The higher derivatives of motion are rarely discussed in the teaching of classical mechanics of rigid bodies; nevertheless, we experience the effect not only of acceleration, but also of jerk and snap. In this paper we will discuss the third and higher order derivatives of displacement with respect to time, using the trampolines and theme park roller coasters to illustrate this concept. We will also discuss the effects on the human body of different types of acceleration, jerk, snap and higher derivatives, and how they can be used in physics education to further enhance the learning and thus the understanding of classical mechanics concepts.

With the advent of quantum technology, the need for affordable, flexible and robust laboratory experiments not only for students, but also at high school level is increasing. Here, for the first time, we report on a simple modular 3D printed low-cost (<250 €) setup which fulfils these needs for quantum sensing experiments based on nitrogen-vacancy centers in diamonds. Commercially available setups for optically detected magnetic resonance in microdiamonds used as quantum sensor for magnetic fields are not only beyond the reach of any high school (>10 000 €), but also have shortcomings from a didactical point of view, as all the components of the setup are hidden within a 'black box', doomed to be successful 'plug and play'. In contrast, our open-source experimental kit consists of optical components that are placed inside 3D printed open-framed cubes, that can be arranged freely on a grid. This modular and flexible design can provide an inquiry-based learning experience both at undergraduate and high school level.

This comment addresses the Jiménezet al paper (2008Eur. J. Phys.29 163–75). We correct these authors' expressions for the electric field and the Poynting vector, and also correct their expressions for the magnetic field. Contrary to the authors, we find that the contribution of the second term in Jefimenko's formula for the electric field is not zero. These corrections in no way affect the main points made by the authors of the aforementioned paper, and help strengthen their work.

The paper aims to fulfil three main functions: (1) to serve as an introduction for the physics education community to the functioning of large language models (LLMs), (2) to present a series of illustrative examples demonstrating how prompt-engineering techniques can impact LLMs performance on conceptual physics tasks and (3) to discuss potential implications of the understanding of LLMs and prompt engineering for physics teaching and learning. We first summarise existing research on the performance of a popular LLM-based chatbot (ChatGPT) on physics tasks. We then give a basic account of how LLMs work, illustrate essential features of their functioning, and discuss their strengths and limitations. Equipped with this knowledge, we discuss some challenges with generating useful output withChatGPT-4 in the context of introductory physics, paying special attention to conceptual questions and problems. We then provide a condensed overview of relevant literature on prompt engineering and demonstrate through illustrative examples how selected prompt-engineering techniques can be employed to improveChatGPT-4's output on conceptual introductory physics problems. Qualitatively studying these examples provides additional insights into ChatGPT's functioning and its utility in physics problem-solving. Finally, we consider how insights from the paper can inform the use of LLMs in the teaching and learning of physics.

This work originated as a project in experimental physics conducted by students in the Laboratory of Mechanics and Thermodynamics, a course designed for first-year Physics bachelors. The students were tasked with studying the motion of rigid bodies while having only been introduced to the laws of point-mass dynamics in their theoretical classes. In the proposed experiment, students discovered that bodies with circular symmetry and identical shapes take the same time to roll down an inclined plane. By appropriately fitting the experimental data, they observed that the expression for travel time is equivalent to that of a point mass, except for a multiplicative factor unique to each category of objects. This factor depends on the body's geometry and is directly related to its moment of inertia. Furthermore, we discuss how a similar result can be derived using scaling analysis, illustrating the power of this tool for students early in their physics education. This work may serve as an effective introduction for first-year Physics students to the moment of inertia, as it naturally emerges from a classic physics experiment: the inclined plane.

