Table of contents

A simple strobe setup with the potential to study higher-order eigenmodes and multifrequency oscillations in micromechanical resonators is described. It requires standard equipment, commonly found in many laboratories, and it can thus be employed for public demonstrations of mechanical resonances. Moreover, the work presented here can be used by undergraduate students and/or teachers to prepare practical work in laboratory courses at physics or engineering universities. The dynamics of a micromachined cantilever is analysed as an example. In fact, using our stroboscopic setup, the first and second flexural eigenmodes, as well as a multifrequency oscillation composed by a superposition of both modes, have been successfully filmed with a conventional optical microscope equipped with a digital camera.

We present a graphical interface designed to demonstrate the techniques of radio interferometry used by telescopes like ALMA, e-Merlin, the JVLA and SKA, in a manner accessible to the general public. Interferometry is an observational technique used by astronomers to combine the signal from a few to tens to hundreds of individual small antennas to achieve high resolution images at radio and millimetre wavelengths. This graphical interface demonstrates how the number of antenna, their position relative to one another and the rotation of the Earth allow astronomers to create highly detailed images at long wavelengths.

We use homogenization theory to derive asymptotic solutions of the Schrödinger equation for periodic potentials. This approach provides a rigorous framework in which the key concepts in solid-state physics naturally arise (Bloch waves, band gaps, effective mass, and group velocity). We solve the resulting spectral cell problem using numerical spectral methods, and validate our solution in an analytically-solvable case. Finally, we briefly discuss the convergence of our asymptotic approach and we prove that the ground-state k = 0 effective mass is never less than the ordinary inertial mass.

Conceptual understanding is one of the main topics in science and physics education research. In the majority of conceptual understanding studies, students' understanding levels were categorized dichotomously, either as alternative or scientific understanding. Although they are invaluable in many ways, namely developing new instructional materials and assessment instruments, students' alternative understandings alone are not sufficient to describe students' conceptual understanding in detail. This paper introduces an example of a study in which a method was developed to assess and describe students' conceptual understanding beyond alternative and scientific understanding levels. In this study, six undergraduate students' conceptual understanding levels of direct current electricity concepts were assessed and described in detail by using their answers to qualitative problems. In order to do this, conceptual understanding indicators are described based on science and mathematics education literature. The students' understanding levels were analysed by assertion analysis based on the conceptual understanding indicators. The results indicated that the participants demonstrated three intermediate understanding levels in addition to alternative and scientific understanding. This paper presents the method and its application to direct current electricity concepts.

A simple and inexpensive experiment requiring nonlinear data transformation for straightforward fitting gives a complete overview of basic data treatment techniques in a way suitable for a beginners on a physics laboratory course. A satisfactory estimate of the relative dielectric constant of a material is thereby obtained. Methodological issues and pedagogical aspects of the experiment are discussed in detail.

The shapes of two wires in a vertical plane with the same starting and ending points are described as complementary curves of descent if beads frictionlessly slide down both of them in the same time, starting from rest. Every analytic curve has a unique complement, except for a cycloid (solution of the brachistochrone problem), which is self complementary. A striking example is a straight wire whose complement is a lemniscate of Bernoulli. Alternatively, the wires can be tracks down which round objects undergo a rolling race. The level of presentation is appropriate for an intermediate undergraduate course in classical mechanics.

We emphasize that it can be didactically very useful for students to realize how a space–time diagram of an observer, moving with a constant velocity with respect to another observer, can be obtained easily by means of a standard matrix of rotation, without recourse to imaginary axes and angles. These diagrams were introduced for the first time by Loedel and their main advantage over Minkowski diagrams is that a scale factor is not necessary to convert the units of an observer to those of another observer. We show this well-known property of Loedel diagrams using a new geometric approach.

In one dimension, we investigate dynamics using special relativity only, but make no statement regarding any particular insight into the nature of the spacetime continuum. Indeed, we do away completely with any kind of formal tensor algebra or differential topology, but do incorporate scaling into our calculations. The purpose here is primarily pedagogical, as an extension to the standard undergraduate treatment of special relativity, but it is also a calculational tool, to be able to determine the motion of objects in a one-dimensional case even with accelerations and forces, without having to use the full general relativistic treatment. To illustrate this, we directly compare relativistic to Newtonian or non-relativistic motions for particularly instructive selected systems.

