This paper theoretically studies a simple system of two identical linear springs connected symmetrically to a mass in a V-shaped configuration, with an additional adjustable external force applied to the mass. As this force is varied, under certain conditions the equilibrium position of the mass demonstrates strong dependence on the history of changes in the external force, exhibiting hysteresis. Mathematically

An ellipse on a non-slip inclined plane can rock, roll, or jump. Below a threshold energy, it rocks about its static equilibrium configuration. Above this energy, it rolls, and, if it rolls, it will eventually jump off the incline after some number of rolls. It is shown that jumping can only occur in certain configurations of the ellipse.

The static and dynamic properties of an asymmetric rigid body are studied using a Lagrangian approach. The body consists of a thin-walled cylinder balanced by a point mass attached to its inner surface. When resting on an inclined plane, stable and unstable equilibrium points are found, and oscillations about the stable point are studied for different angles of inclination and different values of the

Physics textbooks generally present the equation for universal gravitation as if it sprang fully formed from the head of Isaac Newton, which it most certainly did not. This analysis traces the development of Classical/Newtonian gravity as it evolved from Aristotle to Copernicus to Kepler, Horrocks, Roberval, Boulliau, Baliani, Borelli, Hooke, Halley, Wren, Newton, and Cavendish, and then finally on

In this paper, we use a straightforward numerical method to solve scattering models in one-dimensional lattices based on a tight-binding band structure. We do this by using the wave packet approach to scattering, which presents a more intuitive physical picture than the traditional plane wave approach. Moreover, a general matrix diagonalization method that is easily accessible to undergraduate students

In 2015, the first observation of gravitational waves marked a breakthrough in astrophysics and in technological research and development. The discovery of a gravitational-wave signal from the collision of two black holes, a billion light-years away, received considerable interest from the media and public. We describe the development of a purpose-built exhibit explaining this new area of research

Traditional introductory physics laboratories typically focus on guided experimentation while neglecting essential aspects of the scientific practice including computational reasoning. While there is a broad consensus among physics faculty on the need to introduce undergraduate students to computation, its integration in physics curricula is far from adequate. In this article, we document the process

We use coherence theory to explain why it is necessary to modify the conventional setup of a Michelson interferometer to obtain Haidinger rings with an extended source of white light. The modification consists of introducing a glass slide into one of the two arms of the interferometer. This insertion circumvents the drastic restriction imposed by the low temporal coherence of white light, which prevents

The data that students collect from the typical advanced undergraduate experiment on saturated absorption spectroscopy (SAS) of rubidium can be used to measure the isotope shift and thus lead to an estimate of the isotopic ground-state energy shift. This helps students refine their “picture” of the atomic ground state. We describe theoretically why SAS works well with free-running laser diodes, demonstrate

The naïve Bohr quantization condition mvr = n ℏ is applied to arbitrary spherically symmetric power-law potentials. The dependence of the energy eigenvalues on the principal quantum number n agrees with fully quantum mechanical results either exactly (for the hydrogen atom, the harmonic oscillator, and the infinite square well) or asymptotically (the linear potential). This naïve treatment can be used

The problem of rendezvous, the meeting of spacecraft in orbit, is an important aspect of mission planning. We imagine a situation where a chaser craft, initially traveling on the same circular orbit as its target and separated from it by a known distance, must select an initial thrust vector that will allow it to meet the target (interception) followed by a second thrust vector that will allow it to

It is a common experience that water shoots higher when we block a garden hose outlet by our thumb. But what causes this? How high does the water go? Does water from our neighbor's garden hose reach the same height? Is there an optimum outlet blockage that results in the greatest height that water can reach? Here, we show that a competition between viscous friction along the hose and the viscous dissipation

We consider sinusoidal undulations which appear on certain garden hoses under normal use. We propose a model, using linear elasticity, explaining this phenomenon, and make a connection with biological structures as well as self-buckling. We compare observations with model predictions and suggest potential applications in the area of shape-changing materials.

