Gravity
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Gravity

George Gamow

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eBook - ePub

Gravity

George Gamow

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About This Book

A distinguished physicist and teacher, George Gamow also possessed a special gift for making the intricacies of science accessible to a wide audience. In Gravity, he takes an enlightening look at three of the towering figures of science who unlocked many of the mysteries behind the laws of physics: Galileo, the first to take a close look at the process of free and restricted fall; Newton, originator of the concept of gravity as a universal force; and Einstein, who proposed that gravity is no more than the curvature of the four-dimensional space-time continuum.
Graced with the author's own drawings, both technical and fanciful, this remarkably reader-friendly book focuses particularly on Newton, who developed the mathematical system known today as the differential and integral calculus. Readers averse to equations can skip the discussion of the elementary principles of calculus and still achieve a highly satisfactory grasp of a fascinating subject.
Starting with a chapter on Galileo’s pioneering work, this volume devotes six chapters to Newton's ideas and other subsequent developments and one chapter to Einstein, with a concluding chapter on post-Einsteinian speculations concerning the relationship between gravity and other physical phenomena, such as electromagnetic fields.

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Year
2013
ISBN
9780486317113

Chapter 1

HOW THINGS FALL

The notion of “up” and “down” dates back to time immemorial, and the statement that “everything that goes up must come down” could have been coined by a Neanderthal man. In olden times, when it was believed that the world was flat, “up” was the direction to Heaven, the abode of the gods, while “down” was the direction to the Underworld. Everything which was not divine had a natural tendency to fall down, and a fallen angel from Heaven above would inevitably finish in Hell below. And, although great astronomers of ancient Greece, like Eratosthenes and Aristarchus, presented the most persuasive arguments that the Earth was round, the notion of absolute up-and-down directions in space persisted through the Middle Ages and was used to ridicule the idea that the Earth could be spherical. Indeed, it was argued that if the Earth were round, then the antipodes, the people living on the opposite side of the globe, would fall off the Earth into empty space below, and, far worse, all ocean water would pour off the Earth in the same direction.
When the sphericity of the Earth was finally established in the eyes of everyone by Magellan’s round-the-world trip, the notion of up-and-down as an absolute direction in space had to be modified. The terrestrial globe was considered to be resting at the center of the Universe while all the celestial bodies, being attached to crystal spheres, circled around it. This concept of the Universe, or cosmology, stemmed from the Greek astronomer Ptolemy and the philosopher Aristotle. The natural motion of all material objects was toward the center of the Earth, and only Fire, which had something divine in it, defied the rule, shooting upward from burning logs. For centuries Aristotelian philosophy and scholasticism dominated human thought. Scientific questions were answered by dialectic arguments (i.e., by just talking), and no attempt was made to check, by direct experiment, the correctness of the statements made. For example, it was believed that heavy bodies fall faster than light ones, but we have no record from those days of an attempt to study the motion of falling bodies. The philosophers’ excuse was that free fall was too fast to be followed by the human eye.
The first truly scientific approach to the question of how things fall was made by the famous Italian scientist Galileo Galilei (1564–1642) at the time when science and art began to stir from their dark sleep of the Middle Ages. According to the story, which is colorful but probably not true, it all started one day when young Galileo was attending a Mass in the Cathedral of Pisa, and absent-mindedly watched a candelabrum swinging to and fro after an attendant had pulled it to the side to light the candles (Fig. 1). Galileo noticed that although the successive swings became smaller and smaller as the candelabrum came to rest, the time of each swing (oscillation period) remained the same. Returning home, he decided to check this casual observation by using a stone suspended on a string and measuring the swing period by counting his pulse. Yes, he was right; the period remained almost the same while the swings became shorter and shorter. Being of an inquisitive turn of mind, Galileo started a series of experiments, using stones of different weights and strings of different lengths. These studies led him to an astonishing discovery. Although the swing period depended on the string’s length (being longer for longer strings), it was quite independent of the weight of the suspended stone. This observation was definitely contradictory to the accepted dogma that heavy bodies fall faster than light ones. Indeed, the motion of a pendulum is nothing but the free fall of a weight deflected from a vertical direction by a restriction imposed by a string, which makes the weight move along an arc of a circle with the center in the suspension point (Fig. 1).
image
Fig. 1. A candelabrum (a) and a stone on a rope (b) swing with the same period if the suspensions are equally long.
