Applications of the Scientific Method paper 3-5 pages
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ISBN-10: ISBN-13:
2013
1121838936 9781121838932
Contents
1. The Scientific Method 1 2. Section for Chapter 1 27 3. Motion 29 4. Section for Chapter 2 65 5. Energy 68 6. Section for Chapter 3 97
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Credits
1. The Scientific Method: Chapter 1 from The Physical Universe, 15th Edition by Krauskopf, Beiser, 2014 1 2. Section for Chapter 1: Chapter from The Physical Universe, 15th Edition by Krauskopf, Beiser, 2014 27 3. Motion: Chapter 2 from The Physical Universe, 15th Edition by Krauskopf, Beiser, 2014 29 4. Section for Chapter 2: Chapter from The Physical Universe, 15th Edition by Krauskopf, Beiser, 2014 65 5. Energy: Chapter 3 from The Physical Universe, 15th Edition by Krauskopf, Beiser, 2014 68 6. Section for Chapter 3: Chapter from The Physical Universe, 15th Edition by Krauskopf, Beiser, 2014 97
iv
Hell
I Sphe re of the Moon
II Sphe re of Mercury
III Sph ere of Venus
IV Sph ere of the Sun
V Spher e of Mars
VI Spher e of Jupiter
of SaturnVI II Sph
ere of the fixed stars. The Zodiac
IX Cry stalline sphere. Primum Mobile
VII Sphe re
Purgatory
He mis
pher e
of wa
ter
The D ark
W
oo d
Ai r
Jerusalem
Earthly Paradise
H em
isphere
of Earth
Fire
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How Scientists Study Nature 1.1 The Scientific Method
Four Steps • What the scientific method is. • The difference between a law and a
theory. • The role of models in science.
1.2 Why Science Is Successful Science Is a Living Body of Knowledge, Not a Set of Frozen Ideas
• Why the scientific method is so success- ful in understanding the natural world.
The Solar System 1.3 A Survey of the Sky
Everything Seems to Circle the North Star
• Why Polaris seems almost stationary in the sky.
• How to distinguish planets from stars without a telescope.
1.4 The Ptolemaic System The Earth as the Center of the Universe
• How the ptolemaic system explains the astronomical universe.
1.5 The Copernican System A Spinning Earth That Circles the Sun
• How the copernican system explains the astronomical system.
1.6 Kepler’s Laws How the Planets Actually Move
• The significance of Kepler’s laws. 1.7 Why Copernicus Was Right
Evidence Was Needed That Supported His Model While Contradicting Ptolemy’s Model
• How parallax decides which system provides the best explanation for what we see.
Universal Gravitation 1.8 What Is Gravity?
A Fundamental Force • Why gravity is a fundamental force.
1.9 Why the Earth Is Round The Big Squeeze
• What keeps the earth from being a perfect sphere.
1.10 The Tides Up and Down Twice a Day
• The origin of the tides. • The difference between spring and
neap tides and how it comes about.
1.11 The Discovery of Neptune Another Triumph for the Law of Gravity
• The role of the scientific method in finding a hitherto unknown planet.
How Many of What 1.12 The SI System
All Scientists Use These Units • How to go from one system of units to
another. • The use of metric prefixes for small and
large quantities. • What significant figures are and how to
calculate with them.
CHAPTER OUTLINE AND GOALS
Your chief goal in reading each section should be to understand the important findings and ideas indicated (•) below.
The Scientific Method
Medieval picture of the universe.
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All of us belong to two worlds, the world of people and the world of nature. As mem- bers of the world of people, we take an interest in human events of the past and present and find such matters as politics and economics worth knowing about. As members of the world of nature, we also owe ourselves some knowledge of the sciences that seek to understand this world. It is not idle curiosity to ask why the sun shines, why the sky is blue, how old the earth is, why things fall down. These are serious ques- tions, and to know their answers adds an important dimension to our personal lives.
We are made of atoms linked together into molecules, and we live on a planet circling a star—the sun—that is a member of one of the many galaxies of stars in the universe. It is the purpose of this book to survey what physics, chemistry, geology, and astronomy have to tell us about atoms and molecules, stars and galaxies, and everything in between. No single volume can cover all that is significant in this vast span, but the basic ideas of each science can be summarized along with the raw mate- rial of observation and reasoning that led to them.
Like any other voyage into the unknown, the exploration of nature is an adven- ture. This book records that adventure and contains many tales of wonder and dis- covery. The search for knowledge is far from over, with no end of exciting things still to be found. What some of these things might be and where they are being looked for are part of the story in the chapters to come.
Every scientist dreams of lighting up some dark corner of the natural world—or, almost as good, of finding a dark corner where none had been suspected. The most careful observations, the most elaborate calculations will not be fruitful unless the right questions are asked. Here is where creative imagination enters science, which is why many of the greatest scientific advances have been made by young, nimble minds.
Scientists study nature in a variety of ways. Some approaches are quite direct: a geologist takes a rock sample to a laboratory and, by inspection and analysis, finds out what it is made of and how and when it was probably formed. Other approaches are indirect: nobody has ever visited the center of the earth or ever will, but by com- bining a lot of thought with clues from different sources, a geologist can say with near certainty that the earth has a core of molten iron.
No matter what the approaches to particular problems may be, however, the work scientists do always fits into a certain pattern of steps. This pattern, a general scheme for gaining reliable information about the universe, has become known as the scientific method.
1.1 The Scientific Method Four Steps We can think of the scientific method in terms of four steps: (1) formulating a problem, (2) observation and experiment, (3) interpreting the data, and (4) testing the interpre- tation by further observation and experiment to check its predictions. These steps are often carried out by different scientists, sometimes many years apart and not always in this order. Whatever way it is carried out, though, the scientific method is not a mechanical process but a human activity that needs creative thinking in all its steps. Looking at the natural world is at the heart of the scientific method, because the results of observation and experiment serve not only as the foundations on which scientists build their ideas but also as the means by which these ideas are checked ( Fig. 1-1 ).
1. Formulating a problem may mean no more than choosing a certain field to work in, but more often a scientist has in mind some specific idea he or she wishes to investigate. In many cases formulating a problem and interpreting the data overlap.
HOW SCIENTISTS STUDY NATURE
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The scientist has a speculation, perhaps only a hunch, perhaps a fully developed hypothesis, about some aspect of nature but cannot come to a definite conclusion without further study.
2. Observation and experiment are carried out with great care. Facts about nature are the building blocks of science and the ultimate test of its results. This insis- tence on accurate, objective data is what sets science apart from other modes of intellectual endeavor.
3. Interpretation may lead to a general rule or law to which the data seem to con- form. Or it may be a theory, which is a more ambitious attempt to account for what has been found in terms of how nature works. In any case, the interpreta- tion must be able to cover new data obtained under different circumstances. As put forward orginally, a scientific interpretation is usually called a hypothesis.
4. Testing the interpretation involves making new observations or performing new experiments to see whether the interpretation correctly predicts the results. If the results agree with the predictions, the scientist is clearly on the right track. The new data may well lead to refinements of the original idea, which in turn must be checked, and so on indefinitely.
The Laws of Nature The laws of a country tell its citizens how they are supposed to behave. Different countries have different laws, and even in one country laws are changed from time to time. Furthermore, though he or she may be caught and pun- ished for doing so, anybody can break any law at any time.
The laws of nature are different. Everything in the universe, from atoms to gal- axies of stars, behaves in certain regular ways, and these regularities are the laws of nature. To be considered a law of nature, a given regularity must hold everywhere at all times within its range of applicability.
The laws of nature are worth knowing for two reasons apart from satisfying our curiosity about how the universe works. First, we can use them to predict phenomena not yet discovered. Thus Isaac Newton’s law of gravity was applied over a century ago to apparent irregularities in the motion of the planet Uranus, then the farthest known planet from the sun. Calculations not only showed that another, more distant planet should exist but also indicated where in the sky to look for it. Astronomers who looked there found a new planet, which was named Neptune.
Figure 1-1 The scientific method. No hypothesis is ever final because future data may show that it is incorrect or incomplete. Unless it turns out to be wrong, a hypothesis never leaves the loop of experiment, interpretation, testing. Of course, the more times the hypothesis goes around the loop successfully, the more likely it is to be a valid interpretation of nature. Experiment and hypothesis thus evolve together, with experiment having the final word. Although a hypothesis may occur to a scientist as he or she studies experimental results, often the hypothesis comes first and relevant data are sought afterward to test it.
Observation and Experiment
Collecting the data that bear upon the problem
Testing the Interpretation
Predicting the results of new experiments on the basis of the hypothesis
Interpretation
Explaining the data in terms of a hypothesis about how nature works
Statement of Problem What is the question being asked of nature? Are there any preliminary hypotheses?
Finding the Royal Road
Hermann von Helmholtz, a nine- teenth century German physicist and biologist, summed up his experience of scientific research in these words: “I would compare myself to a mountain climber who, not knowing the way, ascends slowly and toilsomely and is often compelled to retrace his steps because his progress is blocked; who, sometimes by rea- soning and sometimes by acci- dent, hits upon signs of a fresh path, which leads him a little farther; and who, finally, when he has reached his goal, discov- ers to his annoyance a royal road which he might have followed if he had been clever enough to find the right starting point at the beginning.”
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Second, the laws of nature can give us an idea of what goes on in places we cannot examine directly. We will never visit the sun’s interior (much too hot) or the interior of an atom (much too small), but we know a lot about both regions. The evidence is indirect but persuasive.
Theories A law tells us what; a theory tells us why. A theory explains why cer- tain events take place and, if they obey a particular law, how that law originates in terms of broader considerations. For example, Albert Einstein’s general theory of relativity interprets gravity as a distortion in the properties of space and time around a body of matter. This theory not only accounts for Newton’s law of gravity but goes further, including the prediction—later confirmed—that light should be affected by gravity.
As the French mathematician Henri Poincaré once remarked, “Science is built with facts just as a house is built with bricks, but a collection of facts is not a science any more than a pile of bricks is a house.”
Models It may not be easy to get a firm intellectual grip on some aspect of nature. Therefore a model —a simplified version of reality—is often part of a hypothesis or theory. In developing the law of gravity, Newton considered the earth to be perfectly round, even though it is actually more like a grapefruit than like a billiard ball. New- ton regarded the path of the earth around the sun as an oval called an ellipse, but the actual orbit has wiggles no ellipse ever had. By choosing a sphere as a model for the earth and an ellipse as a model for its orbit, Newton isolated the most important fea- tures of the earth and its path and used them to arrive at the law of gravity.
If Newton had started with a more realistic model—a somewhat squashed earth moving somewhat irregularly around the sun—he probably would have made little progress. Once he had formulated the law of gravity, Newton was then able to explain how the spinning of the earth causes it to become distorted into the shape of a grape- fruit and how the attractions of the other planets cause the earth’s orbit to differ from a perfect ellipse.
1.2 Why Science Is Successful Science Is a Living Body of Knowledge, Not a Set of Frozen Ideas What has made science such a powerful tool for investigating nature is the constant testing and retesting of its findings. As a result, science is a living body of information and not a collection of dogmas. The laws and theories of science are not necessarily the final word on a subject: they are valid only as long as no contrary evidence comes to light. If such contrary evidence does turn up, the law or theory must be modified or even discarded. To rock the boat is part of the game; to overturn it is one way to win. Thus science is a self-correcting search for better understanding of the natural world, a search with no end in sight.
Experiment Is the Test
A master of several sciences, Michael Faraday is best remem- bered for his discoveries in electricity and magnetism (see biography in Sec. 6.18). This statement appears in the entry for March 19, 1849 in his labora- tory notebook: “Nothing is too wonderful to be true if it be con- sistent with the laws of nature, and . . . experiment is the best test of such consistency.”
Faraday was a Fellow of Brit- ain’s Royal Society, which was founded in 1660 to promote the use of observation and experi- ment to study the natural world. The oldest scientific organiza- tion in the world, the Royal Society has as its motto Nullus in Verba —Latin for “Take nobody’s word for it.” On its 350th anni- versary, the Royal Society held a celebration of “the joy and vital- ity of science, its importance to society and culture, and its role in shaping who we are and who we will become.”
the point is that it is a large-scale framework of ideas and relationships.
To people ignorant of science, a theory is a suggestion, a proposal, what in science is called a hypothesis. For instance, believers in creationism, the unsupported notion that all living things simultaneously appeared on
earth a few thousand years ago, scorn Darwin’s theory of evolution (see Sec. 16.8) as “just a theory” despite the wealth of evidence in its favor and its bedrock position in modern biology. In fact, few aspects of our knowledge of the natural world are as solidly established as the theory of evolution.
In science a theory is a fully developed logical structure based on general principles that ties together a variety of observations and experimental findings and permits as-yet-unknown phenomena and connections to be predicted. A theory may be more or less speculative when proposed, but
Theory
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Scientists are open about the details of their work, so that others can follow their thinking and repeat their experiments and observations. Nothing is accepted on any- body’s word alone, or because it is part of a religious or political doctrine. “Com- mon sense” is not a valid argument, either; if common sense were a reliable guide, we would not need science. What counts are definite measurements and clear reasoning, not vague notions that vary from person to person.
The power of the scientific approach is shown not only by its success in under- standing the natural world but also by the success of the technology based on sci- ence. It is hard to think of any aspect of life today untouched in some way by science. The synthetic clothing we wear, the medicines that lengthen our lives, the cars and airplanes we travel in, the telephone, Internet, radio, and television by which we communicate—all are ultimately the products of a certain way of thinking. Curiosity and imagination are part of that way of thinking, but the most important part is that nothing is ever taken for granted but is always subject to test and change.
Religion and Science In the past, scientists were sometimes punished for daring to make their own interpretations of what they saw. Galileo, the first modern scientist (see biography in Sec. 2.5), was forced by the Roman Catholic Church in 1633 under threat of torture to deny that the earth moves about the sun. Even today, attempts are being made to compel the teaching of religious beliefs—for instance, the story of the Creation as given in the Bible—under the name of science. But “creation science” is a contradiction in terms. The essence of science is that its results are open to change in the light of new evidence, whereas the essence of creationism is that it is a fixed doctrine with no basis in observation. The scientific method has been the means of liberating the world from ignorance and superstition. To discard this method in favor of taking at face value every word in the Bible is to replace the inquiring mind with a closed mind.
Those who wish to believe that the entire universe came into being in 6 days a few thousand years ago are free to do so. What is not proper is for certain politi- cians (whom Galileo would recognize if he were alive today) to try to turn back the intellectual clock and compel such matters of faith to be taught in schools along- side or even in place of scientific concepts, such as evolution (see Sec. 16.8), that have abundant support in the world around us. To anyone with an open mind, the evidence that the universe and its inhabitants have developed over time and con- tinue to do so is overwhelming, as we shall see in later chapters. Nothing stands still. The ongoing evolution of living things is central to biology; the ongoing evo- lution of the earth is central to geology; the ongoing evolution of the universe is central to astronomy.
Many people find religious beliefs important in their lives, but such beliefs are not part of science because they are matters of faith with ideas that are meant to be accepted without question. Skepticism, on the other hand, is at the heart of science. Science follows where evidence leads; religion has fixed principles. It is entirely possible—and indeed most religious people do this—to consult sacred texts for inspiration and guidance while accepting that observation and reason represent the path to another kind of understanding. But religion and science are not inter- changeable because their routes and destinations are different—which means that science classrooms are not the place to teach religion. To mix the religious and the scientific ways of looking at the world is good for neither, particularly if compul- sion is involved.
Advocates of creationism (or “intelligent design”) assert that evolution is an atheistic concept. Yet religious leaders of almost all faiths see no conflict between evolution and religious belief. According to Cardinal Paul Poupard, head of the Roman Catholic Church’s Pontifical Council for Culture, “we . . . know the dan- gers of a religion that severs its links with reason and becomes prey to fundamen- talism. The faithful have the obligation to listen to that which secular modern science has to offer.”
Degrees of Doubt
Although in principle every- thing in science is open to ques- tion, in practice many ideas are not really in doubt. The earth is certainly round, for instance, and the planets certainly revolve around the sun. Even though the earth is not a perfect sphere and the planetary orbits are not per- fect ellipses, the basic models will always be valid.
Other beliefs are less firm. An example is the current picture of the future of the universe. Quite convincing data suggest that the universe has been expand- ing since its start in a “big bang” about 13.7 billion years ago. What about the future? It seems likely from the latest measurements that the expansion will continue forever, but this conclusion is still tentative and is under active study by astronomers today.
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Each day the sun rises in the east, sweeps across the sky, and sets in the west. The moon, planets, and most stars do the same. These heavenly bodies also move relative to one another, though more slowly.
There are two ways to explain the general east-to-west motion. The most obvi- ous is that the earth is stationary and all that we see in the sky revolves around it. The other possibility is that the earth itself turns once a day, so that the heavenly bodies only appear to circle it. How the second alternative came to be seen as correct and how this finding led to the discovery of the law of gravity are important chapters in the history of the scientific method.
1.3 A Survey of the Sky Everything Seems to Circle the North Star One star in the northern sky seems barely to move at all. This is the North Star, or Polaris, long used as a guide by travelers because of its nearly unchanging position. Stars near Polaris do not rise or set but instead move around it in circles ( Fig. 1-2 ). These circles carry the stars under Polaris from west to east and over it from east to west. Farther from Polaris the circles get larger and larger, until eventually they dip below the horizon. Sun, moon, and stars rise and set because their circles lie partly below the horizon. Thus, to an observer north of the equator, the whole sky appears to revolve once a day about this otherwise ordinary star.
Why does Polaris occupy such a central position? The earth rotates once a day on its axis, and Polaris happens by chance to lie almost directly over the North Pole. As the earth turns, everything else around it seems to be moving. Except for their circular motion around Polaris, the stars appear fixed in their positions with respect to one another. Stars of the Big Dipper move halfway around Polaris between every sunset and sunrise, but the shape of the Dipper itself remains unaltered. (Actually, as discussed later, the stars do change their relative positions, but the stars are so far away that these changes are not easy to detect.)
Constellations Easily recognized groups of stars, like those that form the Big Dip- per, are called constellations ( Fig. 1-3 ). Near the Big Dipper is the less conspicuous Little Dipper with Polaris at the end of its handle. On the other side of Polaris from
Figure 1-2 Time exposure of stars in the northern sky. The trail of Polaris is the bright arc slightly to the left of the center of the larger arcs. The dome in the foreground houses one of the many telescopes on the summit of Mauna Kea, Hawaii. This location is favored by astronomers because observing conditions are excellent there. The lights of cars that moved during the exposure are responsible for the yellow traces near the dome.
THE SOLAR SYSTEM
What the Constitution Says
The founders of the United States of America insisted on the separation of church and state, a separation that is part of the Con- stitution. What happens in coun- tries with no such separation, in the past and in the present, testi- fies to the wisdom of the founders.
In 1987 the U.S. Supreme Court ruled that teaching cre- ationism in the public schools is illegal because it is a purely reli- gious doctrine. In response, the believers in creationism changed its name to “intelligent design” without specifying who the designer was or how the design was put into effect. Their sole argument is that life is too com- plex and diverse to be explained by evolution, when in fact this is precisely what evolution does with overwhelming success. Nev- ertheless, attempts have continued to be made to include intelligent design in science classes in public schools. All such attempts have been ruled illegal by the courts. (For more, see Sec. 1.2 at www. mhhe.com/krauskopf .)
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the Big Dipper are Cepheus and the W-shaped Cassiopeia, named, respectively, for an ancient king and queen of Ethiopia. Next to Cepheus is Draco, which means dragon.
Elsewhere in the sky are dozens of other constellations that represent animals, heroes, and beautiful women. An especially easy one to recognize on winter eve- nings is Orion, the mighty hunter of legend. Orion has four stars, three of them quite bright, at the corners of a warped rectangle with a belt of three stars in line across its middle ( Fig. 1-4 ). Except for the Dippers, a lot of imagination is needed to connect a given star pattern with its corresponding figure, but the constellations nevertheless are useful as convenient labels for regions of the sky.
Sun and Moon In their daily east-west crossing of the sky, the sun and moon move more slowly than the stars and so appear to drift eastward relative to the con- stellations. In the same way, a person on a train traveling west who walks toward the rear car is moving east relative to the train although still moving west relative to the ground. In the sky, the apparent eastward motion is most easily observed for the moon. If the moon is seen near a bright star on one evening, by the next evening it will be some distance east of that star, and on later nights it will be farther and farther to the east. In about 4 weeks the moon drifts eastward completely around the sky and returns to its starting point.
The sun’s relative motion is less easy to follow because we cannot observe directly which stars it is near. But if we note which constellations appear where the sun has just set, we can estimate the sun’s location among the stars and follow it from day to day. We find that the sun drifts eastward more slowly than the moon, so slowly that the day-to-day change is scarcely noticeable. Because of the sun’s motion each constellation appears to rise about 4 min earlier each night, and so, after a few weeks or months, the appearance of the night sky becomes quite different from what it was when we started our observations.
By the time the sun has migrated eastward completely around the sky, a year has gone by. In fact, the year is defined as the time needed for the sun to make such an apparent circuit of the stars.
Figure 1-4 Orion, the mighty hunter. Betelgeuse is a bright red star, and Bellatrix and Rigel are bright blue stars. Stars that seem near one another in the sky may actually be far apart in space. The three stars in Orion’s belt, for instance, are in reality at very different distances from us.
Figure 1-3 Constellations near Polaris as they appear in the early evening to an observer who faces north with the figure turned so that the current month is at the bottom. Polaris is located on an imaginary line drawn through the two “pointer” stars at the end of the bowl of the Big Dipper. The brighter stars are shown larger in size.
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Figure 1-5 Apparent path of a planet in the sky looking south from the northern hemisphere of the earth. The planets seem to move eastward relative to the stars most of the time, but at intervals they reverse their motion and briefly move westward. Apparent path
of a planet
Planets Five other celestial objects visible to the naked eye also shift their positions with respect to the stars. These objects, which themselves resemble stars, are planets (Greek for “wanderer”) and are named for the Roman gods Mercury, Venus, Mars, Jupiter, and Saturn. Like the sun and moon, the planets shift their positions so slowly that their day-to-day motion is hard to detect. Unlike the sun, they move in complex paths. In general, each planet drifts eastward among the stars, but its relative speed var- ies and at times the planet even reverses its relative direction to head westward briefly. Thus the path of a planet appears to consist of loops that recur regularly, as in Fig. 1-5 .
1.4 The Ptolemaic System The Earth as the Center of the Universe Although the philosophers of ancient Greece knew that the apparent daily rotation of the sky could be explained by a rotation of the earth, most of them preferred to regard the earth as stationary. The scheme most widely accepted was originally the work of Hippar- chus. Ptolemy of Alexandria ( Fig. 1-6 ) later included Hipparchus’s ideas into his Almagest, a survey of astronomy that was to be the standard reference on the subject for over a thou- sand years. This model of the universe became known as the ptolemaic system.
The model was intricate and ingenious ( Fig. 1-7 ). Our earth stands at the center, motionless, with everything else in the universe moving about it either in circles or in combinations of circles. (To the Greeks, the circle was the only “perfect” curve, hence the only possible path for a celestial object.) The fixed stars are embedded in a huge crystal sphere that makes a little more than a complete turn around the earth each day. Inside the crystal sphere is the sun, which moves around the earth exactly once a day. The dif- ference in speed between sun and stars is just enough so that the sun appears to move eastward past the stars, returning to a given point among them once a year. Near the earth in a small orbit is the moon, revolving more slowly than the sun. The planets Venus and Mercury come between moon and sun, the other planets between sun and stars.
To account for irregularities in the motions of the planets, Ptolemy imagined that each planet moves in a small circle about a point that in turn follows a large circle about the earth. By a combination of these circular motions a planet travels in a series of loops. Since we observe these loops edgewise, it appears to us as if the planets move with variable speeds and sometimes even reverse their directions of motion in the sky.
From observations made by himself and by others, Ptolemy calculated the speed of each celestial object in its assumed orbit. Using these speeds he could then figure out the location in the sky of any object at any time, past or future. These calcu- lated positions checked fairly well, though not perfectly, with positions that had been recorded centuries earlier, and the predictions also agreed at first with observations made in later years. So Ptolemy’s system fulfilled all the requirements of a scientific theory: it was based on observation, it accounted for the celestial motions known in his time, and it made predictions that could be tested in the future.
1.5 The Copernican System A Spinning Earth That Circles the Sun By the sixteenth century it had become clear that something was seriously wrong with the ptolemaic model. The planets were simply not in the positions in the sky predicted for them. The errors could be removed in two ways: either the ptolemaic
Figure 1-6 Ptolemy ( A.D. 100–170).
The Temple of the Sun
Here is how Copernicus summed up his picture of the solar system: “Of the moving bodies first comes Saturn, who completes his circuit in 30 years. After him Jupiter, moving in a 12-year revolution. Then Mars, who revolves bien- nially. Fourth in order an annual cycle takes place, in which we have said is contained the earth, with the lunar orbit as an epicy- cle, that is, with the moon mov- ing in a circle around the earth. In the fifth place Venus is carried around in 9 months. Then Mer- cury holds the sixth place, circu- lating in the space of 80 days. In the middle of all dwells the Sun. Who indeed in this most beauti- ful temple would place the torch in any other or better place than one whence it can illuminate the whole at the same time?”
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system could be made still more complicated, or it could be replaced by a different model of the universe.
Nicolaus Copernicus, a versatile and energetic Pole of the early sixteenth century, chose the second approach. Let us consider the earth, said Copernicus, as one of the planets, a sphere rotating once a day on its axis. Let us imagine that all the planets, including the earth, circle the sun ( Fig. 1-8 ), that the moon circles the earth, and that the stars are all far away. In this model, it is the earth’s rotation that explains
Figure 1-7 The ptolemaic system, showing the assumed arrangement of the members of the solar system within the celestial sphere. Each planet is supposed to travel around the earth in a series of loops, while the orbits of the sun and moon are circular. Only the planets known in Ptolemy’s time are shown. The stars are all supposed to be at the same distance from the earth.
Figure 1-8 The copernican system. The planets, including the earth, are supposed to travel around the sun in circular orbits. The earth rotates daily on its axis, the moon revolves around the earth, and the stars are far away. All planets in the solar system are shown here. There are also a number of dwarf planets, such as Pluto; see Sec. 17.11. The actual orbits are ellipses and are not spaced as shown here, though they do lie in approximately the same plane.
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the daily rising and setting of celestial objects, not the motions of these objects. The apparent shifting of the sun among the stars is due to the earth’s motion in its orbit. As the earth swings around the sun, we see the sun changing its position against the background of the stars. The moon’s gradual eastward drift is mainly due to its orbital motion. Apparently irregular movements of the planets are really just combinations of their motions with our own shifts of position as the earth moves.
The copernican system offended both Protestant and Catholic religious lead- ers, who did not want to see the earth taken from its place at the hub of the universe. The publication of Copernicus’s manuscript began a long and bitter argument. To us, growing up with the knowledge that the earth moves, it seems odd that this straight- forward idea was so long and so violently opposed. But in the sixteenth century good arguments were available to both sides.
Consider, said supporters of Ptolemy, how fast the earth’s surface must move to complete a full turn every 24 h. Would not everything loose be flung into space by this whirling ball, just as mud is thrown from the rim of a carriage wheel? And would not such dizzying speeds produce a great wind to blow down buildings, trees, plants? The earth does spin rapidly, replied the followers of Copernicus, but the effects are counterbalanced by whatever force it is that holds our feet to the ground. Besides, if the speed of the earth’s rotation is a problem, how much more of a problem would be the tremendous speeds of the sun, stars, and planets if they revolve, as Ptolemy thought, once a day around a fixed earth?
1.6 Kepler’s Laws How the Planets Actually Move Fortunately, improvements in astronomical measurements—the first since the time of the Greeks—were not long in coming. Tycho Brahe (1546–1601), an astronomer working for the Danish king, built an observatory on the island of Hven near Copen- hagen in which the instruments were remarkably precise ( Fig. 1-9 ). With the help of these instruments, Brahe, blessed with exceptional eyesight and patience, made thousands of measurements, a labor that occupied much of his life. Even without the
B I O G R A P H Y
currency reform, but much of his time was devoted to developing the idea that the planets move around the sun rather than around the earth. The idea was not new—the ancient Greeks were aware of it—but Copernicus went fur- ther and worked out the planetary orbits and speeds in detail. Although a summary of his results had been circulated in manuscript form earlier, not until a few weeks before his death was Copernicus’s De Revolutionibus Orbium Coelestium published in book form.
Today De Revolutionibus is recog- nized as one of the foundation stones of modern science, but soon after its appearance it was condemned by the Catholic Church (which did not lift its ban until 1835) and had little
impact on astronomy until Kepler further developed its concepts over a half century later.
When Columbus made his first voy- age to the New World Copernicus was a student in his native Poland. In the years that followed intellec- tual as well as geographical horizons receded before eager explorers. In 1496 Copernicus went to Italy to learn medicine, theology, and astron- omy. Italy was then an exciting place to be, a place of business expansion and conflicts between rival cities, great fortunes and corrupt govern- ments, brilliant thinkers and inspired artists such as Leonardo da Vinci and Michelangelo.
After 10 years in Italy Copernicus returned to Poland where he prac- ticed medicine, served as a canon in the cathedral of which his uncle was the bishop, and became involved in
Nicolaus Copernicus (1473–1543)
Leap Years
A day is the time needed for the earth to make a complete turn on its axis, and a year is the time it needs to complete an orbit around the sun. The length of the year is slightly less than 365 days and 6 hours. Thus adding an extra day to February every 4 years (namely those years evenly divisible by 4, which are accordingly called leap years ) keeps the seasons from shifting around the calendar.
The remaining discrepancy adds up to a full day too much every 128 years. To take care of most of this discrepancy, century years not divisible by 400 will not be leap years; thus 2000 was a leap year but 2100 will not be one.
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telescope, which had not yet been invented, Tycho’s observatory was able to deter- mine celestial angles to better than 1 __ 100 of a degree.
At his death in 1601, Brahe left behind his own somewhat peculiar model of the solar system, a body of superb data extending over many years, and an assistant named Johannes Kepler. Kepler regarded the copernican scheme “with incredible and ravishing delight,” in his words, and fully expected that Brahe’s improved figures would prove Copernicus correct once and for all. But this was not the case; after 4 years of work on the orbit of Mars alone, Kepler could not get Brahe’s data to fit any of the models of the solar system that had by then been proposed.
