1. Galileo's experiments with the rolling balls
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 Ч a large
vessel that emptied through a thin tube into a glass. After each run he would
weigh the water that had flowed out Ч his measurement of elapsed time Ч 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= gЈsinα / (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.
Iball = 2mR²/5 = 0.40ЈmR²
Icylinder = mR²/2 = 0.50ЈmR² (solid cylinder)
Isphere = 2mR²/3 = 0.67ЈmR² (sphere with thin wall)
Itube = mR² = 1.0ЈmR² (tube with thin wall)
|Let us consider two solid cylinders moving down on a inclined plane: one cylinder is rolling, while the other one is just sliding. In the case of sliding there is no rotation and a= gЈsinα. For this reason sliding cylinder will reach the end of the plane first. Tr/Ts = (1+I/mR²)1/2 = 1.22|
|If two cylindrical bodies are rolling down on a inclined plane then body which has the smaller inertia moment will reach the end of the plane first. So, in the pair solid cylinder vs. tube with thin wall the tube will "win" and will reach the end of the chute first. Tt/Tc = (2.0/1.5)1/2 = 1.15|
|Next we shall substitute a tube by a ball of the same radius. In this case the time difference will be rather small because inertia moments of a ball and a cylinder are very similar: Iball = 0.40ЈmR² while Icylinder = 0.50ЈmR² Let us calculate which body will reach the finish first. Tc/Tb = (1.5/1.4)1/2 = 1.035 That is the ball will reach the "finish" first though the time difference is very small. Ball is just 3.5% quicker.|
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