In its usual form, the loop-the-loop (LtL) problem involves a uniform solid sphere rolling from rest down a linear ramp that transitions into a circular loop. The task is to find the minimum height from which the ball must be released in order to roll completely around the loop without breaking contact. The answer, found using the conservation of mechanical energy and Newton's second law, is invariably

The pandemic-induced lock-down has resulted in better air quality and visibility, with reported sightings of Himalayan peaks from hundreds of kilometers away. During the 18th and 19th centuries, good visibility was commonplace and there exist many such accounts by reputable Orientalists such as Henry Colebrooke and Sir William Jones. These sightings invite two questions. Is a line of sight to the peak

Modeling is an essential skill for the career physical scientist or engineer. One approach that attempts to integrate this element of critical thinking into the laboratory environment is the modeling framework lab. In this style of lab exercise, students construct a model of both the measurement apparatus and the phenomenon under investigation. A key challenge to implementing this approach in introductory

The operational principle of twisted nematic displays involves the dielectric anisotropy of nematics. This crucial property was discovered about a hundred years ago by Jeżewski and Kast who used a so-called resonance method in which the frequency of an LC tank circuit was set by the capacitance of a capacitor filled with a nematic liquid crystal. Jeżewski and Kast observed that the resonance frequency

Of the many topics generally taught in undergraduate or graduate optics courses, the propagation of light through crystals with electrical anisotropy is one of the most difficult topics for students and for teachers who do not work in this field. In particular, the mathematics and equations are complex vectorial equations, and this complicates the visualization of the implications of the theory. Also

The development of programming languages for quantum computing has increased rapidly over the last few years, making it practical for a hands-on approach in teaching quantum computation. In this paper, I introduce the standard textbook example of searching for one item out of four using Grover's search algorithm and extend it by including a quantum database. In addition to explaining how to include

Time evolution of quantum systems is normally taught using the Schrödinger equation. Here, we use the Heisenberg approach to consider three problems that are exactly solvable in the Schrödinger picture when the initial wavepacket is Gaussian: the free particle, the simple harmonic oscillator and a free particle with a spin-orbit interaction. We show that complete solutions can be obtained for the free

We demonstrate that the de Broglie wave of a particle in a gravitational field turns toward the region of lower gravitational potential, causing the particle to fall. This turning is caused by clocks running slower in the smaller potential. We use the analogy of ocean waves that are slower in shallower water and turn toward beaches. This approach implies that the motion is along a geodesic and explains

The detection of gravitational waves is possible thanks to a multidisciplinary approach involving different disciplines such as astrophysics, physics, engineering, and quantum optics. Consequently, it is important today for teachers to introduce the basic features of gravitational waves science in the undergraduate curriculum. The usual approach to gravitational wave physics is based on the use of

We describe an undergraduate physics laboratory experiment that uses a 2-inch diameter NaI gamma detector to measure natural radiation in soils. Students first calibrate the detector for the source-detector geometry, energy-dependent detection efficiency, and sample self-absorption. Then, the activity of the 238U and 232Th decay series, as well as the activity of 40K in the soil, are measured. The

We develop a direct geometric method to determine the orbital parameters and mass of a planet, and we then apply the method to Neptune using high-precision data for the other planets in the solar system. The method is direct in the sense that it does not involve curve fitting. This paper, thereby, offers a new pedagogical approach to orbital mechanics that could be valuable in a physics classroom.

A ball placed inside a hollow cylinder and spun around an axis perpendicular to the cylinder wall will roll back and forth along the cylinder wall, as expected, but it will also oscillate along the axial direction of the cylinder. These axial oscillations occur in horizontal, tilted, and even vertical cylinders. Here, the equations of motion for this setup are derived and discussed, and experimental

In this work, we present in detail, in an accessible manner for undergraduate and graduate physics students, the model of spontaneous curvature, due to Helfrich, that quantitatively explains why the red blood cells in their natural state adopt a biconcave shape. The main hypothesis is that the equilibrium cell shape satisfies the principle of minimum free energy. Therefore, in the model, an expression

Science students must deal with the errors inherent to all physical measurements and be conscious of the need to express them as a best estimate and a range of uncertainty. Errors are routinely classified as statistical or systematic. Although statistical errors are usually dealt with in the first years of science studies, the typical approaches are based on manually performing repetitive observations

We present a computational method to directly calculate and visualize the directional components of the Coulomb, radiation, and total electromagnetic fields, as well as the scalar and vector potentials generated by moving point charges in arbitrary motion with varying speeds. Our method explicitly calculates the retarded time of the point charge along a discretized grid, which is then used to determine

In the quasistatic limit, a time-varying magnetic field inside a conductor is governed by the diffusion equation. Despite the occurrence of this scenario in many popular physics demonstrations, the concept of magnetic diffusion appears not to have garnered much attention itself as a subject for teaching. We employ the model of a thin conducting tube in a time-varying axial field to introduce magnetic

