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This model applies to a single point |
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Circular Motion with Constant Speed
System:
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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 and r = R. | Info |
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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. |
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.
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\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{R}\right) = \sin^{-1}\left(\frac{y_{i}}{R}\right) \]\end{large} |
A single point particle. 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. | Section |
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\begin{large}\[ x(t) = R\cos\left(\frac{2\pi Rt}{v} + \phi\right)\]\end{large} |
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\begin{large}\[ y(t) = R\sin\left(\frac{2\pi Rt}{v} + \phi\right)\]\end{large} |
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\begin{large}\[ \vec{a}_{\rm c} = -\frac{v^{2}}{R} \hat{r}\]\end{large} |
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Examples Involving Uniform Circular Motion | Cloak |
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| | 50falsetrueANDexample_problem,uniform_circular_motion |
Examples Involving Non-Uniform Circular Motion | Cloak |
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| | 50falsetrueANDexample_problem,circular_motion,centripetal_acceleration |
All Examples Using the Model | Cloak |
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| | 50falsetrueANDexample_problem,uniform_circular_motion 50falsetrueANDexample_problem,circular_motion,centripetal_acceleration |
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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.
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.
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.
Law of Change (*describe the change of the state variables)*
where are vectors of constant magnitude and rotates with a constant angular velocity w.
Definitions and procedures:
Angular velocity w (rad/sec)
- Cartesian and Polar representation of position and velocity.
- Cartesian: x(t) = R cos (wt + fo), y(t) = sin (wt + fo) ...
- 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.
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