What is the shape that results when a flat rubber sheet is warped by placing a heavy ball upon it? We show that, at distance R far from the center of a ball of mass M, the height h of the surface above the ball’s center is given by where A is a constant determined by the stretchiness of the rubber and the earth’s gravitational constant. This happy result allows one to analyze the orbits of marbles and coins as they roll across the surface in some detail, providing very nice analogues for a wealth of topics in celestial mechanics, from Kepler’s laws to tides and the Roche limit.
REFERENCES
1.
A. P. French, Newtonian Mechanics (Norton, New York, 1971), pp. 531–537 is one text that explains the tides in a careful fashion; he computes the expected height of the tides from elementary considerations.
2.
Gary
White
, Tony
Mondragon
, David
Slaughter
, and Dorothy
Coates
, “Modelling Tidal Effects
,” Am. J. Phys.
61
(4
), 367
–371
(1993
). This article describes two models for tides, one using the Spandex and another using magnets, a turntable, and ball bearings; it also includes several references to earlier theoretical descriptions of the tides from this journal.3.
For example, undergraduates R. Gauthier and C. Gresham presented their Spandex resonance demonstration exhibiting the cause of Cassini’s Division at the AAPT 1995 summer meeting in College Park, winning the Society of Physics Students’ Lecture Demonstration Award. Later, at the 1996 summer meeting in Denver, C. Gresham and students won the AAPT Video Competition with their presentation of the Kepler’s third law analog using the Spandex. This paper, in fact, is a summary of the experimental results obtained by students (from junior high to college) over the last few years, with a little theory added.
4.
Frank Morgan, The Math Chat Book (The Mathematical Association of America (MAA), Washington, DC, 2000). Professor Morgan was kind enough to explain to GDW the difference between soap bubble films and the Spandex (soap bubble films do not obey Hooke’s law, but rather the tensile force is constant with extension) after his entertaining presentation at the 2001 Louisiana Mississippi regional MAA meeting. In this book, and especially in his recent article, “Proof of the Double Bubble Conjecture,” Am. Math. Monthly 8, 3 (March 2001), he describes his undergraduate students’ remarkable contributions to the solution of this problem.
5.
A. E. H. Love, A Mathematical Treatise on the Mathematical Theory of Elasticity (Dover, New York, 1944), 4th ed., p. 475.
6.
Kirk T.
McDonald
, “A mechanical model that exhibits a gravitational critical radius
,” Am. J. Phys.
69
(5
), 617
–618
(2001
).7.
8.
Aaron K.
Grant
and Jonathan L.
Rosner
, “Classical orbits in power-law potentials
,” Am. J. Phys.
62
(4
), 310
–315
(1994
).9.
Ralph
Baierlein
, “Rubber asteroids: Some orbits in introductory physics
,” Am. J. Phys.
62
(4
), 378
–379
(1994
).10.
These last paragraphs express strongly held opinions of the authors, but the origin and actual documentation of the value of “spiralling back” and of inaccurate physical models is provided in the text by Arnold Arons, Teaching Introductory Physics (Wiley, New York, 1996), along with a lot more good information for science teachers.
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© 2002 American Association of Physics Teachers.
2002
American Association of Physics Teachers
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