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FLOATING OBJECTS WITH A COMMENT ON CALCULUS           12/1/1994

The articles suggested by J. Stewart (Am. J. of Phys. April 1992, p 335-356)
are very relevant. Here is the essence of the first article as it applies to
our problem. Instead of using traditional (forces and torques) approaches,
look at potential energy. Suppose a wooden block is placed in a large body of
water. As it starts to sink, its potential energy, VC, decreases (lowering
its center of mass). At the same time the displaced water is "squeezed" out
of its depth to a higher elevation. (For a large body of water, such as pond
or lake, the squized water forms an infinitesimally thin layer on top of the
reservoir.) Thus the potential energy of water, VB, increases. The block and
water will "find a position" corresponding to the minimum of VC+VB. It turns
out that VB is affected not only by the immersion depth of the block but on
its orientation as well. Equilibrium configurations of solid prisms in water
(already studied by Huygens) are mathematically analyzed in the article.

The potential energy principle can be used to explain why floating positions
are "sidewise" for long cylinders and why they are "vertical" for short ones.
BY THE WAY, THIS IS A GOOD CASE FOR USING CALCULUS, EITHER ANALYTICAL OR
NUMERICAL. I AM REFERRING TO THE ONGOING EXCHANGE OF IDEAS ABOUT THE WAYS OF
MAKING CALCULUS MEANINGFUL TO STUDENTS. GIVE THEM PROBLEMS WHICH CAN NOT BE
SOLVED EASILY WITHOUT CALCULUS AND THEY WILL APPRECIATE IT. BUT BE PREPARED
TO HELP THEM WHEN THEY ARE NOT READY. THOSE WHO DEVELOPED CALCULUS WERE NOT
SADISTS; DO NOT USE MATHEMATICS AS A TOOL FOR TORTURING.

For example, ask students to find the length at which sidewise stability
changes to vertical stability. (You can do this experimentally first, to
motivate them. Buy a long cylindrical rod, cut it into pieces of different
length and see what happens). Please, do not assume that I have done this, or
that I have a solution. I plan to play with the ideas when time becomes
available. Do not wait, do your own calculations and enjoy them. But do not
forget to share your ideas with the rest of us. I am sure that subscribers
who followed the "floating objects" discussion will be happy to read your
short messages.

HAPPY HOLIDAYS TO EVERYBODY  Ludwik Kowalski   kowalski@apollo.montclair.edu