Rotation and Translation of a Rigid Body

Keys to Applicability

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.

Assumed Knowledge

Prior Models

• Point Particle Dynamics

Vocabulary

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

Model Specification

System Structure

[Constituents]:  One rigid body.

Interactions:  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.

Descriptors

[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.).

[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 Equations

Mathematical Statement of the Model

This model implies the simultaneous equations:

Relevant Examples