Theoretical and experimental studies of the Wilberforce pendulum

This work focuses on exploring the phenomenon of Wilberforce pendulum, a research problem for the International Young Physicists' Tournament in 2021. A Wilberforce pendulum is a system which has a mass hanging vertically from the bottom of a long helical string. The mass is allowed to undergo both rotational motion about and translational motion along its vertical axis with the spring. Under appropriate initial conditions, the pendulum can exhibit coupled oscillations, where the mass switches between vertical oscillations (where it bobs up and down) and azimuthal oscillations (periodic rotations about the vertical axis). This paper aims to explore this phenomenon in three stages: firstly, an analysis on the mechanics of helical springs for an intuitive explanation for the onset of coupled oscillations; secondly, a comprehensive Lagrangian formulation which models the 'beating' motion path of the pendulum and its lightly damped oscillations; thirdly, an experimental verification of our model and the conditions required for clear coupling between the two modes of oscillations. The findings presented in this paper reveal some interesting features of the Wilberforce pendulum and help us gain some understanding of its physical properties. This phenomenon can also be regarded as an extremely insightful problem for introductory college physics courses focussed on the subject of oscillations and resonance, as its construction is easy and inexpensive, and it is a clear demonstration of coupled oscillatory motion.