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This paper is the author‘s attempt to generalize classical (Newtonian) mechanics to take into account
relatively recent gyroscopic work due to first Alexander Charles Jones closely followed by Eric Laithwaite and later Harvey Fiala and others. Of course, Newton himself did not have any gyroscopic motion as data except the rotation of moons and planets, the orbits of comets, and the like, and -- needless to say -- such involved extremely small rpm's. Thus, he could not have discovered the gyroscopic effects that Laithwaite and the others stumbled upon even with all his brilliance. But it seems clear that his mechanics must be almost correct except in the case of very rotational motion, and so the approach of this paper will be to apply as small a "band aid" to Newtonian theory as possible and still explain the new gyro data that does not quite fit. Thus we retain his three laws, but in their original form where the force is not the mass times the acceleration but rather the time derivative of the momentum. We also retain his (implicit) vector addition of forces but not the equivalence of gravitational and inertial mass [11] … as we assume variable inertial mass. In fact, that is the only one of his basic assumptions that we do alter in that we do not assume the inertial mass of an (electrically neutral) particle is constant for non-relativistic velocities and so inertial mass may vary with motion. The paper first develops a generalized neo-Newtonian mechanical theory and then applies the theory to the general case of a gyroscope (having a thin and hollow disc rotor) with a fixed pivot point, which steadily precesses slowly, but does not nutate, and then moves on to investigate the case of the precession angular velocity suddenly experiencing constant deceleration until the precession halts. These gyroscopic examples involve anomalous behavior and so are elementary examples of the gyroscopic results mentioned above, and their analysis will hopefully point the way to successful prediction of anomalous gyroscopic phenomena. The author hopes that his results will "get the researchers thinking in the right direction" as far as obtaining a rigorous generalization of Newton's theory capable of handling all gyroscopic data as well as that known to him from astronomy. (We, of course, restrict ourselves to considering only very non-relativistic velocities here.)

This paper is the author’s attempt to generalize classical (Newtonian) mechanics to take into account relatively recent gyroscopic work due to first Eric Laithwaite and later Harvey Fiala and others.  Of course, Newton himself did not have any gyroscopic motion as data except the rotation of moons and planets, the orbits of comets, and the like, and -- needless to say -- such involved extremely small rpm's.  Thus, he could not have discovered the gyroscopic effects that Laithwaite and the others stumbled upon even with all his brilliance.  But it seems clear that his mechanics must be almost correct except in the case of very rotational motion, and so the approach of this paper will be to apply as small a "band aid" to Newtonian theory as possible and still explain the new gyro data that does not quite fit.  Thus we retain his three laws, but in their original form where the force is not the mass times the acceleration but rather the time derivative of the momentum.  We also retain his (implicit) vector addition of forces but not the equivalence of gravitational and inertial mass {11} … as we assume variable inertial mass.  In fact, that is the only one of his basic assumptions that we do alter in that we do not assume the inertial mass of an (electrically neutral) particle is constant for non-relativistic velocities and so inertial mass may vary with motion.  The paper first develops a generalized neo-Newtonian theory and then applies the theory to the general case of a gyroscope (having a thin and hollow disc rotor) with a fixed pivot, which steadily precesses (not  necessarily slowly), but does not nutate, and then moves on to investigate the case of the precession angular velocity suddenly experiencing constant deceleration until the precession halts. These gyroscopic examples involve anomalous behavior and so are elementary examples of the gyroscopic results mentioned above, and their analysis will hopefully point the way to successful prediction of anomalous gyroscopic phenomena.  The author hopes that his results will "get the researchers thinking in the right direction" as far as obtaining a rigorous generalization of Newton's theory capable of handling all gyroscopic data as well as that known to him from astronomy.  (We, of course, restrict ourselves to considering only non-relativistic velocities here.)

This paper endeavors to show (within the realm of Newtonian mechanics [6]) that the mass of a gyroscope consisting of a thin cylindrical rotor and weightless shaft having a fixed pivot and steadily precessing (without nutation) in a horizontal plane appears to gain mass if the rotor is spinning compared to the same situation but with non-spinning rotor in that if the precession is stopped dead (not altering the rotor spin angular velocity), then the impact is considerably greater in the spinning rotor case than in the case where the gyroscope is constrained to move freely in the horizontal plane with the same angular velocity as the former precessional velocity just prior to impact.  This is surprising because French [5] (in his discussion of a steadily precessing gyroscope) shows that the centrifugal force of a gyroscope in the same situation except that it is allowed to precess steadily, and is not stopped dead, exhibits no apparent change of mass within the realm of Newtonian mechanics in that the spinning rotor exhibits the same centrifugal force independent of spin, the angular velocity of the non-spinning gyroscope (for comparison’s sake) which is constrained to move freely in a horizontal plane being the same.  But, experimentally [2, 3], both in the impact case and also in the steadily precessing case which were just mentioned, these Newtonian results are not, in general, observed … even to first order!  Also, it is shown that Newtonian mechanics is either discontinuous or paradoxical.

The structure and internal motions of ground-state electrons are obtained from a physical Spinning Charged Ring (SCR) model. This new electrodynamical model accurately yields the fundamental and structural properties of the electron, including the exact distribution of charge density inside the ring. Equilibrium of charge distributed throughout the ring interior is a result of electromagnetic self-forces that control the structure and internal motions of charge. The model yields the electron rest-mass energy and the electron-positron annihilation energy with energies 510,999 electron-Volts. The model also yields the actual, non-anomalous radius and non-anomalous magnetic moment of the electron. The model predicts the gyromagnetic ratio of a free electron. A current issue in physics is resolved ? how ?potential energy internal to a particle system come[s] into the picture? ? by treating the electron as a system of potential energies and real mass-energies.

Equilibrium of charge distributed throughout the electron ring interior is a result of electromagnetic self-forces that control the structure and internal motion of charge. The exact distribution of charge density inside the ring-shaped electron is described by equations and graphs. The dimensions of ground-state electrons are obtained from a physical spinning charged ring (SCR) model. The model yields the large radius and small radius of the electron ring. And the model also yields the actual non-anomalous magnetic moment of the electron. Thus, the SCR Model predicts that the diameter of a free electron is finite and physically related to electron properties such as its rest-mass, spin, magnetic moment, line spectra, and wavelength.

In this paper, we first show by extending the proof of B. J. Flehinger that the integers have the first digit property that the primes represented to the base ten also have the first digit property. We note that R. E. Whitney has also proven this using the logarithmic matrix method of summability. We then abstract from these two proofs in view of the Peano Axioms to obtain a (new) definition of what it means to sum a sequence in the spirit of the Peano axioms for the positive integers (which includes Flehinger's and Whitney's methods as special cases) and conjecture any method of summation of this very general type assigns the limit log₁₀((A+ 1)/A) to the two sequences s_n and t_n where

1. s_n = 1 if n has first digit equal to A else 0, and
2. t_n = 1 if the n^th prime has first digit A else 0.

The conjecture, if true, yields a new explanation of the first digit phenomenon by reducing it to its first cause: the basic well ordering of the positive integers.

A law of electrodynamics which was formulated by H. Aspden in the late 50's is examined, and a new field system based on this law is investigated. Maxwell's equations are not affected by this, but the Lorentz force law is modified, and the existence of a new type of radiation is considered.