Plane polar coordinates are a natural way to describe circular motion.
Instead of describing the position of an object using x and y coordinates,
this system uses two coordinates that locate the object's position using
its distance from the origin (called "r" for radius) and its angle (theta)
from the positive x-axis . Angles measured counterclockwise from the x-axis
are defined to be positive angles, and angles measured clockwise from the
x-axis are negative angles. The distance from the origin is ALWAYS
positive.
There is an easy way to convert from the polar (r-theta) coordinate system to the Cartesian (x-y) system. Looking at this diagram you can see, using basic trigonometry, that:
Using these relationships, if you know the object's radius and angle,
you can quickly determine its Cartesian coordinates.
So how do you go from Cartesian to polar coordinates? That's not hard,
either. From the distance formula (the Pythagorean theorem), the square
of r must equal the square of x plus the square of y. The angle theta is
simply the tangent of y/x.
By now you should be able to figure out why we use polar coordinates
to describe circular motion. As the object moves in a circle the radius
stays constant! The only coordinate that changes is the angle theta, which
varies as a relatively simple function of time. If we tried to use Cartesian
coordinates to describe the same motion, BOTH the x and y positions
would be MUCH more complicated functions of time.