Problems In Classical Mechanics Research Articles

Short cracks and V-notches: Finite Fracture Mechanics vs. Cohesive Crack Model

In recent years, Finite Fracture Mechanics has proven to be an effective tool to estimate the strength of mechanical components, allowing fast strength predictions suitable for preliminary sizing and optimization of structures. In the present paper, we intend to corroborate the Finite Fracture Mechanics approach by showing that failure load estimates are very close to the ones provided by the well-established Cohesive Crack Model. To this aim, we consider two classical fracture mechanics problems, i.e. short cracks and V-notches. In the latter case, we believe to be of relevance also the Cohesive Crack Model semi-analytical solution herein provided.

Vínculos dependentes de velocidades e condição de integrabilidade de Frobenius

Sistemas mecânicos sujeitos a vínculos dependentes de velocidades ocorrem com frequência. O teorema de Frobenius permite decidir de forma inequívoca se vínculos que dependem linearmente das velocidades são ou não integráveis. Mostramos como se formula a condição de integrabilidade de Frobenius por meio da teoria das formas diferenciais exteriores e ilustramos sua aplicação a problemas interessantes de mecânica clássica. Por fim, discutimos a importância prática de identificar a natureza dos vínculos no que concerne à integrabilidade.

An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics

We describe a problem that can be tackled more-or-less routinely using the ideas of classical fluid mechanics, but it is a complex flow and even the linearised problem involves considerable algebraic complexity. The presentation here emphasises the approach that we adopt in order to formulate an accessible model of such a flow. The physical background is carefully described: the Equatorial Undercurrent, a particular phenomenon of the Pacific Ocean, the thermocline and the waves on both the free surface and on the thermocline. One of the results of this careful approach, coupled with a solution of the linearised problem for arbitrary wave numbers, is that we are able to provide well-grounded explanations for many of the fundamental processes that are relevant, and observed, in this region of the oceans; the result is a successful application of elementary principles. In the context of this system, we can describe the various types of wave dynamics, depending on wavelength, and also the differences between eastward and westward propagation. It is gratifying that the results of such a simple theory correspond closely to the observations reported in the literature. Of more interest, perhaps, from the theoretical-fluids viewpoint, is that the development leads directly to the prediction of critical layers and to a procedure for their analysis, which we outline here (and critical layers and their associated flows have been observed in the Pacific Ocean). Further, other possible rôles of nonlinearity are immediately accessible, such as wave evolution, for which we provide only an introduction, but this is sufficient to hint at tantalising prospects for further work.

Metric Relativity and the Dynamical Bridge: Highlights of Riemannian Geometry in Physics

We present an overview of recent developments concerning modifications of the geometry of space-time to describe various physical processes of interactions among classical and quantum configurations. We concentrate in two main lines of research: the Metric Relativity and the Dynamical Bridge. We describe the notion of equivalent (dragged) metric $ \widehat {g}_{\mu \nu }$ which is responsible to map the path of any accelerated body in Minkowski space-time onto a geodesic motion in such associated $\widehat {g}$ geometry. Only recently, the method introduced by Einstein in general relativity was used beyond the domain of gravitational forces to map arbitrary accelerated bodies submitted to non-Newtonian attractions onto geodesics of a modified geometry. This process has its roots in the very ancient idea to treat any dynamical problem in Classical Mechanics as nothing but a problem of static where all forces acting on a body annihilates themselves including the inertial ones. This general procedure, that concerns arbitrary forces—beyond the uses of General Relativity that is limited only to gravitational processes—is nothing but the relativistic version of the d’Alembert method in classical mechanics and consists in the principle of Metric Relativity. The main difference between gravitational interaction and all other forces concerns the universality of gravity which added to the interpretation of the equivalence principle allows all associated geometries—one for each different body in the case of non-gravitational forces—to be unified into a unique Riemannian space-time structure. The same geometrical description appears for electromagnetic waves in the optical limit within the context of nonlinear theories or material medium. Once it is largely discussed in the literature, the so-called analogue models of gravity, we will dedicate few sections on this emphasizing their relation with the new concepts introduced here. Then, we pass to the description of the Dynamical Bridge formalism which states the dynamic equivalence of nonlinear theories (driven by arbitrary scalar, spinor or vector fields) that occur in Minkowski background to theories described in associated curved geometries generated by each one of these fields. We shall see that it is possible to map the dynamical properties of a theory, say Maxwell electrodynamics in Minkowski space-time, into Born-Infeld electrodynamics described in a curved space-time the metric of which is defined in terms of the electromagnetic field itself in such way that it yields the same dynamics. It is clear that when considered in whatever unique geometrical structure, these two theories are not the same; they do not describe the same phenomenon. However, we shall see that by a convenient modification of the metric of space-time, an equivalence appears that establishes a bridge between these two theories making they represent the same phenomenon. This method was recently used to achieve a successful geometric scalar theory of gravity. At the end, we briefly review the proposal of geometrization of quantum mechanics in the de Broglie-Bohm formulation using an enlarged non-Riemannian (Weyl) structure.

