Reply to ‘Comment on “Approximation for the large-angle simple pendulum period”’

Abstract

In their comment, Qing-Yin and Pei derived another analytical approximate expression for the large-angle pendulum period, which they compare with other expressions previously published. Most of these approximate formulas are based on the approximation of the original nonlinear differential equation for the simple pendulum motion. However, we point out that it is possible another procedure to obtain an approximate expression for the period. This procedure is based on the approximation of the exact period formula —which is expressed in terms of a complete elliptic integral of the first kind— instead of the approximation of the original differential equation. This last procedure is used, for example, by Carvalhaes and Suppes using the arithmetic-geometric mean.