Note that there is a nice linear relationship between the angular position of the wheel and the horizontal position? The slope of this line is 0.006 meters per degree. If you had a wheel with a larger radius, it would move a greater distance for each rotation – so it seems clear that this slope has something to do with the radius of the wheel. Let’s write it as the following expression.
In this comparison, s is the distance that the center of the wheel moves. The radius is r and the angular position is θ. It just stays k—It’s just a proportional constant. Since s vs. θ is a linear function, kr must be the slope of that line. I know all the value of this slope and I can measure the radius of the wheel at 0.342 meters. With that I have a k value of 0.0175439 with units of 1 / degree.
Big deal, right? No it is. Look at this. What happens if you multiply the value of k with 180 degrees? For my value of k, I get 3.15789. Yes, it is indeed very close to the value of pi = 3.1415 … (at least it is the first five digits of pi). It k is a way of converting angular units of degrees to a better unit to measure angles – we call this new unit the radial. If the wheel angle is measured in radians, k is equal to 1 and you get the next sweet relationship.
This comparison has two things that are important. First, there is technically a pi there, because the angle is in radial (yes, for Pi Day). Second, this is how a train stays on track. Serious.