Rolling and slipping down Galileo’s inclined plane: Rhythms of the spheres

Abstract

In ‘‘Two New Sciences’’ (TNS) Galileo presents a number of theorems and propositions for smooth solid spheres released from rest and rolling a distance d in time t down an incline of height H and length L. We collect and summarize his results in a single grand proportionality P: d1/d2=(t21/t22)(H/L)1/(H/L)2. (P) From what he writes in TNS it is clear that what we call P is assumed by Galileo to hold for all inclinations including vertical free fall with H/L=1. But in TNS he describes only experiments with gentle inclinations H/L<1/2. Indeed he cannot have performed the vertical free fall (H=L) experiment, because we (moderns) know that as we increase H/L, P starts to break down when H/L exceeds about 0.5, because the sphere, which rolls without slipping for small H/L, starts to slip, whence d starts to exceed the predictions of P, becoming too large by a factor of 7/5 for vertical free fall at H/L=1. In 1973 Drake and in 1975 Drake and MacLachlan published their analysis of a previously unpublished experiment that Galileo performed that (without his realizing it) directly compared rolling without slipping to free fall. In the experiment, a sphere that has gained speed v1 while rolling down a gentle incline is deflected so as to be launched horizontally with speed v1 into a free fall orbit discovered by Galileo to be a parabola. The measured horizontal distance X2 traveled in this parabolic orbit (for a given vertical distance fallen to the floor) was smaller than he expected, by a factor 0.84. But that is exactly what we (moderns) expect, since we know that Galileo did not appreciate the difference between rolling without slipping, and slipping on a frictionless surface. We therefore expect him to predict X2 too large by a factor (7/5)1/2=1/0.84. He must have been puzzled. Easy ‘‘home experiments’’ with simple apparatus available to Galileo (no frictionless air tracks, strobe lamps, or electronic timers!) allow the student to use his/her musical ear (for rhythm and tempo) to study vertical free fall as well as balls rolling down steep or gentle inclines, with or without slipping, and perhaps appreciate Galileo’s dilemma.