If the facts do not agree with the theory, then the scientific method requires that the theory, no matter how attractive, must be modified or discarded. Kepler then began to look for a new cosmic design that would fit Brahe’s observations better.
The First Law After considering every possibility, which meant years of drudg- ery in making calculations by hand, Kepler found that circular orbits for the planets were out of the question even when modified in various ways. He abandoned circular orbits reluctantly, for he was something of a mystic and believed, like Copernicus and the Greeks, that circles were the only fitting type of path for celestial bodies. Kepler then examined other geometrical figures, and here he found the key to the puzzle ( Fig. 1-10 ). According to Kepler’s first law:
The Second Law Even this crucial discovery was not enough, as Kepler realized, to establish the courses of the planets through the sky. What was needed next was a way to relate the speeds of the planets to their positions in their elliptical orbits. Kepler could not be sure a general relationship of this kind even existed, and he was overjoyed when he had figured out the answer, known today as Kepler’s second law:
The paths of the planets around the sun are ellipses with the sun at one focus.
Figure 1-9 A 1598 portrait of Tycho Brahe in his observatory. The man at the right is determining the position of a celestial body by shifting a sighting vane along a giant protractor until the body is visible through the aperture at upper left. There were four of each kind of instrument in the observatory, which were used simultaneously for reliable measurements.
Figure 1-10 To draw an ellipse, place a loop of string over two tacks a short distance apart. Then move a pencil as shown, keeping the string taut. By varying the length of the string, ellipses of different shapes can be drawn. The points in an ellipse corresponding to the positions of the tacks are called focuses; the orbits of the planets are ellipses with the sun at one focus, which is Kepler’s first law.
A planet moves so that its radius vector sweeps out equal areas in equal times.
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The radius vector of a planet is an imaginary line between it and the sun. Thus in Fig. 1-11 each of the shaded areas is covered in the same period of time. This means that each planet travels faster when it is near the sun than when it is far away. The earth, for instance, has a speed of 30 km/s when it is nearest the sun and 29 km/s when it is farthest away, a difference of over 3 percent.
The Third Law A great achievement, but Kepler was not satisfied. He was obsessed with the idea of order and regularity in the universe, and spent 10 more years making calculations. It was already known that, the farther a planet is from the sun, the longer it takes to orbit the sun. Kepler’s third law of planetary motion gives the exact relationship:
In equation form, this law states that
(Period of planet)2
___________________ (Average orbit radius)3
5 same value for all the planets
The period of a planet is the time needed for it to go once around the sun; in the case of the earth, the period is 1 year. Figure 1-12 illustrates Kepler’s third law. Table 17-1 gives the values of the periods and average orbit radii for the planets.
At last the solar system could be interpreted in terms of simple motions. Plan- etary positions computed from Kepler’s ellipses agreed not only with Tycho’s data but also with observations made thousands of years earlier. Predictions could be made of positions of the planets in the future—accurate predictions this time, no longer
The ratio between the square of the time needed by a planet to make a revolution around the sun and the cube of its average distance from the sun is the same for all the planets.
Figure 1-11 Kepler’s second law. As a planet goes from a to b in its orbit, its radius vector (an imaginary line joining it with the sun) sweeps out the area A. In the same amount of time the planet can go from c to d, with its radius vector sweeping out the area B, or from e to f, with its radius vector sweeping out the area C. The three areas A, B, and C are equal.
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party given by the Emperor of Bohe- mia), Kepler replaced him at the observatory and gained access to all of Brahe’s data, the most complete and accurate set then in existence.
Kepler felt that the copernican model of the solar system was not only capable of better agreement with the data than had yet been achieved but also contained within it yet- undiscovered regularities. Many years of labor resulted in three laws of plan- etary motion that fulfilled Kepler’s vision and were to bear their ultimate fruit in Newton’s law of gravity.
Kepler also found time to prepare new tables of planetary positions, to explain how telescopes produce magnified images, to father 13 chil- dren, and to prepare horoscopes for the Emperor of Bohemia, the main reason for his employment (as it had been for Brahe). In 1619 he suggested
that comet tails point away from the sun because of a “solar breeze,” a good guess (see Sec. 17.2) though he could not know its nature. A year later Kepler’s mother was accused of being a witch, but he was able to get her acquitted.
As a child, Kepler, who was born in Germany, was much impressed by seeing a comet and a total eclipse of the moon. In college, where astron- omy was his worst subject, Kepler concentrated on theology, but his first job was as a teacher of mathematics and science in Graz, Austria. There he pondered the copernican system and concluded that the sun must exert a force (which he later thought was magnetic) on the planets to keep them in their orbits.
Kepler also devised a geometri- cal scheme to account for the spacing of the planetary orbits and put all his ideas into a book called The Cosmic Mystery. Tycho Brahe, the Danish astronomer, read the book and took Kepler on as an assistant in his new observatory in Prague in what was then Bohemia. Upon Brahe’s death (the result of drinking too much at a
Johannes Kepler (1571–1630)
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approximations. Furthermore, Kepler’s laws showed that the speed of a planet in dif- ferent parts of its orbit was governed by a simple rule and that the speed was related to the size of the orbit.
1.7 Why Copernicus Was Right Evidence Was Needed That Supported His Model While Contradicting Ptolemy’s Model It is often said that Kepler proved that Copernicus was “right” and that Ptolemy was “wrong.” True enough, the copernican system, by having the planets move around the sun rather than around the earth, was simpler than the ptolemaic system. As modified by Kepler, the copernican system was also more accurate. However, the ptolemaic system could also be modified to be just as accurate, though in a very much more complicated way. Astronomers of the time squared themselves both with the practical needs of their profession and with the Church by using the copernican system for calculations while asserting the truth of the ptolemaic system.
Figure 1-12 Kepler’s third law states that the ratio T 2 / R 3 is the same for all the planets.
Example 1.1
Kepler’s laws should be obeyed by all satellite systems, not just the solar system. In the seventeenth century the French astronomer Cassini discovered four of Saturn’s satellites (more have been discovered since). The names, periods, and orbit radii of these satellites are as follows: Tethys 1.89 days Rhea 4.52 days
2.95 3 105 km 5.27 3 105 km Dione 2.74 days Iapetus 79.30 days
3.77 3 105 km 35.60 3 105 km Verify that Kepler’s third law holds for these satellites.
Solution What we must do is calculate the ratio T 2 / R 3 for each satellite. The result for Tethys is
(1.89 days)2
______________ (2.95 3 105 km)3
5 1.40 3 10216 days2/km3
The ratio turns out to be the same for the other satellites as well, so we conclude that Kepler’s third law holds for this satellite system. [Calculations that involve powers of 10 are discussed in the Math Refresher at the end of this book. We note that (10 5 ) 3 5 10 3(5) 5 10 15 and 1/10 15 5 10 2 15 .]
Occam’s Razor
In science, as a general rule, the simplest explanation for a phe- nomenon is most likely to be cor- rect: less is more. This principle was first clearly expressed by the medieval philosopher William of Occam (or Ockham), who was born in England in 1280. In 1746 the French philosopher Etienne de Condillac called the prin- ciple Occam’s razor, an elegant metaphor that suggests cutting away unnecessary complications to get at the heart of the matter. Copernicus was one of many successful users of Occam’s razor. To be sure, as when shaving with an actual razor, it is possible to go too far; as the mathematician Alfred Whitehead said, “Seek simplicity, and distrust it.”
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The copernican system is attractive because it accounts in a straightforward way for many aspects of what we see in the sky. However, only observations that contradict the ptolemaic system can prove it wrong. The copernican system is today considered correct because there is direct evidence of various kinds for the motions of the planets around the sun and for the rotation of the earth. An example of such evidence is the change in apparent position of nearby stars relative to the background of distant ones as the earth revolves around the sun ( Fig. 1-13 ), an effect called parallax; see Sec. 18.8. Shifts of this kind are small because all stars are far away, but they have been found.
As we know from everyday experience, and as we shall learn in a more precise way in Chap. 2, a force is needed to cause something to move in a curved path ( Fig. 1-14 ). The planets are no exception to this rule: a force of some kind must be acting to hold them in their orbits around the sun. Three centuries ago Isaac Newton had the inspired idea that this force must have the same character as the familiar force of gravity that pulls things to the earth’s surface.
1.8 What Is Gravity? A Fundamental Force Perhaps, thought Newton, the moon revolves around the earth much as the ball in Fig. 1-14 revolves around the hand holding the string, with gravity taking the place
UNIVERSAL GRAVITATION
Figure 1-13 As a consequence of the earth’s motion around the sun, nearby stars shift in apparent position relative to distant stars. The effect is known as parallax.
Distant stars
Nearby star
Sun
As seen from earth
1
2
E1
E2
Until only a few hundred years ago, astronomy was almost entirely in the service of astrology. The wealth of precise astronomical measurements that ancient civilizations compiled had as their purpose interpreting the ways of the gods.
Almost nobody today takes seri- ously the mythology of old. Although the basis of the connection has dis- appeared, however, some people still believe that the position in the sky of various celestial bodies at certain
times controls the world we live in and our individual destinies as well.
It does not seem very gracious for contemporary science to dismiss astrology in view of the great debt astronomy owes its practitioners of long ago. However, it is hard to have confidence in a doctrine that, for all its internal consistency and often delight- ful notions, nevertheless lacks any basis in scientific theory or observation and has proved no more useful in predict- ing the future than a crystal ball.
To our ancestors of thousands of years ago, things happened in the world because gods caused them to hap- pen. Famine and war, earthquake and eclipse—any conceivable catastrophe— all occurred under divine control. In time the chief gods were identified with the sun, the moon, and the five planets visible to the naked eye: Mer- cury, Venus, Mars, Jupiter, and Saturn. Early observers of the sky were primar- ily interested in finding links between celestial events and earthly ones, a study that became known as astrology.
Astrology
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of the pull of the string. In other words, perhaps the moon is a falling object, pulled to the earth just as we are, but moving so fast in its orbit that the earth’s pull is just enough to keep the moon from flying off ( Fig. 1-15 ). The earth and its sister planets might well be held in their orbits by a stronger gravitational pull from the sun. These notions turned out to be true, and Newton was able to show that his detailed theory of gravity accounts for Kepler’s laws.
It is worth noting that Newton’s discovery of the law of gravity depended on the copernican model of the solar system. “Common sense” tells us that the earth is the stationary center of the universe, and people were once severely punished for believing otherwise. Clearly the progress of our knowledge about the world we live in depends upon people, like Copernicus, who are able to look behind the screen of appearances that make up everyday life and who are willing to think for themselves.
Fundamental Forces Gravity is a fundamental force in the sense that it can- not be explained in terms of any other force. Only four fundamental forces are known: gravitational, electromagnetic, weak, and strong. These forces are respon- sible for everything that happens in the universe. Gravitational forces act between all bodies everywhere and hold together planets, stars, and the giant groups of stars called galaxies. Electromagnetic forces, which (like gravity) are unlimited in range, act between electrically charged particles and govern the structures and behav- ior of atoms, molecules, solids, and liquids. When a bat hits a ball, the interaction between them can be traced to electromagnetic forces. The weak and strong forces have very short ranges and act inside atomic nuclei. (Fundamental forces are fur- ther discussed in Sec. 8.14.)
The Law of Gravity Is the Same Everywhere How can we be sure that New- ton’s law of gravity, which fits data on the solar system, also holds throughout the rest of the universe? The evidence for this generalization is indirect but persua- sive. For instance, many double stars are known in which each member of the pair revolves around the other, which means some force holds them together. Through- out the universe stars occur in galaxies, and only gravity could keep them assem- bled in this way.
But is the gravity that acts between stars the same as the gravity that acts in the solar system? Analyzing the light and radio waves that reach us from space shows that the matter in the rest of the universe is the same as the matter found on the earth. If we are to believe that the universe contains objects that do not obey Newton’s law of grav- ity, we must have evidence for such a belief—and there is none. This line of thought may not seem as positive as we might prefer, but taken together with various theoreti- cal arguments, it has convinced nearly all scientists that gravity is the same everywhere.
Figure 1-14 An inward force is needed to keep an object moving in a curved path. The force here is provided by the string. If no force acts on it, a moving object will continue moving in a straight line at constant speed. (This is Newton’s first law of motion and is discussed in Sec. 2.7.)
Figure 1-15 The gravitational pull of the earth on the moon causes the moon to move in an orbit around the earth. If the earth exerted no force on the moon, the moon would fly off into space. If the moon had no orbital motion, it would fall directly to the earth.
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1.9 Why the Earth Is Round The Big Squeeze A sign of success of any scientific theory is its ability to account for previously myste- rious findings. One such finding is the roundness of the earth ( Fig. 1-16 ), which was known by the Greeks as long ago as the fifth century b.c. ( Fig. 1-17 ). Early thinkers believed the earth was round because a sphere is the only “perfect” shape, a vague idea that actually explains nothing. In fact, the earth is round because gravity squeezes it into this shape.
As shown in Fig. 1-18 , if any part of the earth were to stick out very much, the gravitational attraction of the rest of the earth would pull downward on the projec- tion. The material underneath would then flow out sideways until the projection became level or nearly so. The downward forces around the rim of a deep hole would similarly cause the surrounding material to flow into it. The same argument applies to the moon, the sun, and the stars.
Such irregularities as mountains and ocean basins are on a very small scale compared with the earth’s size. The total range from the Pacific depths to the summit of Everest is less than 20 km, not much compared with the earth’s radius of 6400 km.
Figure 1-16 Astronauts in the Apollo 11 spacecraft saw this view of the earth as they orbited the moon, part of whose bleak landscape appears in the foreground. The earth is indeed round.
B I O G R A P H Y
tea, under some apple trees . . . he told me he was just in the same situation when the notion of gravitation came into his mind. ‘Why should that apple always descend perpendicularly to the ground,’ thought he to himself.”
When Cambridge University reopened, Newton went back and 2 years later became professor of mathematics there. He lived qui- etly and never married, carrying out experimental as well as theoretical research in many areas of physics; a reflecting telescope he made with his own hands was widely admired.
Especially significant was New- ton’s development of the laws of motion (see Chap. 2), which showed exactly how force and motion are related, and his application of them to a variety of problems. Newton collected the results of his work on mechanics in the Principia, a sci- entific classic that was published in 1687. A later book, Opticks, summa- rized his efforts in this field. Newton also spent much time on chemistry, though here with little success.
After writing the Principia, New- ton began to drift away from science.
He became a member of Parliament in 1689 and later an official, eventu- ally the Master, of the British Mint. At the Mint Newton helped reform the currency (one of Kepler’s interests, too) and fought counterfeiters. New- ton’s spare time in his last 30 years was mainly spent in trying to date events in the Bible. He died at 85, a figure of honor whose stature remains great to this day.
Although his mother wanted him to stay on the family farm in England, the young Newton showed a talent for science and went to Cambridge Uni- versity for further study. An outbreak of plague led the university to close in 1665, the year Newton graduated, and he returned home for 18 months.
In that period Newton came up with the binomial theorem of algebra; invented calculus, which gave science and engineering a new and powerful mathematical tool; discovered the law of gravity, thereby not only showing why the planets move as they do but also providing the key to understand- ing much else about the universe; and demonstrated that white light is a composite of light of all colors—an amazing list. As Newton later wrote, “In those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since.”
Legend has it that Newton’s inter- est turned to gravity when he was struck on the head by a falling apple. Newton’s own recollection was given to a visiting friend, who reported that “We went into the garden and drank
Isaac Newton (1642–1727)
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Figure 1-17 In the distant past evidence for the spherical shape of the earth came from travelers who found that, when they went north, more stars stayed above the horizon all night, and that, when they went south, additional stars became visible. Eratosthenes (276–194 B.C. ) determined the earth’s size with remarkable accuracy by comparing the length of the sun’s shadow at noon on the same day in two places on the same north-south line.
Polaris
Horizon
Horizon
View from B
View from A
Polaris
H orizon A
Apparent paths of stars
Horizon B
Figure 1-18 Gravity forces the earth to be round. (a) How a large bump would be pulled down. (b) How a large hole would be filled in.
The earth is not a perfect sphere. The reason was apparent to Newton: since the earth is spinning rapidly, inertia causes the equatorial portion to swing outward, just as a ball on a string does when it is whirled around. As a result the earth bulges slightly at the equator and is slightly flattened at the poles, much like a grapefruit. The total distortion is not great, for the earth is only 43 km wider than it is high ( Fig. 1-19 ). Venus, whose “day” is 243 of our days, turns so slowly that it has almost no distortion. Saturn, at the other extreme, spins so rapidly that it is almost 10 per- cent out of round.
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Figure 1-19 The influence of its rotation distorts the earth. The effect is greatly exaggerated in the figure; the equatorial diameter of the earth is actually only 43 km (27 mi) more than its polar diameter.
Actual shape of earth Perfect sphere 1.10 The Tides
Up and Down Twice a Day Those of us who live near an ocean know well the rhythm of the tides, the twice-daily rise and fall of water level. Usually the change in height is no more than a few meters, but in some regions—the Bay of Fundy in eastern Canada is one—the total range can be over 20 m. The gravitational pull of the moon is what causes the advance and retreat of the oceans on this grand scale.
The moon gravitationally attracts different parts of the earth to different extents. In Fig. 1-20 the moon’s tug is strongest at A, which is closest, and weakest at B, which is far- thest away. Also, the rotation of the moon around the earth is too simple a picture—what actually happens is that both bodies rotate around the center of mass (CM) of the earth- moon system. (Think of the earth and the moon as opposite ends of a dumbbell. The CM is the balance point of the dumbbell; it is inside the earth 4700 km from its center.)
As it wobbles around the CM, the solid earth is pulled away from the water at B, where the moon’s tug is weakest, to leave the water there heaped up in a tidal bulge. At A, the greater tug of the moon dominates to cause a tidal bulge there as well. The bulges stay in place as the earth revolves under them to produce two high tides and two low tides at a given place every day ( Fig. 1-21 ).
Figure 1-20 The origin of the tides. The moon’s attraction for the waters of the earth is greatest at A, least at B. As the earth and moon rotate around the center of mass of the earth-moon system, which is located inside the earth, water is heaped up at A and B. The water bulges stay in place as the earth turns on its axis to produce two high and two low tides every day. As the earth turns under the bulges, friction between the oceans and the ocean floors slows down the earth’s rotation. As a result the tidal bulges lag slightly behind the earth-moon line. One effect of tidal friction is thus to lengthen the day. The rate of increase is a mere 1 s per day in every 43,500 years, but it adds up. Measurements of the daily growth markings on fossil corals show that the day was only 22 h long 380 million years ago. The other effect is that, because the tidal bulge facing the moon is behind the earth-moon line, the bulge exerts a force on the moon that pulls it slightly forward in its orbit, which causes the orbit to grow larger. As a result the moon is moving away from the earth at about 4 cm per year.
Figure 1-21 High and low water in the Bay of Fundy at Blacks Harbour in New Brunswick, Canada. Two tidal cycles occur daily.
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Spring and Neap Tides There is more to the story. The sun also affects the waters of the earth, but to a smaller extent than the moon even though the gravitational tug of the sun exceeds that of the moon. The reason is that what is involved in the tides is the difference between the attractions on the near and far sides of the earth, and this difference is greater for the moon because it is closer to the earth than the sun. About twice a month—when the sun, moon, and earth are in a straight line—solar tides add to lunar tides to give the especially high (and low) spring tides; see Fig. 1-22 . When the line between moon and earth is perpendicular to that between sun and earth, the tide-raising forces partly cancel to give neap tides, whose range is smaller than average.
1.11 The Discovery of Neptune Another Triumph for the Law of Gravity In Newton’s time, as in Ptolemy’s, only six planets were known: Mercury, Venus, Earth, Mars, Jupiter, and Saturn. In 1781 a seventh, Uranus, was identified. Measure- ments during the next few years enabled astronomers to work out details of the new planet’s orbit and to predict its future positions in the sky. To make these predictions, not only the sun’s attraction but also the smaller attractions of the nearby planets Jupiter and Saturn had to be considered. For 40 years, about half the time needed for Uranus to make one complete revolution around the sun, calculated positions of the planet agreed well with observed positions.
Then a discrepancy crept in. Little by little Uranus moved away from its predicted path among the stars. The calculations were checked and rechecked, but no mistake could be found. There were two possibilities: either the law of gravity, on which the
Figure 1-22 Variation of the tides. Spring tides are produced when the moon is at M 1 or M 2 , neap tides when the moon is at M 3 or M 4 . The range between high and low water is greatest for spring tides.
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calculations were based, was wrong, or else some unknown body was pulling Uranus away from its predicted path.
So firmly established was the law of gravitation that two young men, Urbain Leverrier in France and John Couch Adams in England, set themselves the task of calculating the orbit of an unknown body that might be responsible for the discrepan- cies in Uranus’s position. Adams sent a sketchy account of his studies to George Airy, England’s Astronomer Royal. Because the calculations were incomplete, although later found to be correct as far as they went, Airy asked for further details. Adams (who later blamed habitual lateness and a dislike of writing) did not respond.
A year later, in 1846, Leverrier, with no knowledge of Adams’s work, went further and proposed an actual position in the sky where the new planet should be found. He sent his result to a German astronomer, Johan Gottlieb Galle, who turned his telescope to the part of the sky where the new planet should appear. Very close to the position predicted by Leverrier, Galle found a faint object, which had moved slightly by the following night. This was indeed the eighth member of the sun’s family and was called Neptune. The theory of gravity had again successfully gone around the loop of the scientific method shown in Fig. 1-1 . In 2011 Neptune completed the first circuit of the sun since its discovery 165 years earlier.
When we say that the distance between Chicago and Minneapolis is 405 miles, what we are really doing is comparing this distance with a certain standard length called the mile. Standard quantities such as the mile are known as units. The result of every measurement thus has two parts. One is a number (405 for the Chicago-Minneapolis distance) to answer the question “How many?” The other is a unit (the mile in this case) to answer the question “Of what?”
1.12 The SI System All Scientists Use These Units The most widely used units today are those of the International System, abbreviated SI after its French name Système International d’Unités. Examples of SI units are the meter (m) for length, the second (s) for time, the kilogram (kg) for mass, the joule (J) for energy, and the watt (W) for power. SI units are used universally by scien- tists and in most of the world in everyday life as well. Although the British system of units, with its familiar foot and pound, remains in common use only in a few English- speaking countries, it is on the way out and eventually will be replaced by the SI. Since this is a book about science, only SI units will be used from here on.
The great advantage of SI units is that their subdivisions and multiples are in steps of 10, 100, 1000, and so on, in contrast to the irregularity of British units. In the case of lengths, for instance ( Fig. 1-23 ),
1 meter (m) 5 100 centimeters (cm) 1 kilometer (km) 5 1000 meters
whereas
1 foot (ft) 5 12 inches (in.) 1 mile (mi) 5 5280 feet
Table 1-1 lists the most common subdivisions and multiples of SI units. Each is designated by a prefix according to the corresponding power of 10. (Powers of 10 are widely used in science. What they mean and how to make calculations with them are reviewed in the Math Refresher at the back of the book starting on p. A-1.)
HOW MANY OF WHAT
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How Many of What 21
Table 1-2 contains conversion factors for changing a length expressed in one sys- tem to its equivalent in the other. (More conversion factors are given inside the back cover of this book.) We note from the table that there are about 2 1 _ 2 cm in an inch, so a centimeter is roughly the width of a shirt button; a meter is a few inches longer than 3 feet; and a kilometer is nearly 2 _ 3 mile. Example 1.3 shows how conversion factors are used.
The meter and gram were new units. The ancient division of a day into 24 hours, an hour into 60 minutes, and a minute into 60 seconds was kept for the definition of the second as 1/(24)(60)(60) 5 1/86,400 of a day.
As more and more precision became needed, these definitions were modified several times. Today the second is specified in terms of the microwave radiation given off under certain circumstances by one type of cesium atom, 133Cs: 1 s equals the time needed for 9,192,631,770 cycles of this radiation to be emitted.
The meter, which for convenience had become the distance between two scratches on a platinum-iridium bar
kept at Sèvres, France, is now the dis- tance traveled in 1/299,792,458 s by light in a vacuum. There are approxi- mately 3.28 feet in a meter.
The kilogram (km) is the mass of a platinum-iridium cylinder 39 mm in diameter and 39 mm high at Sèvres. Despite much effort, a unit of mass based on a physical property measurable anywhere has not proved practical as yet. As dis- cussed in Sec. 2.10, mass and weight are not the same. The weight of a given mass is the force with which gravity attracts it to the earth; the weight of 1 kg is 2.2 pounds on the earth’s surface and decreases with altitude (see Fig. 2-38).
SI units are derived from the units of the older metric system. This system was introduced in France at the end of the eighteenth century to replace the hodgepodge of traditional units, often different in different countries and even in different parts of the same country, that was making commerce and industry difficult.
The meter (m), the standard of length, was originally defined as one ten-millionth of the distance from the equator to the North Pole. The gram (g), the standard of mass, was defined as the mass of 1 cubic centi- meter (cm3) of water; 1 cm3 is the vol- ume of a cube 1 cm (0.01 m) on each edge, and 1 kilogram 5 1000 grams.
Meter, Kilogram, Second
Figure 1-23 There are 1000 meters in a kilometer and 100 centimeters in a meter.
Table 1-1 Subdivisions and Multiples of SI Units (The Symbol m Is the Greek Letter “mu”)
Prefix Power of 10 Abbreviation Pronunciation Common Name
Pico- 10−12 p pee’ koe Trillionth Nano- 10−9 n nan’ oe Billionth Micro- 10−6 m my’ kroe Millionth Milli- 10−3 m mil’ i Thousandth Centi- 10−2 c sen’ ti Hundredth Hecto- 102 h hec’ toe Hundred Kilo- 103 k kil’ oe Thousand Mega- 106 M meg’ a Million Giga- 109 G ji’ ga Billion Tera- 1012 T ter’ a Trillion
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22 Chapter 1 The Scientific Method
Significant Figures In Example 1-3 the distance expressed in centimeters is 8.78 3 10 4 cm. Does this mean that d 5 87,800 cm exactly?
The answer is, not necessarily. We are only sure of the three digits 8, 7, and 8. By writing d 5 8.78 3 10 4 cm we can see just how precisely the distance is being expressed. If we needed less precision, we could round off the value of d to 8.8 3 10 4 cm. How- ever, we could not write d 5 8.780 3 10 4 cm because this implies greater precision than that of the original statement.
The accurately known digits in a number, plus one uncertain digit, are its significant figures. When quantities are used in calculations, the result is no more accurate than the quantity with the largest uncertainty. Suppose a 65-lb girl picks up a 0.23-lb apple. The total weight of the girl 1 apple is still 65 lb because all we know about the girl’s weight is that it is somewhere between 64.5 and 65.5 lb, which means an uncertainty greater than the apple’s weight. If the girl’s weight is instead known to be 65.0 lb, the mass of girl 1 apple is 65.2 lb; if her weight is instead known to be 65.00 kg, the weight of girl 1 apple is 65.23 kg. Thus
65 lb 1 0.23 lb 5 65 lb 65.0 lb 1 0.23 lb 5 65.2 lb
65.00 lb 1 0.23 lb 5 65.23 lb
Significant figures must be taken into account in multiplication and division also. For example, in part (b) of Example 1-3, the actual result of the calculation is
d 5 (0.878 km)(0.621 mi/km) 5 0.545238 mi
Example 1.2
How many nanometers are in a kilometer?
Solution A nanometer is a billionth (10 2 9 ) of a meter and a kilometer is a thousand (10 3 ) meters. Hence
kilometer _________ nanometer 5 103 m _______
1029 m 5 1012
There are 10 12 —a trillion—nanometers in a kilometer. [We note from the Math Refresher that 10 n /10 m 5 10 n 2 m , so here 10 3 /10 2 9 5 10 3 2 ( 2 9) 5 10 3 1 9 5 10 12 .]
Table 1-2 Conversion Factors for Length
Multiply a Length Expressed in By
To Get the Same Length Expressed in
Centimeters 0.394 in. ___ cm Inches
Meters 39.4 in. ___ m Inches
Meters 3.28 ft __ m Feet
Kilometers 0.621 mi ___ km
Miles
Inches 2.54 cm ___ in. Centimeters
Feet 30.5 cm ___ ft
Centimeters
Feet 0.305 m __ ft
Meters
Miles 1.61 km ___ mi Kilometers
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Important Terms and Ideas 23
Because both the initial numbers had only three significant figures, the result can only have three also, so it was given as d 5 0.545 mi.
When a calculation has several steps, it is a good idea to keep an extra digit in the intermediate steps. Then, at the end, the final result can be rounded off to the correct number of significant figures.
For simplicity, in this book zeros after the decimal point have usually been omit- ted from values given in problems. For instance, it should be assumed that when a length of 7 m is stated, what is really meant is 7.000 . . . m.
Example 1.3
A few years ago a NASA official quoted a distance of 878 m to a reporter and added, “I don’t know what this is in terms of kilometers or miles.” Let us help him.
Solution (a) Since 1 km 5 10 3 m 5 1000 m, the distance in kilometers is
d 5 878 m__________ 1000 m/km
5 0.878 km
We note that 1______
m/km 5 km___m
and therefore m______
m/km 5
(m)(km)________ m 5 km
If instead we wanted this distance in centimeters, we would proceed in this way:
d 5 (878 m)(102 cm/m) 5 878 3 102 cm 5 (8.78 3 102)(102) cm 5 8.78 3 10212 cm 5 8.78 3 104 cm
This is the usual way such a quantity would be expressed. The Math Refresher at the end of the book might come in handy here. (b) From Table 1-2 the conversion factor we need is 0.621 mi/km, so
d 5 (0.878 km) ( 0.621 mi___km ) 5 0.545 mi
forces the sun exerts on the planets are what hold them in their orbits. Kepler’s laws are explained by the law of gravity.
The tides are periodic rises and falls of sea level caused by differences in the gravitational pulls of the moon and sun. Water facing the moon is attracted to it more than the earth itself is, and the earth moves away from water on its far side. The correspond- ing effect of the sun is smaller than that of the moon and acts to increase or decrease tidal ranges, depending on the relative posi- tions of the moon and sun.
To measure something means to compare it with a standard quantity of the same kind called a unit. The SI system of units is used everywhere by scientists and in most of the world in every- day life as well. The SI unit of length is the meter (m).
