Chapter01 - CHAPTER INTRODUCTION TO THE MECHANICAL UNIVERSE...

Unformatted text preview: CHAPTER INTRODUCTION TO THE MECHANICAL UNIVERSE ln the center of all the celestial bodies rests the sun, For who could in this most beautiful temple place this lamp in another or better place than that from which it can illuminate everything at the same time? Indeed, it is not unsuitable that some have called it the light of the world; others, its mind, and still others, its ruler. Trlsmegistus calls it the visible God; Sophocles’ Electra, the all-seeing. So indeed, as it sitting on a royal throne. the Sun rules the family of the stars which surround it. Nicolaus Copernicus, De Flevo/ution/‘bus Orb/um Coe/est/um (1543) 1.1 THE COPERNICAN REVOLUTION We find it difficult to imagine the frame of mind of people who once firmly believed the earth to be the immovable center of the universe, with all the heavenly bodies revolving harmoniously around it. It is ironic that this view, inherited from the Middle Ages and handed down by the Greeks, particularly Greek thought frozen in the writings of Plato and Aristotle, was one designed to illustrate our insignificance amid the grand scheme of the universe — even while we resided at its center. INTRODUCTION TO THE MECHANICAL UNIVERSE Aristotle’s world consisted of four fundamental elements — fire, air, water, and earth — and each element was inclined to seek its own natural place. Flame leapt through air, bubbles rose in water, rain fell from the heavens, and rocks fell to earth: the world was ordered. Each element strove to return to its sphere surrounding the center of the universe. But even as Aristotle ordered the world, he did not deem it perfect. It was subject to death and decay just as were its inhabitants. Perfection was reserved for the heavens alone, which were serene and immutable. Above the sphere of fire were the crystalline spheres of the moon, planets, sun, and the stars beyond. Each heavenly body was fixed in its orbiting sphere, traveling across the sky in a circle — the perfect shape Plato deemed as the ideal path all cosmic bodies should follow. Thus conceived, the universe was so simple that it was fully and adequately represented in great clocks constructed and painted by craftsmen of the Middle Ages. And the motions of the heavenly bodies were like the inner workings of clocks — regular, predictable, and in the mind of man, free of earthly decay. This scheme, this grand plan, was an effort to describe the environment as it presented itself to the human senses. It was an attempt to find one simple, all-encompassing explanation for natural phenomena. The modern era began when people began to ask questions that Aristotle’s world View could not answer. Our purpose is to understand what has come to be known as classical mechanics — the science that arose to answer those new questions. No discovery in human thought is more important. Through it the temple of Aristotelian thought collapsed and no less than a new view of our place in the universe gradually rose from its ashes to replace it. So before we start to study physics, let’s introduce some of the principal heroes of the story that we’re going to see unfold. Figure 1.1 Nicolaus Copernicus (1473—1543). (Courtesy of the Polish Cultural Institute in London.) N icolaus Copernicus, a timid monk who lived from 1473 to 1543, began the revolution with his book De Revolutionibus Orbium Coelestium, or On the Revolutions of the THE COPERNICAN REVOLUTION 3 Celestial Spheres, published in the year of his death, 1543. In this work, Copernicus froze the sun in the sky and set the earth in motion around it. In an effort to simplify the Aristotelian model of the universe, with its circles upon circles to describe the complex motion of planets, Copernicus placed the sun at the center of the universe. In doing so, the earth was reduced to a common planet orbiting the sun, just like the five other known planets. This theory so upset the academic world that the word revolution has come to be associated with radical change. Thus we have not only the revolution of the earth around the sun, but also that of the colonies against Great Britain. Figure 1.2 Diagram of the sun-centered system from Copemicus’s 0n Revolutions. (Courtesy of the Archives, California Institute of Technology.) Although Copemicus’s ideas ultimately changed the Western view of the world, his book was at first largely ignored, and later considered heresy. One of the first scientists to seize upon the monk’s revolutionary ideas was Johannes Kepler. Born a generation after the death of Copernicus and living from 1571 to 1630, Kepler ardently believed in the sun-centered system. In a battle to fit the best data on the seemingly wandering motions of the planets to the Copernican model, Kepler arrived at a set of mathematical equations that precisely described the motions. His results were as startling as they were elegant: planets tirelessly move around the sun in elliptical orbits. At the beginning