The value of equivalent models in physics teaching: a case study of the effective length of a large-amplitude pendulum

Equivalent modelling is a fundamental reasoning tool in physics education, allowing complex systems to be approximated by simpler, familiar forms while preserving quantitative accuracy. In this paper, we take the simple pendulum as an example to address the failure of the small-angle approximation at larger amplitudes. We propose an effective length model: define an amplitude-dependent length L eff ( θ 0 ) such that the period of a large-amplitude pendulum can still be written as T = 2 π L eff / g . This preserves the structure of the familiar simple pendulum formula and translates the effect of amplitude into an intuitive geometric parameter (a ‘longer’ effective length). Using the exact elliptic integral solution, we compute the effective length factor f ( θ 0 ) = L eff / L for amplitudes from 0 ∘ to 90 ∘ and provide simple fitting formulas suitable for classroom use. We then discuss five instructional activities that highlight the pedagogical value of equivalent models, drawing on recent research in physics education on responsive teaching, epistemic agency, and curriculum design. The case study illustrates how equivalent models can help students smoothly transition from idealized descriptions to more realistic physical situations, and it contributes to the broader discussion of how theoretical commitments shape instructional decisions.