In this paper, we present an approach to learning about statistical distributions of quantum particles suitable for Advanced Placement high school courses and early-year undergraduate physics and chemistry courses. The procedure developed here uses selected tools known from phenomenological thermodynamics, particularly the concepts of chemical potential and chemical equilibrium, which allow us to circumvent the extensive mathematical apparatus traditionally employed for obtaining equilibrium conditions. To this end, we (a) introduce the notion ofsite, i.e. an abstract 'place' that is able to accept particles; (b) assume that sites having different occupation numbers can be treated as differentelementary substances; (c) consider the change of occupation number as a reaction between elementary substances; and (d) derive quantum distributions by assuming that at equilibrium the driving force, i.e. the difference of chemical potentials of 'educts' and 'products' vanishes. Assuming that the available sites can be occupied either by at most one particle or by any number of them, we obtain the Fermi–Dirac and the Bose–Einstein distributions, respectively. We illustrate a number of examples, including the Planck distribution for the blackbody radiation and the pressure of the degenerate electron gas in white dwarfs. The paper ends with a comparison of quantum and classical Boltzmann distributions and with some remarks from an educational perspective.

Language is an important resource for physicists and learners of physics to construe physical phenomena and processes, and communicate ideas. Moreover, any physics-related instructional setting is inherently language-bound, and physics literacy is fundamentally related to comprehending and producing both physics-specific and general language. Consequently, characterizing physics language and understanding language use in physics are important goals for research on physics learning and instructional design. Qualitative physics education research offers a variety of insights into the characteristics of language and language use in physics such as the differences between everyday language and scientific language, or metaphors used to convey concepts. However, qualitative language analysis fails to capture distributional (i.e. quantitative) aspects of language use and is resource-intensive to apply in practice. Integrating quantitative and qualitative language analysis in physics education research might be enhanced by recently advanced artificial intelligence-based technologies such as large language models, as these models were found to be capable to systematically process and analyse language data. Large language models offer new potentials in some language-related tasks in physics education research and instruction, yet they are constrained in various ways. In this scoping review, we seek to demonstrate the multifaceted nature of language and language use in physics and answer the question what potentials and limitations artificial intelligence-based methods such as large language models can have in physics education research and instruction on language and language use.

The quintessential example of a fluid flow problem with a known solution is the drag force required to move a sphere through a stationary fluid. While the equation describing this Stokes drag is simple, deriving it from the Stokes equations requires several pages of mathematics. In this paper, I present an alternative, more intuitive approach, based on the Oseen tensor, which gives the fluid flow due to an applied point force. For completeness, we first derive the expression for the Oseen tensor, then use it to re-derive Stokes law. As an additional application, we also study fluid flow near a wall, using a mirror images technique similar to the one used in electrostatics problems.

The emergence of conversational natural language processing models presents a significant challenge for Higher Education. In this work, we use the entirety of a UK Physics undergraduate (BSc with Honours) degree including all examinations and coursework to test if ChatGPT (GPT-4) can pass a degree. We adopt a 'maximal cheating' approach wherein we permit ourselves to modify questions for clarity, split question up into smaller sub-components, expand on answers given—especially for long form written responses, obtaining references, and use of advanced coaching, plug-ins and custom instructions to optimize outputs. In general, there are only certain parts of the degree in question where GPT-4 fails. Explicitly these include compulsory laboratory elements, and the final project which is assessed by a viva. If these were no issue, then GPT-4 would pass with a grade of an upper second class overall. In general, coding tasks are performed exceptionally well, along with simple single-step solution problems. Multiple step problems and longer prose are generally poorer along with interdisciplinary problems. We strongly suggest that there is now a necessity to urgently re-think and revise assessment practice in physics—and other disciplines—due to the existence of AI such as GPT-4. We recommend close scrutiny of assessment tasks: only invigilated in-person examinations, vivas, laboratory skills testing (or 'performances' in other disciplines), and presentations are not vulnerable to GPT-4, and urge consideration of how AI can be embedded within the disciplinary context.

How far can we see with the naked eye at night? Many celestial objects like stars and galaxies as well as transient phenomena such as comets and supernovae can be observed in the night sky. We discuss the furthest distances of such objects and phenomena observable with the naked eye during the night-time for Earth-bound observers. The physics of night-time visual ranges differs from that of daytime observations because human vision shifts from cones to rods. In addition, mostly point sources are observed due to the large distances involved. Whether celestial objects and phenomena can be detected depends on the contrast of their radiation and the background sky luminance. We present a concise overview of how far we can see at night by first discussing the effects of the Earth's atmosphere. This includes attenuation of transmitted radiation as well as its role as a source of background radiation. Disregarding the attenuation of light due to interstellar and intergalactic dust, simple maximum night-time visual range estimates are based on the inverse square law, which can be easily verified by laboratory and demonstration experiments. From the respective calculations, we find that individual stars within the Milky Way galaxy of up to 15 000 light years are observable. Even further away are observable galaxies with several billion stars. The Andromeda galaxy can be observed with the naked eye at a distance of around 2.5 million light years. Similarly, the observability of supernovae also allows a visual range beyond the Milky Way galaxy. Finally, gamma ray bursts as the most energetic events in the universe are discussed concerning naked eye observations.

