Classical Mechanics Course Research Articles

Why the action?

In this note, I present two variations of the principle of stationary action in order to provide alternative ways for students to think about the action that they may find more intuitive than the traditional approach. These methods involve Lagrange multiplier techniques and are, therefore, best suited for students who are encountering the stationary action principle for a second or third time, perhaps in an advanced undergraduate mathematical methods course or a graduate classical mechanics course.

Studying the rotational dynamics of a meter stick using the Arduino board and 3D printed parts

We present a low-cost experimental apparatus designed for studying the rotational dynamics of a meter stick. This apparatus uses everyday laboratory supplies, 3D printed parts, and an Arduino board for data acquisition. It consists of a common bearing fixed to a vertical axle, connected to the meter stick through a 3D printed spindle adapter. A thread attached to a hanging weight and winded around the spindle applies a constant external torque to the stick. The moment of inertia of the apparatus may be changed by incorporating weights into the apparatus through 3D printed adapters. An Arduino board records the time interval between consecutive half turns of the stick, allowing the calculation of the angular displacement over time. The data collected from experiments is compared with theoretical models. Among others, the effect of the applied torque, air resistance and moment of inertia of the stick is studied. This experimental apparatus is particularly suitable for undergraduate students enrolled in a classical mechanics course, offering them a practical, hand-on experience while remaining cost-effective.

Some remarks on projectile motion with a linear resistance force

In this article, we revisit projectile motion assuming a retarding force proportional to the velocity, . We obtain an analytical expression for the set of maxima of the trajectories, in Cartesian coordinates, without using the Lambert W function. Also, we investigate the effect of parameter k on the radial distance of the projectile showing that the radial distance oscillates from a certain critical launch angle and find an approximate expression for it. In our analysis, we consider the impact of parameter k in the kinetic energy, the potential energy, the total energy, the rate of energy loss, and the phase space. Our results can be included in an intermediate-level classical mechanics course.

Using a Simple Pendulum to Calculate the Distance Between Two Train Stations

In this article, we demonstrate the use of a simple pendulum to explore the concepts of kinematics and dynamics. A simple homemade pendulum and a phone-based accelerometer are used to determine, at various points in time, the acceleration of a moving train. The dynamical and kinematics data from the homemade pendulum and the accelerator can then be compared, and this in turn highlights the relationship between dynamical and kinematics quantities. This project is originally part of a coursework requirement assigned to the first four authors for their introductory classical mechanics course at a university in the summer of 2020. Similar ideas have been explored earlier independent of this work, but this work will focus on the detailed description of our approach and using updated technologies as our benchmark. The project is one that can be implemented readily as an activity in high school physics classes.

Analysis of Implementation Project-Based Learning Model of Teaching Integrated with Computer Programming in Improving Computational Thinking Skills in a Classical Mechanics Course

This research aims to explore student computational thinking skills in implementing a project-based learning model of teaching integrated with computer programming in classical mechanics course in projectile motion topic. The research design uses one group pretest and post-test design. The computational thinking skills have five indicators: abstraction, generalization, decomposition, algorithm, and debugging. The computational thinking indicator was analyzed from the result of Pretest and post-test scores and a comparison between manual solution and numerical solution from computer programming. The instruments used in this study were tasks, rubrics, and questionnaires. The result shows the average score of the Pretest is 53.05, and the post-test score is 80.22. The student computational thinking skills in algorithm and debugging in Pretest are 29.70% and 24.30% and 59.00%, and 54.00% in the post-test stage. This result indicates the implementation of PBL model of Teaching integrated with computer programming has a significant impact on student computational thinking skills

Three experiments on the Einstein’s lift

We propose a didactic model of Einstein's lift used to show the physics in a freefalling frame of reference. We discuss three experiments in Einstein's lift: the spring-mass system, the two repelling magnets system and the pendulum oscillations in free fall. Measurements are carried out through analysing videos acquired using a digital camera by means of open-access video analysis software which allows to describe the motion in both the inertial and moving frames of reference. We also show some videos edited with the software Adobe After Effects which enable us to see what happens in a freefalling frame of reference, and a video acquired by a GoPro camera falling integral with the lift. The topic, the accessibility of the experiments and the level of analysis are appropriate for an undergraduate classical mechanics course.

