## Calculate number of turns in Archimedes spiral

Hi,

I'm an engineer designing a spring system for a garage roller door. I need to know the number of turns of the door for all the size combinations.

I've found this page which gives a good equation for finding the length if you know the number of turns, starting radius and gap between spirals:

The equation of the spiral is r=x+yθ, so x=starting radius, y=gap/2∏, and to find L we're taking the integral from a=0 to b=2∏n (where n=turns).

When you know n, this is straightfoward, and even I could work that out. But it's been a decade since I've done anything like this, so I was wondering if anyone could help me find an expression for n in this:

L=$^{2∏n}_{0}$ $\sqrt{(a+bθ)^2+b^2}dθ$

Lord help me, my way of solving this is to find L for n=1,2,3,4,5 etc, graph it in excel and use "find trendline" to get an equation. Any help appreciated, thanks.
 Hey! If you're only looking for the answer, you can use this. For methodology, you can do two substitutions: first $v=a+b\theta$ Then you need to make a second substitution v=b*sinh(u). Donīt forget the derivatives. For more information you can visit this topic: http://www.physicsforums.com/showthread.php?t=157980 hope that helps a bit!
 Ok, I'll see how I go. Thanks a lot.