**November 2024 **

**Volume 92, Issue No. 11**

Teaching labs for blind students: Equipment to measure the inertia of simple objects

This article describes a laboratory experiment for blind students to measure the moment of inertia of simple objects; in this case, that of a disc about its axis of symmetry. We adapted our usual lab to modify the data collection process, using an open-source electronic platform to convert visual signals into acoustic signals. This allowed blind students at our University to participate alongside their classmates in the mechanics lab.

#### EDITORIAL

In this issue: November 2024 by John Essick; Jesse Kinder; Claire A. Marrache-Kikuchi; Beth Parks; B. Cameron Reed; Todd Springer. DOI: 10.1119/5.0240446

#### PAPERS

On cat–human interaction from the viewpoint of physics: An equation of motion by Anxo Biasi. DOI: 10.1119/5.0158200

Editor's note: Since Erwin Schrödinger, cats have become the unofficial mascots of quantum mechanics. But cats can be useful in classical mechanics too. This paper shows that typical feline behaviors, such as their preferred resting position (on your lap or, more likely, far away), ignoring calls, and purring, can actually be explained by classical mechanics. Even zoomies, those periods of time when cats are frantic, can also be modeled as a stochastic process. So, it turns out your cat might just be the perfect example for most of an undergraduate mechanics course!

The hardest-hit home run? by Donald C. Warren. DOI: 10.1119/5.0219325

Editor's Note: The physics of baseball is endlessly compelling. In this paper, the author analyzes a May 22, 1963, home-run hit by Mickey Mantle which struck a facade 118 feet above ground level at Yankee Stadium and nearly exited the park. Witnesses claimed that the ball was still rising when it struck the facade. This paper takes a numerical-analysis approach to attempting to determine the initial launch speed and angle of the ball, considering conditions of air density, temperature, relative humidity, wind, and spin. This is a fascinating case study of forensic physics subject to multiple uncertainties, and the reader's enjoyment is not diminished by the conclusion that witnesses were likely mistaken about the ball still rising. Appropriate for upper-level dynamics/computational physics students.

Elastic collisions on a simulated circular air track by C. B. Price; M. L. Pethybridge. DOI: 10.1119/5.0125335

Editor's Note: Students in introductory courses often analyze one-dimensional elastic collisions between carts on wheels or gliders on an air track. If the track possesses end stops, and the system is left undisturbed after the initial contact, multiple collisions will occur. The analysis of repeated collisions can be somewhat complicated due to the end-stop reflections. In this article, the authors consider a clever simplification involving a circular track without end stops. This renders the mathematical analysis more tractable for students, and the system presents a variety of interesting behaviors to explore. Students and instructors will enjoy this entertaining application of linear algebra and can freely make use of online computer simulations provided as supplementary material.

Playing with active matter by Angelo Barona Balda; Aykut Argun; Agnese Callegari; Giovanni Volpe. DOI: 10.1119/5.0125111

Editor's Note: Active matter consists of particles (e.g., birds, cells, or synthetic objects) that can self-propel. These systems are currently being researched not only because they can display collective behaviors and some level of self-organization but also because of their potential applications such as in drug delivery. Both in the classroom and in research labs, toy robots such as Hexbugs can be used to model active matter. This article shows that, by playing with these toys, undergraduate students can visualize the concepts they're being taught in advanced thermodynamics or soft matter classes: chiral and non-chiral active Brownian motion, interaction of the particles with their environment, and particle sorting. Time to play!

Kramers–Kronig relations via Laplace formalism and L1 integrability by Marco Prevedelli; Alessio Perinelli; Leonardo Ricci. DOI: 10.1119/5.0217609

Editor's Note The Kramers–Kronig relations are fundamental to optics, linking dispersion to absorption. While incomplete proofs are found in many textbooks, the rigorous mathematical proof that requires the minimal assumption (that the transfer function is square-integrable) is quite challenging and, therefore, rarely taught. In this paper, the authors show that requiring a slightly stricter criterion for the transfer function (L1 = Lebesgue-integrable) allows the proof to be much more accessible.

