Table of contents

The 'twin paradox' of special relativity offers the possibility of making interstellar flights within a lifetime. For very long journeys with velocities close to the speed of light, however, we have to take into account the expansion of the universe. Inspired by the work of Rindler on hyperbolic motion in curved spacetime, we study the worldline of a uniformly accelerated observer in de Sitter spacetime and the communication between the travelling observer and an observer at rest. This paper is intended to give graduate students who are familiar with special relativity and have some basic experience of general relativity a deeper insight into accelerated motion in general relativity, into the relationship between the proper times of different observers and the propagation of light signals between them, and into the use of compactification to describe the global structure of a relativistic model.

We discuss carefully the blackbody approximation, stressing what it is (a limit case of radiative transfer), and what it is not (the assumption that the body is perfectly absorbing, namely black). Furthermore, we derive the Planck spectrum without enclosing the field in a box, as is done in most textbooks. Although convenient, this trick conceals the nature of the idealization expressed in the concept of a blackbody: first, the most obvious examples of approximate blackbodies, stars, are definitely not enclosed in boxes; second, the Planck spectrum is continuous, while the stationary modes of radiation in a box are discrete. Our derivation, although technically less elementary, is conceptually more consistent, and brings the opportunity to introduce to advanced undergraduate and graduate students the important concept of the local density of states via the resolvent formalism.

It is shown that the well-known linear variation of p–n diode terminal voltage with temperature at different fixed forward currents allows easy and accurate determination of the semiconductor ideality factor and bandgap from only two data points. This is possible if the temperature difference required to maintain the same diode voltage drop can be measured. The results for silicon and germanium bandgap energy using this approach are in excellent agreement with the literature. The method therefore provides a fresh and original insight into the derivation of the bandgap using data from a popular experiment. The experiment is suitable for undergraduate laboratories on semiconductors.

We show that in the grounded conducting sphere image problem, all the necessary information about the image charge can be found from a mirror equation and a magnification formula. Then, we propose a method to solve the image problem for an extended charge distribution near a grounded conducting sphere.

The rotational dynamics was studied from the point of view of Rodrigues' vector. This vector is defined here by its connection with other forms of parametrization of the rotation matrix. The rotation matrix was expressed in terms of this vector. The angular velocity was computed using the components of Rodrigues' vector as coordinates. It appears to be a fundamental matrix that is used to express the components of the angular velocity, the rotation matrix and the angular momentum vector. The Hamiltonian formalism of rotational dynamics in terms of this vector uses the same matrix. The quantization of the rotational dynamics is performed with simple rules if one uses Rodrigues' vector and similar formal expressions for the quantum operators that mimic the Hamiltonian classical dynamics.

A description is given of a series of recent experiments using a rotating magnetic circuit comprising a permanent magnet ring and yoke, and a stationary conductor in the air gap between the ring and yoke. The EMF induced in this case cannot be described by a simple application of Faraday's flux law. This is because the magnetic flux in the air gap and the area of the gap both remain constant. The experimental results are best explained by the fact that the magnetic field itself rotates with the rotating magnet. This is controversial in the scientific and educational literature, as shown by citations from various authors (e.g. Feynman, Tamm and Landau all disagree, and with each other). However, these experiments, which may be readily reproduced, do in fact settle the question.

We propose a simple and fascinating experiment for studying diffusion in gels using a pH-sensitive dye. By doping agar with methyl red, we obtain a gel which rapidly reacts to changes in pH by changing its absorption spectrum. The pH gradients can be followed using a digital camera, and we demonstrate here that the pH-sensitive colour changes can be used to print colour patterns in the gel which due to diffusion of ions may disappear entirely.

Here, we present complex resonance states (or Siegert states) that describe the tunnelling decay of a trapped quantum particle from an intuitive point of view that naturally leads to the easily applicable Siegert approximation method. This can be used for analytical and numerical calculations of complex resonances of both the linear and nonlinear Schrödinger equations. This approach thus complements other treatments of the subject that mostly focus on methods based on continuation in the complex plane or on semiclassical approximations.

