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h2. Description and Assumptions

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

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 α=0 and _r_ = _R_.


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. Initial conditions:

{latex}\begin{large}\[ x_{0} = x(t=0)\]
\[ y_{0} = y(t=0) \]
\[\theta_{0} = \theta(t=0)\]\end{large}{latex}

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}

h4. Laws of Change
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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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h2. Diagrammatical 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.

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h2. Relevant Examples

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