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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

$\alpha=0$

and r = R.

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

\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

\begin{large}\[ x(t) = R\cos\left(\frac{2\pi Rt}{v} + \phi\right)\]\end{large}

\begin{large}\[ y(t) = R\sin\left(\frac{2\pi Rt}{v} + \phi\right)\]\end{large}

\begin{large}\[ \vec{a}_{\rm c} = -\frac{v^{2}}{R} \hat{r}\]\end{large}

Diagrammatic Representations

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

Relevant Examples

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Examples Involving Uniform Circular Motion

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Examples Involving Non-Uniform Circular Motion

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All Examples Using the Model

Photos courtesy:

• Wikimedia Commons by David Burton
• NASA Johnson Space Center - Earth Sciences and Image Analysis