Equations, as mathematical models—used to explain facts—are simplified, especially when learning physics at a basic level, to help in the understanding of complex systems that depend on many variables. These simplifications are idealizations in the literature, where abstractions and approximations are also given for the same concept. When reading about a model, knowing that it is idealized, and realizing

Typically, the format of online physics educational videos is closer in style to direct instruction rather than being interactive. One simple way of making videos more interactive is asking questions in one video, providing solutions in the second and seeking student reflection. This study utilises two physics education videos by Derek Muller (Veritasium) which were deployed with 508 first year physics

In theorem 1 of his celebrated Principia , Newton showed that if one assumes a central force, Kepler’s area law follows. In this work, we report a study where two different ways of deriving this relationship, one geometrical ( à la Newton) and the other analytical (using vector calculus), were presented to pre-service physics teachers. The participants were asked to formulate the essential arguments

We analyze the problem of a quantum particle falling under the influence of a one-dimensional constant gravitational field, also known as the bouncing ball , employing a semiclassical approach for the effective equations of motion for the quantum system. In this formalism, the quantum evolution is described through a dynamical system of infinite dimensions for the position, the momentum, and all dispersions

The incident neutron energy dependence on the neutron multiplication rate constant α is studied for 235 U- and 239 Pu-based nuclear explosions. At high neutron energy, α comes out to be about 10 generations/shake for both 235 U and 239 Pu. In addition, the photon gas model seems to hold well for the very early stage of a nuclear explosion as one is able to reproduce the yield (20.4 ± 2.4) kiloton for

In an attempt to regularize a previously known exactly solvable model (Yang and Zhang 2019 Eur. J. Phys. 40 035401), we find yet another exactly solvable toy model. The interesting point is that while the Hamiltonian of the model is parameterized by a function f ( x ) defined on [0, ∞), its spectrum depends only on the end values of f , i.e., f (0) and f (∞). This model can serve as a good exercise

Semi-classical path integral expression for a quantum system coupled to a harmonic bath is derived based on the stationary phase condition. It is discovered that the system path is non-Markovian. Most strikingly, the system path not only couples to its past, but also to its future, i.e. the equation of motion for the system is an integro-differential equation that involves all times. Numerical examples

The force at the pivot of a simple pendulum and the acceleration of the bob have been the subject of a number of papers. The present paper extends this analysis to a compound pendulum, which requires an extra torque to rotate the pendulum about its center of mass. The dependence of the pivot force on κ = k / a , the ratio of the radius of gyration k to the center of mass position a , is investigated

Undergraduate experiments provide practical demonstration of both measurement techniques and the underpinning science. The radioactive source 44 Ti is particularly useful due to the range of experimental measurements it allows. Five experiments utilising 44 Ti, suitable for undergraduate laboratories, are described in this paper. These illustrate several topics in nuclear physics and a number of experimental

We propose, simulate, and experimentally demonstrate a circuit analogue of a special relativity phenomenon known as relativistic aberration of light (RAL) by using a surprisingly simple, low-cost, and easily accessible electronic circuit-based all-pass filter. This work is useful for two audiences: (i) physicists who are interested in research on circuit analogues; and (ii) physics educators who are

In 1905, Einstein carried out his first derivation of the mass–energy equivalence by studying in different reference frames the energy balance of a body emitting electromagnetic radiation and assuming special relativity as a prerequisite. In this paper, we prove that a general mass–energy relationship can be derived solely from very basic assumptions, which are the same made in Einstein’s first derivation

The non-uniqueness of Lagrange functions is a standard topic in classical mechanics. However, it is still troublesome in undergraduate and graduate teaching. In this paper, we show that this non-uniqueness is equivalent to gauge freedom, and thus the gauge, canonical, and Galilean transformations are equivalent. In this sense, we also discuss how to solve the problem of selecting one out of many gauge-equivalent

There are some theoretical arguments about possible variations of fundamental constants with cosmic time. We review the fact that all conversion factors depend on these quantities and consider how their variations may affect transformations of units. We deal with the relevance of the issue to the cosmological constant problem.

A straightforward derivation of the integral governing the trajectory of light around a massive object is presented. By a proper choice of coordinate transformation the singularity in the integrand of this equation is removed, which makes it more suitable for numerical calculations of trajectories. An improved approximate formula for the deflection angle is presented.

