The application of linear regression to data analysis is traditionally a tedious calculation delegated to a computer. This article presents a physical model to facilitate an intuitive way of thinking about the slope of a fit. In this simple model, data points are envisioned as beads applying torques on a massless, rigid, horizontal rod. To illustrate how this model can help colleagues directly address

Captain Einstein is a virtual reality (VR) movie that takes you on a boat trip in a world with a slow speed of light. This allows for a direct experience of the theory of special relativity, much in the same spirit as in the Mr. Tompkins adventure by George Gamow (1939). In this paper, we go through the different relativistic effects (e.g., length contraction, time dilation, Doppler shift, light aberration)

Since the discovery of the first planet outside the solar system in 1995, the detection of exoplanets has been an attractive and engaging scientific field. This article intends to present briefly the radial velocity method for detecting the presence of an unseen planet orbiting a star. Based on an experimental setup, the presentation resorts to the analogy between sound waves and light waves. In particular

This article outlines a generalization of the thermal resistance concept used to model and quantify the rates of spontaneous heat transfer. This leads to the natural corollary that a Carnot cycle possesses zero thermal resistance and provides restatements of the Clausius and Kelvin statements of the second law of thermodynamics. Subsequently, basic aspects of a general thermal engine are modeled by

We present a set of kinesthetic activities that utilize a local positioning system to teach kinematics in the physics classroom or laboratory. The activities build on previously reported activities in scope and complexity, incorporating two-dimensional motion and the simultaneous motions of multiple bodies. In these activities, students act out motions illustrated in graphs of kinematic quantities

The Siegert relation relates electric field and intensity correlations of light, under given assumptions. After a brief derivation of the relation, we present an experimental setup that can be implemented in a student laboratory: it allows measuring both field and intensity correlations at the same time, thus providing a direct test of the Siegert relation. Some experimental results are presented when

New technology like the Arduino microcontroller platform presents an opportunity to transform Beyond the First Year (BFY) physics labs to better prepare physics students for work in research labs and beyond. The flexibility, low cost, and power of these devices provide an attractive way for students to learn to use and master research-grade instrumentation. Therefore, we introduced new technology,

Here the buckling of inextensible rods due to axial body forces is mapped to 1D, nonrelativistic, time-independent quantum mechanics. Focusing on the pedagogical case of rods confined to 2D, three simple and physically realizable applications of the mapping are given in detail; the quantum counterparts of these are particle in a box, particle in a delta-function well, and particle in a triangular well

The largest animals are the rorquals, a group of whales which rapidly engulf large aggregations of small-bodied animals along with the water in which they are embedded, with the latter subsequently expulsed via filtration through baleen. Represented by species like the blue, fin, and humpback whales, rorquals can exist in a wide range of body lengths (8–30 m) and masses (4000–190,000 kg). When feeding

The past is always with us, not only the physics history that we revere and try to put into our classroom presentations. Two hundred years of teaching natural philosophy (physics) have left us a significant amount of physics history in the form of early apparatus. In this article, I will tell stories about how I have become involved in the study of early physics apparatus. Because of my website on

An approximate treatment of neutron slowing by elastic collisions with nuclei of a moderating material in a nuclear reactor is developed on the basis of a compact treatment of elastic collisions previously published in this journal [B. C. Reed, Am. J. Phys. 86(8), 622 (2018)]. The analysis is straightforward enough to be presented to lower-level students and gives results in good accord with more sophisticated

In the article “The equivalence principle in the Schwarzschild geometry” [Am. J. Phys. 62, 1037–1040 (1994)], some manifestation of the equivalence principle in Schwarzschild spacetime was studied. We point out that the result for the Riemann-tensor component in this article is incorrect because only the first order terms in the metric expansion are taken into account. We show the corrected result

We present a mathematically simple procedure for explaining and visualizing the dynamics of quantized transport in topological insulators. The procedure serves to illustrate and clarify the dynamics of topological transport in general, but for the sake of concreteness, it is phrased here in terms of electron transport in a charge-ordered chain, which may be mapped exactly onto transport between edge

Most elementary numerical schemes found useful for solving classical trajectory problems are canonical transformations. This fact should be made more widely known among teachers of computational physics and Hamiltonian mechanics. From the perspective of advanced mechanics, unlike that of numerical schemes, there are no bewildering number of seemingly arbitrary elementary schemes based on Taylor's expansion

