Description and Assumptions
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.
Problem Cues
This model is useful for a stationary object (the special case of statics). In that case, both the linear acceleration a and the angular acceleration α are zero, and there is the additional freedom that the axis can be placed at any point in the object. For accelerating objects, the model is commonly used in cases where a single object is placed in a situation where the forces are well understood, such as a cylinder rolling down an inclined plane or a sphere rolling along level ground. Often, the linear and angular accelerations will be related by the rolling without slipping condition.
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Prerequisite Knowledge
Prior Models
Vocabulary
- force
- free body diagram
- [torque (one-dimensional)]
Compatible Systems
One rigid body.
Relevant 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.
Model
Laws of Change
This model implies the simultaneous equations:
\begin
[ \sum \vec
^
= m\vec
_
]
[ \sum \tau_
= I_
\alpha_
]\end
Diagrammatic Representations
Relevant Examples