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Why is an empty shampoo bottle so easy to knock over?:

It becomes annoyingly easy to knock over a shampoo bottle when it's nearly empty. This is an easily observed and curiosity-provoking phenomenon that, according to Lehigh University physics professor Jerome Licini, yields insights into center-of-mass and impacts.

"The physics of that is pretty interesting and easy to understand," says Licini who, along with first-year physics major Allen Zijun Yuan, wrote a paper on the phenomenon that was recently published in The Physics Teacher. In the paper, they find the center-of-mass of a shampoo bottle, discuss its stability on a tilted surface, and demonstrate its sensitivity to impacts using a simple experiment involving a tennis ball and a protractor.

They write: "An object placed on a tilted surface and released will tip over if the horizontal position of the center of mass lies outside the geometric outline of the base of the object. This means that an object with a low center of mass will be stable for larger incline angles. Students are often surprised to see that the altitude of the center of mass of a shampoo bottle is a nonlinear function of the fraction occupied by the contents. This can, however, be seen in a straightforward manner by recognizing that for a plastic bottle, the mass of the liquid contents is usually far greater than the mass of the bottle. When the bottle is either completely full or completely empty, the center of mass must be approximately at the geometric center of the bottle, but the center of mass gets significantly lower in altitude for small liquid levels in between those two extremes."

Yuan developed a demonstration to show the effect of impact and, it turns out, even more extreme than just looking at angle.

Reference:
Why is an empty shampoo bottle so easy to knock over? The Physics Teacher (DOI: 10.1119/1.5145426^[$]) (free access link)

Original Submission