variables

Time (_t_), radius of circle (_r_), tangential speed (_v_), angular position (θ), angular velocity (ω).

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h2. Interactions

h4. Relevant Types

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. Interaction Variables

Centripetal acceleration (_a_~c~).

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h2. Model

h4. Relevant Definitions

h5. Centripetal acceleration:
\\
{latex}\begin{large}\[ a_{c} = \frac{v^{2}}{r}\]\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. Laws of Change

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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}
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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:
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{latex}\begin{large}\[ v(t) = -\omega x_{i}\sin(\omega t) + v_{i}\cos(\omega t)\]\end{large}{latex}
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or, equivalently:
\\
{latex}\begin{large}\[ v(t) = -A\omega\sin(\omega t + \phi)\]\end{large}{latex}
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h5. Acceleration:
\\
{latex}\begin{large}\[ a(t) = -\omega^{2} x_{i}\cos(\omega t) - \omega v_{i} \sin(\omega t) = -\omega^{2} x \]\end{large}{latex}
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or, equivalently:
\\
{latex}\begin{large}\[ a(t) = -\omega^{2}A\cos(\omega t+\phi) = -\omega^{2} x\]\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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