Mechanics and Relativity

quantum mechanics and general relativity equations, difference between newtonian mechanics and relativity and comparison between newtonian mechanics and relativity
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Dr.LeonBurns,New Zealand,Researcher
Published Date:21-07-2017
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Fall semester, 2011 (revised version) Oberlin College Physics 110 Notes for Mechanics and Relativity There is, in nature, perhaps nothing older than motion, concerning which books written by philosophers are neither few nor small; never- theless I have discovered by experiment some properties of it which are worth knowing. . . Galileo Galilei Two New Sciences (1638)Chapter 1 Introduction Teachers: Daniel F. Styer Wright Laboratory room 215 775-8183, Dan.Styeroberlin.edu Oce hours: Tuesday 2:303:30 pm, Friday 10:0011:00 am, and by appointment Home telephone 281-1348 (2:30 pm to 9:00 pm only) Aaron Santos Wright Laboratory room 210 775-8566, Aaron.Santosoberlin.edu Oce hours: Tuesday 10:0011:00 am and Thursday 9:0010:00 am Course web page: http://www.oberlin.edu/physics/dstyer/MechAndRel/ I will post handouts, problem assignments, and model solutions here. The word \physics" derives from the Greek word for nature. As such, the domain of physics stretches from atoms, molecules, and nuclei to baseballs, trees, and persons to planets, stars, and galaxies. Anyone who is interested in the universe, and in his or her own place in the universe, is interested in physics. This course introduces you to physics through the topics of mechanics (motion, force, momentum, and energy) and relativity (motion, force, momentum, and energy at very high speeds). 12 CHAPTER 1. INTRODUCTION Course goals: This course aims to broaden, deepen, and sharpen your scienti c thinking skills. Such improvement cannot, however, be done in the abstract it must be carried by a vehicle of speci c topics and skills. Knowledge: Introductory classical mechanics Special relativity Skills: Problem solving Reasoning from observations and experiment \Reading an equation" Working with equipment and with phenomena Course themes: Applications from atoms to automobiles to galaxies Problem solving Confront misconceptions Confront the problems of everyday terms adopted as technical language Mutually supporting qualitative and quantitative insight Intellectual rigor use reason, don't quote authority Science is about nature, not about vocabulary or memorization Outline of Questions and Topics What is time? Motion in one dimension What causes motion? Force in one dimension Can this be generalized? Motion and force in three dimensions How can you measure the e ect of a force? Impulse and momentum How else can you measure the e ect of a force? Work and energy What are more applications? Orbits, weightlessness, oscillators, waves, rigid bodies, gyroscopes What happens if you move quite quickly? Special relativity: Time dilation, length contraction, relativity of simultaneity Lorentz transformation Relativistic momentum and energy3 Format This course contains both lectures and workshops. Each week, the entire class meets for two lectures (one hour each), and it breaks into small groups for two workshops (two hours each). The workshops include laboratory experiments, discussion, small group work, development of problem-solving skills, help with assigned homework, and other teaching that doesn't t well into a large group format. Approach There are a lot of details in this course you should not memorize them. Instead, you should be able to work with the information, to make inferences from the data, to solve problems that require an understanding of the material. In the words of Charles W. Misner: \The equationF =ma is easy to memorize, hard to use, and even more dicult to understand." Problem solving: It will not take you long to notice that the problems and exams in this course or any other physics course exercise not only your knowledge of physics but also your skills in solving problems. One of the speci c goals for this course is to teach not just about the content of mechanics and relativity, but also about problem solving. You will nd many hints for honing your problem-solving skills in the books by Elby, Browne, and P olya (on reserve see below). The course web site includes tips for solving problems. And we will bring up suggestions in both lectures and workshops whenever the opportunity arises. Reading an equation: An equation may appear to be brief, and it may appear to be just a jumble of symbols. But there is a meaning, a story, behind every equation. One of the course goals is to help you uncover these meanings to learn to \read an equation" in the same way that a connoisseur can \read a painting" to uncover meaning that lies beneath the