Projectile Motion: The “Coming and Going” Phenomenon

During elementary physics courses, many examples in mechanics are studied without considering air resistance effects. We can mention, for example, projectile motion and free fall, among many others. Only brief studies of such systems in classical mechanics courses consider linear air resistance (Fdrag ∝ v), which is the simplest model to account for the drag effect by modeling objects moving in viscous media with small velocities (more precisely, small Reynolds number). Quadratic air resistance (Fdrag ∝ v2), which applies for objects moving in resistive media with high velocities (high Reynolds number), is hardly mentioned, but it can be found in many papers. However, in nature air resistance plays a big role in the motion of bodies and so it must be included when studying any kind of phenomena related to mechanics.

Smartphones and Gravitational Acceleration I: Overview

Geodesy is a very active and essential research discipline in geophysics but it is not a commonly studied subject at the secondary school or junior post-secondary levels. Far more frequently, gravity and gravitational acceleration are discussed, to some extent, in elementary kinematics or classical mechanics courses. This often takes the form of the force acting on a body or bodies due to gravity, or that the acceleration (agrav) of a free-falling body is 9.8(1) m/s2—which implies the setting of the question is at Earth’s surface. While the latter is a reasonable and practical approximation, agrav observed over the surface of Earth varies and is dependent on several factors. These are normally related to elevation and latitude variations caused by Earth’s rotation. Earth’s rotation contributes negatively to agrav at the equator due to the centrifugal force outward and equatorial bulge, which makes the equatorial radius larger than the polar radius. Offsetting this partially is the positive contribution to agrav at the equator caused by the equatorial bulge because of the extra mass comprising the bulge.

Buquoy’s problem in an introductory physics course

The Buquoy’s problem represents an interesting example of a one-dimensional motion in which a uniform thin fibre is pulled upwards from a horizontal plane by a constant vertical force exerted against the homogeneous gravitational field. The problem includes several interesting concepts that are developed within the introductory university physics course, like motion with a variable mass, the use of the effective potential and a solution of a non-linear ODE of the second order, which provides an opportunity for some numerical modelling of the motion. It is a specific example of damped oscillations and according to our experience it can be discussed e.g. within a course in classical mechanics. Our contribution concentrates on the correspondence between a theoretical model and the experimental realization of this problem, which can be demonstrated in a standard classroom. In our set, the upward force is represented by buoyancy acting on a helium filled balloon. The experiment helps to develop some other competences of the students and naturally leads to a little bit more general mathematical model with an equation of motion that includes the constant mass of the helium ball and the dumping resistance force.

Oscillations of a plate made of A4 sheets of paper

An inexpensive rectangular plate made of sheets of A4 paper is made to oscillate around an axis perpendicular to its own plane or parallel to its minor side. The plate is connected to a rotary motion sensor so that its angular position can be measured as a function of time. The data collected from experiments is compared with solutions of the equation of motion of the plate. Among others, the effect of the amplitude on the period of oscillation of the plate and the effect of different types of damping is studied. The experiments reported here are appropriate for undergraduate students taking a course in classical mechanics.

Duality symmetries behind solutions of the classical simple pendulum

Describing the motion of the classical simple pendulum is one of the aims in every undergraduate classical mechanics course. Its analytical solutions are given in terms of elliptic functions, which are doubly periodic functions in the complex plane. The independent variable of the solutions is time and it can be considered either as a real variable or as a purely imaginary one, which introduces a rich symmetry structure in the space of solutions. When solutions are written in terms of the Jacobi elliptic functions this symmetry is codified in the functional form of its modulus, and is described mathematically by the six dimensional coset group Γ=Γ(2) where Γ is the modular group and Γ(2) is its congruence subgroup of second level. A discussion of the physical consequences that this symmetry has on the motions of the simple pendulum is presented in this contribution and it is argued they have similar properties to the ones termed as duality symmetries in other areas of physics, such as field theory and string theory. Thus by studying deeper a very familiar mechanical system, it is possible to get an insight to more abstract physical and mathematical concepts. In particular a single solution of pure imaginary time for all allowed values of the total mechanical energy is given and obtained as the S-dual of a single solution of real time, where S stands for the S generator of the modular group.

