A new Visual approach to pendulum period determination

TL;DR

This paper introduces a geometric and algebraic method for deriving the pendulum period formula, making the concept more accessible to students without advanced calculus knowledge.

Contribution

It presents an alternative derivation of the pendulum period using geometry and physical intuition, avoiding complex differential equations.

Findings

Derivation based on infinitesimal inclined planes simplifies understanding.

Method accessible to students with basic geometry and algebra.

Provides an intuitive approach to classical physics formula.

Abstract

The period of oscillation of a simple pendulum ($T = 2\pi\sqrt{l/g}$) is a familiar formula to the average first-year physics student. However, deriving this expression from first principles involves solving a non-linear differential equation using the small angle approximation, which may appear obscure to a student in the early stages of learning calculus. Therefore, we propose an alternative approach to the derivation of this formula employing geometry, algebra, and physical intuition. Our method follows the ideas of the fathers of calculus, replacing the circular path of the pendulum with a successive collection of infinitesimal inclined planes and summing the travel times of each plane in the limit as the number of planes becomes very large. Notably, the evaluation of this limit relies solely on geometry, making it accessible to any student, even those not yet familiar with calculus techniques.