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h2. Description and Assumptions

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

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]

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h2. System


h4. Constituents

[Rigid bodies|rigid body] and/or [point particles|point particle].

h4. State Variables

In general, system constituents have angular momentum due to their rotation and their translation:
* For the rotational part, the angular velocity (ω) and the moment of inertia with respect to the center of mass (_I{_}{~}cm~) are needed.
* For translation, the mass (_m{_}) and the combination
{latex}\[\vec{r}_{cm,a}\times \vec{v}_{cm}\]{latex} where _r_~cm,a~ is the position of the center of mass measured from the chosen axis.

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h2. Interactions


h4. Relevant Types

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.)

h4. Interaction Variables

External torques about the chosen axis (τ{~}a~).

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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

h5. Differential Form

{latex}\begin{large}\[ \frac{dL^{\rm sys}_{a}}{dt} = \sum \tau_{a} \]\end{large}{latex}

h5. Integral Form

{latex}\begin{large}\[ L^{\rm sys}_{a,f} = L^{\rm sys}_{a,i} + \int \:\sum \tau_{a} \:dt \]\end{large}{latex}
where the last term is called the "angular impulse"

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h2. Diagrammatical 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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