Catenoid soap bubbles

[Albert Chan]

At the end of video, Matt showed that bubble collapsed when H/R = 0.66274 ...
Let x = H/r, k = H/R

(R/r) = cosh(H/r)       → x/k = cosh(x)

In other words, if k above critical constant (0.66274 ...), equation failed to give a solution.

This is one way to solve for the constant, by geometry

Critical point (at x0) is when cosh(x) tangent line is y = x/k
Matching tangent line, we have:

y = sinh(x0)*(x-x0) + cosh(x0) = sinh(x0)*x + (cosh(x0)-x0*sinh(x0)) = x/k

constant term → x0*tanh(x0) = 1 → x0 ≈ 1.19967864026
linear term     → sinh(x0) = 1/k    → k ≈ 0.662743419349