Many experimenters, starting with Ernst Mach in 1883, have reported that if a device alternately sucks in and then expels a surrounding fluid, it moves in the same direction as if it only expelled fluid. This surprising phenomenon, which we call Machian propulsion, is explained by conservation of momentum: the outflow efficiently transfers momentum away from the device and into the surrounding medium, while the inflow can do so only by viscous diffusion. However, many previous theoretical discussions have focused instead on the difference in the shapes of the outflow and the inflow. Whereas the argument based on conservation is straightforward and complete, the analysis of the shapes of the flows is more subtle and requires conservation in the first place. Our discussion covers three devices that have usually been treated separately: the reverse sprinkler (also called the inverse, or Feynman sprinkler), the putt–putt boat, and the aspirating cantilever. We then briefly mention some applications of Machian propulsion, ranging from microengineering to astrophysics.

In this paper, we present a laboratory activity in computed tomography (CT) primarily composed of a photogate and a rotary motion sensor that can be assembled quickly and partially automates data collection and analysis. We use an enclosure made with a light filter that is largely opaque in the visible spectrum but mostly transparent to the near IR light of the photogate (880 nm) to scan objects hidden from the human eye. This experiment effectively conveys how an image is formed during a CT scan and highlights the important physical and imaging concepts behind CT such as electromagnetic radiation, the interaction of light and matter, artefacts and windowing. Like our setup, previous undergraduate level laboratory activities which teach the basics of CT have also utilized light sources rather than x-rays; however, they required a more extensive setup and used devices not always easily found in undergraduate laboratories. Our setup is easily implemented with equipment found in many teaching laboratories.

A chain assumes the well-known shape known as a catenary when it hangs loosely from two points in a gravitational field. The correct solution of the catenary was one of the early triumphs of the newly invented calculus of variations at the end of the 17th century. Here we revisit the catenary and show that, for a chain hanging from a horizontal rod, three new and distinct configurations are possible if a soap film covers the area bounded by the chain and the rod. We first review the general problem and discuss the conditions under which the chain assumes a concave, triangular or convex configuration. The deciding factor is the strength of surface tension relative to the gravitational force per unit length of the chain. The conditions under which the chain assumes the shape of a perfect triangle are discussed in greater detail and analysed to obtain the tension along the chain. The triangular configuration is especially intriguing to undergraduates and may be used as a simple experiment to obtain the surface tension of the soap solution by measuring just one angle of the triangle.

We report a simple experiment that clearly demonstrates a common error in the explanation of the classic experiment where a small piece of paper is put over a book and the system is let fall. This classic demonstration is used in introductory physics courses to show that after eliminating the friction force with the air, the piece of paper falls with acceleration g. To test if the paper falls behind the book in a nearly free fall motion or if it is dragged by the book, we designed a version of this experiment that includes a ball and a piece of paper over a book that is forced to fall using elastic cords. We recorded a video of our experiment using a high-speed video camera at 300 frames per second that shows that the book and the paper fall faster than the ball, which falls well behind the book with an acceleration approximately equal to g. Our experiment shows that the piece of paper is dragged behind the book and therefore the paper and book demonstration should not be used to show that all objects fall with acceleration g independently of their mass.

In this paper, coupled pendulums with different lengths are studied. Through steel magnets, each pendulum is coupled with others, and a stepping motor is used to drive the whole system. To record the data automatically, we designed a data acquisition system with a CCD camera connected to a computer. The coupled system shows in-phase, locked-phase and anti-phase synchronizations when the driving frequency and the coupling strength are changed. With background knowledge from general physics and the simplicity of the equipment, this experiment is easy to implement and would be of interest to undergraduate students.

Much effort over many years has been devoted to the reform of the teaching of physics. This has led to many new and imaginative approaches in the content and delivery of material. Great strides have been made in the delivery, and the content has been continually supplemented. However, attempts to modernize the basic structure of the presentation have faced resistance, and the majority of introductory physics textbooks in wide adoption today have a general structure that has changed little in over 60 years. Thus, in comparison to biology, chemistry, geology, etc, physics is unique in that its introductory course is not a survey of the current status of the field. In an attempt to circumvent this problem in a tractable way, we have developed a qualitative front-end course designed to create a 21st century perspective that can be embedded into the beginning of a standard introductory physics sequence.

The Schrödinger equation for the ground state of a hydrogen atom confined at the centre of an impenetrable cavity is treated using variational perturbation theory. Energies calculated from variational perturbation theory are comparable in accuracy to the results from a direct numerical solution. The goal of this exercise is to introduce the student to the effects of confinement on atomic systems using a tractable problem from which insight into variational perturbation theory may be gained.

