AJP July 2023 coverJuly 2023 Issue,

Volume 91, No. 7


Using rotation matrices to calculate the locations of sunrise and sunset, the hours of daylight, observed path of the Sun, and its elevation angle for any latitude

During the course of one year, I photographed 95 sunrises along the east bank of the Columbia River from my home on the west bank in Richland, WA (46.3° N latitude). I then calculated the seasonal phenomena listed in the title with the intention of explaining the variation of the photographed sunrises. The calculations use a simplified model of the Sun-Earth system and employ rotation matrices to predict the path of the Sun, as observed at any location in the northern hemisphere, throughout the year. These predictions are in good agreement with those listed by NOAA and also with the photographic data. The analysis presented here provides a novel way to calculate and understand the seasonal variations of visible sunlight.



In this Issue: July 2023 by Joseph C. Amato; Mario Belloni; Harvey Gould; Claire A. Marrache-Kikuchi; Beth Parks; B. Cameron Reed; Donald Salisbury; Jan Tobochnik. DOI: 10.1119/5.0159258


Resource Letter MP-4: The Manhattan Project and related nuclear research  by B. Cameron Reed. DOI: 10.1119/5.0149901
The legacy of the Manhattan Project extends beyond the creation of the atomic bombs that ended WWII to the technology that became the basis of numerous peaceful applications such as nuclear power and nuclear medicine. It is no wonder that the story of the Manhattan Project continues to fascinate the public, as shown by the release of the summer 2023 film, “Oppenheimer.” Adding to the first three Resource Letters on the Manhattan Project, this new Resource Letter includes over 140 new sources (books, journal articles, videos, and websites) that describe the history, physics, personalities, and educational materials related to the Manhattan Project.


Memories of a mid-20th century electrical measurements laboratory and its instrumentation by Thomas B. Greenslade, Jr. DOI: 10.1119/5.0155766
In the mid-20th century, the junior year E&M course typically had a rigorous laboratory component in which students conducted precision electrical measurements using basic–but quite elegant–apparatus: galvanometers, bridges, standard cells, resistors, capacitors, etc. Return to that time with the author, as he describes these instruments and fondly recalls the joys and challenges of using them as a student in the 1950s, and later as a young physics professor. The E&M lab may have disappeared long ago, replaced by courses in electronics and computing, but have we lost something valuable in the transition?

Using rotation matrices to calculate the locations of sunrise and sunset, the hours of daylight, observed path of the Sun, and its elevation angle for any latitude by  Margaret Stautberg Greenwood. DOI: 10.1119/5.0095598
Understanding the apparent movement of the Sun can motivate students to use new mathematical tools to describe their world.  In this paper, rotation matrices are used to predict the daily motion of the Sun and the seasonal variations of sunlight.  The calculations are then compared to observations of sunrises made by the author over the course of a year.

Analysis of ill-conditioned cases of a mass moving on a sphere with friction  by Terry W. McDaniel. DOI: 10.1119/5.0063834
What happens when a body slides down a spherical surface? In the absence of friction, the motion is easy to solve for, and students are often asked to do so in introductory mechanics. But if friction is present, the motion is much more complex, and under the right conditions, exhibits an unusual behavior: the sliding body slows to a near halt before speeding up again, creating a nearly flat “plateau” in a plot of position vs. time. A full solution requires a suite of numerical tools, providing upper-level students with a rich mathematical exercise in an easy-to-visualize physical situation.

Oblique collision of a soft rubber disk with a rigid surface  by Rod Cross. DOI: 10.1119/5.0135633
The collision of deformable ball or disk against a rigid surface can be a complicated interaction involving an initial sliding or gripping phase, tangential deformation of the disk, and then a reverse sliding phase which imparts spin to the disk as it leaves the surface. In this paper, a high-speed video camera is used to record the impact of a rubber disk with a rigid surface. The collision is modeled with empirically determined vertical and horizontal effective spring constants and a coefficient of sliding friction. Results are in reasonable qualitative accord with the model. Appropriate for an upper-level dynamics or properties of materials class.

Analytical calculation of plasmonic resonances in metal nanoparticles: A simple guide  by Marco Locarno; Daan Brinks. DOI: 10.1119/5.0094967
If you want your students to understand how butterflies have those impressively beautiful colored wings, how to design an invisibility cloak, or more generally to understand how light interacts with an assembly of nanoparticles, you should start with this article. In the 2000s, plasmonics emerged as a field in its own right, dealing with the propagation of optical signals at the interface between a metallic surface and a dielectric. This paper gives an undergraduate-level presentation of the bases behind plasmonic resonances that occur when a metallic nanoparticle is submitted to an oscillating electric field, thus affecting its optical properties. It also examines how one can engineer the shape, the size, and the composition of the nanoparticles to obtain the desired optical effect. This paper can serve both as an introduction to the field of plasmonics and as a nice undergraduate E&M problem.

Wigner rotation and Euler angle parametrization by Leehwa Yeh (葉李華). DOI: 10.1119/5.0111222
A more familiar representation of rotations in space through Euler angles has suggested to this author an analogous representation of Lorentz transformations in flat spacetime. This results in a new enlightening perspective on the rotational differences between Lorentz boosts performed in opposite order. This could be introduced in upper-level undergraduate mathematical methods courses.

Simple realization of the polytropic process with a finite-sized reservoir by Yu-Han Ma. DOI: 10.1119/5.0104382
Take an ideal gas with pressure P and volume V. Polytropic processes are thermodynamic transformations for which PVξ = constant. The isothermal (ξ = 1) or adiabatic (ξ = γ, with γ being the heat capacity ratio) transformations are examples of such processes. If you want to learn what polytropic processes with other values of ξ are, and what kind of experimental systems exhibit those (spoiler: The size of the heat bath with which the gas is in contact plays a crucial role), you should consider reading this paper. Appropriate for undergraduate thermodynamics classes or projects.


Comment on “Thermodynamics of Benford's first digit law” [Am. J. Phys. 87, 787–790 (2019)]  by Xi-Jun Ren. DOI: 10.1119/5.0149874
The Newcomb-Benford law states that, in an ensemble of random numbers, the probability of having the numbers starting with digit d (d = 1, 2, …, 9) is not 1/9 as one could expect, but that it follows a logarithmic distribution (specifically p(d) = log (⁠ 1+1d ⁠)). This article builds on a recent paper by Don Lemons to draw an analogy between a number N and a system containing I radiation modes. Using the Bose-Einstein statistics and the grand canonical ensemble, Ren shows that one can recover the first digit law in the thermodynamic limit and calculate the corresponding entropy. This example could nicely fit into an undergraduate statistical physics course.


The game of life as a species model  by David A. Faux; Peter Bassom. DOI: American Journal of Physics 91, 561 (2023) doi: https://doi.org/10.1119/5.0150858
This paper discusses a generalization of John Conway's classic Game of Life that can be used to simulate the growth and decay of animal species, including extinction and recovery. A special property of the model is the use of qubits (from quantum computing) at each site, instead of the usual ones and zeros of the classical Game of Life. Related problems that could be used for student assignments are included.

Additional Resources