before. The hailstones falling in the piazza are not all the same size. Some are small. Others are large. But since the storm began, both small and large hailstones have been falling together, and hitting the ground simultaneously. He’s puzzled by this, because all year long his professors at the university have been teaching that the speed of a falling object depends solely on its weight – that heavy objects fall faster than lighter ones. But now, watching the hailstones he wonders. If this is true shouldn’t he be seeing all the large hailstones falling first, and then all the small ones? His professors at the University seem comfortable with things as they are. But he is no longer quite so sure ...
It is now 1604. Twenty one years have passed since that afternoon storm, but in all this time Galileo has not forgotten what he saw in the Piazza della Signoria. Now chair of the mathematics department at the University of Padua, he is still consumed by the desire to understand the motion of falling bodies. As he sees it, there is nothing more essential and more profound in all of nature than how objects move. This is what he desperately wants to know: How an arrow flies from the bowstring. How a hailstone falls from the sky. Even how the planets in the heavens make their celestial rounds. But most importantly, he wants to be able to describe that motion precisely – with mathematics. But how will he get there? How will he do this? Correctly reasoning that rolling is a form of falling, and that with a slowed descent he will be able to observe the acceleration of a descending object, Galileo constructs a gently inclined plane. On this plane his chosen object, a metal ball, will be able to “ fall” more slowly towards the ground. Galileo is thrilled with his ingenuity but quickly realizes that he has only partially solved his problem. Why? Because he still has no way to accurately measure the increasing speed at which the ball descends. He can see the acceleration but he can’t yet measure it.
Teachers examining inclined plane at Hofstra’s IDEAS Institute. Benjamin Wolff in background.
As he mulls over this problem he again goes to his lute – now it is his inspiration, his favorite muse. He gazes at its fingerboard. He admires the wonderful patterns of horizontal strings and vertical frets. And then, bit by bit, the solution begins to come to him. He will
treat his inclined plane just like the fingerboard of a lute and attach a similar series of moveable frets. As the rolling ball passes over each fret it will hop ever so slightly, and then make an audible “click” as it falls back to the surface of the plane. However, instead of placing the frets on his plane an equal distance from one another, he will move the frets further and further apart – just like the frets on a lute. How far apart will he move them? Well, his musician’s ear will guide him. He will roll ball after ball and, listening carefully, adjust the spacing of the frets after each roll so that the clicks made by the accelerating ball will be even and steady as the ball accelerates down the plane. An accomplished lute player and trained musician, Galileo knows that he can keep a steady beat. He has been cultivating this rhythmic skill since he was a child taking music lessons from his father. Back then, of course, he never would have imagined it, but that skill, that ability to keep a steady beat will now make it possible for him to determine precisely the properties of acceleration. Previously he could only see that the ball speeds up, but now, thanks to his frets, he can tell that it gets faster gradually and continuously. By adding frets to his inclined plane Galileo has found an ingenious way to divide continuous time into distinct units. And this is what he finds: From the time the ball is released until it passes over the first fret it travels a certain distance. When it reaches the second fret he finds that it has traveled four times as far. By the time he hears it passing the third fret it has gone nine times the distance. At the fourth fret it has traveled sixteen
fall 2009
HOFSTRA horizons
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