Analysis of High School Students' Difficulties Related to Procedural Knowledge in Solving Classical Mechanics Problems

Analysis of High School Students' Difficulties Related to Procedural Knowledge in Solving Classical Mechanics Problems

Degradation process simulation of computer components of discrete devices

Formation and research of the materials with predetermined properties is one of the leaders in the research field of physical degradation. During the process of creating computer components we have to combine different types of homogeneous materials. Through a combination of these materials it is obtained composite materials, for which the important problem is durability and reliability. The process of material degradation of discrete device is slowly in relation to the depreciation time of the device. Physical degradation of computer components leads to malfunction of discrete devices. Preliminary simulation of the physical change of the material under certain boundary conditions allows pre-assess the risks in the process of creating discrete device, and in the process of checking reliability. The combination zones of two different materials are attended in the article. Physical and chemical processes that cause the appearance of interphase entities have different origins. Hence the need for a model development of specific class of problems in mechanics of interfacial interaction is occurred. It is simulated the physical behavior of materials of computer components of discrete devices and investigated the conditions of their degradation. The material behavior is conducted in the environment Matlab, simulating the behavior of the stress function from values of which is depended a strength (brittleness) of the material. Process simulation allows carrying out preliminary assessment of the behavior of composite components of discrete device that accelerates the research process of degradation over time. The brittleness of the material causes a malfunction of the device due to improper current flow in the transitions and reduces their reliability.

Algebraic techniques for diagonalization of a split quaternion matrix in split quaternionic mechanics

In the study of the relation between complexified classical and non-Hermitian quantum mechanics, physicists found that there are links to quaternionic and split quaternionic mechanics, and this leads to the possibility of employing algebraic techniques of split quaternions to tackle some problems in complexified classical and quantum mechanics. This paper, by means of real representation of a split quaternion matrix, studies the problem of diagonalization of a split quaternion matrix and gives algebraic techniques for diagonalization of split quaternion matrices in split quaternionic mechanics.

Dark energy in the two-body problem: The local group of galaxies

The two-body problem of classical mechanics can be extended in a natural way by introducing a universal dark-energy background, which acts as a third dynamical factor. For real systems of galaxies, the corresponding additional acceleration is described by general relativity in the Newtonian limit, in which deviations from the Minkowski metric are very small. It is shown that this acceleration has the same form in different inertial and non-inertial reference frames. The invariance of the acceleration produced by dark energy reflects the fact that, according to its mechanical properties, dark energy is a vacuum that is comoving with any motion. In this generalized formulation, as in the classical case, the two-body problem with a dark-energy background reduces to the motion of a single body in a central field. Two problems of this kind are considered for the Local Group of galaxies. The first, “internal,” problem concerns the binary system formed by the main bodies of the Local Group: our galaxy and M31. The subject of the second, “external,” problem is the binary system formed by the Local Group as a whole and its closest neighbor, the Virgo Cluster. In the internal problem, the effect of the dark energy is that the binary system is not bound if its mass does not exceed 3 × 1012 M⊙, which is allowed by the observational data. The external problem demonstrates the possibility of an evolutionary scenario in which a group could initially be located in the volume of a cluster, but then leave it and, moving away with an acceleration created by dark energy, arrive at the observed distance from the cluster.

