Rather than using xy-coordinates, it will be more convenient to describe the position of a particle in circular motion by its distance r from the center of the circle and its angle $\theta$ from the positive x-axis. This is shown in FIGURE 4.22. The angle $\theta$ is the angular position of the particle.

FIGURE 4.22 A particle’s position is described by distance $r$ and angle $\theta$.

We can distinguish a position above the x-axis from a position that is an equal angle below the x-axis by defining $\theta$ to be positive when measured counterclockwise (ccw) from the positive x-axis. An angle measured clockwise (cw) from the positive x-axis has a negative value. “Clockwise” and “counterclockwise” in circular motion are analogous, respectively, to “left of the origin” and “right of the origin” in linear motion, which we associated with negative and positive values of x. A particle $30\text{°}$ below the positive x-axis is equally well described by either $\theta =-30\text{°}$ or $\theta =+330\text{°}$. We could also describe this particle by $\theta ={\displaystyle \frac{11}{12}}\text{rev,}$ where revolutions are another way to measure the angle.

Although degrees and revolutions are widely used measures of angle, mathematicians and scientists usually find it more useful to measure the angle $\theta$ in FIGURE 4.22 by using the arc length s that the particle travels along the edge of a circle of radius r. We define the angular unit of radians such that

The radian, which is abbreviated rad, is the SI unit of angle. An angle of 1 rad has an arc length s exactly equal to the radius r.

The arc length completely around a circle is the circle’s circumference $2\pi r$. Thus the angle of a full circle is

This relationship is the basis for the well-known conversion factors

As a simple example of converting between radians and degrees, let’s convert an angle of 1 rad to degrees:

Thus a rough approximation is $1\text{rad}\approx 60\text{°}$. We will often specify angles in degrees, but keep in mind that the SI unit is the radian.

An important consequence of Equation 4.22 is that the arc length spanning angle $\theta$ is

This is a result that we will use often, but it is valid only if $\theta$ is measured in radians and not in degrees. This very simple relationship between angle and arc length is one of the primary motivations for using radians.