I have very little experience in this - I'm trying to determine how length changes with torsion of two strings. You have a single loop of string that's looped through two pulleys. There's no tension. You put a rod through the center. You begin to rotate the rod as to cause the loops to rap around its self. The more you rotate the more tension increases. What is the formula for decrease in length per rotation?
Best to work it out empirically (ie twist the string and see), because it will depend a lot on the particular string used. But making a few assumptions, I think it should depend only on the string's effective thickness and the distance from the rod to the pulleys. Bear in mind that for any real string the effective thickness will change (probably non-linearly) as you twist them up. I'll post a simple formula later (my in-laws have arrived for tea).
Assumptions: The rod is thin, the pulleys are small and far apart. The string diameter is constant under linear tension and lateral compression. (This is not a realistic assumption.) The length and diameter of an individual string is not changed when it is twisted. The twists are evenly spread between pulley and rod (again, probably not realistic, but might not make a big difference anyway.) r = string radius L = distance from rod to pulley 2L = distance between pulleys When the rod has been turned through n complete revolutions, the length of a rod to pulley segment of the string can be calculated from Pythagorus's theorem: \(\sqrt {L^2 + (2n\pi r)^2}\) This is similar to the increase in length that you get by pulling the strings apart in the middle, but the sideways part of the triangle is wrapped around the other string instead of sticking out where you can see it.
