Answer to Question 03/99

HALF-EMPTY BOTTLE

The question was:

You are riding in a train and have just opened a bottle of beer. How much of the beer should you drink so that after putting your partially empty bottle on a table (which is shaking) it would be as stable as possible?

(3/99) The problem has been solved by Eran Toledo (e-mail toledo@post.tau.ac.il), a Ph.D. student at Tel Aviv University, and by Andreas Wendler (e-mail andreas.wendler@jos.de) from JENOPTIK SYSTEMHAUS, Jena, Germany. We have also been told that the problem and its solution appear as problem 2.3.2 in Probleme aus der Physik by H. Vogel (Springer-Verlag).


The solution:

(3/99) The most stable condition of the bottle is when its center of mass is lowest. Let the mass of the empty bottle be M, the height of its center of mass H, and the cross-sectional area A, while the density of liquid is d and the height of the liquid is h. Then the height of the center of mass of the bottle with the liquid is

(M H+d A h2/2)/(M+d A h),

since the mass of the liquid is d A h and the height of the center of mass of the liquid is h/2. By taking the derivative with respect to h and equating it to zero we find the condition for minimizing the height of the joint center of mass. The solution is

h=[M/(d A)] (sqrt(1+2 d A H/M)-1)

If the bottle is heavy (M >> d A H) this expression has a very simple approximate form: The optimal height h is simply equal to H, i.e., the level of the liquid should be at the position of the center of mass of the bottle.

It is interesting to note that whatever the relative weight of the bottle and the liquid, when the level of the beer h is optimal it coincides with the height of the center of mass of the entire system (bottle+beer).

(2/2000) Peter W. Martin from Bureau International des Poids et Mesures (e-mail pmartin@bipm.fr) suggested a very nice and simple argument proving that the total center of mass will be at minimum when the level of the beer coincides with the level of the total center of mass. His agument is independent of the shape of the bottle and goes as follows:

Imagine filling an empty bottle. Clearly the center of gravity of the system will start to decrease in height because we are adding matter below the total center of gravity. This will continue until the center of gravity reaches the level of the liquid. At that point the addition of more liquid will then cause the center of gravity to rise.
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