Rolling and slipping down Galileo{close_quote}s inclined plane: Rhythms of the spheres
- Physics Department and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 (United States)
In ``Two New Sciences`` (TNS) Galileo presents a number of theorems and propositions for smooth solid spheres released from rest and rolling a distance {ital d} in time {ital t} down an incline of height {ital H} and length {ital L}. We collect and summarize his results in a single grand proportionality {ital P}: {ital d}{sub 1}/{ital d}{sub 2}=({ital t}{sup 2}{sub 1}/{ital t}{sup 2}{sub 2})({ital H}/{ital L}){sub 1}/({ital H}/{ital L}){sub 2}. ({ital P}) From TNS it is clear that what we call {ital P} is assumed by Galileo to hold for all inclinations including vertical free fall with {ital H}/{ital L}=1. But in TNS he describes only experiments with gentle inclinations {ital H}/{ital L}{lt}1/2. Indeed he cannot have performed the vertical free fall ({ital H}={ital L}) experiment, because we (moderns) know that as we increase {ital H}/{ital L}, {ital P} starts to break down when {ital H}/{ital L} exceeds about 0.5, because the sphere, which rolls without slipping for small {ital H}/{ital L}, starts to slip, whence {ital d} starts to exceed the predictions of {ital P}, becoming too large by a factor of 7/5 for vertical free fall at {ital H}/{ital 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 In 1973 Drake and in 1975 Drake and MacLachlan published their analysis of a previously unpublished experiment that Galileo performed that (without his realizing it) directly compared rolling without slipping to free fall.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 {ital v}{sub 1} while rolling down a gentle incline is deflected so as to be launched horizontally with speed {ital v}{sub 1} into a free fall orbit discovered by Galileo to be a parabola. The measured horizontal distance {ital X}{sub 2} 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.
- OSTI ID:
- 283557
- Journal Information:
- American Journal of Physics, Vol. 64, Issue 5; Other Information: PBD: May 1996
- Country of Publication:
- United States
- Language:
- English
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