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{td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]*
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h2. Description and Assumptions

{excerpt}This model applies to a single [point particle] moving in a circle with constant speed.{excerpt}


h2. 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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|| Page Contents ||
| {toc:style=none|indent=10px} |

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h2. Prerequisite Knowledge

h4. Prior Models

* [1-D Motion (Constant Velocity)]
* [1-D Motion (Constant Acceleration)]

h4. Vocabulary and Procedures

* [tangential acceleration]
* [centripetal acceleration]
* [angular frequency]

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h2. System

h4. Constituents

A single [point particle|point particle].

h4. State Variables

Time (_t_), position (_x_) and velocity (_v_).

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h2. Interactions

h4. Relevant Types

The system must be subject to a net force that is directed _radially inward_ to the center of the circular path, with no tangential component.

h4. Interaction Variables

Centripetal acceleration (_a_~c~).

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h2. Model

h4. Relevant Definitions

h5. Amplitude of motion:
\\
{latex}\begin{large}\[ A = \sqrt{x_{i}^{2} + \left(\frac{v_{i}}{\omega}\right)^{2}}\]\end{large}{latex}

h5. Phase:
\\
{latex}
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System: One point particle constrained to move in a circle at constant speed. — Interactions: Centripetal acceleration.
Introduction to the Model
Description and Assumptions
This model applies to a single point particle moving in a circle of fixed radius (assumed to lie in the xy plane with its center at the origin) with constant speed. It is a subclass of the Rotational Motion model defined by 
 Latex
$\alpha=0$
 and r = R.
 Info
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.
Learning Objectives
Students will be assumed to understand this model who can:
  • Explain why an object moving in a circle at constant speed must be accelerating, and why that acceleration will be centripetal.
  • Give the relationship between the speed of the circular motion, the radius of the circle and the magnitude of the centripetal acceleration.
  • Define the period of circular motion in terms of the speed and the radius.
  • Describe the relationship of the centripetal acceleration to the forces applied to the object executing circular motion.
Relevant Definitions
 
Phase
 

 Latex
\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{
A
R}\right) = \sin^{-1}\left(\frac{
v
y_{i}}{
\omega A
R}\right) \]\end{large}
{latex}

h4. Laws of Change

\\
h5. Position:
\\
{latex}
S.I.M. Structure of the Model
Compatible Systems
A single point particle.
Relevant Interactions
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.
Laws of Change
Mathematical Representation
 Section
 Column
 
Position
 

 Latex
\begin{large}\[ x(t) = 
x_{i}
R\cos\left(\
omega t) + \
frac{
v_{i}}{\omega}\sin(\omega t)\]\end{large}{latex}
\\
or, equivalently
\\
{latex}\begin{large}\[ x(t) = A\cos(\omega t 
2\pi Rt}{v} + \phi\right)
 
\]\end{large}
{latex}
\\
h5. Velocity:
\\
{latex}\

 Latex
\begin{large}\[ 
v
y(t) = 
-\omega x_{i}\sin(\omega t) + v_{i}\cos(\omega t)\]\end{large}{latex}
\\
or, equivalently:
\\
{latex}\begin{large}\[ v(t) = -A\omega\sin(\omega t + \phi
R\sin\left(\frac{2\pi Rt}{v} + \phi\right)\]\end{large}
{latex}
\\
h5. Acceleration:
\\
{latex}
 Column
________
 Column
 
Centripetal Acceleration
 

 Latex
\begin{large}\[ 
a(t) = -\omega^{2} x_{i}\cos(\omega t) - \omega v_{i} \sin(\omega t)
\vec{a}_{\rm c} = -\
omega^
frac{v^{2}
 x \]\end{large
}{
latex}
\\
or, equivalently:
\\
{latex}\begin{large}\[ a(t) = -\omega^{2}A\cos(\omega t+\phi) = -\omega^{2} x\]\end{large}{latex}

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h2. Diagrammatical Representations

* Acceleration versus time graph.
* Velocity versus time graph.
* Position versus time graph.

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h2. Relevant Examples

None yet.
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| !copyright and waiver^copyrightnotice.png! | RELATE wiki by David E. Pritchard is licensed under a [Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License|http://creativecommons.org/licenses/by-nc-sa/3.0/us/]. |
R} \hat{r}\]\end{large}
Diagrammatic Representations
Relevant Examples
 
 Toggle Cloak
iduni
 Examples Involving Uniform Circular Motion
 
 Cloak
iduni
 50falsetrueANDexample_problem,uniform_circular_motion
 
 Toggle Cloak
idinst
 Examples Involving Non-Uniform Circular Motion
 
 Cloak
idinst
 50falsetrueANDexample_problem,circular_motion,centripetal_acceleration
 
 Toggle Cloak
idall
 All Examples Using the Model
 
 Cloak
idall
 50falsetrueANDexample_problem,uniform_circular_motion 50falsetrueANDexample_problem,circular_motion,centripetal_acceleration



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