Instructor's Manual to Accompany Physics for Poets Second Edition

INSTRUCTOR'S MANUAL TO ACCOMPANY

SECOND EDITION

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INSTRUCTOR'S MANUAL TO ACCOMPANY

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SECOND ED ITION

'B/;PAr IJ. A1a!1.Ck Univers i ty of Wi sconsin,

Madison, Wisconsin

McGraw-Hili Book Company New York St. Louis San Francisco Auckland Bogota Dusseldorf

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TABLE OF CONTENTS

INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

PACING THE COURSE AND TRUNCATED VERSIONS..............

3

CHAPTER-BY-CHAPTER CLASSROOM SUGGESTIONS..............

4

EXAMINATIONS AND TERM PAPERS . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Sample Examination Questions ....................... 17

Semi-Take-Home Exam................................ 26

Abstracts of Term Papers ........................... ' 30

Instructor's Manual to accompany

PHYSICS FOR POETS

Second Edition

Copyright@ 1978 by McGraw-Hili, Inc, All rights reserved, Printed in the United States of America, The contents, or parts thereof, may be reproduced for use with PHYSICS FOR POETS Second Edition by Robert H, March provided such reproductions bear copyright notice, but may not be reproduced in any form for any other purpose without permisSion of the publisher, 0-07-040244-2 1234567890 "kh'£4;-~'·z i""~' ~"'._

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HOMEWORK ASSIGNMENTS.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

ANSWERS TO EXERCISES IN THE TEXT . . . . . . . . . . . . . . . . . . . . . .

34

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I N T ROD U C T ION

This manual gives some suggestions for the design of a course using the text Physics for Poets, based on experi­ ence with the University of Wisconsin course on which the text is based. The text is explicitly aimed at students majoring in the humanities and social sciences, that is to say, stu­ dents who have no professional need to understand physics, who will probably never take a physics course again, and who will probably never make any practical use of what they learn in this course. Keeping this in mind, remember that the students in this course have no real need to know any particular part of the subject matter of the course. What would be far more significant is for them to come out of the course with a more accurate idea of the nature of science, with a less fearful attitude toward physics, and with a better idea of what a scientist does for a living. If, even for a portion of the course, a student gets really "turned on" to the subject, gets deeply enough involved to start thinking in unfamiliar ways, it is possible the course will achieve its goal. Just exactly what is likely to turn students on varies widely with both the student's own interest and the skill and enthusiasm of the professor. But, in general, it is appropriate in this course to judge the student on the basis of his best rather than his average performance. The section of the course on relativity requires a considerable dose of hard and deep thinking. Unless you have had a great deal of experience in teaching relativ­ ity, you may find that you yourself have difficulty un­ derstanding relativity on the "gut" level required to teach it in this course, when you first start teaching from this book. The chapters on quantum theory demand much less depth but require a mind capable of keeping track of more loose ends. Instructors who are not familiar with the basic elements of quantum field theory, or who have not kept up with recent developments in high energy physics, will I

PACING THE COURSE AND TRUNCATED VERSIONS ha.ve some difficulty teaching chapter 19, and might be well advised to omit it. Very few students seem to be "turned on" by classical mechanics. But we have made attempts at Wisconsin to move directly into relativity without systematic devel­ opment of the classical background, filling it in as needed. This experiment was not too successful, and is not recommended by the author. The most important thing to remember in this course is not to give the student an excuse to "cop out," to decide early in the course that physics is beyond him and stop trying. To avoid this it helps to start slowly, to give assignments that are so easy the student can hardly miss, to build up his confidence. Before long even the most timid student will find himself handling topics he would have been afraid to think about a few months before. The sections that follow give suggestions on the use of this text based on my experience at Wisconsin, where the course has been taught since 1963. In this time it has grown from a cozy gathering of 15 students to a full-dress lecture of 380. Throughout this period the students have remained the same -- an above-average but not exceptional group of humanities and social science majors from a first-class but not ~litist state univer­ sity. Other schools using the text may have better or weaker students or a different classroom situation, and the suggestions offered in this manual may be of limited utility or validity for many schools.

2

The full content of this book represents a relatively challenging one-semester course for the format used at Wisconsin (3 lectures, one discussion per week). The number of lectures devoted to each topic in this format are indicated in the section that follows. Instructors operating in a shorter format, or in schools on the quarter system, may wish to consider several possible stratagems for truncating the course: 1. Eliminate Chapter 19.

The book reaches a reasonably satisfying conclusion with non-relativistic quantum

theory at Chapter 18. This is especially recom­

mended if the instructor is not familiar with Feynman diagrams and the quark model.

2.

Assign one or more chapters for independent reading. Chapters 13 and 18 are quite suitable for this pur­ pose. It is also possible to use Chapter 5 in this fashion.

3. Skip either quantum mechanics or relativity.

The former route terminates with Chapter 12. The

latter uses chapters 1 through 7 and 13 through 19. In this case, it will be necessary to include a lec­ ture on the meaning of E = mc 2 , which is needed in order to understand Chapter 19.

3

C HAP T E R - B Y - C HAP T E R

C LAS S ROO M S U G G EST ION S

involved in falling-body motion. The "punch line" is that what Galileo is saying is that falling-body motion would depend on none of these variables, were it not for the effects of the air.