A catenary is the shape that an electric cable takes under its own weight if suspended only at its ends between two pylons. A rope hung between two masts is also described with a catenary equation which is based on the hyperbolic cosine function. We propose here to approach the hyperbolic cosine function with two simple experiments using a beaded chain. We first study the static form of a chain whose

In this paper the classical well-known solution of the elementary problem of the sphere loop in the presence of friction is studied. Also, a modification of the problem shows a possible inaccuracy involved in the well-known solution found in many references. Finally, a revised solution of the problem is presented.

In the present work, we propose a procedure to allow undergraduate students to obtain two experimental values of the Mercury–Sun distance, using geometric compositions of photographs of Mercury’s transits at its perihelion and its aphelion. From those distances, the model also allows students to calculate the value of the eccentricity of Mercury’s orbit. Our procedure is validated against published

In this paper, approximate bright-soliton solutions of the higher-order nonlinear Schrödinger equation are constructed by treating the higher-order terms as small perturbations. The first-, second-, and third-order asymptotic solutions are obtained. The errors between the asymptotic solutions and the numerical/analytical solutions are discussed, which gives a high accuracy of the approximate solutions

We investigate the behavior of two-dimensional resistor networks, with finite sizes and different kinds (rectangular, hexagonal, and triangular) of lattice geometry. We construct a network by having a network unit repeat itself L x times in the x -direction and L y times in the y -direction. We study the relationship between the effective resistance ( R eff ) of the network on dimensions L x and L

One of the more interesting applications of the integral form of Faraday’s law of electromagnetic induction is to an open surface whose contour is changing with time. An often-used theoretical example for this is a track formed from two parallel, conducting rails. The planar surface between the rails is bounded by a voltmeter at one end and a moving, conducting bar across and in contact with the rails

The deadlift is a popular weightlifting exercise and a common everyday move. Despite this, there are few weightlifting examples in the physics education literature. We calculated the force that the lifter applies to the bar using video analysis and accelerometers, with an inverse dynamics procedure. The presented example is safe, practical, affordable and applicable to any classroom. We compared a

In college physics experiments, a pendulum is often applied to measure the moment of inertia ( J ) of a rigid body. In this paper, we demonstrate a trifilar pendulum to determine not only J , but also the mass ( m ) of an irregular object. To this end, a standard weight is used to balance the pendulum, and a laser pointer is used to observe the amount of imbalance. According to the parallel axis theorem

Astrophysical topics can be treated with the methods of different areas of physics, mathematics and other natural sciences. The integration of these topics into such areas could improve students’ motivation and strengthen the connections between different fields of the mentioned sciences. In this article, an example of the connection between the physics of the stars and that of their statistical distributions

For quantum or other waves that are band-limited, the quantum potential in the Madelung–Bohm representation vanishes on the boundaries of regions where the waves are superoscillatory (i.e. where they vary faster than any of their Fourier components). This connection is illustrated by calculations of the quantum potential zeros for a superoscillatory superposition of plane waves, and by Aharonov–Bohm

Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes

Precision measurements of the pendulum period as a function of the amplitude can now be made with a variety of instruments including MEMs gyro/accelerometers, and thus theoretical expressions are required for comparison. Unfortunately exact solution of the pendulum equation involves elliptic integrals, which cannot be expressed in terms of elementary functions, and therefore a wide variety of approximations

After a brief history and critique of some older instruments, several new Levitron geometries are described. As a result of their greater stability these devices can be used as analogues of a number of phenomena and applications, including magnetic resonance techniques, atom traps and accelerator rings. In particular, the notion of the spinning magnet (or spignet) in a linear trap is similar to the

Wave phenomena are fundamental for many branches of physics, not only for mechanics. Thus, students at university level should become familiar with the underlying theory, and especially with solutions of the wave equation. This paper refers to a multimedia program for PCs, tablets or smartphones, and introduces and discusses several animated illustrations. The introduced program can illustrate the

Computers and computer simulations are no longer new tools in education. The question is whether computer simulations can partially or even completely replace the traditional approaches. In this paper, we describe the outcome of our research, where computer simulations as teaching tools were a competitor to the traditional approach at local grammar schools. Selected factors that may influence the results

We present pygiftgenerator , a python module for systematically preparing a large number of numerical and multiple-choice questions for Moodle-based quizzes oriented to students’ formative evaluation. The use of the module is illustrated by means of examples provided with the code and drawn from different topics, such as mechanics, electromagnetism, thermodynamics and modern physics. The fact that

We study the motional emf generated by squeezing an elliptical conducting loop traversed by a constant magnetic field. The circumference of the loop remains constant in the elastic regime, acting as a non-trivial constraint. We numerically find the dependence of one axis of the ellipse on the other and hence calculate the rate of change in area and the ensuing emf. We discuss possible applications