If light and heavy objects suspended on strings of equal length and deflected by the same angle take equal time to come down, then they should also take equal time to come down if dropped simultaneously from the same height. To prove this fact to the adherents of the Aristotelian school, Galileo climbed the Leaning Tower of Pisa or some other tower (or perhaps deputized a pupil to do it) and dropped two weights, a light and a heavy one, which hit the ground at the same time, to the great astonishment of his opponents (Fig. 2).
There seems to be no official record concerning this demonstration, but the fact is that Galileo was the man who discovered that the velocity of free fall does not depend on the mass of the falling body. This statement was later proved by numerous, much more exact experiments, and, 272 years after Galileo’s death, was used by Albert Einstein as the foundation of his relativistic theory of gravity, to be discussed later in this book.
It is easy to repeat Galileo’s experiment without visiting Pisa. Just take a coin and a small piece of paper and drop them simultaneously from the same height to the floor. The coin will go down fast, while the piece of paper will linger in the air for a much longer period of time. But if you crumple the piece of paper and roll it into a little ball, it will fall almost as fast as the coin. If you had a long glass cylinder evacuated of air, you would see that a coin, an uncrumpled piece of paper, and a feather would fall inside the cylinder at exactly the same speed.
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Fig. 2. Galileo’s experiment in Pisa.
The next step taken by Galileo in the study of falling bodies was to find a mathematical relation between the time taken by the fall and the distance covered. Since the free fall is indeed too fast to be observed in detail by the human eye, and since Galileo did not possess such modern devices as fast movie cameras, he decided to “dilute” the force of gravity by letting balls made of different materials roll down an inclined plane instead of falling straight down. He argued correctly that, since the inclined plane provides a partial support to heavy objects placed on it, the ensuing motion should be similar to free fall except that the time scale would be lengthened by a factor depending on the slope. To measure time he used a water clock, a device with a spigot that could be turned on and off. He could measure intervals of time by weighing the amounts of water that poured out the spigot in different intervals. Galileo marked the successive position of the objects rolling down an inclined plane at equal intervals of time.
You will not find it difficult to repeat Galileo’s experiment and check on the results he obtained.* Take a smooth board 6 feet long and lift one end of it 2 inches from the floor, placing under it a couple of books (Fig. 3a). The slope of the board will be
image
, and this will also be the factor by which the gravity force acting on the object will be reduced. Now take a metal cylinder (which is less likely to roll off the board than a ball) and let it go, without pushing, from the top end of the board. Listen to a ticking clock or a metronome (such as music students use) and mark the position of the rolling cylinder at the end of the first, second, third, and fourth seconds. (The experiment should be repeated several times to get these positions exact.) Under these conditions, consecutive distances from the top end will be 0.53, 2.14, 4.82, 8.5, and 13.0 inches. We notice, as Galileo did, that distances at the end of the second, third, and fourth seconds are respectively 4, 9, 16, and 25 times the distance at the end of the first second. This experiment proves that the velocity of free fall increases in such a way that the distances covered by a moving object increase as the squares of the time of travel. (4 = 22; 9 = 32; 16 = 42; 25 = 52) Repeat the experiment with a wooden cylinder, and a still lighter cylinder made of balsa wood, and you will find that the speed of travel and the distances covered at the end of consecutive time intervals remain the same.
image
Fig. 3. (a) A rolling cylinder on an inclined plane; (b) Galileo’s method of integration.
The problem that then faced Galileo was to find the law of change of velocity with time, which would lead to the distance-time dependence stated above. In his book Dialogue Concerning Two New Sciences Galileo wrote that the distances covered would increase as the squares of time if the velocity of motion was proportional to the first power of time. In Fig. 3b we give a somewhat modernized form of Galileo’s argument. Consider a diagram in which the velocity of motion v is plotted against time t. If v is directly proportional to t we will obtain a straight line running from (o;o) to (t;υ). Let us now divide the time interval from o to t into a large number of very short time intervals and draw vertical lines as shown in the figure, thus forming a large number of thin tall rectangles. We now can replace the smooth slope corresponding to the continuous motion of the object with a kind of staircase representing a jerky motion in which the velocity abruptly changes by small increments and remains constant for a short time until the next jerk takes place. If we make the time intervals shorter and shorter and their number larger and larger, the difference between the smooth slope and the staircase will become less and less noticeable and will disappear when the number of divisions becomes infinitely large.
During each short time interval the motion is assumed to proceed with a constant velocity corresponding to that time, and the distance covered is equal to this velocity multiplied by the time interval. But since the velocity is equal to the height of the thin rectangle, and the time interval to its base, this product is equal to the area of the rectangle.
Repeating the same argument for each thin rectangle, we come to the conclusion that the total distance covered during the time interval (o,t) is equal to the area of the staircase or, in the limit, to the ar...

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