of the seventeenth century the discovery of the telescope drove the last nail in the coffin of the Aristotelian world view. Galileo Galilei, who lived from 1564 until 1642, explored the sky with the newly invented telescope. There, among other unexpected wonders, he found moons revolving around Jupiter. This sighting gave direct proof that the earth did not have to be the center of all heavenly motions. Through his INTRODUCTION TO THE MECHANICAL UNIVERSE Figure 1.3 Johannes Kepler (1571—1630). (Courtesy of the Archives, California Institute of Technology.) fertile experimentation and keen insight into the character of natural phenomena, Galileo’s genius solidified the work that started with Copernicus’s theory and hastened the destruc- tion and replacement of Aristotelian phenomenology by the science of mechanics. Figure 1.4 Galileo Galilei (1564—1642). (SCALA/Art Resource, N .Y.) Like the early Greeks, the scientists of the new mechanics were intent on inventing an all-encompassing theory that would explain and describe every aspect of observable phenomena. Towering above all other scientists in the history of mechanics is Sir Isaac Newton. Born in the year Galileo died, 1642, Newton composed the grand synthesis that 1.2 UNITS AND DIMENSIONS 5 brought together the laws of the heavens and the laws of the earth. The physics that he established stood unchallenged until the beginning of the twentieth century. Figure 15 Sir Isaac Newton (1642—1727). (Courtesy of the Mansell Collections.) 1.2 UNITS AND DIMENSIONS One of the themes in the history of science is the great discovery that there is a connection between what goes on in the world — what we can observe — and mathematics. The discovery of the connection between mathematics and how the world works was first made by the Pythagoreans — followers of the Greek philosopher Pythagoras, in the fifth century B.C., a time when the Iliad and the Odyssey were cast into their final form, when Confucius walked the earth, and when the Greeks began to question nature rather than oracles for answers. The connection was preserved through time principally by astronomers who knew that the planets and stars followed courses that could be predicted by mathematical formulas and tables. However, it was believed during all of this time that the laws of the heavens, whatever they were, were in no way connected with the laws that govern the earth. And so even though the Pythagoreans knew that events on Earth obeyed mathe- matical laws, this idea was later forgotten as the views of Aristotle and Plato came to dominate Greek thought — and for that matter, all of Western thought — for nearly 2000 years. Quantification in the natural sciences, the description of natural phenomena in math- ematical terms, began to arise again at the end of the Middle Ages, about the same time as the invention of double—entry bookkeeping, an essential device for keeping track of the blossoming commerce of the period. Scholars still debate which of these two great discoveries inspired the other. No matter which came first, it is certainly true that com- merce and science share a common need for standardized units of measure. Throughout history, establishing common units of time, distance, and weight for the sake of orderly agriculture and commerce has been one of the principal responsibilities of government, INTRODUCTION TO THE MECHANICAL UNIVERSE and the degradation of these standards, usually for the purpose of increasing effective taxes, has been one of the best-known symptoms of corrupt government. The nursery rhyme “Jack and Jill” started as a popular jibe at the inflation of standards of measure (a gill is still a unit of volume; up and down the hill has obvious meaning; keeping the units constant were the responsibility of “the crown”; and so on). Honest or not, units of measure were seldom the same in different political juris- dictions, and were generally based on some convenient or traditional magnitude. For example, still preserved by use in the United States today, the mile (from milia, thousand) was once 1000 standard strides of a Roman legion; the yard the distance from one’s nose to one’s outstretched fingers (see Fig. 1.6); the foot is obvious enough; and the inch was one thumb — from joint to tip. Fathom “\ -" Stride = 1/2 pace 1000 paces = 1 mile Figure 1.6 Illustration of ancient basic units of the British system. Example 1 Determine the number of inches in 24 yards. In order to convert from yards to inches we use a trick; we multiply 24 yards by 1, written in a creative way: “Ydfiwlmlxllfjél 24 yd 01' 864 in. 12 UNITS AND DIMENSIONS 7 Each of the terms in brackets is equal to l by conversion. By canceling the units, we are left with inches as the final unit. Working scientists use this cancelation method to get their conversions straight. One of the legacies of Napoleon’s conquests in Europe was a new system of units, not based on tradition and whimsy, but on cool, precise French logic and on the decimal system. It