### Volume 91, No. 5

Paragliders' launch trajectory is universal

We designed and built a reduced-scale model experiment to study the paragliding inflation and launching phase at given traction force. We show that the launch trajectory of a single skin glider is universal, that is, independent of the exerted force. As a consequence, the length of the take-off run required for the glider to reach its “ready to launch” vertical position is also universal. We successfully compare our results to full-scale experiments and show that such universality can be understood through a simple theoretical model.

#### EDITORIAL

In this issue: May 2023 by John Essick; Claire A. Marrache-Kikuchi; Beth Parks; B. Cameron Reed; Keith Zengel. DOI: 10.1119/5.0151421

#### PAPERS

On the bifurcation behavior of a folded notebook page by Zhang Chenguang (张晨光). DOI: 10.1119/5.0097340

What do the budworm population, the distribution of debris of the Sagittarius dwarf, coupled quantum wells, or the non-linear Chua’s circuit have in common? These are all bifurcating systems. To this large variety of systems, you can now add the singly folded notebook page. Indeed, the present paper shows that the maximum length a notebook page can extend outside the notebook area is a bifurcating quantity that only depends on the aspect ratio of the page. This nice example shows, using only elementary undergraduate calculus, that bifurcations can be found everywhere in nature, even in the classroom. It is suitable for an undergraduate calculus course, a computer-based problem-solving course, or an undergraduate introduction to phase transitions.

Paragliders' launch trajectory is universal by Quentin Da Cruz Lopes; Sophie Ramananarivo; Caroline Cohen; Michael Benzaquen. DOI: 10.1119/5.0100959
We often tell our students that a potential energy function U(x) can be thought of as representing a particle sliding along a track in a vertical plane, with peaks and valleys corresponding to points of stable and unstable equilibrium. However, motion along a track of varying height z(x) occurs in two dimensions, so the analogy is not exact. In this paper, the authors show that an analysis of the horizontal motion of a particle along a 2-D track in a uniform gravitational field can be made equivalent to that of a particle in 1-D motion by developing an “analogous potential energy function” that can be used to characterize the static and dynamic equilibrium points and their corresponding oscillatory behaviors. In general, a single track corresponds to many such analogous functions, each with a different total energy. Suggestions for associated student projects are given. The level is appropriate for intermediate-level dynamics students.

Sliding and rolling along circular tracks in a vertical plane by Rod Cross. DOI: 10.1119/5.0107553
Many lecture demonstrations and lab activities involve sliding or rolling an object along a curved ramp. When students ask why the object does not return to its initial height on the far side of the ramp, we tell them that it is because the object lost energy to friction along the ramp. But how exactly does this happen? This paper explores in detail the energy losses experienced by a billiard ball rolling along a curved track and a nut sliding along a curved wire and offers recommendations on ways to reduce energy losses in typical lecture demonstrations.

Heavy symmetric tops and the Hannay angle by Changsoo Park. DOI: 10.1119/5.0101149
When a spinning top experiences a precessional motion such that the period of the precession is much longer than that of the top, the periodic motion of the top is almost preserved during a full cycle of the precession. It can be shown that after one period of the precessional motion, the angle variable (the conjugate quantity to the action variable) of the top acquires a phase shift composed of two parts, termed the dynamical angle and geometric angle. The dynamical angle depends on the period, but the geometric angle is determined by the geometric nature of the precession. This geometric contribution to the total phase shift of the angle variable is called the Hannay angle after J. Hannay, who showed that this angle it is identical to the solid angle subtended by the loop swept out by the symmetry axis of the top. Proofs of this, however, typically invoke quite advanced mathematics. In this paper, the author shows with standard spherical-coordinate dynamics that the Hannay angle can also be described by the angle between two radial vectors on the disk of the top, respectively corresponding to the pure spinning motion and the coupled motion of spin and precession for one period of the precession. Appropriate for advanced dynamics students.

