Article

A sequence of good approximations for the period of a pendulum with large amplitude

Mathematical Scientist
(2016)
Abstract

We present three elementary approximate formulas for the period of a pendulum which starts at rest from a large angle of displacement. The first of these formulas is known, but the other two may be new. These three formulas result from taking the first three partial products of a new infinite product of nested radicals for the complete elliptic integral of the first kind that gives the exact period. Thus, more elementary approximations can be obtained from this exact product, but they become increasingly complex. Therefore, we stopped at three. We give a detailed table clearly displaying the accuracy of the approximations over the full range of possible initial angles of displacement. This infinite product of nested radicals is a special case of a new infinite product for the arithmeticgeometric mean that has appeared recently.

Keywords

- approximation of elliptic integrals,
- Arithmetic-geometric mean,
- infinite products,
- nested radicals,
- period of pendulum

Disciplines

Publication Date

June, 2016
Citation Information

Thomas J. Osler and Jesse M. Kosior. "A sequence of good approximations for the period of a pendulum with large amplitude" *Mathematical Scientist*Vol. 41 Iss. 1 (2016) p. 40 - 44 ISSN: 0312-3685

Available at: http://works.bepress.com/thomas-osler/4/