Notice 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 travel a greater distance with each rotation – so it seems obvious that this slope has something to do with the radius of the wheel. Let’s write this as the following expression.
In this equation, s is the distance that the center of the wheel moves. The radius is r and the angular position is θ. That just goes away k– this is just a constant of proportionality. Since s vs. θ is a linear function, kr must be the slope of that 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 in units of 1 / degree.
Big deal, right? No it is. Take a look at this. What if you change the value of k with 180 degrees? For my worth of k, I am getting 3.15789. Yes, that’s VERY close to the value of pi = 3.1415 … (at least that’s the first 5 digits of pi). This k is a way to convert angle units from degrees into a better unit for measuring angles – we call this new unit the radian. If the wheel angle is measured in radians, k equals 1 and you get the next beautiful relationship.
This equation has two things that are important. First, it technically has a pi in it because the angle is in radians (yes for Pi day). Second, this is how a train stays on track. Serious.