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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 the origin.

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

This model applied to a single rigid body or to a single point particle constrained to move in a circular path.


Interactions

The system will be subject to a position-dependent centripetal acceleration, and may also be subject to an angular (or equivalently, tangential) acceleration.


Model

Relevant Definitions

Relationships between Angular and Tangential Quantities


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[ \vec

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_

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= \vec

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\times \vec

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= \omega r \;\hat

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]
[ \vec

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_

= \vec

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\times \vec

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= \alpha r \;\hat

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]\end


Centripetal Acceleration


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= -\frac{v_

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^{2}}

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\hat

= -\omega^

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r\;\hat

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]\end


Magnitude of Total Acceleration


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{2}} = r\sqrt{\alpha

+\omega^{4}} ]\end


By definition, every point in an object undergoing pure rotation will have the same value for all angular quantities (θ, ω, α). The linear quantities (r, v, a), however, will vary with position in the object.

Laws of Change

Differential Form


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[\frac

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= \omega]
\end


Integral Form


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[ \omega_

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[ \theta_

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+\int_{t_{i}}^{t_{f}} \omega\;dt]\end

Note the analogy between these Laws of Change and those of the One-Dimensional Motion (General) model. Thus, for the case of constant angular acceleration, the integral form of these Laws are equivalent to:

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[ \theta_

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= \theta_

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(\omega_

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[ \theta_

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[ \omega_

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)]\end


Diagrammatic Representations

  • Angular position versus time graph.
  • Angular velocity versus time graph.

Relevant Examples

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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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