The significant figures in a number are its accurately known digits. When numbers are combined arithmetically, the result has as many significant figures as those in the number with the fewest of them.
The scientific method of studying nature has four steps: (1) formulating a problem; (2) observation and experiment; (3) interpreting the results; (4) testing the interpretation by further observation and experiment. When first proposed, a scientific interpretation is called a hypothesis. After thorough checking, it becomes a law if it states a regularity or relationship, or a theory if it uses general considerations to account for specific phenomena.
Polaris, the North Star, lies almost directly above the North Pole. A constellation is a group of stars that form a pattern in the sky. The planets are heavenly bodies that shift their positions regularly with respect to the stars.
In the ptolemaic system, the earth is stationary at the center of the universe. In the copernican system, the earth rotates on its axis and, with the other planets, revolves around the sun. Obser- vational evidence supports the copernican system.
Kepler’s laws are three regularities that the planets obey as they move around the sun.
Newton’s law of gravity describes the attraction all bod- ies in the universe have for one another. The gravitational
Important Terms and Ideas
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24 Chapter 1 The Scientific Method
c. combinations of circles forming looped orbits d. the same distance apart from one another
12. The speed of a planet in its orbit around the sun a. is constant b. is highest when the planet is closest to the sun c. is lowest when the planet is closest to the sun d. varies, but not with respect to the planet’s distance from
the sun 13. According to Kepler’s third law, the time needed for a planet
to complete an orbit around the sun a. is the same for all the planets b. depends on the planet’s size c. depends on the planet’s distance from the sun d. depends on how fast the planet spins on its axis
14. The law of gravity a. applies only to large bodies such as planets and stars b. accounts for all known forces c. holds only in the solar system d. holds everywhere in the universe
15. The earth bulges slightly at the equator and is flattened at the poles because
a. it spins on its axis b. it revolves around the sun c. of the sun’s gravitational pull d. of the moon’s gravitational pull
16. The usual tidal pattern in most parts of the world consists of a. a high tide one day and a low tide on the next b. one high tide and one low tide daily c. two high tides and two low tides daily d. three high tides and three low tides daily
17. Tides are caused a. only by the sun b. only by the moon c. by both the sun and the moon d. sometimes by the sun and sometimes by the moon
18. High tide occurs at a given place a. only when the moon faces the place b. only when the moon is on the opposite side of the earth
from the place c. both when the moon faces the place and when the moon
is on the opposite side of the earth from the place d. when the place is halfway between facing the moon and
being on the opposite side of the earth from the moon 19. The prefix micro stands for
a. 1/10 c. 1/1000 b. 1/100 d. 1/1,000,000
20. A centimeter is a. 0.001 m c. 0.1 m b. 0.01 m d. 10 m
21. Of the following, the shortest is a. 1 mm c. 0.001 m b. 0.01 in. d. 0.001 ft
1. The “scientific method” is a. a continuing process b. a way to arrive at ultimate truth c. a laboratory technique d. based on accepted laws and theories
2. A scientific law or theory is valid a. forever b. for a certain number of years, after which it is retested c. as long as a committee of scientists says so d. as long as it is not contradicted by new experimental
findings 3. A hypothesis is
a. a new scientific idea b. a scientific idea that has been confirmed by further
experiment and observation c. a scientific idea that has been discarded because it
disagrees with further experiment and observation d. a group of linked scientific ideas
4. The ongoing evolution of living things a. is one of the basic concepts of biology b. is one of the basic concepts of “intelligent design” c. has no basis in observation d. is in conflict with all religions
5. The object in the sky that apparently moves least in the course of time is
a. Polaris c. the sun b. Venus d. the moon
6. A constellation is a. an especially bright star b. an apparent pattern of stars in the sky c. a group of stars close together in space d. a group of planets close together in space
7. Which of the following is no longer considered valid? a. the ptolemaic system b. the copernican system c. Kepler’s laws of planetary motion d. Newton’s law of gravity
8. A planet not visible to the naked eye is a. Mars c. Mercury b. Venus d. Neptune
9. The planet closest to the sun is a. earth c. Mars b. Venus d. Mercury
10. Leap years are needed because a. the earth’s orbit is not a perfect circle b. the length of the day varies c. the length of the year varies d. the length of the year is not a whole number of days
11. Kepler modified the copernican system by showing that the planetary orbits are
a. ellipses b. circles
Multiple Choice
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22. Of the following, the longest is a. 1000 ft c. 1 km b. 500 m d. 1 mi
23. A person is 180 cm tall. This is equivalent to a. 4 ft 6 in. c. 5 ft 11 in. b. 5 ft 9 in. d. 7 ft 1 in.
24. The measurements of a room are given as length 5 5.28 m and width 5 3.1 m. Since (5.28)(3.1) 5 16.368, the room’s area is correctly expressed as
a. 16 m 2 c. 16.4 m 2 b. 16.0 m 2 d. 16.368 m 2
complete circuit; (c) the eastward drift of the moon rela- tive to the stars that takes about 4 weeks for a complete circuit.
16. What do you think is the reason scientists use an ellipse rather than a circle as the model for a planetary orbit?
17. The average distance from the earth to the sun is called the astronomical unit (AU). If an asteroid is 4 AU from the sun and its period of revolution around the sun is 8 years, does it obey Kepler’s third law?
1.7 Why Copernicus Was Right
18. As the earth revolves around the sun, some stars seem to shift their positions during the year relative to other stars. How is this effect (called parallax) explained in the ptolemaic system? In the copernican system?
1.8 What Is Gravity?
19. Why is gravity considered a fundamental force whereas the force a bat exerts on a ball is not?
1.9 Why the Earth Is Round
20. What, if anything, would happen to the shape of the earth if it were to rotate on its axis faster than it does today?
1.10 The Tides
21. What is the difference between spring and neap tides? Under what circumstances does each occur?
22. The length of the day has varied. When did the longest day thus far occur?
23. The earth takes almost exactly 24 h to make a complete turn on its axis, so we might expect each high tide to occur 12 h after the one before. However, the actual time between high tides is 12 h 25 min. Can you account for the difference?
24. Does the sun or the moon have the greater influence in causing tides?
1.12 The SI System
25. In the following pairs of length units, which is the shorter: inch, centimeter? Yard, meter? Mile, kilometer?
26. A European driving from Paris to Brussels finds she has covered 291 km. How many miles is this?
27. The world’s tallest tree is a sequoia in California 368 ft high. How high is this in meters? In kilometers?
28. The diameter of an atom is roughly 10 4 times the diam- eter of its nucleus. If the nucleus of an atom were 1 mm across, how many feet across would the atom be?
1.2 Why Science Is Successful
1. What role does “common sense” play in the scientific method?
2. What is the basic distinction between the scientific method and other ways of looking at the natural world?
3. What is the difference between a hypothesis and a law? Between a law and a theory?
4. Scientific models do not correspond exactly to reality. Why are they nevertheless so useful?
5. According to the physicist Richard Feynman, “Science is the culture of doubt.” Does this mean that science is an unreliable guide to the natural world?
1.3 A Survey of the Sky
6. What does a year correspond to in terms of observations of the sun and stars?
7. You are lost in the northern hemisphere in the middle of nowhere on a clear night. How could you tell the direc- tion of north by looking at the sky?
8. In terms of what you would actually observe, what does it mean to say that the moon apparently moves eastward among the stars?
9. What must be your location if the stars move across the sky in circles centered directly overhead?
1.5 The Copernican System
10. How do leap years fit into the ptolemaic system? Into the copernican system?
11. From observations of the moon, why would you conclude that it is a relatively small body revolving around the earth rather than another planet revolving around the sun?
12. The sun, moon, and planets all follow approximately the same path from east to west across the sky. What does this suggest about the arrangement of these members of the solar system in space?
13. What is the basic difference between the ptolemaic and copernican models? Why is the ptolemaic model consid- ered incorrect?
14. Ancient astronomers were troubled by variations in the brightnesses of the various planets with time. Does the ptolemaic or the copernican model account better for these variations?
15. Compare the ptolemaic and copernican explanations for (a) the rising and setting of the sun; (b) the eastward drift of the sun relative to the stars that takes a year for a
Exercises
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29. How many square feet are there in an area of 1.00 square meters? Use the proper number of significant figures in the answer.
30. A swimming pool is 20.00 m long, 7.00 m wide, and 2.00 m deep. What is its volume in cubic feet to 3 significant figures?
31. The speedometer of a European car gives its speed in kilometers per hour. What is the car’s speed in miles per hour when the speedometer reads 80?
32. A horse galloped a mile in 2 min 35 s. What was its aver- age speed in km/h?
33. How many microphones are there in a megaphone?
34. Use the proper number of significant figures to express the values of
a. 47.2 1 9.11 2 14 b. (3.58 3 10 2 )(2.1 3 10 3 )
c. 7.8 3 10 3 ___________
3.21 3 1022 1 5.4 3 104
35. Use the proper number of significant figures to express the values of
a. 6.23 3 10 7 1 4.71 3 10 8 1 3.04 3 10 9 b. (9.003 3 10 3 )(2.41 3 10 4 ) c. 9.36/4.0
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“Both defendants and many of the leading proponents of ID make a bedrock assumption which is utterly false. Their presupposition is that evolutionary theory is antithetical to a belief in the existence of a supreme being and to religion in general. Repeatedly in this trial, plaintiffs’ scientific experts testified that the theory of evolution represents good science, is overwhelmingly accepted by the scientific community, and that it in no way conflicts with, nor does it deny, the existence of a divine creator.
“To be sure, Darwin’s theory of evolution is imperfect. However, the fact that a scientific theory cannot yet render an explanation on every point should not be used as a pre- text to thrust an untestable alterna- tive hypothesis grounded in religion into the science classroom or to mis- represent well-established scientific propositions.
“The citizens of the Dover area were poorly served by the members of the board who voted for the ID policy. It is ironic that several of these indi- viduals, who so staunchly and proudly touted their religious convictions in public, would time and again lie to cover their tracks and disguise the real purpose behind the ID policy. . . .
“The breathtaking inanity of the board’s decision is evident when con- sidered against the factual backdrop which has now been fully revealed through this trial. The students, parents, and teachers of the Dover Area School District deserved better than to be dragged into this legal maelstrom, with its resulting utter waste of monetary and personal resources.”
Dover is not the only place in the United States where science teaching is in danger. In order to impose their personal religious beliefs, such as ID, on its school system, the members of the Kansas State Board of Education not long ago rewrote its official defini- tion of science. The original definition called science “the human activity of seeking natural explanations for what we observe in the world around us.” The new definition, called “lunacy” by The New York Times, omitted the word “natural.” Supernatural explanations were to be acceptable in Kansas science classrooms. Did the parents of Kansas really want their children to believe that “what we observe in the world around us” is the work of witches and warlocks, ghosts and goblins, pixies and the Tooth Fairy? They did not: a later Board rebottled the genie of unreason.
In 2004 the school board of Dover, Pennsylvania, required that Intelli- gent Design (ID) be introduced in the science classes of its high school. A lawsuit was then filed by 11 alarmed parents who accused the board of violating the First Amendment to the U.S. Constitution, which has been ruled to prohibit public officials from pursuing religious agendas in their work. After a 6-week trial, federal judge John E. Jones III agreed that the board’s action was indeed illegal. “We conclude that the religious nature of intelligent design would be readily apparent to an objective observer, adult or child. The writ- ings of leading ID proponents reveal that the designer postulated by their arguments is the God of Christianity.” Here are some further excerpts from his 137-page opinion [U.S. District Court for the Middle District of Pennsylvania, Kitzmiller, et al. v. Dover Area School District, et al., Case no. O4cv2688]:
“In making this determination, we have addressed the seminal ques- tion of whether ID is science. We have concluded that it is not, and moreover that ID cannot uncouple itself from its creationist, and thus religious, antecedents.
Section 1.2: Intelligent Design in Court
1-1 1
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Astronaut Bruce McCandless near the orbiting Space Shuttle Challenger.
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2
Describing Motion 2.1 Speed
How Fast Is Fast • The difference between instantaneous
and average speeds. • How to solve problems that involve
speed, distance, and time. 2.2 Vectors
Which Way as Well as How Much • The difference between scalar and vec-
tor quantities. • How to use the Pythagorean theorem
to add two vector quantities of the same kind that act at right angles to each other.
2.3 Acceleration Vroom!
• What is meant by acceleration. • How to solve problems that involve
speed, acceleration, and time. 2.4 Distance, Time, and Acceleration
How Far? • How to solve problems that involve
speed, acceleration, time, and distance.
Acceleration Due to Gravity 2.5 Free Fall
A Downward Acceleration • What is meant by the acceleration of
gravity. • How to use the horizontal and verti-
cal components of the velocity of an object to determine its motion.
2.6 Air Resistance Why Raindrops Don’t Kill
• How air resistance affects falling objects.
Force and Motion 2.7 First Law of Motion
Constant Velocity Is as Natural as Being at Rest
• What is meant by force. 2.8 Mass
A Measure of Inertia • The relationship between mass and
inertia. • How friction differs from inertia.
2.9 Second Law of Motion Force and Acceleration
• How to use the formula F 5 ma to solve problems that involve force and motion.
2.10 Mass and Weight Weight Is a Force
• The difference between mass and weight and how they are related.
2.11 Third Law of Motion Action and Reaction
• The relationship between action and reaction forces.
Gravitation 2.12 Circular Motion
A Curved Path Requires an Inward Pull • The significance of centripetal force in
motion along a curved path. 2.13 Newton’s Law of Gravity
What Holds the Solar System Together • How the gravitational force between
two objects depends on their masses and their separation.
2.14 Artificial Satellites Thousands Circle the Earth
• How a satellite can move around the earth in a stable orbit.
• What is meant by escape speed.
CHAPTER OUTLINE AND GOALS
Your chief goal in reading each section should be to understand the important findings and ideas indicated (•) below.
Motion
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28 Chapter 2 Motion
Everything in the universe is in nonstop movement. Whatever the scale of size, from the tiny particles inside atoms to the huge galaxies of stars far away in space, motion is the rule, not the exception. In order to understand the universe, we must begin by understanding motion and the laws the scientific method shows it to obey.
The laws of motion that govern the behavior of atoms and stars apply just as well to the objects of our daily lives. Engineers need these laws to design cars and airplanes, machines of all kinds, even roads—how steeply to bank a highway curve is calculated from the same basic formula that Newton combined with Kepler’s findings to arrive at the law of gravity. Terms such as speed and acceleration, force and weight, are used by everyone. Let us now see exactly what these terms mean and how the quantities they refer to are related.
When an object goes from one place to another, we say it moves. If the object gets there quickly, we say it moves fast; if the object takes a long time, we say it moves slowly. The first step in analyzing motion is to be able to say just how fast is fast and how slow is slow.
2.1 Speed How Fast Is Fast The speed of something is the rate at which it covers distance. The higher the speed, the faster it travels and the more distance it covers in a given period of time.
If a car goes through a distance of 40 kilometers in a time of 1 hour, its speed is 40 kilometers per hour, usually written 40 km/h.
What if the time interval is not exactly 1 hour? For instance, the car might travel 60 km in 2 hours on another trip. The general formula for speed is distance divided by time:
Speed 5 distance _______ time
Hence the car’s speed in the second case is
Speed 5 distance _______ time 5 60 km ______
2 h 5 30 km/h
The same formula works for times of less than a full hour. The speed of a car that covers 24 km in half an hour is, since 1 _ 2 h 5 0.5 h,
Speed 5 distance _______ time 5 24 km ______ 0.5 h
5 48 km/h
These speeds are all average speeds, because we do not know the details of how the cars moved during their trips. They probably went slower than the average dur- ing some periods, faster at others, and even came to a stop now and then at traffic lights. What the speedometer of a car shows is the car’s instantaneous speed at any moment, that is, how fast it is going at that moment ( Fig. 2-1 ).
For the sake of convenience, quantities such as distance, time, and speed are often abbreviated and printed in italics:
d 5 distance t 5 time v 5 speed
In terms of these symbols the formula for speed becomes
v 5 d __ t Speed 2-1
DESCRIBING MOTION
Figure 2-1 The speedometer of a car shows its instantaneous speed. This speedometer is calibrated in both mi/h (here MPH) and km/h.
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Distance The previous formula can be rewritten in two ways. Suppose we want to know how far a car whose average speed is v goes in a time t. To find out, we must solve v 5 d / t for a distance d. According to one of the rules of algebra (see the Math Refresher at the back of this book), a quantity that divides one side of an equation can be shifted to multiply the other side. Thus
v 5 d __ t
becomes
v 5 d __ t
vt 5 d
which is the same as
d 5 vt 2-2
Distance 5 (speed)(time)
Time In another situation we might want to know how long it takes something moving at a certain speed to cover a certain distance. In other words, we know v and d and want to find the time t. What we do here is solve d 5 vt for the time t. From basic algebra we know that something that multiplies one side of an equation can be shifted to divide the other side. What we do, then, is shift the v in the formula d 5 vt to divide the d:
d 5 v t
d __ v 5 t
which is the same as
t 5 d __ v 2-3
Time 5 distance _______ speed
Frame of Reference
When we say something is mov- ing, we mean that its position relative to something else—the frame of reference —is chang- ing. The choice of an appropriate frame of reference depends on the situation. In the case of a car, for instance, the obvious frame of reference is the road it is on.
In other cases things may not be so straightforward. If we use the earth as our frame of ref- erence, the sun and planets move relative to us in complicated paths, as in Figs. 1-5 and 1-7. On the other hand, if we use the sun as our frame of reference, the earth and the other planets move relative to it in simple paths, as in Fig. 1-8. Newton was able to interpret these motions in terms of the gravitational pull of the sun, whereas he would not have been able to discover the law of gravity if he had used the earth as his frame of reference.
Figure 2-2 A car whose average speed is 40 km/h travels 240 km in 6 hours.
Example 2.1
How far does a car travel in 6 hours when its average speed is 40 km/h?
Solution We put v 5 40 km/h and t 5 6 h into Eq. 2-2 to find that ( Fig. 2-2 )
d 5 vt 5 ( 40 km ___ h ) (6 h) 5 240 ( km ___ h ) (h) 5 240 km We see that, since h/h 5 1, the hours cancel out to give just kilometers in the answer.
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2.2 Vectors Which Way as Well as How Much Some quantities need only a number and a unit to be completely specified. It is enough to say that the area of a farm is 600 acres, that the frequency of a sound wave is 440 cycles per second, that a lightbulb uses electric energy at the rate of 75 watts. These are examples of scalar quantities. The magnitude of a quantity refers to how large it is. Thus the magnitudes of the scalar quantities given above are respectively 600 acres, 440 cycles per second, and 75 watts.
A vector quantity, on the other hand, has a direction as well as a magnitude asso- ciated with it, and this direction can be important. Displacement (change in position) is an example of a vector quantity. If we drive 1000 km north from Denver, we will end up in Canada; if we drive 1000 km south, we will end up in Mexico.
Force is another example of a vector quantity. Applying enough upward force to this book will lift it from the table. Applying a force of the same magnitude downward on the book will press it harder against the table, but the book will not move.
Speed and Velocity The speed of a moving object tells us only how fast the object is going, regardless of its direction. Speed is therefore a scalar quantity. If we are told that a car has a speed of 40 km/h, we do not know where it is headed, or even if it is moving in a straight line—it might well be going in a circle. The vector
Example 2.2
You are standing 100 m north of your car when an alligator appears 20 m north of you and begins to run toward you at 8 m/s, as in Fig. 2-3 . At the same moment you start to run toward your car at 5 m/s. Will you reach the car before the alligator reaches you?
Solution You are 100 m from your car and so would need
t1 5 d1 __ v1 5
100 m ______ 5 m/s 5 20 s
to reach it. The alligator is 120 m from the car but would need only
t2 5 d2 __ v2 5
120 m ______ 8 m/s 5 15 s
to reach it. Hence the alligator would overtake you before you get to the car. Too bad.
Figure 2-3 Watch out for alligators.
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quantity that includes both speed and direction is called velocity. If we are told that a car has a constant velocity of 40 km/h toward the west, we know all there is to know about its motion and can easily figure out where it will be in an hour, or 2 hours, or at any other time.
Vectors A handy way to represent a vector quantity on a drawing is to use a straight line called a vector that has an arrowhead at one end to show the direction of the quantity. The length of the line is scaled according to the magnitude of the quantity. Figure 2-4 shows how a velocity of 40 km/h to the right is represented by a vector on a scale of 1 cm 5 10 km/h. All other vector quantities can be pictured in a similar way.
Vector quantities are usually printed in boldface type ( F for force, v for veloc- ity). Italic type is used for scalar quantities ( f for frequency, V for volume). Italic type is also used for the magnitudes of vector quantities: F is the magnitude of the force F; v is the magnitude of the velocity v. For instance, the magnitude of a velocity v of 40 km/h to the west is the speed v 5 40 km/h. A vector quantity is usually indicated in handwriting by an arrow over its symbol, so that
_ › F means the same thing as F.
Adding Vectors To add scalar quantities of the same kind, we just use ordinary arithmetic. For example, 5 kg of onions plus 3 kg of onions equals 8 kg of onions. The same method holds for vector quantities of the same kind whose directions are the same. If we drive north for 5 km and then continue north for another 3 km, we will go a total of 8 km to the north.
What if the directions are different? If we drive north for 5 km and then east for 3 km, we will not end up 8 km from our starting point. The vector diagram of Fig. 2-5 that shows these displacements provides the answer. To add the vectors A and B, we draw B with its tail at the head of A. Connecting the tail of A with the head of B gives us the vector C, which corresponds to our net displacement from the start of our trip to its finish. The length of C tells us that our displacement was slightly less than 6 km. Any number of vectors of the same kind (for instance, velocity) can be added in this way by stringing them together tail to head and then joining the tail of the first with the head of the last one.
Pythagorean Theorem A right triangle is one in which two of its sides are per- pendicular, that is, meet at a 90 8 angle. The Pythagorean theorem is a useful rela- tionship that holds in such a triangle. This theorem states that the sum of the squares of the short sides of a right triangle is equal to the square of its hypotenuse (longest side). For the triangle of Fig. 2-5 ,
A2 1 B 2 5 C 2 Pythagorean theorem 2-4
where A, B, and C are the respective magnitudes of the vectors A, B, and C. We can therefore express the length of any of the sides of a right triangle in terms
of the other sides by solving Eq. 2-4 accordingly:
A 5 √ _______
C2 2 B2 2-5
B 5 √ _______
C2 2 A2 2-6
C 5 √ _______
A2 1 B2 2-7
Figure 2-4 The vector v represents a velocity of 40 km/h to the right. The scale is 1 cm 5 10 km/h as shown at the bottom of the figure.
Figure 2-5 Adding vector B (3 km east) to vector A (5 km north) gives vector C whose length corresponds to 5.83 km. According to the Pythagorean theorem, A 2 1 B 2 5 C 2 in any right triangle.
A
B
C
N
S
W E
0 1 2 3 4 5 6 km
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2.3 Acceleration Vroom! An accelerated object is one whose velocity is changing. As in Fig. 2-6 , the change can be an increase or a decrease in speed—the object can be going faster and faster, or slower and slower ( Fig. 2-7 ). A change in direction, too, is an acceleration, as discussed later. Acceleration in general is a vector quantity. For the moment, though, we will stick to straight-line motion, where acceleration is the rate of change of speed. That is,
a 5 v2 2 v1 ______ t Straight-line motion 2-8
Acceleration 5 change in speed
_____________ time interval
where the symbols mean the following:
a 5 acceleration t 5 time interval v 1 5 speed at start of time interval 5 initial speed v 2 5 speed at end of time interval 5 final speed
Not all accelerations are constant, but a great many are very nearly so. In what follows all accelerations are assumed to be constant. In Sec. 2-7 we will see why accel- eration is such an important quantity in physics.
Figure 2-6 Three cases of accelerated motion, showing successive positions of a body after equal periods of time. ( a ) The intervals between the positions of the body increase in length because the body is traveling faster and faster. ( b ) The intervals decrease in length because the body is slowing down. ( c ) Here the intervals are the same in length because the speed is constant, but the direction of motion is constantly changing.
(a )
(b )
(c )
Figure 2-7 Express elevators in tall buildings have accelerations of no more than 1 m/s 2 to prevent passenger discomfort. One of the world’s fastest elevators, in the Yokohama Landmark Tower (Japan’s highest building), climbs 69 floors in 40 s, but only 5 s is spent at its top speed of 12.5 m/s (28 mi/h). The elevator reaches this speed at the 27th floor and then begins to slow down at the 42nd floor.
Example 2.3
Use the Pythagorean theorem to find the displacement of a car that goes north for 5 km and then east for 3 km, as in Fig. 2-5 .
Solution Here we let A 5 5 km and B 5 3 km. From Eq. 2-7 we have
C 5 √ ________
A2 1 B2 5 √ _______________
(5 km)2 1 (3 km)2 5 √ ___________
(25 1 9) km2
5 √ ______
34 km2 5 5.83 km
This method evidently gives a more accurate result than using a scale drawing.
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Suppose we know the acceleration of a car (or anything else) and want to know its speed after it has been accelerated for a time t. What we do is first rewrite Eq. 2-8 in the form
at 5 v2 2 v1
which gives us what we want,
v2 5 v1 1 at Final speed 2-9
Final speed 5 initial speed 1 change in speed
Example 2.4
The speed of a car changes from 15 m/s (about 34 mi/h) to 25 m/s (about 56 mi/h) in 20 s when its gas pedal is pressed hard ( Fig. 2-8 ). Find its acceleration.
Solution Here
v1 5 15 m/s v2 5 25 m/s t 5 20 s
and so the car’s acceleration is
a 5 v2 2 v1
______ t 5 25 m/s 2 15 m/s ______________ 20 s 5
10 m/s ______ 20 s 5 0.5 m/s ______ s 5 0.5 m/s
2
This result means that the speed of the car increases by 0.5 m/s during each second the acceleration continues. It is customary to write (m/s)/s (meters per second per second) as just m/s 2 (meters per second squared) since
m/s ____ s 5 m _____
(s)(s) 5 m __
s2
Figure 2-8 A car whose speed increases from 15 m/s to 25 m/s in 20 s has an acceleration of 0.5 m/s 2 .
Example 2.5
A car whose brakes can produce an acceleration of 2 6 m/s 2 is traveling at 30 m/s when its brakes are applied. (a) What is the car’s speed 2 s later? (b) What is the total time needed for the car to come to a stop?
Solution (a) From Eq. 2-9,
v2 5 v1 1 at 5 30 m/s 1 (26 m/s2)(2 s) 5 (30 2 12) m/s 5 18 m/s
(b) Now v 2 5 0 5 v 1 1 at, so at 5 2 v 1 and
t 5 2 v1 __ a 5 2
30 m/s _______ 26 m/s2
5 5 s
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Not all accelerations increase speed. Something whose speed is decreasing is said to have a negative acceleration. For instance, when the brakes of a car are applied, its acceleration might be 2 6 m/s 2 , which means that its speed drops by 6 m/s in each second that the acceleration continues (see Example 2.5). (Sometimes a negative acceleration is called a deceleration. )
2.4 Distance, Time, and Acceleration How Far? An interesting question is, how far does something, say a car, go when it is accelerated from speed v 1 to speed v 2 in the time t?
To find out, we begin by noting that the car’s average speed _ v during the accelera- tion (assumed uniform) is
_ v 5 v1 1 v2 ______ 2 Average speed 2-10
The car moves exactly as far in the time t as if it had the constant speed v equal to its average speed
_ v . Therefore the distance the car covers in the time t is
d 5 _ v t 5 ( v1 1 v2 ______ 2 ) t 5 v1t ___ 2 1 v2t ___ 2 The value of v 2 , the car’s final speed, is given by Eq. 2-9, which means that
d 5 v1t ___ 2 1 ( v1 1 at ______ 2 ) t 5 v1t ___ 2 1 v1t ___ 2 1 at2 ___ 2 5 v1t 1 1 __ 2 at
2 Distance under constant 2-11 acceleration
If the car is stationary at the start of the acceleration, v 1 5 0, and
d 5 1 __ 2 at 2 Distance starting from rest 2-12
Example 2.6
How far did the car of Example 2-5 go while coming to a stop?
Solution Here v 1 5 30 m/s, a 5 2 6 m/s 2 , and t 5 5 s, so
d 5 v1t 1 1 __ 2 at 2 5 (30 m/s)(5 s) 1 1 __ 2 ( 26 m/s
2)(5 s)2
5 150 m 2 75 m 5 75 m
Example 2.7
An airplane needed 20 s to take off from a runway 500 m long. What was its acceleration (assumed constant)? Its final speed?
Solution Since the airplane started from rest, v 1 5 0 and d 5 1 __ 2 at
2. Therefore its acceleration was
a 5 2d ___ t2
5 2 (500 m) ________ (20 s)2
5 2.5 m/s2
The airplane’s final speed was
v2 5 v1 1 at 5 0 1 (2.5 m/s2)(20 s) 5 50 m/s
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As we can see, by defining certain quantities (here speed and acceleration) and relating them to each other and to directly measurable quantities (here distance and time) we can build up a structure of equations that enables us to answer questions with a pencil and paper that otherwise would need separate, perhaps difficult, obser- vations on real objects.
Drop a stone, and it falls. Does the stone fall at a constant speed, or does it go faster and faster? Does the stone’s motion depend on its weight, or its size, or its shape?
Before Galileo, philosophers tried to answer such questions in terms of suppos- edly self-evident principles, concepts so seemingly obvious that there was no need to test them. This was the way in which Aristotle (384–322 b.c. ), the famous thinker of ancient Greece, approached the subject of falling bodies. To Aristotle, every kind of material had a “natural” place where it belonged and toward which it tried to move. Thus fire rose “naturally” toward the sun and stars, whereas stones were “earthy” and so fell downward toward their home in the earth. A big stone was more earthy than a small one and so, Aristotle thought, ought to fall faster.
The trouble with these ideas, and many others like them, is that they are wrong— only the scientific method, not unsupported speculation, can provide reliable infor- mation about how the universe works.