is, nevertheless, firmly based on human magnitudes and on the properties of water, an essential ingredient of life. For example, the central unit of length is the meter, which is roughly a yard. But instead of dividing it into feet and thumbs, it is divided into tenths (decimeters), hundredths (centimeters), thousandths (millimeters), and so on. The unit of mass, the gram, is the mass of one cubic centimeter of water; the unit of volume, the liter, is 1000 cubic centimeters, so that a liter of water has a mass of one kilogram; and so on. The definitions of these quantities no longer vary from one country to another; they are fixed by international treaty, and are used almost everywhere except in the United States. This system is formally known as the Syste‘me International d’ Unités, or SI for short. Americans who wish to learn science or engineering, or just to shop in Europe (or Canada) must learn to convert from units based on the size of a dead king’s foot to units based on the distance between two scratch marks on a platinum—iridium bar kept in a refrigerated vault in Sévres, France. Example 2 Determine the number of centimeters in a football field. We perform a conversion of 100 yd into centimeters using the cancelation method: 3 ft 12 in. 2.54 cm 100d=100dx—x x—, y y [lyd] [1ft] [1m] or, multiplying the numbers and canceling the units, 100 yd = 9144 cm. Remembering that there are 2.54 cm in 1 in. is handy. Mastering the flow of time and dividing it into units seems to have been a part of the growth of every civilization on earth. Astronomer—priests of agricultural societies were responsible for deciding when to begin the annual cycle of tilling, planting, and harvesting. Smaller divisions of time corresponded, at least roughly, to the death and rebirth of the moon (months) and, of course, the daily cycle of light and dark. Intermediate Clusters of days, 5 or 10 or, by Roman times, 7 days per week, also came up. Dividing time into units smaller than a day proved more difcult, since it involved inventing time- keeping devices rather than mere counting. Hours, minutes, and seconds are relatively recent inventions, as is the idea that these units should have the same duration all year, regardless of the proportion of daylight and darkness in each day. Fortunately, unlike INTRODUCTION TO THE MECHANICAL UNIVERSE the units of length and mass, the same units of time are used everywhere, even in the United States. Units of one second and longer have the traditional names (minute, hour, day, week, month, year, century, millennium), whereas shorter times get metric-style prefixes (millisecond, microsecond, nanosecond, etc.). Example 3 How many seconds are there in a fortnight? To answer this, we again use the cancelation method: 1 f m1. ht X 2 weeks x 7 days X 24 h X 3600 s o 1 —— _ ’ g fortnight week day h l fortnight = 1,209,600 s. 1 fortnight OI‘ Quantities like seconds, grams, and meters not only have units, which may vary from one jurisdiction to another, but also dimensions, meaning, respectively, time, mass, and distance. Quantities with the same dimensions, but different units, can easily be compared (see Examples 1—3): an inch is bigger than two centimeters, but less than a Table 1.1 Metric Prefixes and Abbreviations Multiple Prefix Abbreviation 1012 tera T 109 giga G 106 mega M 103 kilo k 10 ‘2 centi c0 10 _ 3 milli m 10'6 micro [.L 10 '9 nano n 10— '2 pico p 10‘ 15 femto f 1.2 UNITS AND DIMENSIONS 9 light-year. But quantities with different dimensions cannot be compared at all, regardless of their units: a kilogram is neither bigger nor smaller than an hour or a yard. For many years after the beginning of the quantification of nature, all quantification consisted of comparing quantities of the same dimension. The idea of compounding quantities of different dimensions — say, dividing a distance by a time to form a speed — is a rather recent but richly useful invention. A compound quantity like speed has a unique dimension (distance divided by time), but various units (cm/s, furlongs/fortnight, etc.). We can write this as an equation of words, distance traveled a = ———-—— . (1.1) time elapsed The bar over the v reminds us that we are talking about an average speed during that time interval. If distance is measured in either feet or meters and time is measured in seconds, the units of speed are feet per second, written ft/s, or meters per second, written m/s. The important lesson is that units can be compounded when new physical quantities are defined. Another physical quantity that we will encounter is acceleration. We often hear about cars that can speed up, say from 0 to 60 mi/h, in so many seconds. That’s an example of acceleration. The average acceleration during a time interval is the change in speed during that interval divided by the time: a 2 change in speed . (1.2) time elapsed Just as speed can be measured in ft/s or m/s, the units of acceleration can be (ft/s)/s (read feet per second per second), usually abbreviated ft/s2 (read feet per second squared), or (m/s)/s, abbreviated m/sz. Example 4 What is the speed limit of 55 mi/h in m/s? Using the conversion that there are 1.6 km in 1 mi, we can write the following equation: mi mi 1.6 km 1000 m 1 h l min 55 — = 55 —— >< . X X . X 9 h h 1 m1 1 km 60 mm 60 s 55 mi/h = 24 m/s. or Questions 1. Determine which of the following distances is greatest: (a) 560 yd (b) 0.3 mi (c) 0.5 km (d) 723,000 cm. 10 INTRODUCTION TO THE MECHANICAL UNIVERSE 2. Convert the following quantities into ft/s: (a) 23 km/h (b) 88 nm/ms (c) 40 cm/yr (d) 3 furlongs/for’tnight (a furlong is one—eighth of a mile). 