Rotation and Translation of a Rigid Body

Description and Assumptions

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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.

This model is applicable to a single rigid body that is both rotating and translating in such a way that its angular momentum is a one-dimensional vector (usually taken to lie along the z-axis). It is a subclass of the [1-D Angular Momentum and Torque] model defined by the constraint that the system consists of only one rigid body which has a fixed mass and a fixed moment of inertia for rotations about its center of mass.

Problem Cues

This model is useful for a stationary object (the special case of statics). In that case, both the linear acceleration a and the angular acceleration α are zero, and there is the additional freedom that the axis can be placed at any point in the object. For accelerating objects, the model is commonly used in cases where a single object is placed in a situation where the forces are well understood, such as a cylinder rolling down an inclined plane or a sphere rolling along level ground. Often, the linear and angular accelerations will be related by the rolling without slipping condition.

Page Contents
Description and Assumptions Problem Cues Prerequisite Knowledge Prior Models Vocabulary System Constituents State Variables Interactions Relevant Types Interaction Variables Model Laws of Change Diagrammatical Representations Relevant Examples

Prerequisite Knowledge

Prior Models

Point Particle Dynamics

Vocabulary

force
free body diagram
[torque (one-dimensional)]

System

Constituents

One rigid body.

State Variables

Mass (m) and moment of inertia about the center of mass (I_cm). Both mass and moment of inertia must be constant to apply this model. The moment of inertia may not be explicitly given, but there must be sufficient information to calculate it (often the mass plus some parameters of the rigid body's shape such as radius, length, etc.).

Interactions

Relevant Types

Forces must be specified not only by their magnitude and direction, but also by either their point of application or moment arm with respect to the center of mass of the rigid body.

Interaction Variables

Acceleration of the center of mass (a_cm), angular acceleration about the center of mass (α_cm), external forces (F^ext) and torques about the center of mass (τ_cm).

Model

Laws of Change

This model implies the simultaneous equations:

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\begin

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[ \sum \vec

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^

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= m\vec

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_

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]
[ \sum \tau_

= I_