2.5 Free Fall A Downward Acceleration Almost two thousand years later the Italian physicist Galileo, the first modern scien- tist, found that the higher a stone is when it is dropped, the greater its speed when it reaches the ground ( Figs. 2-9 and 2-10 ). This means the stone is accelerated. Fur- thermore, the acceleration is the same for all stones, big and small. For more accuracy with the primitive instruments of his time, Galileo measured the accelerations of balls rolling down an inclined plane rather than their accelerations in free fall, but his con- clusions were perfectly general; modern experiments have verified them to at least 1 part in 10 12 (a trillion!).
Galileo’s experiments showed that, if there were no air for them to push their way through, all falling objects near the earth’s surface would have the same acceleration of 9.8 m/s 2 . This acceleration is usually abbreviated g:
Acceleration due to gravity 5 g 5 9.8 m/s2
Ignoring for the moment the effect of air resistance, something that drops from rest has a speed of 9.8 m/s at the end of the first second, a speed of (9.8 m/s 2 ) (2 s) 5 19.6 m/s at the end of the next second, and so on ( Fig. 2-11 ). In general, under these circumstances
vdownward 5 gt Object falling from rest 2-13
How Far Does a Falling Object Fall? Equation 2-9 tells us the speed of a falling object at any time t after it has been dropped from rest (and before it hits the ground, of course). To find out how far h the object has fallen in the time t, we refer back to Eq. 2-12 for accelerated motion starting from rest. Here the distance is d 5 h and the acceleration is that of gravity, so a 5 g, which gives
h 5 1 __ 2 gt 2 Object falling from rest 2-14
ACCELERATION DUE TO GRAVITY
Figure 2-9 Falling bodies are accelerated downward. A stone dropped from a height of 5 m strikes the ground with a speed more than double that of a stone dropped from a height of 1 m.
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The t 2 factor means that h increases with time much faster than the object’s speed v, which is given by v 5 gt. Figure 2-12 shows h and v for various times of fall. At t 5 10 s, the object’s speed is 10 times its speed at t 5 1 s, but the distance it has fallen is 100 times the distance it fell during the first second.
Thrown Objects The downward acceleration g is the same whether an object is just dropped or is thrown upward, downward, or sideways. If a ball is held in the air and dropped, it goes faster and faster until it hits the ground. If the ball is thrown
Figure 2-11 All falling objects near the earth’s surface have a downward acceleration of 9.8 m/s 2 . (The distance an object will have fallen in each time interval is not shown to scale here.)
Figure 2-10 The Leaning Tower of Pisa, which is 58 m high, was begun in 1174 and took over two centuries to complete. During its construction the tower started to sink into the clay soil under its south side, and corrections were made to the upper floors to try to make them level. According to legend, Galileo dropped a bullet and a cannonball from the tower to show that all objects fall with the same acceleration. For centuries the Pisa tower had the greatest tilt (5.5 8 ) of any in the world, but recent work has reduced the tilt to only 3.9 8 . Several other towers lean more than that; the current record seems to be 5.2 8 , held by a tower next to a church in Suurhusen, Germany.
Figure 2-12 Downward speed and distance fallen in the first 10 seconds after an object is dropped from rest. Air resistance (Sec. 2.6) is ignored here. Because h 5 1 _ 2 at
2, the distance fallen increases with time at a greater rate than the downward speed. For instance, 1 s after being dropped, an object has fallen through 4.9 m and has a speed of 9.8 m/s, but 10 s after being dropped, it has fallen through 490 m but its speed has increased to only 98 m/s.
100
80
60
40
20
0 0 2 4 6 8 10
Time, s
D ow
nw ar
d sp
ee d,
m /s
500
400
300
200
100
0 0 2 4 6 8 10
Time, s
D is
ta nc
e fa
lle n,
m
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B I O G R A P H Y
doctrine of the Catholic Church, the Holy Office of the Inquisition told Galileo to stop advocating the con- trary, and he obeyed. Then, when a friend of his became pope in 1631, Galileo resumed teaching the mer- its of the copernican system. But the pope turned against him, and in 1633, when he was 70, Galileo was con- victed of heresy by the Inquisition.
Although he escaped being burnt at the stake, the fate of other heretics, Gali- leo was sentenced to house arrest for the remainder of his life and was forced to publicly deny that the earth moves. (According to legend, Galileo then mut- tered, “Yet it does move.”) Galileo’s con- demnation almost extinguished Italian science for over a century.
It would be nice to think that today, four centuries after Galileo’s time, the age of reason has become firmly established. But religious fun- damentalists who seek to replace the findings of science by their own interpretations of the Bible are again on the march. Do we really want a return to the blind ignorance of the past? Evolution in biology, geology, and astronomy is now under attack, and a movement to return to “Bibli- cal astronomy,” with the earth at the center of the universe, has actually started. The struggle to keep rea- son in the life of the mind is never- ending, but it is an essential struggle.
Galileo fathered modern science by clearly stating the central idea of the scientific method: the study of nature must be based on observation and experiment. He also pioneered the use of mathematical reasoning to interpret and generalize his findings.
Galileo was born in Pisa, Italy; following a local custom, his given name was a variation of his family name. Although his father thought that medicine would be a more sen- sible career choice, Galileo studied physics and mathematics and soon became a professor at Pisa and after- ward at Padua. His early work was on accelerated motion, falling bodies, and the paths taken by projectiles.
Later, with a telescope he had built, Galileo was the first person to see sunspots, the phases of Venus, the four largest satellites of Jupiter, and the mountains of the moon. Turning his telescope to the Milky Way, he found that it consisted of individual stars. To Galileo, as to his contempo- raries, these discoveries were “infi- nitely stupendous.”
By then famous, in 1610 Galileo went to Florence as court mathema- tician to Cosimo II dé Medici, Duke of Tuscany. Here Galileo expounded the copernican model of the universe and pointed out that his astronomical observations supported this model. Galileo discovered Jupiter’s four larg- est satellites in 1610 and concluded from his observations, shown here, that they revolve around Jupiter.
Because the ptolemaic model with the earth as the stationary cen- ter of the universe was part of the
Galileo Galilei (1564–1642)
horizontally, we can imagine its velocity as having two parts, a horizontal one that stays constant and a vertical one that is affected by gravity. The result, as in Fig. 2-13 , is a curved path that becomes steeper as the downward speed increases.
When a ball is thrown upward, as in Fig. 2-14 , the effect of the downward accel- eration of gravity is at first to reduce the ball’s upward speed. The upward speed decreases steadily until finally it is zero. The ball is then at the top of its path, when the ball is at rest for an instant. The ball next begins to fall at ever-increasing speed, exactly as though it had been dropped from the highest point.
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Example 2.8
A stone dropped from a bridge strikes the water 2.2 s later. How high is the bridge above the water?
Solution Substituting in Eq. 2-14 gives
h 5 1 __ 2 gt 2 5 1 __ 2 (9.8 m/s
2)(2.2 s)2 5 24 m
Example 2.9
An apple is dropped from a window 20 m above the ground. (a) How long does it take the apple to reach the ground? (b) What is its final speed?
Solution (a) Since h 5 1 __ 2 gt
2,
t 5 √ ___
2h ___ g 5 √ _________
(2)(20 m) ________ 9.8 m/s2
5 2.0 s
(b) From Eq. 2-13 the ball’s final speed is
v 5 gt 5 19.6 m/s
Example 2.10
An airplane is in level flight at a velocity of 150 m/s and an altitude of 1500 m when a wheel falls off. What horizontal distance will the wheel travel before it strikes the ground?
Solution The horizontal velocity of the wheel does not affect its vertical motion. The wheel there- fore reaches the ground at the same time as a wheel dropped from rest at an altitude of 1500 m, which is
t 5 √ ___
2h ___ g 5 √ __________
(2)(1500 m) __________ 9.8 m/s2
5 17.5 s
In this time the wheel will travel a horizontal distance of
d 5 vhorizt 5 (150 m/s)(17.5 s) 5 2625 m 5 2.63 km
Figure 2-13 The acceleration of gravity does not depend upon horizontal motion. When one ball is thrown horizontally from a building at the same time that a second ball is dropped vertically, the two reach the ground at the same time because both have the same downward acceleration.
Figure 2-14 When a ball is thrown upward, its downward acceleration reduces its original speed until it comes to a momentary stop. At this time the ball is at the top of its path, and it then begins to fall as if it had been dropped from there. The ball is shown after equal time intervals.
Interestingly enough, something thrown upward at a certain speed will return to its starting point with the same speed, although the object is now moving in the opposite direction.
What happens when a ball is thrown downward? Now the ball’s original speed is steadily increased by the downward acceleration of gravity. When the ball reaches the ground, its final speed will be the sum of its original speed and the speed increase due to the acceleration.
Projectile Motion When a ball is thrown upward at an angle to the ground, the result is a curved path called a parabola ( Fig. 2-15 ). The maximum range (horizontal distance) for a given initial speed occurs when the ball is thrown at an angle of 45 8 above the ground. At higher and lower angles, the range will be shorter. As the figure shows, for every range up to the maximum there are two angles at which the ball can be thrown and land in the same place.
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2.6 Air Resistance Why Raindrops Don’t Kill Air resistance keeps falling things from developing the full acceleration of gravity. Without this resistance raindrops would reach the ground with bullet-like speeds and even a light shower would be dangerous.
In air, a stone falls faster than a feather because air resistance affects the stone less. In a vacuum, however, there is no air, and the stone and feather fall with the same acceleration of 9.8 m/s 2 ( Fig. 2-16 ).
The faster something moves, the more the air in its path resists its motion. At 100 km/h (62 mi/h), the drag on a car due to air resistance is about 5 times as great as the drag at 50 km/h (31 mi/h). In the case of a falling object, the air resistance increases with speed until it equals the force of gravity on the object. The object then contin- ues to drop at a constant terminal speed that depends on its size and shape and on how heavy it is ( Table 2-1 ). A person in free fall has a terminal speed of about 54 m/s (120 mi/h), but with an open parachute the terminal speed of only about 6.3 m/s (14 mi/h) permits a safe landing ( Fig. 2-17 ).
Air resistance reduces the range of a projectile. Figure 2-18 shows how the path of a ball is affected. In a vacuum, as we saw in Fig. 2-15 , the ball goes farthest when it is thrown at an angle of 45 8 , but in air (that is, in real life), the maximum range occurs for an angle of less than 45 8 . For a baseball struck hard by a bat, an angle of 40 8 will take it the greatest distance.
What can make something originally at rest begin to move? Why do some things move faster than others? Why are some accelerated and others not? Questions like these led Isaac Newton to formulate three principles that summarize so much of the
FORCE AND MOTION
Figure 2-15 In the absence of air resistance, a ball travels farthest when it is thrown at an angle of 45 8 .
Example 2.11
A stone thrown upward reaches its highest point 2.2 s later. (a) How high did it go? (b) What was its initial speed?
Solution (a) The height h is the same as that from which the stone would have been dropped to reach
the ground in 2.2 s. Hence
h 5 1 __ 2 gt 2 5 (9.8 m/s
2)(2.2 s)2 ______________ 2 5 23.7 m
(b) The upward speed was the same as the downward speed the stone would have had 2.2 s after being dropped, so
v 5 gt 5 (9.8 m/s2)(2.2 s) 5 21.6 m/s
Object Terminal Speed
16-lb shot 145 m/s Baseball 42 Golf ball 40 Tennis ball 30 Basketball 20 Large raindrop 10 Ping-Pong ball 9
Table 2-1 Some Terminal Speeds (1 m/s 5 2.2 mi/h)
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Figure 2-16 In a vacuum all bodies fall with the same acceleration.
Figure 2-17 The terminal speeds of sky divers are greatly reduced when their parachutes open, which permits them to land safely.
Figure 2-18 Effect of air resistance on the path of a thrown ball. An angle of less than 45 8 now gives the greatest range. In air
In vacuum
45°
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behavior of moving bodies that they have become known as the laws of motion. Based on observations made by Newton and others, these laws are as valid today as they were when they were set down about three centuries ago.
2.7 First Law of Motion Constant Velocity Is as Natural as Being at Rest Imagine a ball lying on a level floor. Left alone, the ball stays where it is. If you give it a push, the ball rolls a short way and then comes to a stop. The smoother the floor, the farther the ball rolls before stopping. With a perfectly round ball and a perfectly smooth and level floor, and no air to slow down its motion, would the ball ever stop rolling?
There will never be a perfect ball and a perfect surface for it to roll on, of course. But we can come close. The result is that, as the resistance to its motion becomes less and less, the ball goes farther and farther for the same push. We can reasonably expect that, under ideal conditions, the ball would keep rolling forever.
This conclusion was first reached by Galileo. Later it was stated by Newton as his first law of motion:
According to this law, an object at rest never begins to move all by itself—a force is needed to start it off. If it is moving, the object will continue going at constant velocity unless a force acts to slow it down, to speed it up, or to change its direction. Motion at constant velocity is just as “natural” as staying at rest.
Force In thinking about force, most of us think of a car pulling a trailer or a per- son pushing a lawn mower or lifting a crate. Also familiar are the force of gravity, which pulls us and things about us downward, the pull of a magnet on a piece of iron, and the force of air pushing against the sails of a boat. In these examples the central idea is one of pushing, pulling, or lifting. Newton’s first law gives us a more precise definition:
When we see something accelerated, we know that a net force —a force not bal- anced out by one or more other forces ( Fig. 2-19 )—must be acting upon it. Only a net, or unbalanced, force can accelerate something. A force always has a direction associated with it, so force is a vector quantity.
Although it seems simple and straightforward, the first law of motion has far- reaching implications. For instance, until Newton’s time most people believed that the orbits of the moon around the earth and of the planets around the sun were “nat- ural” ones, with no forces needed to keep these bodies moving as they do. However, because the moon and planets move in curved paths, their velocities are not constant, so according to the first law, forces of some kind must be acting on them. The search for these forces led Newton to the law of gravitation.
An object continues to be accelerated only as long as a net force acts upon it. An ideal car on a level road would therefore need its engine only to be accelerated to a particular speed, after which it would keep moving at this speed forever with the engine turned off. Actual cars are not so cooperative because of
If no net force acts on it, an object at rest remains at rest and an object in motion remains in motion at constant velocity (that is, at constant speed in a straight line).
A force is any influence that can change the speed or direction of motion of an object.
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the retarding forces of friction and air resistance, which require counteracting by a force applied by the engine to the wheels.
2.8 Mass A Measure of Inertia The reluctance of an object to change its state of rest or of uniform motion in a straight line is called inertia.
When you are in a car that starts to move, you feel yourself pushed back in your seat ( Fig. 2-20 ). What is actually happening is that your inertia tends to keep your body where it was before the car started moving. When the car stops, on the other hand, you feel yourself pushed forward. What is actually happening now is that your inertia tends to keep your body moving while the car comes to a halt.
The name mass is given to the property of matter that shows itself as inertia. The inertia of a bowling ball exceeds that of a basketball, as you can tell by kicking them in turn, so the mass of the bowling ball exceeds that of the basketball. Mass may be thought of as quantity of matter: the more mass something has, the greater its inertia and the more matter it contains ( Fig. 2-21 ).
Figure 2-19 When several forces act on an object, they may cancel one another out to leave no net force. In this case the object is not accelerated.
Figure 2-20 ( a ) When a car suddenly starts to move, the inertia of the passengers tends to keep them at rest relative to the earth, and so their heads move backward relative to the car. ( b ) When the car comes to a sudden stop, inertia tends to keep the passengers moving, and so their heads move forward relative to the car.
Figure 2-21 The more mass an object has, the greater its resistance to a change in its state of motion, as this shot-putter knows.
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The SI unit of mass is the kilogram (kg). A liter of water, which is a little more than a quart, has a mass of 1 kg ( Fig. 2-22 ). Table 2-2 lists a range of mass values.
2.9 Second Law of Motion Force and Acceleration Throw a baseball hard, and it leaves your hand going faster than if you toss it gently. This suggests that the greater the force, the greater the acceleration while the force acts. Experiments show that doubling the net force doubles the acceleration, tripling the net force triples the acceleration, and so on ( Fig. 2-23 ).
Do all balls you throw with the same force leave your hand with the same speed? Heave an iron shot instead, and it is clear that the more mass something has, the less its acceleration for a given force. Experiments make the relationship precise: for the same net force, doubling the mass cuts the acceleration in half, tripling the mass cuts the acceleration to one-third its original value, and so on.
Newton’s second law of motion is a statement of these findings. If we let F 5 net force and m 5 mass, this law states that
to stick together because of attractive forces between their respective atoms and molecules, as described in Sec. 11.3.
Sometimes friction is welcome. The fastening abil- ity of nails and screws and the resistive action of brakes depend on friction, and walking would be impossible with- out it. In other cases friction means wasted effort, and to reduce it lubricants (oil and grease) and rollers or wheels are commonly used. About half the power of a car’s engine is lost to friction in the engine itself and in its drive train. The joints of the human body are lubricated by a substance called synovial fluid, which resembles blood plasma.
Friction is a force that acts to oppose the relative motion between two surfaces in contact. The harder the surfaces are pressed together, the stronger the frictional force.
Friction is an actual force, unlike inertia. Even a small net force can accelerate an object despite its inertia, but friction may prevent a small force from pushing one object across another.
Friction has two chief causes. One is the interlocking of irregularities in the two surfaces, which prevents one surface from sliding smoothly past the other. The second cause is the tendency for materials in very close contact
Friction
Figure 2-22 A liter, which is equal to 1.057 quarts, represents a volume of 1000 cubic centimeters (cm 3 ). One liter of water has a mass of 1 kg.
The sun 2 3 10 30 kg The earth 6 3 10 24 Large tanker 4 3 10 8 747 airliner (at takeoff) 4 3 10 5 Large car 2 3 10 3 165-lb person 75 This book 1.4 Pencil 3 3 10 2 3 Postage stamp 3 3 10 2 5 Smallest known bacterium 1 3 10 2 19 Oxygen molecule 5 3 10 2 26 Electron 9 3 10 2 31
Table 2-2 Some Approximate Mass Values
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a 5 F __ m Second law of motion 2-15
Acceleration 5 force _____ mass
Another way to express the second law of motion is in the form of a definition of force:
F 5 ma 2-16
Force 5 (mass)(acceleration)
The second law of motion was experimentally verified down to an acceleration of 5 3 10 2 14 m/s 2 in 2007.
An important aspect of the second law concerns direction because both force and acceleration are vector quantities. The direction of the acceleration is always the same as the direction of the net force. A car is going faster and faster—therefore the net force on it is in the same direction as that in which the car is headed. The car then slows down—therefore the net force on it is now in the direction opposite that in which it is headed ( Fig. 2-24 ).
Thus we can say that
In terms of vectors, the second law of motion becomes
F 5 ma
An animal whose length is L has muscles that have cross-sectional areas and hence strengths roughly propor- tional to L2. The mass of the animal, however, depends on its volume, which is roughly proportional to L3. Therefore the larger an animal is, in general, the weaker it is relative to its mass. This is obvious in nature. For instance, even though insect muscles are intrinsically weaker than human muscles, many insects can carry loads several times their weights, whereas animals the size of humans are limited to loads comparable with their weights. But a human-size insect would be a rather feeble creature.
The forces an animal exerts result from contractions of its skeletal muscles, which occur when the muscles are electri- cally stimulated by nerves. The maximum force a muscle can exert is proportional to its cross-sectional area and can be as much as 70 N/cm2 (100 lb/in.2).
An athlete might have a biceps muscle in his arm 8 cm across, so it could produce up to 3500 N (790 lb) of force. This is a lot, but the geometry of an animal’s skeleton and muscles favors range of motion over force. As a result the actual force a person’s arm can exert is much smaller than the forces exerted by the arm muscles themselves, but the person’s arm can move through a much greater distance than the amount the muscles contract.
Muscular Forces
Figure 2-23 Newton’s second law of motion. When different forces act upon identical masses, the greater force produces the greater acceleration. When the same force acts upon different masses, the greater mass receives the smaller acceleration.
2F
F
M
M
F
F
M
2M
Figure 2-24 The direction of a force is significant. A force applied to a car in the direction in which it is moving (for instance by giving more fuel to its engine) produces a positive acceleration, which increases the speed of the car. A force applied opposite to the direction of motion (for instance by using the brakes) produces a negative acceleration, which decreases the speed of the car until it comes to a stop. An acceleration that reduces the speed of a moving object is sometimes called a deceleration.
The net force on an object equals the product of the mass and the acceleration of the object. The direction of the force is the same as that of the acceleration.
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Figure 2-26 A force of 1 newton gives a mass of 1 kilogram an acceleration of 1 m/s 2 .
Figure 2-25 The relationship between force and acceleration means that the less acceleration something has, the smaller the net force on it. If you drop to the ground from a height, as this jumper has, you can reduce the force of the impact by bending your knees as you hit the ground so you come to a stop gradually instead of suddenly. The same reasoning can be applied to make cars safer. If a car’s body is built to crumple progressively in a crash, the forces acting on the passengers will be smaller than if the car were rigid.
Example 2.12
When a tennis ball is served, it is in contact with the racket for a time that is typically 0.005 s, which is 5 thousandths of a second. Find the force needed to serve a 60-g tennis ball at 30 m/s.
Solution Since the ball starts from rest, v 1 5 0, and its acceleration when struck by the racket is, from Eq. 2-8,
a 5 v2 2 v1 ______ t 5 30 m/s 2 0 __________ 0.005 s 5 6000 m/s
2
Because the ball’s mass is 60 g 5 0.06 kg, the force the racket must exert on it is
F 5 ma 5 (0.06 kg)(6000 m/s2) 5 360 N
In more familiar units, this force is 81 lb. Of course, it does not seem so great to the person serving because the duration of the impact is so brief ( Fig. 2-27 ).
Figure 2-27 A person serving a tennis ball must exert a force of 360 N on it for the ball to have a speed of 30 m/s if the racket is in contact with the ball for 0.005 s.
The second law of motion is the key to understanding the behavior of mov- ing objects because it links cause (force) and effect (acceleration) in a definite way ( Fig. 2-25 ). When we speak of force from now on, we know exactly what we mean, and we know exactly how an object free to move will respond when a given force acts on it.
The Newton The second law of motion shows us how to define a unit for force. If we express mass m in kilograms and acceleration a in m/s 2 , force F is given in terms of (kg)(m/s 2 ). This unit is given a special name, the newton (N). Thus
1 newton 5 1 N 5 1 (kg)(m/s2)
When a force of 1 N is applied to a 1-kg mass, the mass is given an acceleration of 1 m/s 2 ( Fig. 2-26 ).
In the British system, the unit of force is the pound (lb). The pound and the new- ton are related as follows:
1 N 5 0.225 lb
1 lb 5 4.45 N
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2.10 Mass and Weight Weight Is a Force The weight of an object is the force with which it is attracted by the earth’s gravita- tional pull. If you weigh 150 lb (668 N), the earth is pulling you down with a force of 150 lb. Weight is different from mass, which refers to how much matter something contains. There is a very close relationship between weight and mass, however.
Let us look at the situation in the following way. Whenever a net force F is applied to a mass m, Newton’s second law of motion tells us that the acceleration a of the mass will be in accord with the formula
F 5 ma
Force 5 (mass)(acceleration) In the case of an object at the earth’s surface, the force gravity exerts on it is its weight w. This is the force that causes the object to fall with the constant acceleration g 5 9.8 m/s 2 when no other force acts. We may therefore substitute w for F and g for a in the formula F 5 ma to give w 5 mg Weight and mass 2-17
Weight 5 (mass)(acceleration of gravity) The weight w of an object and its mass m are always proportional to each other: twice the mass means twice the weight, and half the mass means half the weight.
In the SI system, mass rather than weight is normally specified. A customer in a French grocery might ask for a kilogram of bread or 5 kg of potatoes. To find the weight in newtons of something whose mass in kilograms is known, we simply turn to w 5 mg and set g 5 9.8 m/s 2 . Thus the weight of 5 kg of potatoes is
w 5 mg 5 (5 kg)(9.8 m/s2) 5 49 N This is the force with which the earth attracts a mass of 5 kg.
At the earth’s surface, the weight of a 1-kg mass in British units is 2.2 lb. The weight in pounds of 5 kg of potatoes is therefore 5(2.2 lb) 5 11 lb. The mass that cor- responds to a weight of 1 lb is 454 g.
Your Weight Elsewhere in the Solar System
The more mass a planet has and the smaller it is, the greater the acceleration of gravity g at its sur- face. The values of g for the vari- ous planets are listed in Table 17-1. From these values, if you know your mass you can figure out your weight on any planet using the formula w 5 mg. In familiar units, if you weigh 150 lb on the earth, here is what you would weigh on the planets and the moon:
Mercury 57 lb Saturn 180 lb Venus 135 Uranus 165 Earth 150 Neptune 180 Mars 57 Moon 25 Jupiter 390
Figure 2-28 The net upward force on an elevator of mass 600 kg is 9120 N when its supporting cable exerts a total upward force of 15,000 N.
Fmax = 15,000 N
F = 9120 N
m = 600 kg
w = 5880 N
Example 2.13
An elevator whose total mass is 600 kg is suspended by a cable that can exert a maximum upward force of Fmax 5 15,000 N. What is the greatest upward acceleration the elevator can have? The greatest downward acceleration?
Solution When the elevator is stationary (or moving at constant speed) the upward force the cable exerts is just the elevator’s weight of
w 5 mg 5 (600 kg)(9.8 m/s2) 5 5880 N
To accelerate the elevator upward, an additional upward force F is needed ( Fig. 2-28 ), where
F 5 Fmax 2 w 5 15,000 N 2 5880 N 5 9120 N
The elevator’s upward acceleration when this net force acts on it is
a 5 F__m 5 9120 N______ 600 kg
5 15.2 m/s2
For the elevator to have a downward acceleration of more than the acceleration of gravity g, a downward force besides its own weight is needed. The cable cannot push the eleva- tor downward, so its greatest downward acceleration is g 5 9.8 m/s 2 . For a discussion of apparent weight in an elevator, see Sec. 2.10 at mhhe.com/krauskopf.
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Weight Varies with Location The mass of something is a more basic property than its weight because the pull of gravity on it is not the same everywhere. This pull is less on a mountaintop than at sea level and less at the equator than near the poles because the earth bulges slightly at the equator. A person who weighs 200 lb in Lima, Peru, would weigh nearly 201 lb in Oslo, Norway. On the surface of Mars the same person would weigh only 76 lb, and he or she would be able to jump much higher than on the earth. However, the person would not be able to throw a ball any faster: because the force F the person exerts on the ball and the ball’s mass m are the same on both planets, the acceleration a would be the same, too. Out in space, far away from planets and stars, the person would weigh nothing.
2.11 Third Law of Motion Action and Reaction Suppose you push against a heavy table and it does not move. This must mean that the table is resisting your push on it. The table stays in place because your force on it is matched by the opposing force of friction between the table legs and the floor. You don’t move because the force of the table on you is matched by a similar opposing force between your shoes and the floor.
Now imagine that you and the table are on a frozen lake whose surface is so slip- pery on a warm day that there is no friction. Again you push on the table, which this time moves away as a result ( Fig. 2-29 ). But you can stick to the ice no better than the table can, and you find yourself sliding backward. No matter what you do, pushing on the table always means that the table pushes back on you.
Considerations of this kind led Newton to his third law of motion:
No force ever occurs singly. A chair pushes downward on the floor; the floor presses upward on the chair ( Fig. 2-30 ). The firing of a rifle exerts a force on the bul- let; at the same time the firing exerts a backward push (recoil) on the rifle. A pear falls from a tree because of the earth’s pull on the pear; there is an equal upward pull on the earth by the pear that is not apparent because the earth has so much more mass than the pear, but this upward force is nevertheless present.
Action and Reaction Forces Newton’s third law always applies to two different forces on two different objects—the action force that the first object exerts on the second, and the opposite reaction force the second exerts on the first.
Figure 2-29 Action and reaction forces act on different bodies. Pushing a table on a frozen lake results in person and table moving apart in opposite directions.
Figure 2-30 Some examples of action-reaction pairs of forces.
When one object exerts a force on a second object, the second object exerts an equal force in the opposite direction on the first object.
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The third law of motion permits us to walk. When you walk, what is actually pushing you forward is not your own push on the ground but instead the reac- tion force of the ground on you ( Fig. 2-31 ). As you move forward, the earth itself moves backward, though by too small an amount (by virtue of its enormous mass) to be detected.
Sometimes the origin of the reaction force is not obvious. A book lying on a table exerts the downward force of its weight; but how can an apparently rigid object like the table exert an upward force on the book? If the tabletop were made of rubber, we would see the book push it down, and the upward force would result from the elasticity of the rubber. A similar explanation actually holds for tabletops of wood or metal, which are never perfectly rigid, although the depressions made in them may be extremely small.
It is sometimes arbitrary which force of an action-reaction pair to consider action and which reaction. For instance, we can’t really say that the gravitational pull of the earth on a pear is the action force and the pull of the pear on the earth is the reac- tion force, or the other way around. When you push on the ground when walking, however, it is legitimate to call this force the action force and the force with which the earth pushes back on you the reaction force.
Left to itself, a moving object travels in a straight line at constant speed. Because the moon circles the earth and the planets circle the sun, forces must be acting on the moon and planets. As we learned in Chap. 1, Newton discovered that these forces are the same in nature as the gravitational force that holds us to the earth. Before we consider how gravity works, we must look into exactly how curved paths come about.
2.12 Circular Motion A Curved Path Requires an Inward Pull Tie a ball to the end of a string and whirl the ball around your head, as in Fig. 2-32 . What you will find is that your hand must pull on the string to keep the ball moving in a circle. If you let go of the string, there is no longer an inward force on the ball, and it flies off to the side.
Centripetal Force The force that has to be applied to make something move in a curved path is called centripetal (“toward the center”) force:
The centripetal force always points toward the center of curvature of the object’s path, which means the force is at right angles to the object’s direction of motion at each moment. In Fig. 2-32 the ball is moving in a circle, so its velocity vector v is always tangent to the circle and the centripetal force vector F c is always directed toward the center of the circle.
A detailed calculation shows that the centripetal force F c needed for something of mass m and speed v to travel in a circle of radius r has the magnitude
Fc 5 mv2 ____ r Centripetal force 2-18
This formula tells us three things about the force needed to cause an object to move in a circular path: (1) the greater the object’s mass, the greater the force; (2) the faster the object, the greater the force; and (3) the smaller the circle, the greater the force ( Fig. 2-33 ).
GRAVITATION
Centripetal force 5 inward force on an object moving in a curved path
Figure 2-31 When these girls push backward on the ground with their feet, the ground pushes forward on them. The latter reaction force is what leads to their forward motion.