For an oscillating electric dipole in the shape of a small, solid, uniformly polarized, spherical particle, we compute the self-field as well as the radiated electromagnetic field in the surrounding free space. The assumed geometry enables us to obtain the exact solution of Maxwell's equations as a function of the dipole moment, the sphere radius, and the oscillation frequency. The self-field, which

An American football is a rotationally symmetric object, which, when well-thrown, spins rapidly around its symmetry axis. In the absence of aerodynamic effects, the football would be a torque-free gyroscope and the symmetry/spin axis would remain pointing in a fixed direction in space as the football moved on its parabolic path. When a pass is well-thrown through the atmosphere, however, the symmetry

The second-order perturbative Stark effect on the ground state of hydrogen is a typical example presented in many standard texts on quantum mechanics. Some texts miss the fact that the scattering states are significant contributors to the perturbative energy correction. The inclusion of scattering states has wider applicability than to just the Stark effect. An explicit calculation involving a finite-square

A tutorial description of plasma waves in a cold plasma, with emphasis on their application in plasma-based electron accelerators, is presented. The basic physics of linear plasma oscillations and waves and the principle of electron acceleration in a plasma wave are discussed without assuming any previous knowledge of plasma physics. It is shown that estimating key parameters for plasma acceleration

Undergraduate research has become an essential mode of engaging and retaining students in physics. At Loyola University Chicago, first-year physics students have been participating in the Freshman Projects program for over twenty years, which has coincided with a period of significant growth for our department. In this paper, we describe how the Freshman Projects program has played an important role

We use a Michelson interferometer to explore the spectroscopic properties of laser diode emission over a wide range of operational conditions. By studying how the interferogram changes with drive current, we demonstrate the relationship between coherence length and spectral bandwidth. At low injection current, the laser source operates more like an ordinary light-emitting diode (LED), generating a

The quantum mechanical bound states of the −α/x2 potential are truly anomalous. We revisit this problem by adopting a slightly modified version of this potential, one that adopts a cutoff in the potential arbitrarily close to the origin. The resulting solutions are completely well-defined and “normal,” and provide us with additional insight into the solutions of the “bare” −α/x2 potential. We present

In a recent paper, Price, Moss, and Gay have given a simple explanation of a paradox in the flight of a football, why the long axis of the football, contrary to intuition but in good agreement with experience, turns so that it is tangent to the path of the football. Here, we add to the analysis the assumption of only first-order differences between the direction of the velocity, the orientation of

In this study, a simple asymmetrical double torsion pendulum is built and operated to study coupled harmonic motion. The setup, which consists of two circular inertia members suspended horizontally at different locations on a vertical guitar wire, has a close mechanical similarity to a wall-spring-mass-spring-mass system. The restoring torque of the twisted guitar wire drives the two inertia members

The distribution of eigenvalues of the wave equation in a bounded domain is known as Weyl's problem. We describe several computational projects related to the cumulative state number, defined as the number of states having wavenumber up to a maximum value. This quantity and its derivative, the density of states, have important applications in nuclear physics, degenerate Fermi gases, blackbody radiation

The appearance of the coronavirus (COVID-19) in late 2019 has dominated the news in the last few months as it developed into a pandemic. In many mathematics and physics classrooms, instructors are using the time series of the number of cases to show exponential growth of the infection. In this manuscript, we propose a simple diffusion process as the mode of spreading infections. This model is less

In introductory physics courses, simple arguments based on traveling waves on a string are used to relate the frequency of standing waves to boundary conditions, e.g., the fixed ends of the string. Here, we extend that approach to two-dimensional waves such as the oscillations of a rectangular membrane with edges fixed at the boundary. This results in a graphical method that uses only simple geometry

This paper describes time-dependent phase modulation experiments using an optical Michelson interferometer. Changes in the path length of the interferometer arms induce phase shifts in time and, thanks to a loudspeaker, make the interferometric signal audible. We produce time-fluctuating refractive indices and mechanical motions of one reflecting mirror. A sinusoidal motion is applied, thus creating