Analytical solution of the Sommerfeld–Page equation by Zurab K. Silagadze. DOI: 10.1119/5.0156685

Editor's Note: The Sommerfeld–Page equation governs the classical dynamics of a uniformly charged sphere, such as an electron, when considered to be a classical particle of finite size. It was introduced in relation to the radiation reaction problem in classical electron theory: If the electron has a finite size and since electromagnetic interactions can only propagate at the speed of light, does part of the electron exert a back-action on the other parts? This paper proposes an analytical solution to this complex problem, obtained by solving delay differential equations, which is appropriate for advanced electromagnetism or mathematical physics classes.

Revisiting Sommerfeld's atomic model using Euler–Lagrange dynamics by Wilder R. Cardoso; Mariana C. Nakagaki. DOI: 10.1119/5.0219344

Editor's Note: The authors present Sommerfeld s atomic model as a valuable teaching tool, both because it incorporates many important topics in physics instruction and because it helps students appreciate the development of scientific ideas. Readers will appreciate the careful presentation of the physics.

A simple model of a gravitational lens from geometric optics by Bogdan Szafraniec; James F. Harford. DOI: 10.1119/5.0157513

Editor's Note: This article describes a simple theoretical model for gravitational lensing. The authors analyze a graded index of refraction that reproduces the behavior for light passing near the event horizon of a black hole. The mathematical simplicity of the model permits exploration of the effects of gravitational lensing—including bending, reflection, and the formation of Einstein rings—using only integral calculus and Fermat's principle. The authors illustrate many interesting lensing phenomena with 2D and 3D graphics. The model described in this paper could be introduced as a “theoretical toy model” to complement classroom demonstrations of gravitational lensing such as a “logarithmic lens” or the stem of a wine glass, making gravitational lensing and its use in modern astrophysics accessible to introductory physics students.

#### NOTES AND DISCUSSIONS

Why the action? by Keith Zengel. DOI: 10.1119/5.0199741

Editor's Note: Students of advanced dynamics encounter the principle of stationary action, wherein the time-integral of the Lagrangian L = T - U is extremized; T and U are the kinetic and potential energies of a system. At first glance, extremizing the difference between T and U may seem strange; the usual justification is that it works in the sense of being equivalent to Newton's second law. In this paper, the author offers two alternative approaches to the least-action principle: extremizing the time-averaged kinetic energy or the product of the time-averaged kinetic and potential energies. The stationary paths are established, and it is shown how these include the customary energy-conserving paths and how they can be isolated. The motivation is to offer instructors options for ``deriving'' the principle of stationary action. Appropriate for upper/graduate-level dynamics students.

#### INSTRUCTIONAL LABORATORIES AND DEMONSTRATIONS

Teaching labs for blind students: Equipment to measure the inertia of simple objects by Alfredo Lisboa; Francisco J. Peña. DOI: 10.1119/5.0152980

Editor's Note: As educators, we value inclusion, although we sometimes struggle with how to implement it in real life. This paper presents a practical realization of an undergraduate mechanics labs designed to determine the moment of inertia of a disk rotating about its axis, specifically tailored to be accessible for blind students.

An instructional lab apparatus for quantum experiments with single nitrogen-vacancy centers in diamond by Zhiyang Yuan; Sounak Mukherjee; Jeff D. Thompson; Nathalie P. de Leon; Aedan Gardill; Shimon Kolkowitz. DOI: 10.1119/5.0216511

Editor's Note: This paper describes an experimental setup for performing measurements on the nitrogen-vacancy (NV) center in diamond. While three recent AJP papers describe experimental setups for taking measurement on an ensemble of NV centers, this paper takes such studies to the single NV center level. After a review of the basic physics of the NV center, a comprehensive and clear description of a challenging but important set of undergraduate-accessible experiments is presented. These experiments investigate topics including single-qubit initialization, rotation, and measurement as well as advanced studies on electron–nuclear spin interactions. This work will be of interest to those wishing to introduce quantum-related experiments on a forefront research topic in their advanced physics instructional laboratory courses or independent student projects.