A pendulum bob has been constructed to experimentally probe the adiabatic transfer in the normal modes of a Wilberforce pendulum. The rotational inertia of the pendulum bob can be changed during its motion in order to follow the gradual transition from a pure translational to a pure rotational oscillation. The reported extension of the conventional use of the Wilberforce pendulum helps understand and demonstrate the concept of normal modes, coupled oscillators, beating, and avoided crossing in undergraduate university classes.
References
1.
L. R.
Wilberforce
, “On the vibrations of a loaded spiral spring
,” Philos. Mag. J. Sci.
38
, 386
–392
(1894
).2.
A.
Sommerfeld
, Mechanics of Deformable Bodies: Lectures on Theoretical Physics
(Academic
, New York
, 1964
), Vol. 2
.3.
R.
Geballe
, “Statics and dynamics of a helical spring
,” Am. J. Phys.
26
, 287
–290
(1958
).4.
U.
Köpf
, “Wilberforce's pendulum revisited
,” Am. J. Phys.
58
, 833
–837
(1990
).5.
R. E.
Berg
and T. S.
Marshall
, “Wilberforce pendulum oscillations and normal modes
,” Am. J. Phys.
59
, 32
–38
(1991
).6.
7.
M.
Mewes
, “The slinky wilberforce pendulum: A simple coupled oscillator
,” Am. J. Phys.
82
, 254
–256
(2014
).8.
E.
Debowska
, S.
Jakubowicz
, and Z.
Mazur
, “Computer visualization of the beating of a wilberforce pendulum
,” Eur. J. Phys.
20
, 89
–95
(1999
).9.
D. E.
Boucher
, “A visual and acoustic demonstration of beats and interference
,” Phys. Teach.
37
, 177
–178
(1999
).10.
G.
Kerschen
, A. F.
Vakakis
, Y. S.
Lee
, D. M.
Mcfarland
, J. J.
Kowtko
, and L. A.
Bergman
, “Energy transfers in a system of two coupled oscillators with essential nonlinearity: 1:1 resonance manifold and transient bridging orbits
,” Nonlinear Dyn.
42
, 283
–303
(2005
).11.
D. J.
D'Orazio
, M. J.
Pearson
, J. T.
Schultz
, D.
Sidor
, M. W.
Best
, K. M.
Goodfellow
, R. E.
Scholten
, and J. D.
White
, “Measuring the speed of light using beating longitudinal modes in an open-cavity HeNe laser
,” Am. J. Phys.
78
, 524
–528
(2010
).12.
H.
Zhan
, Y.
Gu
, and H. S.
Park
, “Beat phenomena in metal nanowires, and their implications for resonance-based elastic property measurements
,” Nanoscale
, 4
, 6779
–6785
(2012
).13.
M.
McDonald
, J.
Ha
, B. H.
McGuyer
, and T.
Zelevinsky
, “Visible optical beats at the hertz level
,” Am. J. Phys.
82
, 1003
–1005
(2014
).14.
T. S.
Wilhelm
, J.
Orndorff
, and D. A. V.
Baak
, “The avoided crossing in the normal-mode frequencies of a wilberforce pendulum
,” <https://www.joshorndorff.com/research>.15.
R. W.
Claassen
, “Vibrations of a rectangular cantilever plate
,” J. Aerospace Sci.
29
, 1300
–1305
(1962
).16.
M.
Petyt
and C.
Fleischer
, “Free vibration of a curved beam
,” J. Sound Vibr.
18
, 17
–30
(1971
).17.
P.
Nair
and S.
Durvasula
, “On quasi-degeneracies in plate vibration problems
,” Int. J. Mech. Sci.
15
, 975
–986
(1973
).18.
J. L.
du Bois
, S.
Adhikari
, and N. A.
Lieven
, “Eigenvalue curve veering in stressed structures: An experimental study
,” J. Sound Vibr.
322
, 1117
–1124
(2009
).19.
C. G.
Cooley
and R. G.
Parker
, “Eigenvalue sensitivity and veering in gyroscopic systems with application to high-speed planetary gears
,” Eur. J. Mech. - A/Solids
67
, 123
–136
(2018
).20.
F. S.
Crawford
, “Elementary examples of adiabatic invariance
,” Am. J. Phys.
58
, 337
–344
(1990
).21.
B. W.
Shore
, M. V.
Gromovyy
, L. P.
Yatsenko
, and V. I.
Romanenko
, “Simple mechanical analogs of rapid adiabatic passage in atomic physics
,” Am. J. Phys.
77
, 1183
–1194
(2009
).22.
L.
Novotny
, “Strong coupling, energy splitting, and level crossings: A classical perspective
,” Am. J. Phys.
78
, 1199
–1202
(2010
).23.
M.
Meissner
and H.-J.
Schocht
, Metallfedern: Grundlagen, Werkstoffe, Berechnung, Gestaltung und Rechnereinsatz
(Springer
, Berlin, Heidelberg
, 2007
).24.
F. W.
Sears
, “A demonstration of the spring-mass correction
,” Am. J. Phys.
37
, 645
–648
(1969
).25.
J. G.
Fox
and J.
Mahanty
, “The effective mass of an oscillating spring
,” Am. J. Phys.
38
, 98
–100
(1970
).26.
T. W.
Edwards
and R. A.
Hultsch
, “Mass distribution and frequencies of a vertical spring
,” Am. J. Phys.
40
, 445
–449
(1972
).27.
E. E.
Galloni
and M.
Kohen
, “Influence of the mass of the spring on its static and dynamic effects
,” Am. J. Phys.
47
, 1076
–1078
(1979
).28.
G.
Torzo
and M.
D'Anna
, “The Wilberforce Pendulum: A complete analysis through RTL and modelling
,” Quality Development in Teacher Education and Training
(Forum
, Editrice Universitaria Udinese srl
, 2004
), pp. 579
–584
.29.
See Supplementary Material at https://doi.org/10.1119/1.5051179 for a video file that illustrates the adiabatic transfer in the Wilberforce pendulum normal modes. Engineering detail drawings of the experimental setup of all pendulum components together with data acquisition software will be provided upon request.
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2018
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