Angular Kinematics

Abstract

Two-dimensional angular motions of bodies are commonly described in terms of a pair of parameters, r and θ (theta), which are called the polar coordinates. Polar coordinates are particularly well suited for analyzing motions restricted to circular paths. As illustrated in Fig. 9.1, let O and P be two points on a two-dimensional surface. The location of P with respect to O can be specified in many different ways. For example, in terms of rectangular coordinates, P is a point with coordinates x and y. Point P is also located at a distance r from point O with r making an angle θ with the horizontal. Both x and y, and r and θ specify the position of P with respect to O uniquely, and O forms the origin of both the rectangular and polar coordinate systems. Note that these pairs of coordinates are not mutually independent. If one pair is known, then the other pair can be calculated because they are associated with a right triangle: r is the hypotenuse, θ is one of the two acute angles, and x and y are the lengths of the adjacent and opposite sides of the right triangle with respect to angle θ. Therefore:

$$ x=r \cos \theta . $$

$$ y=r \sin \theta . $$

Expressing r and θ in terms of x and y:

$$ r=\sqrt{x^2+{y}^2,} $$

$$ \theta = \arctan \frac{y}{x}. $$