It is also interesting to repeat Galileo's inclined CHAPTER 1 (3 lectures)

plane experiment. You need a very flat, rigid, grooved Topics: Introduction to the concepts of velocity and ac­

board or metal beam at least 10 feet long, set at such an celeration; Galileo's description of falling body angle that a ball takes about 10 seconds to roll the full motion as an example of the scientific method. length. A large coffee pot or picnic jug with a spigot, and a graduated cylinder, make a reasonable water clock. This is a very difficult chapter. If it is not treated with a little practice you can achieve an accuracy level with great care and gone through slowly, you may lose some of about a quarter of a second this way. of the students from the outset. What makes it difficult is the concept of acceleration, which may be the most confusing concept in the course for students with a weak mathematical background. It is best to put it across with a lot of examples, emphasizing the sign rather than the magnitude of velocity and accelera­ tion. For example, when a car is braking, acceleration is negative, velocity is positive, and so on through many such cases. It also sometimes helps to give examples of second de­ rivatives from areas outside physics. For example, the stock market rose 5 points today and 15 the day before; thus, the market is rising (positive velocity) but the "boom" is tapering off (negative acceleration). It may also help to cover this topic through interpre­ tation of graphs, pointing out the relation between slope and velocity, curvature and acceleration. But at least one class period will have to be devoted to this topic alone. It is much easier to drive home the scientific point raised in the chapter -- that while Galileo was only try­ ing to describe the motion of a falling body, even that simple process is a pretty abstract business. Here a few extremely simple demonstrations can be very helpful. For example, demonstrate the fall of various objects, such as coins, crumpled-up paper, etc. The difference between a balloon inflated and the same balloon deflated, a paper crumpled and the same paper flat, etc., can persuade the student that a lot more variables than mere weight are 4

CHAPTER 2 (2 lectures) Topics: Projectile motion; momentum conservation (two­ bodies, one-dimensional motion)

projectile motion is analyzed using three concepts: the principle of inertia and the mechanical principle of superposition, introduced in this chapter, and the de­ scription of falling body motion from the preceding chap­ ter. Be sure those principles get across and emphasize that while it is possible to get a complete descripti'on of the combined motion as a parabola, in practical terms one need not do this -- it is sufficient to treat the horizontal and vertical motions separately. If an appa­ ratus that produces a collision between a projectile and a freely falling ball released at the same instant is available, this makes a very convincing demonstration. Demonstrations of momentum conservation with an air

table are also useful in this portion of the course.

CHAPTER 3 (2 lectures) Topic: Newton's laws

No new mathematical concepts are introduced in this section. Emphasize that Newton's crucial contribution was the realization that change in motion results only from an interaction of two objects, and that the differ­ ence in how each of them is affected results from a dif­ ference in mass rather than any asymmetry in the inter­ action. 5

Another point that deserves emphasis is that Newton's Laws are not so much generalizations from experience as the adoption of a scheme in which to interpret that ex­ perience. Apart from the assumption that mass is an in­ trinsic quality of a body which does not change with the situation, they have no empirically falsifiable content. It is the possibility of discovering laws of force that is the true test of the Newtonian scheme. If you are intending to cover chapter 19, it is a good idea to stress at this point that Newton's laws can be regarded as a recipe for calculating momentum transfer, which permits a neat comparison with Feynman diagrams. In circular motion, the crucial task is to convince the student intuitively that a change of direction is an acceleration. One example that helps some students see this is to describe a right-angle turn first as bringing an object to a stop, then bringing it back up to speed at right angles. Curved motion is then doing the two simultaneously rather than sequentially. Qualitative arguments of this sort are more important than emphasizing the formula a = v 2 /R. However, students tend to have strong intuitions about this formula based on their experience as drivers, and it sometimes pays to draw on this experience.

CHAPTER 4 (2 lectures) Topics: Gravitation; Kepler's laws; the acceleration of the moon This chapter builds toward the climax of the computa­ tion of the moon's acceleration. The section on Kepler's laws shows that the motion of the planets can be accounted for by an inverse-square force directed toward the sun. That this has anything at all to do with gravitation on the surface of the earth is based solely on the fact that the moon's acceleration turns out right.

masses of the bodies involved had little empirical sup­ port until the Cavendish experiment.

CHAPTER 5 (3 lectures) Topic: Energy conservation in mechanics and in physics as a whole Energy, more than any other concept, is the unifying feature of physical science, and thus this chapter is rather full. Fortunately, it is one in which the stu­ dent's intuition is often well-developed. Some instructors may be disturbed by my contention that energy is not merely an extension of Newtonian mech­ anics, but a conceptually new approach to motion that can be connected to the Newtonian scheme through the concept of work as a measure of energy transfer. I have tried not to lean too heavily on this point, and thus you may feel free to adopt a different point of view. Students in the humanities, especially literature, tend to get quite interested in this chapter, because of the connection with romanticism. Most of the difficulties with this chapter lie in the concept of potential energy. Some students find it too abstract; it may help to tell them that the concept of field, introduced in the next chapter, will make it seem a bit more real. Others are disturbed by the arbitrari­ ness of the reference point for zero potential energy. For these students, emphasize that potential energy is never directly observed, but only inferred from changes in kinetic energy. Finally, the reference point at in­ finity for action-at-a-distance forces disturbs many students, as does the accompanying result that potential energy for an attractive force is always negative. Here one can cite the argument that no force and no potential energy should come at the same place in a "natural" scheme of things.

It is important to emphasize that the success of the moon calculation, plus the neat way the theory fits into Newtonian mechanics, made believers out of the whole sci­ enti~ic community, even though the claim that gravity was a universal force proportional to the product of the 6

7

CHAPTER 6 (2 lectures)

Topics: Electricity and magnetism; the concept of fields;

philosophical consequences of deterministic laws in physics There is nothing terribly difficult in this chapter. By now most students should be sufficiently conditioned to the physicist's point of view to at least be tolerant of the argument that if the field has to take up momentum and energy to save the conservation laws for these quan­ tities and Newton's laws, then the field must in some sense be "real." Some of the better students may find the rather sketchy introduction to electricity a bit too open-ended to be satisfying. It might not hurt to give such students sup­ plementary reading in a conventional physics text. The usual amber rod, cat's fur, and pith-ball demon­ strations of electrostatics can be effective here.