A common formulation of the law of increase of entropy, found in textbooks on thermodynamics, states that in a process taking place in a completely isolated system the entropy of the final equilibrium state cannot be smaller than that of the initial equilibrium state. This statement does not specify that thermal isolation is all that is needed for its validity, with no need for mechanical isolation. For the purpose of illustrating this situation, we exhibit examples of thermodynamic processes carried out with thermally isolated—although not mechanically isolated—systems, which we know to be allowed by the second law because the entropy of the system increases. We believe that the analysis presented in this paper may be useful in a first undergraduate course on thermodynamics.

We present the basics of a powerful contemporary statistical mechanical technique that can be used by students to explore first-order phase transitions by themselves and for models of their own construction. The technique is a generalization of the well-known Peierls argument and is applicable to various models on a lattice. We illustrate the technique with the help of two simple models that were recently used to simulate phase transitions on surfaces.

We analyze quantitatively the problem of exchanging the axes in a linear fit. We show that the inverse slope of the line obtained after inverting the axes is not necessarily compatible, within standard error, with the original slope.

The two-electron configuration in the helium atom is known to very high precision. Yet, we tend to refer to this configuration as a 1s↑1s↓ singlet, where the designations refer to hydrogen orbitals. The high precision calculations utilize basis sets that are suited for high accuracy and ease of calculation, but do not really aid in our understanding of the electron configuration in terms of product states of hydrogen orbitals. Since undergraduate students are generally taught to think of helium, and indeed the rest of the periodic table, in terms of hydrogenic orbitals, we present in this paper a detailed spectral decomposition of the two-electron ground state for helium in terms of these basis states. The 1s↑1s↓ singlet contributes less than 93% to the ground state configuration, with other contributions coming from both bound and continuum hydrogenic states.

Hooke's name is familiar to students of mechanics thanks to the law of force that bears his name. Less well-known is the influence his findings had on the founder of mechanics, Isaac Newton. In a lecture given some twenty years ago, W Arnol'd pointed out the outstanding contribution to science made by Hooke, and also noted the controversial issue of the attribution of important discoveries to Newton that were actually inspired by Hooke. It therefore seems ironic that the two most famous force laws, named after Hooke and Newton, are two geometrical aspects of the same law. This relationship, together with other illuminating aspects of Newtonian mechanics, is described in Arnol'd's book and is worth remembering in standard physics courses. In this didactical paper the duality of the two forces is expounded and an account of the more recent contributions to the subject is given.

It is pointed out that recent cosmological findings seem to support the view that the mass/energy distribution of the universe defines the Newtonian inertial frames, as originally suggested by Mach. The background concepts of inertial frame, Newton's second law and fictitious forces are clarified. A precise definition of Mach's principle is suggested. Then, an approximation to general relativity discovered by Einstein, Infeld and Hoffmann is used and it is found that this precise formulation of Mach's principle is realized provided the mass/energy density of the universe has a specific value. This value turns out to be twice the critical density. The implications of this approximate result are put into context.

We study the motion of charged particles constrained to arbitrary two-dimensional curved surfaces but interacting in three-dimensional space via the Coulomb potential. To speed up the interaction calculations, we use the parallel compute capability of the Compute Unified Device Architecture of today's graphics boards. The particles and the curved surfaces are shown using the Open Graphics Library. This paper is intended to give graduate students, who have basic experiences with electrostatics and the Lagrangian formalism, a deeper understanding of charged particle interactions and a short introduction of how to handle a many particle system using parallel computing on a single home computer.

Gauge field theory approaches have been used extensively in recent times to discuss the physical consequences of spin–orbit interactions in condensed matter physics. An SU(2)×U(1) gauge theory is very naturally borne out and provides an illustrative example of classical Yang–Mills field theory at work. This approach may serve as an exemplification of non-Abelian field theories for students in a general physics curriculum. It allows one to introduce discussions on fundamental ideas like Noether currents, the gauge symmetry principle, gauge symmetry breaking and non-linear Yang–Mills equations in very concrete physical situations that make them accessible to a broad audience.