The present study explored the role of first-year physics students' sense of belonging to physics and how it relates to their learning progression in higher education-level physics. Conducted at Paderborn University, this study examined how students' sense of belonging to physics influences their academic outcomes in an introductory experimental physics course. Additionally, we investigated how physics students' engagement in thePhysiktreff—a holistic support program for first-year physics students to help them cope with academic and social challenges during their studies—impacts the development of their sense of belonging to physics over time. Our findings indicated that students with a stronger sense of belonging to physics performed better academically. Moreover, students who actively participated in the support program experienced a positive shift in their sense of belonging to physics. However, our findings also revealed that physics students with a higher initial sense of belonging to physics tended to experience a decline in their sense of belonging to physics during their first semester. These results underscore the importance of fostering a sense of belonging to physics within higher education, particularly during the introductory phase of students' studies.

Mirror astigmatism is analysed with different approaches in the two orthogonal planes, in a way that is easier to follow and understand than the classic textbooks usually referred to in research papers.

A sphere dropped from rest in a stationary fluid will fall under the combined forces of gravity, buoyancy, and drag if its average density is larger than that of the fluid. Introductory physics courses typically model the drag force as varying either with the first power of the sphere's speed (at low Reynolds numbers) or with its second power (at high Reynolds numbers). The proportionality constants depend on parameters of the sphere and of the fluid that are in principle all known. It is then straightforward to algebraically deduce the terminal speed of the falling sphere. In a calculus-based course, with a bit more effort one can analytically find the complete mathematical dependence of the sphere's speed and distance of descent as a function of time for either linear or quadratic drag. An issue which is often glossed over, however, is what happens if the Reynolds number varies over a range from low to high values. Then it would seem reasonable to sum together the linear and quadratic drag forces. This paper compares such a two-term model both to experimental data from previous investigators and to a more detailed model which empirically describes the variation in the drag coefficient over a large range of Reynolds numbers.

Internal friction is an important tool to understand the movement of atoms and molecules in solid materials. It is widely used to study the evolution of microstructures, significantly contributing to our understanding of the dynamic relaxation processes affecting the atomic-scale structure of solids. This paper derives the formula for internal friction in solid mechanics by examining the energy and stress–strain hysteresis curves. It subsequently outlines the research frontiers related to the mechanical relaxation modes of crystalline and amorphous solids, highlighting the intrinsic connection between internal friction and the relaxation of microstructural defects in solids. Additionally, this work shows how to employ molecular dynamics simulation methods to quantify the internal friction and to directly observe the evolution of structural and dynamic characteristics. Based on the authors teaching experience and recent research findings regarding the theory of solid internal friction, this paper presents a course program with various dimensions, including content, methods, and instructional flow. The proposed teaching strategies aim to serve as a reference for other educators involved in courses of solid mechanics.

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 quantityβ (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 byβ = 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, thenr(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.

It is universally believed that with his 1905 paper 'Does the inertia of a body depend on its energy content?' Einstein first demonstrated the equivalence of mass and energy by making use of his special theory of relativity. In the final step of that paper, however, Einstein equates the kinetic energy of a body to its Newtonian value, indicating that his result is at best a low-velocity approximation. Today, several characters debate whether a mid-nineteenth century physicist, employing only physics available at the time, could plausibly arrive at the celebrated result. In other words, is Einsteinian relativity necessary to derive?