CHAPTER 7 (1 lecture) Topics: Wave pulses; wave superposition; periodic waves; standing waves; two-slit interference This chapter stands alone as an introduction to waves. It makes no reference to periodic motion, nor does it mention the trigonometric functions, an omission which a few superior students may find dissatisfying. The emphasis of the chapter is on wave laws themselves, independent of the underlying dynamics of the wave-propa­ gating medium. Thus, such traditional topics as the distinction between longitudinal and transverse waves are also omitted. The major goal of the chapter is for the student to understand standing waves and two-slit interference. At Wisconsin we have found the text material is rea­ sonably self-explanatory. The best use of lecture time is for demonstrations. For two-slit interference, a pair of loudspeakers about six feet apart driven by the same monotone audio

source gives a striking effect. A low-wattage laser pro­ duces spectacular two-slit fringe patterns. A ripple tank can also be effective, but to make the effect really convincing takes a good ripple tank and a reasonable amount of practice. A "slinky" spring toy mounted between two fixed posts is a good way to demonstrate standing waves. But far and away the most popular and effective demonstration we have used at Wisconsin is the observation of a mechanically driven vibrating rope in various modes with a strobe light. This both enables the students to really see standing wave patterns and convinces them that standing waves are beautiful, which is a great motivational aid.

CHAPTER 8 (2 lectures)

Topic: The Michelson-Morley experiment

This is the first of four chapters on relativity and, as was the case with the first of the chapters on clas­ sical mechanics, contains most of the math needed to un­ derstand the subject. Continually emphasize to the stu­ dent that if he has a feeling for how y varies with vic he will need no further mathematical skills to follow the remaining three chapters on relativity. At the risk of boring the better students, devoting one class session to a slow, careful review of the derivation, with careful and repeated explanations of the motivation for every step, seems to help reassure the students that relativity will not prove mathematically beyond them.

CHAPTER 9 (3 lectures)

Topics: Non-quantitative arguments for time dilation;

the FitzGerald contraction; relativity of simultaneity; and uniqueness of the speed of light It is in this chapter that the battle to teach relativ­ ity is won or lost, and it calls for all of the teacher's skill. The method used in the text is the analysis of gedanken experiments, and the best use of classroom time is to repeat these examples, answer questions about them, and add further examples. I have found that every stu­ dent seems to have a different point at which it suddenly hits him what relativity is all about; each example makes a few new "converts."

8

9

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The first step is to convince the student that if two observers relatively in motion are to agree on the speed of one and the same light signal, they must obviously disagree about some of the things that go into measuring that speed. For the time being, one must "suspend dis­ belief," as in the theatre; one must not inquire how it is possible for two observers to disagree on such elemen­ tary matters, but merely whether it is possible to live with these disagreements. To provide reassurance to the students, continually remind them that relativistic disagreements apply only to remote events, events displaced from one another along the line of relative motion. Furthermore, two observers at the same point will always agree on what they are see­ ing at that instant; it is when they try to interpret the past phenomena responsible for what they now observe that disagreement arises. Finally, each is perfectly capable of reconstructing the other's point of view, so there is no "communication gap." Lest this make it seem as if relativistic effects are merely illusory, the last ex­ ample offered in the chapter is that of the "garage para­ dox." The most useful gedanken experiment to add to those in the text is Einstein's own original one, that of a train that is struck at both ends by lightning flashes, simul­ taneously in the reference frame of the train. A moving and a stationary observer, both of whom are at the center of the train when the flashes arrive, agree that the flashes appear to be simultaneous. The observer on the ground concludes that since the lightning bolt at the front of the train was closer, it must have come later. The analysis can be extended. Suppose that clocks synchronized in the train's rest frame are placed at either end of the train and are stopped by the lightning bolts. Again, the observers agree that the two clocks stopped at the same setting. After all, they are no longer running and can be brought to the same point and compared directly. The observer on the train feels that this is because well-synchronized clocks were hit by truly simultaneous lightning bolts. The observer on the ground feels that unsynchronized clocks were stopped by non-simultaneous lightning bolts. 10 ~\ol;;U

If the lightning bolts leave marks on the railroad ties, the stationary observer feels they are farther apart than the length of the train. Again, the observer on the train agrees, but he sees it as a consequence of the fact that the rails had shrunk, whereas the observer on the ground sees it as a consequence of the fact that the lightning bolts were non-simultaneous, which more than compensates for the shrinkage of the train. To emphasize the contrast with the expected non­ relativistic behavior, point out that the thunder claps produced by these flashes do not arrive simultaneously at the center of the train, and there is no disagreement be­ tween the moving and the stationary observer on this point. CHAPTER 10 (3 lectures) Topics: Quantitative basis for relativistic effects; space-time; the twin paradox

The first part of this chapter simply demonstrates that the factor Y derived in Chapter 8 gives the correct quantitative result for the time dilation and FitzGerald contraction. Thus, if the proper groundwork has been laid, the mathematics in this chapter will present no difficulties. A more rigorous derivation of the clock-setting prob­ lem can be done as follows: consider the measurement of the speed of a light signal moving the length of a train, using clocks at both ends of the train. After taking into account the shrinkage of the train and the clock slowdown, a stationary observer still finds a discrepancy, as a result of the ~t. Be prepared to go over the twin paradox carefully; most students are particularly intrigued by this example. CHAPTER 11 (2 lectures)

Topics: Relativistic mass increase and the survival of

Newton's laws; mass-energy equivalence; experimental confirmation of special relativity

