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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}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{R}\right) = \sin^{-1}\left(\frac{y_{i}}{R}\right) \]\end{large}
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}\[ 
 Wiki Markup
h1. Circular Motion with Constant Speed


h3. *System:*

Point particle moving in a circle of radius R at constant angular speed {latex} \omega {latex} . (Requires a net force of constant magnitude and direction radially inwards to the circle, i.e  no force component in the direction tangent to the velocity.) 

h3. *Description of the system:*

\- *Object in the system*: point particle.

\- *State variables:*  .

\- *Environment:* external agents interacting with the particle which are the responsible of the *real* forces acting *on* the particle. * *

h3. *Description of the Interactions:*

\- Because we are describing the motion of a point particle we only consider force from outside the interactions as the cause of the acceleration. The total force acting on the point particle has a constant magnitude and direction pointing towards the center of the circle. \\

h3. *Multiple Representations and geometric description.*

\- Position of the particle with respect to a reference frame, in general the center of the circle:  or _q{_}_(t)_. Use of Cartesian and polar coordinates system.

\- Motion Diagrams, tables, equations, vectors.
\\

h3. *Law of Change (*describe the change of the state variables*)\*
\\

where  are vectors of constant magnitude and rotates with a constant angular velocity w.  \\

h3. *Definitions and procedures:*     

  Angular velocity _w_ (rad/sec)

\-          Cartesian and Polar representation of position and velocity.

\-          Cartesian:  _
x(t) = R
 
\cos
 
\left(
_{_}w{_}{_}t +_ _f{_}{_}{~}o{~}{_}_),
\frac{2\pi Rt}{v} + \phi\right)\]\end{large}

 Latex
\begin{large}\[ y(t) = R\sin
 (_{_}w{_}{_}t +_ _f{_}{_}{~}o{~}{_}_)_ ...

\-          Differentiating Cartesian and Polar representation of position and velocity, and implications of the derivative of a vector with constant magnitude but a direction that changes with time.

\-          In uniform circular motion the acceleration points toward the center, the velocity is tangent to the circle. 
\left(\frac{2\pi Rt}{v} + \phi\right)\]\end{large}
 Column
________
 Column
 
Centripetal Acceleration
 

 Latex
\begin{large}\[ \vec{a}_{\rm c} = -\frac{v^{2}}{R} \hat{r}\]\end{large}
Diagrammatic Representations
Relevant Examples
 
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iduni
 Examples Involving Uniform Circular Motion
 
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iduni
 50falsetrueANDexample_problem,uniform_circular_motion
 
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idinst
 Examples Involving Non-Uniform Circular Motion
 
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idinst
 50falsetrueANDexample_problem,circular_motion,centripetal_acceleration
 
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idall
 All Examples Using the Model
 
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idall
 50falsetrueANDexample_problem,uniform_circular_motion 50falsetrueANDexample_problem,circular_motion,centripetal_acceleration



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