A not-so-expected Cartesian feature of the simple pendulum

This brief study suggests the pedagogical value of analysing the simple pendulum somewhat beyond the standard linear small-angle approximation. By employing video tracking techniques, we examine the Cartesian components of the pendulum’s motion for relatively small amplitudes, revealing that while the horizontal displacement follows a linear harmonic motion, the vertical component exhibits a quadratic dependence on the angular displacement. This dual-frequency motion, with the vertical oscillation occurring at twice the frequency of the horizontal one, provides a tangible example for students to understand the implications of linear and quadratic approximations in physical systems. The experiment underscores the importance of considering higher-order terms in series expansions to accurately describe physical phenomena, thereby enhancing students’ comprehension of approximation methods in physics.