October 2024 Issue
Volume 92, No. 10
What is a degree of freedom? Configuration spaces and their topology
Understanding degrees of freedom in classical mechanics is fundamental to characterizing physical systems. Counting them is usually easy, especially if we can assign them a clear meaning. However, the precise definition of a degree of freedom is not usually presented in first-year physics courses since it requires mathematical knowledge only learned in more advanced courses. In this paper, we use a pedagogical approach motivated by simple but non-trivial mechanical examples to define degrees of freedom and configuration spaces. We highlight the role that topology plays in understanding these ideas.
EDITORIAL
In this issue: October 2024 by Mario Belloni; John Essick; Harvey Gould; Jesse Kinder; Claire A. Marrache-Kikuchi; Raina Olsen; Beth Parks; B. Cameron Reed; Donald Salisbury; Todd Springer; Jan Tobochnik; Keith Zengel. DOI: 10.1119/5.0234511
AWARDS
2024 Lillian McDermott medal lecture: You only get one—My life pursuing excellence in physics education by Stephanie Viola Chasteen. DOI: 10.1119/5.0234015
Editor's Note: This paper is the text of a plenary talk given by the author at the AAPT Summer 2024 meeting when she accepted the Lillian McDermott Medal.
RESOURCE LETTERS
Resource Letter: Synthesis of the elements in stars Editor’s Pick by Artemis Spyrou. DOI: 10.1119/5.0209176
Editor's Note: For many years the source of the energy in stars was a mystery since conventional explanations (gravity and chemical reactions) yielded stellar lifetimes that were too short. With the development of nuclear physics and the interdisciplinary field of nuclear astrophysics, not only was this mystery solved, but others, like the production of elements heavier than helium, were as well. This Resource Letter describes resources that provide a guide to the field of nuclear astrophysics and the origin of the elements.
PAPERS
The physics of “everesting” on a bicycle by Martin Bier. DOI: 10.1119/5.0131679
Editor's Note: For cyclists, “everesting” consists of cumulating the climbs of a single hill or mountain many times until the total cycled height is equivalent to that of Mount Everest. This has become a fashionable challenge for amateur and professional cyclists alike, and many physics-based recipes to improve the overall time have been proposed. This paper establishes the equations that can be used to model everesting and examines the influence of the cyclist's weight and power production, as well as the cyclist's aerodynamical coefficient during both the ascent and the descent. This makes for a nice example of undergraduate mechanics, or an illustration of how physics can help athletes improve their performance.
What is a degree of freedom? Configuration spaces and their topology by Juan Margalef-Bentabol; D. Leigh Herman; Ivan Booth. DOI: 10.1119/5.0151379
Editor's Note Landau and Lifshitz defined a degree of freedom as “the number of independent quantities which must be specified in order to define uniquely the position of any system.” Undergraduate physics students may find this concept obvious when a system is simple, such as a particle free to move in three dimensions, but struggle with the concept when faced with a more complex system, especially one with constraints. This paper shows how to present the concept of “degrees of freedom” in a mathematically rigorous but accessible way using examples like the coupled double pendulum.
Elongation of a ferrofluid droplet near a permanent magnet: A tidal or magnetic energy effect? by Zoe Boekelheide. DOI: 10.1119/5.0207189
Editor's Note: Have you ever heard of a ferrofluid? They are colloidal suspensions of ferromagnetic particles in a liquid, allowing them to act almost as a deformable superparamagnet. Among their many interesting properties is a distinctive elongation when droplets of ferrofluid fall toward a magnet. This paper was motivated by the question of whether the elongation is due to tidal forces (a force gradient due to the non-uniform magnetic field) or due simply to the lower magnetic self-energy for this shape. Spoiler alert: both contribute. Learn more by reading this paper, which could be used in an upper-level electromagnetism course or could motivate student projects.
Heat engines with finite reservoirs by Randall D. Knight. DOI: 10.1119/5.0220126
Editor's Note: Most textbook thermodynamics problems assume thermal reservoirs of infinite heat capacity and consequently unchanging temperatures. Analyses involving engines operating between finite reservoirs are not unknown, but they typically assume that the temperature change per cycle is infinitesimal and that the engine stops operating when the hot reservoir has come to the temperature of the cold reservoir. This paper considers a Carnot-like engine operating between a finite hot reservoir and an infinite cold one, showing that the problem can be solved exactly with no requirement for infinitesimal changes and also that when the reservoir temperatures converge, the device simply transitions from being a heat engine to a refrigerator. Explicit expressions are developed for the limiting initially-hot reservoir temperature and the total efficiency. Appropriate for upper-level thermodynamics courses.
