{excerpt:hidden=true}*[System|system]:* One [rigid body] rotating about a fixed axis or rotating and translating such that its angular momentum is constrained to one-dimension and the moment of inertia about its center of mass is constant. --- *[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
This model is applicable to a _single_ [rigid body|rigid body] that is rotating about a fixed axis or else both rotating and translating in such a way that its [angular momentum|angular momentum about a single axis] is a one-dimensional vector (usually taken to lie along the z-axis). As a special case, this model is also useful to constrain the forces acting on a static rigid body. It is a subclass of the [Angular Momentum and External Torque about a Single Axis] model defined by the constraint that the system consists of only one rigid body which has a fixed mass and a fixed moment of inertia for rotations about the fixed axis or its center of mass.
h5. Learning Objectives
Students will be assumed to understand this model who can:
* State appropriate choices for the [axis of rotation] for cases of a static object, an object rotating about a fixed axle, and an object that is both rotating and translating.
* Calculate the [moment of inertia] of a [rigid body] constructed of simple shapes like spheres and cylinders about any specified [axis of rotation].
* Compute the [torque|torque (single-axis)] resulting from a [force] applied at a known position about an arbitrary [axis of rotation].
* Relate the net [torque|torque (single-axis)] acting on a [rigid body] to its [angular acceleration].
h4. S.I.M. Structure of the Model
h5. Compatible Systems
One [rigid body|rigid body] whose moment of inertia calculated about the chosen [axis of rotation] is constant.
h5. Relevant Interactions
Only external torques need be considered, as internal torques do not produce angular accelerations. Since torque is needed, forces must be specified not only by their magnitude and direction, but also by either their point of application or [moment arm] with respect to the [axis of rotation]. Allowed choices of the [axis of rotation] will depend upon the specific
circumstances of the problem:
* If the body is [static|statics], the [axis of rotation] may be placed at _any_ location.
* If the body is rotating about a [fixed axis], the [axis of rotation] must be chosen to coincide with that axis.
* If the body is free to translate as well as rotate, the axis must pass through the body's [center of mass].
h4. Laws of Change
h5. Mathematical Representation
{latex}\begin{large} \[ \sum \tau_{a}^{\rm ext} = I_{a}\alpha\]\end{large}{latex}
h5. Diagrammatic Representations
* [force diagram]
h4. Relevant Examples
h6. {toggle-cloak:id=fix} Examples Involving a Fixed Axis
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{contentbylabel:example_problem,fixed_axis|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
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h6. {toggle-cloak:id=trans} Examples Involving a Rotating and Translating Rigid Body
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{contentbylabel:rotation_translation,example_problem|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
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h6. {toggle-cloak:id=stat} Examples Involving a Static Rigid Body
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{contentbylabel:example_problem,statics|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
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h6. {toggle-cloak:id=all} All Examples Using this Model
{cloak:id=all}
{contentbylabel:rotation_translation,statics,fixed_axis|showSpace=false|showLabels=true|excerpt=true|operator=OR|maxResults=50}
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