Do you notice that there is a good 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. We write this as the following expression.
In this equation, s is the distance the center of the wheel moves. The radius is r and the angular position is θ. That just comes out k—This is just a constant of proportionality. From s θ is a linear function, kr must be the slope of this line. I already know the value of this slope and I can measure the radius of the wheel at 0.342 meters. With that, I have one k value of 0.0175439 with units of 1 / degree.
Great thing, right? No, that’s right. Look at this. What happens if you multiply the value of k by 180 degrees? For my value of k, I have 3.15789. Yes, it is really VERY close to the value of pine = 3.1415 … (at least the first 5 digits of pine). That k it is a way to convert from angular units of degrees to a better unit for measuring angles; we call this new unit the radian. If the angle of the wheel is measured in radians, k is equal to 1 and you will get the next lovely relationship.
This equation has two things that are important. First, there is technically a pine there, as the angle is in radians (per day Pi). Second, this keeps a train on track. Really.