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In this paper, the analytical and numerical solutions of a non-steady state mathematical model are developed and analyzed. The mathematical model development of a non-steady state heat transport for a one-dimensional system is shown, and the analytical solution of the model is presented. The numerical solution of the model, using the finite element method (FEM), is compared to its analytical solution, proving its consistency. One of the advantages of using numerical tools is that more complex solutions can be obtained, even if the corresponding analytical solution does not exist or is not known, which is useful for engineering students. To demonstrate the applications and possibilities of this work, it is shown that changing the boundary conditions, geometry, or dimension in the system and the mathematical model, it can be solved through a numerical solution method. This is easier and more comprehensive for students rather than facing the complexity of the analytical solutions. The paper shows that it is possible to use the FEM in a university teaching context to complementarily explain the underlying physical phenomena of an engineering problem, here applied to a heat transfer problem in a vegetable.

The mechanically controllable break junction (MCBJ) experimental setup is one of the main techniques employed in the study of electronic transport properties at the atomic and molecular scales. This work presents the construction of an inexpensive and simple but robust setup that shows the emergence of conductance quantization as a macroscopic gold wire is pulled to atomic dimensions. The homemade device is based on the MCBJ principle and allows repeatedly forming and breaking the metallic contact to perform a statistical analysis of the data extracting the most frequent electron transport structure. The histogram built from conductance measurements, at room temperature in air, show that the quality of the MCBJ equipment developed here is comparable to that of more sophisticated devices used in research laboratories. It is able to resolve up to three conductance peaks associated with gold nanowires reported in the literature. Such an experiment is suggested to be implemented as a powerful pedagogical tool in modern undergraduate physics labs. We are grateful for all the support of Generalitat Valenciana through PROMETEO2017/139 and GENT (CDEIGENT2018/028). This work was supported by the CEPRA Grant XII-2018-06 ‘Mechanical Spectroscopy’.

This paper makes use of a smartphone's ambient light sensor to analyse a system of two coupled springs undergoing either simple or damped oscillatory motion. The period, frequency and stiffness of the spring, together with the damping constant and extinction time, are extracted from light intensity curves obtained using a free Android application. The results demonstrate the instructional value of mobile phone sensors as a tool in the physics laboratory., The authors would like to thank the Institute of Education Sciences, Universitat Politecnica de Valencia (Spain) for the support of the Teaching Innovation Groups, e-MACAFI and MoMa.

In this paper, smartphone acceleration sensors were used to perform a quantitative analysis of mechanical coupled oscillations. Symmetric and asymmetric normal modes were studied separately in the first two experiments. In the third, a coupled oscillation was studied as a combination of the normal modes. Results indicate that acceleration sensors of smartphones, which are very familiar to students, represent valuable measurement instruments for introductory and first-year physics courses., The authors would like to thank the Institute of Education Sciences, Universitat Politecnica de Valencia (Spain), for the support of the Teaching Innovation Group, MoMa. We would also like to thank Dr Juan Angel Sans for his contribution in the measurement of the acceleration using his smartphone and Dr Michael Devereux for kindly revising the manuscript as a native English speaker.

An experimental apparatus was designed and built to allow students to carry out heat conduction experiments in hollow cylinders made of different materials, as well as to determine the thermal conductivity of these materials. The evolution of the temperature difference between the inner and outer walls of the cylinder as a function of time is analysed, and when the process reaches the steady state regime the thermal conductivity can be easily calculated. Several materials such as wood, plastic and metals are considered and the values of their thermal conductivities, obtained experimentally, are compared with those given in the reference list. This work was supported by the ‘Vicerrectorado de Tecnología e Innovación Educativa’ of the University of Alicante, Spain (GITE-09006-UA) and by the Generalitat Valenciana of Spain under project PROMETEO/2011/021.

[EN] The experimental analysis of three-dimensional (3D) movement is of great importance to introduce students to knowledge of the actual physical world. However, the difficulties of collecting 3Ddatamake it little suitable for beginner physics students. In this work, we propose an experiment to analyse 3D motion in stationary and rotating systems. A projectile is launched from a rotating platform. The tracking of the projectile is carried out with the assistance of a stereoscopic camera and simplified object tracking techniques. The experiment can be modelled through the well-known transformation equations between an inertial reference system and a non-inertial one with constant angular velocity. The simplicity of the method and the agreement between the experimental data and the theoretical model make the experiment accessible to undergraduate physics students. In this way, they can easily measure a 3D movement and verify its theoretical equations. © 2011 IOP Publishing Ltd.

A closed-form approximate expression for the solution of a simple pendulum in terms of elementary functions is obtained. To do this, the exact expression for the maximum tension of the string of the pendulum is first considered and a trial approximate solution depending on some parameters is used, which is substituted in the tension equation. We obtain the parameters for the approximate by means of a term-by-term comparison of the power series expansion for the approximate maximum tension with the corresponding series for the exact one. We believe that this letter may be a suitable and fruitful exercise for teaching and better understanding nonlinear oscillations of a simple pendulum in undergraduate courses on classical mechanics. This work was supported by the “Vicerrectorado de Tecnología e Innovación Educativa” of the University of Alicante, Spain (GITE-09006-UA).

