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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}

h1. Angular Momentum and External Torque about a Single Axis

h4. 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.

h4. 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_ multi-object 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 an [Atwood's Machine] in which the pulley has [mass].


h4. 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.

h1. Model

h4. 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.


h4. 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. 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. Laws of Change

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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. Diagrammatic Representations

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


h1. Relevant Examples

h4. {toggle-cloak:id=cons} Examples Involving Constant Angular Momentum
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h4. {toggle-cloak:id=rws} Examples Involving Rolling without Slipping
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h4. {toggle-cloak:id=par} Examples Involving the Parallel Axis Theorem

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h4. {toggle-cloak:id=all} All Examples Using this Model
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!skater.jpg!
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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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