How Pi Keeps Train Wheels on Track


Illustration: Rhett Allain

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 bigger 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 this as the following expression.

Illustration: Rhett Allain

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 leaves k—this is just a proportionality constant. 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 to be 0.342 meters. With that, I have a k value of 0.0175439 with units of 1/degree.

Big deal, right? No, it is. Check this out. What happens if you multiply the value of k by 180 degrees? For my value of k, I get 3.15789. Yes, that is indeed 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 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 following lovely relationship.

Illustration: Rhett Allain

This equation has two things that are important. First, there’s technically a pi in there since the angle is in radians (yay for Pi Day). Second, this is how a train stays on the track. Seriously.

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