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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} = 
h2. Description and Assumptions

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{td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]*
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{excerpt:hidden=true}*System:* Any number of [rigid bodies|rigid body] or [point particles|point particle] whose angular momentum is constrained to one dimension.  --- *Interactions:* Any that respect the one-dimensional angular momentum.{excerpt}

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.

h2. Problem Cues

Systems involving several rigid bodies that interact.  The integral form of this model is used in essentially all problems involving a collision where at least one body can rotate (e.g. a person jumping onto a rotating merry-go-round, a rotating disk falling onto another rotating object) or that involve a changing moment of inertia (spinning skater pulling her arms into her body).  The differential form is useful in situations that involve the acceleration of a _system_  that involves rotation and acceleration and for which the forces are well understood (a single object can be treated with the simpler [Rotation and Translation of a Rigid Body]).  For example, it could be used to solve for the acceleration of a modified Atwood's machine which involves a massive pulley that accelerates.

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h2. Prerequisite Knowledge


h4. Prior Models

* [Momentum and External Force]
* [Point Particle Dynamics]
* [Rotation and Translation of a Rigid Body]is helpful, but not necessary

h4. Vocabulary and Procedures

* [torque (one-dimensional)]
* [angular momentum (one-dimensional)]
* [moment of inertia]

----
h2. System

The system can be composed of any number of [rigid bodies|rigid body] and [point particles|point particle].

----
h2. Interactions

External interactions must be explicitly given as torques, or as forces with their point of application or [moment arm] about the chosen axis specified along with their magnitude and direction.  (Internal interactions do not change the angular momentum of the system.)

----
h2. Model


h4. Relevant Definitions

{latex}\begin{large}\[ L^{\rm sys}_{a} = \sum_{\rm constituents} \left(
I_{cm}\omega + m\vec{r}_{{\rm cm},a}\times \vec{v}_{{\rm cm}}
\right)
 \]\end{large}
{latex}

h4. Laws of Change

{section}{column}
h5. Differential Form

{latex}
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}\[ \
frac{dL^
sum_{\rm 
sys
system}\frac{dL_{a}}{dt} = \sum_{\rm external} \tau_{a} \]\end{large}
{latex}
{column}{column}

h5. Integral Form

{latex}
 Column

Integral Form

 Latex
\begin{large}\[ 
L^
\sum_{\rm 
sys
system}L_{a,f} = 
L^
\sum_{\rm 
sys
system}L_{a,i} + \int \:\sum_{\rm external} \tau_{a} \:dt \]\end{large}
{latex}

where 
 
 the 
 
 last 
 
 term 
 
 is 
 
 called 
 
 the 
 
 "angular 
 
 impulse"

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h2. Diagrammatic Representations

* [Force diagram|force diagram].
* [Initial-state final-state diagram|initial-state final-state diagram].

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h2. Relevant Examples

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Diagrammatic Representations
Relevant Examples
 
 Toggle Cloak
idcons
 Examples Involving Constant Angular Momentum
 
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idcons
 falsetruetrue50constant_angular_momentum 

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

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

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


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