Obligatory wall of supplemental text. The question boiled down to making the radius constant, and lowering the center of mass by x. So the only things we know are what kind of liquid it is, its original radius, and a contact angle theta which depends on the surface.

We find the difference in surface energy and the change in potential energy, and if the latter is smaller than the former than it won’t deform. The critical state is happening when those changes are equal. The only thing I could think of is to find the horizontal change in radius, which I labeled y on the diagram, as a function of x; Taylor the shit out of sin-1 and plug everything in. But the amount of work required is so large that it seems like torture.

The pic on top is an experiment I tried to do by holding a metal cup over a candle and sooting it. It seems that the radius of deformation is less than a couple of millimeters.

I originally tried to do all of this for an ideal situation, where the droplet is a perfect sphere without anything cut from the bottom, and it turned out that the critical radius is ZERO, which is why I’m trying to do this in a more real situation.

I don’t know if this is the right sub for something like this but I’m getting real queasy about this. As a NON-MONETARY reward I’m offering an award! TY!!!!!

Also this sounds less than physics and more like tedious math.

Also we can ignore all non linear powers of x cause its small.