This question has been on my mind lately, is it possible to find the radius of a tube with paper wrapped around it that's unrolled a given amount knowing the starting radius, thickness of the paper wrapped around it, and the initial radius before it was unrolled?
For a specific example, a tube of toilet paper. In a typical roll of toilet paper, it technically wouldn't be cylindrical, every subsequent wrapping around the cylinder would have a small "step up" where the paper is tangent to the circle which would affect calculations, and the paper could be compressed or stretched, but these can be ignored to simplify the problem, as without these assumptions it would quickly become a mess of unknowns.
if the inner radius of the tube is r(1), the initial outer radius of the entire tube including the paper is r(2), and the thickness of one sheet of paper is t(p) how would one go about approaching this? Given the assumptions from earlier, this problem could be expressed as tubes within tubes, each tube having a radius of +t(p) more than the one inside of it. The length of one of these tubes (n, where n = 1 is the tube touching the inner piece of cardboard of the toilet paper roll) when unrolled could then be given by π(r(1)+(n-1)(t(p))+(1/2)(t(p). So if you know how to find the length of a given unrolled tube n, and you know the overall length unrolled, is there any way to find the number of tubes that have been unrolled? Because then it's simply the radius of the unrolled roll = r(2) - x(t(p)) where x is the number of tubes unrolled.
Forgive me if this is a very simple question, or if my problem is poorly formatted. It's my first time posting on this site but I can't get this equation of my mind and don't have anyone to ask IRL. I feel like the answer is right in front of my nose yet I just can't see it. Thanks for any help!