A aprendizagem da linguagem física em um curso de introdução à Mecânica Clássica

Este estudo aborda a aprendizagem da linguagem física, segundo os pressupostos estabelecidos por Thomas Kuhn em A Estrutura das Revoluções Científicas e outros escritos. Trata-se de um estudo de caso, onde investigamos a aprendizagem do estudante A5 durante um curso introdutório de Mecânica Clássica. Por meio de seus relatos e resoluções de problemas foi possível concluir que as proposições de Kuhn são confirmadas no que diz respeito ao aprendizado da ciência Física e que estão relacionados aos efeitos e à utilização da linguagem pelo estudante. O principal destaque no processo de aprendizagem de A5 refere-se à expansão de seu léxico ou vocabulário. Com a análise dos dados coletados pudemos constatar que essa expansão se deu durante as aulas por meio de procedimentos didáticos instaurados pelo docente, que se pautava em explicações simples e diretas e na utilização contínua de exemplos. Por conseguinte, com as resoluções dos problemas propostos pelo professor, o estudante A5 demonstrou em suas manifestações e registros os processos de interpretação, principalmente durante o módulo de Lagrange. Fica evidente nesse processo que a Mecânica Lagrangiana, por exemplo, se enquadrou como um novo vocabulário adquirido por A5.

Minimum descent time along a set of connected inclined planes

The time required for a particle to slide frictionlessly down a set of ramps connected end to end can be minimized numerically as a function of the coordinates of the connection points between ramps and compared to the exact cycloidal solution of the brachistochrone problem. It is found that a set of just three joined ramps over a large range of geometrical aspect ratios has a descent time within 5% of the optimal cycloid. The special case where the particle starts and ends at the same height has sufficient symmetry that it can be analyzed analytically using algebra alone. The level of analysis is appropriate to an undergraduate classical mechanics course.

The shape of the surface of a rotating mass of water as a variational problem

Variational methods have a long and remarkable role in theoretical physics. Few of our students when first exposed to them fail to admire their elegance and efficacy in the formulation and solution of physical problems. In this paper we apply the variational approach that leads to the Euler-Lagrange equations to the determination of the shape of the surface of a mass of water that partially fills a cylindrical bucket that rotates with constant angular velocity (Newton's bucket). Here this approach will lead us to the principle of minimization of the effective potential energy associated with the system. The effect of an external pressure on the equilibrium shape is also taken into account and two models, the constant pressure model and the linear model are discussed. The level of the discussion is kept accessible to undergraduates taking an intermediate level course in classical mechanics.

Damping mechanisms of a pendulum

In this paper, we study the damping mechanisms of a pendulum. The originality of our setup is the use of a metal strip suspension and the development of extremely sensitive electric measurements of the pendulum velocity and position. Their sensitivity is absolutely necessary for a reliable measurement of the pendulum damping time constant because this measurement is possible only for very low oscillation amplitudes, when air friction forces quadratic in velocity have a negligible contribution to the observed damping. We have thus carefully studied damping by air friction forces, which is the dominant mechanism for large values of the Reynolds number Re but which is negligible in the Stokes regime, . In this last case, we have found that the dominant damping is due to internal friction in the metal strip, a universal effect called anelasticity, and, for certain frequencies, to resonant coupling to the support of the pendulum. All our measurements are well explained by theory. We believe this paper would be of interest to students in an undergraduate classical mechanics course.

Levitating a strip of paper by blowing over it

It is shown that if you blow vigorously over a curved strip of paper, it levitates into the shape of a catenary. This result quantifies a common classroom demonstration and is a pedagogically useful addition to other studies of catenaries in an intermediate classical mechanics course.