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Figure 2-32 A centripetal force is necessary for circular motion. An inward centripetal force F c acts upon every object that moves in a curved path. If the force is removed, the object continues moving in a straight line tangent to its original path.
Figure 2-33 The centripetal force needed to keep an object moving in a circle depends upon the mass and speed of the object and upon the radius of the circle. The direction of the force is always toward the center of the circle. (Centripetal force is the name given to any force that is always directed toward a center of motion. It is not a distinct type of force, such as gravity or friction.)
Example 2.14
Find the centripetal force needed by a 1000-kg car moving at 5 m/s to go around a curve 30 m in radius, as in Fig. 2-34 .
Solution The centripetal force needed to make the turn is
Fc 5 mv2 ____ r 5
(1000 kg)(5 m/s)2 ______________ 30 m 5 833 N
This force (184 lb) is easily transferred from the road to the car’s tires if the road is dry and in good condition. However, if the car’s speed were 20 m/s, the force needed would be 16 times as great, and the car would probably skid outward.
To reduce the chance of skids, particularly when the road is wet and therefore slip- pery, highway curves are often banked so that the roadbed tilts inward. A car going around a banked curve has an inward reaction force on it provided by the road itself, apart from friction ( Fig. 2-35 ).
[For another example, see Sec. 2.12 at www.mhhe.com/krauskopf .]
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From the formula for F c we can see why cars rounding a curve are so difficult to steer when the curve is sharp (small r ) or the speed is high (a large value for v means a very large value for v 2 ). On a level road, the centripetal force is supplied by friction between the car’s tires and the road. If the force needed to make a particular turn at a certain speed is more than friction can supply, the car skids outward.
The acceleration a c caused by a centripetal force is called centripetal acceleration. From the second law of motion, F c 5 ma c , and so
ac 5 Fc __ m 5
mv2 ____ rm 5 v2 __ r Centripetal acceleration 2-19
2.13 Newton’s Law of Gravity What Holds the Solar System Together Newton used Kepler’s laws of planetary motion and Galileo’s findings about falling bodies to establish how the gravitational force between two objects
Figure 2-35 A wall of snow provides this bobsled with the centripetal force it needs to round the turn.
Figure 2-34 A centripetal force of 833 N is needed by this car to make the turn shown.
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depends on their masses and on the distance between them. His conclusion was this:
In equation form, Newton’s law of gravity states that the force F that acts between two objects whose masses are m 1 and m 2 is
Gravitational force 5 F 5 Gm1m2 ______ R2
Law of gravity 2-20
Here R is the distance between the objects and G is a constant of nature, the same number everywhere in the universe. The value of G is 6.670 3 10 2 11 N · m 2 /kg 2 .
Center of Mass The point in an object from which R is to be measured depends on the object’s shape and on the way in which its mass is distributed. The center of mass of a uniform sphere is its geometric center ( Fig. 2-36 ).
The inverse square—1/ R 2 —variation of gravitational force with distance R means that this force drops off rapidly with increasing R ( Fig. 2-37 ).
Figure 2-38 shows how this variation affects the weight of a 61-kg astronaut who leaves the earth on a spacecraft. At the earth’s surface she weighs 600 N (135 lb); that is, the gravitational attraction of the earth on her is 600 N. When she is 100 times farther from the center of the earth, her weight is 1/100 2 or 1 ____ 10,000 as great, only 0.06 N—the weight of a cigar on the earth’s surface.
2.14 Artificial Satellites Thousands Circle the Earth The first artificial satellite, Sputnik I, was launched by the Soviet Union in 1957. Since then thousands of others have been put into orbits around the earth, most of them by the United States and the former Soviet Union. Men and women have been in orbit
Every object in the universe attracts every other object with a force propor- tional to both of their masses and inversely proportional to the square of the distance between them.
Figure 2-36 For computing gravitational effects, spherical bodies (such as the earth and moon) may be regarded as though their masses are located at their geometrical centers, provided that they are uniform spheres or consist of concentric uniform spherical shells.
Figure 2-37 The gravitational force between two bodies depends upon the square of the distance between them. The gravitational force on a planet would drop to one-fourth its usual amount if the distance of the planet from the sun were to be doubled. If the distance is halved, the force would increase to 4 times its usual amount.
Figure 2-38 The weight of a person near the earth is the gravitational force the earth exerts upon her. As she goes farther and farther away from the earth’s surface, her weight decreases inversely as the square of her distance from the earth’s center. The mass of the person here is 61 kg.
600 N 150 N 67 N 0.06 N
6400 km 12,800 km 19,200 km 640,000 km
R
R earth
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regularly since 1961, when a Soviet cosmonaut circled the globe at an average height of 240 km.
Over a thousand active satellites are now in orbit ( Fig. 2-40 ). (Over 2000 more are inactive. The average active lifetime of a satellite is 15 years.) About half belong to the United States, 10 percent to Russia, and 4 percent to China, with the rest distributed
Example 2.15
On the basis of what we know already, we can find the mass of the earth. This sounds, perhaps, like a formidable job, but it is really fairly easy to do. It is worth following as an example of the indirect way in which scientists go about performing such seemingly impossible feats as “weighing” the earth, the sun, other planets, and even distant stars.
Solution Let us focus our attention on an apple of mass m on the earth’s surface. The downward force of gravity on the apple is its weight of mg:
Weight of apple 5 F 5 mg
We can also use Newton’s law of gravity to find F, with the result
Gravitational force on apple 5 F 5 GmM _____ R2
Here M is the earth’s mass and R is the distance between the apple and the center of the earth, which is the earth’s radius of 6400 km 5 6.4 3 106 m (Fig. 2-39). The two ways to find F must give the same result, so
GmM _____ R2
5 mg We note that the apple’s mass m appears on both sides of this equation, hence it cancels out. Solving for the earth’s mass M gives
M 5 gR2
___ G 5 (9.8 m/s2)(6.4 3 106 m)2 ____________________ 6.67 3 10211 N ∙ m2/kg2
5 6 3 1024 kg
The number 6 3 1024 is 6 followed by 24 zeros! Enormous as it is, the earth is one of the least massive planets: Saturn has 95 times as much mass, and Jupiter 318 times as much. The sun’s mass is more than 300,000 times that of the earth.
Figure 2-39 The gravitational force of the earth on an apple at the earth’s surface is the same as the force between masses M and m the distance R apart. This force equals the weight of the apple.
Figure 2-40 An earth satellite is always falling toward the earth. As a result, an astronaut inside feels “weightless,” just as a person who jumps off a diving board feels “weightless.” But a gravitational force does act on both people—what is missing is the upward reaction force of the ground, the diving board, the floor of a room, the seat of a chair, or whatever each person would otherwise be pressing on. In the case of an astronaut, the floor of the satellite falls just as fast as he or she does instead of pushing back.
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Gravitation 53
among more than 50 other countries. The closest satellites, from 80 to 2000 km above the earth, are mostly “eyes in the sky” that map the earth’s surface; survey it for military purposes; provide information on weather and resources such as mineral deposits, crops, and water; and carry out various scientific studies. A group of 66 satellites in low-earth orbit supports the worldwide Iridium telephone system.
Twenty-seven satellites (3 of them spares) at an altitude of 20,360 km are used in the Global Positioning System (GPS) developed by the United States ( Fig. 2-41 ). GPS receivers, some hardly larger than a wristwatch, enable users to find their positions, including altitude, anywhere in the world at any time with uncertainties of only a few meters ( Fig. 2-42 ). A similar system, called Glonass, is operated by Russia; China has a system called Beidou (“compass”); and the European Union is developing yet another system, called Galileo.
The most distant satellites, nearly half the total, circle the equator exactly once a day at an altitude of 35,800 km, so they remain in place indefinitely over a particular location on the earth. A satellite in such a geostationary orbit can “see” a large area of the earth’s surface. Most satellites in geostationary orbits are used to relay com- munications of all kinds from one place to another, which is often cheaper than using cables between them.
Seeing Satellites
On a dark night many satellites, even quite small ones, are vis- ible to the naked eye because of the sunlight they reflect. The best times to look are an hour or so before sunrise and an hour or so after sunset. High-altitude satellites can be seen for longer periods because they spend more time outside the earth’s shadow. When conditions are just right, the International Space Station, which is 109 m across, is about as bright as the brightest stars. The ISS nearly rivals Venus, and except for the moon is the most brilliant object in the night sky.
Figure 2-41 In the Global Positioning System (GPS), each of a fleet of orbiting satellites sends out coded radio signals that enable a receiver on the earth to determine both the exact position of the satellite in space and its exact distance from the receiver. Given this information, a computer in the receiver then calculates the circle on the earth’s surface on which the receiver must lie. Data from three satellites give three circles, and the receiver must be located at the one point where all three intersect. The satellites provide extremely accurate time signals used for many purposes, for instance to synchronize the alternating currents from different power plants that feed electricity to a grid (see Sec. 6.18).
In addition, millions of smaller bits and pieces are out there, many of them able to damage spacecraft windows, solar cells, and other relatively fragile components. Even encoun- tering something the size of a pea would be an unwelcome event when its impact speed is 15 to 45 times the speed of a bullet. This is no idle worry: in 2009, the three astronauts in the International Space Station had to temporarily retreat into its “lifeboat,” a Russian Soyuz spacecraft, when warned that a 9-mm metal fragment might be on a collision course. In fact, the fragment, whose relative speed was about 32,000 km/h, missed, but by less than 4.5 km. An even closer approach took place in 2011 when a piece of debris moving at 47,000 km/h came within 335 m of the ISS and its six occupants. In both cases the warning came too late for the ISS to move out of the way, as it has on over a dozen other occasions.
A remarkable amount of debris is in orbit around the earth, relics of the 5000 or so rockets that have thus far gone into space. Most of the debris orbits are 700 to 1000 km above the earth. The U.S. Space Surveillance Network uses radar to track more than 20,000 objects over 10 cm across, which range from dropped astronaut tools all the way up to discarded rocket stages and “dead” satellites, that might collide catastrophically with spacecraft. Unfortunately this system, together with a private one that supplements it, cannot foresee every potential collision, even two satel- lites smashing together. In 2009 such a catastrophe actu- ally occurred between two communications satellites at an altitude of 780 km, adding over 2000 more pieces of debris large enough for radars on earth to track those already in orbit.
Space Junk
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Why Satellites Don’t Fall Down What keeps all these satellites up there? The answer is that a satellite is actually falling down but, like the moon (which is a natural satellite), at exactly such a rate as to circle the earth in a stable orbit. “Stable” is a rela- tive term, to be sure, since friction due to the extremely thin atmosphere present at the altitudes of actual satellites will eventually bring them down. Satellite lifetimes in orbit range from a matter of days to hundreds of years.
Let us think about a satellite in a circular orbit. The gravitational force on the satellite is its weight mg, where g is the acceleration of gravity at the satellite’s altitude (the value of g decreases with increasing altitude). The centripetal force a satellite of speed v needs to circle the earth at the distance r from the earth’s center is mv 2 / r. Since the earth’s gravity is providing this centripetal force,
Centripetal force 5 gravitational force
mv 2 ____ r 5 mg
v2 5 rg
Satellite speed 5 v 5 √ __
rg 2-21
The mass of the satellite does not matter. For an orbit a few kilometers above the earth’s surface, the satellite speed turns
out to be about 28,400 km/h. Anything sent off around the earth at this speed will become a satellite of the earth. (Of course, at such a low altitude air resistance will soon bring it down.) At a lower speed than this an object sent into space would sim- ply fall to the earth, while at a higher speed it would have an elliptical rather than a circular orbit ( Fig. 2-43 ). A satellite initially in an elliptical orbit can be given a circu- lar orbit if it has a small rocket motor to give it a further push at the required distance from the earth ( Fig. 2-44 ).
Escape Speed If its original speed is high enough, at least 40,000 km/h, a space- craft can escape entirely from the earth. The speed required for something to leave the gravitational influence of an astronomical body permanently is called the escape speed. Readers of Through the Looking Glass may recall the Red Queen’s remark, “Now, here, you see, it takes all the running you can do to stay in the same place. If you want to get somewhere else, you must run at least twice as fast as that!”
Figure 2-42 An automobile navigation system uses GPS data to show the location of a car on a displayed map and can give visual or verbal directions to a destination. The map can be a normal top-down view or, in some models, a perspective view.
Figure 2-43 The minimum speed an earth satellite can have is 28,400 km/h. The escape speed from the earth is 40,000 km/h.
Figure 2-44 This Landsat satellite circles the earth at an altitude of 915 km. The satellite carries a television camera and a scanner system that provides images of the earth’s surface in four color bands. The data radioed back provide information valuable in geology, water supply, agriculture, and land-use planning. Today the cost of placing a satellite in earth orbit is between about $10,000 and $20,000 per kg of payload, but new rocket designs are hoped to do this for much less.
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The ratio between escape speed and minimum orbital speed for any planet is actually √
__ 2 , about 1.41, so that
vescape 5 √ � 2 rg
where r and g are their values at the planet’s surface. Escape speeds for the planets are listed in Table 17-1.
It is worth keeping in mind that escape speed is the initial speed needed for something to leave a planet (or other astronomical body, such as a star) permanently when there is no further propulsion. A spacecraft whose motor runs continuously can go out into space without ever reaching the escape speed—even a snail’s pace would be enough, given sufficient time.
A force is any influence that can cause an object to be accel- erated. The unit of force is the newton (N). The weight of an object is the gravitational force with which the earth attracts it.
Newton’s first law of motion states that, if no net force acts on it, every object continues in its state of rest or uniform motion in a straight line. Newton’s second law of motion states that when a net force F acts on an object of mass m, the object is given an acceleration of F / m in the same direction as that of the force. Newton’s third law of motion states that when one object exerts a force on a second object, the second object exerts an equal but opposite force on the first. Thus for every action force there is an equal but opposite reaction force.
The centripetal force on an object moving along a curved path is the inward force needed to cause this motion. Centripetal force acts toward the center of curvature of the path. This force produces a centripetal acceleration in the object.
Newton’s law of gravity states that every object in the uni- verse attracts every other object with a force directly proportional to both their masses and inversely proportional to the square of the distance separating them.
When we say something is moving, we mean that its position relative to something else—the frame of reference —is changing. The choice of an appropriate frame of reference depends on the situation.
The speed of an object is the rate at which it covers dis- tance relative to a frame of reference. The object’s velocity speci- fies both its speed and the direction in which it is moving. The acceleration of an object is the rate at which its speed changes. Changes in direction are also accelerations.
A scalar quantity has magnitude only; mass and speed are examples. A vector quantity has both magnitude and direction; force and velocity are examples. An arrowed line that represents the magnitude and direction of a quantity is called a vector.
The acceleration of gravity is the downward accelera- tion of a freely falling object near the earth’s surface. Its value is g 5 9.8 m/s 2 .
The inertia of an object is the resistance the object offers to any change in its state of rest or motion. The property of matter that shows itself as inertia is called mass; mass may be thought of as quantity of matter. The unit of mass is the kilogram (kg).
Important Terms and Ideas
3. A box suspended by a rope is pulled to one side by a horizontal force. The tension in the rope
a. is less than before b. is unchanged c. is greater than before d. may be any of the above, depending on how strong the
force is
1. Which of the following quantities is not a vector quantity? a. velocity c. mass b. acceleration d. force
2. Which of the following statements is incorrect? a. All vector quantities have directions. b. All vector quantities have magnitudes. c. All scalar quantities have directions. d. All scalar quantities have magnitudes.
Multiple Choice
Pythagorean theorem for right triangle ( C is longest side): A 2 1 B 2 5 C 2
Important Formulas
Second law of motion: F 5 ma Weight: w 5 mg
Centripetal force: Fc 5 mv2 ____ r
Centripetal acceleration: ac 5 v2 __ r
Law of gravity: F 5 Gm1m2 ______ R2
Speed: v 5 d __ t
Acceleration: a 5 v2 2 v1 ______ t
d 5 v1t 1 1 __ 2 at 2
h 5 1 __ 2 gt 2
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c. less mass and more weight d. the same mass and less weight
13. The earth and the moon exert equal and opposite forces on each other. The force the earth exerts on the moon
a. is the action force b. is the reaction force c. can be considered either as the action or as the reaction
force d. cannot be considered as part of an action-reaction pair
because the forces act in opposite directions 14. A car that is towing a trailer is accelerating on a level road.
The magnitude of the force the car exerts on the trailer is a. equal to the force the trailer exerts on the car b. greater than the force the trailer exerts on the car c. equal to the force the trailer exerts on the road d. equal to the force the road exerts on the trailer
15. When a boy pulls a cart, the force that causes him to move forward is
a. the force the cart exerts on him b. the force he exerts on the cart c. the force he exerts on the ground with his feet d. the force the ground exerts on his feet
16. In order to cause something to move in a circular path, it is necessary to provide
a. a reaction force b. an inertial force c. a centripetal force d. a gravitational force
17. An object is moving in a circle with a constant speed. Its acceleration is constant in
a. magnitude only b. direction only c. both magnitude and direction d. neither magnitude nor direction
18. A car rounds a curve on a level road. The centripetal force on the car is provided by
a. inertia b. gravity c. friction between the tires and the road d. the force applied to the steering wheel
19. The centripetal force that keeps the earth in its orbit around the sun is provided
a. by inertia b. by the earth’s rotation on its axis c. partly by the gravitational pull of the sun d. entirely by the gravitational pull of the sun
20. The gravitational force with which the earth attracts the moon
a. is less than the force with which the moon attracts the earth b. is the same as the force with which the moon attracts the
earth c. is more than the force with which the moon attracts the earth d. varies with the phase of the moon
21. The speed needed to put a satellite in orbit does not depend on a. the mass of the satellite c. the shape of the orbit b. the radius of the orbit d. the value of g at the orbit
4. The sum of two vectors is a minimum when the angle between them is
a. 0 c. 90 8 b. 45 8 d. 180 8
5. In which of the following examples is the motion of the car not accelerated?
a. A car turns a corner at the constant speed of 20 km/h. b. A car climbs a steep hill with its speed dropping from
60 km/h at the bottom to 15 km/h at the top. c. A car climbs a steep hill at the constant speed of 40 km/h. d. A car climbs a steep hill and goes over the crest and
down on the other side, all at the same speed of 40 km/h. 6. Two objects have the same size and shape but one of them
is twice as heavy as the other. They are dropped simultane- ously from a tower. If air resistance is negligible,
a. the heavy object strikes the ground before the light one b. they strike the ground at the same time, but the heavy
object has the higher speed c. they strike the ground at the same time and have the
same speed d. they strike the ground at the same time, but the heavy
object has the lower acceleration because it has more mass
7. The acceleration of a stone thrown upward is a. greater than that of a stone thrown downward b. the same as that of a stone thrown downward c. less than that of a stone thrown downward d. zero until it reaches the highest point in its path
8. You are riding a bicycle at constant speed when you throw a ball vertically upward. It will land
a. in front of you b. on your head c. behind you d. any of the above, depending on the ball’s speed
9. When an object is accelerated, a. its direction never changes b. its speed always increases c. it always falls toward the earth d. a net force always acts on it
10. If we know the magnitude and direction of the net force on an object of known mass, Newton’s second law of motion lets us find its
a. position b. speed c. acceleration d. weight
11. The weight of an object a. is the quantity of matter it contains b. is the force with which it is attracted to the earth c. is basically the same quantity as its mass but is expressed
in different units d. refers to its inertia
12. Compared with her mass and weight on the earth, an astronaut on Venus, where the acceleration of gravity is 8.8 m/s 2 , has
a. less mass and less weight b. less mass and the same weight
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34. When a net force of 1 N acts on a 1-kg body, the body receives a. a speed of 1 m/s b. an acceleration of 0.1 m/s 2 c. an acceleration of 1 m/s 2 d. an acceleration of 9.8 m/s 2
35. When a net force of 1 N acts on a 1-N body, the body receives a. a speed of 1 m/s b. an acceleration of 0.1 m/s 2 c. an acceleration of 1 m/s 2 d. an acceleration of 9.8 m/s 2
36. A car whose mass is 1600 kg (including the driver) has a maximum acceleration of 1.20 m/s 2 . When a certain pas- senger is also in the car, its maximum acceleration becomes 1.13 m/s 2 . The mass of the passenger is
a. 100 kg c. 112 kg b. 108 kg d. 134 kg
37. A 300-g ball is struck with a bat with a force of 150 N. If the bat was in contact with the ball for 0.020 s, the ball’s speed is
a. 0.01 m/s c. 2.5 m/s b. 0.1 m/s d. 10 m/s
38. A bicycle and its rider together have a mass of 80 kg. If the bicycle’s speed is 6 m/s, the force needed to bring it to a stop in 4 s is
a. 12 N c. 120 N b. 53 N d. 1176 N
39. The weight of 400 g of onions is a. 0.041 N c. 3.9 N b. 0.4 N d. 3920 N
40. A salami weighs 3 lb. Its mass is a. 0.31 kg c. 6.6 kg b. 1.36 kg d. 29.4 kg
41. An upward force of 600 N acts on a 50-kg dumbwaiter. The dumbwaiter’s acceleration is
a. 0.82 m/s 2 c. 11 m/s 2 b. 2.2 m/s 2 d. 12 m/s 2
42. The upward force the rope of a hoist must exert to raise a 400-kg load of bricks with an acceleration of 0.4 m/s 2 is
a. 160 N c. 3760 N b. 1568 N d. 4080 N
43. The radius of the circle in which an object is moving at constant speed is doubled. The required centripetal force is
a. one-quarter as great as before b. one-half as great as before c. twice as great as before d. 4 times as great as before
44. A car rounds a curve at 20 km/h. If it rounds the curve at 40 km/h, its tendency to overturn is
a. halved c. tripled b. doubled d. quadrupled
45. A 1200-kg car whose speed is 6 m/s rounds a turn whose radius is 30 m. The centripetal force on the car is
a. 48 N c. 240 N b. 147 N d. 1440 N
22. An astronaut inside an orbiting satellite feels weightless because
a. he or she is wearing a space suit b. the satellite is falling toward the earth just as fast as the
astronaut is, so there is no upward reaction force on him or her
c. there is no gravitational pull from the earth so far away d. the sun’s gravitational pull balances out the earth’s
gravitational pull 23. A bicycle travels 12 km in 40 min. Its average speed is
a. 0.3 km/h c. 18 km/h b. 8 km/h d. 48 km/h
24. Which one or more of the following sets of displacements might be able to return a car to its starting point?
a. 5, 5, and 5 km c. 5, 10, and 10 km b. 5, 5, and 10 km d. 5, 5, and 20 km
25. An airplane whose airspeed is 200 km/h is flying in a wind of 80 km/h. The airplane’s speed relative to the ground is between
a. 80 and 200 km/h c. 120 and 200 km/h b. 80 and 280 km/h d. 120 and 280 km/h
26. A ship travels 200 km to the south and then 400 km to the west. The ship’s displacement from its starting point is
a. 200 km c. 450 km b. 400 km d. 600 km
27. How long does a car whose acceleration is 2 m/s 2 need to go from 10 m/s to 30 m/s?
a. 10 s c. 40 s b. 20 s d. 400 s
28. A ball is thrown upward at a speed of 12 m/s. It will reach the top of its path in about
a. 0.6 s c. 1.8 s b. 1.2 s d. 2.4 s
29. A car that starts from rest has a constant acceleration of 4 m/s 2 . In the first 3 s the car travels
a. 6 m c. 18 m b. 12 m d. 172 m
30. A car traveling at 10 m/s begins to be accelerated at 12 m/s 2 . The distance the car covers in the first 5 s after the accelera- tion begins is
a. 60 m c. 150 m b. 80 m d. 200 m
31. A car with its brakes applied has an acceleration of 2 1.2 m/s 2 . If its initial speed is 10 m/s, the distance the car covers in the first 5 s after the acceleration begins is
a. 15 m c. 35 m b. 32 m d. 47 m
32. The distance the car in Multiple Choice 31 travels before it comes to a stop is
a. 6.5 m c. 21 m b. 8.3 m d. 42 m
33. A bottle falls from a blimp whose altitude is 1200 m. If there was no air resistance, the bottle would reach the ground in
a. 5 s c. 16 s b. 11 s d. 245 s
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48. A man whose weight is 800 N on the earth’s surface is also in the spacecraft of Multiple Choice 47. His weight there is
a. 200 N c. 800 N b. 400 N d. 1600 N
46. If the earth were 3 times as far from the sun as it is now, the gravitational force exerted on it by the sun would be
a. 3 times as large as it is now b. 9 times as large as it is now c. one-third as large as it is now d. one-ninth as large as it is now
47. A woman whose mass is 60 kg on the earth’s surface is in a spacecraft at an altitude of one earth’s radius above the sur- face. Her mass there is
a. 15 kg c. 60 kg b. 30 kg d. 120 kg
13. A car whose acceleration is constant reaches a speed of 80 km/h in 20 s starting from rest. How much more time is required for it to reach a speed of 130 km/h?
14. The brakes of a car are applied to give it an acceleration of 2 3 m/s 2 . The car comes to a stop in 5 s. What was its speed when the brakes were applied?
15. A car starts from rest and reaches a speed of 40 m/s in 10 s. If its acceleration remains the same, how fast will it be moving 5 s later?
16. The brakes of a car moving at 14 m/s are applied, and the car comes to a stop in 4 s. (a) What was the car’s accelera- tion? (b) How long would the car take to come to a stop starting from 20 m/s with the same acceleration? (c) How long would the car take to slow down from 20 m/s to 10 m/s with the same acceleration?
2.4 Distance, Time, and Acceleration 17. A car is moving at 10 m/s when it begins to be accelerated at
2.5 m/s 2 . (a) How long does the car take to reach a speed of 25 m/s? (b) How far does it go during this period?
18. The driver of a train moving at 20 m/s applies the brakes when it passes an amber signal. The next signal is 1 km down the track and the train reaches it 75 s later. Find the train’s acceleration and its speed at the next signal.
19. A car starts from rest and covers 400 m (very nearly 1 _ 4 mi) in 20 s. Find the average acceleration of the car and its final speed.
2.5 Free Fall 20. Is it true that something dropped from rest falls three
times farther in the second second after being let go than it does in the first second?
21. A rifle is aimed directly at a squirrel in a tree. Should the squirrel drop from the tree at the instant the rifle is fired or should it remain where it is? Why?
22. The acceleration of gravity on the surface of Venus is 8.9 m/s 2 . Would a ball thrown upward on Venus return to the ground sooner or later than a ball thrown upward with the same speed on the earth?
23. When a football is thrown, it follows a curved path through the air like the ones shown in Fig. 2-18 . Where in its path is the ball’s speed greatest? Where is it least?
24. A crate is dropped from an airplane flying horizontally at constant speed. How does the path of the crate appear to somebody on the airplane? To somebody on the ground?
2.1 Speed
1. A woman standing before a cliff claps her hands, and 2.8 s later she hears the echo. How far away is the cliff? The speed of sound in air at ordinary temperatures is 343 m/s.
2. The starter of a race stands at one end of a line of runners. What is the difference in time between the arrival of the sound of his pistol at the nearest runner and at the most dis- tant runner 10 m farther away? (In a sprint, 0.01 s can mean the difference between winning and coming in second.)
3. In 1977 Steve Weldon ate 91 m of spaghetti in 29 s. At the same speed, how long would it take Mr. Weldon to eat 5 m of spaghetti?
4. A snake is slithering toward you at 1.5 m/s. If you start walk- ing when it is 5 m away, how fast must you go so that the snake will not overtake you when you have gone 100 m?
5. A woman jogs for 2 km at 8 km/h and then walks for 2 km at 6 km/h. What is her average speed for the entire trip?
2.2 Vectors
6. Three forces, each of 10 lb, act on the same object. What is the maximum total force they can exert on the object? The minimum total force?
7. Is it correct to say that scalar quantities are abstract, idealized quantities with no precise counterparts in the physical world, whereas vector quantities properly repre- sent reality because they take directions into account?
8. A man is rowing at 8 km/h in a river 1.5 km wide in which the current is 5 km/h. (a) In what direction should he head in order to get across the river in the shortest possible time? (b) How much time will he take if he goes in this direction? (c) How far downstream will the boat have gone when it reaches the opposite side?
9. A woman walks 70 m to an elevator and then rises upward 40 m. What is her displacement from her starting point?
10. Two tugboats are towing a ship. Each exerts a horizontal force of 5 tons and the angle between their towropes is 90 8 . What net force is exerted on the ship?
2.3 Acceleration
11. Can a rapidly moving object have the same acceleration as a slowly moving one?
12. The acceleration of a certain moving object is constant in magnitude and direction. Must the path of the object be a straight line? If not, give an example.
Exercises
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39. A bullet is fired horizontally from a rifle at 200 m/s from a cliff above a plain below. The bullet reaches the plain 5 s later. (a) How high was the cliff? (b) How far from the cliff did the bullet reach the plain? (c) What was the bul- let’s speed when it reached the plain?
40. An airplane whose speed is 60 m/s is flying at an altitude of 500 m over the ocean toward a stationary sinking ship. At what horizontal distance from the ship should the crew of the airplane drop a pump into the water next to the ship?
41. A person at the masthead of a sailboat moving at con- stant speed in a straight line drops a wrench. The mast- head is 20 m above the boat’s deck and the stern of the boat is 10 m behind the mast. Is there a minimum speed the sailboat can have so that the wrench will not land on the deck? If there is such a speed, what is it?
2.9 Second Law of Motion
42. Compare the tension in the coupling between the first two cars of a train with the tension in the coupling between the last two cars when (a) the train’s speed is constant and (b) the train is accelerating. (The tension is the magnitude of the force either end of the coupling exerts on the car it is attached to.)
43. In accelerating from a standing start to a speed of 300 km/h (186 mi/h—not its top speed!), the 1900-kg Bugatti Veyron sports car exerts an average force on the road of 9.4 kN. How long does the car take to reach 300 km/h?
44. A 12,000-kg airplane launched by a catapult from an aircraft carrier is accelerated from 0 to 200 km/h in 3 s. (a) How many times the acceleration due to gravity is the airplane’s acceleration? (b) What is the average force the catapult exerts on the airplane?
45. The brakes of a 1200-kg car exert a force of 4 kN. How long will it take for them to slow the car to a stop from an initial speed of 24 m/s?