Although synchronization effects play an important role in many areas of basic and applied science, their treatment in undergraduate physics courses requires more attention. Based on acoustic experiments with a driven organ pipe, the article proposes analytical, numerical and qualitative approaches to this universal phenomenon, suitable for introductory teaching. The Adler equation is developed, a first-order nonlinear differential equation describing the phase dynamics of driven self-sustained oscillations in the weak coupling limit. Analytical solutions, intuitive mechanical analogues and properties of the resulting comb spectra are discussed. The underlying phase model is paradigmatic for synchronization-based self-organization phenomena in a wide range of fields, from physics and engineering to life and social sciences.

The use of augmented reality (AR) allows for the integration of digital information onto our perception of the physical world. In this article, we present a comprehensive review of previously published literature on the implementation of AR in physics education, at the school and the university level. Our review includes an analysis of 96 papers from the Scopus and Eric databases, all of which were published between 1st January 2012 and 1st January 2023. We evaluated how AR has been used for facilitating learning about physics. Potential AR-based learning activities for different physics topics have been summarized and opportunities, as well as challenges associated with AR-based learning of physics have been reported. It has been shown that AR technologies may facilitate physics learning by providing complementary visualizations, optimizing cognitive load, allowing for haptic learning, reducing task completion time and promoting collaborative inquiry. The potential disadvantages of using AR in physics teaching are mainly related to the shortcomings of software and hardware technologies (e.g. camera freeze, visualization delay) and extraneous cognitive load (e.g. paying more attention to secondary details than to constructing target knowledge).

Language is an important resource for physicists and learners of physics to construe physical phenomena and processes, and communicate ideas. Moreover, any physics-related instructional setting is inherently language-bound, and physics literacy is fundamentally related to comprehending and producing both physics-specific and general language. Consequently, characterizing physics language and understanding language use in physics are important goals for research on physics learning and instructional design. Qualitative physics education research offers a variety of insights into the characteristics of language and language use in physics such as the differences between everyday language and scientific language, or metaphors used to convey concepts. However, qualitative language analysis fails to capture distributional (i.e. quantitative) aspects of language use and is resource-intensive to apply in practice. Integrating quantitative and qualitative language analysis in physics education research might be enhanced by recently advanced artificial intelligence-based technologies such as large language models, as these models were found to be capable to systematically process and analyse language data. Large language models offer new potentials in some language-related tasks in physics education research and instruction, yet they are constrained in various ways. In this scoping review, we seek to demonstrate the multifaceted nature of language and language use in physics and answer the question what potentials and limitations artificial intelligence-based methods such as large language models can have in physics education research and instruction on language and language use.

This pedagogical article elucidates the fundamentals of trapped-ion quantum computing, which is one of the potential platforms for constructing a scalable quantum computer. The evaluation of a trapped-ion system's viability for quantum computing is conducted in accordance with DiVincenzo's criteria.

Modeling the hydrogen atom is often considered an essential milestone in an introductory quantum mechanics course. Mathematically, the instructional path is clear: from the square potential well, through the harmonic potential, the 3-dimensional box, and the spherical well potential prior to the 1/r Coulomb potential. However, there are few experimental examples to help visualize and ground the key physical concepts. Here, we present an experiment where the light scattering of an evaporating, optically levitating water droplet can be used as a quantum analogy to a quantum spherical well potential. We have distilled two important concepts in this analogy: angular momentum and tunneling. This approach is akin to the analogy between a square well potential and a Fabry-Pérot cavity. It can be used as a visual example on the path toward the hydrogen atom in undergraduate-level quantum mechanics courses.&#xD;

Particle physics, when taught in the classroom or lecture theatre, suffers from a lack of practical experience by students. Thus, we describe the construction of a fully working small particle physics detector using state of the art detector technology for demonstration in educational context. Most of our setup can be constructed with relatively moderate effort, given that a home-level 3D-printer, a photosensor and readout electronics (at least an oscilloscope) are available.

We use a Legendre polynomial expansion to find the electrostatic potential of a uniformly charged disk.&#xD;We then use the potential to find the electric field of the disk.