Analyzing the many skills involved in solving complex physics problems

We have empirically identified over 40 distinct sub-skills that affect a person's ability to solve complex problems in many different contexts. The identification of so many sub-skills explains why it has been so difficult to teach or assess problem solving as a single skill. The existence of these sub-skills is supported by several studies comparing a wide range of individuals' strengths and weaknesses in these sub-skills, their “problem solving fingerprint,” while solving different types of problems including a classical mechanics problem, quantum mechanics problems, and a complex trip-planning problem with no physics. We see clear differences in the problem solving fingerprint of physics and engineering majors compared to the elementary education majors that we tested. The implications of these findings for guiding the teaching and assessing of problem solving in physics instruction are discussed.

Analysis of High-school Students' Difficulty Related to Conceptual Knowledge in Solving Problems in Classical Mechanics

Analysis of High-school Students' Difficulty Related to Conceptual Knowledge in Solving Problems in Classical Mechanics

A note on the breathing mode of an elastic sphere in Newtonian and complex fluids

Experiments on the acoustic vibrations of elastic nanostructures in fluid media have been used to study the mechanical properties of materials, as well as for mechanical and biological sensing. The medium surrounding the nanostructure is typically modeled as a Newtonian fluid. A recent experiment however suggested that high-frequency longitudinal vibration of bipyramidal nanoparticles could trigger a viscoelastic response in water-glycerol mixtures [Pelton et al., “Viscoelastic flows in simple liquids generated by vibrating nanostructures,” Phys. Rev. Lett. 111, 244502 (2013)]. Motivated by these experimental studies, we first revisit a classical continuum mechanics problem of the purely radial vibration of an elastic sphere, also called the breathing mode, in a compressible viscous fluid and then extend our analysis to a viscoelastic medium using the Maxwell fluid model. The effects of fluid compressibility and viscoelasticity are discussed. Although in the case of longitudinal vibration of bipyramidal nanoparticles, the effects of fluid compressibility were shown to be negligible, we demonstrate that it plays a significant role in the breathing mode of an elastic sphere. On the other hand, despite the different vibration modes, the breathing mode of a sphere triggers a viscoelastic response in water-glycerol mixtures similar to that triggered by the longitudinal vibration of bipyramidal nanoparticles. We also comment on the effect of fluid viscoelasticity on the idea of destroying virus particles by acoustic resonance.

Weyl-Euler-Lagrange Equations of Motion on Flat Manifold

This paper deals with Weyl-Euler-Lagrange equations of motion on flat manifold. It is well known that a Riemannian manifold is said to be flat if its curvature is everywhere zero. Furthermore, a flat manifold is one Euclidean space in terms of distances. Weyl introduced a metric with a conformal transformation for unified theory in 1918. Classical mechanics is one of the major subfields of mechanics. Also, one way of solving problems in classical mechanics occurs with the help of the Euler-Lagrange equations. In this study, partial differential equations have been obtained for movement of objects in space and solutions of these equations have been generated by using the symbolic Algebra software. Additionally, the improvements, obtained in this study, will be presented.

Dynamics Equations with Almost Complex Structures on Contact 5-Manifolds

Contact geometry is the odd-dimensional analogue of symplectic geometry with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter it originated in questions of classical and analytical mechanics. Contact geometry has, as does symplectic geometry, broad applications in mathematical physics, geometrical optics, classical mechanics, analytical mechanics, mechanical systems, thermodynamics, geometric quantization and applied mathematics such as control theory. On the other hand, one way of solving problems in classical mechanics is with the help of the Euler-Lagrange and the Hamilton equations. In this study, Euler-Lagrange mechanical equations as representing the motion of the body were found on contact 5-manifolds. Also, closed solutions of the differential equations found in this study are solved by symbolic computation program.

