Notice that there is a beautiful 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 is related to the radius of the wheel. Let’s write this as the next expression.
In this equation, S is the distance the center of the wheel moves. The ray is R and the angular position is θ. That’s just leaving k– this is just a proportionality constant. Since S vs. θ is a linear function, Cr it must be the slope of that line. I already know the value of this slope and can measure the radius of the wheel at 0.332 meters. With that, I have a k value of 0.0175439 with units of 1 / degree.
Great deal, right? Is not. Check this out. What happens if you multiply the value k by 180 degrees? For my value of k, I receive 3.15789. Yes, it is really VERY close to the value of pi = 3.1415 … (at least these are the first 5 digits of pi). This k is a way to convert from angular units of degrees to a better unit to measure angles – we call this new unit the radian. If the wheel angle is measured in radians, k is equal to 1 and you get the next wonderful relationship.
This equation has two important things. First, there is a technical point of view there, because the angle is in radians (for Pi day). Secondly, that’s how a train stays. Seriously.