The Wilberforce pendulum illustrates important properties of coupled oscillators including normal modes and beat phenomena. When helical spring is attached to a mass to create the Wilberforce pendulum, the longitudinal and torsional oscillations are coupled. A Wilberforce can be constructed simply from a standard laboratory spring, and a smartphone's accelerometer and gyroscope can be used to monitor the oscillations. We show that the resulting time-series data match theoretical predictions, and we share the procedures for observing both normal modes and beats.
REFERENCES
1.
Lionel Robert
Wilberforce
, “
XLIV. On the vibrations of a loaded spiral spring
,” London, Edinburgh, Dublin Philos. Mag. J. Sci.
38
(233
), 386
–392
(1894
).2.
E.
Debowska
,
S.
Jakubowicz
, and
Z.
Mazur
, “
Computer visualization of the beating of a wilberforce pendulum
,” Eur. J. Phys.
20
(2
), 89
–95
(1999
).3.
T.
Greczylo
and
E.
Debowska
, “
Using a digital video camera to examine coupled oscillations
,” Eur. J. Phys.
23
(4
), 441
–447
(2002
).4.
Giacomo
Torzo
and
Michele D'
Anna
, “
The Wilberforce pendulum: A complete analysis through RTL and modelling
,” in Quality Development in Teacher Education and Training
(Forum Editrice Universitaria Udinese srl
, Udine, Italy
, 2004
), pp. 579
–585
.5.
Pierre
Devaux
,
Valentin
Piau
,
Ombeline
Vignaud
,
Guillaume
Grosse
,
Renato
Olarte
, and
Alexis
Nuttin
, “
Cross-camera tracking and frequency analysis of a cheap slinky Wilberforce pendulum
,” Emergent Scientist
3
, 1
(2019
).6.
Matthew
Mewes
, “
The Slinky Wilberforce pendulum: A simple coupled oscillator
,” Am. J. Phys.
82
(3
), 254
–256
(2014
).7.
Miro
Plavcic
and
Pasko
Zupanovic
, “
The resonance of the Wilberforce pendulum and the period of beats
,” Latin-Am. J. Phys. Educ.
3
(3
), 8
(2009
).8.
C. J.
Pereyra
,
M.
Osorio
,
A.
Laguarda
, and
D. L.
Gau
, “
Fourier analysis of a vibrating string through a low-cost experimental setup and a smartphone
,” Phys. Educ.
53
(4
), 045019
(2018
).9.
M.
Osorio
,
C. J.
Pereyra
,
D. L.
Gau
, and
A.
Laguarda
, “
Measuring and characterizing beat phenomena with a smartphone
,” Eur. J. Phys.
39
(2
), 025708
(2018
).10.
Ulrich
Köpf
, “
Wilberforce's pendulum revisited
,” Am. J. Phys.
58
(9
), 833
–837
(1990
).11.
Álvaro
Suárez
,
Daniel
Baccino
,
Martín
Monteiro
, and
Arturo C.
Martí
, “
Normal coordinates in a system of coupled oscillators and influence of the masses of the springs
,” Eur. J. Phys.
42
(1
), 015003
(2020
).12.
Martin
Monteiro
,
Cecilia
Stari
,
Cecilia
Cabeza
, and
Arturo C.
Marti
, “
Physics experiments using simultaneously more than one smartphone sensors
,” J. Phys.: Conf. Ser.
1287
(1
), 012058
(2019
).13.
Martín
Monteiro
,
Cecilia
Stari
,
Cecilia
Cabeza
, and
Arturo C.
Marti
, “
A bottle of tea as a universal helmholtz resonator
,” Phys. Teach.
56
(9
), 644
–645
(2018
).14.
Jochen
Kuhn
and
Patrik
Vogt
, “
Analyzing spring pendulum phenomena with a smart-phone acceleration sensor
,” Phys. Teach.
50
(8
), 504
–505
(2012
).15.
David
Weiler
and
Arne
Bewersdorff
, “
Superposition of oscillation on the metapendulum: Visualization of energy conservation with the smartphone
,” Phys. Teach.
57
(9
), 646
–647
(2019
).16.
Katrin
Hochberg
,
Jochen
Kuhn
, and
Andreas
Müller
, “
Using smartphones as experimental tools–Effects on interest, curiosity, and learning in physics education
,” J. Sci. Educ. Technol.
27
(5
), 385
–403
(2018
).17.
Gallot
Thomas
(2022
), “Github repository for data analysis and simulation of the wilbuerforce pendulum,” Github
, https://github.com/RodrigoGarciaTejera/WilberforcePendulum.18.
Richard E.
Berg
and
Todd S.
Marshall
, “
Wilberforce pendulum oscillations and normal modes
,” Am. J. Phys.
59
(1
), 32
–38
(1991
).19.
See supplementary material online for an enhanced understanding of the coupled pendulum’s theoretical physics.
20.
Christoph
Stampfer
,
Heidrun
Heinke
, and
Sebastian
Staacks
, “
A lab in the pocket
,” Nat. Rev. Mater.
5
(3
), 169
–170
(2020
).21.
Chris Isaac
Larnder
, “
A purely geometrical method of locating a smartphone accelerometer
,” Phys. Teach.
58
(1
), 52
–54
(2020
).22.
The scipy.signal has a detrend function, a Butterworth filter coefficient design (butter), and sosfiltfilt to apply the filter in the time domain.
23.
Christian
Hill
(2015
). “
Scipython: The willburforce pendulum
,” https://scipython.com/blog/the-wilberforce-pendulum/.© 2023 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2023
Author(s)
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.
