Physics of a Particle on a Rotating Hoop
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Physics of a Particle on a Rotating Hoop∗ Experiment and Theory Kushal Lodha, Anushree Roy and Sayan Kar
The simple textbook problem of a particle on a vertical, rotating hoop is analysed both in theory and through experiments. We begin by detailing out a somewhat generalised theory, where the effect of dry friction as well as the possibility of a shift in the vertical axis of rotation are incorporated. The bifurcation curves (plots of the angular position of stability versus the angular frequency of rotation) are obtained for all cases (i.e. with and without friction and a shift of the axis). Thereafter we present the experimental set-up fabricated by us and elaborate on the various measurements performed. Finally, we demonstrate through our experiments how well the theoretical results on the bifurcation curves tally with the experimental findings. The match between theory and experiment is found to be reasonably satisfactory. We conclude by mentioning how various aspects of this simple problem as well as its generalisations and extensions, are linked with different advanced areas of physics.
Kushal Lodha is a fifth year undergraduate student in the Integrated M.Sc. (Physics) programme at IIT Kharagpur.
Anushree Roy and Sayan Kar are faculty members in the
1. Introduction Among the chapter-end problems given in Chapter 2 of the famous textbook on Classical Mechanics by Goldstein [1] (now by Goldstein, Poole and Safko) we find the problem of a particle on a rotating hoop. The same system is also discussed in detail, in many texts on nonlinear dynamics, for example, in the book by Strogatz [2]. It is quite amazing how this rather simple problem has attracted the attention of many students and teachers over the last several decades. In addition, as we note at the end of our article, this problem and its generalisations can serve as a starting ∗
Department of Physics, IIT Kharagpur.
Keywords Classical mechanics, bifurcation curves.
Vol.25, No.9, DOI: https://doi.org/10.1007/s12045-020-1044-5
RESONANCE | September 2020
1261
GENERAL ARTICLE
We take a re-look at the physics of a particle on a rotating hoop, both theoretically and experimentally.
One of our primary objectives in this work is to verify experimentally, a general relation between the equilibrium points and a control parameter of a given system, known as a bifurcation curve.
point to many an advanced concept in physics. In this article, we plan to take a re-look at the physics of a particle on a rotating hoop, both theoretically and experimentally. The theory is well-known. But actual experiments are few in number. Moreover, the theory we outline later, is somewhat recent and the experiment, as well as its comparison with theory has never been attempted. In that sense, though simple, our results are indeed new. One of our primary objectives in this work is to verify experimentally, a general relation between the equilibrium points and a control parameter of a given system, known as a bifurcation curve. As the name suggests, a bifurcation curve for a
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