We use the arithmetic-geometric mean to derive approximate solutions for the period of the simple pendulum. The fast convergence of the arithmetic-geometric mean yields accurate solutions. We also discuss the invention of the pendulum clock by Christiaan Huygens in 1656–1657.
REFERENCES
1.
D.
Halliday
, R.
Resnick
, and J.
Walker
, Fundamentals of Physics
, 7th ed. (Wiley
, Hoboken, NJ
, 2004
).2.
S.
Thornton
and J.
Marion
, Classical Dynamics of Particles and Systems
, 4th ed. (Saunders College
, Fort Worth, TX
, 1995
).3.
4.
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
).5.
W. P.
Ganley
, “Simple pendulum approximation
,” Am. J. Phys.
53
, 73
–76
(1985
).6.
L. H.
Cadwell
and E. R.
Boyko
, “Linearization of the simple pendulum
,” Am. J. Phys.
59
, 979
–981
(1991
).7.
M. I.
Molina
, “Simple linearization of the simple pendulum for any amplitude
,” Phys. Teach.
35
, 489
–490
(1997
).8.
R. B.
Kidd
and S. L.
Fogg
, “A simple formula for the large-angle pendulum period
,” Phys. Teach.
40
, 81
–83
(2002
).9.
L. E.
Millet
, “The large-angle pendulum period
,” Phys. Teach.
41
, 162
–163
(2003
).10.
R. R.
Parwani
, “An approximate expression for the large angle period of a simple pendulum
,” Eur. J. Phys.
25
, 37
–39
(2004
).11.
G. E.
Hite
, “Approximations for the period of a simple pendulum
,” Phys. Teach.
43
, 290
–292
(2005
).12.
A.
Beléndez
, T.
Hernández
, A.
Beléndez
, and C.
Neipp
, “Analytical approximations for the period of a nonlinear pendulum
,” Eur. J. Phys.
27
, 539
–551
(2006
).13.
F. M. S.
Lima
and P.
Arun
, “An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime
,” Am. J. Phys.
74
, 892
–895
(2006
).14.
M.
Abramowitz
, Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables
(Dover
, New York
, 1974
).15.
E.
Salamin
, “Computation of using arithmetic-geometric mean
,” Math. Comput.
30
, 565
–570
(1976
).16.
J. M.
Borwein
and P. B.
Borwein
, “The arithmetic-geometric mean and fast computation of elementary functions
,” SIAM Rev.
26
, 351
–366
, 1984
.17.
G.
Almkvist
and B.
Berndt
, “Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses , and the ladies diary
,” Am. Math. Monthly
95
, 585
–608
(1988
).18.
G. E.
Andrews
, R.
Askey
, and R.
Roy
, Special Functions
(Cambridge University Press
, New York
, 1999
).19.
S.
Bullett
and J.
Stark
, “Renormalizing the simple pendulum
,” SIAM Rev.
35
, 631
–640
(1993
).20.
S.
Siboni
, “Superlinearly convergent homogeneous maps and period of the pendulum
,” Am. J. Phys.
75
, 368
–373
(2007
).21.
N.
Aggarwal
, N.
Verma
, and P.
Arun
, “Simple pendulum revisited
,” Eur. J. Phys.
26
, 517
–523
(2005
).22.
D.
Stevenson
and others, “An American national standard: IEEE standard for binary float point arithmetic
,” ACM SIGPLAN Notices
22
, 9
–25
(1987
).23.
R. H.
Good
, “Elliptic integrals, the forgotten functions
,” Eur. J. Phys.
22
, 119
–126
(2001
).24.
R. G.
Forbes
, “Simple good approximations for the special elliptic functions in standard Fowler–Nordheim tunneling theory for a Schottky–Nordheim barrier
,” Appl. Phys. Lett.
89
, 113122
–1
(2006
).25.
B. A.
Cooke
, “Some ideas for using spreadsheets in physics
,” Phys. Educ.
32
, 80
–87
(1997
).26.
J. E.
Baker
and S. J.
Sugden
, “Spreadsheets in education: The first
,” Spread. Educ.
1
, 18
–43
(2003
).27.
M.
Fowler
, “Using Excel to simulate pendulum motion and maybe understand calculus a little better
,” Sci. Educ.
13
, 791
–796
(2004
).28.
C.
Huygens
, Horologium Oscillatorium sive De Motu Pendulorum ad Horologia Aptato Demonstrationes Geometric
, Paris, 1673;translated into English by
R. J.
Blackwell
as The Pendulum Clock, or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks
(Iowa State University Press
, Ames
, 1986
).29.
J.-L.
Lagrange
, “Sur une nouvelle méthode de calcul intégral pour les différentielles affectées d’un radical carré sous lequel la variable ne passe pas le quatrième degré
,” Mem. l’Acad. Roy. Sci. Turin, 2 (1784–85
);Reprinted in
Oeuvres de Lagrange, tome 2
(Gauthier-Villars
, Paris
, 1868
), pp. 251
–312
.30.
K. F.
Gauss
, Arithmetisch Geometrisches Mittel. Werke 3
, 361
–432
(1799
) (K. Gesellschaft der Wissenschaften zu Göttingen
, Leipzig-Berlin, 1866
).© 2008 American Association of Physics Teachers.
2008
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