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 |
View:
Downloads:
Received: 8 July 2026
Accepted: 20 September 2026
Published Online: 8 October 2026
Funding
| 1. | Gansu Provincial University Teachers Innovation Fund Project (Contract No. 2025B-056) |
References
| [1] | Nelson R A and Olsson M G 1986 The pendulum—rich physics from a simple system Am. J. Phys. 54, 112–121. https://doi.org/10.1119/1.14703 |
| [2] | Oliveira V 2016 Measuring g with a classroom pendulum using changes in the pendulum string length Phys. Educ. 51, 063007. https://doi.org/10.1088/0031-9120/51/6/063007 |
| [3] | Oliveira V 2022 Limits of the simple pendulum formula for classroom use Phys. Educ. 57, 045016. https://doi.org/10.1088/1361-6552/ac5da0 |
| [4] | Oliveira V 2026 Influence of the l/R ratio on the experimental determination of g using a pendulum Phys. Educ. 61, 023007. https://doi.org/10.1088/1361-6552/ae4e0d |
| [5] | Fulton E L and Gay T J 2024 Does a physical pendulum ever act like a simple pendulum? Eur. J. Phys. 45, 025001. https://doi.org/10.1088/1361-6404/ad1f95 |
| [6] | Hinrichsen P F 2021 Review of approximate equations for the pendulum period Eur. J. Phys. 42, 015005. https://doi.org/10.1088/1361-6404/abad10 |
| [7] | Bentley S J 2026 Teaching entanglement: overcoming common misconceptions Phys. Educ. 61, 025032. https://doi.org/10.1088/1361-6552/ae4cea |
| [8] | Cross R 2026 A simplified Newton’s cradle Phys. Educ. 61, 013001. https://doi.org/10.1088/1361-6552/ae187e |
| [9] | Cross R 2026 Collision of a paddle with a pickleball Phys. Educ. 61, 025014. https://doi.org/10.1088/1361-6552/ae3b87 |
| [10] | Cross R 2026 Kinetic energy of a rolling bicycle wheel Phys. Educ. 61, 023006. https://doi.org/10.1088/1361-6552/ae4c12 |
| [11] | Lustig F, Brom P and Hejnová E 2024 Remote physics experiment: Mathematical pendulum as an attractive alternative to traditional laboratory exercises J. Phys.: Conf. Ser. 2715, 012020. https://doi.org/10.1088/1742-6596/2715/1/012020 |
| [12] | Santosa I E 2025 Exploring students’ experience through gravitational acceleration experiments J. Phys.: Conf. Ser. 3139, 012081. https://doi.org/10.1088/1742-6596/3139/1/012081 |
| [13] | Mohanty S, Tripathy P P, Das R, and Prusty S 2025 Integrating artificial neural networks into undergraduate physics laboratory: A compound pendulum case study arXiv:2503.23506. https://doi.org/10.48550/arXiv.2503.23506 |
| [14] | Wahab D S, Hamsa B, Sina T A, et al 2024 Experimental study of gravity measurement with a video-based laboratory pendulum with Tracker software J. Novel Eng. Sci. Technol. 1(1) 1–11. https://doi.org/10.56741/jnest.v4i01.644 |
| [15] | Redish E F 2021 Using math in physics: Overview Phys. Teach. 59, 314-318. https://doi.org/10.1119/5.0021129 |
| [16] | Suarez A, Baccino D and Martí A C 2020 An experiment to address conceptual difficulties in slipping and rolling problems Phys. Educ. 55, 013002. https://doi.org/10.1088/1361-6552/ab4d1e |
Join the Editorial Board
We invite active, qualified and high profile scientists and researchers to join as Editorial Board Members.
Join UsBecome a Reviewer
Scholars with a strong interest in reviewing are invited to join the reviewer panel to ensure the quality of the research to be published.
Join Us