46. A force of 20 N gives an object an acceleration of 5 m/s 2 . (a) What force would be needed to give the same object an acceleration of 1 m/s 2 ? (b) What force would be needed to give it an acceleration of 10 m/s 2 ?
47. A bicycle and its rider together have a mass of 80 kg. If the bicycle’s speed is 6 m/s, how much force is needed to bring it to a stop in 4 s?
48. A 430-g soccer ball at rest on the ground is kicked with a force of 600 N and flies off at 15 m/s. How long was the toe of the person kicking the ball in contact with it?
49. A car and driver with a total mass of 1600 kg has a maxi- mum acceleration of 1.2 m/s 2 . If the car picks up three 80-kg passengers, what is its maximum acceleration now?
50. Before picking up the passengers, the driver of the car of Exercise 49 shifts into neutral when the car is mov- ing at 80 km/h and finds that its speed has dropped to 65 km/h after 10 s. What was the average drag force act- ing on the car?
2.10 Mass and Weight
51. Consider the statement: Sara weighs 55 kg. What is wrong with the statement? Give two ways to correct it.
52. When a force equal to its weight is applied to an object free to move, what is its acceleration?
25. A stone is thrown horizontally from a cliff and another, identical stone is dropped from there at the same time. Do the stones reach the ground at the same time? How do their speeds compare when they reach the ground? Their accelerations?
26. (a) Imagine that Charlotte drops a ball from a window on the twentieth floor of a building while at the same time Fred drops another ball from a window on the nineteenth floor of that building. As the balls fall, what happens to the distance between them (assuming no air resistance)? (b) Next imagine that Charlotte and Fred are at the same window on the twentieth floor and that Fred drops his ball a few seconds after Charlotte drops hers. As the balls fall, what happens to the distance between them now (again assuming no air resistance)?
27. A person in a stationary elevator drops a coin and the coin reaches the floor of the elevator 0.6 s later. Would the coin reach the floor in less time, the same time, or more time if it were dropped when the elevator was (a) falling at a con- stant speed? (b) falling at a constant acceleration? (c) rising at a constant speed? (d) rising at a constant acceleration?
28. How fast must a ball be thrown upward to reach a height of 12 m?
29. A person dives off the edge of a cliff 33 m above the sur- face of the sea below. Assuming that air resistance is neg- ligible, how long does the dive last and with what speed does the person enter the water?
30. A ball dropped from the roof of a building takes 4 s to reach the street. How high is the building?
31. A ball is thrown downward at 12 m/s. What is its speed 1.0 s later?
32. When will a stone thrown vertically upward at 9.8 m/s reach the ground?
33. A ball is thrown upward from the edge of a cliff with an initial speed of 6 m/s. (a) How fast is it moving 0.5 s later? In what direction? (b) How fast is it moving 2 s later? In what direction? (Consider upward as 1 and downward as 2 ; then v 1 5 1 6 m/s and g 5 2 9.8 m/s 2 .)
34. The air resistance experienced by a falling object is not an important factor until a speed of about half its terminal speed is reached. The terminal speed of a golf ball is 40 m/s. How much time is needed for a dropped golf ball to reach a speed of half this? How far does it fall in this time?
35. A ball is thrown vertically upward with an initial speed of 30 m/s. (a) How long will it take the ball to reach the highest point in its path? (b) How long will it take the ball to return to its starting place? (c) What will the ball’s speed be there?
36. A rifle is aimed directly at the bull’s-eye of a target 50 m away. If the bullet’s speed is 350 m/s, how far below the bull’s-eye does the bullet strike the target?
37. An airplane is in level flight at a speed of 100 m/s and an altitude of 1200 m when a windshield wiper falls off. What will the wiper’s speed be when it reaches the ground? (Hint: A vector calculation is needed here and in Exercises 38 and 39.)
38. A ball is thrown horizontally from the roof of a building 20 m high at 30 m/s. At what speed will the ball strike the ground?
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2.12 Circular Motion
68. Where should you stand on the earth’s surface to experi- ence the most centripetal force? The least?
69. Under what circumstances, if any, can something move in a circular path without a centripetal force acting on it?
70. A person swings an iron ball in a vertical circle at the end of a string. At what point in the circle is the string most likely to break? Why?
71. A car makes a clockwise turn on a level road at too high a speed and overturns. Do its left or right wheels leave the road first?
72. When you whirl a ball at the end of a string, the ball seems to be pulling outward away from your hand. When you let the string go, however, the ball moves along a straight path perpendicular to the direction of the string at the moment you let go. Explain each of these effects.
73. A 40-kg crate is lying on the flat floor of a truck moving at 15 m/s. A force of 150 N is needed to slide the crate against the friction between the bottom of the crate and the floor. What is the minimum radius of a turn the sta- tion wagon can make if the box is not to slip?
74. The greatest force a level road can exert on the tires of a certain 2000-kg car is 4 kN. What is the highest speed the car can round a curve of radius 200 m without skidding?
75. Find the minimum radius at which an airplane flying at 300 m/s can make a U-turn if the centripetal force on it is not to exceed 4 times the airplane’s weight.
76. Some people believe that aliens from elsewhere in the uni- verse visit the earth in spacecraft that travel faster than jet airplanes and can turn in their own lengths. Calculate the centripetal force on a 100-kg alien in a spacecraft moving at 500 m/s (1120 mi/h) while it is making a turn of radius 30 m. How many times the weight of the alien is this force? Do you think such stories can be believed?
77. The 200-g head of a golf club moves at 40 m/s in a circu- lar arc of 1.2 m radius. How much force must the player exert on the handle of the club to prevent it from flying out of her hands at the bottom of the swing? Ignore the mass of the club’s shaft.
78. An airplane flying at a constant speed of 160 m/s pulls out of a dive in a circular arc. The 80-kg pilot presses down on his seat with a force of 3000 N at the bottom of the arc. What is the radius of the arc?
2.13 Newton’s Law of Gravity
79. A track team on the moon could set new records for the high jump or pole vault (if they did not need space suits, of course) because of the smaller gravitational force. Could sprinters also improve their times for the 100-m dash?
80. If the moon were half as far from the earth as it is today, how would the gravitational force it exerts on the earth compare with the force it exerts today?
81. Compare the weight and mass of an object at the earth’s surface with what they would be at an altitude of two earth’s radii.
82. A hole is bored to the center of the earth and a stone is dropped into it. How do the mass and weight of the stone
53. A person weighs 85 N on the surface of the moon and 490 N on the surface of the earth. What is the accelera- tion of gravity on the surface of the moon?
54. A mass of 8 kg and another of 12 kg are suspended by a string on either side of a frictionless pulley. Find the acceleration of each mass.
55. An 80-kg man slides down a rope at constant speed. (a) What is the minimum breaking strength the rope must have? (b) If the rope has precisely this strength, will it support the man if he tries to climb back up?
56. How much force is needed to give a 5-kg box an upward acceleration of 2 m/s 2 ?
57. A parachutist whose total mass is 100 kg is falling at 50 m/s when her parachute opens. Her speed drops to 6 m/s in 2 s. What is the total force her harness had to withstand? How many times her weight is this force?
58. A person in an elevator suspends a 1-kg mass from a spring balance. What is the nature of the elevator’s motion when the balance reads 9.0 N? 9.8 N? 10.0 N?
59. A person stands on a scale in an elevator. When the ele- vator is at rest, the scale reads 700 N. When the elevator starts to move, the scale reads 600 N. (a) Is the elevator going up or down? (b) Is it accelerated? If so, what is the acceleration?
60. A 60-kg person stands on a scale in an elevator. How many newtons does the scale read (a) when the elevator is ascending with an acceleration of 1 m/s 2 ; (b) when it is descending with an acceleration of 1 m/s 2 ; (c) when it is ascending at a constant speed of 3 m/s; (d) when it is descending at a constant speed of 3 m/s; (e) when the cable has broken and the elevator is
descending in free fall? 2.11 Third Law of Motion
61. Since the opposite forces of the third law of motion are equal in magnitude, how can anything ever be accelerated?
62. What is the relationship, if any, between the first and sec- ond laws of motion? Between the second and third laws of motion?
63. A book rests on a table. (a) What is the reaction force to the force the book exerts on the table? (b) To the force gravity exerts on the book?
64. A car with its engine running and in forward gear goes up a hill and then down on the other side. What forces cause it to move upward? Downward?
65. An engineer designs a propeller-driven spacecraft. Because there is no air in space, the engineer includes a supply of oxygen as well as a supply of fuel for the motor. What do you think of the idea?
66. Two children wish to break a string. Are they more likely to succeed if each takes one end of the string and they pull against each other, or if they tie one end of the string to a tree and both pull on the free end? Why?
67. When a 5-kg rifle is fired, the 9-g bullet is given an accel- eration of 30 km/s while it is in the barrel. (a) How much force acts on the bullet? (b) Does any force act on the rifle? If so, how much and in what direction?
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Exercises 61
the lead exert on 2 kg of cheese placed on the pan if the centers of mass of the lead and cheese are 0.3 m apart? Compare this force with the weight of 1 g of cheese to see if putting the lead under the scale was worth doing.
88. A bull and a cow elephant, each of mass 2000 kg, attract each other gravitationally with a force of 2 3 10 2 5 N. How far apart are they?
2.14 Artificial Satellites
89. An airplane makes a vertical circle in which it is upside down at the top of the loop. Will the passengers fall out of their seats if there is no belt to hold them in place?
90. Two satellites are launched from Cape Canaveral with the same initial speeds relative to the earth’s surface. One is sent toward the west, the other toward the east. Will there be any difference in their orbits? If so, what will the difference be and why?
91. Is an astronaut in an orbiting spacecraft actually “weightless”?
92. With the help of the data in Table 17-1, find the mini- mum speed artificial satellites must have to pursue stable orbits about Mars.
at the earth’s center compare with their values at the earth’s surface?
83. Is the sun’s gravitational pull on the earth the same at all seasons of the year? Explain.
84. The centripetal force that keeps the moon in its orbit around the earth is provided by the gravitational pull of the earth. This force accelerates the moon toward the earth at 2.7 3 10 2 3 m/s 2 , so that the moon is continually “falling” toward the earth. How far does the moon fall toward the earth per second? Per year? Why doesn’t the moon come closer and closer to the earth?
85. According to Kepler’s second law, the earth travels fastest when it is closest to the sun. Is this consistent with the law of gravitation? Explain.
86. A 2-kg mass is 1 m away from a 5-kg mass. What is the gravitational force (a) that the 5-kg mass exerts on the 2-kg mass, and (b) that the 2-kg mass exerts on the 5-kg mass? (c) If both masses are free to move, what are their respective accelerations if no other forces are acting?
87. A dishonest grocer installs a 100-kg lead block under the pan of his scale. How much gravitational force does
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dropping just as fast as you are, and your apparent weight is then zero.
Suppose you are standing on a scale in an elevator, as in Figure 1 . When the elevator’s upward accel- eration is a, the upward force F Y on you is the sum of your actual weight mg and the force that accelerates you upward, which is ma. Hence
Fy 5 mg 1 ma The reaction force to F y has the same magnitude and is the force you exert on the scale, which is your apparent weight w app . Hence
wapp 5 mg 1 ma
Apparent weight Actual weight
Apparent weight
If the acceleration a is positive, cor- responding to an upward accelera- tion, then w app > mg, and the scale reading will be greater than your actual weight. If a is negative, cor- responding to a downward accelera- tion, w app < mg, and the scale reading will be less than your actual weight. If the cable that supports the eleva- tor breaks and the elevator falls freely, a 5 2 g and w app 5 mg 2 mg 5 0. When the elevator is at rest or moving at constant speed up or down, a 5 0 and w app 5 mg.
Jump off a diving board into a pool and you feel “weightless” until you hit the water. In reality, a gravitational force is acting on you during your fall. What is missing is the upward reac- tion force provided by whatever sup- ports your weight at other times—the seat of a chair, the floor of a room, the diving board, the water.
It is useful to distinguish between the actual weight of an object, which is the gravitational force that acts on it, and its apparent weight, which is the force the object exerts on what- ever it rests on. You can think of your apparent weight as the reading of a bathroom scale you are standing on. If you are falling freely, the scale is
Section 2.10: Apparent Weight
At rest or constant velocity
a = 0 a a g
700 N
Upward acceleration
Downward acceleration
Free fall
800 N 600 N 0
FIGURE 1 The actual weight of this person is 700 N. When the elevator is accelerated upward, her apparent weight is greater, and when the elevator is accelerated downward, her apparent weight is smaller. In free fall, her apparent weight is zero.
2-1 1
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2 2-2
Section 2.10 Example
You are standing on a scale in an elevator. When the elevator is at rest on the fourteenth floor, the scale reads 700 N. When the elevator starts to move, the scale reads 600 N. Describe the motion of the elevator.
Solution Here your actual weight is 700 N and your apparent weight is 600 N. Because wapp and mg are different, the elevator must be accelerating, and because wapp < mg, the acceleration must be negative, that is, downward.
Your mass is
m 5 mg
___ g 5 700 N ________
9.80 m/s2 5 71.4 kg
Solving the apparent-weight equation for the elevator’s acceleration gives
a 5 wapp ____ m 2 g 5
600 N ______ 71.4 kg
2 9.80 m/s2 5 (8.40 2 9.80) m/s2 5 2 1.40 m/s2
Section 2.12: Example
A road has a hump 12 m in radius. What is the minimum speed at which a car will leave the road at the top of the hump?
Solution The car will leave the hump when the required centripetal force mv2/r is more than the car’s weight of mg, since it is this weight that is providing the centripetal force. Hence
mg 5 mv 2 ____ r , g 5
v2 __ r , v 5 √ __
rg
We note that the mass of the car does not matter here. (We will find the same formula later as the orbital speed a satellite of the earth must have.) Substituting for r and g gives
v 5 √ ______________
(12 m)(9.8 m/s2) 5 10.8 m/s
This is about 24 mi/h. Driving faster over the hump is not recommended.
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In a pole vault, the athlete’s energy of motion while running is first transformed into energy of position at the top of the vault, then back into energy of motion while falling, and finally into work done when landing.
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62
3
Work 3.1 The Meaning of Work
A Measure of the Change a Force Produces
• Why work is an important quantity. 3.2 Power
The Rate of Doing Work • How work and power are related.
Energy 3.3 Kinetic Energy
The Energy of Motion • Why the kinetic energy of a moving
object depends more strongly on its speed than on its mass.
3.4 Potential Energy The Energy of Position
• The basic distinction between kinetic and potential energies.
• There are different ways in which something can have potential energy.
• What the gravitational potential energy of an object depends on.
3.5 Conservation of Energy A Fundamental Law of Nature
• The meaning of a conservation law. • How energy is conserved in various
situations. • How to use the work-energy theorem.
3.6 Mechanical Advantage How to Change the Magnitude of a Force
• Why a simple machine can increase or decrease the magnitude of a force while conserving energy.
• What mechanical advantage is. • How to find the mechanical advantage
of a simple machine. 3.7 The Nature of Heat
The Downfall of Caloric • Evidence that shows heat to be a
form of energy rather than an actual substance.
Momentum 3.8 Linear Momentum
Another Conservation Law • Similarities and differences between
linear momentum and kinetic energy. 3.9 Rockets
Momentum Conservation Is the Basis of Space Travel
• How momentum conservation explains what happens when an object breaks apart or collides with another object.
3.10 Angular Momentum A Measure of the Tendency of a Spinning Object to Continue to Spin
• How conservation of angular momentum affects the motions of footballs, skaters, and planets.
Relativity 3.11 Special Relativity
Things Are Seldom What They Seem • Some relativistic effects and why they
are not conspicuous in everyday life. 3.12 Rest Energy
Matter Is a Form of Energy • Where the energy given off in chemical
and physical reactions comes from. 3.13 General Relativity
Gravity Is a Warping of Spacetime • The nature of gravity and how it affects
light.
CHAPTER OUTLINE AND GOALS
Your chief goal in reading each section should be to understand the important findings and ideas indicated (•) below.
Energy
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The word energy has become part of everyday life. We say that an active person is energetic. We hear a candy bar described as being full of energy. We com- plain about the cost of the electric energy that lights our lamps and turns our motors. We worry about some day running out of the energy stored in coal and oil. We argue about whether nuclear energy is a blessing or a curse. Exactly what is meant by energy?
In general, energy refers to an ability to accomplish change. When almost any- thing happens in the physical world, energy is somehow involved. But “change” is not a very precise notion, and we must be sure of exactly what we are talking about in order to go further. Our procedure will be to begin with the simpler idea of work and then use it to relate change and energy in the orderly way of science.
Changes that take place in the physical world are the result of forces. Forces are needed to pick things up, to move things from one place to another, to squeeze things, to stretch things, and so on. However, not all forces act to produce changes, and it is the distinction between forces that accomplish change and forces that do not that is central to the idea of work.
3.1 The Meaning of Work A Measure of the Change a Force Produces Suppose we push against a wall. When we stop, nothing has happened even though we exerted a force on the wall. But if we apply the same force to a stone, the stone flies through the air when we let it go ( Fig. 3-1 ). The difference is that the wall did not move during our push but the stone did. A physicist would say that we have done work on the stone, and as a result it was accelerated and moved away from our hand.
Or we might try to lift a heavy barbell. If we fail, the world is exactly the same afterward. If we succeed, though, the barbell is now up in the air, which repre- sents a change ( Fig. 3-2 ). As before, the difference is that in the second case an object moved while we exerted a force on it, which means that work was done on the object.
To make our ideas definite, work is defined in this way:
If nothing moves, no work is done, no matter how great the force. And even if some- thing moves, work is not done on it unless a force is acting on it.
What we usually think of as work agrees with this definition. However, we must be careful not to confuse becoming tired with the amount of work done. Pushing against a wall for an afternoon in the hot sun is certainly tiring, but we have done no work because the wall didn’t move.
In equation form,
W 5 Fd Work 3-1
Work done 5 (applied force)(distance through which force acts)
WORK
Figure 3-1 Work is done by a force when the object it acts on moves while the force is applied. No work is done by pushing against a stationary wall. Work is done when throwing a ball because the ball moves while being pushed during the throw.
The work done by a force acting on an object is equal to the magnitude of the force multiplied by the distance through which the force acts when both are in the same direction.
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Direction The direction of the force F is assumed to be the same as the direction of the displacement d. If not, for example in the case of a child pulling a wagon with a rope not parallel to the ground, we must use for F the magnitude F d of the projection of the applied force F that acts in the direction of motion ( Fig. 3-3 ).
A force that is perpendicular to the direction of motion of an object can do no work on the object. Thus gravity, which results in a downward force on everything near the earth, does no work on objects moving horizontally along the earth’s surface. However, if we drop an object, work is definitely done on it as it falls to the ground.
The Joule The SI unit of work is the joule (J), where one joule is the amount of work done by a force of one newton when it acts through a distance of one meter. That is,
1 joule (J) 5 1 newton-meter (N ∙ m)
The joule is named after the English scientist James Joule and is pronounced “jool.” To raise an apple from your waist to your mouth takes about 1 J of work. Since 1 N 5 1 kg · m/s 2 , the joule can also be expressed as 1 J 5 1 N ∙ m 5 1 kg · m 2 /s 2 , which is more convenient in some problems.
Work Done Against Gravity It is easy to find the work done in lifting an object against gravity near the earth’s surface. The force of gravity on the object is its weight of mg. In order to raise the object to a height h above its original position ( Fig. 3-4 a ), we need to apply an upward force of F 5 mg. With F 5 mg and d 5 h, Eq. 3-1 becomes
W 5 mgh Work done against gravity 3-2 Work 5 (weight)(height)
Only the total height h is involved here: the particular route upward taken by the object is not significant. Excluding friction, exactly as much work must be done when you climb a flight of stairs as when you go up to the same floor in an elevator ( Fig. 3-5 )—though the source of the work is not the same, to be sure.
If an object of mass m at the height h falls, the amount of work done by gravity on it is given by the same formula, W 5 mgh ( Fig. 3-4 b ).
3.2 Power The Rate of Doing Work The time needed to carry out a job is often as important as the amount of work needed. If we have enough time, even the tiny motor of a toy train can lift an elevator as high as we like. However, if we want the elevator to take us up fairly quickly, we
Figure 3-2 Work is done when a barbell is lifted, but no work is done while it is being held in the air even though this can be very tiring.
Figure 3-3 When a force and the distance through which it acts are parallel, the work done is equal to the product of F and d. When they are not in the same direction, the work done is equal to the product of d and the magnitude F d of the projection of F in the direction of d.
Figure 3-4 (a) The work a person does to lift an object to a height h is mgh. (b) If the object falls through the same height, the force of gravity does the work mgh.
(a) (b)
F = mg
W = work done by person = mgh
h mg
W = work done by gravity = mgh
h
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must use a motor whose output of work is rapid in terms of the total work needed. Thus the rate at which work is being done is significant. This rate is called power : The more powerful something is, the faster it can do work.
If the amount of work W is done in a period of time t, the power involved is
P 5 W ___ t Power 3-3
Power 5 work done ___________ time interval
The SI unit of power is the watt (W), where
1 watt (W) 5 1 joule/second (J/s)
Figure 3-5 Neglecting friction, the work needed to raise a person to a height h is the same regardless of the path taken. W = mgh W = mgh
h
Example 3.1
(a) A horizontal force of 100 N is used to push a 20-kg box across a level floor for 10 m. How much work is done? (b) How much work is needed to raise the same box by 10 m?
Solution (a) The work done in pushing the box is
W 5 Fd 5 (100 N)(10 m) 5 1000 J The mass of the box does not matter here. What counts is the applied force, the distance through which it acts, and the relative directions of the force and the displacement of the box. (b) Now the work done is
W 5 mgh 5 (20 kg)(9.8 m/s2)(10 m) 5 1960 J The work done in this case does depend on the mass of the box.
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Thus a motor with a power output of 500 W is capable of doing 500 J of work per second. The same motor can do 250 J of work in 0.5 s, 1000 J of work in 2 s, 5000 J of work in 10 s, and so on. The watt is quite a small unit, and often the kilowatt (kW) is used instead, where 1 kW 5 1000 W.
A person in good physical condition is usually capable of a continuous power output of about 75 W, which is 0.1 horsepower. A runner or swimmer during a dis- tance event may have a power output 2 or 3 times greater. What limits the power output of a trained athlete is not muscular development but the supply of oxygen from the lungs through the bloodstream to the muscles, where oxygen is used in the metabolic processes that enable the muscles to do work. However, for a period of less than a second, an athlete’s power output may exceed 5 kW, which accounts for the feats of weightlifters and jumpers.
Force, Speed, and Power Often a force performs work on an object moving at constant speed in the same direction as that of the force. The work done is W 5 Fd when the force F acts over the distance d in the time t, so that in this situation Eq. 3-3 becomes
P 5 W ___ t 5 Fd ___ t
But d / t equals v, the object’s constant speed, so
P 5 Fv 3-4 Power 5 (force)(speed)
Suppose we want to know how much thrust a 5000-kW (6700 hp) aircraft engine produces when the speed of the airplane it powers is 240 m/s (537 mi/h). The answer comes in one step: F 5 P / v 5 21 kN (2.3 tons).
We now go from the straightforward idea of work to the complex and many-sided idea of energy :
When we say that something has energy, we mean it is able, directly or indirectly, to exert a force on something else and perform work. When work is done on something, energy is added to it. Energy is measured in the same unit as work, the joule.
ENERGY
The Horsepower
The horsepower (hp) is the traditional unit of power in engi- neering. The origin of this unit is interesting. In order to sell the steam engines he had perfected two centuries ago, the Scottish engineer James Watt (1736–1819) had to compare their power out- puts with that of a horse, a source of work his customers were familiar with.
After various tests, Watt found that a typical horse could perform work at a rate of 497 W for as much as 10 hours per day. To avoid any disputes, he increased this figure by one-half to establish the unit he called the horsepower. Watt’s horsepower therefore represents a rate of doing work of 746 W:
1 horsepower (hp) 5 746 W 5 0.746 kW
1 kilowatt (kW) 5 1.34 hp
Few horses can develop this much power for very long. The early steam engines ranged from 4 to 100 hp, with the 20-hp model being the most popular.
Example 3.2
A 15-kW electric motor provides power for the elevator of a building. What is the mini- mum time needed for the elevator to rise 30 m to the sixth floor when its total mass when loaded is 900 kg?
Solution The work that must be done to raise the elevator is W 5 mgh. Since P 5 W/t, the time needed is
t 5 W___P 5 mgh____
P 5 (900 kg)(9.8 m/s2)(30 m)_____________________
15 3 103 W 5 17.6 s
Energy is that property something has that enables it to do work.
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3.3 Kinetic Energy The Energy of Motion Energy occurs in several forms. One of them is the energy a moving object has because of its motion. Every moving object has the capacity to do work. By striking something else, the moving object can exert a force and cause the second object to shift its position, to break apart, or to otherwise show the effects of having work done on it. It is this property that defines energy, so we conclude that all moving things have energy by virtue of their motion.
The energy of a moving object is called kinetic energy (KE). (“Kinetic” is a word of Greek origin that suggests motion is involved.)
The kinetic energy of a moving thing depends upon its mass and its speed. The greater the mass and the greater the speed, the more the KE. A train going at 30 km/h has more energy than a horse galloping at the same speed and more energy than a similar train going at 10 km/h. The exact way KE varies with mass m and speed v is given by the formula
KE 5 1 _ 2 mv 2 Kinetic energy 3-5
Energy and Speed The v 2 factor means the kinetic energy increases very rapidly with increasing speed. At 30 m/s a car has 9 times as much KE as at 10 m/s—and requires 9 times as much force to bring to a stop in the same distance ( Fig. 3-6 ). The fact that KE, and hence the ability to do work (in this case, dam- age), depends upon the square of the speed is what is responsible for the sever- ity of automobile accidents at high speeds. The variation of KE with mass is less marked: a 2000-kg car going at 10 m/s has just twice the KE of a 1000-kg car with the same speed. (For a derivation of Eq. 3-5 and a sidebar on running speeds, see Sec. 3.3 at www.mhhe.com/krauskopf .)
3.4 Potential Energy The Energy of Position When we drop a stone, it falls faster and faster and finally strikes the ground. If we lift the stone afterward, we see that it has done work by making a shallow hole in the
Figure 3-6 Kinetic energy is proportional to the square of the speed. A car traveling at 30 m/s has 9 times the KE of the same car traveling at 10 m/s.
m = 1000 kg, v = 30 m/s, KE = 450,000 J
m = 1000 kg, v = 10 m/s, KE = 50,000 J
Example 3.3
Find the kinetic energy of a 1000-kg car when its speed is 10 m/s.
Solution From Eq. 3-5 we have
KE 5 1 _ 2 mv 2 5 ( 1 _ 2 ) (1000 kg)(10 m/s)2
5 ( 1 _ 2 ) (1000 kg)(10 m/s)(10 m/s) 5 50,000 J 5 50 kJ In order to bring the car to this speed from rest, 50 kJ of work had to be done by its engine. To stop the car from this speed, the same amount of work must be done by its brakes.
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ground. In its original raised position, the stone must have had the capacity to do work even though it was not moving at the time and therefore had no KE.
The amount of work the stone could do by falling to the ground is called its potential energy (PE). Just as kinetic energy may be thought of as energy of motion, potential energy may be thought of as energy of position ( Fig. 3-8 ).
Examples of potential energy are everywhere. A book on a table has PE since it can fall to the floor. A skier at the top of a slope, water at the top of a waterfall, a car at the top of the hill, anything able to move toward the earth under the influence of gravity has PE because of its position. Nor is the earth’s gravity necessary: a stretched spring has PE since it can do work when it is let go, and a nail near a magnet has PE since it can do work in moving to the magnet ( Fig. 3-9 ).
Figure 3-8 A raised stone has potential energy because it can do work on the ground when dropped.
Example 3.4
Have you ever wondered how much force a hammer exerts on a nail? Suppose you hit a nail with a hammer and drive the nail 5 mm into a wooden board ( Fig. 3-7 ). If the ham- mer’s head has a mass of 0.6 kg and it is moving at 4 m/s when it strikes the nail, what is the average force on the nail?
Solution The KE of the hammer head is 1 _ 2 mv
2, and this amount of energy becomes the work Fd done in driving the nail the distance d 5 5 mm 5 0.005 m into the board. Hence
KE of hammer head 5 work done on nail
1 _ 2 mv 2 5 Fd
and F 5 mv 2 ____
2d 5 (0.6 kg)(4 m/s)2
_____________ 2(0.005 m)
5 960 N
This is 216 lb—watch your fingers!
m
v
F
d
Figure 3-7 When a hammer strikes this nail, the hammer’s kinetic energy is converted into the work done to push the nail into the wooden board.
Figure 3-9 Two examples of potential energy.
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Gravitational Potential Energy When an object of mass m is raised to a height h above its original position, its gravitational potential energy is equal to the work that was done against gravity to bring it to that height ( Fig. 3-10 ). According to Eq. 3-2 this work is W 5 mgh, and so
PE 5 mgh Gravitational potential energy 3-6
This result for PE agrees with our experience. Consider a pile driver ( Fig. 3-11 ), a machine that lifts a heavy weight (the “hammer”) and allows it to fall on the head of a pile, which is a wooden or steel post, to drive the pile into the ground. From the formula PE 5 mgh we would expect the effectiveness of a pile driver to depend on the mass m of its hammer and the height h from which it is dropped, which is exactly what experience shows.
PE Is Relative It is worth noting that the gravitational PE of an object depends on the level from which it is reckoned. Often the earth’s surface is convenient, but some- times other references are more appropriate.
Suppose you lift this book as high as you can above the table while remaining seated. It will then have a PE relative to the table of about 12 J. But the book will have a PE relative to the floor of about twice that, or 24 J. And if the floor of your room is, say, 50 m above the ground, the book’s PE relative to the ground will be about 760 J.
Figure 3-10 The increase in the potential energy of a raised object is equal to the work mgh used to lift it.
Figure 3-11 In the operation of a pile driver, the gravitational potential energy of the raised hammer becomes kinetic energy as it falls. The kinetic energy in turn becomes work as the pile is pushed into the ground.
Example 3.5
Find the potential energy of a 1000-kg car when it is on top of a 45-m cliff.