Educational versions of Millikan's oil-drop experiment have frequently been criticized; suggestions for improvement either focus on technical innovations of the setup or on replacing the experiment by other approaches of familiarization, such as computer simulations. In our approach, we have analysed experimental procedures. In doing so, we were able to identify several sources of error and took measures to minimize their influence. At the same time, we attempted to minimize the standard deviation of each individual series of measurements. Our paper describes how we developed criteria which helped to stabilize the data produced in the following series of measurements. The final series of measurements results in data which demonstrate the atomic structure of electricity and enable a demonstration of the elementary charge.

Riding a bicycle on the newest form of indoor training, rollers, presents a unique experiment on bicycle stability. The stability factors eliminated by riding on rollers are discussed in terms of refined handling and control of the centre of mass on a bicycle. This paper is intended for undergraduate physics majors as well as any other general readership interested in the dynamics of bicycle stability.

An approximate scheme for obtaining the period of a simple pendulum for large-amplitude oscillations is analysed and discussed. When students express the exact frequency or the period of a simple pendulum as a function of the oscillation amplitude, and they are told to expand this function in a Taylor series, they always do so using the oscillation amplitude as the variable, without considering that if they change the variable (in this paper to the new variable m), a different Taylor series expansion may be performed which is in addition more accurate than previously published ones. Students tend to believe that there is one and only one way of performing a Taylor series expansion of a specific function. The approximate analytical formula for the period is obtained by means of a Taylor expansion of the exact frequency taking into account the Kidd–Fogg formula for the period. This approach based on the Taylor expansion of the frequency about a suitable value converges quickly even for large amplitudes. We believe that this method may be very useful for teaching undergraduate courses on classical mechanics and helping students understand nonlinear oscillations of a simple pendulum.

Gamow's explanation of the exponential decay law uses complex 'eigenvalues' and exponentially growing 'eigenfunctions'. This raises the question, how Gamow's description fits into the quantum mechanical description of nature, which is based on real eigenvalues and square integrable wavefunctions. Observing that the time evolution of any wavefunction is given by its expansion in generalized eigenfunctions, we shall answer this question in the most straightforward manner, which at the same time is accessible to graduate students and specialists. Moreover, the presentation can well be used in physics lectures to students.

The Machian effect of distant masses of the universe in the frame of reference of the rotating Earth is demonstrated using the gravitomagnetic approach of general relativity. This effect appears in the form of a gravitomagnetic Lorentz force acting on moving bodies on the Earth. The gravitomagnetic field of the universe—deduced from a simple model—exerts a gravitomagnetic Lorentz force on moving bodies, a force parallel to and with comparable strength to the Coriolis force observed on the rotating Earth. It seems after simple considerations that the Coriolis force happens to be the gravitomagnetic Lorentz force exerted by the mass of a black hole universe. The description of the phenomenon is simpler using the gravitomagnetic approach than the standard formulation of general relativity, so the method relying on gravitomagnetism is advisable in lectures intended for master's degree level physics students and advanced undergraduates.

We present a derivation of the paraxial geometrical laws starting from a wave-optics approach, in particular by using simple continuity conditions of paraxial spherical waves at boundaries (discontinuities) between optical media. Paraxial geometrical imaging and magnification laws, under refraction and reflection at boundaries, are derived for several instructive cases and without using Fresnel diffraction theory. The primary aim is to provide a complementary insight into the standard axiomatic approach of paraxial geometrical optics and likewise to allow the introduction of some wave imaging concepts, such as the transmittance function, with a notable didactic interest for advanced subjects such as Fourier optics. This approach provides a more homogeneous vision of classical optics in which the use of the optical field continuity conditions at a boundary is a usual requirement as is clearly seen, for example, in the case of the derivation of Fresnel formulas. The work is particularly intended for university physics teachers and pregraduate and first year postgraduate students.

Is it possible to compare approximately inertial frames in the inertial property? If this is the case, the inertial property becomes a measurable quantity. We give a positive answer to this question, and discuss the general principle of design of devices for making the required measurements. This paper is intended for advanced undergraduate and graduate students in high energy physics and relativity. Our aim is twofold: (i) to provide a deeper insight into the essentials of classical dynamics, and (ii) to give impetus to ingenious young people to devise new clever, useful and highly sensitive tools for measuring the inertial property following the pattern outlined in the present discussion.

We use light deflection by a Coulomb field, due to nonlinear quantum electrodynamics effects, as an opportunity for a pedagogical discussion of the electrodynamical analogue of the Aichelburg–Sexl ultraboost.

