- #1

BP Leonard

- 4

- 2

s = r theta

usually with the stipulation that "theta must be in radians." So let's say that r = 0.1 m and theta = 30 degrees. To find s, we first convert the angle to radians. Since 360 degrees = 2 pi radians (= one revolution), we have 30 degrees = pi/6 radians. Then, using the above well-known formula:

s = (0.1 m) (pi/6 rad) =(pi/60) m rad

Since we know that s is a length (to be expressed in metres), the "rad" in the answer seems anomalous. We therefore "

**get rid of rad**"--using a variety of justifications and explanations. [Note that this is equivalent to

**dividing**the post-calculation expression by 1 radian.] The (correct) answer is then s = pi/60 m.

The inverse problem--finding theta, given s and r--gives an angle that appears to be dimensionless. We must then "

**insert rad**" in order to get the correct answer. [Equivalent to

**multiplying**the post-calculation expression by 1 radian.] The angle (now expressed in radians) can then be converted to other angle units using well-known formulas.

This deletion or insertion of rad (however justified and explained) causes widespread confusion. The problem stems from the fundamental equation, "s = r theta," which is dimensionally inconsistent. A dimensionally consistent relationship must be written:

s = const (r theta)

where, if

**A**represents the dimension "angle," then the constant must have the dimension

**1**/

**A**. We can evaluate this constant by setting s = 2 pi r and theta = 1 rev (one complete revolution), giving:

const = (2 pi)/rev

Since rev/(2 pi) is one radian, rad, we see that const = 1/rad. Therefore, the dimensionally correct fundamental relationship is:

s = r theta/rad

where theta can be expressed in

*any*angle units. In other words, we have (correctly) explicitly

**divided**the right-hand side by rad before the calculation--rather than post calculation. Similarly for the inverse relationship, theta = (s/r) rad, where we have (correctly) multiplied the RHS by rad before the calculation--rather than post calculation.

The correct equation, s = r theta/rad and its differential vector form d

**s**= (d

**theta**/rad)

**x r**(noting the order of factors), has far-reaching consequences in rotational kinematics and dynamics, trigonometry, vibrations, and many other areas. In other words, if "rad" is correctly expressed in fundamental equations, it does not have to be deleted or inserted post calculation--thereby avoiding the multiple and varied (and very confusing) justifications and explanations that seem to abound in textbooks and on-line tutorials.