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{excerpt:hidden=true}*System:* One [rigid body] in [pure rotation] or one [point particle] constrained to move in a circle. --- *Interactions:* Any [angular acceleration]. --- *Warning:* The constraint of rotational motion implies [centripetal acceleration] may have to be considered.{excerpt}

h1. Rotational Motion

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

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This model applies to a [rigid body] which is executing [pure rotation] confined to the _xy_ plane about the origin.

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

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Problems in rotational motion often feature an object which is constrained to rotate about some axle or pivot point.  Additionally, the motion of any rigid body which can be treated using the [1-D Angular Momentum and External Torque about a Single Axis] model can be described as translation of the center of mass plus pure rotation about the center of mass.

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

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* [Uniform Circular Motion]
* [One-Dimensional Motion with Constant Acceleration|1-D Motion (Constant Acceleration)]

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

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

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h2. Model

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

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This model applied to a single [rigid body] or to a single [point particle] constrained to move in a circular path.

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

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The system will be subject to a position-dependent [centripetal acceleration], and may also be subject to an angular (or equivalently, [tangential|tangential acceleration]) acceleration.

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

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h5. Relationships between Angular and Tangential Quantities
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{latex}\begin{large}\[ \vec{v}_{\rm tan} = \vec{\omega} \times \vec{r} = \omega r \;\hat{\theta}\]
\[ \vec{a}_{\rm tan} = \vec{\alpha}\times \vec{r} = \alpha r \;\hat{\theta}\]\end{large}{latex}
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h5. Centripetal Acceleration
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{latex}\begin{large}\[ \vec{a}_{c} = -\frac{v_{\rm tan}^{2}}{r}\hat{r} = -\omega^{2}r\;\hat{r}\]\end{large}{latex}
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h5. Magnitude of Total Acceleration
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{latex}\begin{large}\[ a = \sqrt{a_{tan}^{2}+a_{c}^{2}} = r\sqrt{\alpha^{2}+\omega^{4}} \]\end{large}{latex}
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{note}By definition, _every point_ in an object undergoing [pure rotation] will have the same value for all _angular_ quantities (θ, ω, α).  The linear quantities (_r_, _v_, _a_), however, will vary with position in the object.{note}

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

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h5. Differential Form
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{latex}\begin{large}\[ \frac{d\omega}{dt} = \alpha \]
\[\frac{d\theta}{dt} = \omega\]
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h5. Integral Form
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{latex}\begin{large}\[ \omega_{f} = \omega_{i} +\int_{t_{i}}^{t_{f}} \alpha \;dt\]
\[ \theta_{f} = \theta_{i} +\int_{t_{i}}^{t_{f}} \omega\;dt\]\end{large}{latex}
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{note}Note the analogy between these Laws of Change and those of the [One-Dimensional Motion (General)] model.  Thus, for the case of *constant angular acceleration*, the integral form of these Laws are equivalent to:
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{latex}\begin{large}\[ \omega_{f} = \omega_{i} + \alpha(t_{f}-t_{i})\]
\[ \theta_{f} = \theta_{i} + \frac{1}{2}(\omega_{i}+\omega_{f})(t_{f}-t_{i})\]
\[ \theta_{f} = \theta_{i} + \omega_{i}(t_{f}-t_{i}) +\frac{1}{2}\alpha(t_{f}-t_{i})^{2}\]
\[ \omega_{f}^{2} =\omega_{i}^{2} + 2\alpha(\theta_{f}-\theta_{i})\]\end{large}{latex} {note}

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h4. {toggle-cloak:id=diag} {color:red}Diagrammatic Representations{color}

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* Angular position versus time graph.
* Angular velocity versus time graph.

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h2. Relevant Examples

h4. {toggle-cloak:id=all} All Examples Using the Model

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