Angular Momentum and External Torque about a Single Axis

Introduction to the Model

Description and Assumptions

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.

Learning Objectives

Students are assumed to understand this model who can:

• Describe the conditions that must be satisfied for the valid selection of an axis of rotation in a physics problem.
• Cacluate the moment of inertia of a system composed purely of basic objects like rods and spheres.
• Calculate the angular momentum of a rigid body rotating about a fixed axle.
• Calculate the angular momentum of a rotating and translating rigid body about any axis parallel to the body's angular velocity about its center of mass.
• Determine the net external torque on a system.
• Describe the conditions for angular momentum to be conserved.
• Describe how internal changes to the configuration of a system will affect its angular velocity.
• Analyze collisions involving rotational and translational motion of the participants.

Relevant Definitions

Angular momentum about axis a:

S.I.M. Structure of the Model

Compatible Systems

The system can be composed of any number of rigid bodies and point particles. The system must either be constrained to move in such a way that the angular momentum will be one-dimensional, or else the symmetries of the situation (system plus interactions) must guarantee that the angular momentum will remain one dimensional.

Relevant Interactions

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

Laws of Change

Mathematical Representation

Differential Form

Integral Form

where the last term is called the "angular impulse"

Diagrammatic Representations

• Force diagram.
• Initial-state final-state diagram.

Relevant Examples

Examples Involving Constant Angular Momentum

Examples Involving Rolling without Slipping

Examples Involving the Parallel Axis Theorem

All Examples Using this Model

Pictures courtesy of:
Wikimedia Commons user Dobromila
Wikimedia Commons user Vmenkov