Monday, April 19, 2021

How Pi keeps the train wheels on track

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Illustration: Rhett Allain

Do you 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 move a greater distance for each spin – so it seems clear that this slope has something to do with the radius of the wheel. Let’s write this like the following expression.

Illustration: Rhett Allain

In this equation, s is the distance traveled by the center of the wheel. The radius is r and the angular position is θ. That just leaves k– it is only a constant of proportionality. Since s vs 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 this I have a k value of 0.0175439 with units of 1 / degree.

A big deal, right? No it’s. Check it out. What happens if you multiply the value of k 180 degrees? For my worth 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 5 digits of pi). This k is a way to convert angular units of degrees to a better unit for measuring 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 nice relation.

Illustration: Rhett Allain

This equation has two important things. First, there is 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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