[Model Hierarchy]
Description and Assumptions
This model applies to a single point particle moving in a circle with constant speed.
Problem Cues
Usually uniform circular motion will be explicitly specified if you are to assume it. (Be especially careful of vertical circles, which are generally nonuniform circular motion because of the effects of gravity. Unless you are specifically told the speed is constant in a vertical loop, you should not assume it to be.) You can also use this model to describe the acceleration in instantaneously uniform circular motion, which is motion along a curved path with the tangential acceleration instantaneously equal to zero. This will usually apply, for example, when a particle is at the top or the bottom of a vertical loop, when gravity is not changing the speed of the particle.
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Prerequisite Knowledge
Prior Models
Vocabulary and Procedures
System
Constituents
A single point particle.
State Variables
Time (t), radius of circle (r), tangential speed (v), angular position (θ), angular velocity (ω).
Interactions
Relevant Types
The system must be subject to an acceleration (and so a net force) that is directed radially inward to the center of the circular path, with no tangential component.
Interaction Variables
Centripetal acceleration (ac).
Model
Relevant Definitions
Centripetal acceleration:
\begin
[ a_
= \frac{v^{2}}
]\end
Phase:
\begin
[ \phi = \cos^{-1}\left(\frac{x_{i}}
\right) = \sin^{-1}\left(\frac{v_{i}}
\right)]\end
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 |