surface, or that a white-water kayaker can \read a river" to discern dangerous and safe passages through a rapid before actually running the rapid. I sometimes call this \investing an equation with meaning". Regardless of your future life and career, you will nd this to be a valuable skill. If you are faced in debate with an opponent who writes down an inscrutable equation and claims that this equation proves his point, you should then demand that your opponent describe in words its meaning that he provide the story behind the equation. If your opponent can't do so, then it's as if he had used a ve syllable word for e ect and he didn't know its meaning. Hints for doing well in the course: We recommend that you rst do the readings, then attend the lectures, and then work on the problem assignments. More tips can be found through the course web site, but we cannot overemphasize that we expect you to read the textbook.4 CHAPTER 1. INTRODUCTION Readings Required readings: (To be purchased.) David Halliday, Robert Resnick, and Jearl Walker, Fundamentals of Physics, ninth edition (John Wiley and Sons, New York, 2011). Referred to as \HRW". We will use chapters 1 through 11, plus a little bit of chapters 15 and 37. Either the standard or the extended version is okay. From what I can see, the eighth edition is also all right, although problem numbers might be di erent. Keep your workshop notes in a bound, quadrille-ruled lab book. Supplemental readings: (The following books are on reserve in the science library. They are located on shelves along the south wall, not far to your right as you enter, near some comfy chairs to encourage browsing.) Problem solving tips and techniques: Andrew Elby, The Portable TA: A Physics Problem Solving Guide (Prentice Hall, 1998) Oversize QC32.E56 1998. Be sure to read the introduction (vol. I, page vii), the test taking tips (vol. II, page 327), and the advice on romance (vol. II, page ix). Michael E. Browne, Schaum's Outline of Physics for Engineering and Science (McGraw Hill, 1999) QC21.2.B77 1999. (Some student nd it pro table to purchase one or the other of the above two books.) Sanjoy Mahajan, Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving (MIT Press, 2010). \In problem solving, as in street ghting, rules are for fools." Available through http://mitpress.mit.edu/books/full_pdfs/Street-Fighting_Mathematics.pdf George P olya, How To Solve It (Princeton University Press, 1973) Mudd QA11.P6 1973. What is time? Klaus Mainzer, The Little Book of Time (Copernicus, 2002) QB209.M3513 2002. Scienti c American, special issue on time: September 2002. Not on reserve. Galileo and the origin of mechanics: Galileo Galilei, Two New Sciences (1638; Stillman Drake, translator, University of Wisconsin Press, 1974) QC123.G13 1974. When Galileo produced the new science of mechanics, he couldn't expect people to just take his word for it. Instead, he came up with ingenious, convincing and funny arguments that are still worth reading today. I. Bernard Cohen, The Birth of a New Physics (W.W. Norton, 1985) QC122.C6 1985. The best brief description of the scienti c setting in which Galileo worked. Dava Sobel, Galileo's Daughter (Walker, 1999) QB36.G2 S65 1999. The best brief description of the social setting in which Galileo worked.5 Physics in the everyday world: Jearl Walker, The Flying Circus of Physics (John Wiley and Sons, 1977) QC32.W2 1977. Rain- bows and barking sands, superballs and bicycles. Barry Parker, The Isaac Newton School of Driving: Physics and Your Car (Johns Hopkins University Press, 2003) QC125.2.P37 2003. Nathan A. Unterman, Amusement Park Physics (J. Weston Walch, Portland, Maine, 2001) Over- size QC32.U57 2001. David W. Hafemeister, Physics of Societal Issues: Calculations on National Security, Environ- ment, and Energy (Springer, 2007) QC28.H25 2007. Georg H ahner and Nicholas Spencer, \Rubbing and Scrubbing" Physics Today, volume 51, num- ber 9, pages 2227 (September 1998). A nice introduction to our understanding of friction, from ancient Egypt to present-day research. Not on reserve. Special topics:  Felix Klein and Arnold Sommerfeld, Uber die Theorie des Kreisels (B.G. Teubner, Leipzig, 1897 1910) 531K6722. \On the theory of tops." Check this out to see how so much can be deduced from so little. Various approaches: Paul G. Hewitt, Conceptual Physics (Addison Wesley, San Francisco, 2002) QC23.2.H488 2002 and Conceptual Physics: Practicing Physics (Addison Wesley, San Francisco, 2002) QC23.2.H49 2002. Interesting approach emphasizing topics rather than analysis. Larry Gonick and Art Hu man, The Cartoon Guide to Physics (Harper Collins, New York, 1990) QC24.5.G66 1991b. A surprisingly e ective summary. Eric M. Rogers, Physics for the Inquiring Mind (Princeton University Press, 1960) QC23.R68. An older