{composition-setup}{compostion-setup}

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


{table:rules=cols|cellpadding=8|cellspacing=0|border=1|frame=void}
{tr:valign=top}{td:width=340px|bgcolor=#F2F2F2}
{live-template:Left Column}
{td}
{td}

h1. Angular Momentum and External Torque

h4. {toggle-cloak:id=desc} Description and Assumptions

{cloak:id=desc}
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.
{cloak}

h4. {toggle-cloak:id=cues} Problem Cues

{cloak:id=cues}
Only used in problems involving a gyroscope.
{cloak}

h4. {toggle-cloak:id=pri} Prior Models

{cloak:id=pri}

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

{cloak}

h4. {toggle-cloak:id=vocab} Vocabulary

{cloak:id=vocab}

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

{cloak}

h2. Model

h4. {toggle-cloak:id=sys} Compatible Systems

{cloak:id=sys}
Technically, any number of [rigid bodies|rigid body].  In practice, only used in analyzing gyroscopes (single rigid body with a fixed pivot point).
{cloak}

h4. {toggle-cloak:id=int} Relevant Interactions

{cloak:id=int}
Only external torques need be considered.  Internal torques do not change the system's angular momentum.
{cloak}

h4. {toggle-cloak:id=def} Relevant Definitions

{cloak:id=def}
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})
\\
\\
\\
h6. Angular Frequency of Gyroscopic Precession
\\
{latex}\begin{large}\[\displaystyle  \Omega = \frac{\displaystyle \left(\frac{dL}{dt}\right)}{L} \]\end{large}{latex}
{cloak}

h4. {toggle-cloak:id=law} Law of Change

{cloak:id=law}
h6. Differential Form
\\
{latex}\begin{large}\[\sum_{\rm system}\frac{d\vec{L}}{dt} = \sum_{\rm external}\vec{\tau}\]\end{large}{latex}
\\
{cloak}

h4. {toggle-cloak:id=diag} Diagrammatic Representations

{cloak:id=diag}

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

{cloak}

h2. Relevant Examples

None yet.

\\
\\
{search-box}
\\
\\
{td}
{tr}
{table}
{live-template:RELATE license}