American Journal of Physics March 2024March 2024 Issue,

Volume 92, No. 3

Playing melodies on a single string by exciting harmonics using the Lorentz force

We show how a single metal guitar string of fixed length can produce a musical scale. The string is placed near a permanent magnet, and by applying an AC to the string at the frequency of the desired musical note, the Lorentz force creates vibrations in the string at that frequency. The tension of the string is set so that its harmonics correspond to the desired notes. A one-octave scale can be approximated by using these harmonic frequencies, allowing several melodies to be played using our non-contact monochord. This project could be adopted for demonstration or laboratory projects.


In this issue: March 2024 by John Essick; Jesse Kinder; Claire A. Marrache-Kikuchi; Beth Parks; B. Cameron Reed; Todd Springer. DOI: 10.1119/5.0200617


2024 AAPT award citations at the winter meeting in New Orleans, Louisiana. DOI: 10.1119/5.0200616


Predicting the axial deformation of an extensible garden hose when pressurized by Jordi Renart; Pere Roura-Grabulosa. DOI: 10.1119/5.0086302
If you have ever used a garden hose to water your plants, you may have wondered why the hose does not expand radially under pressure, similarly to, say, party balloons. One possible reason is that you were using a hose lined with a helical reinforcing mesh. In that case, there is almost no radial expansion, but instead the hose expands in length with increasing water pressure. The authors have transformed this common experience into a lab experiment, and they use a multiaxial stress-strain model to discuss their results. You could use this paper as an undergraduate mechanics assignment to introduce students to multiaxial stress-strain relations or as the basis for a mechanics lab.

Playing melodies on a single string by exciting harmonics using the Lorentz force by Yukai Wei; Hao Zhu; Haotian Jiang; Quanxin Luo; Shan Lin; Junqing Li; Yu Zhang; Bibo Zhao. DOI: 10.1119/5.0152828
A familiar technique: when a cello player lightly rests their finger on the middle of the string, we hear the first harmonic: a note one octave higher than the fundamental. But did you know that harmonics can be used not only to play an octave higher but also to play most of the notes of a scale on just a single string? And had you ever thought of exciting these harmonics not through a carefully placed finger, but by applying an ac current at the harmonic frequency in the presence of a dc magnetic field? The authors show how to play melodies using non-contact excitation of a single string, which could be used as a demonstration or could form the basis of a student project.

Demonstration of the propagation of errors using resistors by K. K. Gan. DOI: 10.1119/5.0145005
This short paper proposes an experimental setup to illustrate the propagation of errors in undergraduate statistics (or general physics) labs. Printed circuit boards can be designed to measure a few hundreds of resistors in a reasonable time. Here, they are wired so that students can measure the dispersion in the values of resistances of a single reel and check that their distribution is Gaussian. In a second step, they measure the dispersion in the values of the sum of two such resistances and verify that it is consistent with the propagation of errors.

Voltage decay in an RLC circuit is not what is taught: An advanced laboratory exercise by Frank V. Kowalski; Justin L. Swantek; Tony D'Esposito; Jacob Brannum. DOI: 10.1119/5.0068145
Experienced experimentalists know that real laboratory experiments exhibit subtleties and nuances that are absent from idealized situations. Though practicing scientists grapple with this fact on a regular basis, it is sometimes underemphasized in the undergraduate curriculum. This paper, which will be of interest to undergraduate lab instructors, describes an advanced laboratory experience in which students are confronted with results that are in tension with a familiar idealized model. In their attempts to resolve the discrepancies, students must engage in model building and scientific inquiry. Such activities are in line with the AAPT guidelines for undergraduate lab experiences, and the paper nicely connects this particular laboratory experience with the AAPT recommendations.

The uncertainty principle and quantum wave functions that are their own Fourier transforms by Keith Zengel; Nick DeVitto; Nathanael Hillyer; Jeffrey Rodden; Vinh Vu. DOI: 10.1119/5.0162363
Did you know that the Gaussian function is not the only function that is its own Fourier transform, and that, in fact, one can construct such a function starting with any normalizable wave function? Moreover, using the calculus of variations, one can show that any minimum uncertainty wave function must be its own Fourier transform. In addition to showing new ways to prove theorems about minimum uncertainty states, this paper provides connections between quantum mechanics instruction and the techniques used in classical mechanics.

