Circular Motion with Constant Speed

System:

Point particle moving in a circle of radius R at constant angular speed

. (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 + f_o), y(t) = sin (wt + f_o) ...

-          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.