Equivalent Length of a Physical Pendulum: A Numerical Correction for the Simple Pendulum Formula
College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, People's Republic of China
| Abstract: | The simple pendulum formula is widely used in introductory physics laboratories to determine the gravitational acceleration. However, this formula assumes a point mass and a massless string, an idealization that fails when the pendulum bob has a finite size. This paper introduces the concept of an equivalent length such that the simple pendulum formula exactly reproduces the correct period of a physical pendulum consisting of a uniform spherical bob and a light string. Using elementary rigid-body dynamics, we derive an analytical expression for the correction factor, which depends only on the ratio of the string length to the bob radius. Numerical calculations over a wide range of this ratio show that the correction is below 0.5% when the string length is at least ten times the bob radius, but exceeds 4% when the ratio is two. A detailed lookup table and a classroom procedure are provided, allowing students to continue using the familiar simple pendulum equation while obtaining accurate values of gravitational acceleration. This work bridges the gap between idealized models and real physical systems without requiring advanced mathematics. The approach is purely computational and requires no laboratory equipment, making it suitable as a pre-lab exercise, a classroom demonstration, or an independent student project in numerical methods. By preserving the simple pendulum equation, the method reduces cognitive load and provides a natural bridge to more advanced topics such as moment of inertia and the parallel axis theorem. The results are exact within the stated assumptions and offer immediate practical utility for both high school and introductory university teachers. |
| Keywords: | Simple Pendulum; Physical Pendulum; Equivalent Length; Numerical Correction; Moment of Inertia |
| DOI: | 10.57237/j.edu.2026.05.002 |
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收稿日期: 2026-07-08
接受日期: 2026-09-20
在线出版日期: 2026-10-08
基金
| 1. | Gansu Provincial University Teachers Innovation Fund Project (Contract No. 2025B-056) |
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