The definitions of Q factor of a damped oscillator are analysed and a general expression for this figure of merit is established. Moreover, the circumstances for using the widespread formula for the Q factor are derived.&#xD;

In the advanced Chinese language curriculum, there is a classic prose titled Shi Zhong Shan Ji. This prose documents an interesting phenomenon that water flowing through a hollow and porous boulder will create instrumental sounds. Inspired by these ancient texts, we have conceived an acoustic experiment aimed at investigating the auditory properties of water flowing through porous materials. We designed an experimental setup using commonly available household materials, which ensures both cost-effectiveness and practicality. This device successfully simulates the process described in the prose that lake water flow through a hollow and porous boulder by using water to flush a teacup strainer. Utilizing sound analysis software, we recorded and analyzed the acoustic signals generated by the water passing through the strainer. Our results demonstrate that these sounds exhibit characteristics consistent with the edge tone instruments. This experiment not only provides a practical approach to the study of acoustics, but also seamlessly integrates Chinese classical literature with physics education, allowing students to appreciate the unique charm of Chinese classical culture while exploring acoustics.&#xD;

The present study explored the role of first-year physics students' sense of belonging to physics and how it relates to their learning progression in higher education-level physics. Conducted at Paderborn University, this study examined how students' sense of belonging to physics influences their academic outcomes in an introductory experimental physics course. Additionally, we investigated how physics students' engagement in thePhysiktreff—a holistic support program for first-year physics students to help them cope with academic and social challenges during their studies—impacts the development of their sense of belonging to physics over time. Our findings indicated that students with a stronger sense of belonging to physics performed better academically. Moreover, students who actively participated in the support program experienced a positive shift in their sense of belonging to physics. However, our findings also revealed that physics students with a higher initial sense of belonging to physics tended to experience a decline in their sense of belonging to physics during their first semester. These results underscore the importance of fostering a sense of belonging to physics within higher education, particularly during the introductory phase of students' studies.

Mirror astigmatism is analysed with different approaches in the two orthogonal planes, in a way that is easier to follow and understand than the classic textbooks usually referred to in research papers.

Modeling the hydrogen atom is often considered an essential milestone in an introductory quantum mechanics course. Mathematically, the instructional path is clear: from the square potential well, through the harmonic potential, the 3-dimensional box, and the spherical well potential prior to the 1/r Coulomb potential. However, there are few experimental examples to help visualize and ground the key physical concepts. Here, we present an experiment where the light scattering of an evaporating, optically levitating water droplet can be used as a quantum analogy to a quantum spherical well potential. We have distilled two important concepts in this analogy: angular momentum and tunneling. This approach is akin to the analogy between a square well potential and a Fabry-Pérot cavity. It can be used as a visual example on the path toward the hydrogen atom in undergraduate-level quantum mechanics courses.&#xD;

How far can we see with the naked eye at night? Many celestial objects like stars and galaxies as well as transient phenomena such as comets and supernovae can be observed in the night sky. We discuss the furthest distances of such objects and phenomena observable with the naked eye during the night-time for Earth-bound observers. The physics of night-time visual ranges differs from that of daytime observations because human vision shifts from cones to rods. In addition, mostly point sources are observed due to the large distances involved. Whether celestial objects and phenomena can be detected depends on the contrast of their radiation and the background sky luminance. We present a concise overview of how far we can see at night by first discussing the effects of the Earth's atmosphere. This includes attenuation of transmitted radiation as well as its role as a source of background radiation. Disregarding the attenuation of light due to interstellar and intergalactic dust, simple maximum night-time visual range estimates are based on the inverse square law, which can be easily verified by laboratory and demonstration experiments. From the respective calculations, we find that individual stars within the Milky Way galaxy of up to 15 000 light years are observable. Even further away are observable galaxies with several billion stars. The Andromeda galaxy can be observed with the naked eye at a distance of around 2.5 million light years. Similarly, the observability of supernovae also allows a visual range beyond the Milky Way galaxy. Finally, gamma ray bursts as the most energetic events in the universe are discussed concerning naked eye observations.

