Galileo's writings show a modern approach to nature. In antiquity, observation was appreciated, but the idea of experiments for a particular purpose was unfamiliar. Recall from Chap. 2 that Aristotle insisted that we understand a phenomenon only if we know its cause of a special kind, the final cause. When we know the "motivation," we can tell why something happens. For instance, a stone falls because its goal is to get closer to its natural place, the center of the universe. In Aristotle's approach,
observing such spontaneous, instead of contrived forced processes, was essential for understanding.
In contrast, modern science considers that if one knows the initial state of a system and all the forces present, one can understand the next state without assuming any natural end. This causal relation makes experimentation an efficient way to study nature. By changing the initial state of the experiment, one explores the laws that link cause and effect. An important task for experiments is to test theories that intend to explain phenomena. Experiment and theory also go hand in hand in the sense that a good theory can have practical value since it predicts the course of natural events in different situations. An application, like television, validates the underlying theory every time the "on" button is pushed.
Galileo, the experimenter's, main results in science of dynamics may be stated as a few laws.
I. A free horizontal movement happens at constant speed and without change of direction.
In everyday conditions on Earth, there is always some friction finally stopping any body, e.g., a ball rolling on a plane. However, aided by his experiments and intuition, Galileo could conclude that the ball would never stop if the friction could be totally eliminated, that is, the motion is "free".
II. A freely falling body experiences a constant acceleration.
Acceleration is the change in an object's velocity in a unit interval of time. For a uniformly accelerating object initially at rest, after an interval of time, the velocity v will equal the acceleration a multiplied by the time t(v = at). For a falling object at the Earth's surface, the acceleration is 9.8m/s2. After 1 s, the velocity will be 9.8 m/s, after 2 s, 19.6 m/s and so on for progressively larger times. In studies at the Merton College (Oxford) in the fourteenth century, it was already proposed that the distance s, a uniformly accelerating body travels during a time interval, is equal to one half of the product of the acceleration and the time squared (s = 1/2 at2). Galileo showed that this formula is valid by studying the gentle acceleration of balls rolling down inclined planes. Extrapolating to the case of a vertical plane, he concluded that freely falling bodies have a constant (but greater) acceleration obeying the same law. Recall the 9.8 m/s2 acceleration. After 1 s, the object has fallen 4.4 m. After 2 s, the total distance is 17.6 m, four times that in the first second, and so on.
This result, commonly ascribed to Galileo's dropping objects from the leaning tower of Pisa, was actually arrived at earlier by the Dutch-Belgian mathematician Simon Stevinus. He reported in 1586 that bodies with different masses fall with the same acceleration. Galileo was of the same opinion and may have attempted similar experiments with two dense objects of different masses. Indeed, if one could eliminate air friction, a hammer and a feather dropped simultaneously would both hit the ground at the same moment. Apollo astronauts on the airless surface of our Moon found this to be the case!
IV. Galilean principle of relativity: The trajectory and speed of motion of a body depend on the reference frame relative to which it is observed.
One argument against the revolving Earth was that a body released from the top of a tower would not appear to fall to the point directly beneath because the surface of the rotating Earth would move aside during the fall. The validity of the argument may be studied in an analogous situation, by dropping a stone from the top of the mast of a moving ship. Is the stone's trajectory deflected toward the back of the vessel? The French philosopher Pierre Gassendi (1592-1655) made such tests and found that the stone always hit the deck just beside the foot of the mast and there was no deflection! The object shares the uniform motion of the ship, even while falling. The conclusion made by Galileo was that an observer participating in a uniform motion couldn't detect this motion by free-fall experiments. Interestingly, for an observer standing on the shore the falling stone appears to make a parabolic curved trajectory. Which trajectory is the "true" one, the vertical straight line or the curved parabola? Galileo's answer is that both are correct, as the trajectory depends on the reference frame that may be fixed to the shore or to the uniformly moving vessel depending on the location of the observer.
At the time of Galileo, the significance of these laws of motion was twofold. First, they were clearly contrary to the old conceptions based on Aristotelian physics. Secondly, they helped to understand why the Earth could move without any dramatic consequences other than the regular daily rising and setting of the Sun and other heavenly bodies. The atmosphere can move together with the Earth without high winds or escape into space.
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