**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*·sin*α* / (1+*I*/*mR*²),
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} = 2*mR*²/5 = 0.40·*mR*²
(solid ball)

*I*_{cylinder} = *mR*²/2 = 0.50·*mR*²
(solid cylinder)

*I*_{sphere} = 2*mR*²/3 = 0.67·*mR*²
(sphere with thin wall)

*I*_{tube} = *mR*² = 1.0·*mR*²
(tube with thin wall)