Analytical modeling of temperature distribution in an anisotropic cylinder with circumferentially-varying convective heat transfer

Fluid flow past a cylinder is a classical problem of fluid mechanics and convective heat transfer. In this problem, the local convective heat transfer coefficient on the cylinder surface varies around the cylinder due to boundary layer growth, separation and transition to turbulence. While there is considerable literature on computing the temperature distribution in the fluid, not much work exists on computing the temperature distribution within the solid body, particularly if the solid has anisotropic thermal properties. This paper presents an analytical technique for computing the temperature distribution within a heat-generating cylinder with anisotropic thermal conductivity subjected to spatially varying convective heat transfer coefficient due to fluid flow. As expected, temperature distribution on the cylinder surface exhibits minima and maxima at locations where the convective heat transfer coefficient has maxima and minima respectively. The effect of various parameters, including Reynolds number of the flow and extent of anisotropy are examined. Results presented in this paper contribute towards the fundamental understanding of a classical heat transfer problem . Further, since Li-ion cells that are commonly used for energy conversion and storage exhibit strong thermal conduction anisotropy, these results may be useful for design of convection-based thermal management of Li-ion cells.

Simulations of Interfacial Phenomena in Soft Condensed Matter and Nanoscience

Computation of interfacial free energies between coexisting phases (e.g. saturated vapor coexisting with liquid) is a fundamental problem of classical statistical mechanics: the standard approach (dating back to van der Waals, Ginzburg-Landau, Cahn-Hillard · · ·) is based on the continuation of the free energy of homogeneous states throughout the two phase coexistence region. Beyond mean field this continuation does not exist, nor does an "intrinsic profile" of the interface exist! These problems can be overcome by computer simulation: one popular method is based on sampling the order parameter distribution function in the two-phase coexistence region, which yields information on the surface tension of planar interfaces (from "slab configurations") and of curved interfaces (from states containing "droplets"), elucidating the problem of the "Tolman length". Another method (suitable also for solid-liquid interfaces) analyzes the capillary wave broadening or the capillary wave spectrum; all these methods require a careful assessment of finite size effects. Related problems occur for excess free energies due to walls, needed to describe wetting phenomena, capillary condensation, heterogenous nucleation, etc. As an example, a thermodynamic integration method (based on "mixing" systems with and without walls) will be described, and an application to understand phase separation in nanoconfinement will be mentioned.

Restricted three-body problem in effective-field-theory models of gravity

One of the outstanding problems of classical celestial mechanics was the restricted 3-body prob- lem, in which a planetoid of small mass is subject to the Newtonian attraction of two celestial bodies of large mass, as it occurs, for example, in the sun-earth-moon system. On the other hand, over the last decades, a systematic investigation of quantum corrections to the Newtonian potential has been carried out in the literature on quantum gravity. The present paper studies the effect of these tiny quantum corrections on the evaluation of equilibrium points. It is shown that, despite the extreme smallness of the corrections, there exists no choice of sign of these corrections for which all qualitative features of the restricted 3-body problem in Newtonian theory remain unaffected. Moreover, first-order stability of equilibrium points is characterized by solving a pair of algebraic equations of fifth degree, where some coefficients depend on the Planck length. The coordinates of stable equilibrium points are slightly changed with respect to Newtonian theory, because the planetoid is no longer at equal distance from the two bodies of large mass. The effect is conceptually interesting but too small to be observed, at least for the restricted 3-body problems available in the solar system.

