Description and Assumptions
[Model Hierarchy]
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.
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Prerequisite Knowledge
Prior Models
- Momentum and External Force
- Point Particle Dynamics
- [Rotation and Translation of a Rigid Body]is helpful, but not necessary
Vocabulary and Procedures
- [torque (one-dimensional)]
- [angular momentum (one-dimensional)]
- moment of inertia
System
Constituents
Rigid bodies and/or point particles.
State Variables
In general, system constituents have angular momentum due to their rotation and their translation:
- For the rotational part, the angular velocity (ω) and the moment of inertia with respect to the center of mass (Icm) are needed.
- For translation, the mass (m) and the combination
Unknown macro: {latex}where rcm,a is the position of the center of mass measured from the chosen axis.
[\vec
Unknown macro: {r}_
Unknown macro: {cm,a}\times \vec
Unknown macro: {v}_
Unknown macro: {cm}]
Interactions
Relevant Types
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.)
Interaction Variables
External torques about the chosen axis (τa).
Model
Relevant Definitions
\begin
[ L^
_
= \sum_
\left(I_
\omega + m\vec
_{{\rm cm},a}\times \vec
_\rm cm
\right) ]\end
Laws of Change
Differential Form
\begin
[ \frac{dL^
_{a}}
= \sum \tau_
]\end
Integral Form
\begin
[ L^
_
= L^
_
+ \int \:\sum \tau_
\:dt ]\end
where the last term is called the "angular impulse"
Diagrammatical 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. |