Fourier analysis is a key tool in physics and engineering in the broadest sense and is particularly used in the field of optics to design and study the properties of advanced imaging systems. Fourier optics as a distinct topic is usually taught in final-year undergraduate or first-year postgraduate studies, and a wide range of teaching approaches have been used to describe and explain its key concepts

The azimuthal-radial pendulum is a coupled pendulum-beam system where a pendulum and a supporting string are attached to the free end of a cantilever beam with its other end clamped. Conversion between radial (parallel to cantilever) and azimuthal (perpendicular to cantilever) modes of oscillations is observed when the radial frequency is twice the azimuthal frequency. A theoretical Lagrangian formulation

Dirac's Poisson-bracket-to-commutator analogy for the transition from classical to quantum mechanics assures that for many systems, the classical and quantum systems share the same algebraic structure. The quantum side of the analogy (involving operators on Hilbert space with commutators scaled by Planck's constant ℏ ) not only gives the algebraic structure but also dictates the average values of physical

A new technique is presented for understanding how charges distribute themselves on the surface of a conductor during current flow. The technique uses a set of three-dimensional calculation cells (“pixels”) that cover the conductor's surface and contain internal charge. The pixels have two faces separated by an infinitesimal, but finite, distance, with one face being conductive and the other non-conductive

In 1893, Otto Wiener developed the first optical method for measuring the rate at which two liquids mix by interdiffusion. His elegant experiment is often reproduced at University to illustrate the phenomenon of molecular diffusion. It is based on the property that an interface between two liquids deflects light by refraction because of the composition gradient formed by interdiffusion. In this study

A terrella developed for the undergraduate Advanced Laboratory course in the University of Wisconsin-Madison Physics Department is described. Our terrella consists of a permanent magnet, mounted on a pedestal in a vacuum chamber, surrounded by electrodes that may be biased in various ways. The system can confine a plasma, which may, in some ways, be considered as a toy model of the plasma confined

We present a simple demonstration of the skin effect by observing the current distribution in a wide rectangular strip conductor driven at frequencies in the 0.25–5 kHz range. We measure the amplitude and phase of the current distribution as a function of the transverse position and find that they agree well with numerical simulations: The current hugs the edges of the strip conductor with a significant

In a simple time of flight experiment, the speed of sound is observed to travel about 5% slower inside a corrugated irrigation pipe as compared with measurements conducted in open air. Although some theoretical understanding for this retardation is provided, the focus of this paper concerns how this finding might be incorporated into an undergraduate lab activity that challenges student understanding

We present the design, pedagogical logic, and assessment of a laboratory and supporting materials that integrate a clinical academic cardiologist's understanding of the origins of the electrocardiogram (ECG) with a physics educator's insights into how to teach the underlying physics at the introductory level to life science students. In this article, we explain the choices made throughout the design

Over the past couple of decades, mobile devices have become ubiquitous items in our daily lives. Companies continuously add innovative features that make them more affordable and easy to use. This success in mobile technology has prompted educators and instructors to utilize these gadgets to boost the teaching-learning process by changing the traditional classroom to one that is interactive and engaging

The ability to use concepts learned in one context to solve problems in a different context (i.e., transfer of learning) is often one of the major goals of physics instruction. After all, the ability to transfer learning from one context to another is a prerequisite to recognizing the applicability of compact physical laws to a multitude of contexts and is a hallmark of expertise in physics. The majority

We analyze here the changes that may occur in the length of a rod due to Lorentz contraction, when the rod, initially stationary in one inertial frame, is accelerated so as to come to rest in another inertial frame. The rod that earlier appeared shorter due to the Lorentz contraction in one frame should later appear shorter in the other frame. Has the rod length been reduced during acceleration, as

This article reviews the predictions of quantum mechanics (QM) for one- and two-particle Stern–Gerlach experiments and then frames Bell's results, which rule out hidden-variable alternatives to QM, as attempts by a skeptical Eve to fool Alice and Bob with (first example) classical probability mixtures of non-entangled quantum states and (second example) a classical hidden-variable theory. With hidden

A 1943 graph depicting the time evolution of physical conditions within an exploding nuclear weapon is reconstructed and discussed. The graph is remarkable for its combination of generality, simplicity, and density of information. Analyzing how Robert Serber constructed this graph proves to be a valuable exercise for developing deeper understanding of the physics of a nuclear explosion.