O APRENDIZADO DA SINTAXE DA LINGUAGEM FÍSICA

Vamos discutir neste artigo o processo de aprendizagem de um aluno matriculado em um curso de Introdução à Mecânica Clássica, disciplina ministrada em um curso de pós-graduação em uma universidade do Estado do Paraná, Brasil. Como resultado, apontamos para o fato de que o aluno analisado – formado em Matemática – ainda que tenha participado das aulas, as quais abordaram conteúdos relacionados aos formalismos newtoniano, lagrangeano e hamiltoniano, não conseguiu construir uma aprendizagem significativa da linguagem física, embora tenha sido bem-sucedido na aprendizagem da sintaxe da linguagem física envolvida no curso. Observamos que as dificuldades apresentadas pelo aluno resultaram de: sua falta de referentes e seu baixo conhecimento do vocabulário relacionado ao conteúdo em questão, bem como da falta de subsunçores para esse aprendizado.

Hurricane Balls: A rigid-body-motion project for undergraduates

We discuss a project on rigid-body motion that is appropriate for students in an upper-division course in classical mechanics. We analyze the motion of Hurricane Balls, two spheres that are welded (or glued) together so they act as a single object that can be spun like a top. The steady-state motion consists of purely rotational motion about the center of mass, such that only one ball is in contact with the table as it rolls without slipping. We give a qualitative explanation for why one ball rises into the air, and we theoretically analyze the system using multiple approaches. We also perform a high-speed video analysis to obtain experimental data on how the orientation depends on the spin rate, and find agreement within a few percent of the theory.

Tensor calculus for physics: a concise guide

Understanding tensors is essential for any physics student dealing with phenomena where causes and effects have different directions. A horizontal electric field producing vertical in dielectrics; an unbalanced car wheel wobbling in the vertical plane while spinning about a horizontal axis; an electrostatic field on Earth observed to be a magnetic field by orbiting astronauts-these are some situations where physicists employ tensors. But the true beauty of tensors lies in this fact: When coordinates are transformed from one system to another, tensors change according to the same rules as the coordinates. Tensors, therefore, allow for the convenience of coordinates while also transcending them. This makes tensors the gold standard for expressing physical relationships in physics and geometry. Undergraduate physics majors are typically introduced to tensors in special-case applications. For example, in a classical mechanics course, they meet the inertia tensor, and in electricity and magnetism, they encounter the polarization tensor. However, this piecemeal approach can set students up for misconceptions when they have to learn about tensors in more advanced physics and mathematics studies (e.g., while enrolled in a graduate-level general relativity course or when studying non-Euclidean geometries in a higher mathematics class). Dwight E. Neuenschwander's Tensor Calculus for Physics is a bottom-up approach that emphasizes motivations before providing definitions. Using a clear, step-by-step approach, the book strives to embed the logic of tensors in contexts that demonstrate why that logic is worth pursuing. It is an ideal companion for courses such as mathematical methods of physics, classical mechanics, electricity and magnetism, and relativity.

Magnetic monopoles and dyons revisited: a useful contribution to the study of classical mechanics

Graduate-level physics curricula in many countries around the world, as well as senior-level undergraduate ones in some major institutions, include classical mechanics courses, mostly based on Goldstein’s textbook masterpiece. During the discussion of central force motion, however, the Kepler problem is virtually the only serious application presented. In this paper, we present another problem that is also soluble, namely the interaction of Schwinger’s dual-charged (dyon) particles. While the electromagnetic interaction of magnetic monopoles and electric charges was studied in detail some 40 years ago, we consider that a pedagogical discussion of it from an essentially classical mechanics point of view is a useful contribution for students. Following a path that generalizes Kepler’s problem and Rutherford scattering, we show that they exhibit remarkable properties such as stable non-planar orbits, as well as rainbow and glory scattering, which are not present in the ordinary scattering of two singly charged particles. Moreover, it can be extended further to the relativistic case and to a semi-classical quantization, which can also be included in the class discussion.