[Model Hierarchy]
Description and Assumptions
This model applies to a rigid body which is executing pure rotation confined to the xy plane about the origin.
Problem Cues
Problems in rotational motion often feature an object which is constrained to rotate about some axle or pivot point. Additionally, the motion of any rigid body which can be treated using the [1-D Angular Momentum and Torque] model can be described as translation of the center of mass plus pure rotation about the center of mass.
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Prerequisite Knowledge
Prior Models
Vocabulary and Procedures
- centripetal acceleration
- tangential acceleration
- angular position
- angular frequency
- angular acceleration
System
Constituents
A single rigid body.
State Variables
Time (t), position (r), angular position (θ), tangential velocity (v), angular velocity (ω).
Interactions
Relevant Types
The system will be subject to a position-dependent centripetal acceleration, and may also be subject to an angular (or equivalently, tangential) acceleration.
Interaction Variables
Angular acceleration (α), tangential acceleration (atan) and radial (or centripetal) acceleration (ac).
Model
Relevant Definitions
Relationships between angular and tangential quantities:
\begin
[ \vec
_
= \vec
\times \vec
= \omega r \;\hat
]
[ \vec
_
= \vec
\times \vec
= \alpha r \;\hat
]\end
Centripetal acceleration:
\begin
[ \vec
_
= -\frac{v_
^{2}}
\hat
= -\omega^
r\;\hat
]\end
Magnitude of total acceleration:
\begin
[ a = \sqrt{a_
^
+a_
{2}} = r\sqrt{\alpha
+\omega^{4}} ]\end
By definition, every point in an object undergoing pure rotation will have the same value for all angular quantities (θ, ω, α). The linear quantities (r, v, a), however, will vary with position in the object.
Laws of Change
Position:
\begin
[ x(t) = x_
\cos(\omega t) + \frac{v_{i}}
\sin(\omega t)]\end
or, equivalently
\begin
[ x(t) = A\cos(\omega t + \phi) ]\end
Velocity:
\begin
[ v(t) = -\omega x_
\sin(\omega t) + v_
\cos(\omega t)]\end
or, equivalently:
\begin
[ v(t) = -A\omega\sin(\omega t + \phi)]\end
Acceleration:
\begin
[ a(t) = -\omega^
x_
\cos(\omega t) - \omega v_
\sin(\omega t) = -\omega^
x ]\end
or, equivalently:
\begin
[ a(t) = -\omega^
A\cos(\omega t+\phi) = -\omega^
x]\end
Diagrammatical Representations
- Acceleration versus time graph.
- Velocity versus time graph.
- Position versus time graph.
Relevant Examples
None yet.
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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 |
