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Angular Momentum and External Torque about a Single Axis
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.
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 Single-Axis Rotation 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.
Model
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.
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.)
Angular momentum about axis a:
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where the last term is called the "angular impulse"
Relevant Examples
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