Classical Mechanics

We describe a new curriculum for the final two years of a B.S. program in Physics. Case studies in the junior year provide concrete examples or Paradigms as pillars to support systematic Capstone lectures in the senior year. In each of nine three-week Paradigms, the junior progresses from a descriptive lower-division understanding to an advanced analysis of a topic defined by phenomenon rather than discipline. Students generally view the new format with favor. They are better at visualization and make important connections among physics disciplines. Independent assessment is ongoing.

Logic which ought to help people to debate more coherently has become a largely useless and even harmful discipline. Paradoxes like Russell's Village Barber are solved and examples of logicians misleading the public as in the Four Card Problem are analysed in depth.

In this work a general multibody system theory is implemented within a bond graph modeling framework. In classical mechanics several procedures exist by which differential equations can be derived of a system of rigid bodies. In the case of large systems these procedures are labor-intensive and consequently error-prone, unless they are computerized. The bond graphs formalism allows for a unified modeling of multidisciplinary physical systems. It is well-suited for a modular modeling approach based on physical principles. The theory of multibody system dynamics in terms of bond graphs modeling is revisited with the purpose of designing a multibond graph library for such systems. Several mechanical systems undergoing large 3-dimensional rotations are numerically solved in order to validate this software library written in 20-sim software.   J z J y J x t z y x z y x J z J y J x

An object accelerates while it falls under the influence of the gravitational force. By using two sensors a precise and automated measurement of the velocity can be obtained. The analysis of these measurements may be insufficient if air resistance is important. We discuss how by increasing the number of sensors we can determine the velocity, terminal velocity, and acceleration due to gravity.

Kepler(supposedly) kicked off modern science with the discovery of Planetary Laws. They are the very first empirical laws of Physics. Newton partially identified the universal nature of Kepler's work and mathematically put (Mass of)Sun at the center of the solar system. He connected Kepler's laws with Galileo's observation of the same rate of fall of all objects(independent of Mass) on Earth surface. Since there was no clear grasp of concepts like Energy conservation and Angular momentum conservation at his time, Newton's analytical method remained a bit more complicated than necessary(where it is applicable) and erroneous in most cases. Perhaps, Newton tried hard to conceal that, all we need to do is apply calculus on Kepler's laws to understand the underlying dynamics. If he really did discover calculus then why was he reluctant to show off the far more easier and superior technique?. A point to be noted is, a rudimentary form of Indian(+Arabic?) calculus/number system started floating in European continent from 1100 AD onwards. In this article we see that the first time derivative of Kepler's I law, along with the application of II law gives the Planetary Energy conservation equation. We can use this as the basis to solve the generalized 3-body problem also. Along the way we see that the term Inertia coined by Galileo to explain the height conserving property of balls rolling down inclined plane has to be properly interpreted as energy. That is, Inertia = Energy. Hence we should replace Newton's I law by Energy conservation principle. We also see that F = ma is not a correct definition, the correct definition should be F = Gradient of Kinetic Energy. In certain cases the two definitions match, most often they do not.

Abstract Summary form only given. In a broadcasting problem, a message is sent from a source to all the other nodes in the network. Blind flooding is a classical mechanism for broadcasting, where each node retransmits received message to all its neighbors. We ...

The probability of vibrational energy exchange in a molecular collision can be calculated using (1) a wave-mechanical treatment using the method of ``distorted waves,'' (2) a semiclassical time-dependent perturbation procedure in which the perturbation energy is obtained as a function of time from the classical collision trajectory, and (3) a purely classical calculation of the energy transferred to a classical vibrator. These methods are reviewed, related, and compared.

In this paper we study scalar perturbations of the metric for nonlinear f(R) models. We consider the Universe at the late stage of its evolution and deep inside the cell of uniformity. We investigate the astrophysical approach in the case of Minkowski spacetime background and two cases in the cosmological approach, the large scalaron mass approximation and the quasi-static approximation, getting explicit expressions for scalar perturbations for both these cases. In the most interesting quasi-static approximation, the scalar perturbation functions depend on both the nonlinearity function f(R) and the scale factor a. Hence, we can study the dynamical behavior of the inhomogeneities (e.g., galaxies and dwarf galaxies) including into consideration their gravitational attraction and the cosmological expansion, and also taking into account the effects of nonlinearity. Our investigation is valid for functions f(R) which have stable de Sitter points in future with respect to the present tim...