This chapter is not particularly difficult. Its main objective is to take some of the mystery away from E = mc 2 11

by showing how universally the formula applies. Be sure to emphasize that making mass a function of velocity is the only conceptual modification in Newton's mechanics required by relativity, aside from, of course, using the appropriate space-time coordinates and trans­ formations. The only puzzling point for average students is why rest mass should exist at all -- why should an object have energy simply by virtue of its existence? Examples useful here are those where what appears as rest mass when a system is viewed as a whole from outside becomes partly dynamical when the system is analyzed into its component parts. The best example is binding energy of a nucleus. CHAPTER 12 (4 lectures) Topics: More detail on twin paradox; genera1 relativity; black holes; cosmology

This chapter makes severe demands on the students' power of abstract reasoning. It helps to reassure them that not even experts in the field can truly "visualize" curved space-time. It is also important to stress that this is an alternative to the newtonian approach to motion; where newtonianism calls for force laws, general relativity calls for theories in which fields produce curvature of space-time. Fortunately, you can exploit the students' inherent curiosity about black holes and the big bang. CHAPTER 13 (1 lecture) Topics: Ancient atomic theories and the phases of matter; chemical evidence for atomism; kinetic theory of gases; atomic size; electrochemistry; the discovery of the atom

This chapter provides a very sketchy introduction to the emergence of the atomic theory in classical physics and chemistry. As such, it presents no difficulties to the average student, nor is it important that the material in it be well mastered. Its major purpose is to provide a proper historical starting point for the quantum theory.

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12 _

In a longer course, you may use this chapter as a peg on which to hang a more thorough and general survey of physical science from supplementary materials. CHAPTER 14 (2 lectures)

Topics: Plum-pudding and planetary atomic models; spectra

and spectral laws; the Rutherford-Geiger-Marsden experiments

Here the student is introduced to what modern experi­ mental physics is all about, as the experimental technique and interpretation are fully modern. Some students are puzzled by the l/(sin ~ )4 law, which seems unnecessarily complicated for such a simple situa­ tion to someone without much mathematical experience. There is little value in deriving it, and that is why the derivation is omitted here. But a la-minute rundown of the factors that go into the derivation might remove the mystery, while persuading the student that quite simple situations can quickly get mathematically messy, a valu­ able lesson to learn. Keep in mind that to many students, the process of plotting measurements on a graph and seeing which of two curves fits best is a new experience that may require some explanation.

CHAPTER 15 (3 lectures) Topics: Planck's theory; Einstein's ,theory of the photo­ electric effect; the Bohr model of hydrogen

This chapter depicts the quantum theory in its early years, when it was based on empirically successful but arbitrary assumptions. If a student complains that he doesn't get the connections between all these ideas, point out that this is an accurate reflection of the at­ titude of the physicists he's reading about. -The Planck theory is best sloughed off as quickly as possible. If you wish to go into it at somewhat greater depth, the treatment in Gamow and Cleveland, Physics: Foundations and Frontiers (Prentice-Hall, 1960), p. 378££., is suitable for students on this level. The photoelectric effect is pretty straightforward. The Bohr theory, 13

however, is more difficult and must be gone over slowly. Keep the diagram at the top of page 190 in mind as you plan your lectures, as it is easy for the students to lose the thread of the rather complex paths of reasoning leading to the Bohr theory. It is also wise, in terms of the future development of the theory, to emphasize the difference between the idea of stationary states, which survives in the later versions of the theory, and Bohr's circular orbits, which do not. This is also a good point at which to begin working numerical examples in class to give the students a feel­ ing for the magnitudes of the quantities involved. CHAPTER 16 (3 lectures) Topics: The DeBroglie hypothesis; Shroainger's equation; wave equivalent of Bohr orbits; expansion of the wave packet In this chapter the quantum theory advances one level deeper; the wave theory appears, removing the arbitrary character of the earlier theories, but it still remains to be interpreted. The only difficulty students tend to have with this part of the story of quantum mechanics comes from the fact that it is hard to visualize three-dimensional standing wave patterns such as are obtained in the hydro­ gen atom. We have found the following derr,onstration helps a great deal. Mount on a drum a loose rubber drum­ head. Drive it with a speaker inside the drum and ob­ serve it with a strobe light. The resulting undulations are very striking.

understanding what they are all about. It may be neces­ sary to carefully explain what you mean by "error" and "deviation." The more examples you can give in class, the better. If you have an unusually bright group of students, the wave interpretation of the uncertainty principle can be explored. Show how a finite wave packet can be construc­ ted from a spread of close wavelengths. The easiest way to do this is to start with two close wavelengths. The resulting beat pattern is a "string" of wave packets. The wave halfway between them in wavelength suppresses the odd wave packets, and further "in-between" wave­ lengths suppress others, until but one is left. Then you can relate the spread in wavelength (momentum) to the size of the packet (position). CHAPTER 18 (lor more lectures)

Topics: Quantum-mechanical interpretation of a two-slit

interference experiment with electrons; the Copenhagen interpretation; disagreements with this interpretation The two-slit interference experiment is used as a gedanken experiment to show the distinction made in quan­ tum mechanics between what is knowable in principle and what is actually known from measurement. If you have some philosophically sophisticated students in your class, this can lead to some lively discussions. But the weaker students will simply learn nothing from this chapter. CHAPTER 19 (4 lectures) Topics: Quantum field theory; accelerators; the quark model; cosmological implications

Instructors who are not themselves particle or nuclear CHAPTER 17 (3 lectures) physicists, or who do not at least follow particle Topics: The probability interpretation of the wave physics on a Scientific American level, may have some function; the uncertainty relations; consequences of trouble teaching this chapter and may be well-advised to the uncertainty relation for behavior of free parti­ omit it. The exciting part is the possibility of ex­ cles and electrons in Bohr orbits plaining subsequent developments in the field, giving students a sense of the swiftness of scientific progress This chapter is the real heart of the section on the once a breakthrough is made. The hardest part of the quantum theory. The probability interpretation should chapter is the quantum field theory; the fact that the pose no difficulties, but the uncertainty relations are Heisenberg relations allow the field to "borrow" energy more of a problem, not because they are mathematically to create field quanta. Once over this hump, the rest of difficult, but because students may have a hard time the material is straightforward. 15 14 ~

E X A MIN A T ION S

about some topic in the course.