Kilonovae are cosmic optical flashes produced in the aftermath of the merger of two neutron stars. While the typical radiant flux of a kilonova can be as high as 1034 W, they typically occur at cosmological distances, requiring meter-class or larger telescopes for their observation. Here, we explore how a kilonova would look like from the Earth if it occurred in the Solar System's backyard, 1000 light

We provide an elementary derivation of the one-dimensional quantum harmonic oscillator propagator, using a mix of approaches, such as path integrals, canonical operators, and ladder operators. This way we take the best of each world, and find the propagator with as few tears as possible.

Single-particle energy states for a neutron and a proton are obtained by solving the time-independent Schrödinger equation for the mean-field Woods–Saxon potential along with the spin-orbit term. The wavefunctions are expanded as a linear combination of simple sine-wave basis states, which are eigenfunctions of the infinite spherical-well potential. The requisite algorithm based on matrix diagonalization

We present a simple autocollimator with sub-microradian sensitivity. To demonstrate the capabilities of our autocollimator, we study the simple harmonic motion of a cantilever beam and apply an external force to affect the cantilever's resonant frequency in the context of dynamic force microscopy. Our setup is ideal for the advanced undergraduate instructional laboratory and allows a variety of high-precision

The double slit experiment provides a classic example of both interference and the effect of observation in quantum physics. When particles are sent individually through a pair of slits, a wave-like interference pattern develops, but no such interference is found when one observes which "path" the particles take. We present a model of interference, dephasing, and measurement-induced decoherence in

We have developed a programmable illumination system capable of tracking and illuminating numerous objects simultaneously using only low-cost and reused optical components. The active feedback control software allows for a closed-loop system that tracks and perturbs objects of interest automatically. Our system uses a static stage where the objects of interest are tracked computationally as they move

Dropping objects into a tunnel bored through Earth has been used to visualize simple harmonic motion for many years, and even imagined for use as rapid transport systems. Unlike previous studies that assumed a constant density Earth, here we calculate the fall-through time of polytropes, models of Earth's interior where the pressure varies as a power of the density. This means the fall-through time

The problem of electrostatics in biomolecular systems presents an excellent opportunity for cross-disciplinary science and a context in which fundamental physics is called for to answer complex questions. Due to the large density in biological cells of charged biomacromolecules such as protein factors and DNA, it is challenging to understand quantitatively the electric forces in these systems. Two

Single-molecule-sensitive microscopy and spectroscopy are transforming biophysics and materials science laboratories. Techniques such as fluorescence correlation spectroscopy (FCS) and single-molecule sensitive fluorescence resonance energy transfer (FRET) are now commonly available in research laboratories but are as yet infrequently available in teaching laboratories. We describe inexpensive electronics

Photoacoustic (PA) imaging techniques have recently attracted much attention and can be used for noninvasive imaging of biological tissues. Most PA imaging systems in research laboratories use the time domain method with expensive nanosecond pulsed lasers that are not affordable for most educational laboratories. Using an intensity modulated light source to excite PA signals is an alternative technique

This Resource Letter provides a guide to the literature on optical tweezers, also known as laser-based, gradient-force optical traps. Journal articles and books are cited for the following main topics: general papers on optical tweezers, trapping instrument design, optical detection methods, optical trapping theory, mechanical measurements, single molecule studies, and sections on biological motors

We describe a simple framework for teaching the principles that underlie the dynamical laws of transport: Fick's law of diffusion, Fourier's law of heat flow, the Newtonian viscosity law, and the mass-action laws of chemical kinetics. In analogy with the way that the maximization of entropy over microstates leads to the Boltzmann distribution and predictions about equilibria, maximizing a quantity

This Resource Letter provides a guide to the literature about intelligent life beyond the human sphere of exploration. It offers a starting point for professionals and academics interested in participating in the debate about the existence of other technological civilizations or in the search for extraterrestrial intelligence (SETI). It can also serve as a reference for teaching. This Letter is not

We present a simple application of the three-dimensional harmonic oscillator which should provide a very nice particle physics example to be presented in introductory undergraduate quantum mechanics course. The idea is to use the nonrelativistic quark model to calculate the spin-averaged mass levels of the charmonium and bottomonium spectra.