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{excerpt:hidden=true}*[System|system]:* Any number of [rigid bodies|rigid body] or [point particles|point particle] whose angular momentum is constrained to one dimension.  --- *[Interactions|interaction]:* Any that respect the [one-dimensional angular momentum|angular momentum (one-dimensional)].{excerpt}

h1. One-Dimensional Angular Momentum and Torque  

h4. {toggle-cloak:id=desc} Description and Assumptions

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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.
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h4. {toggle-cloak:id=cues} Problem Cues

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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 [Single-Axis Rotation 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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h4. {toggle-cloak:id=pri} Prior Models

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

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h4. {toggle-cloak:id=vocab} Vocabulary and Procedures

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

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h2. Model

h4. {toggle-cloak:id=sys} {color:red} Compatible Systems {color}

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The system can be composed of any number of [rigid bodies|rigid body] and [point particles|point particle].
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h4. {toggle-cloak:id=int} {color:red} Relevant Interactions {color}

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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.)
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h4. {toggle-cloak:id=def} {color:red} Relevant Definitions {color}

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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}{latex}
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h4. {toggle-cloak:id=laws} {color:red} Laws of Change {color}

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h5. Differential Form
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{latex}\begin{large}\[ \sum_{\rm system}\frac{dL_{a}}{dt} = \sum_{\rm external} \tau_{a} \]\end{large}{latex}
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h5. Integral Form
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{latex}\begin{large}\[ \sum_{\rm system}L_{a,f} = \sum_{\rm system}L_{a,i} + \int \:\sum_{\rm external} \tau_{a} \:dt \]\end{large}{latex}
where the last term is called the "angular impulse"
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h4. {toggle-cloak:id=diag} {color:red} Diagrammatic Representations {color}

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

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h2. Relevant Examples

h4. {toggle-cloak:id=cons} Examples Involving Constant Angular Momentum
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{contentbylabel:constant_angular_momentum|showSpace=false|showLabels=true|excerpt=true|maxResults=50}
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h4. {toggle-cloak:id=rws} Examples Involving Rolling without Slipping
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{contentbylabel:angular_momentum,rolling_without_slipping|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
{contentbylabel:constant_angular_momentum,rolling_without_slipping|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
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h4. {toggle-cloak:id=par} Examples Involving the Parallel Axis Theorem

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{contentbylabel:angular_momentum,parallel_axis|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
{contentbylabel:constant_angular_momentum,parallel_axis|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
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h4. {toggle-cloak:id=all} All Examples Using this Model
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{contentbylabel:angular_momentum,constant_angular_momentum|showSpace=false|showLabels=true|excerpt=true|operator=OR|maxResults=50}
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