Find parametric equations for the path of a particle that...

Find parametric equations for the path of a particle that moves along the circle $x^{2}+(y-1)^{2}=4$ in the manner described. $$\begin{array}{l}{\text { (a) Once around clockwise, starting at }(2,1)} \\ {\text { (b) Three times around counterclockwise, starting at }(2,1)} \\ {\text { (c) Halfway around counterclockwise, starting at }(0,3)}\end{array}$$

Find parametric equations for the path of a particle that
moves along the circle $x^{2}+(y-1)^{2}=4$ in the manner
described.
$$\begin{array}{l}{\text { (a) Once around clockwise, starting at }(2,1)} \\ {\text { (b) Three times around counterclockwise, starting at }(2,1)} \\ {\text { (c) Halfway around counterclockwise, starting at }(0,3)}\end{array}$$

Key Concepts

Scaling the Angular Parameter for Multiple or Partial Rotations

This concept deals with modifying the parameter t so that the motion covers a specified number of rotations or a fraction of a rotation. Multiplying t by a constant factor causes the particle to travel around the circle multiple times (if the factor is greater than 1) or less than one full rotation (if the factor is a fraction). It is essential to adjust the interval over which t varies to match the intended number of complete or partial revolutions along the circle.

Adjusting the Parameter for a Given Starting Point

When the starting point of the motion is specified, the standard parameterization may need a phase shift. The phase shift is determined by finding the angle t for which the cosine and sine values yield the initial coordinates relative to the circle's center. This adjustment ensures that as the parameter t begins its interval, the particle is at the designated start position, thereby aligning the parametric equations with the given initial condition.

Trigonometric Parameterization of a Circle

This concept involves using sine and cosine functions to represent the points on a circle. For a circle with center (h, k) and radius r, the typical parametric equations x = h + r*cos(t) and y = k + r*sin(t) describe the circle as t varies over an interval. This method leverages the periodic nature of trigonometric functions to represent the continuous and closed path of the circle.

Orientation and Direction of Rotation

Understanding orientation is essential when parameterizing a circle, as you must decide whether the particle moves clockwise or counterclockwise. The standard parameterization using (cos(t), sin(t)) typically results in a counterclockwise motion as the parameter increases. To reverse the direction (i.e. get clockwise motion), one can replace t with -t or adjust the trigonometric functions accordingly. This adjustment affects the sign in one of the sine or cosine components, changing the direction of travel along the curve.

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