In this paper we discuss the role of static and kinetic friction in the dynamics of rolling motion. First, the role of kinetic friction in bringing about and maintaining an equilibration, called free rolling, between the translational and rotational motions of a rigid disc on a rigid surface is briefly discussed. We then extend the discussion to real wheels and provide a physical origin of rolling friction by introducing a new concept, quasi-rolling, to explain how pure rolling can exist even in the presence of some sliding. Our treatment is specifically applicable to rolling motion between two hard surfaces, such as ball- and roller-bearings.
The introduction of the electromagnetic field has been a necessity only in time-varying cases. In static or steady-state cases, it has been mostly a convenience. An example is discussed where even in static situations the field is a necessity in order to conserve angular momentum. The results are used to provide a classical picture for the electron spin in terms of the angular momentum of the associated electromagnetic field.
Nondipolar contribution to optical scattering in liquids and nanoparticle suspensions has been discerned for the first time from the dominant electric dipole scattering by assigning the observed polarization and azimuthal angular distribution of scattered polarized light to pure magnetic dipole and/or electric quadrupole radiation and ruling out other (the impurity of laser polarization, multiple scattering, optical activity, and optical anisotropy) explanations. The observed scattering has potential use in the optical study of nanoparticles.
Time-dependent behavior of one-dimensional (1D) many-fermion models is obtained by a method of recurrence relations. The Hilbert space of the density-fluctuation operator is two dimensional (2d), resulting in a time-independent generalized random force. The relevant Hilbert spaces of 2D and 3D many-fermion models, however, are infinite dimensional and the generalized random forces are consequently time dependent. The structure of these Hilbert spaces provides a picture of time-dependent behavior for 1D models which is fundamentally very different from that for 2D or 3D models.The Tomonaga model is perhaps the best known strictly one-dimensional (1D) many-fermion model. ' It is an exactly soluble model. The exact solutions for the Tomonaga model are not generalizable to higher dimensions. Hence this model is not so useful for understanding the importance of dimensionality in many-body problems. The standard electron gas model, when restricted to electron-hole scattering, is often referred to as the Sawada mode. ' The Sawada model is not exactly soluble, but one can show that the Sawada model in 1D reduces to the Tomonaga model if the electron-hole excitations are further confined to the vicinity of the Fermi surface. Hence one may regard the Sawada model as a generalizable version of the Tomonaga model.The time-dependent behavior of the Tomonaga or 1D Sawada model is basically very different from that of the Sawada model in 2D or 3D. If a dense electron gas in the ground state is slightly perturbed momentarily, the system will undergo a relaxation process. In 1D the relaxation process will be purely oscillatory. It will remain oscillatory even when the electron-hole interaction is removed, i.e. , when the system becomes an ideal, degenerate electron gas. This is because the 1D system has only one degree of freedom in momentum space whether there is an interaction or not. In 2D or 3D the system has infinitely many degrees of freedom and its relaxation process is richer, reflecting the two distinctly different single-particle and collective modes. 4 The different time-dependent behavior of the Sawada model in 1D and 2D or 3D is especially apparent if we study the time evolution in this model via the method of recurrence relations. ' Consider the density fluctuations at wave vector k, pk =~C p C&+k P where ck and ck are, respectively, the creation and annihilation operators at wave vector k. %e shaB confine our consideration to~k/kF~((1, where kF is the Fermi wave vector and A =1. The time-dependent behavior of this system can be completely determined by pk(t) and its generalized random force 5'(t). According to the method of recurrence relations, ps(t) and 9q(t) are given by(lb) 5"+ta"+t(t) = -a"(t) +a" t(t), 0~v~d -1, (2) where a I~O, a"=da"/dt, /s, "=(f",f")/(f" t, f" t) with 50~1. The inner product denotes the Kubo scalar product. We shall term 4" the vth recurrant. Also, {b"} satisfies exactly the same recurrence relation but starting with v=1 and b0=0. Hence dl does not appear in the recurrence relation for b...
Examination of a photograph of a rolling bicycle wheel shows blurring of its spokes which seems to be distributed in an apparently irregular pattern. Around certain points, however, there is no blurring at all. These points have no transverse velocities and are only moving radially inward or outward. On further analysis it is found that they form a circle whose equation, in polar coordinates, is given by , where a is the wheel radius and lies between and . Comparison with the photograph shows a very good agreement. Other detailed aspects of the photograph are discussed. Zusammenfassung. Die Untersuchung einer Fotografie eines rollenden Rades eines Fahrrades zeigt eine Verwischung der Speichen, die in einem scheinbar unregelmäßigen Muster angeordnet ist. Um gewisse Punkte herum tritt jedoch keinerlei Verwischung auf. Diese Punkte besitzen keine transversalen Geschindigkeiten und bewegen sich nur radial nach innen oder außen. Bei einer weiteren Analyse findet man, daß sie einen Kreis mit der Gleichung , in Polarkoordinaten bilden, wobei a der Radius des Rades ist, und zwischen und liegt. Ein Vergleich mit der Fotografie zeigt eine sehr gute Übereinstimmung. Auch andere Detailaspekte der Fotografie werden diskutiert.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.