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{}{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}
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System: Any. — Interactions: Any. — Note: This difficult model is only used for gyroscopes.
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Angular Momentum and External Torque
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 Description and Assumptions
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This model is generally applicable, but mathematically very complicated. In introductory mechanics it will only be used to describe the motion of a gyroscope.
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idcues
 Problem Cues
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Only used in problems involving a gyroscope.
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idpri
 Prior Models
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idvocab
 Vocabulary
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Model
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 Compatible Systems
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Technically, any number of rigid bodies. In practice, only used in analyzing gyroscopes (single rigid body with a fixed pivot point).
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idint
 Relevant Interactions
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Only external torques need be considered. Internal torques do not change the system's angular momentum.
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iddef
 Relevant Definitions
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idlaw
 Law of Change
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 Diagrammatic Representations
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  • A delta-L diagram analogous to the  of .
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 Relevant Examples
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Photos courtesy:
 Wikimedia Commons by user Kiko2000
 NASA
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