Intuition suggests that twisting a cylinder will shorten it, but here it is shown that for a cylinder of an incompressible material, like rubber, twisting will always produce elongation.

We describe a simple classroom demonstration of a fluid-dynamic instability. The demonstration requires only a bucket of water, a piece of string, and some used tealeaves or coffee grounds. We argue that the mechanism for the instability, at least in its later stages, is two-dimensional barotropic (shear-flow) instability and we present evidence in support of this. We show results of an equivalent

The teaching of structural stiffness is one of the keystones of the undergraduate curriculum in mechanics and the strength of materials. Standard linear theory, going back to Hooke's law, has proven to be very successful in predicting the performance of elastic structures under load. Many courses in basic mechanics have a conventional laboratory component often involving a universal testing machine

The purpose of this work is to study the phenomenon of tidal locking in a pedagogical framework by analyzing the effective gravitational potential of a two-body system with two spinning objects. It is shown that the effective potential of such a system is an example of a fold catastrophe. In fact, the existence of a local minimum and saddle point, corresponding to tidally locked circular orbits, is

With developments in modern instrumentation such as microelectromechanical gyro/accelerometers, high speed video analysis, and precision shaft encoders, there is an increased interest in the study of the large angle oscillations of pendulums as an example of nonlinear dynamics. The solution to the equation of motion for the non-linear pendulum cannot be expressed in terms of elementary functions and

We discuss an extension of the velocity Verlet method that accurately approximates the kinetic-energy-conserving charged particle motion that comes from magnetic forcing. For a uniform magnetic field, the method is shown to conserve both particle kinetic energy and magnetic dipole moment better than midpoint Runge–Kutta. We then use the magnetic velocity Verlet method to generate trapped particle trajectories

The illuminating role of differential forms in electromagnetism (EM) is seldom discussed in the classroom. It is the aim of this article to bring forth some of the relevant insights that can be learnt from a differential forms approach to EM. The article is self-contained in that no previous knowledge of forms is needed to follow it. The effective polarization of the classical vacuum due to a uniform

We developed a modified version of a conventional (BB84) quantum key distribution protocol that can be understood and implemented by students at a pre-university level. We intentionally introduce a subtle but critical simplification to the original protocol, allowing the experiment to be assembled at the skill level appropriate for the students, at the cost of creating a security loophole. The security

Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrödinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this

A prototypical model of a one-dimensional metallic monatomic solid containing noninteracting electrons is studied, where the argument of the cosine potential energy, periodic with the lattice, contains the first reciprocal lattice vector G 1 = 2 π / a, where a is the lattice constant. The time-independent Schrödinger equation can be written in reduced variables as a Mathieu equation for which numerically

Spatial filtering is a commonly deployed technique to improve the quality of laser beams by optically filtering the noise. In the “textbook” example, the noise is usually assumed to be high frequency and the laser beam, Gaussian. In this case, the filtering is achieved by a simple pinhole placed at the common focal plane of two lenses. Here, we explain how to generalize the concept of spatial filtering

Optical coherence tomography, or in short OCT, is a measurement technique established in the early 1990s for the non-invasive imaging of interfaces in the bulk of biological tissues or other samples. A full-field OCT setup is built from a microscope combined with a Michelson interferometer, where the mirror in one arm is replaced by the sample. Using white light, which is temporally partially coherent

Charles Édouard Guillaume (1861–1928) was a Swiss physicist who received the 1920 Nobel Prize in physics for his precision measurements and discovery of anomalies in nickel steel alloys. In this work, we present a complete and commented translation of his remarkable article of 1896 on the temperature of interstellar space. The importance of this work is that it is the oldest estimate known to us of

An elementary, analytically soluble example of a pair of isospectral potentials is discussed. This example was developed in response to an inquiry by an undergraduate student who was intrigued by the widely used (but not often discussed in undergraduate courses) Rydberg–Klein–Rees (RKR) procedure for inverting rovibrational data to determine potential functions of diatomic molecules, which is based

An approximate solution is presented for simple harmonic motion in the presence of damping by a force which is a general power-law function of the velocity. The approximation is shown to be quite robust, allowing for a simple way to investigate amplitude decay in the presence of a general type of weak, nonlinear damping.

