Experimental Investigation of the Dynamics of a Wilberforce Pendulum

The Wilberforce pendulum provides an important qualitative demonstration of coupled harmonic oscillators, but quantitative measurements are not typically performed. The present work examines the dynamics of a simple pendulum comprising a steel spring with attached mass using an ultrasonic motion detector. A series of measurements allowed determination of the coupling constant between the longitudinal and torsional modes of the spring-mass system. The motion depends on the moment of inertia, which could be varied by about 20%. The pendulum motion was recorded at six values of moment of inertia, and the frequencies of the two normal modes were determined with Fourier transform techniques. The coupling constant was measured to be (5.49± 0.38× 10−5 for the 20 mm diameter steel spring used in the experiment. Correspondence of the measured mode frequency to that predicted by the model was qualitatively excellent, but exhibited a systematic shift to lower frequency that merits further study.