Experimental and Theoretical Physics Studies
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Abstract
Abstract
We present a magnetic torsional pendulum that provides a unified experimental platform for investigating forced resonance, parametric resonance, and degenerate parametric amplification in the undergraduate laboratory. The system consists of a permanent magnet suspended by thin wires and driven by externally applied magnetic fields generated by Helmholtz coils. By independently controlling a direct driving field and a periodically modulated bias field, the apparatus can realize ordinary forced oscillations, parametric excitation, and phase-sensitive parametric amplification within the same physical system. A miniature wireless gyroscope embedded in the pendulum bob enables direct measurement of the angular velocity and provides convenient real-time acquisition of quantitative dynamical data. A unified equation of motion is derived to describe all three operating regimes. Experimental studies of forced resonance, parametric resonance, and phase-sensitive parametric amplification are compared with theoretical predictions and numerical simulations. The measurements reproduce the characteristic features of all three phenomena and illustrate the influence of nonlinear effects on the system dynamics. The apparatus combines a simple mechanical design, low-cost instrumentation, and highly visible motion. By allowing direct comparison of different resonance mechanisms and their underlying energy-transfer processes,, it provides an accessible platform for studying oscillation theory, nonlinear dynamics, and parametric phenomena in advanced undergraduate laboratories.
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@article{Xie2026Magnetic,
title = {A Magnetic Torsional Pendulum for Exploring Forced Resonance, Parametric Resonance, and Parametric Amplification},
author = {Wenqing Xie and Jiahao Wu and Y Shi},
journal = {arXiv (Cornell University)},
year = {2026},
doi = {10.48550/arxiv.2606.13103},
url = {https://doi.org/10.48550/arxiv.2606.13103}
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