Motion, central force

The relationship (equation (5.81)) between M and L depends only on fundamental constants, the electronic mass and charge, and does not depend on any of the variables used in the derivation. Although this equation was obtained by applying classical theory to a circular orbit, it is more generally valid. It applies to elliptical orbits as well as to classical motion with attractive forces other than dependence. For any orbit in any central force field, the angular... 

Tn the Rohr model of the hydrogen atom, the proton is a massive positive point charge about which the electron moves. By placing quantum mechanical conditions upon an otherwise classical planetary motion of the electron, Bohr explained the lines observed in optical spectra as transitions between discrete quantum mechanical energy states. Except for hvperfine splitting, which is a minute decomposition of spectrum lines into a group of closely spaced lines, the proton plays a passive role in the mechanics of the hydrogen atom, It simply provides the attractive central force field for the electron,... 

Rotation of the positron-electron pair is produced by the mutual Coulomb central force. Hence, the torque is zero.12 It follows that motion is in a plane perpendicular to orbital angular momentum L. Furthermore, the magnitude of L is a constant. 

Consider the two-dimensional motion of a particle in a central force field. The Lagrange function in Cartesian coordinates is... 

A simple example is provided by the coordinates that may be used to describe the motion of a particle in a plane. As generalized coordinates one can use either Cartesian coordinates qi = x, q2 = y, or plane polar coordinates qi = r, q2 = central force the coordinate p becomes cyclic, while both x and y are non-cyclic. The number of cyclic coordinates thus depends on the choice of generalized coordinates, and for each problem there may be one particular choice for which all coordinates are cyclic. 

Since the fictitious particle moves in a central force field described by a spherically symmetric potential function U(r), its angular momentum is conserved. Therefore, the motion of the fictitious particle will be in a plane defined by the velocity and the radius vectors. The Lagrangian may then be conveniently expressed in polar coordinates as... 

Solids in a gas-free orbit around a star will follow Keplerian orbits, meaning that their orbital velocity is found by balancing the centrifugal force of their motions with the central force of gravity from the star, or ... 

Note that for two spherical molecules with central force fields, the relative motion will take place in a plane. This plane will be defined by the centers of the two molecules and the direction of their relative velocity vector, which is parallel to the plane. No forces act perpendicularly to the plane. 

To extend the usefulness of the model to permit a description of chemical reactions, we must introduce another parameter, the effective duration of a collision. The rectangular well or central force models do this automatically by permitting molecular interaction over a range of distances. However, they are both more complex than the hard sphere model. We can rescue the hard sphere model by specifying a parameter era, the effective diameter for chemical interaction, while keeping hard sphere core diameter. When the centers of two identical molecules are a distance effective reaction volume is 7r([Pg.155]

For the more complicated molecular models such as, for example, those that assume central forces, we replace the above set of parameters by a new set involved in defining the force field. If we add to this the problem of complex molecules (i.c., those with internal structure), then there is the additional set of parameters needed to define the interactions between the internal molecular motions and the external force fields. From the point of view of the hard sphere model this would involve the definition of still more accommodation coefficients to describe the efficiency of transfer of internal energy between colliding molecules. 

The MD calculation to study supercritical fluid extraction from ceramics was performed with the XDORTO program developed by Kawamura[12]. The Verlet algorithm was used for the calculation of atomic motions, while the Ewald method was applied to the calculation of electrostatic interactions. Temperature was controlled by means of scaling the atom velocities under 3-dimensional periodic boundary condition. The calculations were made for 40000 time steps with the time increment of 2.5 10 seconds. The two body central force interaction potential, as shown in equation (3) was used for all the calculations ... 

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,... 

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