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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}

h4. Introduction to the Model

h5. Description and Assumptions

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

h5. Learning Objectives

Students will be assumed to understand this model who can:

* Describe what it means for a system to execute pure rotation.
* Convert from tangential (linear) quantities to the corresponding angular quantities using the radius of the motion.
* Explain the dependence of angular quantities and of tangential quantities describing the motion of a point on the radius of the point from the [axis of rotation].
* Define tangential and centripetal acceleration for an object in rotational motion.
* Relate centripetal acceleration to angular velocity.
* Give an expression for the total [acceleration] of any point in a [rigid body] executing rotational motion in terms of the [angular acceleration] of the body, the [angular velocity] of the body and the radius of the point from the [axis of rotation].
* Summarize the analogies between angular motion with constant angular acceleration and linear motion with constant (linear) acceleration.

h5. Relevant Definitions

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{panel:title=Relationship between Angular and Tangential Quantities|bgColor=#FFFFFF}
{center}{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}{center}{panel}
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{panel:title=Centripetal Acceleration|bgColor=#FFFFFF}
{center}{latex}\begin{large}\[ \vec{a}_{c} = -\frac{v_{\rm tan}^{2}}{r}\hat{r} = -\omega^{2}r\;\hat{r}\]\end{large}{latex}{center}{panel}
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{panel:title=Magnitude of Total Acceleration|bgColor=#FFFFFF}
{center}{latex}\begin{large}\[ a = \sqrt{a_{tan}^{2}+a_{c}^{2}} = r\sqrt{\alpha^{2}+\omega^{4}} \]\end{large}{latex}{center}
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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}

h4. S.I.M. Structure of the Model

h5. Compatible Systems

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

h5. Relevant Interactions 

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.

h4. Laws of Change

h5. Mathematical Representation

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{panel:title=Differential Form|bgColor=#FFFFFF}
{center}{latex}\begin{large}\[ \frac{d\omega}{dt} = \alpha \]
\[\frac{d\theta}{dt} = \omega\]
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{panel:title=Integral Form|bgColor=#FFFFFF}
{center}{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}{center}
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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:
\\
{center}{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}{center} {note}

h5. Diagrammatic Representations

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

h4. Relevant Examples

h6. {toggle-cloak:id=all} All Examples Using the Model

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