The authors thank to Claudio G. Carvalhaes for send us their papers about the nonlinear pendulum and the arithmetic-geometric mean. In their comment, Qing-Yin and Pei derived another analytical approximate expression for the large-angle pendulum period, which they compare with other expressions previously published. Most of these approximate formulas are based on the approximation of the original nonlinear differential equation for the simple pendulum motion. However, we point out that it is possible another procedure to obtain an approximate expression for the period. This procedure is based on the approximation of the exact period formula —which is expressed in terms of a complete elliptic integral of the first kind— instead of the approximation of the original differential equation. This last procedure is used, for example, by Carvalhaes and Suppes using the arithmetic-geometric mean. This work was supported by the “Vicerrectorado de Tecnología e Innovación Educativa” of the University of Alicante, Spain (GITE-09006-UA).

A linearization method of the nonlinear differential equation for conservative nonlinear oscillators is analyzed and discussed. This scheme is based on the Chebyshev series expansion of the restoring force which allows us to obtain a frequency-amplitude relation which is valid not only for small but also for large amplitudes and, sometimes, for the complete range of oscillation amplitudes. Some conservative nonlinear oscillators are analyzed to illustrate the usefulness and effectiveness of the technique.

The classical problem of the deflection of a cantilever beam of linear elastic material, under the action of an external vertical concentrated load at the free end, is analysed. We present the differential equation governing the behaviour of this physical system and show that this equation, although straightforward in appearance, is in fact rather difficult to solve due to the presence of a non-linear term. In this sense, this system is similar to another well known physical system: the simple pendulum. An approximation of the behaviour of a cantilever beam for small deflections was obtained from the equation for large deflections, and we present various numerical results for both cases. Finally, we compare the theoretical results with the experimental results obtained in the laboratory.

This is a reply to the comment by Iqbal and Anwar on our recently published work. First of all, the authors of ‘Measurement of the magnetic field of small magnets with a smartphone: a very economical laboratory practice for introductory physics courses’ greatly appreciate the comments of Iqbal and Anwar. One of us (AB) is very grateful to the Vicerectorship of Information Technologies of the University of Alicante (Spain) the help of GITE-09006-UA and to the Generalitat Valenciana (Spain), project PROMETEOII/2015/015. IE is grateful to the Ministry of Economy and Competitiveness the help provided through the project DPI2012-32994.

In this work, we propose an inexpensive laboratory practice for an introductory physics course laboratory for any grade of science and engineering study. This practice was very well received by our students, where a smartphone (iOS, Android, or Windows) is used together with mini magnets (similar to those used on refrigerator doors), a 20 cm long school rule, a paper, and a free application (app) that needs to be downloaded and installed that measures magnetic fields using the smartphone's magnetic field sensor or magnetometer. The apps we have used are: Magnetometer (iOS), Magnetometer Metal Detector, and Physics Toolbox Magnetometer (Android). Nothing else is needed. Cost of this practice: free. The main purpose of the practice is that students determine the dependence of the component x of the magnetic field produced by different magnets (including ring magnets and sphere magnets). We obtained that the dependency of the magnetic field with the distance is of the form x-3, in total agreement with the theoretical analysis. The secondary objective is to apply the technique of least squares fit to obtain this exponent and the magnetic moment of the magnets, with the corresponding absolute error. AB thanks the Vicerectorship of Information Technologies of the University of Alicante (Spain) and the help of GITE-09006-UA. IE is grateful to the Ministry of Economy and Competitiveness for help provided through the project DPI2012-32994.

In this work we propose using phase diagrams to explain the dynamical behaviour of simple mechanical systems. First the motion of the system x (t) is experimentally measured, and then the derivatives, v(t) and a(t), are obtained from it and the motion equation f (x, v, a) = 0 is represented graphically. This idea is applied to the study of a system with linear viscous drag, explaining the evolution of the system towards the dynamical equilibrium point corresponding to the limit velocity. The phase diagrams of the viscous drag are compared with those of the Coulomb drag, which is not continuous and does not necessarily lead to a uniformly accelerated motion. The method is illustrated by an experiment in a dynamic track with magnetic damping. The use of phase diagrams allows for the checking the linearity of this damping. Moreover it allows for the identification of the existence of a small Coulomb drag between the track and the cart that appears as a small discontinuity of the function a (v) when the direction of the movement changes.

We re-derive a general procedure to substitute any rigid body by an equivalent system of exactly four masses, located at the vertices of an irregular tetrahedron.

In this work, the change of frequency of a sound wave due to the Doppler effect has been measured using a smartphone. For this purpose, a speaker at rest and a smartphone placed on a cart on an air track were used. The change in frequency was measured by using an application for AndroidTM, `Frequency Analyzer¿, which was developed by us speci¿cally for this work. This made it possible to analyze four types of mechanical motions: uniform linear motion, uniform accelerated linear motion, harmonic oscillations and damped harmonic oscillations. These experiments are suitable for undergraduate students. The main novelty of this work was the possibility of measuring the instantaneous frequency as a function of time with high precision. The results were compared with alternative measurements yielding good agreement., The authors would like to thank the Institute of Education Sciences, Universitat Politecnica de Valencia (Spain) for the support of the Teaching Innovation Groups, e-MACAFI and MoMa.

In this paper, we present a simple experiment to introduce the nonlinear behaviour of oscillating systems in the undergraduate physics laboratory. The transverse oscillations of a spring allow reproduction of three totally different scenarios: linear oscillations, nonlinear oscillations reducible to linear for small displacements, and intrinsically nonlinear oscillations. The chosen approach consists of measuring the displacements using video photogrammetry and computing the velocities and the accelerations by means of a numerical differentiation algorithm. In this way, one can directly check the differential equation of the motion without having to integrate it, or perform an experimental study of the potential energy in each of the analysed scenarios. This experiment allows first year students to reflect on the consequences and the limits of the linearity assumption for small displacements that is so often made in technical studies.