Hey /sci/ I've been struggling with this problem for a few weeks. I've been wanting to come up with a way to find out the maximum angle a cup can tilt without losing it's liquid. To my knowledge, the only inputs required are: Dimensions of the cup, and the volume of the water, and output is simply some angle.

I believe you can solve this completely 2-dimensionally. I have no way of proving that but it's kind of just intuition tell me that so I hope it's right.

My initial thought was to vertically integrate a tilted cup (giving us a function with Y's and thetas in it) to tell us how much liquid is in the cup, set it equal to the volume we poured in, then set Y equal to the height of the cup times cos(theta) leaving us with an equation of only constants, and theta, then solving away. I'm running into countless problems and I'm starting to think it's just a bad approach.

How would /sci/ go about this? Can/should integration even be used or is there a much more trivial solution? I'm curious to hear your ideas.

I was going to upload my work, but it's so scattered on paper, and a video of me running through it would probably be just as awful, but I can try if you guys are interested.

pic not my work, just found it on google because it's related. I have no clue if that solution is accurate or even how they got to that point.