A new class of linear multistep methods is proposed for the solution of the equations of motion of certain dynamical systems encountered in celestial mechanics and astrodynamics. These methods are distinguished from the classical predictor-corrector methods in that they permit ‘back-corrections’ of the solution to be made. As the integration advances in time, the numerical solution is corrected or improved at certain points in the past. The enhanced numerical stability of these methods allows the meaningful application of high-order algorithms. Consequently, stepsizes larger than those attainable with the classical methods may be adopted and thus greater over-all efficiency may be realized. The application of these methods to the problem of determining the orbit of an artificial satellite is accomplished and the results are compared with those obtained using classical methods.

In this presentation the general motion of an asymmetrical homogeneous rigid body in a fluid environment is studied. By means of a defined new theorem – The Theorem for changing the rigid body general impulse and new form of Lagrange equations, called Lagrange condensed equations, the differential equations describing the three-dimensional motion of the body in a matrix form is obtained. With a new theorem for the fluid flow of an asymmetrical rigid body, the presence of a destabilizing aerodynamic moment is proved. A new complete matrix of aerodynamic forces is defined. The spherical component of body general motion is counted with Cardan corners. The paper is entirely theoretical. It is the basis on which a second paper on the motion of an ideal rigid ellipsoid in a fluid environment is developed.

A biomechanical study of sprinting is an interesting and attractive practical investigation task for undergraduates and pre-university students who have already done a fair amount of mechanics and calculus. These students can work with authentic real data about sprinting and carry out practical investigations similar to the way sports scientists do research. Student research activities are viable when the students are familiar with tools to collect and work with data from sensors and video recordings, and with modeling tools for comparison of simulation results with experimental results. This article presents a multipurpose system, named coach, which is rather unique in the sense that it offers students and teachers a versatile integrated set of tools for learning, doing, and teaching mathematics and science in a computer-based inquiry approach. Automated tracking of reference points and correction of perspective distortion in video clips, state-of-the-art algorithms of data smoothing and numerical differentiation, and graphical system dynamics based modeling, are some of the built-in techniques that are very suitable for motion analysis. Their implementation in coach and their application in student activities involving mathematical models of running are discussed.

An approach to the catenary along the lines found in [1] which we frequently discuss in lectures on intermediate level Newtonian mechanics is presented. Some aspects of the calculation in particular the detailed solution of the basic equations and the connection between the parabolic approximation to the catenary and the static limit of the the wave equation are also discussed.

Via decrement of mass of planets, we can send the entire planets to far space orbital allocations. We can convert physical matter into energy, it can either get irradiated to outer space, get transmitted back to Earth's potential energy, get used as a self-propellant, or it can get used in a complex model of these systems; it decreases the mass of the planet. By conversion of the matter to energy, Earth will lose some mass that decreases the gravitational fields for Earth; the formulas of the current research are deduced to control the movements of the planets. Celestial bodies, like any other mechanical systems which follow, and are based on, the physical laws of mechanics and dynamical systems, follow these laws. So since the celestial object “A” exerts the “F” force on the celestial object “B”, the celestial object “B” exerts an interactive force “F” on the celestial object “A” also. All celestial objects exert gravitational influences on each other. Scientists believe, once upon a time, the Sun would be much hotter than what it is today. By that point, this high temperature leads to extinction of the entire existence of life on Earth. When the gravitational force changes, a space particle may either get departed from the other particle or come closer to the other particle.

The analytic solution of a body falling an arbitrary distance toward a gravitational source is presented. This problem has apparently received little attention in textbooks. The solution can be extended smoothly through the singularity at the origin to form a class of trajectories that we call singular orbits. Trajectories in phase space have singular branches, yet cycle in a finite time. We calculate the period of oscillation about the singularity confirming that the singular orbits obey Kepler's third law. Viewed as geodesics in a Newtonian 1 + 1 spacetime, the geodesic deviation equation is derived and solved analytically, providing an excellent example of curvature physics in Newtonian spacetime. The results are used to estimate the duration during which a freely falling local frame can be considered inertial. A numerical investigation with damping included shows that a final state is reached in which the particle is confined to the origin, but acquires infinite speed during each pass. We show via some examples, the pedagogical applications of the solution.