Considering that classical electrodynamics is essentially a covariant theory, we propose a simple heuristic approach to nonlinear Maxwell equations where certain scalar fields are coupled to the electromagnetic (EM) field. This leads us to introduce a new (nonlinear) version of the Poynting theorem. This model seems to have solutions in which coupled scalar fields and EM waves propagate. This work

The time dependence of the temperature distribution in cooling/heating bodies is studied with a simple, direct visual measurement of infrared radiation. The adoption of a smartphone thermal camera allows one to verify some essential features of theoretical models which describe the temperature field in a cylindrical symmetric configuration for both lumped (concentrated) and non-lumped (distributed)

Introductory mechanics courses use the bouncing ball model to familiarize students with the principles of binary inelastic collisions. Nonetheless, in undergraduate courses, the modeling of binary systems typically disregards the time of contact and the effects of gravity, which yields to a constant coefficient of restitution (COR) and, as a consequence, prevents students from elucidating the real

In a previous paper (2019 Eur. J. Phys. 40 035005) we showed how to design a discrete brachistochrone with an arbitrary number of segments. We have proved, numerically and graphically, that in the limit of a large number of segments, N ≫ 1, the discrete brachistochrone converges into the continuous brachistochrone, i.e. into a cycloid. Here we show this convergence analytically, in two different ways

Experimental analysis of the motion in a system of two coupled oscillators with arbitrary initial conditions was performed and the normal coordinates were obtained directly. The system consisted of two gliders moving on an air track, joined together by a spring and joined by two other springs to the fixed ends. From the positions of the center of mass and the relative distance, acquired by analysis

This paper presents an approach based on a laser interferometer set-up to perform an experimental study of Hooke’s law for the case of the bending of a cantilever beam. We used weights and a pulley to deform the beam laterally and measured displacement with an electronic speckle pattern interferometer (ESPI). The ESPI set-up was built using items commonly found in an undergraduate optics laboratory

Einstein’s special theory of relativity is presented in a Minkowski four-vector formalism integrating mechanics and thermodynamics at a sophomore level, allowing the solution of undergraduate exercises in linear translation requiring both. This relativistic formalism directly incorporates the mechanics (Newton’s second law) and the thermodynamics (first law of thermodynamics) of a process in a four-vector

In this study, we show the results of some traditional experiments, performed by using an inexpensive dual-trace cathode ray oscilloscope (CRO), combined with an automated data extraction tool (based on image processing techniques) to learn about their fundamentals. This tool, called a WebPlotDigitizer (WPD) is an opensource software program ( https://automeris.io/WebPlotDigitizer/download.html [https://automeris

How do engineering physics students come to understand and share their physics learnings as a result of careful integration of oral communication with engineering skills like computer aided design and 3D-printing technology? Based in a sociocognitive theory of situated communication pedagogy, the action research conducted in this study set out to answer this research question in an introductory first-year

We present a new proof of Cramer’s rule by interpreting a system of linear equations as a transformation of n -dimensional Cartesian-coordinate vectors. To find the solution, we carry out the inverse transformation by convolving the original coordinate vector with Dirac delta functions and changing integration variables from the original coordinates to new coordinates. As a byproduct, we derive a generalized

The blue color perception of Cherenkov radiation is often intuitively attributed to its spectrum. However observations usually take place through several meters of water which strongly modifies the original spectrum. A quantitative description of color allows to judge the relative importance of source spectrum versus water influence. This physics example nicely illustrates the scientific method of

Wave mechanics triumphed when Schrödinger published his now famous equation and showed how to describe hydrogen-like atoms using it. However, while looking for the right equation, Schrödinger first explored, but did not publish, the equation that we today call the Klein–Gordon equation. An alternative possible choice is explored in this work. It is shown a quasi-relativistic wave equation which solutions

We make a multicontextual and critical review of widely used undergraduate physics textbooks, regarding Pascal’s principle, and compare them with the original work of Blaise Pascal. Our analysis shows that there are inconsistencies throughout almost all the books considered, both conceptual and historical. We provide suggestions on how to overcome or fix these inconsistencies, so Pascal’s principle

This article explains and illustrates the use of a set of coupled dynamical equations, second order in a fictitious time, which converges to solutions of stationary Schrödinger equations with additional constraints. In fact, the method is general and can solve constrained minimization problems in many fields. We present the method for introductory applications in quantum mechanics including three qualitative

