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 with the previous post, we can imagine the unspooling
of the string in a coordinate system which keeps the string pulling straight down. Starting from position (1,0), the string extends in the -y direction. Additionally this whole coordinate system rotates clockwise. The whole time, the tension is kept on the string at right angle to the origin.
The walker's new coordinate system (x', y') is given in the standard way:
x' = x cos(theta) - y sin(theta)
y' = x sin(theta) + y cos(theta)
Here the string unspools 1 circumference (pi D = 2 pi) in 2n seconds:
x = 1 (always)
y = - (2 pi/2n) t = - (pi/n) t
Likewise, the walker goes around the origin 2 pi in angle every 2n seconds:
theta = (2 pi/2n) t = (pi/n) t
So,
x'(t) = cos[(pi/n) t] + (pi/n) t sin[(pi/n) t]
y'(t) = sin[(pi/n) t] - (pi/n) t cos[(pi/n) t]
The axis crossings times are solved by setting x' and y' in turn to 0.
The x' and y'-axis crossing points result.
TBC (with plots!)
Edited on December 3, 2021, 12:16 am
soln (cont.)
