We report on theoretical and experimental results for a ball that rolls without slipping on a surface of revolution, whose symmetry axis is aligned with a uniform gravitational field, particularly investigating both near-circular orbits and scattering-type orbits in cones. The experimental data give support for the theoretical treatment, a non-trivial application of Newton's second law that expands on our previous work and related work of others. Our findings refine those from a recent article in this journal, and largely replicate those obtained from an earlier Lagrangian approach, adding some new details and commentary. While the orbits of marbles rolling in cones do not match inverse-square-law orbits quantitatively (e.g., instead of Kepler's 3rd law, we have T2R), we argue that students should experience these qualitative phenomena—precession of orbits, escape velocity behavior, spin-orbit coupling, conservation laws for angular momentum, energy, and spin projection—as much for the fun and kinesthetic impressions as for the raw learning. We also report on a heretofore largely ignored variable in the exploration of rolling orbits in a gravity well: the marble's spin about its own axis as it rolls. Experimenters can, intentionally or not, vary this initial condition and produce different orbital periods for a given orbital radius—a distinctly non-celestial behavior. Careful selection of the initial spin direction and speed for a particular cone can result in marble orbits that mimic the planetary ellipses.

1.
The author is an unabashed advocate of using spandex to model potential wells so that students can learn about various celestial phenomena in weak gravitational fields in a visceral way, and because they are tangible and meaningful representations of Wheeler's oft-cited quote, “Matter tells space how to curve. Space tells matter how to move.” However, it is important to note that these potential wells are conceptually quite distinct from “embedding diagrams,” often used visualize the nature of curved space-time in strong gravitational fields.
2.
Gary D.
White
and
Michael
Walker
, “
The shape of ‘the Spandex’ and orbits upon its surface
,”
Am. J. Phys.
70
(
1
),
48
52
(
2002
).
3.
Don S.
Lemons
and
T. C.
Lipscombe
, “Comment on “The shape of `the Spandex' and the orbits upon its surface”,” A
m. J. Phys.
70
,
1056
1058
(
2002
).
4.
Gary
White
,
Tony
Mondragon
,
David
Slaughter
, and
Dorothy
Coates
, “
Modelling Tidal Effects
,”
Am. J. Phys.
61
(
4
),
367
371
(
1993
).
5.
L. Q.
English
and
A.
Mareno
, “
Trajectories of rolling marbles on various funnels
,”
Am. J. Phys.
80
(
11
),
996
1000
(
2012
).
6.
See, for example,
Landau
and
Lifschitz
,
Mechanics
(
Pergamon Press
,
Oxford
,
1960
), pp.
122
123
, or
E. A.
Milne
,
Vectorial Mechanics
(
Interscience Publishers Inc.
,
New York
,
1948
), pp.
298
309
.
Also, see
Tadashi
Tokieda
, “
Roll Models
,”
Am. Math. Monthly
120
(
3
),
265
282
(
2013
), for a refreshing take on many of these considerations.
7.
R.
Lopez-Ruiz
and
A. F.
Pacheco
, “
Sliding on the inside of a cone
,”
Eur. J. Phys.
23
,
579
589
(
2002
).
8.
I.
Campos
,
J. L.
Fernandez-Chapou
,
A. L.
Salas-Brito
, and
C. A.
Vargas
, “
A sphere rolling on the inside surface of a cone
,”
Eur. J. Phys.
27
,
567
576
(
2006
).
9.
J. L.
Fernandez-Chapou
, “
The motion of a sphere on a surface of revolution: A geometric approach
,”
Canadian Mathematical Society Conference Proceedings
, Vol. 25, pp.
199
220
(
1998
).
10.
Fowles
and
Cassidy
,
Analytical Mechanics
, 6th ed. (
Saunders College Publishing
,
Ft. Worth, TX
,
1999
), pp.
320
321
; especially note that this equation applies even if the center-of-mass frame is not an inertial frame.
11.
H.
Goldstein
,
Classical Mechanics
, 2nd ed. (
Addison-Wesley Publishing
,
Reading, MA
,
1980
), pp.
45
50
.
12.

Reference 1 documents how experimenters rolling marbles in circular orbits on our un-stretched spandex gravity wells (wells whose profile satisfies z(ρ)ρ2/3 and thus have. tanθ=dz/dρρ1/3 obtained the result T3R2, reversing the usual Kepler exponents. It is relatively straightforward to show that Eq. (34) confirms this result theoretically for spandex wells.

13.
This circumstance provides impetus for constructing cones of this angle, and some with different angles, so that the comparison between how the planets behave is more tangible. In addition, by adjusting B and the angles α and θ, other closed, near-circular planet-like orbits (when the ratio of the two frequencies in Eq. (34) is unity) and other closed resonant orbits (when the ratio of the two frequencies in Eq. (34) is a square of a rational number, not merely unity) can be constructed. For example, choosing α=θ=sin1(14/19)59.1° and B = 7/5 gives an example where the frequency ratio is unity, but for a solid marble rather than a slider; rolling in this particular cone with the spin axis aligned with the cone surface will result in an “elliptical” orbit when perturbed slightly away from a circle, mimicking the planets. Incidentally, near-circular orbits are not only a topic of interest for planetary and marble systems; similar considerations are invoked when designing concentric cylindrical and hemispherical electrostatic lenses to focus ion beams. See, for example,
D.
Briggs
and
M.
Seah
,
Practical Surface Analysis
(
John Wiley & Sons
,
Chichester
,
1990
) or
F.
Hadjarab
and
J. L.
Erskine
, “
Image properties of the hemispherical analyzer applied to multichannel energy detection
,”
J. Electron. Spectrosc. Relat. Phenom.
36
(
3
),
227
243
(
1985
).
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