Riassunto. Il paranco semplice è spesso trascurato nei corsi di meccanica delle scuole superiori e della stessa università. Nella sua configurazione più generale questa macchina mostra un comportamento complesso e un suo modello matematico è assente nei manuali di fisica. In questo lavoro si ottiene una descrizione soddisfacente utilizzando le leggi della dinamica e il calcolo vettoriale. Le espressioni ma-tematiche trovate sono poste a confronto con i dati sperimentali e brevemente discusse. Abstract. The gun tackle is often neglected in the introductory courses of classical mechanics. In its general configuration this machine shows a non trivial behavior and a mathematical model is missing in physics handbooks. In this paper we obtain a satisfying description by means of the laws of dynamics and the vector algebra. The mathematical expressions one finds are then compared with experimental data and briefly discussed. 1. Introduzione Si chiama paranco una macchina costituita da due o più pulegge (almeno una delle quali fissa) collegate da un filo continuo, utilizzata di solito per sollevare o tirare un carico. Quando le pulegge sono più di due, esse possono essere montate su un unico asse in modo da formare blocchi. Questi ultimi sono poi accoppiati tra loro in modo che (almeno) uno di essi sia fisso e (almeno) un altro si muova insieme al carico. Il paranco semplice è costituito da una sola coppia di blocchi, ciascuno dei quali possiede una sola puleggia e il carico viene sostenuto da due tratti di filo (vedi Figura 1). Se il sistema è tale da non dissipare energia, esso costituisce una macchina vantaggiosa il cui rendimento dipende dal numero delle pulegge utilizzate e dalla loro reciproca disposizione. La teo-ria di questa macchina nella sua forma più semplice (che è quella più o meno esplicitamente pre-sentata nella gran parte dei manuali di fisica) assume che le pulegge e il filo abbiano massa trascu-rabile, che quest'ultimo sia perfettamente flessibile e inestensibile e che non vi siano interazioni di contatto tra esso e le pulegge. In conseguenza di ciò, il filo può scivolare liberamente nelle gole delle carrucole e queste ultime non sono poste in rotazione dal moto del filo Si considerano inoltre trascurabili gli attriti (volventi) negli assi di rotazione delle pulegge e le interazioni tra i diversi corpi e il mezzo in cui sono immersi. Questo modello di paranco semplice si dice solitamente idea-le. Separando le due pulegge del paranco e disponendole in modo tale che i tre tratti di filo siano tra loro paralleli, si ottiene lo schema di corpo libero mostrato nella Figura 2. F è la forza applicata

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.

In this paper, analytical expressions for stress concentration factors in pipes subjected to internal pressure and axial force are derived for a number of design cases based on classical shell theory. The effect of fabrication tolerances in simple butt welds is assessed. Analyses based on classical mechanics are compared with results from axisymmetric finite element analyses for verification of the presented methodology. Stress concentration factors are presented for circumferential butt welds in pipes welded together from pipes with different thicknesses, welds at buckling arrestors, welds at flanged connections in pipelines, and welds at ring stiffeners on the inside and the outside of the pipes. It also includes stress concentration factors at end closures in pipes for gas storage. Larger pipes are fabricated from plates with a longitudinal weld. This fabrication process introduces out-of-roundness in the pipes. The actual out-of-roundness is a function of internal pressure. An analytical expression for the bending stress in the pipe wall due to this out-of-roundness is presented. The derived stress concentration factors can be used together with a hot spot stress S-N curve for calculation of fatigue damage.

Let a ladder of length "l" and mass "m" be sliding against a wall. There is no friction anywhere. Initially, the ladder was at an angle from the ground. At a critical angle, the ladder loses contact with the wall as it slides down. This can be seen clearly as when the ladder falls completely, there is a displacement between the ladder and the wall (they are not in contact). Determine the angle at which the ladder loses its contact with the vertical wall.

We study new Legendre transforms in classical mechanics and investigate some of their general properties. The behaviour of the new functions is analysed under coordinate transformations. When invariance under different kinds of transformations is considered the new formulation is found to be completely equivalent to the usual Lagrangian formulation, recovering well-established results such as conservation of angular momentum. Furthermore, a natural generalization of the Poisson bracket is found to be inherent to the formalism introduced. On the other hand, we find that with a convenient redefinition of the Lagrangian, L = −L, it is possible to establish an exact one-to-one mathematical correspondence between the thermodynamic potentials and the new potentials of classical mechanics.