Finding and improving bounds of real functions by thermodynamic arguments by Andrés Vallejo. DOI: 10.1119/5.0121919
That thermal interactions cause the entropy of the Universe to increase is a tenet of classical thermodynamics. This paper uses this principle to show how analytic expressions for bounds on mathematical functions such as log(1 + x) can be tightened by computing the entropy change associated with changing the temperature of a system to a desired final state by bringing it into contact with an increasing number of heat reservoirs of temperatures between that of the object and the desired final temperature. By having the object initially hotter or colder than the desired final temperature, both upper and lower bounds can be expressed. This shows students how a physical principle can be used to establish results typically treated as arising from pure mathematics. Appropriate for intermediate-level students of thermal physics.

Introducing quantum mechanics through its historical roots: The hydrogen Rydberg atom viewed through the lens of the old quantum theory by M. G. Littman; E. Gordis; P. Zhelnin; J. Arnold. DOI: 10.1119/5.0094860
It’s well known that the Bohr model’s stability condition is consistent with using the de Broglie wavelength to restrict the possible radii of circular orbits. This paper shows how the more general Bohr-Sommerfeld planetary orbits can also be understood using de Broglie waves, providing a link between quantum and classical physics that may help students better understand both aspects of physics.

Applications of the eikonal approximation in quantum mechanical scattering by Barry R. Holstein. DOI: 10.1119/5.0077649
Scattering amplitudes in quantum mechanics are often calculated using the Born approximation, which does not embed general properties such as unitarity, analyticity, or the optical theorem. The paper presents the less frequently discussed eikonal approximation with examples for electromagnetic and gravitational interactions. This straightforward modification is suitable for inclusion when scattering is introduced in typical quantum mechanics courses.

The discovery of a supermassive black hole at the center of the Milky Way galaxy by Bryanne McDonough; Paul Withers. DOI: 10.1119/5.0086222
The 2020 Nobel Prize in Physics was awarded for the discovery of a supermassive compact object at the center of our galaxy as well as for the theoretical work showing that black holes are a robust prediction of the general theory of relativity. Now your students can tie their classroom learning to this Nobel Prize. The observations of stars orbiting near the galactic center allow students who have learned about Keplerian orbits to quantify its mass. More advanced mechanics students can assess whether the mass could be concentrated in a sphere or a disk. Students who have learned about the Boltzmann constant can determine that a central dust cloud of the necessary mass would not be stable against collapse, and students who have learned about collisions can rule out the possibility of a cluster of compact objects. The data and models presented in this paper could be used individually as examples in physics courses on many levels, or they could be combined in an upper-level course to help students understand the evidence for the presence of a black hole at the galactic center.

#### INSTRUCTIONAL LABORATORIES AND DEMONSTRATIONS

Flexible, low-cost phase-sensitive detection for the undergraduate laboratory with a Teensy microcontroller by Jerome Fung; Christopher L. Weil. DOI: 10.1119/5.0126691
A low-cost setup to implement and explore phase-sensitive detection is presented. The setup is based on the Teensy 3.5 microcontroller and is controlled via user-written Python software, which includes a graphical user interface. The performance of the setup is evaluated and a successful laboratory measurement—measuring the distance dependence of the intensity of a light-emitting diode in the presence of environmental lighting—is demonstrated. The reasons for the choice of the Teensy 3.5 device, in comparison with the popular Arduino Uno, are given as well as a description of the limitations of the setup (i.e., no front-end amplification as well as lack of synchronous lock in the external reference mode). This setup is a useful tool for teaching advanced physics students about the phase-sensitive detection experimental technique, and it can be a viable low-cost alternative to commercial lock-in amplifiers for certain research situations.

A low-cost confocal microscope for the undergraduate lab by A. Reguilon; W. Betharda; E. Brekke. DOI: 10.1119/5.0128277
This paper presents a confocal microscope setup intended for pedagogical purposes. The simplicity and cost-efficient design of the setup make it an excellent avenue for developing student understanding of confocal microscopy and the physics underlying this widely used experimental technique. Experimental results acquired using the setup--measuring the thickness and surface variation of a microscope slide--are given. In addition, suggestions for several further investigations as well as expansions of the setup to improve its range of application are provided. This work provides an accessible and affordable introduction to the topic of confocal microscopy for optics, biophysics, and advanced instructional laboratory instructors and students.