Wigner versus Stark: Connecting quantum to classical in a tunnel ionization process by Seyedmohammad Yusofsani; Miroslav Kolesik. DOI: 10.1119/5.0077113
In the Stark effect, an electron, initially attached to an atom, tunnels through the atomic confinement potential into vacuum when submitted to a constant external electric field. This paper re-examines this problem via three different methods to determine where and when the electron comes out of the tunneling barrier. First, the authors propose an easily implementable numerical resolution of the time-dependent Schrödinger equation. Second, the state-expansion method enables them to derive an analytical solution to the problem. Third, they use Wigner's trajectories which are established by assuming that the wavefunction is dominated by the contribution of a single eigen-energy, and by working out the locations of the maximum amplitude of the wavefunction. One or several of these methods could be used in an advanced quantum mechanics course. This paper can also trigger discussions on unsettled issues about the tunneling process – is it possible to define a specific location or a specific time at which the electron exits the barrier? – and on the connection between the quantum and classical descriptions of a particle.

On the normalization and density of 1D scattering states by Chris L. Lin. DOI: 10.1119/5.0082650
In introductory quantum physics courses, the analysis of a quantum particle in the presence of a one-dimensional potential V(x) is often separated into two seemingly disconnected problems. The first involves determining the bound states and their energies while the second analyzes scattering and the probability of reflection and transmission. This paper highlights an important link between these two topics - the number of bound states is related to the normalization of the scattering states. After deriving the general result, examples (in familiar one-dimensional potentials) are worked out, and an application to statistical mechanics is discussed. Instructors of courses on quantum mechanics or statistical physics will find this paper pedagogically useful as it may help students to connect the bound state sector to the scattering sector.

Event-based analysis of convergence to energy equipartition by Arnaldo Spalvieri. DOI: 10.1119/5.0142173
The concept of equipartition is hugely important to physics, and yet its mechanism may seem mysterious to students. This paper shares a relatively simple way to understand the process of reaching energy equipartition through collisions in an ideal gas.

Thermal damping of the motion of a piston: Any irreversibility implies dissipation by David S. Corti; Joshua A. Ciesar; Juan M. Vazquez. DOI: 10.1119/5.0166800
Thermodynamics classes often consider ideal gases confined within cylinders with movable pistons. If the gas and outside atmosphere are at different pressures, the piston accelerates when released. The system might oscillate initially, but it is commonly assumed that it eventually comes to a final equilibrium state due to frictional damping. In this paper, the authors explore a system where no mechanical dissipation occurs, only thermal dissipation in the form of irreversible heat transfer between the gas and the piston. Thermodynamic analysis and numerical simulations show that the motion of the piston is damped and eventually comes to rest with a corresponding increase in the entropy of the Universe despite the lack of mechanical dissipation. Appropriate for intermediate-level thermodynamics students.


Designer spectrographs for applications in the advanced undergraduate instructional lab by Timothy T. Grove; C. Daly; Naomi Jacobs. DOI: 10.1119/5.0173768
The paper reports a novel approach to the design and construction of a spectrograph for upper-level undergraduate laboratory work. The innovative design for the 3-D-printed, low-budget instrument is explained in detail, allowing one to construct spectrographs for specific applications, where narrower spectral wavelength range yields higher wavelength precision, hence the name designer spectrograph. To demonstrate the utility of this approach, designer spectrographs are constructed, then used to obtain the spectra of sodium doublets as well as the isotope shift in the Balmer alpha line. This paper will be of interest not only to undergraduate instructors, but also to laboratory scientists and engineers.


A diagrammatic representation of entropy production by Andrés Vallejo. DOI: 10.1119/5.0167570
In this Note, the authors present a graphical approach to calculating the entropy produced in thermodynamic processes. They illustrate the method for the heating or cooling of an incompressible solid and for the Brayton cycle. This diagrammatic approach complements the familiar analytic approach and clearly illustrates the Second Law of Thermodynamics. The approach may interest instructors of introductory physics and thermodynamics who wish to offer students multiple representations of entropy production in their first encounter with the topic.

Virtual lab: Using spreadsheets to simulate experiments measuring viscosity, BJT NPN curves, and RL circuits by Ahmed Ali Rajput. DOI: 10.1119/5.0185096
This Note points readers to Excel spreadsheets that may be useful for virtual labs or visualizations.

Comment on “Experimental determination of heat capacities and their correlation with theoretical predictions” [Am. J. Phys. 79(11), 1099–1103 (2011)] by Matt Beekman; Allison M. Phillips; Muhammad Sabieh Anwar. DOI: 10.1119/5.0166198
This Comment corrects an error in a previous paper, showing how students can measure heat capacities as a function of temperature.

Comment on “Spectral shifts in general relativity,” [Am. J. Phys. 62(10), 903–907 (1994)] by Joseph D. Romano; Teviet Creighton. DOI: 10.1119/5.0195020
This comment corrects a pair of compensating errors that could confuse readers of this classic paper.

Erratum: “All objects and some questions” [Am. J. Phys. 91, 819–825 (2023)] by Charles H. Lineweaver; Vihan M. Patel. DOI: 10.1119/5.0198864
This very popular paper attracted significant reader feedback, which led to corrections of some minor errors in the figures.


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