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Description and Assumptions

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[Model Hierarchy]

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1-D Angular Momentum and Torque is a subclass of the general Angular Momentum and External Torque model in which a system of rigid bodies is constrained to move only in a plane (usually taken to be the xy plane) with each body's angular momentum therefore directed along an axis perpendicular to the plane (along the z-axis). Under these conditions, the angular momentum is a one-dimensional vector, and the directional subscript (z) is generally omitted.

Problem Cues

Systems involving several rigid bodies that interact.  The integral form of this model is used in essentially all problems involving a collision where at least one body can rotate (e.g. a person jumping onto a rotating merry-go-round, a rotating disk falling onto another rotating object) or that involve a changing moment of inertia (spinning skater pulling her arms into her body). The differential form is useful in situations that involve the acceleration of a system  that involves rotation and acceleration and for which the forces are well understood (a single object can be treated with the simpler [Rotation and Translation of a Rigid Body]). For example, it could be used to solve for the acceleration of a modified Atwood's machine which involves a massive pulley that accelerates.

Page Contents

Prerequisite Knowledge

Prior Models

Vocabulary and Procedures

System

The system can be composed of any number of rigid bodies and point particles.

Interactions

External interactions must be explicitly given as torques, or as forces with their point of application or moment arm about the chosen axis specified along with their magnitude and direction.  (Internal interactions do not change the angular momentum of the system.)

Model

Relevant Definitions

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\begin

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[ L^

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_

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= \sum_

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\left(I_

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\omega + m\vec

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_{{\rm cm},a}\times \vec

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_\rm cm\right) ]\end

Laws of Change

Differential Form
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[ \frac{dL^

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_{a}}

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= \sum \tau_

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]\end

Integral Form
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[ L^

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_

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= L^

_

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+ \int \:\sum \tau_

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where the last term is called the "angular impulse"

Diagrammatic Representations

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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