Solution From Eq. 3-6 the car’s potential energy is
PE 5 mgh 5 (1000 kg)(9.8 m/s2)(45 m) 5 441,000 J 5 441 kJ
This is less than the KE of the same car when it moves at 30 m/s (Fig. 3-6). Thus a crash at 30 m/s into a wall or tree will yield more work—that is, do more damage—than dropping the car from a cliff 45 m high.
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What is the book’s true PE? The answer is that there is no such thing as “true” PE. Gravitational PE is a relative quantity. However, the difference between the PEs of an object at two points is significant, since it is this difference that can be changed into work or KE.
3.5 Conservation of Energy A Fundamental Law of Nature According to a conservation law that applies to a system of any kind, a certain quan- tity keeps the same value it originally had no matter what changes the system under- goes. The great success of physics and chemistry in understanding the natural world is largely due to the various conservation laws that have been discovered, several of which are discussed in this and later chapters.
The law of conservation of energy states that
What this means is that the total amount of energy in a system isolated from the rest of the universe remains constant, although transformations from one form of energy to another may occur within the system. This law has the widest application in sci- ence, applying equally to distant stars and to biological processes in living cells. No violation of it has ever been found.
Examples of energy conservation are found in the motion of a planet in its orbit around the sun ( Fig. 3-12 ) and in the motion of a pendulum ( Fig. 3-13 ). The orbits of the planets are ellipses with the sun at one focus (Fig. 1-10), and each planet is there- fore at a constantly varying distance from the sun. At all times the total of its potential and kinetic energies remains the same. When close to the sun, the PE of a planet is low and its KE is high. The additional speed due to increased KE keeps the planet from being pulled into the sun by the greater gravitational force on it at this point in its path. When the planet is far from the sun, its PE is higher and its KE lower, with the reduced speed exactly keeping pace with the reduced gravitational force.
A pendulum ( Fig. 3-13 ) consists of a ball suspended by a string. When the ball is pulled to one side with its string taut and then released, it swings back and forth. When it is released, the ball has a PE relative to the bottom of its path of mgh. At its lowest point all this PE has become kinetic energy 1 _ 2 mv
2. After reaching the bottom, the ball continues in its motion until it rises to the same height h on the opposite side from its initial position. Then, momentarily at rest since all its KE is now PE, the ball begins to retrace its path back through the bottom to its initial position.
Energy cannot be created or destroyed, although it can be changed from one form to another.
Figure 3-12 Energy transformations in planetary motion. The total energy (KE 1 PE) of the planet is the same at all points in its orbit. (Planetary orbits are much more nearly circular than shown here.)
Figure 3-13 Energy transformations in pendulum motion. The total energy of the ball stays the same but is continuously exchanged between kinetic and potential forms.
Example 3.6
A girl on a swing is 2.2 m above the ground at the ends of her motion and 1.0 m above the ground at the lowest point. What is the girl’s maximum speed?
Solution The maximum speed v will occur at the lowest point where her potential energy above this point has been entirely converted to kinetic energy. If the difference in height is h 5 (2.2 m) 2 (1.0 m) 5 1.2 m and the girl’s mass is m, then
Kinetic energy 5 change in potential energy 1 _ 2 mv2 5 mgh
v 5 √ ____
2gh 5 √ _______________
2(9.8 m/s2)(1.2 m) 5 4.8 m/s
The girl’s mass does not matter here.
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Other Forms of Energy Energy can exist in a number of forms besides those already described. Although all of them can ultimately be classed as either kinetic or potential, they are often thought of as separate kinds. Figure 3-9 showed examples of magnetic and elastic potential energies. Chemical energy becomes the energy released when the gasoline in our cars reacts with oxygen, and electric energy turns motors in homes, vehicles, and factories. Radiant energy in sunlight does work in evaporat- ing water from the earth’s surface, in producing differences in air temperature that cause winds, and in powering chemical reactions in plants that end up as our food. As we shall find in Sec. 3.12, matter itself must also be considered as a form of energy, and its energy equivalent follows the law of energy conservation in processes such as nuclear reactions where mass changes occur.
Heat A skier slides down a hill and comes to rest at the bottom. What became of the potential energy he or she had at the top? The engine of a car is shut off while the car is allowed to coast along a level road. Eventually the car slows down and comes to a stop. What became of its original kinetic energy?
All of us can give similar examples of the apparent disappearance of kinetic or potential energy. What these examples have in common is that heat is always pro- duced in an amount just equivalent to the “lost” energy ( Fig. 3-14 ). One kind of energy is simply being converted to another; no energy is lost, nor is any new energy created. We can regard the skier and the car as having done work against friction as they moved, work that became heat.
Work-Energy Theorem If we apply the law of conservation of energy to the motion of an object, we have the work-energy theorem:
Winput 5 DKE 1DPE 1 Woutput Work-energy theorem 3-7
where W input 5 work done on object ΔKE 5 change in object’s kinetic energy ΔPE 5 change in object’s potential energy W output 5 work done by object
(The symbol Δ, the Greek capital letter delta, is often used in science to mean “change in”.) The KE and PE of the object may each either increase or decrease. Some or all of the work done by the object may become heat, as in the cases of the skier and the car mentioned earlier.
Example 3.7
A 40-kg girl slides down a playground slide 5 m long whose top is 2 m above its lowest point. Her speed at the bottom is 4 m/s. (a) How much energy was lost to friction? (b) With how much force did friction slow her down?
Solution (a) Since no work was done on the girl during her slide, W input 5 0. The changes in her kinetic
and potential energies were
DKE 5 KEfinal 2 KEinitial 5 1 _ 2 mv
2 2 0 5 1 _ 2 mv2
DPE 5 PEfinal 2 PEinitial 5 0 2 mgh 5 2 mgh and W output 5 W friction . Hence, from Eq. 3-7,
0 5 1 _ 2 mv 2 2 mgh 1 Wfriction
Wfriction 5 mgh 2 1 _ 2 mv 2 5 (40 kg)(9.8 m/s)(2 m) 2 1 _ 2 (40 kg)(4 m/s)
2
5 784 J 2 320 J 5 464 J
(b) According to Eq. 3-1, W friction 5 F friction d, so the frictional force on the girl was
Ffriction 5 Wfriction _______ d 5
464 J _____ 5 m 5 92.8 N (about 21 lb)
Figure 3-14 The potential energy of these skiers at the top of the slope turns into kinetic energy and eventually into heat as they slide downhill.
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3.6 Mechanical Advantage How to Change the Magnitude of a Force A simple machine is a device that changes the magnitude of a force while transmit- ting it for a particular purpose. An example is a pulley system like the one shown in Fig. 3-15 . This system increases the magnitude F in of the input force pulling on the rope to give a larger output force F out lifting the load. The increase in force comes with a price: whatever is exerting the input force must move through a longer dis- tance s in than the distance s out through which the load moves.
The mechanical advantage MA of any machine is the ratio between the output force and the input force:
Mechanical advantage 5 MA 5 Fout ____ Fin
If friction is negligible, energy conservation requires that the output work W out 5 F out s out be equal to the input work W in 5 F in s in , so that
Fout sout 5 Fin sin and
MA 5 Fout ____ Fin
5 sin ___ sout 5 input distance
_____________ output distance
Mechanical advantage 3-8
Equation 3-8 is a general conclusion. In the pulley system of Fig. 3-15 , when the free end of the rope is pulled through a distance s in , the movable pair of pulleys is raised through a height of sout 5 1 _ 4 sin because there are four strands that must be shortened. Hence in this case
MA 5 sin ___ sout 5 sin ___ 1 _ 4 sin
5 4
For any pulley system, the MA is equal to the number of strands of rope that support the movable pulley or pulleys and thereby the load. The strand to which the input force is applied does not contribute to supporting the load; the uppermost pulley merely changes the direction of the input force.
The Lever Another example of a simple machine is the lever of Fig. 3-16 , which is pivoted at a point called the fulcrum. From the figure, we see that, because both ends of the lever move through the same angle u , the arcs s in and s out are respectively proportional to the lever arms L in and L out . Thus
MA(lever) 5 sin ___ sout 5 Lin ____ Lout
3-9
Figure 3-15 The mechanical advantage of a pulley system is MA 5 s in / s out which is equal to the number of strands of rope that support the load, here 4. Fin
Fin
sin
sout
Fout
Fout
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The larger the ratio of the lever arms, the greater the output force for a given input force. Sometimes an MA less than 1 is useful. An example is a pair of scissors, for which the output distance is greater than the input distance to give a larger range of motion to its blades at the expense of a reduced output force.
Figure 3-17 shows examples of the three kinds of lever. Belt and gear drives are developments of the lever that can be used on a continuous basis, unlike the lever, and one or both of them are found in most motor-driven machines.
Figure 3-16 A lever. Because the angles marked θ are equal, the ratios sin/Lin and sout/Lout are also equal, so the mechanical advantage of the lever is MA 5 Lin/Lout.
Fout
LoutL in
Lin Lout
Fout Fin
θ
= = =
θ
sin
sout
F in Fulcrum
MA sin sout
Lin Lout
Figure 3-17 Examples of the three classes of lever. In each case, the input distance is measured from the input force to the fulcrum and the output distance is measured from the output force to the fulcrum. The mechanical advantage of the human arm is less than 1, which enables a large range of motion at the expense of a correspondingly large input muscular force.
Fout
Fout
Fout
Fout
Fout
Fin
Fin
Fin
Fin
F in
LoutL in
Fout
F in Lout
L in
Fin
Fout L in
Lout
Class I lever
Class II lever
Class III lever
Fulcrum
Fulcrum
Fulcrum
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The Inclined Plane Another type of simple machine is the inclined plane. Choosing to walk up a gradual slope of a hill instead of a steep slope is, though we might not realize it, based on the principle of the inclined plane. Figure 3-18 a shows a box of weight w being pushed up an inclined plane by a force of magnitude F. Here s in 5 L and s out 5 h, so
MA(inclined plane) 5 sin ___ sout 5 L __ h 3-10
The screw ( Fig. 3-18 b ) is a development of the inclined plane. The smaller the dis- tance between its threads and the larger the handle of the screwdriver (or the longer the wrench) used to tighten the screw, the greater the MA. The MA of a screw can be very great even when reduced by the considerable friction between its threads and the material they are in contact with. Actually, this friction is often an advantage as it helps prevent the screw from backing out under load.
Figure 3-18 (a) An inclined plane. The theoretical mechanical advantage will be reduced by friction between the load and the ramp. (b) A screw is an inclined plane wrapped around a cylinder.
L
F
h
(a)
(b)
w w
F = =MA L
h
Example 3.8
A class II lever (see Fig. 3-17 ) whose length is 5 m is to be used to lift a 20-kg box whose center is located on the lever 1.2 m from its lower end. What is the magnitude of the force needed?
Solution The output force is the weight mg of the box, which is
Fout 5 mg 5 (20 kg)(9.8 m/s2) 5 196 N
Here L in 5 5 m and L out 5 1.2 m, so that, from Eqs. 3-8 and 3-9,
MA 5 Fout ____ Fin
5 Lin ____ Lout
Fin 5 Lin ____ Lout
Fout 5 ( 1.2 m _____ 5 m ) (196 N) 5 47 N The mechanical advantage is 5/1.2 5 4.2.
3.7 The Nature of Heat The Downfall of Caloric Although it comes as little surprise to us today to learn that heat is a form of energy, in earlier times this was not so clear. Less than two centuries ago most scientists regarded heat as an actual substance called caloric . Absorbing caloric
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caused an object to become warmer; the escape of caloric caused it to become cooler. Because the weight of an object does not change when the object is heated or cooled, caloric was considered to be weightless. It was also supposed to be invisible, odorless, and tasteless, properties that, of course, were why it could not be observed directly.
Actually, the idea of heat as a substance was fairly satisfactory for materials heated over a flame, but it could not account for the unlimited heat that could be generated by friction. One of the first to appreciate this difficulty was the American Benjamin Thompson ( Fig. 3-19 ), who had supported the British during the Revo- lutionary War and thought it wise to move to Europe afterward, where he became Count Rumford.
One of Rumford’s many occupations was supervising the making of cannon for a German prince, and he was impressed by the large amounts of heat given off by friction in the boring process. He showed that the heat could be used to boil water and that heat could be produced again and again from the same piece of metal. If heat was a fluid, it was not unreasonable that boring a hole in a piece of metal should allow it to escape. However, even a dull drill that cut no metal produced a great deal of heat. Also, it was hard to imagine a piece of metal as con- taining an infinite amount of caloric, and Rumford accordingly regarded heat as a form of energy.
Joule James Prescott Joule ( Fig. 3-20 ) was an English brewer who performed a classic experiment that settled the nature of heat once and for all. Joule’s experi- ment used a small paddle wheel inside a container of water ( Fig. 3-21 ). Work was done to turn the paddle wheel against the resistance of the water, and Joule measured exactly how much heat was supplied to the water by friction in this process. He found that a given amount of work always produced exactly the same amount of heat. This was a clear demonstration that heat is energy and not some- thing else.
Joule also carried out chemical and electrical experiments that agreed with his mechanical ones, and the result was his announcement of the law of conservation of energy in 1847, when he was 29. Although Joule was a modest man (“I have done two or three little things, but nothing to make a fuss about,” he later wrote), many honors came his way, including naming the SI unit of energy after him.
Figure 3-19 Count Rumford (1753–1814).
Figure 3-20 James Prescott Joule (1818–1889).
Figure 3-21 Joule’s experimental demonstration that heat is a form of energy. As the weight falls, it turns the paddle wheel, which heats the water by friction. The potential energy of the weight is converted first into the kinetic energy of the paddle wheel and then into heat.
What Is Heat?
As we shall learn in Chap. 5, the heat content of a body of mat- ter consists of the KE of random motion of the atoms and molecules of which the body consists. The greater the average KE of each of its atoms and molecules, the higher the temperature of the body.
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Because the universe is so complex, a variety of different quantities besides the basic ones of length, time, and mass are useful to help us understand its many aspects. We have already found velocity, acceleration, force, work, and energy to be valuable, and more are to come. The idea behind defining each of these quantities is to single out something that is involved in a wide range of observations. Then we can boil down a great many separate findings about nature into a brief, clear statement, for example, the law of conservation of energy. Now we shall learn how the concepts of linear and angular momenta can give us further insights into the behavior of moving things.
3.8 Linear Momentum Another Conservation Law As we know (Sec. 2.7), a moving object tends to continue moving at constant speed along a straight path. The linear momentum of such an object is a measure of this tendency. The more linear momentum something has, the more effort is needed to slow it down or to change its direction. Another kind of momentum is angular momentum , which reflects the tendency of a spinning body to continue to spin. When there is no question as to which is meant, linear momentum is usually referred to simply as momentum.
The linear momentum p of an object of mass m and velocity v (we recall that velocity includes both speed and direction) is defined as
p 5 mv Linear momentum 3-11 Linear momentum 5 (mass)(velocity)
The greater m and v are, the more difficult it is to change the object’s speed or direction. This definition of momentum is in accord with our experience. A baseball hit
squarely by a bat (large v ) is more difficult to stop than a baseball thrown gently (small v ). The heavy iron ball used for the shotput (large m ) is more difficult to stop than a baseball (small m ) when their speeds are the same ( Fig. 3-22 ).
Conservation of Momentum Momentum considerations are most useful in situ- ations that involve explosions and collisions. When outside forces do not act on the objects involved, their combined momentum (taking directions into account) is con- served, that is, does not change:
This statement is called the law of conservation of momentum . What it means is that, if the objects interact only with one another, each object can have its momentum
MOMENTUM
In the absence of outside forces, the total momentum of a set of objects remains the same no matter how the objects interact with one another.
Figure 3-22 The linear momentum m v of a moving object is a measure of its tendency to continue in motion at constant velocity. The symbol > means “greater than.”
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changed in the interaction, provided that the total momentum after it occurs is the same as it was before.
Momentum is conserved when a running girl jumps on a stationary sled, as in Fig. 3-23 . Even if there is no friction between the sled and the snow, the combination of girl and sled moves off more slowly than the girl’s running speed. The original momentum, which is that of the girl alone, had to be shared between her and the sled when she jumped on it. Now that the sled is also moving, the new speed must be less than before in order that the total momentum stays the same.
Figure 3-23 When a running girl jumps on a stationary sled, the combination moves off more slowly than the girl’s original speed. The total momentum of girl 1 sled is the same before and after she jumps on it.
m2
m2
Example 3.9
Let us see what happens when an object breaks up into two parts. Suppose that an astro- naut outside a space station throws away a 0.5-kg camera in disgust when it fails to work ( Fig. 3-24 ). The mass of the spacesuited astronaut is 100 kg, and the camera moves off at 6 m/s. What happens to the astronaut?
Solution The total momentum of the astronaut and camera was zero originally. According to the law of conservation of momentum, their total momentum must therefore be zero after- ward as well. If we call the astronaut A and the camera C, then
Momentum before 5 momentum afterward 0 5 mAvA 1 mCvC
Hence mAvA 5 2mCvC
where the minus sign signifies that v A is opposite in direction to v C . Throwing the camera away therefore sets the astronaut in motion as well, with camera and astronaut moving in opposite directions. Newton’s third law of motion (action-reaction) tells us the same thing, but conservation of momentum enables us to find the astronaut’s speed at once:
vA 5 2 mCvC _____ mA 5 2
(0.5 kg)(6 m/s) _____________
100 kg 5 20.03 m/s
After an hour, which is 3600 s, the camera will have traveled v C t 5 21,600 m 5 21.6 km, and the astronaut will have traveled v A t 5 108 m in the opposite direction if not tethered to the space station.
Figure 3-24 The momentum m C v C to the right of the thrown camera is equal in magnitude to the momentum m A v A to the left of the astronaut who threw it away. vA 5 20.03 m/s
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3.9 Rockets Momentum Conservation Is the Basis of Space Travel The operation of a rocket is based on conservation of linear momentum. When the rocket stands on its launching pad, its momentum is zero. When it is fired, the momentum of the exhaust gases that rush downward is balanced by the momen- tum in the other direction of the rocket moving upward. The total momentum of the entire system, gases and rocket, remains zero, because momentum is a vector quan- tity and the upward and downward momenta cancel ( Fig. 3-27 ).
Thus a rocket does not work by “pushing” against its launching pad, the air, or anything else. In fact, rockets function best in space where no atmosphere is present to interfere with their motion.
The ultimate speed a rocket can reach is governed by the amount of fuel it can carry and by the speed of its exhaust gases. Because both these quantities are lim- ited, multistage rockets are used in the exploration of space. The first stage is a large rocket that has a smaller one mounted in front of it. When the fuel of the first stage has burnt up, its motor and empty fuel tanks are cast off. Then the second stage is fired. Since the second stage is already moving rapidly and does not have to carry the motor and empty fuel tanks of the first stage, it can reach a much higher final speed than would otherwise be possible.
Applying the law of conservation of momentum to collisions gives some interesting results. These are shown in Fig. 3-25 for an object of mass m and speed v that strikes a stationary object of mass M and does not stick to it. Three situations are possible: 1. The target object has more mass, so that M > m. What happens here is that the
incoming object bounces off the heavier target object and they move apart in opposite directions.
2. The two objects have the same mass, so that M 5 m. Now the incoming object stops and the target object moves off with the same speed v the incoming one had.
3. The target object has less mass, so that m > M. In this case the incoming object continues in its original direction after the impact but with reduced speed while the target object moves ahead of it at a faster pace. The greater m is compared with M, the closer the target object’s final speed is to 2v.
The third case corresponds to a golf club striking a golf ball (Fig. 3-26). This suggests that the more mass the clubhead has for a given speed, the faster the ball will fly off when struck. However, a heavy golf club is harder to swing fast than a light one, so a compro- mise is necessary. Experience has led golfers to use clubheads with masses about 4 times the 46-g mass of a golf ball when they want maximum distance. A good golfer can swing a clubhead at over 50 m/s.
Collisions
Figure 3-26 The speed of a golf ball is greater than the speed of the clubhead that struck it because the mass of the ball is smaller than that of the clubhead.
m M
(1) (2) (3)
Figure 3-25 How the effects of a head-on collision with a stationary target object depend on the relative masses of the two objects.
Figure 3-27 Rocket propulsion is based upon conservation of momentum. If gravity is absent, the downward momentum of the exhaust gases is equal in magnitude and opposite in direction to the upward momentum of the rocket at all times.
v1
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Depending upon the final speed needed for a given mission, three or even four stages may be required. The Saturn V launch vehicle that carried the Apollo 11 space- craft to the moon in July 1969 had three stages. Just before takeoff the entire assembly was 111 m long and had a mass of nearly 3 million kg ( Fig. 3-28 ).
3.10 Angular Momentum A Measure of the Tendency of a Spinning Object to Continue to Spin We have all noticed the tendency of rotating objects to continue to spin unless they are slowed down by an outside agency. A top would spin indefinitely but for friction between its tip and the ground. Another example is the earth, which has been turning for billions of years and is likely to continue doing so for many more to come.
The rotational quantity that corresponds to linear momentum is called angular momentum , and conservation of angular momentum is the formal way to describe the tendency of spinning objects to keep spinning.
The precise definition of angular momentum is complicated because it depends not only upon the mass of the object and upon how fast it is turning, but also upon how the mass is arranged in the body. As we might expect, the greater the mass of a body and the more rapidly it rotates, the more angular momentum it has and the more pronounced is its tendency to continue to spin. Less obvious is the fact that, the farther away from the axis of rotation the mass is distributed, the more the angular momentum.
An illustration of both the latter fact and the conservation of angular momentum is a skater doing a spin ( Fig. 3-30 ). When the skater starts the spin, she pushes against the ice with one skate to start turning. Initially both arms and one leg are extended, so that her mass is spread as far as possible from the axis of rotation. Then she brings her arms and the outstretched leg in tightly against her body, so that now all her mass is as close as possible to the axis of rotation. As a result, she spins faster. To make up for the change in the mass distribution, the speed must change as well to conserve angular momentum.
Figure 3-28 Apollo 11 lifts off its pad to begin the first human visit to the moon. The spacecraft’s final speed was 10.8 km/s, which is equivalent to 6.7 mi/s. Conservation of linear momentum underlies rocket propulsion.
The conservation principles of energy, linear momentum, and angular momen- tum are useful because they are obeyed in all known processes. They are signifi- cant for another reason as well. In 1917 the German mathematician Emmy Noether (Fig. 3-29) proved that: 1. If the laws of nature are the same at all times, past, present, and future, then
energy must be conserved. 2. If the laws of nature are the same everywhere in the universe, then linear
momentum must be conserved. 3. If the laws of nature do not depend on direction, then angular momentum
must be conserved. All other conservation principles in physics, for instance conservation of elec-
tric charge (Sec. 6.1), can also be traced to similar general regularities in the uni- verse. Thus the existence of these principles testifies to a profound order in the universe, despite the irregularities and randomness of many aspects of it, a truly remarkable finding. In 1933 Noether moved to the United States where, after a period at the Institute for Advanced Study in Princeton, she became a professor at Bryn Mawr.
Conservation Principles
Figure 3-29 Emmy Noether (1882–1935).
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Spin Stabilization Like linear momentum, angular momentum is a vector quan- tity with direction as well as magnitude. Conservation of angular momentum there- fore means that a spinning body tends to maintain the direction of its spin axis in addition to the amount of angular momentum it has. A stationary top falls over at once, but a rapidly spinning top stays upright because its tendency to keep its axis in the same orientation by virtue of its angular momentum is greater than its tendency to fall over ( Fig. 3-31 ). Footballs and rifle bullets are sent off spinning to prevent them from tumbling during flight, which would increase air resistance and hence shorten their range ( Fig. 3-32 ).
Figure 3-30 Conservation of angular momentum. Angular momentum depends upon both the speed of turning and the distribution of mass. When the skater pulls in her arms and extended leg, she spins faster to compensate for the change in the way her mass is distributed.
Axis Axis
Fast spinSlow spin
Figure 3-31 The faster a top spins, the more stable it is. When all its angular momentum has been lost through friction, the top falls over.
Figure 3-32 Conservation of angular momentum keeps a spinning football from tumbling end-over-end, which would slow it down and reduce its range.
Planetary Motion
Kepler’s second law of planetary motion (Fig. 1-11) has an origin similar to that of the changing spin rate of a skater. A planet moving around the sun has angu- lar momentum, which must be the same everywhere in its orbit. As a result the planet’s speed is greatest when it is close to the sun, least when it is far away.
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Relativity 81
RELATIVITY
In 1905 a young physicist of 26 named Albert Einstein published an analysis of how measurements of time and space are affected by motion between an observer and what he or she is studying. To say that Einstein’s theory of relativity revolutionized science is no exaggeration.
Relativity links not only time and space but also energy and matter. From it have come a host of remarkable predictions, all of which have been confirmed by experi- ment. Eleven years later Einstein took relativity a step further by interpreting grav- ity as a distortion in the structure of space and time, again predicting extraordinary effects that were verified in detail.
3.11 Special Relativity Things Are Seldom What They Seem Thus far in this book no special point has been made about how such quantities as length, time, and mass are measured. In particular, who makes a certain measure- ment would not seem to matter—everybody ought to get the same result. Suppose we want to find the length of an airplane when we are on board. All we have to do is put one end of a tape measure at the airplane’s nose and look at the number on the tape at the airplane’s tail.
But what if we are standing on the ground and the airplane is in flight? Now things become more complicated because the light that carries information to our instruments travels at a definite speed. According to Einstein, our measurements from the ground of length, time, and mass in the airplane would differ from those made by somebody moving with the airplane.
Postulates of Relativity Einstein began with two postulates. The first concerns frames of reference , which were mentioned in Sec. 2.1. Motion always implies a frame of reference relative to which the location of something is changing. A passen- ger walking down the aisle moves relative to an airplane, the airplane moves relative to the earth, the earth moves relative to the sun, and so on ( Fig. 3-33 ).
Figure 3-33 All motion is relative to a chosen frame of reference. Here the photographer has turned the camera to keep pace with one of the cyclists. Relative to him, both the road and the other cyclists are moving. There is no fixed frame of reference in nature, and therefore no such thing as “absolute motion”; all motion is relative.
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If we are in the windowless cabin of a cargo airplane, we cannot tell whether the airplane is in flight at constant velocity or is at rest on the ground, since without an external frame of reference the question has no meaning. To say that something is moving always requires a frame of reference. From this follows Einstein’s first postulate:
If the laws of physics were different for different observers in relative motion, the observers could find from these differences which of them were “stationary” in space and which were “moving.” But such a distinction does not exist, hence the first postulate.
The second postulate, which follows from the results of a great many experi- ments, states that
The speed of light in free space is always c 5 3 3 10 8 m/s, about 186,000 mi/s.
Length, Time, and Kinetic Energy Let us suppose I am in an airplane moving at the constant velocity v relative to you on the ground. I find that the airplane is L 0 long, that it has a mass of m, and that a certain time interval (say an hour on my watch) is t 0 . Einstein showed from the above postulates that you, on the ground, would find that
1. The length L you measure is shorter than L 0 . 2. The time interval t you measure is longer than t 0 . 3. The kinetic energy KE you determine is greater than 1 _ 2 mv
2.
That is, to you on the ground, the airplane appears shorter than to me and to have more KE, and to you, my watch appears to tick more slowly.
The differences between L and L 0 , t and t 0 , and KE and 1 _ 2 mv 2 depend on the ratio
v/c between the relative speed v of the frames of reference (here the speed of the airplane relative to the ground) and the speed of light c. Because c is so great, these differences are too small to detect at speeds like those of airplanes. However, they must be taken into account in spacecraft flight. And, at speeds near c, which often occur in the subatomic world of such tiny particles as electrons and protons, relativ- istic effects are conspicuous. Although at speeds much less than c the formula 1 _ 2 mv
2 for kinetic energy is still valid, at high speeds the theory of relativity shows that the KE of a moving object is higher than 1 _ 2 mv
2 ( Fig. 3-34 ).
The laws of physics are the same in all frames of reference moving at constant velocity with respect to one another.
The speed of light in free space has the same value for all observers.
Figure 3-34 The faster an object moves relative to an observer, the more the object’s kinetic energy KE exceeds 1 _ 2 mv
2 . This effect is only conspicuous at speeds near the speed of light c 5 3 3 10 8 m/s, which is about 186,000 mi/s. Because an object would have an infinite KE if v 5 c, nothing with mass can ever move that fast or faster.
K in
et ic
E ne
rg y
0 0.2c 0.4c 0.6c 0.8c c
mv2
KE
Speed v
1 2
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Relativity 83
The Ultimate Speed Limit As we can see from the graph, the closer v gets to c, the closer KE gets to infinity. Since an infinite kinetic energy is impossible, this conclusion means that nothing can travel as fast as light or faster: c is the absolute speed limit in the universe. The implications of this limit for space travel are dis- cussed in Chap. 19.
Einstein’s 1905 theory, which led to the above results among others, is called special relativity because it is restricted to constant velocities. His later theory of general relativity , which deals with gravity, includes accelerations.
3.12 Rest Energy Matter Is a Form of Energy The most far-reaching conclusion of special relativity is that mass and energy are related to each other so closely that matter can be converted into energy and energy into matter. The rest energy of a body is the energy equivalent of its mass. If a body has the mass m, its rest energy is
E0 5 mc2 Rest energy 3-12 Rest energy 5 (mass)(speed of light)2
Experiments show that this formula is accurate to at least 0.00004 percent. The rest energy of a 1.4-kg object, such as this book, is
E0 5 mc2 5 (1.4 kg)(3 3 108 m/s)2 5 1.26 3 1017 J
quite apart from any kinetic or potential energy it might have. If liberated, this energy would be more than enough to send a million tons to the moon. By contrast, the PE of this book on top of Mt. Everest, which is 8850 m high, relative to its sea-level PE is 1.21 3 10 5 J, about 10 12 —a trillion—times smaller.
How is it possible that so much energy can be bottled up in even a little bit of matter without anybody having known about it until Einstein’s work? In fact, we do see matter being converted into energy around us all the time. We just do not nor- mally think about what we find in these terms. All the energy-producing reactions of chemistry and physics, from the lighting of a match to the nuclear fusion that pow- ers the sun and stars, involve the disappearance of a small amount of matter and its reappearance as energy. The simple formula E 0 5 mc 2 has led not only to a better
Example 3.10
How much mass is converted into energy per day in a 100-MW nuclear power plant?
