April 2025
Volume 93, Issue No. 4
Interplay between Airy and Coriolis precessions in a real Foucault pendulum
We study the precession of a Foucault pendulum using a new approach. We characterize the support anisotropy by the difference between the maximum and minimum periods of the pendulum along the principal axes of the support. Then, we compute the total precession rate, taking into account both the Airy precession of a spherical pendulum and the Coriolis precession due to the Earth's rotation. To study the resulting motion, we developed a calculation loop, period after period, that describes the movement of the oscillatory trajectory of the bob. To test our model, we mounted a test pendulum of 480.3 cm length and measured its periods and precession. The rate of precession is sensitive to the dimensions of the pendulum, the anisotropy of the support, and the initial conditions. We find that for certain amplitudes, the precession can stop entirely while the pendulum continues to oscillate. It is also possible to obtain continuous precession at lower oscillation amplitudes. We give an upper bound for this critical oscillation amplitude. We close with a discussion of the implications of our findings for the design of Foucault pendulums used in demonstrations and lab experiments.
EDITORIAL
In this issue: April 2025 by Jesse Kinder; Claire A. Marrache-Kikuchi; Beth Parks; B. Cameron Reed; Todd Springer; Keith Zengel. DOI: 10.1119/5.0267685
LETTERS TO THE EDITOR
State reduction? by Alan Macdonald. DOI: 10.1119/5.0235813
Comment on “Atom-emitting-a-photon solved on the back of an envelope” [Am. J. Phys. 91, 576 (2023)] by Mathieu Beau. DOI: 10.1119/5.0235401
A simple derivation of relativistic energy and momentum by Ajoy Ghatak; Beth Parks. DOI: 10.1119/5.0251790
Editor's Note: Instructors have many choices for deriving the relativistic energy and momentum expressions E = γmc2 and p = γmu for an object moving with respect to an observer, where u is the observed speed. This brief paper adds another option by considering a particle within a moving train car that decays into two oppositely moving photons that are observed from both within the car and outside. The derivation invokes the Planck relation E = hν for photons, the Doppler effect for light, and the energy/momentum relationship for photons, E = pc. The derivation is brief enough to easily be covered within a class period.
AWARDS
2025 Melba Newell Phillips Medal award talk: Who's your wingman? by Karen Jo Matsler. DOI: 10.1119/5.0260093
Editor's Note: This paper is the text of a plenary talk given by the author at the AAPT Winter 2025 meeting when she accepted the Melba Newell Phillips Medal.
PAPERS
Interplay between Airy and Coriolis precessions in a real Foucault pendulum by N. N. Salva; H. R. Salva. DOI: 10.1119/5.0208092 Editor's Note: Imagine setting up a Foucault pendulum to demonstrate the rotation of the earth to a full lecture hall, pulling it back, letting it go … and observing no precession. In this article, the authors illustrate how the intrinsic precession of a spherical pendulum, anisotropy in the support, the rotation of the earth, and the choice of the initial plane of oscillation could conspire to produce such a “null result.” Their minimal model can be used to evaluate any pendulum after measuring its periods of oscillation along two principal axes. They apply their model to their own experiments on a real pendulum as well as Foucault's original design for the 67-m pendulum at the Pantheon of Paris in 1851, providing interesting historical notes and technical details on the not-so-simple dynamics of a real pendulum.
A derivation of precessional effects in the Foucault pendulum using complex numbers by Morgan Facchin. DOI: 10.1119/5.0237736
Editor's Note: The Foucault pendulum is a fascinating dynamical system, but somewhat notorious for the difficulty of its analytic formulation, particularly if it is desired to include the effect of any anharmonicity of the restoring force and demonstrate that this can be canceled by using permanent magnets. In this paper, the author shows how the analysis of the Foucault pendulum can be simplified by treating the pendulum's position as a complex number that evolves in the complex plane. Appropriate for students of advanced dynamics and mathematical physics.
Unveiling the interdisciplinary character of negative pressure by Francisco F. Barbosa; Lucas Squillante; Roberto E. Lagos-Monaco; Mariano de Souza; Luciano Ricco; Antonio C. Seridonio. DOI: 10.1119/5.0186499
Editor's Note: The concept of a negative pressure may seem puzzling and paradoxical. Even though it is non-intuitive, negative pressures can be found across scientific disciplines and in diverse contexts such as chemical mixtures, transpiration in plants, and even the expansion of the universe. This article successfully introduces and vividly explores these varied topics and more. For readers who want to take a deeper dive, the article's extensive references provide an entryway into the rich literature on these topics.
