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Circular
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Motion
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with
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Constant
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Speed
System:
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Point
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particle
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moving
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in
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a
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circle
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of
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radius
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R
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at
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constant
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angular
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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.