Heisenberg's 1939 reactor theory, Serber's 1943 Los Alamos Primer, and Heisenberg's 1945 Farm Hall critical mass calculation by Joseph L. McCauley. DOI: 10.1119/5.213080
Editor's Note: Historians debate whether Heisenberg was aware, before the launching of the American Manhattan project, of the possible characteristics of a U-235 bomb. This paper presents convincing evidence that in 1939 Heisenberg had already performed the calculation that was presented in 1943 by Serber to the Los Alamos group that predicted a critical radius. This paper will be of interest to those who teach about the development of the atomic bomb.
A simplified relativity experiment by David P. Jackson; Fedya Grishanov; Noah Lape; Emma Lothrop. DOI: 10.1119/5.0218373
Editors Note: Special relativity is one of those theories that we expect our students to learn without ever performing their own experimental test. Though they find the concepts fascinating, they usually don't get to see them in action the way they get to see classical mechanics, optics, electromagnetism, and the rest of physics. That is what makes this article so delightful: The authors have found a simple relativity experiment that is feasible for most physics departments, presents a decisive test between classical and relativistic predictions of particle trajectories, and motivates students to refine their theoretical model to account for systematic effects. The link to the paper will also take readers to a video abstract.
Gravitational waves without general relativity redux by Robert C. Hilborn. DOI: 10.1119/5.0175979
Editor's Note: A deep understanding of gravitational waves requires knowledge of general relativity (which is notoriously difficult). Over the decades, and for various reasons, alternative theories/models of gravity have been developed. Even though such alternatives were eventually found to conflict with observations, they still can be used for pedagogical purposes. In this article, the author develops a vector theory of gravitational waves, reminiscent of electromagnetism, which leads to many of the same predictions as general relativity. Many calculations proceed by analogy with electromagnetism, and the context of this model may help learners build their intuition for gravitational radiation. Readers familiar with electromagnetic radiation will find the model presented here to be a helpful stepping stone to the full theory of general relativity.
Demonstrating two-particle interference with a one-dimensional delta potential well by Zhi Jiao Deng; Xin Zhang; Yong Shen; Wei Tao Liu; Ping Xing Chen. DOI: 10.1119/5.0176364
Editor's Note: In this article, the authors explore the role of particle statistics in the scattering of noninteracting particles. They illustrate how a one-dimensional delta-function well can play the role of a beam splitter in a Hong–Ou–Mandel experiment and then compare and contrast the scattering of pairs of bosons, fermions, and distinguishable particles. The authors introduce graphical methods for visualizing the scattering process alongside a comprehensive analysis of the evolution of the wave function. The model is accessible to students in an introductory quantum mechanics course and connects one-dimensional scattering theory to contemporary experiments with photons and cold atoms.
Numerical simulation projects in micromagnetics with Jupyter by Martin Lonsky; Martin Lang; Samuel Holt; Swapneel Amit Pathak; Robin Klause; Tzu-Hsiang Lo; Marijan Beg; Axel Hoffmann; Hans Fangohr. DOI: 10.1119/5.0149038
Editor's Note: Acquiring computational skills has become an important goal in STEM education. Computational literacy can be achieved through standard programming or numerical methods courses, but it can also be promoted in other courses, for instance through student projects. This paper describes this approach within the context of micromagnetics, making use of project Jupyter. The authors discuss how the Python-based open-source simulation software package they have developed facilitates student access to specialized numerical simulations. This paper should interest educators who try to implement numerical methods into their courses or labs.
COMPUTATIONAL PHYSICS
Cognitive biases can move opinion dynamics from consensus to signatures of transient chaos by Emily Dong; Sarah Marzen. DOI: 10.1119/5.0220792
Editor's Note: Computer models have recently been used to help understand opinion formation. This paper describes a simple agent-based model which uses Bayesian updates of the agents' opinions and incorporates the effects of confirmation and in-group biases. The model may be used in a computational physics course, where students will enjoy seeing how it provides insight into the formation of consensus and polarization in society.
INSTRUCTIONAL LABORATORIES AND DEMONSTRATIONS
Flipping the electronics lab: Learning upper division electronics at home by Brian Rasnow. DOI: 10.1119/5.0206534
Editor's Note: In this paper, the author describes a junior-level flipped electronics instructional laboratory that combines electronics and computer interfacing instruction in one course. Students are provided with a low-cost electronics kit (including an Arduino microcontroller, a handheld multimeter, and various circuit components), along with a MATLAB license, from which they build a versatile electronics workbench at home. Many of the labs involve building permanent apparatus, including a programmable DC power supply and function generator, a spectrum analyzer, and an oscilloscope, that the students use to measure I–V characteristic curves, characterize transistors, work with Fourier transforms, Bode plots, and so on. Using their workbench, each student can work on their lab projects outside of the classroom whenever and wherever it is convenient for them. Various supplementary materials, including code, are provided, which are useful for those wishing to implement this curriculum at their own institution.
BOOK REVIEWS
A Standard Model Workbook by Flip Tanedo. DOI: 10.1119/5.0235428
Additional Resources
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