Versions Compared

Old Version 88

changes.mady.by.user Andrew E Pawl

Saved on

compared with

New Version Current

changes.mady.by.user Andrew E Pawl

Saved on

Key

  • This line was added.
  • This line was removed.
  • Formatting was changed.
Comment: Migration of unmigrated content due to installation of a new plugin
Composition Setup
 HTML Table
border1
cellpadding8
cellspacing0
rulescols
framevoid
 Table Row (tr)
valigntop
 Table Cell (td)
 Excerpt
hiddentrue
System: Any number of rigid bodies or point particles whose angular momentum is constrained to lie along a certain axis. — Interactions: Any that respect the one-dimensional angular momentum.
Introduction to the Model
Description and Assumptions
1-D Angular Momentum and Torque is a subclass of the general Angular Momentum and External Torque model in which a system of rigid bodies is constrained to move only in a plane (usually taken to be the xy plane) with each body's angular momentum therefore directed along an axis perpendicular to the plane (along the z-axis). Under these conditions, the angular momentum is a one-dimensional vector, and the directional subscript (z) is generally omitted.
Learning Objectives
Students are assumed to understand this model who can:
Relevant Definitions 
Angular momentum about axis a:
 Latex
\begin{large}\[ L_{a} = I_{cm}\omega + m\vec{r}_{{\rm cm},a}\times \vec{v}_{{\rm cm}} \]\end{large}
S.I.M. Structure of the Model
Compatible Systems 
The system can be composed of any number of rigid bodies and point particles. The system must either be constrained to move in such a way that the angular momentum will be one-dimensional, or else the symmetries of the situation (system plus interactions) must guarantee that the angular momentum will remain one dimensional.
Relevant Interactions 
External interactions must be explicitly given as torques, or as forces with their point of application or moment arm about a chosen axis of rotation specified along with their magnitude and direction.  (Internal interactions do not change the angular momentum of the system.)
Laws of Change
Mathematical Representation
 Section
 Column

Differential Form

 Latex
\begin{large}\[ \sum_{\rm system}\frac{dL_{a}}{dt} = \sum_{\rm external} \tau_{a} \]\end{large}
 Column

Integral Form

 Latex
\begin{large}\[ \sum_{\rm system}L_{a,f} = \sum_{\rm system}L_{a,i} + \int \:\sum_{\rm external} \tau_{a} \:dt \]\end{large}
where the last term is called the "angular impulse"
Diagrammatic Representations
Relevant Examples
 
 Toggle Cloak
idcons
 Examples Involving Constant Angular Momentum
 
 Cloak
idcons
 falsetruetrue50constant_angular_momentum 

 
 Toggle Cloak
idrws
 Examples Involving Rolling without Slipping
 
 Cloak
idrws
 falsetruetrueAND50angular_momentum,rolling_without_slipping falsetruetrueAND50constant_angular_momentum,rolling_without_slipping 

 
 Toggle Cloak
idpar
 Examples Involving the Parallel Axis Theorem
 
 Cloak
idpar
 falsetruetrueAND50angular_momentum,parallel_axis falsetruetrueAND50constant_angular_momentum,parallel_axis 

 
 Toggle Cloak
idall
 All Examples Using this Model
 
 Cloak
idall
 falsetruetrueOR50angular_momentum,constant_angular_momentum 


 Search Box


 Table Cell (td)
width235
 

  

 Pictures courtesy of:
 Wikimedia Commons user Dobromila 
 Wikimedia Commons user Vmenkov
 Wiki Markup
{html}
<script type="text/javascript">
var gaJsHost = (("https:" == document.location.protocol) ? "https://ssl." : "http://www.");
document.write(unescape("%3Cscript src='" + gaJsHost + "google-analytics.com/ga.js' type='text/javascript'%3E%3C/script%3E"));
</script>
<script type="text/javascript">
try {
var pageTracker = _gat._getTracker("UA-11762009-2");
pageTracker._trackPageview();
} catch(err) {}</script>
{html}