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

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

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

h2. Problem Cues

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 Torque] model can be described as translation of the center of mass plus pure rotation about the center of mass.

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h2. Prerequisite Knowledge

h4. Prior Models

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

h4. Vocabulary and Procedures

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

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h2. System

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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h2. 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.

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h2. Model

h4. Relevant Definitions

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h5. Relationships between Angular and Tangential Quantities
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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.
Introduction to the Model
Description and Assumptions
This model applies to a rigid body which is executing pure rotation confined to the xy plane about the origin.
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.
Relevant Definitions
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titleRelationship between Angular and Tangential Quantities
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\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}
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h5. Centripetal Acceleration
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titleCentripetal Acceleration
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\begin{large}\[ \vec{a}_{c} = -\frac{v_{\rm tan}^{2}}{r}\hat{r} = -\omega^{2}r\;\hat{r}\]\end{large}
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h5. Magnitude of Total Acceleration
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titleMagnitude of Total Acceleration
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\begin{large}\[ a = \sqrt{a_{tan}^{2}+a_{c}^{2}} = r\sqrt{\alpha^{2}+\omega^{4}} \]\end{large}
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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. Laws of Change

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h5. Differential Form
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{latex}
 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.
S.I.M. Structure of the Model
Compatible Systems
This model applied to a single rigid body or to a single point particle constrained to move in a circular path.
Relevant Interactions 
The system will be subject to a position-dependent centripetal acceleration, and may also be subject to an angular (or equivalently, tangential) acceleration.
Laws of Change
Mathematical Representation
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titleDifferential Form
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\begin{large}\[ \frac{d\omega}{dt} = \alpha \]
\[\frac{d\theta}{dt} = \omega\]
\end{large}
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h5. Integral Form
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titleIntegral Form
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\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}
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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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 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
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\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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h2. Diagrammatic Representations

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

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h2. Relevant Examples

{contentbylabel:constant_angular_acceleration|maxResults=50|showSpace=false|excerpt=true}

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| !copyright and waiver^copyrightnotice.png! | RELATE wiki by David E. Pritchard is licensed under a [Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License|http://creativecommons.org/licenses/by-nc-sa/3.0/us/]. |
Diagrammatic Representations
  • Angular position versus time graph.
  • Angular velocity versus time graph.
Relevant Examples
 
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