Schrödinger's equation can be considered as a diffusion equation with a diffusion coefficient β 2 = ℏ / 2 m. In this article, we explore the implications of this view. Rewriting the wave function in a polar form and transforming Schrödinger's equation into two real equations, we show that one of them reduces to the continuity equation, and the other, a nonlinear dynamical equation for the probability

We present and discuss numerical solutions to a two-body quantum bound state problem closely related to that of the hydrogen atom and the deuterium nucleus. The forces binding the particles of our system are Yukawa forces, which fall off with distance faster than the Coulomb force and arise for non-relativistic particles whose interactions are mediated by massive scalar or vector particles; the Coulomb

The identification of particle velocity with the velocity of a wave group was a crucial assumption that resolved important inconsistencies in the theoretical developments of Louis de Broglie and Erwin Schrödinger. Interestingly, this was one of the few common aspects of their work. In this paper, we present a reconstruction of how group velocity became essential for both de Broglie and Schrödinger

The finite rectangular well (FRW) has been a staple of quantum mechanics texts and classes for decades and is the subject of a rich literature. Despite being a problem about which there would apparently be not much more to be said, the FRW continues to serve as a system for introducing students to various analytic techniques and has numerous connections to current technology and research. This paper

By finding that the projection operators of the basic quantum mechanical representations can be written as operator delta functions, we derive the explicit forms of quantum-mechanical fundamental representations in Fock space by directly carrying out the integrals within the normally ordered product of operators.

We have designed and experimentally studied several systems of standard coaxial cables with different impedances which mimic the operation of so-called photonic structures like coupled photonic crystal microcavities. Using elementary cells of half-meter long coaxial cables, we got resonances around 100 MHz, a range of frequencies that can be easily studied with standard teaching laboratory apparatus

Modern mobile phones contain a three-axis microelectromechanical system (MEMS) gyroscope, capable of taking accurate measurements of the angular velocity along the three principal axes of the phone with a sampling rate of 100 Hz or better. If the phone is tossed in the air, then, neglecting air resistance, it is in free rotation (rotation in the absence of a torque) with respect to its centre of mass

In this paper, the torque-free rotational motion of a general rigid body is developed analytically and is applied to the flipping motion of a T-handle spinning in zero gravity that can be seen in videos on the internet. This flipping motion is known both as the Dzhanibekov effect (after the cosmonaut who reported it) and more recently the tennis racket effect. The presentation is self-contained, accessible

Solving the equations of motion for a cylinder with elliptical cross section rolling down an inclined plane constitutes a rather challenging problem. Using Lagrangian mechanics, we have derived the equations of motion analytically and the resulting differential equations are solved numerically. Using a CNC milling machine, we manufactured a solid elliptic cylinder and recorded its dynamics using a

We devise an undergraduate experiment to evaluate the wind turbine power coefficient. The apparatus features a commercial miniature rotor hub that allows both blade replacement and adjustment of the pitch angle. The blades are 3D printed. Three pedagogical objectives are emphasized: (1) the demonstration of a practical application of rotational mechanics, (2) the illustration of wind harvesting by

Measurements of capacitors made with paper sheets reveal a significant decrease in capacitance with increasing frequency from 10 to 100,000 Hz, offering a simple demonstration of complex dielectric phenomena using common equipment.

A simple and inexpensive method to generate plasma using a kitchen microwave oven is described in this paper. The microwave-generated plasma is characterized by spectroscopic analysis and compared with the absorption spectra of a gas discharge tube. A Paschen-like curve is observed as the microwave plasma initiation time is plotted as a function of the pressure of the plasma chamber. We have also demonstrated

We investigate acoustic levitation in a vertical standing wave in an attempt to understand the basic physical mechanism responsible for this phenomenon. We find that a description in terms of a simple pressure force leads to the prediction of stable equilibria that occur slightly below the anti-nodes of the standing pressure wave. We then demonstrate that such a prediction is at odds with experimental

Reasoning by analogy is powerful in physics for students and researchers alike, a case in point being electronics and hydraulics as analogous studies of electric currents and fluid flows. Around 100 years ago, Nikola Tesla proposed a flow control device intended to operate similarly to an electronic diode, allowing fluid to pass easily in one direction but providing high resistance in reverse. Here

Distance education has expanded significantly over the last decade, but the natural sciences have lagged in the implementation of this instructional mode. The abrupt onset of the COVID-19 pandemic left educational institutions scrambling to adapt curricula to distance modalities. With projected effects lasting through the 2020–2021 academic year, this problem will not go away soon. Analysis of the

We present an important and forgotten result of fundamental quantum mechanics obtained in 1930 by Enrico Persico, consisting of an eigenvalue equation for minimum uncertainty states.

