Table of contents

A relativistic point charge moving in a Coulomb potential does not admit a conserved Hamilton vector. Despite this fact, a Hamilton-like vector may be developed that proves useful in the derivation and analysis of the particle's orbit.

Momentum is commonly defined as 'mass times velocity'. However, this cannot be a general definition, since it does not include other types of momentum, including macroscopic radiation momentum or the momentum of a photon. Rankine coined a general definition for energy. Is it possible to describe a corresponding definition of momentum?

'There is a part of everything which is unexplored...Even in the smallest thing there is something in it which is unknown. We must find it'.

Gustav Flaubert (1821–1880) quoted by Guy de Maupassant (1850–1893) [144].

We discuss the classical motion of a finite mass spring coupled to two pointlike masses fixed at its ends. A general approach to the problem is presented and some general results are obtained. Examples for which a simple elastic function can be inferred are discussed and the normal modes and normal frequencies obtained. An approximation procedure to the evaluation of the normal frequencies in the case of uniform elastic function and mass density is also discussed.

The horizontal deflection in free fall has been traditionally measured and calculated relative to the bob of a plumb line, with the object being released at the suspension point. A modern experiment could measure the full 3D trajectory, in which case the definition of deflection relative to a plumb line might not be appropriate. A logical choice would be to define the deflection relative to the initial acceleration, i.e. relative to the line of effective gravity at the release point. With this line as the vertical (z) axis, it is shown that the meridional (x) component of the Coriolis force is cancelled by the leading order variation of effective gravity due to a spherically symmetric earth. The meridional projection of the trajectory is thus a parabola, with curvature which is independent of earth's rotation, and which depends only on the deviation of gravity from its spherical earth value. The traditional meridional deflection, calculated relative to the bob of a plumb line, is seen to be largely due to the tilt of the plumb line with respect to effective gravity at the point of release. An experiment is proposed to measure the full free fall trajectory and not just the displacement at the landing point.

This paper describes a laboratory work that permits physics students to determine the moment of inertia for different objects. This experiment is computerized by using the Science Workshop data-acquisition system with the Datastudio software. The experimental results of the moment of inertia were in good agreement with the theoretical value.

A simple treatment of the important old problem of the torque-induced rotation of a spinning symmetrical top is suggested. Our discussion is appropriate for teaching introductory mechanics and general physics to undergraduate students and is free from the difficulties of a traditional approach to the problem. The origin of nutation that accompanies forced precession is explained in detail, with an accent on its relationship to the torque-free precession of a symmetrical body. A small simulation program is developed that visualizes the investigated motion and illustrates its principal features. The program facilitates understanding about the counterintuitive behaviour of a gyroscope on a qualitative level.

In this paper a laboratory experiment is proposed to infer Poiseuille's law. A simple set-up based on two flasks joined by a detachable tube allows one to measure using tubes of different radii and different lengths. One of the flasks is connected to a vacuum pump to control the pressure differential between the tube extremes. The influence on the flow of different radii, lengths, pressures and viscosities can be studied in a didactic way by measuring the flow rate for each of these variables. The experiment can be performed getting together the students in groups, so that each group concentrates on the effect on the flow of a specific variable, leaving the rest fixed. After putting together all these results the law of Poiseuille can be obtained as a very good approach.

Ever since Johann Bernoulli put forward the challenge "Problema novum ad cujus solutionem Mathematice invitantur" in Acta Eruditorum Lipsiae of June, 1696, of finding the minimum time trajectory (the brachistochrone) described by an object moving from one point to another (not directly behind the first one) in a constant uniform gravitational field, many works have been published on this subject, and some books mention it as part of the applications of the Euler–Lagrange formalism. However, we have found only one reference of the problem related to the general inhomogeneous inverse square gravitational field (Supplee and Schmidt 1991 Am. J. Phys.59 467). Even in this reference, the problem is treated for particular initial conditions. In this work, we develop a simplified method to arrive to the equation of the brachistochrone curve for an arbitrary potential and also to the inverse formulation of the problem: what type of potential energy function is associated with a specified brachistochrone curve?

