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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}
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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
 Section
 Column
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bgColor#FFFFFF
titleRelationship between Angular and Tangential Quantities
 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}
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{panel:title=Centripetal Acceleration|bgColor=#FFFFFF}
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titleCentripetal Acceleration
 Center
 Latex
\begin{large}\[ \vec{a}_{c} = -\frac{v_{\rm tan}^{2}}{r}\hat{r} = -\omega^{2}r\;\hat{r}\]\end{large}
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{panel:title=Magnitude of Total Acceleration|bgColor=#FFFFFF}
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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. 
 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


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


{section}{column}
{panel:title=Differential Form|bgColor=#FFFFFF}
{center}{latex}
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titleDifferential Form
 Center
 Latex
\begin{large}\[ \frac{d\omega}{dt} = \alpha \]
\[\frac{d\theta}{dt} = \omega\]
\end{large}
{latex}{center}
{panel}
{column}{column}
{panel:title=Integral Form|bgColor=#FFFFFF}
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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}
{latex}{center}
{panel}
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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}
 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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{contentbylabel:constant_angular_acceleration|maxResults=50|showSpace=false|excerpt=true}
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Diagrammatic Representations
  • Angular position versus time graph.
  • Angular velocity versus time graph.
Relevant Examples
 
 Toggle Cloak
idall
 All Examples Using the Model
 
 Cloak
idall
 50falsetrueconstant_angular_acceleration


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