book, and not targeted speci cally to audience of this course, but nevertheless soothing and worthwhile. David Halliday, Robert Resnick, and Jearl Walker, Fundamentals of Physics, ninth edition (Wiley, New York, 2010) QC21.3.H35 2011. If you nd yourself growing bored in this course, then dip into any of these books for more elaborate (or more idiosyncratic) treatments: A.P. French, Newtonian Mechanics (Norton, 1971) QC125.2.F74. Particularly good on orbits. A.P. French, Vibrations and Waves (Norton, 1971) QC235.F74. Charles Kittel, Walter D. Knight, and Malvin A. Ruderman, Mechanics (Berkeley physics course; McGraw-Hill, 1965) 530B455 vol. 1. Daniel Kleppner and Robert J. Kolenkow, An Introduction to Mechanics (McGraw-Hill, 1973) QA805.K62. Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics, volume 1 (Addison-Wesley, 1963) 530F438F vol. 1.6 CHAPTER 1. INTRODUCTION If, on the other hand, you nd yourself getting lost in the course, then try these books for a more algorithmic point of view: F.W. Sears, M.W. Zemansky, and H.D. Young, University Physics, sixth edition (Addison-Wesley, 1983) QC21.2.S36 1983. Daniel Kleppner and Norman Ramsey, Quick Calculus: A Self-Teaching Guide (Wiley, 1985) QA303.K665 1985. Relativity has its own raft of books: D.F. Styer, Relativity for the Questioning Mind (Johns Hopkins University Press, 2011) QC173.55.S79 2011. At a di erent level from this course but (as you might guess from the author's name) with an approach similar to the one I will use. A.P. French, Special Relativity (Norton, 1968) 530.11F887S. N. David Mermin, Space and Time in Special Relativity (McGraw-Hill, 1968) QC6.M367 1989. Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics, second edition (W.H. Freeman, 1992) QC173.65.T37 1992. Robert Resnick, Introduction to Special Relativity (John Wiley, 1968) 530.11R312I. Jones Hammond Smith, Introduction to Special Relativity (Benjamin, 1965) 530.11Sm61I. John B. Kogut, Introduction to Relativity (Harcourt, 2001) QC173.55.K64 2001. Yuan Chung Chang, Special Relativity and its Experimental Foundations (World Scienti c, Sin- gapore, 1997) QC173.65.C465 1997. James B. Hartle, Gravity: An Introduction to Einstein's General Relativity (Addison Wesley, 2003) QC173.6.H38 2003. Nice discussion of gravitational time dilation on page 116. There are a lot of supplemental readings here and I don't expect you to read all of them, or even any of them Many of these topics could make good winter term projects. . . either of us would be happy to sponsor winter term projects inspired through this course. Nuts and Bolts Background: This course assumes a knowledge of the calculus at the level of Mathematics 133: Calculus I: Limits, Continuity, Di erentiation, Integration, and Applications or the equivalent. If you lack this mathematical background then you will nd this course to be excruciatingly dicult: you should either put o the course for a year while you study calculus or else enroll instead in the algebra/trigonometry- based course Physics 103: Elementary Physics. (It is also a good idea for you to currently be taking Mathematics 134: Calculus II: Special Functions, Integration Techniques, and Power Series.)7 Arrowhead conventions: This course provides many occasions to draw di erent vector quantities. I will usually distinguish them by drawing a velocity with an open arrowhead, an acceleration with a one-sided arrowhead, and a force with a closed arrowhead. velocity: acceleration: force: or Course assignments: Readings from texts and from Galileo's Two New Sciences. Weekly problem assignments. Workshops. Two one-hour exams and one two-hour nal exam. \Minute papers": You must submit to me a written reaction at the end of each lecture. Problem assignments: The problem assignments in this course are not a dry appendage designed to keep you indoors on sunny days. Instead, the problems are central to your learning in the course. Problem solving is a more active, and hence more e ective, way to learn than reading text or listening to lecture. Problem assignments will be posted on the course web site every Wednesday, and are due at the beginning of class the following Wednesday unless there is an exam. My model solutions will be posted at the end of this class, so late assignments cannot usually be accepted. (I may make an exception in the case of a medical or family emergency, but in most cases it is to your advantage to move on to the next assignment rather than to let old work pile up.) In writing your solutions, do not just write down the nal answer. Show your reasoning and your intermediate steps. Describe (in words) the thought that went into your work as well as describing (in equations) the mathematical manipulations involved. I encourage you to collaborate or to seek printed help in working the problems, but