The one-dimensional hydrogen atom is an intriguing quantum mechanics problem that exhibits several properties which have been continually debated. In particular, there has been variance as to whether or not even-parity solutions exist, and specifically whether or not the ground state is an even-parity state with infinite negative energy. We study a “regularized” version of this system, where the potential

This paper is intended to serve as a bridge between introductory textbooks on quantum mechanics, which typically do not cover the Hartree-Fock self-consistent-field method, and more advanced ones which treat this important computational method for fermionic many-body systems in an abstract and formal way. We derive the Hartree-Fock equation for the 1s orbital of a realistic two-electron atom. By employing

We discuss computer simulations of a particle that hops forward and backward randomly on a one-dimensional lattice. To track the motion of the particle, we use a data acquisition protocol that mimics a virtual digital optical imaging system. Plots of the data extracted from these images yield a pattern that is similar to the patterns observed in real imaging of a moving particle in a living cell. To

A highly reflective sail provides a way to propel a spacecraft out of the solar system using solar radiation pressure. The closer the spacecraft is to the Sun when it starts its outward journey, the larger the radiation pressure and so the larger the final velocity. For a spacecraft starting on the Earth's orbit, closer proximity can be achieved via a retrograde impulse from a rocket engine. The sail

We discuss a simple experiment investigating the shrinkage of surface soap bubbles sitting on a thin solid plate with a circular orifice located under the apex of the bubble. We identify three different shrinking regimes, the occurrence of which depends on a combination of key parameters that include the ratio between initial bubble and orifice sizes and physicochemical properties of the fluid system

High-speed (2873 fps) visualization of soap film bursting dynamics has been performed by means of equipment available to a high school laboratory. This paper presents the geometry of the experiment and description of the analysis method of the video frames. We carried out the experiments on the bursting of vertical plane soap films using a circular frame with single central and non-central puncture

Many textbooks dealing with surface tension favor the thermodynamic approach (minimization of some thermodynamic potential such as free energy) over the mechanical approach (balance of forces) to describe capillary phenomena, stating that the latter is flawed and misleading. Yet, a mechanical approach is more intuitive for students than free energy minimization, and does not require any knowledge of

When a falling chain strikes a surface, it can accelerate downwards faster than free-fall. This counterintuitive effect occurs when a tension is created in the chain above where it strikes the surface. The size of this tension, and how it is produced, depend on the type of chain used. For a chain made of rods that are slightly tilted from horizontal, the impact-induced tension is readily observable

The distribution of string tension on the contact line between an ideal string and a massive pulley is a frequently discussed but incompletely posed problem that confronts students in introductory mechanics. We highlight ambiguities in the usual presentation of this problem by the massive Atwood's machine and discuss two compact resolutions that treat situations where the pulley or the string elastically

This article considers an example of eddy current behaviour that has not been described previously in textbooks and physics education journals. This phenomenon involves a sharp rise in the active resistance of a multilayer coil with increasing alternating current frequency. This effect is not explained by the classic skin effect and is not related to losses in a core. The results of measurements of

The question of how much surplus of electric charge (“surcharge”) fits on an object is generally very difficult to answer. Here, it is shown that it is easy to answer when the object is a failed white dwarf star (a brown dwarf in its ground state) made of protons and electrons: Given the number of protons, how many electrons can there be? Surprisingly, the answer (in the form: as few as A and as many

In this work, we present a master equation approach to simulating DC transport through single electron transistors and quantum dots suitable for an upper-division undergraduate computational physics project. After introducing the basic theory describing transport through quantum dots, we present a simulation of the simple case of a metallic dot including effects due to the finite temperature of the

Interactive simulations and visualizations augment the teaching and learning of quantum mechanics by making equations and concepts come to life. However, graphical visualizations are nearly always limited to a set of hard-coded functionalities. Text-based codes can offer a higher degree of flexibility, but only at the expense of steep learning curves or time investments. We introduce Quantum Composer

This article describes how the author successfully adapted techniques drawn from the literature on active learning for use in a graduate-level course on quantum field theory. Students completed readings and online questions ahead of each class and spent class time working through problems that required them to practice the decisions and skills typical of a theoretical physicist. The instructor monitored

Many different formalisms exist for computing the phase of a matter-wave interferometer. However, it can be challenging to develop physical intuition about what a particular interferometer is actually measuring or about whether a given classical measurement provides equivalent information. Here, we investigate the physical content of the interferometer phase through a series of thought experiments