AND

The technique used was to leave the topic open but in­ form the student that the paper was supposed to show that he could incorporate something from the course into his own personal frame of reference. This is too vague a charge for most students, so it was illustrated by giving out abstracts of some of the more successful term papers from previous years as examples. A group of such ab­ stracts appears after the sample exam questions.

T E R M PAP E R S

For the type of student who takes this course, the traditional physics examination consisting exclusively of mathematical problems is simply not suitable. While such students can often work quite challenging problems, they can rarely do so within the time limits imposed by an exam. Most problems sufficiently simple to put on an exam for this course test very little of significance. When the course at Wisconsin is not too large, we use exams consisting of a mixture of three kinds of questions. First, there are problems, usually closely related to ones given as homework and broken down into steps to help lead the student to the correct route to solution. Then there are short-answer, multiple-choice, etc., questions designed to test qualitative understanding of the predic­ tions of physical laws or the logical interrelationships of those laws. Finally, there are short essay questions, similar to those given in the Appendix of the text. Of course, the grading of such questions tends to be rather subjective. In a school with an honor system, take-home exams are a useful device. At Wisconsin we have evolved a "semi­ take-home" exam. In this type of exam, the student is given a set of problems to solve or questions to answer. He then is given a multiple-choice or short-answer exam in class to test his knowledge of the areas he has studied. Regardless of the type of exam used, it is the author's personal inclination to make all examinations open-book, if only to persuade the student that learning physics is not just a matter of memory work. When the course is not too large at Wisconsin, we assign term papers. At our institution, most humanities and social science majors do a great deal of writing for courses in their own discipline and feel confident of their abiLity to tackle such projects. The term paper also serves to stimulate the student to think more deeply 16

SAMPLE EXAM QUESTIONS

The questions below have all been used with success at Wisconsin. Five to seven such questions, suitably bal­ anced for difficulty, constitute an hour-long exam. We generally strive for an exam which, given generous partial credit, gives average scores of 70 to 75 -- high enough to avoid discouraging most students, while low enough to make the better students stand out. Some of the questions and exercises from the text have been used as examination questions also. Classical Mechanics

1) Ga1i1eo found a ball rolled down an inclined plane a distance proportional to __~___________________ whereas had Aristotle been correct, the distance would have been proportional ~o (2) Newton's law of gravitation can be reduced to the

following four statements:

(a) A falling body experiences a force proportional to its mass. (b) And also proportional to the mass of the body with which it is interacting. (c) The force acts along the line joining the cen­ ters of the objects. (d) And is inversely proportional to the square of the distance. Ga1i1eo's law of falling body motion supports statement 17

Kepler's laws of planetary motion support state­ ments and The agreement of the moon's observed acceleration with that predicted from the acceleration of fall­ ing bodies supports I and

(a) The velocity is positive between and , and also between and (b) The acceleration is positive at and (c) The acceleration is negative at and (d) The motion (instantaneously) comes to a halt at and (e) The highest velocity is found at

--~--~----

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(3) Consider the formula

1 2

mv

2

+ mgh

(6) The Fairmont Hotel in San Francisco has an outside elevator with one transparent wall. Suppose that while the elevator is rising at constant speed, a passenger drops a cigarette lighter, which drops straight to the floor of the elevator. Describe the motion of the lighter:

E

This formula represents: (a) The (b) The

law of momentum conservation. law of energy conservation for an object moving subject to gravity. (c) The law of energy conservation for an object moving subject to any form of potential energy. (d) A combined statement of Newton's first and second laws. The term 1 2 is called , and mgh is 2' mv called --~----:--The formula can be used to calculate the maximum height to which an object can rise by setting the variable equal to ( 4)

(a) As seen by the passengers on the elevator. (b) As seen by a nosy resident of an adjoining building. ( 7)

(a) Galileo's assertion that all bodies fall at the same rate in a vacuum. (b) Newton's assertion that the force of gravity on a falling rock is proportional to the mass of the earth.

A falling object has reached terminal velocity when two forces are equal. What are these forces?

(5) Fill in the blanks in the statements below with the letters corresponding to the appropriate points on Graph 1.

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c

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The following two statements could not be directly tested at the time they were originally made. Cite indirect evidence or plausible arguments for their validity:

(8) A ball dropped from a height qf 9 meters rebounds to a height of 4 meters. (a) What fraction of its energy is lost in the re­ bound? (b) Its speed immediately after leaving the floor is what fraction of its speed just before striking the floor?

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(9) A pendulum consisting of a string and a sticky clay ball is raised to a height 4 meters above its normal horizontal position and allowed to swing. At the bottom of its swing it strikes and sticks to an identical clay ball, carrying it up on the other side. (a) Which conservation laws apply during (i) the downswing, (ii) the collision, and (iii) the 19

upswing? (b) How far does the pendulum rise on the upswing?

(3)

.1.'•"., '.. . . .

..

(4)

,

(10) How long does it take a freely falling body to fall 125 meters? How fast is it then going? (Use g = 10 m/sec 2 .) (11) A car drives off a vertical cliff. Two seconds later it hits the ground, 30 meters from the base of the cliff. (Use g ~ 10 m/sec 2 .) (a) How high was the cliff? (b) What speed was the car going? (c) Draw a vector diagram to find the velocity vector for the car at the instant it hit the ground.

The speed of sound is about 330 m/sec. The A below middle C has a frequency of 220 Hz. What is the wavelength of A below middle C?