the nal write-up must be entirely your own: you may not copy word for word or equation for equation. When you do obtain outside help you must acknowledge it. (E.g. \By integrating HRW equation (6-3) I nd that. . . " or \Employing the substitution u = sin(x) (suggested by Carol Hall). . . " or even \In working these problems I bene ted from discussions with Mike Fisher and John Silsbee.") Such an acknowledgement will never lower your grade; it is required as a simple matter of intellectual fairness. Each assignment will be graded by a student grader working under my close supervision. Workshops: You are required to attend all your workshops. For the most part, the Monday or Tuesday workshop focuses on problem solving and conceptual understanding, while the Wednesday or Thursday workshop (the \lab workshop") focuses on interacting with phenomena and conceptual understanding. Many of the lab workshops come with \pre-lab warm-up questions" which must be answered through this course's BlackBoard site to your workshop instructor, who will grade them on a \fail-pass-super" basis. At the end of each lab, take your lab notebook to your instructor, who will take a quick look at it and grade ve facets8 CHAPTER 1. INTRODUCTION of the notebook, again on a three-point basis. For one lab the \simple harmonic oscillator" lab on 34 November you will turn in your lab notebook and your instructor will grade it in detail. Your lab grade is compounded from 1/3 pre-lab questions, 1/3 quick grading on every lab, and 1/3 on the lab graded in detail. Exams: There will be two one-hour exams and one two-hour nal exam. All exams are in-class. I will drop the lowest hour's worth of exam score in determining your grade (i.e. either the score of one hour exam or half the score of the nal). No collaboration is permitted in working the exams. You may consult your 1 textbook (HRW) and your own notes that t on both sides of one 8 by 11 inch page of paper, but no other 2 material. Calculators are permitted and encouraged. Exam questions will come from lecture, workshop, or problem assignment topics I will not quiz you on obscure points deep in the text that I didn't emphasize. Before each exam I will distribute a topics list and a sample exam to give you an idea of what to expect. At each exam you're allowed to bring your text and one sheet of paper with your own notes. (You may also, of course, make notes in the margins of your text.) What are the reasons for these rules? 1. I doubt that you'll use your notes, but the process of making up your notes of deciding which ideas are the most important and the most useful can be very helpful in giving you a clear overview of what's transpired in the course. 2. I doubt that you'll use your textbook, but the fact that you can bring it emphasizes that you're not supposed to memorize equations you're supposed to work with and apply the ideas. Minute papers: I will end every lecture a minute or two early so that you can write a brief (one- or two-sentence) reaction to the state of your knowledge concerning this course. Write this reaction, and your name, on a slip of paper and hand it in to me as you leave class. I will use these reactions to plan the next lecture and the future path of this course. Your most useful reaction would be a speci c question: for example, \How can the net force acting on an object be directed toward the right while the object is moving toward the left?" Other possible reactions would be indications of general interest (\I'd like to learn more about the relation between oscillations and time-keeping.") or general questions (\Why should I care about this stu , anyway?"). Please avoid questions of marginal relevance to this course (\How can I get that cute redhead in the second row to notice me?"). The ability to answer questions is an important skill. The ability to ask them is too. The problem assignments hone your answering skills, and the minute papers hone your asking skills. Guest lectures: The department of physics and astronomy periodically invites visiting scientists to lecture at Oberlin. I will announce these visits in class. If you attend the guest lecture and submit to me a one- paragraph description through this course's BlackBoard site, you will be awarded 20 extra-credit problem-set points. The purpose of the guest lectures is to broaden your horizons: to show you physics as it is done today and to present you with a viewpoint di erent from my own. You will not understand everything that the visiting9 speakers say. . . neither will I One objective of the guest lectures is to show you how to get something out of a talk even when you don't understand everything in the talk. Informal Fridays: From time to time I will announce additional, non-required, talks on Fridays at 9:00 am in Wright 