A simple model of an electromagnetic launcher is introduced in this paper. The proposed model is an electrical one and is suitable for undergraduate students. The work consists of establishing an electrical model whose resolution gives the induced current and the speed of the projectile at a given inductor current and material projectile. In this manner by changing the material of the projectile we can deduce which one is the best. Then, a simple experiment illustrating the principles of a coilgun is described and some test results are given. Therefore, students learn electromagnetic phenomena through an electrical model which is simpler to study than a magnetic one. They have to solve a system of differential equations of the second order using numerical method. And finally, their theoretical study of the coilgun leads to an experimental bench.

The resolution concept in connection with the Fabry–Perot interferometer is difficult to understand for undergraduate students enrolled in physical optics courses. The resolution criterion proposed in textbooks for distinguishing equal intensity maxima and the deduction of the resolving power equation is formal and non-intuitive. In this paper, we study the practical meaning of the resolution criterion and resolution power using a computer simulation of a Fabry–Perot interferometer. The light source in the program has two monochromatic components, the wavelength difference being tunable by the user. The student can also adjust other physical parameters so as to obtain different simulation results. By analysing the images and graphics of the simulation, the resolving power concept becomes intuitive and understandable.

Details of a method for obtaining the instantaneous distance from the earth to the moon to within 5% by direct triangulation are described. Line of sight unit vectors are calculated based on azimuth and zenith angle measurements of the moon made simultaneously with ordinary surveying theodolites from two known points on the surface of the earth. These unit vectors and the baseline distance between the two observation points, typically several thousand kilometres apart, are used to compute the earth–moon distance. Recommendations for college and high school physics classes are included.

Several formulations of the correspondence principle, including both the historical and contemporary ones, are presented and critically reviewed.

Determining the speed of a ball as a function of its height after it is launched straight upward in the absence of fluid resistance is a standard problem treated at all levels of introductory physics. Inclusion of drag in the problem is seldom covered even in the introductory course for majors. But, in fact, this problem can be solved via a straightforward application of the work–energy theorem for either linear or quadratic drag, and instructively plotted on graphs of kinetic versus potential energy.

We give, for the first time, the English translation of a manuscript by Ettore Majorana, which probably corresponds to the text for a seminar lecture delivered at the University of Naples in 1938, where he lectured on theoretical physics. Some passages reveal a physical interpretation of quantum mechanics which anticipates for several years the Feynman approach in terms of path integrals, independent of the underlying mathematical formulation.

Relying on the equivalence principle, a first approach of the general theory of relativity is presented using the spacetime metric of an observer with a constant proper acceleration. Within this non-inertial frame, the equation of motion of a freely moving object is studied and the equation of motion of a second accelerated observer with the same proper acceleration is examined. A comparison of the metric of the accelerated observer with the metric due to a gravitational field is also performed.

It is shown in this paper that for pure substances obeying the van der Waals (VdW) equation of state (EOS) or Redlich–Kwong (RK) type equations of state below the critical temperature, it is possible to obtain with the help of the Maxwell equal areas rule the vapour pressure curve, the difference between the molar volume of the vapour and of the liquid, and the difference between the molar entropies of vapour and liquid; each property is obtained in the first step as a dimensionless expression, function of a dimensionless temperature. Any particular property of a particular fluid obeying VdW EOS or one type of RK EOS can be obtained from the corresponding dimensionless expression by multiplying it with a dimension unit which is a function of the particular temperature-dependent parameter a(T) of the EOS considered for the fluid. In the cases of VdW and the original RK (EOS), the dimensionless functions are unique for all the pure fluids that obey the EOS under consideration. A comparison of VdW, RK and RK–Soave (SRK) predictions with experimental vapour pressure data is exemplified for carbon dioxide.

