A chain of length $k$ is hung between two fixed walls of $2x$ distance apart. How low does the chain hang (at its lowest point)? Assume the chain makes a parabola (not a catenary).
This is the reverse question of the classic problem, which is much easier to solve:
What is the arc length of a parabola $y = a x^2$ from the origin to the point where $x = x_0$?
This version is straightforward, because $$arc\ length = \int_0^x \sqrt{1+4a^2t^2}\ dt$$ which can also be used to solve the top problem numerically.
However, I realized that since all parabolas are similar, the $a$ factor in the equation is a bit of a red herring! We should be able to scale our units so that the parabola $Y = X^2$ has our answer (for some scaling $r$ such that $Y = ry, X = rx$), which should make it possible to solve the top problem.
I also tried tackling
A chain is hung between two fixed walls so that it makes a parabola $y_0 = a_0x^2$ with arc length $s_0$. The chain stretches (e.g. due to heat) by a factor of $h$. What is the new shape - that is, what is $a_h$ if the curve $y_h = a_hx^2$ has arclength $h \cdot s_0$, given that $x$ is fixed.
I thought this would be easier: since the derivative of arc length with respect to $x$ is clearly $\sqrt{1 + 4a^2x^2}$, we should be able to determine the ration between change in arc length and change in $x$. But the resultant change in $a$ threw me for a curve (no pun intended).
Can any of these problems be solved analytically (acknowledging that numeric methods can be used for any of them)?
