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{td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]*
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h2. Description and Assumptions
{excerpt}This model applies to a single [point particle] moving in a circle with constant speed.{excerpt}
h2. 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.
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h2. Prerequisite Knowledge
h4. Prior Models
* [1-D Motion (Constant Velocity)]
* [1-D Motion (Constant Acceleration)]
h4. Vocabulary and Procedures
* [tangential acceleration]
* [centripetal acceleration]
* [angular frequency]
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h2. System
h4. Constituents
A single [point particle|point particle].
h4. State Variables
Time (_t_), position (_x_) and velocity (_v_).
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h2. Interactions
h4. Relevant Types
The system must be subject to a net force that is directed _radially inward_ to the center of the circular path, with no tangential component.
h4. Interaction Variables
Centripetal acceleration (_a_~c~).
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h2. Model
h4. Relevant Definitions
h5. Amplitude of motion:
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{latex}\begin{large}\[ A = \sqrt{x_{i}^{2} + \left(\frac{v_{i}}{\omega}\right)^{2}}\]\end{large}{latex}
h5. Phase:
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{latex}\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{A}\right) = \sin^{-1}\left(\frac{v_{i}}{\omega A}\right)\]\end{large}{latex}
h4. Laws of Change
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h5. Position:
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{latex}\begin{large}\[ x(t) = x_{i}\cos(\omega t) + \frac{v_{i}}{\omega}\sin(\omega t)\]\end{large}{latex}
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or, equivalently
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{latex}\begin{large}\[ x(t) = A\cos(\omega t + \phi) \]\end{large}{latex}
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h5. Velocity:
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{latex}\begin{large}\[ v(t) = -\omega x_{i}\sin(\omega t) + v_{i}\cos(\omega t)\]\end{large}{latex}
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or, equivalently:
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{latex}\begin{large}\[ v(t) = -A\omega\sin(\omega t + \phi)\]\end{large}{latex}
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h5. Acceleration:
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{latex}\begin{large}\[ a(t) = -\omega^{2} x_{i}\cos(\omega t) - \omega v_{i} \sin(\omega t) = -\omega^{2} x \]\end{large}{latex}
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or, equivalently:
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{latex}\begin{large}\[ a(t) = -\omega^{2}A\cos(\omega t+\phi) = -\omega^{2} x\]\end{large}{latex}
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h2. Diagrammatical Representations
* Acceleration versus time graph.
* Velocity versus time graph.
* Position versus time graph.
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h2. Relevant Examples
None yet.
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System:
Point particle moving in a circle of radius R at constant angular speed ω. (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.)
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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