201. These informal talks are designed to introduce you to physics research and the Oberlin Department of Physics and Astronomy. If you attend one of these Friday talks and submit to me a one- paragraph description through this course's BlackBoard site, you will be awarded 20 extra-credit problem-set points. Grading: Your nal numerical grade will be the average compounded of two parts problem assignments, two parts exam, and one part workshop. On a 40-point scale, those with 4033 points earn the grade \A", 3227 points earn the grade \B", 2620 points earn the grade \C", 19 or fewer points earn the grade \F".Chapter 2 Classical Mechanics 2.1 Various Kinds of Clocks pendulum (\grandfather") clock quartz crystal oscillator (used today in most clocks, watches, computers, and cell phones) spring-driven balance wheel watch (the kind that needs to be wound) atomic clock fountain atomic clock (today's world standard) motion of the sun in the sky (that is, the rotation of the earth) (the 19561967 world standard) clepsydra amount of water dripping from a tank through a standard-sized leak such as the \Tower of the Winds" in Athens, built about 50 B.C. tuning fork clock hourglass candle or incense burning clock heartbeat (In many of his experiments regarding motion, Galileo used his own pulse as a clock.) ngernail or hair growth biological aging (e.g. wrinkling of skin, whitening of hair) waterfall wearing away a cli (F.M. Gradstein, J.G. Ogg, and A.G. Smith, A Geologic Time Scale 2004 (Cambridge University Press, Cambridge, UK, 2004). In science library reference section QE508.G3956 2004.) 102.2. DESCRIBING MOTION 11 2.2 Describing Motion To understand motion is to understand nature. Leonardo de Vinci A bicyclist stands next to his bicycle, chatting with a friend near the north edge of Stevenson Hall. He mounts his bike and pedals south at a leisurely pace, until he sees another friend standing near the south edge of Stevenson Hall. He stops the bike to stand and chat with the second friend, then realizes he's late for class and pedals south rapidly. After a little while he realizes that he was wrong today is Tuesday, not Monday, so he doesn't have a 9:00 am class so he turns around and pedals north rapidly to return to the conversation with his second friend. This situation is summarized through the three graphs on the next three pages.12 CHAPTER 2. CLASSICAL MECHANICS Stevenson Hall position (x) in meters 0 10 20 30 40 50 60 70 80 90 0 still 10 20 moving 30 40 50 moving fast 60 70 80 moving fast backwards 90 time (t) in seconds2.2. DESCRIBING MOTION 13 velocity (v) in meters/second 0 2 4 6 -6 -4 -2 0 10 20 30 40 50 60 70 80 90 time (t) in seconds14 CHAPTER 2. CLASSICAL MECHANICS 2 acceleration (a) in meters/second 0 0.4 0.8 1.2 1.6 -1.6 -1.2 -0.8 -0.4 0 10 20 30 40 50 60 70 80 90 time (t) in seconds2.3. THE WORD \ACCELERATION" 15 2.3 The Word \Acceleration" It is a commonplace that one word can have several meanings, and that the meaning used in physics can di er dramatically from the meaning used in everyday speech. For example the word \acceleration" means 1 one thing in physics and a di erent thing in everyday speech. A car has a gas petal and a break petal. The gas petal is called \the accelerator" even though the break petal provides a greater magnitude of acceleration. Many students nd it dicult to accept that a ball, tossed vertically upward, experiences the same acceleration while going up, while going down, and at the instant between when it is going nowhere. Surely part of the reason for this diculty is that they think of the ball as breaking while going up and accelerating only while going down. 2.4 Special Kinds of Motion Motion with constant velocity Suppose the velocity is a constant: v(t) =v . What is the acceleration? What is the position x(t)? 0 The acceleration is easy: dv a(t) = = 0: (2.1) dt The position is somewhat harder. Begin with dx =v : (2.2) 0 dt Integrate each side with respect to time from some initial time t to some nal time t i f Z Z t t f f dx dt = v dt: (2.3) 0 dt t t i i The right hand side is easily integrated: Z Z t t f f t f v dt =v dt =v t =v t t : (2.4) 0 0 0 0 f i t i t t i i The left hand side is evaluated using the fundamental theorem of calculus, namely, that integration \undoes" di erentiation: Z t f dx t f dt = x(t) =x(t )x(t ): (2.5) f i t i dt t i Equating these two gives x(t )x(t ) =v t t (2.6) f i 0 f i or x(t ) =x(t ) +v t t : (2.7) f i 0 f i 1 This was pointed out to me by Elizabeth W. Garbee, class of 2014.16 CHAPTER 2. CLASSICAL MECHANICS It's conventional to set t = 0. . . we say \start the clock at the beginning of the time interval we're i interested in." And since the end of the time interval could be any time at all, it's conventional to call t f simply \t". Finally, it's conventional to call x(0) by