A highly accurate solution of the nonlinear pendulum with small initial amplitude and damping proportional to the velocity is derived using multi-time scale perturbation methods. The nonlinear pendulum has been studied extensively, but its formulation has rarely incorporated damping. This paper extends the body of traditional results by using a mathematical framework that incorporates damping to get a clear, robust solution in which the interplay of the damping and amplitude yields new physical insights into the pendulum's motion.
REFERENCES
1.
David M.
Burton
, The History of Mathematics, an Introduction
, 7th ed. (
McGraw Hill
,
New York
, 2011
).2.
George
Gamow
, The Great Physicists from Galileo to Einstein
(
Dover
,
New York
, 1961
), p. 34
.3.
F. R. S.
Henry Kater
, “
An account of experiments for determining the length of the pendulum vibrating seconds in the latitude of London
,” Philos. Trans. R. Soc. London
108
, 33
–102
(1818
).4.
5.
Robert H.
Cannon
, Jr.
, “
Schuler pendulum
,” AccessScience
(
McGraw-Hill Education
,
New York
, 2014
).6.
Gregory L.
Baker
and
James A.
Blackburn
, The Pendulum, a Case Study in Physics
(
Oxford U. P.
,
New York
, 2005
).7.
S. C.
Zilio
, “
Measurement and analysis of large-angle pendulum motion
,” Am. J. Phys.
50
, 450
–452
(1982
).8.
L. P.
Fulcher
and
B. F.
Davis
, “
Theoretical and experimental study of the motion of the simple pendulum
,” Am. J. Phys.
44
, 51
–55
(1976
).9.
10.
Robert A.
Nelson
and
M. G.
Olsson
, “
The pendulum-rich physics from a simple system
,” Am. J. Phys.
54
, 112
–121
(1986
).11.
Claudio G.
Carvalhaes
and
Patrick
Suppes
, “
Approximations for the period of the simple pendulum based on arithmetic-geometric mean
,” Am. J. Phys.
76
(12
), 1150
–1154
(2008
).12.
Stephen D.
Schery
, “
Design of an inexpensive pendulum for study of large angle motion
,” Am. J. Phys.
44
, 666
–670
(1976
).13.
A.
Belendez
,
C.
Pascual
,
D. I.
Mendez
,
T.
Belendez
, and
C.
Neipp
, “
Exact solution for the nonlinear pendulum
,” Rev. Bras. Ensino Fis.
29
(4
), 645
–648
(2007
).14.
Temple
Fay
, “
The pendulum equation
,” Int. J. Math. Educ. Sci. Tech.
33
(4
), 505
–519
(2002
).15.
Kim
Johannessen
, “
An analytical solution to the equation of motion for the damped nonlinear pendulum
,” Eur. J. Phys.
35
, 035104
(2014
).16.
C.
Henry Edwards
and
David E.
Penney
, Differential Equations and Boundary Value Problems, Computing and Modeling
, 5th ed. (
Pearson
,
Delhi, India
, 2015
), pp. 174
–181
.17.
Horologium Oscillatorium (The Pendulum Clock, or Geometrical Demonstrations concerning the Motion of Pendula as Applied to Clocks)
, translated by
Christiaan
Huygens
and
Richard J.
Blackwell
(
Iowa State U. P.
,
Ames, Iowa
, 1986
), ISBN 0813809339.18.
J.
Kevorkian
and
J. D.
Cole
, Perturbation Methods in Applied Mathematics
(
Springer-Verlag
,
Berlin
, 1981
), pp. 115
–124
.19.
J. A.
Sander
and
F.
Verhulst
, Averaging Methods in Nonlinear Dynamical Systems
(
Springer Verlag
,
Berlin
, 1985
), pp. 189
–194
.20.
Carl M.
Bender
and
Steven A.
Orszag
, Advanced Mathematical Methods for Scientists and Engineers
(
McGraw-Hill
,
New York
, 1978
), pp.549
–560
.21.
Frank S.
Crawford
, “
Damping of a simple pendulum
,” Am. J. Phys.
43
, 276
–277
(1975
).22.
Patrick T.
Squire
, “
Pendulum damping
,” Am. J. Phys.
54
, 984
–991
(1986
).23.
Salvador
Gil
,
Andres E.
Legarreta
, and
Daniel E.
Di Gregorio
, “
Measuring anharmonicity in a large amplitude pendulum
,” Am. J. Phys.
76
(9
), 843
–847
(2008
).24.
Manuel I.
Gonzalez
and
Alfredo
Bol
, “
Controlled damping of a physical pendulum: Experiments near critical conditions
,” Eur. J. Phys.
27
(2
), 257
–264
(2006
).25.
Electric Power Transformer Engineering
, edited by
James H.
Harlow
(
CRC Press
,
FL
, 2004
), p. 216
.26.
Wolfram Research, Inc., mathematica, Version 12.0, Champaign, IL (
2019
).© 2020 American Association of Physics Teachers.
2020
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.
