Abstract This paper offers educational insight into the Dirac equation, examining its historical context and contrasting it with the earlier Schrödinger and Klein-Gordon equations. The comparison highlights their Lorentz transformation symmetry and potential probabilistic interpretations. We explicitly solve the free particle dynamics in Dirac's model, revealing the emergence of solutions with negative energy. In this discussion, we examine the Dirac Sea Hypothesis and explore the inherent helicity of the solutions. Additionally, we demonstrate how the Dirac equation accounts for spin and derive the Pauli equation in the non-relativistic limit. Through the Foldy-Wouthuysen transformation, we reveal how the equation incorporates spin-orbit interaction and other relativistic effects, ultimately leading to the hydrogen fine structure. A section on relativistic covariant notation is included to emphasize the invariance of the Dirac equation, along with more refined formulations of both the KG and Dirac equations. Designed for undergraduate students with an interest in the Dirac equation, this resource provides a historical perspective without being purely theoretical. Our approach underscores the significance of a pedagogical method that combines historical and comparative elements to profoundly understand the role of the Dirac equation in modern physics.

Abstract The Young Physicists’ Tournament (YPT) inspires high-school students to immerse themselves into an inquiry process closely resembling the real physics research. Using questionnaires’ replies from the students and their teachers engaged in YPT, we investigated the perception of teachers and students on how the preparation for YPT contributes to the development of hard skills (physics content knowledge, mathematics, modelling...) and soft skills (communication, team work, organization...) of the students, and how it compares to regular classes and other activities. This comparison shows a positive role of YPT among students, and even a more positive perception among teachers. The significant development of a wide range of advanced skills justifies the substantial effort, resources and dedication of both teachers and students required for the participation in YPT.

Abstract Lato-Lato, a simple toy consisting of two small balls, embodies a wealth of physical principles in its motion dynamics. This paper begins with the fundamental principles of pendulum motion and extends to the nonlinear system of the Lato-Lato. Utilizing Lagrangian mechanics, we explore the conservation of momentum, energy conversion, and resonance phenomena during the oscillation of the Lato-Lato. The differential equations governing the motion of the Lato-Lato are solved, and the Mathieu equation is employed to determine the frequency range for resonance occurrence. Furthermore, the trajectory and swing angle of the Lato-Lato are analyzed. Based on theoretical analysis, this paper proposes a series of recommendations for optimizing the manipulation techniques of the Lato-Lato, including the selection of appropriate Lato-Lato models, control of force and rhythm. The research findings indicate that understanding the scientific principles behind the Lato-Lato, combined with effective practice methods, can significantly enhance manipulation skills, providing a theoretical foundation and practical guidance for the popularization and development of the Lato-Lato.

Abstract In the present work we study single and coupled Helmholtz resonators (HR) using theoretical and experimental methods. On the theoretical side, we present two analytical approaches to obtain the resonant frequencies of some HR configurations: an elementary lumped-parameter model analogous to a system of mechanical oscillators, valid only in the long-wavelength approximation; and a more sophisticated impedance-function technique that takes into account the continuous and distributed nature of the system, and allows to obtain higher resonant frequencies under the assumption of plane-wave propagation, recovering the previous model in the long-wavelength limit. We then experimentally determine the resonant frequencies of the proposed configurations by directly recording with a microphone the sound pressure inside the resonators when excited with white noise. We also characterize the absorption properties of the resonators using a custom-built impedance tube, and find excellent agreement between the directly measured resonance frequencies and the frequencies corresponding to the absorption maxima. With appropriate effective length corrections for the necks and coupling connectors, both theoretical methods reproduce the experimentally found frequencies with relative errors below 5%. The project is suitable for undergraduate students and may appeal to those interested in acoustics, requiring only basic theoretical tools and inexpensive or readily available laboratory materials.

