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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 lie along a certain axis.  --- *[Interactions|interaction]:* Any that respect the [one-dimensional angular momentum|angular momentum about a single axis].{excerpt}

h4. Introduction to the Model

h5. 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.

h5. Learning Objectives

Students are assumed to understand this model who can:

* Describe the conditions that must be satisfied for the valid selection of an [axis of rotation] in a physics problem.
* Cacluate the [moment of inertia] of a [system] composed purely of basic objects like rods and spheres.
* Calculate the [angular momentum|angular momentum about a single axis] of a [rigid body] rotating about a fixed axle.
* Calculate the [angular momentum|angular momentum about a single axis] of a rotating and translating [rigid body] about any [axis|axis of rotation] parallel to the body's [angular velocity] about its [center of mass].
* Determine the net [external|external force] [torque|torque (single-axis)] on a [system].
* Describe the conditions for [angular momentum|angular momentum about a single axis] to be conserved.
* Describe how internal changes to the configuration of a [system] will affect its [angular velocity].
* Analyze collisions involving rotational and translational motion of the participants.

h5. 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}{latex}

h4. S.I.M. Structure of the Model

h5. Compatible Systems 

The [system] can be composed of any number of [rigid bodies|rigid body] and [point particles|point particle].  The system must either be constrained to move in such a way that the [angular momentum|angular momentum about a single axis] will be one-dimensional, or else the symmetries of the situation ([system] plus [interactions|interaction]) must guarantee that the [angular momentum|angular momentum about a single axis] will remain one dimensional.

h5. 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.)

h4. Laws of Change

h5. Mathematical Representation

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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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h5. Diagrammatic Representations

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

h4. Relevant Examples

h6. {toggle-cloak:id=cons} Examples Involving Constant Angular Momentum
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h6. {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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h6. {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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h6. {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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[!skater.jpg|alt="How can a skater change their rate of rotation in mid-spin?  Click the image to investigate"!|Twirling Skater]
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[!well.jpg|alt="How fast does a bucket fall down a well?  Click the image to investigate."!|Down the Well]
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Pictures courtesy of:
[Wikimedia Commons|http://commons.wikimedia.org] user [Dobromila|http://commons.wikimedia.org/wiki/User:Dobromila]
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[Wikimedia Commons|http://commons.wikimedia.org] user [Vmenkov|http://commons.wikimedia.org/wiki/User:Vmenkov]
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