Finding the period of a simple pendulum

TL;DR

This paper explores the differences in period calculations between small-angle and large-angle simple pendulums, deriving formulas from fundamental laws and illustrating the role of elliptic integrals in their analysis.

Contribution

It provides a detailed derivation of pendulum periods for both models and introduces methods to derive elliptic integrals from physical phenomena.

Findings

Derivation of small-angle pendulum period using simple approximation

Exact period derivation for large-angle pendulums involving elliptic integrals

Discussion of the limitations of the small-angle approximation

Abstract

Pendulums have long fascinated humans ever since Galileo theorized that they are isochronic with regards to their swing. While this simplification is useful in the case of small-angle pendulums due to the accuracy of the small-angle approximation, it breaks down for large-angle pendulums and can cause larger problems with the computational modelling of simple pendulums. This paper will examine the differences between the periods of small-angle and large-angle pendulums, offering derivations of the period in both models from the basic laws of nature. This paper also provides a common way of deriving elliptic integrals from physical phenomena, and the period of pendulums has been one of the major building blocks in this new, developing field. Lastly, this paper makes a number of suggestions for extensions into the study of simple pendulums that can be performed. While this paper is not intended as a rigorous mathematical proof, it is designed to illuminate the derivation of the exact periods of simple pendulums and carefully walks through the mathematics involved.