3. Determine which of the following bizarre quantities have units of distance, speed, or acceleration: (a) 2.3 m yr/s (b) 144 m yd s/mi yr2 (6) 4 cm/s century (d) 54.2 mZ/ft s. 1.3 A FINAL WORD The rise of modern science grows out of quantification. But quantification means much more than expressing observations in mathematical form. It is also a turning away from natural philosophy — grand schemes based on aesthetic preference — to detailed and precise observations and measurement. In other words, it is a turning to the accumulation of knowledge by small, detailed increments. “I would rather learn a single fact, no matter how ordinary,” wrote Galileo, “than discourse endlessly on Great Issues.” Before Cop- ernicus, many speculated on a sun—centered universe, but Copernicus took the trouble to do the detailed calculations and produce the astronomical tables that made his system a serious competitor for the very successful Ptolemaic system that came before it. Others before Galileo speculated on the properties of matter in motion, but Galileo based his arguments on detailed observations. His experiments with balls rolling down smooth inclined planes led to the law of falling bodies (Chapter 2), the law of inertia (Chapter 4), and ultimately to the law of conservation of energy (Chapter 13). Before clocks as we know them were invented, he devised means of timing his experiments by weighing how much water flowed through a specially constructed pipe. These measurements were accurate to a tenth of a second. The ultimate result of this kind of careful attention to detail, together with ingenuity, was no less than a new view of the universe. Quantification of physics tends to condense its ideas into mathematical formulas. As Galileo so delightfully expressed it, the great book of Nature lies ever open before our eyes, but it is written in mathematical characters. History teaches us that mathematics helps to advance physics, but it also shows that, like the tides, ideas flow in both directions. New discoveries in one field often lead to improvements in the other. For example, early in the seventeenth century the French mathematician Pierre de Ferrnat devised a crude method for drawing a tangent line to a curve. This gave Newton a hint for determining the velocity of a moving point, and this, in turn, led to Newton’s version of differential calculus. The mathematics in this book is developed in the same spirit. When new mathematical concepts are introduced, such as derivatives, integrals, and vectors, they arise naturally from physical problems; and then these new concepts help us to read the great book of Nature and to write new chapters. After Copernicus and his revolution and the events that led Europe through the years of Kepler and Galileo, and finally Newton, our View of the universe was, and still is today, that we live on a speck of dust in a lost corner, somewhere in the universe. Aristotle and Plato tried to teach us humility by placing us in a lowly sphere, isolated from the 1.3 A FINAL WORD 1‘ serene perfection of the heavens, but never in their wildest dreams could they have imagined the impact of the psychological change that occurred when we first realized that we are not at the center of the universe. When we study history, we learn about kings and queens, about social problems, wars, economics, and so on. All of these things come and go. And when they’re gone the world is pretty much the same as it was before. If you walk through an ancient Roman town today, say the town of Herculaneum in Italy, you can perceive exactly where you are and what the people were doing and why the town was built. The human condition has not changed very much in the past 2000 years. But there is one profound change that has altered the human race forever. That is the discovery of our real place in the universe. Studying history, we learn about the Renaissance and Reformation, the Counter Refor- mation and the Thirty Years’ War — the events that dominated the history of Europe during the time of Copernicus, Kepler, Galileo, and Newton. But those events were minor readjustments in the social fabric compared with this one monumental change that was occurring — new ideas in the history of ideas — that changed the human race absolutely forever. Our job in this volume is to study that story, to see exactly how ...
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