How Pi Keeps Train Wheels on Track

Illustration: Artet Allen

Note that there is a good linear relationship between the angular position and horizontal position of the wheel? The slope of this line is 0.006 m per degree. If you had a wheel with a larger radius, it would cover more distance for each turn – so it seems obvious that this slope has something to do with the radius of the wheel. Let’s write it as the following expression.

Illustration: Artet Allen

In this equation, S The center of the wheel moves is the distance. Is the radius R And the angular position is. He just leaves K—It is only a proportionality constant. since S Vs vs Is a linear function, kr That line must have a slope. I already know the value of this slope and I can measure the radius of the wheel to 0.342 meters. In addition, I am K Value of 0.0175439 with units of 1 / degree.

Big deal, okay? No it is. Check it out What happens if you multiply the value of K 180 degrees? For my price K, I get 3.15789. Yes, it is actually very close to the value of pi = 3.1415 … (at least it is the first 5 digits of pi). this K To measure angles there is a way to convert angular units from degrees to a better unit – we call this new unit radians. If the wheel angle is measured in radians, K Is equal to 1 and you get the following lovely relationship.

Illustration: Artet Allen

There are two things in this equation that are important. First, there is technically a pie because the angle is in radian (yay for pie day). Second, this is how a train stays on track. Seriously.

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