It becomes dramatically and annoyingly easier to knock over a bottle as the last of its contents are removed. This is an easily observed and curiosity-provoking phenomenon that yields insights into center of mass and impacts.
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...