h1. Description and Assumptions

{excerpt:hidden=true}*System:* Any. --- *Interactions:* Any. --- *Note:* This difficult model is only used for [gyroscopes|gyroscope].{excerpt}

This model is [generally applicable|generally applicable model], but mathematically very complicated.  In introductory mechanics it will only be used to describe the motion of a gyroscope.


h1. Problem Cues

Only used in problems involving a gyroscope.

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h1. Prerequisite Knowledge

h3. Prior Models

* [1-D Angular Momentum and Torque]
* [Uniform Circular Motion]


h3. Vocabulary

* [torque (one-dimensional)]
* [angular momentum (one-dimensional)]


h1. Compatible Systems

Technically, any number of [rigid bodies|rigid body].  In practice, only used in analyzing gyroscopes (single rigid body with a fixed pivot point).

h1. Relevant Interactions

Only external torques need be considered.  Internal torques do not change the system's angular momentum.


h1. Model

h3. Definitions

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h6. Gyroscopic Approximation 

{latex}\begin{large}\[ \vec{L} \approx \vec{\omega} I\]\end{large}{latex}
(_I_ is the moment of inertia of the gyroscope about the spin axis {latex}$\hat{\omega}${latex})
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h6. Angular Frequency of Gyroscopic Precession

{latex}\begin{large}\[\displaystyle  \Omega = \frac{\displaystyle \left(\frac{dL}{dt}\right)}{L} \]\end{large}{latex}

h3. Law of Change

h6. Differential Form
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{latex}\begin{large}\[\sum_{\rm system}\frac{d\vec{L}}{dt} = \sum_{\rm external}\vec{\tau}\]\end{large}{latex}
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h1. Diagrammatic Representations

* A delta-L diagram analogous to the [Delta-v diagram] of [Uniform Circular Motion].


h1. Relevant Examples

None yet.

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