Solution There are (60)(60)(24) 5 86,400 s/day, so the energy liberated per day is
E0 5 Pt 5 (102)(106 W)(8.64 3 104 s) 5 8.64 3 1012 J
From Eq. 3-12 the corresponding mass is
m 5 E0 __ c2
5 8.64 3 10 12 J ____________
(3 3 108 m/s)2 5 9.6 3 1025 kg
This is less than a tenth of a gram—not much. To liberate the same amount of energy from coal, about 270 tons would have to be burned.
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understanding of how nature works but also to the nuclear power plants—and nuclear weapons—that are so important in today’s world.
The discovery that matter and energy can be converted into each other does not affect the law of conservation of energy provided we include mass as a form of energy. Table 3-1 lists the basic features of the various quantities introduced in this chapter.
3.13 General Relativity Gravity Is a Warping of Spacetime Einstein’s general theory of relativity, published in 1916, related gravitation to the structure of space and time. What is meant by “the structure of space and time” can
B I O G R A P H Y
or skeptical, even the most unexpected of Einstein’s conclusions were soon confirmed and the development of what is now called modern physics began in earnest. After university posts in Switzerland and Czechoslovakia, in 1913 Einstein took up an appointment at the Kaiser Wilhelm Institute in Ber- lin that left him able to do research free of financial worries and routine duties. His interest was now mainly in gravity, and he began where Newton had left off more than 200 years earlier.
The general theory of relativity that resulted from Einstein’s work provided a deep understanding of
gravity, but his name remained unknown to the general public. This changed in 1919 with the dramatic discovery that gravity affects light exactly as Einstein had predicted. He immediately became a world celebrity, but his well-earned fame did not provide security when Hitler and the Nazis came to power in Ger- many in the early 1930s. Einstein left in 1933 and spent the rest of his life at the Institute for Advanced Study in Princeton, New Jersey, thereby escaping the fate of millions of other European Jews at the hands of the Germans.
Einstein’s last years were spent in a fruitless search for a “unified field the- ory” that would bring together gravita- tion and electromagnetism in a single picture. The problem was worthy of his gifts, but it remains unsolved to this day although progress is being made.
Bitterly unhappy with the rigid disci- pline of the schools of his native Ger- many, Einstein went to Switzerland at 16 to complete his education and later got a job examining patent applications at the Swiss Patent Office in Berne. Then, in 1905, ideas that had been in his mind for years when he should have been paying attention to other mat- ters (one of his math teachers called Einstein a “lazy dog”) blossomed into three short papers that were to change decisively the course of not only phys- ics but modern civilization as well.
The first paper proposed that light has a dual character with par- ticle as well as wave properties. This work is described in Chap. 9 together with the quantum theory of the atom that flowed from it.
The subject of the second paper was brownian motion, the irregu- lar zigzag motion of tiny bits of sus- pended matter such as pollen grains in water ( Fig. 3-35 ). Einstein arrived at a formula that related brownian motion to the bombardment of the particles by randomly moving mol- ecules of the fluid in which they were suspended. Although the molecular theory of matter had been proposed many years before, this formula was the long-awaited definite link with experiment that convinced the remaining doubters that molecules actually exist. The third paper intro- duced the theory of relativity.
Although much of the world of physics was originally either indifferent
Albert Einstein (1879–1955)
Figure 3-35 The irregular path of a microscopic particle bombarded by molecules. The line joins the positions of a single particle observed at constant intervals. This phenomenon is called brownian movement and is direct evidence of the reality of molecules and their random motions. It was discovered in 1827 by the British botanist Robert Brown.
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Relativity 85
be given a quite precise meaning mathematically, but unfortunately no such preci- sion is possible using ordinary language. All the same, we can legitimately think of the force of gravity as arising from a warping of spacetime around a body of matter so that a nearby mass tends to move toward the body, much as a marble rolls toward the depression in a rubber sheet formed by an object placed on it ( Fig. 3-36 ). In an apt formulation, “Matter tells spacetime how to curve, and spacetime tells matter how to move.”
It may seem as though one abstract concept is merely replacing another, but in fact the new point of view led Einstein and other scientists to a variety of remarkable discoveries that could not have come from the older way of thinking.
Gravity and Light Perhaps the most spectacular of Einstein’s results was that light ought to be subject to gravity. The effect is very small, so a large mass, such as that of the sun, is needed to detect the influence of its gravity on light. If Einstein was right, light rays that pass near the sun should be bent toward it by 0.0005 8 —the diameter of a dime seen from a mile away. To check this prediction, photographs were taken of stars that appeared in the sky near the sun during an eclipse in 1919, when they could be seen because the moon obscured the sun’s disk (see Chap. 17). These photographs were then compared with photographs of the same region of the sky taken when the sun was far away ( Fig. 3-37 ), and the observed changes in the apparent positions of the stars matched Einstein’s calculations. Other predictions based on general relativ- ity have also been verified, and the theory remains today without serious rival. (For more, see Sec. 3.13 at www.mhhe.com/krauskopf .)
Quantity Type Symbol Unit Meaning Formula
Work Scalar W Joule (J) A measure of the change produced by a force that acts on something
W 5 Fd
Power Scalar P Watt (W) The rate at which work is being done P 5 W / t Kinetic energy Scalar KE Joule (J) Energy of motion KE 5 1 _ 2 mv
2
Potential energy Scalar PE Joule (J) Energy of position PE gravitational 5 mgh Rest energy Scalar E 0 Joule (J) Energy equivalent of the mass of an object E 0 5 mc 2 Linear momentum Vector p Kg ∙ m/s A measure of the tendency of a moving object
to continue moving in the same straight line at the same speed
p 5 m v
Angular momentum Vector — — A measure of the tendency of a rotating object to continue rotating about the same axis at the same speed
—
Table 3-1 Energy, Power, and Momentum
Figure 3-36 General relativity pictures gravity as a warping of the structure of space and time due to the presence of a body of matter. An object nearby experiences an attractive force as a result of this distortion in spacetime, much as a marble rolls toward the depression formed in a rubber sheet by an object placed on it.
Gravity and Time
Among the unexpected predic- tions of general relativity is that the stronger gravity is in a partic- ular place, the slower the passage of time there. This has been veri- fied by experiment. Our heads are more often farther from the earth than our feet, hence the weaker gravity our heads experience on the average means that they age faster than our feet. However, this is probably not why our minds usually become less sharp as we grow older—the difference is only about 10 2 11 s per day.
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Figure 3-37 Starlight that passes near the sun is deflected by its strong gravitational pull. The deflection, which is very small, can be measured during a solar eclipse when the sun’s disk is obscured by the moon.
The mechanical advantage of a machine is the ratio between the output force it exerts and the input force provided to it.
Linear momentum is a measure of the tendency of a moving object to continue in motion along a straight line. Angular momentum is a measure of the tendency of a rotat- ing object to continue spinning about the same axis. Both are vector quantities. If no outside forces act on a set of objects, then their linear and angular momenta are conserved, that is, remain the same regardless of how the objects interact with one another.
According to the special theory of relativity, when there is relative motion between an observer and what is being observed, lengths are shorter than when at rest, time intervals are longer, and kinetic energies are greater. Nothing can travel faster than the speed of light.
The general theory of relativity, which relates gravitation to the structure of space and time, correctly predicts that light should be subject to gravity.
Work is a measure of the change, in a general sense, that a force causes when it acts upon something. The work done by a force act- ing on an object is the product of the magnitude of the force and the distance through which the object moves while the force acts on it. If the direction of the force is not the same as the direction of motion, the projection of the force in the direction of motion must be used. The unit of work is the joule (J).
Power is the rate at which work is being done. Its unit is the watt (W).
Energy is the property that something has that enables it to do work. The unit of energy is the joule. The three broad catego- ries of energy are kinetic energy, which is the energy something has by virtue of its motion, potential energy, which is the energy something has by virtue of its position, and rest energy, which is the energy something has by virtue of its mass. According to the law of conservation of energy, energy cannot be created or destroyed, although it can be changed from one form to another (including mass).
Important Terms and Ideas
Work: W 5 Fd Power: P 5 W ___ t 5 Fv
Kinetic energy: KE 5 1 _ 2 mv 2
Gravitational potential energy: PE 5 mgh Work-energy theorem: Winput 5 D KE 1 D PE 1 Woutput
(D 5 “change in”)
Mechanical advantage: MA 5 Fout ____ Fin
5 sin ___ sout
Linear momentum: p 5 mv
Rest energy: E0 5 mc2
Important Formulas
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Multiple Choice 87
12. The formula 1 _ 2 mv 2 for kinetic energy
a. is the correct formula if v is properly interpreted b. always gives too high a value c. is the low-speed approximation to the correct formula d. is the high-speed approximation of the correct formula
13. A spacecraft has left the earth and is moving toward Mars. An observer on the earth finds that, relative to measure- ments made when the spacecraft was at rest, its
a. length is shorter b. KE is less than 1 _ 2 mv
2
c. clocks tick faster d. rest energy is greater
14. The upper limit to the speed of an object with mass a. depends on the mass b. corresponds to a KE equal to its rest energy c. is the speed of sound d. is the speed of light
15. It is not true that a. light is affected by gravity b. the mass of a moving object depends upon its speed c. the maximum speed anything can have is the speed of light d. momentum is a form of energy
16. Albert Einstein did not discover that a. the length of a moving object is less than its length at rest b. the acceleration of gravity g is a universal constant c. light is affected by gravity d. gravity is a warping of spacetime
17. The work done in holding a 50-kg object at a height of 2 m above the floor for 10 s is
a. 0 c. 1000 J b. 250 J d. 98,000 J
18. The work done in lifting 30 kg of bricks to a height of 20 m is a. 61 J c. 2940 J b. 600 J d. 5880 J
19. A total of 4900 J is used to lift a 50-kg mass. The mass is raised to a height of
a. 10 m c. 960 m b. 98 m d. 245 km
20. The work a 300-W electric grinder can do in 5.0 min is a. 1 kJ c. 25 kJ b. 1.5 kJ d. 90 kJ
21. A 150-kg yak has an average power output of 120 W. The yak can climb a mountain 1.2 km high in
a. 25 min c. 13.3 h b. 4.1 h d. 14.7 h
22. A 40-kg boy runs up a flight of stairs 4 m high in 4 s. His power output is
a. 160 W c. 40 W b. 392 W d. 1568 W
23. Car A has a mass of 1000 kg and is moving at 60 km/h. Car B has a mass of 2000 kg and is moving at 30 km/h. The kinetic energy of car A is
a. half that of car B b. equal to that of car B c. twice that of car B d. 4 times that of car B
1. Which of the following is not a unit of work? a. newton-meter c. kilogram-meter b. joule d. watt-hour
2. An object at rest may have a. velocity c. kinetic energy b. momentum d. potential energy
3. A moving object must have which one or more of the following?
a. potential energy c. rest energy b. kinetic energy d. momentum
4. When the momentum of a moving object is increased, there must also be an increase in which one or more of the follow- ing of the object’s properties?
a. speed c. kinetic energy b. acceleration d. potential energy
5. The total amount of energy (including the rest energy of matter) in the universe
a. cannot change b. can decrease but not increase c. can increase but not decrease d. can either increase or decrease
6. When the speed of a moving object is halved, a. its KE is halved b. its PE is halved c. its rest energy is halved d. its momentum is halved
7. Two balls, one of mass 5 kg and the other of mass 10 kg, are dropped simultaneously from a window. When they are 1 m above the ground, the balls have the same
a. kinetic energy b. potential energy c. momentum d. acceleration
8. A bomb dropped from an airplane explodes in midair. a. Its total kinetic energy increases b. Its total kinetic energy decreases c. Its total momentum increases d. Its total momentum decreases
9. The operation of a rocket is based upon a. pushing against its launching pad b. pushing against the air c. conservation of linear momentum d. conservation of angular momentum
10. A spinning skater whose arms are at her sides then stretches out her arms horizontally.
a. She continues to spin at the same rate b. She spins more rapidly c. She spins more slowly d. Any of the choices could be correct, depending on how
heavy her arms are 11. According to the principle of relativity, the laws of physics
are the same in all frames of reference a. at rest with respect to one another b. moving toward or away from one another at constant velocity c. moving parallel to one another at constant velocity d. all of these
Multiple Choice
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a. kinetic energy b. potential energy relative to the ground c. rest energy d. momentum
31. A 10,000-kg freight car moving at 2 m/s collides with a sta- tionary 15,000-kg freight car. The two cars couple together and move off at
a. 0.8 m/s c. 1.3 m/s b. 1 m/s d. 2 m/s
32. A 30-kg girl and a 25-kg boy are standing on frictionless roller skates. The girl pushes the boy, who moves off at 1.0 m/s. The girl’s speed is
a. 0.45 m/s c. 0.83 m/s b. 0.55 m/s d. 1.2 m/s
33. An object has a rest energy of 1 J when its mass is a. 1.1 3 10217 kg c. 1 kg b. 3.3 3 1029 kg d. 9 3 1016 kg
34. The smallest part of the total energy of the ball of Multiple Choice 30 is
a. kinetic energy b. potential energy relative to the ground c. rest energy d. momentum
35. The lightest particle in an atom is an electron, whose rest mass is 9.1 3 10231 kg. The energy equivalent of this mass is approximately
a. 10213 J c. 3 3 10223 J b. 10215 J d. 10247 J
24. A 1-kg object has a potential energy of 1 J relative to the ground when it is at a height of
a. 0.102 m c. 9.8 m b. 1 m d. 98 m
25. A 1-kg object has kinetic energy of 1 J when its speed is a. 0.45 m/s c. 1.4 m/s b. 1 m/s d. 4.4 m/s
26. The 2-kg blade of an ax is moving at 60 m/s when it strikes a log. If the blade penetrates 2 cm into the log as its KE is turned into work, the average force it exerts is
a. 3 kN c. 72 kN b. 90 kN d. 180 kN
27. The highest MA that can be obtained by a system of two pulleys is
a. 2 c. 4 b. 3 d. 5
28. A machine has a MA of 6.0. The work input needed by the machine to raise a 40-kg load by 3.0 m is
a. 0.12 kJ c. 1.2 kJ b. 0.20 kJ d. 7.1 kJ
29. A person uses a force of 300 N to pry up one end of a 120-kg box using a lever 2.4 m long. The distance from the end of the box to the fulcrum is
a. 40 cm c. 81 cm b. 49 cm d. 171 cm
30. A 1-kg ball is thrown in the air. When it is 10 m above the ground, its speed is 3 m/s. At this time most of the ball’s total energy is in the form of
20 cm on the earth’s surface, how high could he jump on the surface of Mars?
3.2 Power
8. The kilowatt-hour is a unit of what physical quantity or quantities?
9. The motor of a boat develops 60 kW when the boat’s speed is 15 km/h. With how much force does the water resist the motion of the boat?
10. How much power must the legs of a 70-kg man develop in order to run up a staircase 5 m high in 9 s?
11. A weightlifter raises a 70-kg barbell from the floor to a height of 2.2 m in 1.2 s. What was his average power output during the lift?
12. An escalator 14 m long is carrying a 70-kg person from one floor to another 8 m higher. The escalator has a speed of 1.0 m/s. (a) How much work does the escala- tor do in carrying the person to the top? (b) What is its power output while doing so?
13. A 700-kg horse whose power output is 1.0 hp is pulling a sled over the snow at 3.5 m/s. Find the force the horse exerts on the sled.
14. A person’s metabolic processes can usually operate at a power of 6 W/kg of body mass for several hours at a time. If a 60-kg woman carrying a 12-kg pack is walking uphill with an energy-conversion efficiency of 20 percent, at what rate, in meters/hour, does her altitude increase?
3.1 The Meaning of Work
1. A person holds a 10-kg package 1.2 m above the floor for 1 min. How much work is done?
2. Under what circumstances (if any) is no work done on a moving object even though a net force acts upon it?
3. The sun exerts a gravitational force of 4.0 3 1028 N on the earth, and the earth travels 9.4 3 1011 m in its yearly orbit around the sun. How much work is done by the sun on the earth each year?
4. A crate is pushed across a horizontal floor at constant speed by a horizontal force of 140 N, which is just enough to overcome the friction between the crate and the floor. (a) How much work is done in pushing the crate through 10 m? (b) Rollers are then used under the crate to reduce friction and the same force is applied over the next 10 m. What happens to the work done now? (c) Will there be any change in the speed of the crate?
5. A total of 490 J of work is needed to lift a body of unknown mass through a height of 10 m. What is its mass?
6. A woman eats a cupcake and proposes to work off its 400-kJ energy content by exercising with a 15-kg barbell. If each lift of the barbell from chest height to overhead is through 60 cm and the efficiency of her body under these circumstances is 10 percent, how many times must she lift the barbell?
7. The acceleration of gravity on the surface of Mars is 3.7 m/s2. If an astronaut in a space suit can jump upward
Exercises
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Exercises 89
15. A crane whose motor has a power input of 5.0 kW lifts a 1200-kg load of bricks through a height of 30 m in 90 s. Find the efficiency of the crane, which is the ratio between its output power and its input power.
16. A total of 104 kg of water per second flows over a water- fall 30 m high. If half of the power this flow represents could be converted into electricity, how many 100-W lightbulbs could be supplied?
3.3 Kinetic Energy
17. Which of these energies might correspond to the KE of a person riding a bicycle on a road? 10 J; 1 kJ; 100 kJ
18. What is the speed of an 800-kg car whose KE is 250 kJ? 19. A moving object whose initial KE is 10 J is subject to a
frictional force of 2 N that acts in the opposite direction. How far will the object move before coming to a stop?
20. Is the work needed to bring a car’s speed from 0 to 10 km/h less than, equal to, or more than the work needed to bring its speed from 10 to 20 km/h? If the amounts of work are different, what is the ratio between them?
21. A 1-kg salmon is hooked by a fisherman and it swims off at 2 m/s. The fisherman stops the salmon in 50 cm by braking his reel. How much force does the fishing line exert on the fish?
22. During a circus performance, John Tailor was fired from a compressed-air cannon whose barrel was 20 m long. Mr. Tailor emerged from the cannon (twice on weekdays, three times on Saturdays and Sundays) at 40 m/s. If Mr. Tailor’s mass was 60 kg, what was the average force on him when he was inside the cannon’s barrel?
23. How long will it take a 1000-kg car with a power output of 20 kW to go from 10 m/s to 20 m/s?
3.4 Potential Energy
24. Does every moving body possess kinetic energy? Does every stationary body possess potential energy?
25. As we will learn in Chap. 6, electric charges of the same kind (both positive or both negative) repel each other, whereas charges of opposite sign (one positive and the other negative) attract each other. (a) What happens to the PE of a positive charge when it is brought near another pos- itive charge? (b) When it is brought near a negative charge?
26. A 60-kg woman jumps off a wall 80 cm high and lands on a concrete road with her knees stiff. Her body is compressed by 8 cm at the moment of impact. (a) What was the average force the road exerted on her body? (b) If the woman bent her knees on impact so that she came to a stop over a dis- tance of 40 cm, what would the average force on her body be?
3.5 Conservation of Energy 27. Why does a nail become hot when it is hammered into a
piece of wood? 28. A 3-kg stone is dropped from a height of 100 m. Find its
kinetic and potential energies when it is halfway to the ground.
29. Two identical balls move down a tilted board. Ball A slides down without friction and ball B rolls down. Which ball reaches the bottom first? Why?
30. (a) A yo-yo is swung in a vertical circle in such a way that its total energy KE 1 PE is constant. At what point in the circle is its speed a maximum? A minimum? Why?
(b) If the yo-yo has a speed of 3 m/s at the top of the circle, whose radius is 80 cm, what is its speed at the bottom?
31. A ball is dropped from a height of 1 m and loses 10 per- cent of its kinetic energy when it bounces on the ground. To what height does it then rise?
32. A person sitting under a coconut palm is struck by a 1-kg coconut that fell from a height of 20 m. (a) Find the kinetic energy of the coconut when it reaches the person. (b) Find the average force exerted by the coconut if its impact is absorbed over a distance of 5 cm. (c) What is this force in pounds? Is it a good idea to sit under a coconut palm?
33. A skier is sliding downhill at 8 m/s when she reaches an icy patch on which her skis move freely with negligible friction. The difference in altitude between the top of the icy patch and its bottom is 10 m. What is the speed of the skier at the bottom of the icy patch? Do you have to know her mass?
34. A force of 10 N is used to lift a 600-g ball from the ground to a height of 1.8 m, when it is let go. What is the speed of the ball when it is let go?
3.6 Mechanical Advantage 35. A person uses a force of 49 N to raise a 30-kg load by
1.5 m with a pulley system. How much rope did she pull through the system?
36. A nut is held in the jaws of a pair of pliers. The nut is 25 mm from the pivot of the pliers and a force of 10 N is applied to each handle 150 mm from the pivot. How much force does each jaw exert on the nut?
37. The human forearm is a class III lever. Find the force the biceps muscle of an upper arm must develop to lift further a 5-kg dumbbell held in the hand of a person whose fore- arm is horizontal. How many pounds is this? Assume the force is exerted at a point 2.5 cm in front of the person’s elbow and that the hand is 32 cm in front of the elbow.
38. A ramp 20 m long slopes down 1.2 m to the edge of a lake. A force of 300 N is needed to pull a boat on a trailer at constant speed up the ramp when they are clear of the water. If friction is negligible, what is the combined mass of boat and trailer?
3.7 The Nature of Heat 39. In an effort to lose weight, a person runs 5 km per day
at a speed of 4 m/s. While running, the person’s body processes consume energy at a rate of 1.4 kW. Fat has an energy content of about 40 kJ/g. How many grams of fat are metabolized during each run?
40. An 80-kg crate is raised 2 m from the ground by a man who uses a rope and a system of pulleys. He exerts a force of 220 N on the rope and pulls a total of 8 m of rope through the pulleys while lifting the crate, which is at rest afterward. (a) How much work does the man do? (b) What is the change in the potential energy of the crate? (c) If the answers to these questions are different, explain why.
41. An 800-kg car coasts down a hill 40 m high with its engine off and the driver’s foot pressing on the brake pedal. At the top of the hill the car’s speed is 6 m/s and at the bottom it is 20 m/s. How much energy was converted into heat on the way down?
3.8 Linear Momentum 42. (a) When an object at rest explodes into two parts that fly
apart, must they move in exactly opposite directions? (b) When a moving object strikes a stationary one, must they move off in exactly opposite directions?
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58. A 1000-kg car moving east at 80 km/h collides head-on with a 1500-kg car moving west at 40 km/h, and the two cars stick together. (a) Which way does the wreckage move and with what initial speed? (b) How much KE is lost in the collision?
3.10 Angular Momentum
59. As the polar ice caps melt, the length of the day increases. Why? 60. All helicopters have two rotors. Some have both rotors on
vertical axes but rotating in opposite directions, and the rest have one rotor on a horizontal axis perpendicular to the helicopter body at the tail. Why is a single rotor never used?
61. The earthquake that caused the Indian Ocean tsunami of 2004 (see Fig. 14.40) led to changes in the earth’s crust that reduced its diameter slightly. What effect, if any, do you think this reduction had on the length of the day?
3.11 Special Relativity
62. What are the two postulates from which Einstein devel- oped the special theory of relativity?
63. The theory of relativity predicts a variety of effects that disagree with our everyday experience. Why do you think this theory is universally accepted by scientists?
64. What physical quantity will all observers always find the same value for?
65. The length of a rod is measured by several observers, one of whom is stationary with respect to the rod. What must be true of the value obtained by the stationary observer?
66. Under what circumstances does it become conspicu- ous that the formula KE 5 1 _ 2 mv
2 understates the kinetic energy of a moving object of speed v?
67. Why is it impossible for an object to move faster than the speed of light?
3.12 Rest Energy
68. The potential energy of a golf ball in a hole is negative with respect to the ground. Under what circumstances (if any) is the ball’s kinetic energy negative? Its rest energy?
69. What is the effect on the law of conservation of energy of the discovery that matter and energy can be converted into each other?
70. A certain walking person uses energy at an average rate of 300 W. All of this energy has its ultimate origin in the sun. How much matter is converted to energy in the sun per hour to supply this person?
71. One kilogram of water at 08C contains 335 kJ of energy more than 1 kg of ice at 08C. What is the mass equivalent of this amount of energy?
72. When 1 g of gasoline is burned in an engine, about 48 kJ of heat is produced. How much mass is lost in the process? Do you think this mass change could be directly measured?
73. Approximately 5.4 3 106 J of chemical energy is released when 1 kg of dynamite explodes. What fraction of the total energy of the dynamite is this?
74. Approximately 4 3 109 kg of matter is converted into energy in the sun per second. Express the power output of the sun in watts.
43. A golf ball and a Ping-Pong ball are dropped in a vacuum chamber. When they have fallen halfway to the bottom, how do their speeds compare? Their kinetic energies? Their potential energies? Their momenta?
44. Is it possible for an object to have more kinetic energy but less momentum than another object? Less kinetic energy but more momentum?
45. What happens to the momentum of a car when it comes to a stop?
46. The speed of an airplane doubles in flight. (a) How is the law of conservation of momentum obeyed in this situa- tion? (b) The law of conservation of energy?
47. When the kinetic energy of an object is doubled, what happens to its momentum?
48. What, if anything, happens to the speed of a fighter plane when it fires a cannon at an enemy plane in front of it?
49. A ball of mass m rolling on a smooth surface collides with a stationary ball of mass M. (a) Under what circum- stances will the first ball come to a stop while the second ball moves off? (b) Under what circumstances will the first ball reverse its direction while the second ball moves off in the original direction of the first ball? (c) Under what circumstances will both balls move off in the origi- nal direction of the first ball?
50. A railway car is at rest on a frictionless track. A man at one end of the car walks to the other end. (a) Does the car move while he is walking? (b) If so, in which direc- tion? (c) What happens when the man comes to a stop?
51. An empty dump truck coasts freely with its engine off along a level road. (a) What happens to the truck’s speed if it starts to rain and water collects in it? (b) The rain stops and the accumulated water leaks out. What hap- pens to the truck’s speed now?
52. A boy throws a 4-kg pumpkin at 8 m/s to a 40-kg girl on roller skates, who catches it. At what speed does the girl then move backward?
53. A 30-kg girl who is running at 3 m/s jumps on a station- ary 10-kg sled on a frozen lake. How fast does the sled with the girl on it then move?
54. A 70-kg man and a 50-kg woman are in a 60-kg boat when its motor fails. The man dives into the water with a horizontal speed of 3 m/s in order to swim ashore. If he changes his mind, can he swim back to the boat if his swimming speed is 1 m/s? If not, can the woman change the boat’s motion enough by diving off it at 3 m/s in the opposite direction? Could she then return to the boat herself if her swimming speed is also 1 m/s?
55. The 176-g head of a golf club is moving at 45 m/s when it strikes a 46-g golf ball and sends it off at 65 m/s. Find the final speed of the clubhead after the impact, assuming that the mass of the club’s shaft can be neglected.
56. A 40-kg skater moving at 4 m/s overtakes a 60-kg skater moving at 2 m/s in the same direction and collides with her. The two skaters stick together. (a) What is their final speed? (b) How much kinetic energy is lost?
57. The two skaters of Exercise 56 are moving in opposite directions when they collide and stick together. Answer the same questions for this case.
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d = 1 _ 2 at2 Distance moved during the acceleration
Next we substitute the formulas for F and d into the first equation here to give
KE 5 Fd 5 (ma) ( 1 __ 2 at2 ) 5 1 __ 2 m(at)2 But at is the ball’s speed v when it leaves our hand at the end of the acceleration, as in Fig. 1 c, so that
KE = 1 _ 2 mv2 Kinetic energy of moving ball
which is Eq. 3-5 s.
Here is a simple derivation of the formula KE 5 1 __ 2 mv 2 for
the kinetic energy of a moving object. When we throw a ball, the work we do on it becomes
its kinetic energy KE when it leaves our hand. Suppose we apply a constant force F for a distance d while the ball is in our hand, as in Fig. 1 a. The work we do is W 5 Fd, so the ball’s kinetic energy is
KE 5 Fd Work done on ball
According to the second law of motion,
F 5 ma Force applied to ball
where a is the ball’s acceleration while the force acts on it. If the time during which the force was applied is t, as in Fig. 1 b, Eq. 2-12 gives us the distance d as
Section 3.3: Deriving the Kinetic Energy Formula
FIGURE 1 How the formula KE 5 1 __ 2
mv2 for the kinetic energy of a moving object can be derived.
(c)
m
v
KE = mv2
W = 0
KE = 0
F
(a)
W = Fd
W = Fd
(b)
d
F
m a = d = at2
v
0 t
F m
1
2
1
2
3-1 1
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2 3-2
are roughly proportional to L 2 . The distance through which corresponding muscles act is roughly proportional to L. This means that the quantity Fd / m in the formula for v var- ies with L as ( L 2 )( L )/ L 3 5 1, so that in general v should not vary with L at all! In fact, although different animals have dif- ferent running speeds, there is indeed little correlation with size over a wide span. A fox can run about as fast as a horse.
The relationship Fd 5 1 __ 2 mv 2 between work done and the
resulting kinetic energy can be solved for speed v to give v 5 √
______ 2Fd/m . Let us interpret v as an animal’s running
speed, F as the force its muscles exert over the distance d, and m as its mass.
As mentioned in Sec. 2.9, if L is the animal’s length, its mass is roughly proportional to L 3 and its muscular forces
Section 3.3: Running Speeds
evidence for their existence was discovered in the behavior of a pair of close-together stars that revolve around each other. A system of this kind gives off gravitational waves and slows down as it loses energy to them. This slowing down was indeed observed and agrees well with the theo- retical expectation.
Gravitational waves should have been produced in abundance early in the history of the universe and may be present throughout today’s universe. Ultrasensitive instru- ments are now operating that may be able to pick up such waves directly.
The existence of gravitational waves that travel with the speed of light was the prediction of general relativity that had to wait longest for experimental evidence. To visualize such waves, we can think in terms of the two-dimensional model of Fig. 3-36 by imagining spacetime as a rubber sheet distorted by masses lying on it. If one of the masses vibrates, waves will be sent out in the sheet (like waves on a water surface) that set other masses in vibration.
Gravitational waves—“ripples in spacetime”—are expected to be extremely weak, and none has yet been directly detected. However, in 1974 indirect but strong
Section 3.13: Gravitational Waves
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- The Scientific Method
- Section for Chapter 1
- Motion
- Section for Chapter 2
- Energy
- Section for Chapter 3