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Two hi-fi speakers sounding the same sustained note are 3 m apart. An observer walking along a line 4 m from the speakers hears a maximum when he is halfway between the two speakers, but directly in front of either he hears a minimum. What is the wavelength of the note?

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Relativity

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(See also the "semi-take-home" exam in the next section.)

~

(12) A body of mass 5 kg, speed 6 m/sec strikes a sta­ tionary body of mass 1 kg. This collision slows it down to 4 m/sec. (a) How fast is the 1 kg body moving after colli­ sion? (b) Is the collision elastic? Explain why or why not. (c) Suppose the bodies had instead stuck together. What speed would the combined mass be moving?

( 1)

@

Cross out from the list below those wavelengths that cannot exist as standing waves on a string 1 meter long: 1/4 m 1/3 m 1/2 m

(2)

2/3 m 3/4 m 1 m

11.:! m 2 m 3 m

In the two-slit interference experiment, the point directly opposite the speakers and halfway between them: (a) is always a maximum. (b) is always a minimum. (c) can be either a maximum or minimum, depending on wavelength. (d) cannot be either a maximum or minimum. 20

/'

®

Waves (1)

~~

@ A spaceship passes an ob­ server S at a speed of \ 6/10 the velocity of light. There are three clocks on board the spaceship -- A, B, and C, as shown. They have been synchronized by the crew of the spaceship. As clock C passes the observer, he sets his clock by it. Answer the next three questions from the point of view of observer S, at the instant depicted in the picture. (a) How do the readings of clocks A and B compare? A is faster ----- B is faster same (b) How do the readings of clocks A and C compare? A is faster

C is faster same (c) How do the readings of clock A and S's clock compare? faster --- AS is is faster ---- same

• t:,

~:;

" t,

,

.,.

(2)

On the list below, check those quantities on which S and the spaceship crew agree (there is more than one correct answer). 21

light absorptions, or a mixture?

On the basis of the Ritz principle, the follow­

(b) ing relation holds for the frequency of light in transitions A, B, and F:

_____ The length of the spaceship ---- The width of the spaceship ---- The time elapsed while the spaceship is passing S ____ T he relative speed (0.6 c) rate at which clock S is running --- The The rate at which clocks A, B, and Care --- running velocity of light --- The The mass of the spaceship

'VA

( 2)

One or more of the following statements is an in­ correct application of the mass-energy equivalence. Mark each one "T" or "F" and explain below the flaw in the one or ones marked "F". (a) The combination products of a fire weigh less than the fuel and oxygen that went into it. (b) If a fire takes place in a sealed insulated box so that neither heat nor material can escape, the weight will not change. (c) By virtue of its motion around the sun, the earth appears (to an observer not sharing this motion) heavier than it would if standing still. (d) All objects on the earth share in the mass in­ crease mentioned in (c) above, and a very sensitive scale on the earth could detect this. (e) Light can be used to transport mass from one place to another.

t:~

(5)

(i) Produce one quantum of light. (ii) Create an electron. (iii) Overcome the forces holding the electron. ( 3) J~~I

J

:"~ ',.:

"

:;

Quantum Theory 1) The figure at the right shows six transitions among the first four Bohr or­ bits in the hydrogen atom. Ca) Are the transitions shown light emissions, 22

·i ~

I,

Calculate Y for v = 0.6 c.

r-

- W.

(b) The symbol W stands for the work required to:

l

Explain how Newton's laws must be modified in the light of relativity.

= hv

(a) The symbol E stands for: (i) The average energy of electrons emitted. (ii) The maximum energy of electrons emitted. (iii) The energy of the light quantum.

~i

(

Einstein's formula for the photoelectric effect is E

'< '

(4)

'VB + 'VF

write down at least four more correct relation­ ships of this type.

---

(3)

=

'"

Rank the following developments in the history of the quantum theory in the order they happened (1 for the earliest, etc.). Bohr's theory of the hydrogen atom. ----- The Geiger-Marsden experiments. Planck's theory of incandescent light. DeBroglie's wave-particle hypothesis. Einstein's photoelectric theory.

The Schrodinger equation.

The uncertainty principle.

-----

( 4)

Below is a list of empirical results important in the history of the quantum theory: (a) (b) (c) (d) (e) (f)

The Geiger-Marsden alpha-scattering experiment Millikan's photoelectric experiment Balmer's formula for hydrogen spectrum lines Davisson's experiments with electron scattering The Franck-Hertz experiment The absence of certain emission lines in absorp­ tion line spectra (g) The Ritz principle 23

,

'I'

Mark which of the above results is best described by each statement below:

Quantum Field Theory and Quarks

The confirmation of DeBroglie's hypothesis.

----_____ The first proof that quantization applied to

(1)

a process not involving light. _____ Two minor facts explained naturally by Bohr's theory, but awkward in a plum-pudding atom.

How many baryons are possible?

(Show work or list)

(6) Which of the following features of the Bohr theory of the hydrogen atom was eliminated or signifi­ cantly modified in the Schrodinger picture? (There may be more than one correct answer.) (a) Circular orbits. (b) Use of E = hv to predict energy of the photon. (c) Quantum jump. (d) Energies of electron states.

(8)

Attack or defend the following interpretation of the two-slit interference experiment, taking the usual quantum mechanical point of view: "The electron actually passes through both slits simultaneously."

(9) Show how the "expanding wave packet" of a free particle is explained in terms of the uncertainty relations. (10) If E is the amount of energy required to raise an electron from the lowest Bohr orbit to the second 24

The original Gell-Mann-Zweig theory had only three quark flavors. How many mesons are possible in this theory? (Show work or list)

(5) In Bohr's theory of the hydrogen atom, what physical significance is attached to the integers nand m in the Balmer formula (p. 177 of the text)? (a) The number of photons emitted in the jump that produces the spectrum line. (b) The energies of the Bohr orbits involved in the jump that produces the line. (c) The angular momenta of the Bohr orbits in­ VOlved, in units of~. (d) None; they are pure numbers used to count orbits.

