Robert Weinstock

New Approach to Special Relativity

A development of the theory of special relativity is based on a principle of relativity for inertial frames as applied to mechanical phenomena only. The Lorentz-transformation equations and the familiar mass-variation formula are deduced from the observation that mass does vary with speed; but in these the speed of light c is everywhere replaced by a universal speed k—namely, the speed at which the mass of any particle must become infinite—which arises as a consequence of the assumptions. The experimental result that the free-space speed of light with respect to the earth is equal to k then implies the same free-space speed of light with respect to every inertial frame—the usual starting point of the theory.

Dismantling a centuries‐old myth: Newton’s Principia and inverse‐square orbits

Examination of Newton’s Principia reveals a fallacy in its purported proof of the otherwise well established fact that an inverse‐square central force acting on a particle requires that the particle move in a conic‐section orbit. The great difficulty of reading through the Principia material antecedent to the fallacy is offered as a major reason for the centuries‐long delay in its detection and general acknowledgment. It was Johann Bernoulli, evidently, who in 1710 first proved that the inverse‐square force implies a conic‐section orbit.

Long‐buried dismantling of a centuries‐old myth: Newton’s Principia and inverse‐square orbits

The author’s 1982 report of the fallacy inherent in Newton’s treatment of inverse‐square orbits in the Principia was anticipated by Ferdinand Rosenberger in his 1895 book‐length study of Newton’s life and work [Isaac Newton und Seine Physikalischen Principien (Barth, Leipzig, 1895)]. A new alternative formulation of the case against the Principia treatment is offered as surpassing in effectiveness that of Rosenberger and the essentially equivalent 1982 formulation by the author. Also offered is exposure of a long‐enduring misrepresentation of a portion of the Principia that deals with inverse‐square orbits.

Propagation of a Longitudinal Disturbance on a One-Dimensional Lattice

Propagation of a particular longitudinal disturbance on a semi-infinite one-dimensional lattice of identical mass particles coupled by identical massless ideal springs is studied. With all but one of the particles at rest and all the connecting springs at equilibrium length at t = 0, the end particle is constrained to move with fixed longitudinal velocity for all t ≥ 0. Use is made of a method in which an infinite system of differential equations of motion is replaced by a single ordinary differential equation; the time-dependent Fourier coefficients of the equations solution are the respective displacements from equilibrium of the individual lattice particles. These displacements and the corresponding velocities are expressed in terms of definite integrals. The results of velocity computations are presented along with asymptotic velocity distributions on the lattice and are discussed in relation to a simplified propagation description posited in a recent introductory textbook. The case of a semi-infinit...

Inverse‐square orbits: Three little‐known solutions and a novel integration technique

Three methods, none of them widely known, are presented for determining the orbit of a particle subject solely to an inverse‐square central force: One is by Laplace, another by Jacobi; the third may be here making its first appearance in print. All three differ markedly in major thrust; all of them culminate, of course, in conic‐section orbits. Also included is use of a novel integration technique in the execution of a standard textbook solution of the same inverse‐square orbit problem.

Oscillations of a particle attached to a heavy spring: An application of the Stieltjes integral

The normal‐mode motions of a massive particle and a spring of non‐negligible mass to whose end it is attached are shown to be described by ’’nonorthogonal’’ displacement functions. Two equivalent methods of establishing an orthogonality relation — a specially defined inner product and application of the Riemann‐Stieltjes integral — are introduced. The results lead to the complete solution for the general motion of the spring‐plus‐particle system. This solution is applied to provide rigorous justification of the intuitive method that leads to the familiar rule (’’... add one‐third the mass of the spring ...’’) for obtaining the period of a particle oscillating at the end of a uniform spring whose non‐negligible mass is small in comparison. It is shown that the presence/absence of gravity is irrelevant to the general conclusions reached.

Laws of Classical Motion: What's F? What's m? What's a?

The basic definitions and laws of classical motion are systematically developed in a way which, when suitably amplified by the class-room teacher, can be made clearly understandable to any collection of reasonably intelligent beginning students of physics.

Newton’s Principia and the external gravitational field of a spherically symmetric mass distribution

A detailed analysis is offered—with concomitant clarification, correction, and celebration—of the treatment in Newton’s Principia of the external gravitational field produced by a spherically symmetric mass distribution. It is established, in particular, just how close the Principia comes to achieving the most general result (whereby the sphere can be considered as concentrated at its center); it is also shown how the Principia argument could easily have been extended so as to have gone the whole way.

Normal-Mode Frequencies of Finite One-Dimensional Lattices with Single Mass Defect: Exact Solutions

The normal-mode longitudinal-vibration frequencies are determined for a finite one-dimensional lattice of mass particles joined by identical massless ideal springs, when at most one of the particle masses differs from all the rest. Results are derived for all boundary conditions of common physical interest by a method that is both elementary and straightforward. The results are expressed through solutions of trigonometric equations readily solvable with arbitrary precision by numerical methods. Considerable information about the solutions is presented by means of simple graphical representations.

Spring-Mass Correction in Uniform Circular Motion

A detailed analysis is presented for the problem of uniform circular motion in which the centripetal force is provided by a stretched spring whose mass is not negligible compared with that of the whirling body. It is shown that the force required to stretch the spring statically to the length it has during the rotation exceeds the centripetal force it exerts on the whirling body by an amount depending in a simple way on the spring mass; the excess can be erased by adding to the body mass approximately one-third the spring mass in computing the centripetal force. Some experimental results that tend to support the theory are exhibited and discussed. A computation of radial-oscillation frequencies is included.