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{table:align=right|cellspacing=0|cellpadding=1|border=1|frame=box|width=40%} {tr} {td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]* {td} {tr} {tr} {td} {pagetree:root=Model Hierarchy|reverse=true} {search-box} {td} {tr} {table} h2. Description and Assumptions {excerpt}This model applies to a [rigid body] which is executing [pure rotation] confined to the _xy_ plane about some point in space.{excerpt} h2. Problem Cues Problems in rotational motion often feature an object which is constrained to rotate about some axle or pivot point. Additionally, the motion of any rigid body which can be treated using the [1-D Angular Momentum and Torque] model can be described as translation of the center of mass plus pure rotation about the center of mass. ---- || Page Contents || | {toc:style=none|indent=10px} | ---- h2. Prerequisite Knowledge h4. Prior Models * [Uniform Circular |
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Motion]
* [One-Dimensional Motion with Constant Acceleration|1-D Motion (Constant Acceleration)]
h4. Vocabulary and Procedures
* [centripetal acceleration]
* [tangential acceleration]
* [angular position]
* [angular frequency]
* [angular acceleration]
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h2. System
h4. Constituents
A single [rigid body].
h4. State Variables
Time (_t_), angular position (θ), tangential velocity (_v_), angular velocity (ω).
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h2. Interactions
h4. Relevant Types
The system will be subject to a position-dependent centripetal acceleration, and may also be subject to an angular (or equivalently, tangential) acceleration.
h4. Interaction Variables
Angular acceleration (α), tangential acceleration (_a_~tan~) and radial (or centripetal) acceleration (_a_~c~).
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h2. Model
h4. Relevant Definitions
h5. Amplitude of motion:
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{latex}\begin{large}\[ A = \sqrt{x_{i}^{2} + \left(\frac{v_{i}}{\omega}\right)^{2}}\]\end{large}{latex}
h5. Phase:
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{latex}\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{A}\right) = \sin^{-1}\left(\frac{v_{i}}{\omega A}\right)\]\end{large}{latex}
h4. Laws of Change
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h5. Position:
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{latex}\begin{large}\[ x(t) = x_{i}\cos(\omega t) + \frac{v_{i}}{\omega}\sin(\omega t)\]\end{large}{latex}
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or, equivalently
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{latex}\begin{large}\[ x(t) = A\cos(\omega t + \phi) \]\end{large}{latex}
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h5. Velocity:
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{latex}\begin{large}\[ v(t) = -\omega x_{i}\sin(\omega t) + v_{i}\cos(\omega t)\]\end{large}{latex}
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or, equivalently:
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{latex}\begin{large}\[ v(t) = -A\omega\sin(\omega t + \phi)\]\end{large}{latex}
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h5. Acceleration:
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{latex}\begin{large}\[ a(t) = -\omega^{2} x_{i}\cos(\omega t) - \omega v_{i} \sin(\omega t) = -\omega^{2} x \]\end{large}{latex}
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or, equivalently:
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{latex}\begin{large}\[ a(t) = -\omega^{2}A\cos(\omega t+\phi) = -\omega^{2} x\]\end{large}{latex}
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h2. Diagrammatical Representations
* Acceleration versus time graph.
* Velocity versus time graph.
* Position versus time graph.
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h2. Relevant Examples
None yet.
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{search-box}
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