(7) Explain what is meant by the word "uncertainty" in the term "uncertainty principle."

orbit, how much energy is required to remove the electron from the atom altogether?

(2)

Give the electric charge and particle type (meson, baryon, antibaryon) of the following quark com­ binations. type

charge uc

\

I

sss usd ss Which of the above particles would you expect to last the shortest time before changing to a stable combination or annihilating? Explain why.

25

Which would last the longest time?

In particular, what would happen if the wavelength were 1 meter?

Explain why.

(3)

of electron by positrons between quarks and leptons the known mesons and baryons the IjJ particle

(4)

Now, back to relativity. Consider measuring the speed of light by timing a signal from the rear of the train, off a mirror at the front" and back to the rear, using a single clock at the rear of the train. How is this measurement interpreted by an observer on the ground?

(5)

In non-relativistic mechanics, the same change in momentum results when a body is speeded up from 0 to 0.2 c and from 0.6 to 0.8 c. What does rela­ tivity say? Calculate the ratio of the momentum change from 0.6 to 0.8 c to that from 0 to 0.2 c.

(6)

In the example of the twin paradox given on pages 146-148 of the text, suppose the astronaut is sent 10 radio messages from earth during his voyage, the first sent three years after he leaves, then one every five years thereafter. Divide the voyage up into three parts -- outward trip, turn-around, and return trip. The turn-around takes one day. When does the astronaut actually receive the messages? When does he believe they were sent? Does his brother on earth disagree on these two points?

The ~in clue that quarks are related to leptons is that both (a) have similar masses (b) have the same electric charges (c) are transformed by the weak interaction in similar fashion (d) are found in mesons and baryons

"SEMI-TAKE-HOME" EXAM ON WAVES AND RELATIVITY

The following six questions are handed out in class a week before the exam. (1)

Suppose relativity had been wrong and Maxwell right all along. Consider the example of setting clocks at the ends of a train to a light flash in the middle of the train. The train is moving "ab­ solutely" -- i.e., moving with respect to the aether -- and the ground is "absolutely" at rest. Do the observers disagree about the clock settings? Are the clocks synchronized?

The ~in obstacle to acceptance of the quark theory

has been the

(a) failure to observe free quarks (b) fractional charges of the quarks (c) unresolved relationship between quarks and leptons (d) inability to fit the electron into the theory

(6)

(3)

Which of the following features was not present in

this original version of the theory?--­ (a) Quarks are fundamental, point-like particles (b) Quarks have f~actional electric charges (c) Only (qqq, qqq, qq) combinations are allowed (d) Mesons and baryons have "excited" states

(5)

Newtonian mechanics is based on Newton's three laws plus the definition of mass and the principle of superposition. Consider whether each of these principles is carried over into relativity intact, modified, or discarded.

The original Gell-Mann-Zweig quark theory was in­

vented mainly to explain the

(a) annihilation (b) relationship (c) existence of (d) discovery of

(4)

(2)

Consider problem 11 from the homework in Chapter 8. What WOuld happen if one changed the wavelength? 26

~

,.

The exam itself, given in class, is as follows. Question numbers refer to those given above. 27

(1)

(a) Is it ever possible to get a minimum halfway

between the speakers?

(4)

Yes --- No ---

The clock on the train runs slow. The train is shorter than its "rest" --­ length. The light takes longer to get to the --­ mirror than to return. The light travels farther than it would --­ if the train were not moving. ___ The observer on the train gets the wrong value for c.

--­

(b) What is heard at the points directly in front of the speakers when the wavelength is I m?

--- Maximum --- Minimum Somewhere

between

(c) Are there more or less maxima with I m waves than with 2 m waves? More Same number ___ Less (2)

(b) This gedanken experiment was not used to derive the clock slowdown because it involves one effect not involved in measuring the speed of light across the train. Which effect is that?

Check the appropriate boxes below:

---­ Asynchronism of two clocks on the train.

Unchanged Modified Abandoned

____ Shrinkage of the train. increase of the train. --­ Mass Relativistic velocity limit.

I

Definition of mass Superposition principle ~I_______~________-+________~ Newton's first law Newton's second law Newton's third law (3)

(5)

(a) What does an observer on the train feel about the clock settings?

(a) The momentum change in going from 0.6 to 0.8 c exceeds that from 0 to 0.2 c by a factor of (b) The non-relativistic answer to (a) would be

--- They

are synchronized

Rear clock is ahead

- - - - Front clock is ahead

(c) The difference between your answers to (a) and (b) is solely due to:

--­

(b) Does an observer on the ground agree with the answer to (a) above?

--­

'\

Yes --- No

(a) Check any statement or statements below that do not reflect the belief of an observer witness­ ing the experiment from the ground (more than one answer is possible) .

The failure to momentum conservation. The relativistic mass increase. The relativistic velocity addition law.

(d) In order to answer this problem two of the fol­ lowing formulas were needed. Check them.

---

(c) What happens to the other relativistic effects, such as clock slowdown and shrinkage of the train, in this situation?

--- They are still present, as in relativity. They are present, but become real effects - - - observable to the man on the train. They are not present. 28

---­ E = me ---­ P = mv --­

(6)

m tJ.t

= ymo

Lv/c 2

(a) Give the number of messages received by the astronaut during each phase of the voyage: 29

Outward _ __

Turn-around

Return _ __ -(b) Does his brother on earth agree with these figures? Yes

--- No

---

(c) How does the astronaut feel about how many mes­ sages were sent during each phase? Outward

--Turn-around

--Return

---

(d) Does his brother agree with these figures? Yes

No

ABSTRACTS OF TERM PAPERS These are summaries of some of the more successful

term papers written for the course at Wisconsin.