Frictional losses are experimentally determined for a uniform circular disc exhibiting rotational motion. The clockwise and anticlockwise rotations of the disc, that result when a hanger tied to a thread is released from a certain height, give rise to vertical oscillations of the hanger as the thread winds and unwinds over a pulley attached to the disc. It is thus observed how the maximum height is achieved by the hanger decrements in every bounce. From the decrements, the rotational frictional losses are measured. The precision is enhanced by correlating vertical motion with the angular motion. This method leads to a substantial improvement in precision. Furthermore, the frictional torque is shown to be proportional to the angular speed. The experiment has been successfully employed in the undergraduate lab setting.

We derive the energy levels associated with the even-parity wavefunctions of the harmonic oscillator with an additional delta-function potential at the origin. Our results bring to the attention of students a non-trivial and analytical example of a modification of the usual harmonic oscillator potential, with emphasis on the modification of the boundary conditions at the origin. This problem calls the attention of the students to an inaccurate statement in quantum mechanics textbooks often found in the context of the solution of the harmonic oscillator problem.

We introduce a modification to the matching Ronchi test to visualize lens aberrations with simple and inexpensive equipment available in educational optics labs. This method can help instructors and students to observe and estimate lens aberrations in real time. It is also a semi-quantitative tool for primary tests in research labs. In this work by comparing a single lens with a doublet, we can clearly demonstrate the superior quality of the doublet over the single lens, and estimate their conic constants.

We describe a simple dynamical model of a one-dimensional ideal gas and use computer simulations of the model to illustrate two fundamental results of kinetic theory: the Boltzmann transport equation and the Boltzmann H-theorem. Although the model is time-reversal invariant, both results predict that the behaviour of the gas is time-asymmetric. We show that the assumption of molecular chaos is a necessary condition, but not a sufficient condition, for such time-asymmetric results to correctly describe the model, and we use computer simulations to investigate the conditions under which the assumption of molecular chaos holds.

Kepler's laws of planetary motion are acknowledged as highly significant to the construction of universal gravitation. This paper demonstrates different ways to derive the law of equal areas for the Earth by general geometrical and trigonometric methods, which are much simpler than the original derivation depicted by Kepler. The established law of equal area for the Earth was applied to analyse the angular velocity or the reciprocal of the distance—for the Earth's orbit around the Sun—and can be defined as a periodic function by analysing the available data, which help explain the law of ellipses for the Earth.

We give a brief description of how Maxwell's equations are expressed in the language of differential forms and use this to provide an elegant demonstration of how the method of images (well known in electrostatics) also works for electrodynamics in the presence of an infinite plane conducting boundary.

Learning in the laboratory is different from learning in other contexts because students have to engage with various aspects of the practice of science. They have to use many skills and knowledge in parallel—not only to understand the concepts of physics but also to use the tools and analyse the data. The question arises, how to best guide students' learning in the laboratory. This study is about creating and using questions with a specifically designed framework to aid learning in the laboratory. The concepts, tools and techniques questioning (CTTQ) method was initially designed and used at Mahidol University, Thailand, and was subsequently extended to laboratory pre-work at the University of Sydney. The CTTQ method was implemented in Sydney with 190 first-year students. Three pre-work exercises on a series of electrical experiments were created based on the CTTQ method. The pre-works were completed individually and submitted before the experiment started. Analysed pre-work, surveys and interviews were used to evaluate the pre-work questions in this study. The results indicated that the CTTQ method was successful and the flow in the experiments was better than that in the previous year. At the same time students had difficulty with the last experiment in the sequence and with techniques.

We have recently demonstrated that, when investigating the internal energy and pressure of an ideal boson gas at temperatures below the Bose–Einstein temperature, it is necessary to include the contribution from the kinetic energy of the particles in the condensate. Here we express the same requirement in terms of the constraints of the Heisenberg uncertainty principle applied to the particles confined in the condensate. The problem may be used by undergraduate students as an additional example of the connection between the ground state energy of a system and the Heisenberg uncertainty principle.

An analytic solution can be derived for the angles of two mutually repelling charged pith balls of unequal mass hanging from strings from a common point of attachment. Just as in the equal-mass case, a cubic equation is found for the square of the sine of either angle, and an approximation can be used to avoid Cardano's formula for small angles. These results extend a standard problem treated in introductory undergraduate courses in electricity and magnetism.