The room temperature compatibility of the negatively charged nitrogen-vacancy (NV−) center in diamond makes it the ideal quantum system for a university teaching lab. Here, we describe a low-cost experimental setup for coherent control experiments on the electronic spin state of the NV− center. We implement spin-relaxation measurements, optically detected magnetic resonance, Rabi oscillations, and

The ability to control the frequency of an external-cavity diode laser (ECDL) is an essential component for undergraduate laboratories and atomic physics research. Typically, the housing for the ECDL's diffraction grating and piezoelectric transducer is either purchased commercially or machined from metal. Here, we present an alternative to these commonly used options that utilizes 3D printing, a tool

We describe the design, construction, and operation of a LEGOTM model of a dynamic force microscope, using magnetic forces as an analog of interatomic interactions. The macroscope provides key insights into the operating principles of frequency modulated non-contact atomic force microscopy—currently the scanning probe technique of choice for maximal (i.e., submolecular) spatial resolution—and is, therefore

We construct an electronic circuit for mimicking a single neuron's behavior in connection with the dynamics of the Hodgkin–Huxley mathematical model. Our results show that the electronic neuron, even though it contains binary-state circuitry components, displays a timing interplay between the ion channels, which is consistent with the corresponding timing encountered in the model equations. This is

We present results and lessons learned from a Learning Assistant (LA) program combined with flipped classroom delivery and team learning in our calculus-based, introductory physics (mechanics) courses at California State University, Sacramento. A primarily undergraduate institution, our university serves a largely underprivileged student population, with one half of students coming from low-income

Standard university or high-school physics teaching material on projectile motion is usually based on Newton's second law in vacuum, neglecting aerodynamics. We present a low-cost experiment for teaching projectile motion using the students' cell phones and sports equipment, which allows the students to test theory and numerical simulation against experimental data in the real world. For a shot put

We use videos taken with a mobile phone to study conservation of energy, conservation of momentum, and the work-energy theorem by analyzing the collision of a cue ball and the eight ball. A video of the full time sequence, starting from before the cue ball is struck until well after the collision, is recorded with a mobile phone. The video is imported into Origin (free to teachers and students taking

We give a detailed derivation of the equation describing the elastic properties of the helical springs in the case of their large deformations. It is shown that the Taylor expansion for the restoring force contains, in addition to a linear term, a term quadratic in the strain. We discuss the physical consequences of this term. A surprising example of this is that for large values of the initial pitch

A detailed analysis of pendular motion is presented. Inertial effects, self-oscillation, and memory, together with non-constant moment of inertia, hysteresis, and negative damping are shown to be required for the comprehensive description of the free pendulum oscillatory regime. The effects of very high initial amplitudes, friction in the roller bearing axle, drag, and pendulum geometry are also analyzed

We present a full algebraic derivation of the wavefunctions of a simple harmonic oscillator. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the coordinate-space or momentum-space wavefunctions from the energy eigenvectors. It is simple to incorporate into the undergraduate and graduate curricula. We provide a summary of the history

We examine the behavior of a charged particle in a two dimensional quantum dot in the presence of a magnetic field. Emphasis is placed on the high magnetic field regime. Compared to free space geometry, confinement in a dot geometry provides a more realistic system where edge effects arise naturally. It also serves to remove the otherwise infinite degeneracy due to the magnetic field; nonetheless,

Delay dynamics occur in a wide variety of natural and man-made systems. Even simple delay systems can generate complex dynamics whose exploration is rewarding. To allow such exploration as part of advanced undergraduate laboratory courses and be able to utilize systems that operate at convenient timescales, it is necessary to delay analog signals by several milliseconds. In this paper, we describe

Observation of time-dependent luminescence from excited states with a wide range of lifetimes allows students to explore the connection between selection rules and transition rates. It is fairly simple to measure microsecond and longer lifetimes with equipment common to undergraduate programs, because the instrument response time of even modest bandwidth systems is insignificant on microsecond and

The computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear

The left-to-right motion of a free quantum Gaussian wave packet can be accompanied by the right-to-left flow of the probability density, the effect recently studied by Villanueva. Using the Wigner representation of the wave packet, we analyze the effect in phase space and demonstrate that its physical origin is rooted in classical mechanics.

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