This seems like it should be simple, but I cannot figure it out for the life of me.
I have a thin beam of length L made of some bendy material. One end of the beam is anchored at the origin and is locally vertical at the origin. When bent, this object will always make an arc with a constant radius of curvature, but its arc length will always remain constant (=L). Starting from a straight configuration, what is the equation for the distance moved by the tip of the beam as a function of radius (r) (or curvature (k), where k=1/r, which has the benefit of never being infinite) only?
A little bit of geometry gives me:
$L=r*(theta)$
$x=r-rcos(theta)=r-rcos(L/r)$
$y=rsin(theta)=rsin(L/r)$
So far so good. I should just be able to treat these as parametric equations for x and y, each as a function of r (or easily convert them to k), take the derivatives $dx/dr$ and $dy/dr$, compute the path length of the tip point's motion as $\int_{a}^{b}\sqrt((dx/dr)^2+(dy/dr)^2))dr$ with appropriate values for $b$ and $a$. No problem, I do some simple derivatives and simple algebra, and I've checked these using symbolab online and by graphing, so I'm pretty sure I didn't mess up.
The problem is that I get this: $\int_{a}^{b}\sqrt(2r^4+L^2/r^2-2Lsin(L/r)/r^3-2cos(L/r)/r^4)dr$ which is ugly as hell and I can't solve symbolically. More importantly, if expressed in terms of r, the equation inside the integral (which I'm trying to get the area under) rapidly rises from zero and asymptotes at $0.5*L^2$ (see here), and since a straight beam has an r of infinity, that means the integral is infinite as well. Conversely, if I convert everything to curvature (and thus start from zero), the equation is undefined at zero, and exhibits weird behavior for very small x (x<0.001)(see here). I imagine this would be a moot point of I could symbolically integrate it, but I can't figure out how.
One possibility is that there is already some elegant solution, probably derived by someone 3 centuries ago who's way smarter than me, but my google-fu has failed me.
Another is that there's some trick or technique I've overlooked for symbolically integrating that mess.
It's also possible I screwed up somewhere above and just haven't realized where.
Finally, I might simply have chosen a poor approach, but I can't seem to think of what a better one would be.
Any help?
Edit: Well, I just did the whole thing again in polar coordinates and got the same thing. I'm stumped.