the name x . This gives 0 x(t) =x +v t for motion with constant velocity: (2.8) 0 0 Motion with constant acceleration Suppose the acceleration is a constant: a(t) = a . Now we know that the velocity is not a constant it 0 will grow larger (when a 0) or smaller (when a 0) as time goes on. But how exactly will the velocity 0 0 change? And what is the position x(t)? Begin with the velocity. We know dv a(t) = =a : (2.9) 0 dt Integrate each side with respect to time from some initial time t to some nal time t i f Z Z t t f f dv dt = a dt: (2.10) 0 dt t t i i The right hand side is easily integrated: Z Z t t f f t f a dt =a dt =a t =a t t : (2.11) 0 0 0 0 f i t i t t i i The left hand side is again evaluated using the fundamental theorem of calculus: Z t f dv t f dt = v(t) =v(t )v(t ): (2.12) f i t i dt t i Equating these two gives v(t ) =v(t ) +a t t : (2.13) f i 0 f i Using the conventions described just above equation (2.8), plus the name v(0) =v , we have 0 v(t) =v +a t for motion at constant acceleration: (2.14) 0 0 Okay, that's the velocity. Now what about the position? Write the above equation as dx =v +a t: (2.15) 0 0 dt Integrate each side with respect to time from some initial time t to some nal time t i f Z Z t t f f dx dt = (v +a t)dt: (2.16) 0 0 dt t t i i2.4. SPECIAL KINDS OF MOTION 17 The right hand side is integrated as follows: Z Z Z t t t f f f (v +a t)dt = v dt +a tdt (2.17) 0 0 0 0 t t t i i i t t f 1 2 f = v t +a t (2.18) 0 0 t t i 2 i 1 2 2 = v t t + a t t : (2.19) 0 f i 0 f i 2 Once again, the left hand side falls to the fundamental theorem of calculus: Z t f dx t f dt = x(t) =x(t )x(t ): (2.20) f i t i dt t i Equating these two gives 1 2 2 x(t ) =x(t ) +v t t + a t t : (2.21) f i 0 f i 0 f i 2 Using yet again the conventions described above equation (2.8), plus the name x(0) =x , we have 0 2 1 x(t) =x +v t + a t for motion at constant acceleration: (2.22) 0 0 0 2 There's one more thing we need to know: We have expressions for v(t) and x(t), but what about v(x)? We could just eliminate the variable t between the two equations (2.14) and (2.22), but there's a beautiful slick trick that is even easier. (I didn't think up this beauty: my teacher taught me. And his teacher taught him. I don't know who thought it up in the rst place, but whoever it was deserves a metal.) Let's look at acceleration not just when the acceleration is constant, but in all cases: dv a = . . . use the chain rule to nd. . . (2.23) dt dvdx = . . . then use dx=dt =v to nd. . . (2.24) dx dt dv = v (2.25) dx Now we have an expression for acceleration involving v(x). . . time doesn't enter the picture at all This is called the v(dv=dx) trick (pronounced \vee dee vee dee ex"). Applying this trick to the case of constant acceleration gives dv v =a : (2.26) 0 dx Or (being a bit loose with the notation) vdv =a dx: (2.27) 0 Integrating each side (from v , x to v , x ) gives i i f f Z Z v x f f vdv = a dx (2.28) 0 v x i i v x f f 1 2 v = a x (2.29) v 0 x i i 2 2 2 v v = 2a x x : (2.30) 0 f i f i Using our by-now-familiar conventions of v =v , x =x , v =v, and x =x, we have i 0 i 0 f f 2 2 v =v + 2a (xx ) for motion at constant acceleration: (2.31) 0 0 018 CHAPTER 2. CLASSICAL MECHANICS 2.5 Galileo on Falling Bodies Galileo Galilei's Discourses and Mathematical Demonstrations Concerning Two New Sciences is perhaps the rst book of modern science. It takes the form of a conversation between three gentlemen: Salviati, a supporter of Galileo; Simplicio, a defender of Aristotelian views; and Sagredo, a skeptical but open-minded layman with a deep interest in the world around him. The rst new science concerns the strength of materials what we would today call \materials science." The second new science is mechanics. But the conversation ranges widely over a 2 host of topics: falling bodies, musical instruments, the speed of light, the nature of in nity. It's important to realize that, while the thrust of the book is correct, Galileo also makes several errors. Two New Sciences was written or dictated by Galileo in the nal years of his life, while he was growing blind and was under house arrest by the Holy Oce of the Inquisition for advocating the Copernican view of the solar system. It was published in Leyden in 1638. The following excerpt, from Stillman Drake's 1974 translation (pages 6567), concerns falling bodies. It gives a good picture of Galileo's style of writing and of argumentation. Simplicio. Aristotle. . . assumes that bodies di ering in heaviness are moved in the same medium with unequal speeds, which maintain to one another the same ratio as their weights. Thus, for example, a body ten times as heavy as another, is moved ten times as fast. . . Salviati. I seriously doubt that Aristotle ever tested whether it is true that two stones, one ten times as heavy