Problems Plus 6: Cable wound around a spool

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In this video, I calculate the height covered by a coil of cable of known length around a spool, counting only the portion of the height that includes a full loop. I first note that as a cable is wound in one direction, it forms a consistent angle as it wraps around itself. Unwinding it completely while keeping the angle forms a straight line, which we can then use similar triangles to obtain the vertical distance between two successive loops of the cable. The length of each loop can be obtained by using the arc length formula and parametric equations of the coil. Finally, the total height covered by the cable is the vertical distance between two loops multiplied by the floor function of the total cable length divided by the length of each loop. The floor function is the greatest integer less than a given number, and in our case represents the number of loops completely covered by the cable (i.e. excluding partial loops). This is a fascinating look at the complex physics of winding a simple cable around a spool!

Timestamps

Problem 6: Cable wound around a spool – 0:00
Solution: Cable forms a helix of radius R + r, measured from the centerline of the cable – 0:44
Unwinding the cable while ignoring the initial flat portion – 1:50
Cross-section view of a full loop of cable – 3:33
Similar triangles from the circumference of one loop to obtain vertical distance, h, between two loops – 7:47
Parametrizing the helix – 17:46
z parameter is obtained by taking the integral of the rise over run vertical slope with time – 22:00
Length of one complete cycle via the arc length formula – 24:23
Floor function ⟦x⟧ or ⌊x⌋ is the greatest integer less than x – 36:42
Ceiling function ⟧x⟦ or ⌈x⌉ is the least integer greater than x – 38:19
Number of complete cycles is the floor of the ratio of total cable length to one loop length – 39:35
Equation of the height completely covered by loops of cable around our spool – 43:20