Page History

{composition-setup}{composition-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}

* [Angular Momentum and External Torque about a Single Axis]
* [Uniform Circular Motion]

{cloak}

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

{cloak:id=vocab}

* [torque (single-axis)]
* [angular momentum about a single axis]

{cloak}

h2. Model

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

{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} {color:red}Relevant Interactions{color}

{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} {color:red}Relevant Definitions{color}

{cloak:id=def}
h6. Gyroscopic Approximation 
\\

The *gyroscopic approximation* assumes that the angular momentum due to precession of the gyroscope is negligible compared to the angular momentum of the spinning gyroscope. If Ω is the angular velocity of precession and ω is the angular velocity of the gyroscope's spin, then the gyroscopic approximation holds when

{latex}\begin{large}\[ \Omega \ll \omega \]\end{large}{latex}
and 
{latex}\begin{large}\[ \vec{L} \simeq \vec{\omega} I\]\end{large}{latex}

_I_ is the moment of inertia of the gyroscope about the spin axis {latex}$\hat{\omega}${latex}
See [Spinning Top]  and [Delta-v diagram] for more details.
\\
\\
\\
h6. Angular Frequency of Gyroscopic Precession
\\

Under the gyroscopic approximation, the angular velocity of the precession is given by Ω 

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

This result is independent of the tipping angle of the gyroscope.
{cloak}

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

{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} {color:red}Diagrammatic Representations{color}

{cloak:id=diag}

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

{cloak}

h2. {toggle-cloak:id=examples} Relevant Examples


{cloak:id=examples}
* [Spinning Top]
{cloak}

\\
\\
{td}
{td:width=235px}
!gyro.jpg!
\\
\\
!nasa.jpg!
\\

Photos courtesy:
[Wikimedia Commons|http://commons.wikimedia.org] by user [Kiko2000|http://commons.wikimedia.org/wiki/User:Kiko2000]
[NASA|http://www.nasa.gov]

{td}
{tr}
{table}

{live-template:RELATE license}