An experiment is described in which a small strong cylindrical magnet is levitated by a vertical non-uniform alternating magnetic field. Surprisingly, no superimposed constant field is necessary, but the levitation can be explained when the vertical motion of the magnet is taken into account. The theoretical mean levitation force is (0.26 ± 0.06) N, which is in good agreement with the levitated weight of (0.239 ± 0.001) N. This experiment is suitable for an undergraduate laboratory, particularly as a final year project. Students have found it interesting, and it sharpens up knowledge of basic magnetism.

By considering the extension of Bernoulli's theorem to the case of the isentropic flow of ideal gases we conceive a small-scale wind–energy system able to work in the presence of low wind velocities in any direction. The flow of air inside a hyperbolically shaped pipe is studied using elementary physics concepts. The results obtained show that wind velocity in the system increases for decreasing cross-sectional areas, allowing a lower cut-in wind speed and an increase in the annual energy production of the device.

The one-dimensional quantum harmonic oscillator problem is examined via the Laplace transform method. The stationary states are determined by requiring definite parity and good behaviour of the eigenfunction at the origin and at infinity.

We present an elementary analysis of effects observed when light is reflected from a uniformly moving mirror, using the photon picture of light and the conservation laws of energy and momentum for the system photon–mirror. Such a dynamical approach to the problem seems very suitable for introductory physics courses, not requiring any previous knowledge of wave optics or special relativity.

Dimensional analysis, and in particular the Buckingham Π theorem is widely used in fluid mechanics. In this paper we obtain an expression for the impact parameter from Buckingham's theorem and we compare our result with Rutherford's original discovery of the early twentieth century.

We simplify a recent derivation by Corbò (2010 Eur. J. Phys.31 L55–7) of a torque on a current loop due to the action of a static magnetic field.

The journal has received a range of correspondence on this topic. The Editor-in-Chief would like to thank the contributors for their interest. However, the journal does not extend discussion on a topic beyond the initial comment and reply. As such, further comments on these articles will not be considered for publication; however, developments in this area of physics education will be welcome.

Errors and/or inconsistencies in recent papers on the Clausius inequality and its rectification (2011 Eur. J. Phys.32 279; 2011 Eur. J. Phys.32 845) are pointed out that have to do with the fact that the dissipative work therein is not split into its two components: a part coming from friction and ending up delivered as heat to the system and/or surroundings plus a part corresponding to the extra work that would not be wasted if the transformation were reversible. Also criticized, as potential sources of confusion, are the attempts in that paper to clarify so-called subtleties and misunderstandings related to the definition of heat and the computation of the entropy change of a system.

We address Bizarro's comment on a paper by Anacleto (2011 Eur. J. Phys.32 279). Bizarro claims that (i) Anacleto's approach is either incomplete or incorrect; (ii) one problem is the definition of dissipative work; and (iii) additional ambiguities and misconceptions may stem from his explanations. We contend that (i) both authors present exactly the same definition of dissipative work; and (ii) it is possible to obtain a more general expression to evaluate the entropy change that comprises the expressions developed by both authors—indicating that Anacleto's approach is correct and coherent, and that the criticism of the paper is therefore unfounded.

In an analysis of dissipative work in thermodynamics (2011 Eur. J. Phys.32 37), the irreversible adiabatic processes corresponding to the so-called fire piston and fog bottle are treated numerically, it being claimed that an analytical approach is not possible. It is shown that such a claim is erroneous and that analytical solutions, against which the previous numerical results can be gauged, do exist and are provided here for all thermodynamical quantities pertaining to those two processes. In addition, misconceptions arising therein related to entropy calculations and the use of auxiliary reversible processes and also to the behaviour of heat and work under a system–surroundings interchange are identified and corrected.

We address Bizarro's comment on our paper (2011 Eur. J. Phys.32 37). It is argued that while Bizarro's analytical solution for the adiabatic irreversible processes in our paper is pedagogically relevant, his criticism regarding the behaviour of heat and work for a system–surroundings interchange is unfounded. Some of Bizarro's other statements regarding our paper are also rebutted.

We make some observations regarding a recent paper by Anacleto and Ferreira (2011 Symmetry of the adiabatic condition in the piston problem Eur. J. Phys.32 1625).

We respond to remarks made by Guerra and Abreu (2013 Eur. J. Phys.34 L35) regarding our paper (2011 Eur. J. Phys.32 1625).

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