1. Galileo's experiments with the rolling balls
Galileo
continued to refine his ideas about objects in motion. He took a board 12 cubits
long and half a cubit wide (about 20 feet by 10 inches) and cut a groove, as
straight and smooth as possible, down the center. He inclined the plane and
rolled brass balls down it, timing their descent with a water clock — a large
vessel that emptied through a thin tube into a glass. After each run he would
weigh the water that had flowed out — his measurement of elapsed time — and
compare it with the distance the ball had traveled.
Aristotle would have predicted that the velocity of a rolling ball was constant:
double its time in transit and you would double the distance it traversed.
Galileo was able to show that the distance is actually proportional to the
square of the time: Double it and the ball would go four times as far. The
reason is that it is being constantly accelerated by gravity.
2. Rolling balls, cylinders and tubes down on an inclined plane
Acceleration of a rolling body along the inclined plane equals a= g·sinα / (1+I/mR²), where I is the inertia moment, R is the outer radius, m is the mass. Time of rolling down T = √2L/a ~ a^{1/2}, where L is the length of the plane.
I_{ball} = 2mR²/5 = 0.40·mR²
(solid ball)
I_{cylinder} = mR²/2 = 0.50·mR²
(solid cylinder)
I_{sphere} = 2mR²/3 = 0.67·mR²
(sphere with thin wall)
I_{tube} = mR² = 1.0·mR²
(tube with thin wall)
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