A vibrating hanging tapered string (chord, chain, cable) is studied both analytically and numerically. The proper boundary condition for a tip-mass-less string is derived. It is found that the frequencies depend heavily on the taper, tip mass and the shape of the cross section.

Ten years ago, a book with a title like this would be interesting only to a narrow circle of specialists. Thanks to rapid advances in technology, the price of thermal imaging devices has dropped sharply, so they have, almost overnight, become accessible to a wide range of users. As the authors point out in the preface, the growth of this area has led to a paradoxical situation: now there are probably more infrared (IR) cameras sold worldwide than there are people who understand the basic physics behind them and know how to correctly interpret the colourful images that are obtained with these devices. My experience confirms this. When I started using the IR camera during lectures on the didactics of physics, I soon realized that I needed more knowledge, which I later found in this book.

A wide range of potential readers and topical areas provides a good motive for writing a book such as this one, but it also represents a major challenge for authors, as compromises in the style of writing and choice of topics are required. The authors of this book have successfully achieved this, and indeed done an excellent job. This book addresses a wide range of readers, from engineers, technicians, and physics and science teachers in schools and universities, to researchers and specialists who are professionally active in the field. As technology in this area has made great progress in recent times, this book is also a valuable guide for those who opt to purchase an infrared camera. Chapters in this book could be divided into three areas: the fundamentals of IR thermal imaging and related physics (two chapters); IR imaging systems and methods (two chapters) and applications, including six chapters on pedagogical applications; IR imaging of buildings and infrastructure, industrial applications, microsystems, selected topics in research and industry, and selected applications from other fields. All chapters contain numerous colour pictures and diagrams, and a rich list of relevant literature.

Let's devote a few more words to the section on pedagogical applications. It is the usual perception that the use of IR cameras for educational purposes is limited primarily to help visualize processes in thermodynamics such as heat conduction, evaporation, radiation and convection. In this book the authors show that the range of pedagogical applications of IR cameras is much wider. They describe concrete examples (from the descriptions it is clear that the authors have performed all experiments themselves) from mechanics (friction, inelastic collisions), electromagnetism (eddy currents, thermoelectric effect, analysis of standing waves in the microwave oven), optics (specular and diffuse reflection, wave optics in the IR region) and modern physics (selective absorption in gases). Readers who may want to repeat the experiments will appreciate the colour IR photos that are equipped with temperature scales from which one may learn which settings to use in order to achieve the best visibility of the phenomena to be observed.

As said earlier, the decision to write a book for a wide range of readers requires authors to make certain compromises. The inclusion of interpretations and explanations at a basic level will certainly be welcomed by some readers, but due to the limited space some simplifications of this type of content were inevitable. Readers who might be put off by these simplifications should bear in mind that there are few authors who describe specialized topics such as this one and devote so much space to fundamentals. One can only wish that future authors of similar books will try to meet the standards set by this one.

Seven Tales of the Pendulum is a kind of 'biography' of a unique physical system. We all think we know everything about the pendulum. It is indeed the elementary example in countless textbooks; it is so simple and familiar; it is such an old topic. If this is your opinion, or if you think that the movement of the pendulum is dull and boring, then you should read the book of Gregory Baker; it will come as a surprise.

Inside, you will read about the various avatars of the pendulum in science, comprehending how, down the years, the complexity of the notion of pendulum behaviour developed and deepened, from regularity in classical physics to chaotic motion, and eventually to the probabilistic attitude in quantum physics. You will learn why the pendulum has often been a key element in the history of human intelligence development, and how the concept of the pendulum in science uncovered routes for novel breakthroughs.

Using everything but dry mathematics, Baker uses narrative and figures well to give a vivid review of the physics of the pendulum in the general context of human culture. This is a captivating book, and you will surprise yourself when you ask, 'what will be written on the next page?'.

See PDF file for details.

See PDF file for details.

It has come to the attention of IOP Publishing that this article should not have been submitted for publication because of its substantial replication of an earlier-published paper by a different author (Kiessling M K-H 2003 The 'Jeans swindle' a true story—mathematically speaking Adv. Appl. Math.31 132). Consequently this paper has been retracted by IOP Publishing.

It has come to the attention of IOP Publishing that this article should not have been submitted for publication because of its substantial verbatim replication of a number of other sources, some of them without citation. Consequently, this paper has been retracted by IOP Publishing.