Abstract One of the key skills of any scientist is noticing what’s going on. Both in the experiment one’s performing and in one’s data: is there something interesting, reason to doubt one’s data or suspect that one’s theoretical description is insufficient? Many experiments developed for undergraduate teaching still focus on quantitative evaluation. Here we take a different approach where careful observation identifies the interesting qualitative behaviour of a ball dropped with a water bottle balanced on top of it, but where numerical agreement with a simple theoretical model is impossible. Thus ‘success’ occurs when students are satisfied with their efforts and the development of their experimental process. Laboratory note keeping can also be introduced in a meaningful, non-formulaic way since students are making independent observations and method changes. We describe pedagogical and didactic considerations for implementing this experiment and others like it as part of an introduction to practical work in physics, including variations and extensions, and give examples of experimental outcomes. We suggest that considering qualitative behaviour may be a fruitful strategy for identifying experiments that are both amenable to student autonomy and embedding skills such as laboratory note keeping in a flexible and genuine way.

Abstract The rotary voice coil motor from hard disc drives has a unique structure and can be easily modified to a low-cost forced vibration system. This modified oscillator can demonstrate basic vibration phenomena such as damping vibration, the transient and steady-state process during the forced oscillation with the help of a signal generator, a laser, and a scrollable phosphorescent foil. This device is very suitable for demonstrating the evolution of oscillation states, such as the transient processes and the damped oscillation after turning on and off the signal. This low-cost portable device can be used for physics teaching at the undergraduate level.

Abstract The divergence and curl operators are fundamental mathematical tools introduced in university-level electromagnetism courses. Although students generally understand the operational use of these concepts, developing a clear physical intuition of what divergence and curl are can be challenging. This difficulty often arises due to the superficial or confusing treatment found in many standard textbooks. The goal of this article is to provide a clear demonstration, based on examples, that two-dimensional visualizations can be misleading when used to interpret these operators. A combination of graphical and analytical (symbolic) representations was found to be essential for developing an accurate physical intuition about divergence and curl. A deeper comprehension of these operators not only improves proficiency in electromagnetism, but also benefits a range of other fields, from fluid dynamics to meteorology and to technical applications.

Abstract Radioactive decay has a statistical nature, which implies that the radioactive decay law can be deduced from statistical considerations alone. More precisely, the number of disintegrations for a given interval of time builds up towards a Poisson distribution. However, it can sometimes be difficult to distinguish between Poisson and Gaussian distributions, especially when the mean values are large. On the other hand, it is well known that a Poisson distribution for the number of events occurring during a certain time implies an Erlang distribution for the number of time intervals between two given events. We thus present an experimental method that allows measuring this time interval distribution and we show that this result follows with a huge precision (up to ten orders of magnitude) an Erlang distribution. In particular, this experimental method is addressed to Master physics students to put in evidence the Poisson behaviour of nuclear processes as radiation emissions.

Abstract We reconsider the dynamics of a point charge in the field of a point-like electric dipole. The force is noncentral, so the angular momentum is not conserved. Our discussion is largely based on the third conserved quantity β (in addition to the total energy and the axial component of angular momentum) that we derive in spherical coordinates. The equations of motion admit orbits characterised by β = 0 which are traced on a sphere centred at the dipole. Transitions between orbits on the same sphere demand highly specialised, restricted perturbations of the angular momentum components. If the restrictions are not satisfied, then r(t) eventually goes to zero or becomes unbounded. This is due to the lack of a confining, radial effective potential. Transitions between orbits lying on a sphere are exceedingly unlikely under realistic conditions. The stability properties of this classical system are virtually unobservable and inconsequential from a physical point of view. Our approach clarifies and extends previous discussions of this matter, and should be accessible to intermediate or advanced undergraduate physics students.

Abstract The dynamics of a falling slinky is revisited with the usual discrete model, where the slinky is treated as a chain of small masses connected by small springs. The motion of the falling slinky is described analytically by assuming that the masses accelerate due to the spring force and gravity. They inelastically collide with one another from the top down, forming the falling section, while uncollided masses are still at rest. In this study, the initially tied top turns of the hanging slinky are taken into theoretical account. This sole feature robustly solved several existing problems in a past model (Calkin 1993 Am. J. Phys. 61 261), which were discussed in a different study (Cross and Wheatland 2012 Am. J. Phys. 80 1051). The involved physics principles and the mathematical approach in this study is highly suitable for physics freshmen. The model derives the velocity, position, and acceleration of the slinky’s top as explicit functions of time, which showed good agreement with measurements.