A method is presented which is an extension to negative energies of a spectral integral equation method to solve the Schroedinger equation, developed previously for scattering applications. One important innovation is a re-scaling procedure in order to compensate for the exponential behaviour of the negative energy Green's function. Another is the need to find approximate energy eigenvalues, to serve as starting values for a subsequent iteration procedure. In order to illustrate the new method, the binding energy of the He–He dimer is calculated, using the He–He TTY potential. In view of the small value of the binding energy, the wavefunction has to be calculated to a distance of 3000 au. Two hundred and twenty mesh points were sufficient to obtain an accuracy of three significant figures for the binding energy, and with 320 mesh points the accuracy increased to six significant figures. An application to a potential with two wells, separated by a barrier, is also made.

The connection between entanglement and the speed of quantum evolution (as measured by the time needed to reach an orthogonal state) is discussed in the case of two quantum particles moving in a one-dimensional double well. The aforementioned connection offers an interesting opportunity of discussing the basic features of quantum entanglement within an elementary context, using concepts and methods usually included in university courses of quantum mechanics.

The conservation during elastic collisions of the classical and the relativistic kinetic energy along with its consequences is a study where the use of analogies is the right teaching tool. Accordingly, novel analogies that the classical and the relativistic consequences share are presented as well as a remarkable disanalogy concerning the centre of mass.

An inexpensive Web camera connected to a personal computer with no additional hardware is used to measure position and light intensity. Three experiments: ink diffusion in water, pendulum damped oscillations and light diffraction are described. The measurements are done online and the results are observed on the computer screen in real time. The configuration is simple and cheap, suitable for demonstrations and laboratory exercises.

In this work, we show some diffraction experiments performed with a liquid crystal display (LCD) that shows how useful this device can be to teach and experience diffraction optics and signal processing experiments. The LCD acts as a programmable pixelated diffractive mask. The Fourier spectrum of the image displayed in the LCD is visualized through a simple free propagation diffraction experiment. This optical system allows easy testing of different diffractive elements. As a demonstration we include experimental results with well-known diffractive elements like diffraction gratings or Fresnel lenses, and with more complicated elements like computer-generated holograms.

Light ray paths in an optical fibre with a conic hole at one of its extremes are studied, by means of geometrical optics, in connection with the aim of transferring the maximum amount of radiant energy from a concentrated light source to a solar trap. This particular application is useful in sunlight collecting systems which are able to concentrate light rays in a spot of small dimensions. The geometry of the optical fibre cable is such to allow multiple reflections of light rays on the external surface of the light guide, when its hollow extreme is placed in the proximity of the concentrated light spot. The simplicity of both the structure of the proposed optical system and of the analytic approach followed makes the present work useful in physics education.

In this paper we show how Feynman diagrams, which are used as a tool to implement perturbation theory in quantum field theory, can be very useful also in classical mechanics, provided we introduce also at the classical level concepts such as path integrals and generating functionals.

The absence of any tendency toward rotation in the Trouton–Noble experiment is given a simple explanation.

When Pauli postulated the existence of a new elementary particle, the 'neutron', in the late 1930s, he was introducing an entity to fill the gaps in the physics of the day. There was no direct evidence for such a particle; only the need to account for the missing energy in beta decay processes and to explain the statistics of some nuclei. Thus, Pauli's neutron was called into existence to solve problems in the areas of radioactivity and nuclear physics; but, in time, it would prove useful in many other areas of physics. The aim of this paper is to describe the many uses the neutrino has had in physics since its appearance, uses that have determined the ontological status of this elusive particle. Interestingly, the experimental detection of neutrinos 50 years ago was a minor episode in understanding the nature of such particles. As we shall argue, other moments in the biography of the particle proved more significant in determining the uses of neutrinos and their ontology.

The magnetic field due to a static circular current loop with the origin on the axis of the loop, expressed in terms of spherical harmonics, is found by directly solving the corresponding Maxwell equations for B. The derivation seems to be simpler than that presented in the literature. As a sine qua non, a fresh delta function expansion is proved.

A digital video on forces occurring in rotating frames of reference is presented. It allows students to observe and to analyse the trajectory of one and the same moving ball in both systems, the laboratory system at rest and the rotating system. An application of the video for physics exercises in the first year of physics study is outlined.

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