Abstract:  Wilberforce pendulum is composed of a vertical spiral string and a pendulum bob fixed on the bottom of the string, it can vibrate vertically and rotate back and forth horizontally. In this paper, the reason why the amplitude of vertical oscillation and horizontal oscillation changes periodical is analyzed experimentally from the perspective of mechanics, and the motion of vertical oscillation and horizontal oscillation of Wilberforce pendulum from the perspective of analytical mechanics. The factors affecting the coupled motion state of Wilberforce pendulum are analyzed, and the experiment is used to study the influence of moment of inertia of Wilberforce pendulum on the coupled motion state. Wilberforce pendulum undergoes three motion states with the increase of moment of inertia: strong coupling, weak coupling and approximate decoupling. control variates are used to study the influence of the initial elongation of the string on the motion of Wilberforce pendulum. The results are in good agreement with the theoretical analysis.

Key words: Wilberforce pendulum, moment of inertia, coupled vibrations, Fourier analysis