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Uniform Circular Motion

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

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$\alpha=0$

and r = R.

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.

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.

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.

Relevant Definitions

Phase


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\begin

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[ \phi = \cos^{-1}\left(\frac{x_{i}}

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\right) = \sin^{-1}\left(\frac{y_

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}

\right) ]\end

Laws of Change

Position


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\begin

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[ x(t) = R\cos\left(\frac

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+ \phi\right)]\end


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\begin

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[ y(t) = R\sin\left(\frac

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+ \phi\right)]\end

________

Centripetal Acceleration


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\begin

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[ \vec

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_

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= -\frac{v^{2}}

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\hat

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]\end

Diagrammatic Representations

Relevant Examples

ExamplesInvolvingUniformCircularMotion"> Examples Involving Uniform Circular Motion

ExamplesInvolvingNon-UniformCircularMotion"> Examples Involving Non-Uniform Circular Motion

AllExamplesUsingtheModel"> All Examples Using the Model




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