We present a theoretical and experimental analysis of circular-like orbits made by a marble rolling on a warped spandex fabric. We show that the mass of the fabric interior to the orbital path influences the motion of the marble in a nontrivial way and can even dominate the orbital characteristics. We also compare a Kepler-like expression for such orbits to similar expressions for orbits about a spherically symmetric massive object in the presence of a constant vacuum energy, as described by general relativity.
REFERENCES
1.
James B.
Hartle
, Gravity: An Introduction to Einstein's General Relativity
(Addison-Wesley
, Reading, MA
, 2003
).2.
Sean M.
Carrol
, Spacetime and Geometry: An Introduction to General Relativity
(Addison-Wesley
, Reading, MA
, 2004
).3.
L. Q.
English
and A.
Mareno
, “Trajectories of rolling marbles on various funnels
,” Am. J. Phys.
80
(11
), 996
–1000
(2012
).4.
Gary D.
White
and Michael
Walker
, “The shape of “the Spandex” and orbits upon its surface
,” Am. J. Phys.
70
(1
), 48
–52
(2002
).5.
Don S.
Lemons
and T. C.
Lipscombe
, “Comment on “The shape of ‘the Spandex’ and orbits upon its surface,” by Gary D. White and Michael Walker
,” Am. J. Phys.
70
(10
), 1056
–1058
(2002
).6.
Stephen T.
Thornton
and Jerry B.
Marion
, Classical Dynamics of Particles & Systems
(Harcourt Brace Jovanovich, Inc.
, 1988
).7.
Don S.
Lemons
, Perfect Form: Variational Principles, Methods, and Applications in Elementary Physics
(Princeton U.P.
, Princeton, New Jersey
, 1997
), Chap. 4.8.
Notice that the analysis presented in Ref. 5 allowed for the spandex fabric to be pre-stretched, which dramatically alters the shape of the fabric. In the present work, we ensured that the spandex fabric was given zero pre-stretch.
9.
For more information about Tracker please visit <http://www.cabrillo.edu/~dbrown/tracker>.
10.
We thank Gary White for this useful suggestion of positioning the bulk of the mass below the spandex fabric.
11.
Z.
Stuchlík
and S.
Hledík
, “Some properties of the Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter spacetimes
,” Phys. Rev. D
60
, 044006
–1
(1999
).12.
E.
Hackmann
and C.
Lämmerzahl
, “Geodesic equation in Schwarzschild-(anti-)de Sitter space-times: Analytical solutions and applications
,” Phys. Rev. D
78
, 024035
–1
(2008
).13.
N.
Cruz
, M.
Olivares
, and J.
Villanueva
, “The geodesic structure of the Schwarzschild Anti-de Sitter black hole
,” Class. Quantum Grav.
22
, 1167
–1190
(2005
).14.
As is the case for the Schwarzschild metric, r < 3GM yields no circular orbits.
15.
It may be surprising for the reader to learn that Kepler's third law is of the same form in general relativity as it is in Newtonian gravitation.
© 2014 American Association of Physics Teachers.
2014
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
