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{excerpt:hidden=true}*[System|system]:* One [point particle] constrained to move in a circle at constant speed. --- *[Interactions|interaction]:* [Centripetal acceleration|centripetal acceleration].{excerpt}

h1. Uniform Circular Motion

h4. {toggle-cloak:id=desc} Description and Assumptions

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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}$\alpha=0${latex} and _r_ = _R_.

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h4. {toggle-cloak:id=cues} Problem Cues

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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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h4. {toggle-cloak:id=mod} Prior Models

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

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h4. {toggle-cloak:id=vocab} Vocabulary

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

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h2. Model

h4. {toggle-cloak:id=sys} {color:red} Compatible Systems {color}

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A single [point particle|point particle].

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h4. {toggle-cloak:id=int} {color:red}Relevant Interactions{color}

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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.

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h4. {toggle-cloak:id=def} {color:red} Relevant Definitions{color}

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h5. Initial conditions

{latex}\begin{large}\[ x_{0} = x(t=0)\]
\[ y_{0} = y(t=0) \]
\[\theta_{0} = \theta(t=0)\]\end{large}{latex}
{column}{column}{color:white}________{color}{column}{column}
h5. Centripetal acceleration
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{latex}\begin{large}\[ \vec{a}_{c} = -\frac{v^{2}}{R}\hat{r} = -\omega^{2}R\;\hat{r}\]\end{large}{latex}
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h5. Phase
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 Excerpt
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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}\[ \phi = \cos^{-1}\left(\frac{x_{
0
i}}{R}\right) = \sin^{-1}\left(\frac{y_{
0
i}}{R}\right) \]\end{large}
{latex}
{column}{section}
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h4. {toggle-cloak:id=laws} {color:red}Laws of Change{color}

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{section}{column}
h5. Angular 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}\[ 
\theta
x(t) = 
\theta_{0}+\omega t\]\end{large}{latex}
{column}{column}{color:white}________{color}{column}{column}
h5. Position
\\
{latex}\begin{large}\[ x(t) = R\cos(\omega t + \phi
R\cos\left(\frac{2\pi Rt}{v} + \phi\right)\]\end{large}
{latex}
\\
{latex}

 Latex
\begin{large}\[ y(t) = R\sin\left(\frac{2\
omega t
pi Rt}{v} + \phi\right)\]\end{large}
{latex}
{column}{column}{color:white}
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________
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h5. Velocity
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{latex}
 Column
 
Centripetal Acceleration
 

 Latex
\begin{large}\[ \vec
{v
{a}_{\rm c} = -
R\omega\sin(\omega t+\phi) \;\hat{x} + R\omega\cos(\omega t +\phi)\;\hat{y} = \omega R \hat{\theta
\frac{v^{2}}{R} \hat{r}\]\end{large}
{latex}
{column}{section}

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h4. {toggle-cloak:id=diag} {color:red}Diagrammatic Representations{color}

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* [Free body diagram|free body diagram] (used to demonstrate that a net radial force is present).
* [Delta-v diagram|Delta-v diagram].
* x- and y-position versus time graphs.
* θ versus time graph.

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h2. Relevant Examples

h4. {toggle-cloak:id=uni} Examples Involving Uniform Circular Motion
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Diagrammatic Representations
Relevant Examples
 
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 Examples Involving Uniform Circular Motion
 
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iduni
50falsetrueANDexample_problem,uniform_circular_motion
|maxResults=50|showSpace=false|excerpt=true|operator=AND}
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h4. {
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id
=
inst
} 
 Examples 
 
 Involving 
 
 Non-Uniform 
 
 Circular 
 
 Motion

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50falsetrueANDexample_problem,circular_motion,centripetal_acceleration
|maxResults=50|showSpace=false|excerpt=true|operator=AND}
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h4. {
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id
=
all
} 
 All 
 
 Examples 
 
 Using 
 
 the 
 
 Model

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50falsetrueANDexample_problem,uniform_circular_motion
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example_problem,circular_motion,centripetal_acceleration
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