Investigation of different aspects of laminar horseshoe vortex system using PIV

Juncture flow is a classical fluid mechanics problem having wide applications in both aero and hydro dynamics. The flow separates upstream of the obstacle due to the adverse pressure gradient generated by it, with the formation of the vortical structure called “horseshoe vortex.” The current study is carried out for an elliptical leading edge obstacle placed on a flat plate to investigate the horseshoe vortex for a range of Reynolds number (ReW) based on maximum width (W) for which the incoming boundary layer is laminar. Four different types of horseshoe vortex systems were found: the steady, amalgamation, transition and breakaway. The transition vortex system is one after which the vortex system changes from amalgamation to breakaway. In this phase the vortex system alternatively undergoes both amalgamation and breakaway vortex cycles. The effect of variation in the chord wise shape of the obstacle is investigated. The quantitative measurements of PIV show that the vortex system does not undergo any significant change for different chord lengths of the model with the fixed aspect ratio and maximum width. The most upstream saddle point is also studied for steady horseshoe vortex region and found that it is the “saddle of attachment” where flow attaches to the plate surface instead of separating from it.

Theory and contents of frictional mechanics

Abstract In this paper, we first discuss the development of the field of tribology, and highlight some of the main problems encountered in this area, such as lack of systematicness, loose correlation, and inadequate focus on the microscopic perspective. Then, we provide basic formulas of frictional mechanics while considering the friction effect on classical mechanics formulae. In order to carry out the frictional mechanics analysis, we first classify the interface. According to the size analysis of surface films, the manufacturing roughness of the surface, the contact width, and the roller radius of the rolling contact bearing, frictional mechanics has the features of interface mechanics, while interfaces are classified based on the presence or absence of a medium. Based on the classification, we further analyze the pressure and frictional stress of sliding and rolling friction problems without a medium, such as a slider, wedge key, and V belt. We also analyze problems with a medium, such as journal and rolling contact bearings. By comparing these results with those of classical mechanics without considering friction, we see that (1) friction causes deviations in the result for classical mechanics which does not consider friction, and (2) if the frictional stress and normal pressure affect each other, their interaction should be considered simultaneously. Finally, we summarize the friction problems, namely, sliding and rolling, with and without a medium, and deformed and non-deformed. From our analysis, we propose two conclusions. First, the frictional mechanics problem is a deviation of the classical mechanics problem, and secondly, frictional stress and normal pressure influence each other.

The Spheronic Toy Universe: How Special Relativity may be Visualised to Emerge from a Wave-Nature of Matter

AbstractWe construct an idealised universe for didactic purposes. This universe is assumed to consist of absolute Euclidean space and to be filled with a classical medium which allows for sound waves. A known solution to the wave equation describing the dynamics of the medium is a standing spherical wave. Although this is a problem of classical mechanics, we demonstrate that the Lorentz transformation is required to generate a moving solution from the stationary one. Both solutions are here collectively referred to as “spherons”. These spherons exhibit properties which have analogues in the physical description of matter with rest mass, among them de Broglie like phase waves and at the same time “relativistic” effects such as contraction and a speed limit. This leads to a theory of special relativity by assuming the point of view of an observer made of such spheronic “matter”. The argument made here may thus be useful as a visualisation or didactic approach to the real universe, in which matter has wave-like properties and obeys the laws of special relativity.

Non-Linear Forces and Irreversibility Problem in Classical Mechanics

Restrictions of classical mechanics which take place because of holonomic constraints hypothesis used for obtaining canonical Lagrange equation are analyzed. As it was shown that this hypothesis excludes non-linear terms in the expression for forces which are responsible for energy exchange between different degrees of freedom of a many-body system. An oscillator passing a potential barrier is considered as an example which demonstrated this fact. It was found that the oscillator can pass the barrier even if kinetic energy of its mass center is below the potential barrier’s height due to non-linear terms. This effect is lost because of holonomic constraints hypothesis. We also explained how one can derive a system’s motion equation without the use of holonomic constraints hypothesis. This equation can be used to describe non-linear irreversible processes within the frames of Newton’s laws.