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An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime
1.R. A. Serway and R. J. Beichner, Physics for Scientists and Engineers, 5th ed. (Harcourt Brace, Orlando, FL, 2000), pp. 402–404.
2.S. T. Thornton and J. B. Marion, Classical Dynamics of Particles and Systems, 5th ed. (Brooks/Cole, New York, 2004), pp. 155–158. See also Appendix B for a good discussion of elliptic integrals.
3.It seems the only exception is the pendulum of antique astronomical clocks, whose amplitude is less than , as pointed out in A. Sommerfeld, Mechanics (Academic, New York, 1952), p. 90.
4.L. P. Fulcher and B. F. Davis, “Theoretical and experimental study of the motion of the simple pendulum,” Am. J. Phys.0002-9505 44, 51–55 (1976).
5.N. Aggarwal, N. Verma, and P. Arun, “Simple pendulum revisited,” Eur. J. Phys.0143-0807 26, 517–523 (2005).
9.M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968), p. 589.
10.S. D. Schery, “Design of an inexpensive pendulum for study of large-angle motion,” Am. J. Phys.0002-9505 44, 666–670 (1976).
11.This error depends on implicitly (through ) and its absolute value increases rapidly with it. For instance, underestimates the exact period with an error of 15.3% for an amplitude of .
12.Of course, the cases with are of less interest because most simple pendulum experiments in introductory physics laboratories are done with flexible cords instead of rigid rods, which prevents the pendulum bob from following a circular path soon after it is released. However, our approximate expression is more accurate than other ones even for .
13.The error with respect to the exact period for each amplitude is the quantity to be analyzed here instead of the error with respect to .
14.C. J. Smith, A Degree Physics: Part I. The General Properties of Matter (Edward Arnold, London, 1960).
17.M. I. Molina, “Simple linearization of the simple pendulum for any amplitude,” Phys. Teach.0031-921X 35, 489–490 (1997).
18.R. K. Curtis, “The simple pendulum experiment,” Phys. Teach.0031-921X 19, 36 (1981).
For a version of Ref. 5
that is richer in experimental details, see phys-0409086
, available at arxiv.org
20.H. L. Armstrong, “Effect of the mass of the cord on the period of a simple pendulum,” Am. J. Phys.0002-9505 44, 564–566 (1976).
21.The lengths were measured after tying up the thread firmly to a hook in the ceiling laboratory, at one end, and to a small ring at the top of the lead cylinder, at the other end.
22.T. H. Fay, “The pendulum equation,” Int. J. Math.0129-167X 33, 505–519 (2002).
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