Imagine a pole of unit radius stuck in the ground at the origin. A thin string is wrapped to the left many times around the pole (clockwise direction as seen from above) and you are holding it tightly against the pole at (1,0).
If you hold the string tightly, you can unwind it by walking in a spiral to the right (counterclockwise.)
Task 1: Find parametric equations for your path.
Task 2: If you adjust your speed so the string unwinds at a constant rate of 1 revolution every 2π seconds and graph your position over time, formulate the x- and y-intercepts of this graph.
Continuing from the previous post, we note that the speed of the walker is irrelevant to the problem. The exact same spiral will result, with the same x and y intercepts, no matter what the speed the walker takes. After all, the same length of string (2 pi) is played out on each revolution. The path is an Archimedes' Spiral, as can be seen by plotting the parametric equations of the last post. The radius of the spool sets the scale factor for the spiral, where the general form of the equations is:
x = cos(t) + t sin(t)
y = sin(t) - t cos(t)
To be fair, the very inner part of the spiral is distorted from a purely Archimedean one, since the angle formed by the three points: the string's last contact with the spool, the walker (string end) and the spool center, is not zero. But it approaches zero with each revolution, and the crossing points become spaced evenly at 2 pi intervals (or whatever the circumference of the spool happens to be).
While the first few crossing points could be worked out with numerical methods, such as, for y=0, solving tan(ct)=ct, with ct being slightly larger than multiples of 2 pi, we note that the crossings become uniformly spaced quickly enough.
Edited on December 5, 2021, 3:44 pm
soln (conclusion)
