Motion Under a Central Force

In our discussions of ion-solid interactions, we restrict ourselves to central forces where the potential V is a function of r only (V = K(r)), so that the force is always along r. We need to consider only the problem of a single particle of mass, Mc, moving about a fixed center of force, which will be taken as the origin of the 

In the problem examined in this section, we will assume that, in the laboratory system, one of the particles is practically at rest at the origin, O, while the other one moves with velocity v - a good approximation when the stationary particle is much heavier than the moving particle. 

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The motion of a pendulum in a medium which offers no resistance to its motion, is that of a material particle under the influence of a central force, F, attracting with an intensity which is proportional to the distance of the particle away from the centre of attraction. We shall call F the effective force since this is the force which is effective in producing motion. Consequently,... 

Apply this law to the analysis of the motion of an MP under the action of a central force (Figure 1.29). Let an MP of mass m be under the action of an external force so that in all its positions the line of force action passes through one point (throngh the center of a cir-cleX Then Mp(F) = 0, accordingly (dLldt) = 0 and L = const. It can be seen that if movement takes place under the action of the central force, vector L is fixed, therefore vectors p and L are fixed as well (as [r p] = L). It, in turn, fixes a plane, in which vectors r and p lie. Hence, under the action of the central force the MP (a body) moves along a flat trajectory (circular, elliptic or hyperbolic) so that [r p] = const. (Again it is appropriate to recollect the conversations of Jules Verne s heroes in the projectile in which they tried to reach the moon.) Examples of such movement are the motion of the planets around the Sun (according Kepler s laws) and the electron motion in atoms (within the framework of the Bohr model, refer to Chapter 6, Section 6.7). 

Neither of these vibrations corresponds to stretching vibrations of AH or BH. The antisymmetric vibrational mode represents translational motion in the transition state and has an imaginary force constant. The symmetric transition-state vibration has a real force constant but the vibration may or may not involve motion of the central H(D) atom2,12 13. If the motion is truly symmetric, the central atom will be motionless in the vibration and the frequency of the vibration will not depend on the mass of this atom, i.e. the vibrational frequency will be the same for both isotopically substituted transition states. It is apparent that under such circumstances there will be no zero-point energy difference... 

We proceed now to those spectra not of the hydrogen type. As wc have already mentioned in 21 we endeavour, following Bohr, to ascribe the production of these spectra to transitions between stationary states of the atom, each of these stationary states being characterised essentially by the motion of a single radiating or series electron in an orbit under the influence of the core, which is represented approximately by a central field of force. This conception explains some of the most important regularities of the series of spectra, namely, the existence of several series, each of which is more or less similar to the hydrogen type, and the possibility of combinations between these. 

Historically, one of the central research areas in physical chemistry has been the study of transport phenomena in electrolyte solutions. A triumph of nonequilibrium statistical mechanics has been the Debye—Hiickel—Onsager—Falkenhagen theory, where ions are treated as Brownian particles in a continuum dielectric solvent interacting through Cou-lombic forces. Because the ions are under continuous motion, the frictional force on a given ion is proportional to its velocity. The proportionality constant is the friction coefficient and has been intensely studied, both experimentally and theoretically, for almost 100... 

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