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[Model Hierarchy]

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The root page Model Hierarchy could not be found in space Modeling Applied to Problem Solving.
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Description and Assumptions

This model applies to a rigid body which is executing pure rotation confined to the xy plane about some point in space.

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


Prerequisite Knowledge

Prior Models

Vocabulary and Procedures


System

Constituents

A single rigid body.

State Variables

Time (t), angular position (θ), tangential velocity (v), angular velocity (ω).


Interactions

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.

Interaction Variables

Angular acceleration (α), tangential acceleration (atan) and radial (or centripetal) acceleration (ac).


Model

Relevant Definitions

Amplitude of motion:


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\begin

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[ A = \sqrt{x_

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^

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+ \left(\frac{v_{i}}

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\right)^{2}}]\end

Phase:


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\begin

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[ \phi = \cos^{-1}\left(\frac{x_{i}}

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\right) = \sin^{-1}\left(\frac{v_{i}}

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\right)]\end

Laws of Change


Position:


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\begin

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[ x(t) = x_

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\cos(\omega t) + \frac{v_{i}}

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\sin(\omega t)]\end


or, equivalently

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\begin

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[ x(t) = A\cos(\omega t + \phi) ]\end


Velocity:


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\begin

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[ v(t) = -\omega x_

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\sin(\omega t) + v_

\cos(\omega t)]\end


or, equivalently:

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\begin

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[ v(t) = -A\omega\sin(\omega t + \phi)]\end


Acceleration:


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\begin

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[ a(t) = -\omega^

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x_

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\cos(\omega t) - \omega v_

\sin(\omega t) = -\omega^

x ]\end


or, equivalently:

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\begin

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[ a(t) = -\omega^

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A\cos(\omega t+\phi) = -\omega^

x]\end


Diagrammatical Representations

  • Acceleration versus time graph.
  • Velocity versus time graph.
  • Position versus time graph.

Relevant Examples

None yet.


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RELATE wiki by David E. Pritchard is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.

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