Relativity and Cubism. Compares two comtemporaneous efforts, in science and art, to deal with the concepts of space and time. The key point is that both differ from preceding efforts by concentrating on the observer as ac­ tive rather than passive. Concludes both were influenced by philosophical movements of the time. Da Vinci and Cyrano de Bergerac on Flight. Discusses the obsession with flying machines of these two late­ Renaissance figures. The main point is to show how the artistic, religious, philosophical, scientific, and tech­ nical ideas of these men were unified in a "rational humanistic" view that was favorable both to the emergence of the natural sciences and rapid development of new art forms. Color. The author (an interior designer) outlines the development of color theories by physicists and artists. Shows motivations of the two are different, but they are unified by the problem of obtaining a reproducible de­ scription of the subjective experience of color perception. 30

Similarity of conclusions taken as showing the subject is dominated by problems of perception psychology foreign to both fields. Space and Time in Philosophy and Physics. Traces the development of these concepts in physics (Newton to Ein­ stein) and philosophy (Locke to Kant). Shows the mutual interaction of the two, particularly how Kant's ideas helped create a favorable intellectual climate for the emergence of relativity. Galileo and the Church. Makes the case that.Galileo attempted to defeat his academic rivals by gaining Church acceptance of Copernican cosmology, rather than the con­ ventional view that he merely sought Church neutrality. Shows how his effort was doomed to fail for reasons of politics having to do with the debacle of the Thirty Years' War, despite the fact that the intellectual climate even among orthodox theologians was favorable to such a change. The Uncertainty Principle and Sunday Morning. Shows that the point of view expressed in the celebrated poem by Wallace Stevens bears philosophical similarity to the uncertainty principle. Relates this to the author's own personal beliefs, in particular, to the manner in which she came to accept her own mortality as a blessing rather than a curse. The Physicists and Hiroshima. Argues that the profes­ sional training of physicists hampers them in political action, and thus accounts for the failure of the Franck­ Szilard attempt to prevent the use of the atomic bomb against a civilian target. Einstein and Bohr. Though protagonists in a great de­ bate and dissimilar in outward style, these two physicists. shared humanist beliefs and a willingness to let personal philosophical views influence their scientific work. Concludes that this type of person is most likely to take great strides in the development of new theories. Physics and Sociology. Compares scientific method as outlined in a sociology methods course and in this course. Concludes that the former is too rigidly 31

empirical and dogmatic to yield much progress, and urges sociologists to use more intuitive methods.

HOM E W 0 R K ASS I G N MEN T S

Most students should be able to come up with some sort of answer to the questions in the appendix, but the nu­ merical exercises may prove quite challenging. It is un­ wise to assign these unless the students have some oppor­ tunity to get individual help. Remember that students in a course like this have no professional need for skill at solving physics problems, so the latter should be kept to the minimum required for purposes of illuminating the principles involved. As a general rule, it is a good idea to work out in class at least one problem for everyone assigned. For the first few assignments, it helps to "spoon-feed" the students by working problems identical to those assigned, except for the actual numbers, to build up their confi­ dence. At Wisconsin we have found that one question and two problems per week are more than adequate for the classical portions of the course. The problems are not as useful in the relativity and quantum theory units, so the number assigned should be even less in these sections of the course. The questions are nonetheless still of consid­ erable benefit to the student •

.

"

32

33

A N S WE R S

TOE X E R CIS E S

ChaI2ter 7

1.

34 mi/h

2. 3. 4. 5. 6. 7. S.

(a) 10 mls (b) 2 :41 (a) 90s (b) 33.3 mls 6s 3 mls 2 14 m/s Ss •OS m/s2 or .OOS g

2. 3. 4.

Chapters 9, 10

1­

2. 3. 4. 5.

Chapter 2

2. 3. 4. 5. 6. 7.

2. 3. 4. 5. 6.

7.

2. 3. 4.

.S6 c 16

9 x 10 J

100,000 kg

3.

(a) after 25 yr 10- 5 s, 5 x 10- 10 m

2.2 x 10- 15

Chapter 13

10

-3

m/s

2

(b) 6 x 10

-4

g

(c)

2.7 x 10 7

Chapter 5

1.

5 x 10 -S kg; not measurable

1. 2.

15 N , 5 m/s 2 2 mls 2

(a) 6 x

1.

Chapter 12

Chapter 4

2.

.6 m -12 5 x 10 m 7

.2 c, or 6 x 10 mls

O.S times as fast 3 x 10-6 s

ChaI2ter 11

(a) 60 m (b) 20 m (a) 50 m/s (b) 20 m/s (c) 20 m 6 mls 2 mis, to the right (a) 12 m/s in same direction (b) no (a) yes

Chapter 3

1. 3. 4.

1000 m/s 3 m 6m, 3m, 2m (b) 3

1.

Chapter 1

500 J (a) 50 J (b) 5 mls 100 J kinetic energy both before and after (a) 1250 N (b) 1.25 m/s2 5 m . 240 C 60,000 W or SO hp

1-

2. 3. 4. 5.

273 0 C

Hydrogen 4 times as fast 2 x 10- 7 em

Chapter 14

1. 2.

4000

(a)

4

1.6 x 10- 7 (b) 6.4 x 10-

ChaI2ter 15

1.

2. 34

35 cl to 1 H 293 K

5/27 19 J

(a) 6.4 x 10- 34 J-s (b) 6.4 x 1035

Chapter 16

1.

2. 4.

2: 1

2 x 10- 36 em

(a) 5.4 x 10- 23 kg-mis, 1.6 x 10- 15 j, 1.2 x 10- 1 1 rn (b) photon wavelength 10 times greater

Chapter 17

1.

3 x 10- 20 kg-mis

Chapter 19

2.

36 mesons, 56 baryons

36

\..,