The Mathematics and Physics of Diderot. I. On Pendulums and Air Resistance

In this article Denis Diderot's Fifth Memoir of 1748 on the problem of a pendulum damped by air resistance is discussed. Diderot wrote the Memoir in order to clarify an assumption Newton made without further justification in the first pages of the Principia in connection with an experiment to verify the Third Law of Motion using colliding pendulums. To explain the differences between experimental and theoretical values of momentum in the collision experiments he conducted Newton assumed that the bob was retarded by an air resistance $F_R$ proportional to the velocity $v$. By giving Newton's arguments a mathematical scaffolding and recasting his geometrical reasoning in the language of differential calculus, Diderot provides a step-by-step solution guide to the problem and proposes experiments to settle the question about the appropriate form of $F_R$, which for Diderot quadratic in $v$, that is $F_R \sim v^2$. The solution of Diderot is presented in full detail and his results are compared to those obtained from a Lindstedt-Poincare approximation for an oscillator with quadratic damping. It is shown that, up to a prefactor, both coincide. Some results that one can derive from his approach are presented and discussed for the first time. Experimental evidence to support Diderot's or Newton's claims is discussed together with the limitations of their solutions. Some misprints in the original memoir are pointed out.Comment: 31 pages, 8 figures. Submitted to European Physical Journal

Boltzmann and the art of flying

One of the less known facets of Ludwig Boltzmann was that of an advocate of Aviation, one of the most challenging technological problems of his times. Boltzmann followed closely the studies of pioneers like Otto Lilienthal in Berlin, and during a lecture on a prestigious conference he vehemently defended further investments in the area. In this article I discuss his involvement with Aviation, his role in its development and his correspondence with two flight pioneers, Otto Lilienthal e Wilhelm Kress.Comment: 15 pages, no figure

Coupling between static friction force and torque

We show that the static friction force which must be overcome to render a sticking contact sliding is reduced if an external torque is also exerted. As a test system we study a planar disk lying on horizontal flat surface. We perform experiments and compare with analytical results to find that the coupling between static friction force and torque is nontrivial: It is not determined by the Coulomb friction laws alone, instead it depends on the microscopic details of friction. Hence, we conclude that the macroscopic experiment presented here reveals details about the microscopic processes lying behind friction.Comment: 6 pages, 4 figures, revte