The one-dimensional (1D) hydrogen atom with potential energy V ( q ) = − e 2 /| q |, with e the electron charge and q its position coordinate, has been a source of discussion and controversy for more than 55 years. A number of incorrect claims have been made about its spectrum; for example, that its ground state has infinite binding energy, that bound states associated with a continuum of negative

The infinite square wall model is one of the simplest models to have been introduced in quantum mechanics. We employ this model to illustrate the quantum transitions of particles due to the deconfinement process. With the infinite square wall, we propose that a shift of the wall to a larger x position corresponds to the process of deconfinement with strong interaction. Two limits of the wall’s movement

It is often said, in connection with the Pauli principle of quantum mechanics, that ‘if the spatial wavefunction is symmetric under a coordinate swap, the particles huddle together; if antisymmetric, they spread apart’. While this saying is often true, this essay uncovers a counterexample.

We revisit the concept of optical spatial coherence by considering the case of the complex degree of spatial coherence induced over a plane by a quasi-monochromatic irradiating source. We apply the Van Cittert–Zernike theorem and exploit a Lloyd’s two-beam interferometer setup with its two small coherent light sources, as a way to illustrate the calculation of such complex spatial coherence, and because

We investigate generic heat engines and refrigerators operating between two heat reservoirs, for the condition when their efficiencies are equal to each other. It is shown that the corresponding value of efficiency is given as the inverse of the generalized golden mean, ϕ p , where the parameter p depends on the degrees of irreversibility of both engine and refrigerator. The reversible case ( p = 1)

The effects of speed humps and bumps in slowing vehicles were investigated using a mass suspended on a spring. An impulse was applied to the mass by lowering and raising the top end of the spring by hand. The impulse duration was varied to simulate different driving speeds, and the displacement of the mass vs time was recorded with a video camera. Maximum displacement of the mass was observed when

The launch in 2019 of the new international system of units is an opportunity to highlight the key role that the fundamental laws of physics and chemistry play in our lives and in all the processes of basic research, industry and commerce. The main objective of these notes is to present the new SI in an accessible way for a wide audience. After reviewing the fundamental constants of nature and its

The purpose of this paper is to articulate a coherent and easy-to-understand way of doing quantum mechanics in any finite-dimensional Liouville space, based on the use of a Kronecker product and what we have termed the ‘bra-flipper’ operator. One of the greater strengths of the formalism expatiated on here is the striking similarities it bears with Dirac’s bra-ket notation. For the purpose of illustrating

Molecules and atoms that are normally gaseous at room temperature and pressure will liquefy or solidify as their temperature reduces or their pressure increases. In the apparatus detailed here a commercial cell contains pressurized SF 6 so that it exists in both gaseous and liquid states, with the density of both being similar at room temperature. The critical point on the phase diagram can then be

Analysis of institutional data for physics majors showing predictive relationships between required mathematics and physics courses in various years is important for contemplating how the courses build on each other and whether there is need to make changes to the curriculum for the majors to strengthen these relationships. We used 15 years of institutional data at a US-based large research university

This paper outlines some of the benefits of using virtual reality (VR) in physics education at the university level. Based on the existing experimental infrastructure of the low-background gamma spectroscopy laboratory, a VR experiment has been designed which includes all the stages of the physical measurement process. Following the VR experiment, the students receive pre-prepared experimental data

The mechanically controllable break junction (MCBJ) experimental setup is one of the main techniques employed in the study of electronic transport properties at the atomic and molecular scales. This work presents the construction of an inexpensive and simple but robust setup that shows the emergence of conductance quantization as a macroscopic gold wire is pulled to atomic dimensions. The homemade

We derive here, from first principles, the energy–momentum density of a perfect fluid in the form of an ideal molecular gas, in an inertial frame where the fluid possesses a bulk motion. We begin from the simple expressions for the energy density and pressure of a perfect fluid in the rest frame of the fluid, where the fluid constituents (gas molecules) may possess a random motion, but no bulk motion

We discuss the influence of two factors on the deviations from the model of the magnetic field of a long straight wire. Firstly, we derive the magnetostatic analogue of McCullagh’s formula for the case of the infinitely long straight conductor with an arbitrary cross-section. Using this equation on the basis of its practical significance, we find that the greatest deviations from the long straight

Here we study the physical problem in which a thin bar, such as a strip of paper, is bent under several types of loads and boundary conditions. Initially, we consider lengthwise compression of the strip due to concentrated forces applied on its ends. We present this problem for the three boundary conditions: doubly-clamped, hinged–clamped, and doubly-hinged ends constrained on a flat surface. We also