In this paper, we present an approach to learning about statistical distributions of quantum particles suitable for Advanced Placement high school courses and early-year undergraduate physics and chemistry courses. The procedure developed here uses selected tools known from phenomenological thermodynamics, particularly the concepts of chemical potential and chemical equilibrium, which allow us to circumvent the extensive mathematical apparatus traditionally employed for obtaining equilibrium conditions. To this end, we (a) introduce the notion ofsite, i.e. an abstract 'place' that is able to accept particles; (b) assume that sites having different occupation numbers can be treated as differentelementary substances; (c) consider the change of occupation number as a reaction between elementary substances; and (d) derive quantum distributions by assuming that at equilibrium the driving force, i.e. the difference of chemical potentials of 'educts' and 'products' vanishes. Assuming that the available sites can be occupied either by at most one particle or by any number of them, we obtain the Fermi–Dirac and the Bose–Einstein distributions, respectively. We illustrate a number of examples, including the Planck distribution for the blackbody radiation and the pressure of the degenerate electron gas in white dwarfs. The paper ends with a comparison of quantum and classical Boltzmann distributions and with some remarks from an educational perspective.

We present educational material about Bell inequalities in the context of quantum computing.&#xD;In particular, we provide software tools to simulate their violation, together with a&#xD;guide for the classroom discussion. The material is organized in three modules of increasing&#xD;difficulty, and the relative implementation has been written in Qibo, an open-source software&#xD;suite to simulate quantum circuits with the ability to interface with quantum hardware.&#xD;The topic of inequalities allows not only to introduce undergraduate or graduate students&#xD;to crucial theoretical issues in quantum mechanics – like entanglement, correlations, hidden&#xD;variables, non-locality –, but also to practically put hands on tools to implement a real simulation,&#xD;where statistical aspects and noise coming from current quantum chips also come&#xD;into play.&#xD;

The quintessential example of a fluid flow problem with a known solution is the drag force required to move a sphere through a stationary fluid. While the equation describing this Stokes drag is simple, deriving it from the Stokes equations requires several pages of mathematics. In this paper, I present an alternative, more intuitive approach, based on the Oseen tensor, which gives the fluid flow due to an applied point force. For completeness, we first derive the expression for the Oseen tensor, then use it to re-derive Stokes law. As an additional application, we also study fluid flow near a wall, using a mirror images technique similar to the one used in electrostatics problems.

The 2019 revision of the International System of Units (SI) set exact values for four defining natural constants, allowing all base and derived units to be defined exactly. While the distinction between base and derived units remains valuable, particularly in education where it helps students grasp the SI's structure more easily, we argue for clarifying the classification of these base units. Currently, the SI defines seven base units, but these fall into two fundamentally different subsets. The first subset—the kilogram (kg), meter (m), second (s), and ampere (A)—forms a foundational set of physics units from which all other units can be derived. The second subset—the candela (cd), kelvin (K), and mole (mol)—serves a different role. We propose recognizing this difference by classifying the candela, kelvin, and mole as a distinct category, possibly called 'scale-spanning units'. While such a revision would not affect the accuracy of measurement, it would help students and newcomers to metrology to attain better clarity of understanding of the International System of Units.

An undergraduate physics experiment is described that uses a Fabry–Perot etalon and digital camera to determine the hyperfine frequency shifts in naturally occurring mercury. Radiation at 546 nm emitted from the 7³S₁ state relaxing to the 6³P₂ state in a low pressure Hg lamp is selected by an interference filter to pass through the etalon. This produces a series of concentric rings at the focus of the camera lens that are analyzed using bespoke software written for this experiment. Results from the analysis are compared to theoretical calculations of the hyperfine shifts for both the¹⁹⁹Hg and²⁰¹Hg isotopes.

This work originated as a project in experimental physics conducted by students in the Laboratory of Mechanics and Thermodynamics, a course designed for first-year Physics bachelors. The students were tasked with studying the motion of rigid bodies while having only been introduced to the laws of point-mass dynamics in their theoretical classes. In the proposed experiment, students discovered that bodies with circular symmetry and identical shapes take the same time to roll down an inclined plane. By appropriately fitting the experimental data, they observed that the expression for travel time is equivalent to that of a point mass, except for a multiplicative factor unique to each category of objects. This factor depends on the body's geometry and is directly related to its moment of inertia. Furthermore, we discuss how a similar result can be derived using scaling analysis, illustrating the power of this tool for students early in their physics education. This work may serve as an effective introduction for first-year Physics students to the moment of inertia, as it naturally emerges from a classic physics experiment: the inclined plane.