An Everyday Physics Problem

Four years ago while on a half year break from university, I found myself on a sleeper bus in the hills of north Vietnam. During the trip I came up with a fun physics problem.

At the beginning of the trip, when the bottle in the center of the picture below was full, it kept falling over when going around sharp turns in the mountains. After most of it was gone however (as in the picture), it stabilized.

This felt surprising to me! I expected an empty bottle to fall over as well. Can you figure out why this is the case?

Bottle in Sleeper Buss

Soon I realized what was going on:

Since the bottles surface contact area is constant as it drains, the amount the bottle can tilt without toppling over only depends on the center of mass. Let $m_{b} $ be the mass of the bottle without any water, $c_{b}$ the center of mass (up/down) (without water) and $m_{w}$ the water weight per height unit. If $h$ is the current water height, then the center of mass of the bottle and the water is

$$ c_{total}(h) = \frac{m_{b}c_{b} + hm_{w}\cdot (h/2)}{m_{b} + hm_{w}}. $$

Here is a graph for how the center of mass changes as a function of water height in the bottle ($m_{b} = 1$, $c_{b} = 1$, and $m_{w} = 10$). As you can see, the lowest point is not when it’s full or empty. Play with the graph yourself here.

Perfect material for an exam for high school students or the like.