Swinging and Rolling Revisited: Motion Along Broken Chords and the Pendulum Plane Experiment
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Galileo’s early investigations into the relation of swinging and rolling had led him to the understanding that the time to fall along a polygonal path should ever the closer approach the time to fall along the arc, which he believed was the same as the time to swing along the same arc, the closer the actual path of motion considered approximated the arc. With this understanding, Galileo returned to the measurement data of an experiment in which he had timed the quarter period of a pendulum and had also measured the time taken for a ball to roll down a long plane of modest inclination. The chapter discusses Galileo’s revaluation of the measurement data. From the time measured for fall along the inclined plane, he inferred the time of motion along polygonal paths, closely approximating the arc of pendulum swing. This time, however, it did not correspond to the measured quarter period of the pendulum as Galileo would have expected but differed from it by about 20%. Galileo’s considerations had, moreover, yielded strong indications that this difference would not vanish if polygonal paths composed of an even higher number of sides were considered. The difference finds its explanation in the fact that Galileo, unaware that kinematically rolling and sliding were different, had equated a sliding to a rolling motion. Moreover, Galileo had implicitly compared motions along arcs subtending different angles. From his perspective this should not have mattered, as he believed that motion along the arc, swinging or rolling, was isochronous, i.e., that the time of such motions was not affected by a variation of the angle. Thus Galileo was ultimately unable to accommodate the experimental results with the theoretical expectations based on his conceptualization of the relation of swinging and rolling, and this brought his research agenda to a halt for the time being.
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