Liouville integrability may not be what you think by F. Leyvraz. DOI: 10.1119/5.0237159
Editor's Note: One approach to solving classical mechanics problems is to identify the conserved quantities of the system (sometimes called the integrals of motion) and then use those identities to find expressions for the motion of the system. Systems that have the right number and right kind of integrals of motion for this approach to work are called Liouville integrable. It is sometimes said that a Liouville integrable system is one that can be solved exactly, but Liouville integrability may not mean what it is popularly thought to mean. Here the author provides examples of Liouville integrable systems that are, surprisingly, also inherently chaotic.
Active charge and discharge of a capacitor: Scaling solution and energy optimization by S. Faure; D. Guéry-Odelin; C. A. Plata; A. Prados. DOI: 10.1119/5.0217194
Editor's Note: It is with good reason that the RC circuit is a staple of the undergraduate curriculum. The system can be easily modeled and understood, but there is plenty of room for students to explore extensions and introduce complications. The authors of this article present an interesting extension: connect a time-varying voltage source in series with the resistor and capacitor to accelerate or decelerate the charging process. An interesting optimization problem then arises: given a desired charging time, which voltage profile will minimize the dissipation? Students and instructors will enjoy this surprisingly rich extension to a standard RC circuit lab, which allows for cross-curricular connections to Lagrangian mechanics and thermodynamics.
Solving introductory physics problems recursively using iterated maps by L. Q. English; D. P. Jackson; D. Richeson; W. A. Morgan; K. Sigmon; G. Vollmer; E. Dunkel; J. Zavrel; J. Wiseman. DOI: 10.1119/5.0220986
Editor's Note: In middle school or high school, students sometimes use a method of “guess and check” as a way to find the answer to a problem which they have difficulty solving. In college courses, the “guess and check” method is usually discouraged in favor of algebraic methods that arrive at an exact answer. Yet, this article discusses more advanced and systematic iterative methods and how students can use them to solve classic introductory physics problems. A beautiful analysis is performed studying the conditions under which the iterative methods converge to the correct answer and when they fail. There is something for everyone here, as the article contains pedagogical suggestions for teaching introductory physics and iterative methods, along with more advanced topics of interest to experts in nonlinear dynamics and chaos.
Quantum fields from dimensional analysis: Applications in condensed matter systems by Richard Robinett. DOI: 10.1119/5.0220969
Editor's Note: Students are typically taught dimensional analysis to check the veracity of their results when solving problems. This article takes dimensional analysis to an advanced level, applying it to quantum field theory and using it to predict relationships between different relevant observables. For instance, the critical electric field required to produce electron-positron pairs from vacuum (the Schwinger effect) can be derived in that way. The article explores a range of illustrative examples from particle physics and their condensed matter analogues, thereby illuminating the connections that unify these two fields.
NOTES AND DISCUSSIONS
An electrostatic “proof” that cos36° —cos72°= 1/2 by Christopher Ong. DOI: 10.1119/5.0253077
Editor's Note: In this brief note, the authors describe an unconventional route to geometric proof: electrostatics. By analyzing the electric field of charges placed around a pentagon, the authors demonstrate that cos(36°) is a simple algebraic expression equal to half the golden ratio. Although the result can be proved by other means and dates back to antiquity, this example is simple enough to introduce in an introductory physics course and adds mathematical interest to vector arithmetic while illustrating the power of symmetry arguments and superposition.
Measuring light intensity from a photograph: Comment on “Using a shoebox spectrograph to investigate the differences between reflection and emission” by Timothy T. Grove, Jacob Millspaw, Eric Tomek, Rebeca Manns, and Mark Masters [Am. J. Phys. 86, 594 (2018)] by Alessandro Salmoiraghi; Luigi M. Gratton; Marco Di Mauro; Pasquale Onorato; Timothy T. Grove. DOI: 10.1119/5.0212332
Editor's Note: This brief Comment points out that a previous paper was erroneous in claiming that the JPG black-and-white-format output is linear in light intensity. While that error does not affect the usefulness of the previous paper, the Comment may prevent readers from making mistakes in other work with images.
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