Angular Momentum and External Torque about a Single Axis
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.
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 multi-object system that involves rotation and acceleration and for which the forces are well understood (a single object can be treated with the simpler Single-Axis Rotation of a Rigid Body). For example, it could be used to solve for the acceleration of an Atwood's Machine in which the pulley has mass.
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.
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.)
Relevant Definitions
Angular momentum about axis a:
\begin
[ L_
= I_
\omega + m\vec
_{{\rm cm},a}\times \vec
_\rm cm ]\end
Laws of Change
Diagrammatic Representations
Relevant Examples
ExamplesInvolvingConstantAngularMomentum"> Examples Involving Constant Angular Momentum
ExamplesInvolvingRollingwithoutSlipping"> Examples Involving Rolling without Slipping
ExamplesInvolvingtheParallelAxisTheorem"> Examples Involving the Parallel Axis Theorem
AllExamplesUsingthisModel"> All Examples Using this Model
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user Dobromila
Wikimedia Commons user Vmenkov