as the other, both released at the same instant to fall from a height, say, of two hundred feet, di ered so much in their speeds that upon the arrival of the larger stone upon the ground, the other would be found to have descended no more than twenty feet. Simplicio. But it is seen from his words that he appears to have tested this, for he says \We see the heavier. . . " Now this \We see" suggests that he had made the experiment. Sagredo. But I, Simplicio, who have made the test, assure you that a cannonball that weights one hundred pounds (or two hundred, or even more) does not anticipate by even one span the arrival on the ground of a musket ball of no more than half an ounce, both coming from a height of four hundred feet. Salviati. But even without other experiences, by a short and conclusive demonstration, we can prove clearly that it is not true that a heavier body is moved more swiftly than another, less heavy, these being of the same material, and in a word, those of which Aristotle speaks. Tell me, Simplicio, whether you assume that for every heavy falling body there is a speed determined by nature such that this cannot be increased or diminished except by using force or opposing some impediment to it. Simplicio. There can be no doubt that a given body in a given medium has an established speed determined by nature, which cannot be increased except by conferring on it some new impetus, nor diminished save by some impediment that retards it. Salviati. Then if we had two bodies whose natural speeds were unequal, it is evident that were we to connect the slower to the faster, the latter would be partly retarded by the slower, and this would be partly speeded up by the faster. Do you not agree with me in this opinion? Simplicio. It seems to me that this would undoubtedly follow. 2 In this connection it is worth mentioning that Galileo's father, the composer and music theorist Vincenzo Galilei (c. 1525 1591), was one of the inventors of the art form known today as opera.2.6. GALILEO ON REFERENCE FRAMES 19 Salviati. But if this is so, and if it is also true that a large stone is moved with eight degrees of speed, for example, and a smaller one with four degrees, then joining both together, their composite will be moved with a speed less than eight degrees. But the two stones joined together make a larger stone than that rst one which was moved with eight degrees of speed; therefore this greater stone is moved less swiftly than the lesser one. But this is contrary to your assumption. So you see how, from the supposition that the heavier body is moved more swiftly than the less heavy, I conclude that the heavier moves less swiftly. Simplicio. I nd myself in a tangle. . . 2.6 Galileo on Reference Frames Galileo Galilei's Dialogue Concerning the Two Chief World Systems Ptolemaic and Copernican is a bit earlier than Two New Sciences, and a bit funnier as well. As with Two New Sciences the dialog involves Salviati, Simplicio, and Sagredo. The Dialogue was published in Florence in 1632. The following passage, from Stillman Drake's 1953 translation (pages 125, 186188), treats the question: If the earth is in motion, then why don't we get left behind when we jump up? Simplicio. Aristotle. . . strengthens his arguments that the earth is stationary with a fourth argument taken from experiments with heavy bodies which, falling from a height, go perpendicularly to the surface of the earth. Similarly, projectiles thrown vertically upward come down again perpendicularly by the same line, even though they have been thrown to immense height. These arguments are necessary proofs that their motion is toward the center of the earth, which, without moving in the least, awaits and receives them.. . . Salviati. For a nal indication of the nullity of the experiments brought forth, this seems to me the place to show you a way to test them all very easily. Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some ies, butter ies, and other small ying animals. Have a large bowl of water with some sh in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals y with equal speed to all sides of the cabin. The sh swim indi erently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though there is no doubt that when the ship is standing still everything must happen this way), have the ship proceed with any speed you like, so long as that motion is uniform and not uctuating this way and that. You will discover not the least change in all the e ects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the oor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the oor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping towards the stern, although while the drops are in the air the ship runs many spans. The sh in

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