The question was:
Stanislav Ulam asked (in "The Scottish Book") whether a sphere is the only solid of uniform density which will float in water in any position. We ask a simpler, two-dimensional, question: Consider a long log of circular cross-section; it will, obviously, float in any position without tending to rotate. (Of course, the axis of the log is assumed to be parallel to the water surface.)
(a) Are there any additional cross-section shapes such that the log will float in any position if the density of the floating body is 0.5 gm/cm3?
(b) Are there any shapes for some other (predefined) values of the density? (Of course, we assume that the given density is smaller than the density of water.)
(c) Are there any shapes, besides the circle, which will float in any position independent of the density of the material?
(6/98) Y. Kantor: Many people noted that part (c) of the
question can be easily solved. Omer Edhan (e-mail
edhan@netvision.net.il) was the
first to answer correctly. Most of the solutions provided only a partial argumentation.
The complete and correct argument for part (c) is as follows:
Since the question refers to floating body
of arbitrary density it must be also correct in the limit of
vanishing density. A body is in stable equilibrium when the difference
between the height of the center of mass and the center of buoyancy (center
of mass of the part which is under the water surface) is minimal. In the
vanishing density limit, the center of buoyancy is at the water surface, and
the question is reduced to minimization of the height of the center of mass.
Since the body is supposed to be in equilibrium independently of its orientation
the height of the center of mass must be independent of the orientation,
and therefore the distance between the center of mass and the water must
be independent of orientation.
Therefore the external boundary of the cross section of the log must be a
circle. (L. Montejano, proved this statement in Studies in Applied
Mathematics 53, p. 243 (1974).)
(Of course, one can also use
a log with a hole - log of circular cross section from which a smaller
concentric circle has been removed.)
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