In this paper a novel biomechanical model is presented. The model is based on de Hanavan model and aims at artificial motor control applications. Body segments are represented as links of a cinematic chain, allowing the application of techniques usually adopted in robotics: the Denavit-Hartenberg notation and the Lagrangean formulation of Classical Mechanics. The reference frames attached to each body

Intuition can be seen as the primordial and pre-verbal faculty by means of which the mind gains immediate epistemic access to the phenomena. The concept of structural intuition is premised on the basic principle that an infallible resonance exists between internally intuitive and externally phenomenal structures; it generally proves to be compositional in that it unifies mathematical formality with me- chanical and material phenomenality. The present work is an attempt to extend the doctrine of structural intuition with regard to its epis- temological potential to an epistemic-structural intuition and, accord- ingly, to substantiate it as the characteristic trait of axiomatic intui- tions in the early modern mathematical and material sciences.


Intuition lässt sich als ein primordiales und vorsprachliches Ver- mögen betrachten, anhand dessen der Geist einen unmittelbaren epistemischen Zugang zu den Phänomenen erhält. Die strukturelle Intuition baut auf dem Grundprinzip auf, dass zwischen innerlich intuitiven und äußerlich phänomenalen Strukturen eine unfehlbare Resonanz besteht; sie erweist sich gemeinhin als kompositorisch, in- dem sie die mathematische Formhaftigkeit mit der mechanischen und materiellen Phänomenalität vereinheitlicht. Die vorliegende Abhand- lung ist ein Versuch, die Lehre der strukturellen Intuition hinsichtlich ihres erkenntnistheoretischen Potenzials auf eine epistemisch-struk- turelle Intuition zu erweitern und sie demnach als den Grundzug axiomatischer Intuitionen in den frühneuzeitlichen mathematischen und materiellen Wissenschaften zu begründen.

This textbook covers all the standard introductory topics in classical mechanics, including Newton's laws, oscillations, energy, momentum, angular momentum, planetary motion, and special relativity. It also explores more advanced topics, such as normal modes, the Lagrangian method, gyroscopic motion, fictitious forces, 4-vectors, and general relativity. It contains more than 250 problems with detailed solutions so students can easily check their understanding of the topic. There are also over 350 unworked exercises, which are ideal for homework assignments. Password-protected solutions are available to instructors at www.cambridge.org/9780521876223. The vast number of problems alone makes it an ideal supplementary book for all levels of undergraduate physics courses in classical mechanics. The text also includes many additional remarks which discuss issues that are often glossed over in other textbooks, and it is thoroughly illustrated with more than 600 figures to help demonstrate key concepts.

In this document, I will derive an expression for the work done by the frictional force acting on an object constrained to move along a two-dimensional curved surface. The scenario I am considering assumes that the body is moving only due to the gravitational force. The same calculations, however, allow for the existence of other tangential forces. An example would be to calculate the work done by friction in the case of a roller coaster. Our plan is to determine a mathematical expression for the frictional force and then use the definition of work: F · dr to calculate the work done.

An object accelerates while it falls under the influence of the gravitational force. By using two sensors a precise and automated measurement of the velocity can be obtained. The analysis of these measurements may be insufficient if air resistance is important. We discuss how by increasing the number of sensors we can determine the velocity, terminal velocity, and acceleration due to gravity.

Consider that a particle is moving in a uniform circular motion of radius about the point and with a tangential velocity. We can choose any two random points, and , for reference. As the particle moves from to , it travels along the arc of the circular path, the length of which is. The diagram below shows the slice of the minor sector. is the angular displacement of from , which is the minor angle ∠. Considering is infinitesimal, we can say that the sector is a triangle, of sides , and .

Link to my video about the derivation of the rocket equation.
References:
* SPIEGEL, Murray R. Schaum's outline of theory and problems of theoretical mechanics: with an introduction to Lagrange's equations and Hamiltonian theory. McGraw-Hill Companies, 1967.
*HALLIDAY, David; RESNICK, Robert; WALKER, Jearl. Fundamentals of physics. John Wiley & Sons, 2013.

Riassunto. Il problema sperimentale proposto per i Giochi di Anacleto 2013 viene riesaminato amplian-do le possibilità d'indagine del sistema fisico considerato. Il modello matematico ottenuto è applicato al-le situazioni realizzate in laboratorio e ai dati raccolti da alcune classi di liceo scientifico. Abstract. The experimental proof proposed in the Italian Giochi di Anacleto 2013 is re-analysed to widen the opportunities of inquiry the physical system. The mathematical model one finds is then applied to laboratory situations and data collected by some high school classes.