Matter in Motion

DOCUMENT RESUME SE 009 208 ED 041 755 Detenbeck, Robert W. AUTHOR Cutts, Fobert M.; TITLE Matter inMotion. Commission on Coil.Physics, CollegePark, Md. INSTITUTION Foundation, Washington,D.C. SPONS AGENCY National Science 65 PUB DATE written for theConference on the NOTE 83p.; Monograph (University New InstructionalMaterials in Physics of Washington,Seattle, 1965) MP-SO.50 HC-$4.25 EDRS PRICE EDRS Price Energy, *College Science,*Conservation (Concept), DESCRIPTORS Experiments, *InstructionalMaterials, Laboratory *Motion, *Physics,Resource Materials

ABSTRACT Conference on the This monograph waswritten for the Materials in Physics,held at theUniversity of New Instructicnal intended for collegestudents who Washington in summer,1965. It is professional physicists.The monograph are notpreparing to become inertia for chapters. Chapter1 deals withthe law of contains three Galilean relativityand objects at restand in motion,the theory of law of inertia.The law of momentumconservation deviations from the 2. In chapter3, the applications arediscussed in chapter and its concept of kineticenergy principle of energyconservation and the and problems areincluded in arediscussed. A numberof experiments each chapter. (LC) t. r ! vs-

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COMMISSION ON COLLEGE PHYSICS DEPARTMENT OF PHYSICS AND ASTRONOMY UNIVERSITY OF MARYLAND 4321 HARTWICK ROAD COLLEGE PARK, MD. 20740

ROBERT M. COTTS Cornell University

ROBERT W. DETENBECK University of Maryland GENERAL PREFACE

This monograph was written Icr theConference on the New Instructional Materials in Physics; held at the Universityof Washington in the summer of 1965. The general purposeof the conference was to create effec- tive ways of presenting physics tocollege students who are not pre- paring to become professional physicists.Such an audience might include prospective secondary school physicsteachers, prospective practitioners of other sciences, and those who wishto learn physics as one component of a liberal education.

At the Conference some 40 physicistsand 12 filmmakers and design- ers worked for periodsranging from four to nine weeks.The central task, certainly the one in whichmost physicists participated, wasthe writing of monographs.

Although there was no consensus on asingle approach, many writers felt that their presentationsought to put more than the customary emphasis on physical insightand synthesis. Moreover, thetreatment was consist of sev- to be "multi-level" that is, each monograph would eral sections arranged inincreasing order of sophistication.Such papers, it was hoped,could be readily introducedinto existing courses or provide thebasis for new kinds of courses.

Monographs were written in fourcontent areas: Forces andFields,

Quantum Mechanics, Thermaland Statistical Physics, and theStructure and Properties of Matter.Topic selections and generaloutlines were leave authors only loosely coordinatedwithin each area in order to number of free to invent new approaches.In point of fact, however, a monographs do relate to othersin complementary ways, aresult of their authors' close, informalinteraction.

Because of stringent timelimitations, few of the monographshave been completed, and none hasbeen extensively rewritten.Indeed, most writers feel that they arebarely more than clean firstdrafts. Yet, is because of the highly experimentalnature of the undertaking, it essential that these manuscriptsbe made available forcareful review by other physicists and for trial use with students. Much effort, therefore, has gone into publishing them in a readable format intended to facilitate serious consideration.

So many people have contributed to the project that complete acknowledgement is not possible. The National Science Foundation supported the Confertnco. The staff of the Commission on College Physics, led by E. Leonard Jossem, and that of the University of Washington physics department, led by Ronald Geballe and Ernest M. Henley, carried the heavy burden of organization. Walter C. Michels, Lyman G. Parratt, and George M. Volkoff read and criticized manuscripts at a critical stage in the writing. Judith Bregman, Edward Gerjuoy, Ernest M. Henley, and Lawrence Wilets read manuscripts editorially. Martha Ellis and Margery Lang did the technical editing; Ann Widditsch supervised the initial typing and assembled the final drafts. James Grunbaum designed the format and, assisted in Seattle by Roselyn Pape, directed the art preparation. Richard A. Mould has helkld in all phases of readying manuscripts for the printer. Finally, and crucially, Jay F. Wilson, of the D. Van Nostrani Company, served as Managing Editor. For the hard work and steadfast support of all these persons and many others, I am deeply grateful.

Edward D. Lambe Chairman, Panel on the New Instructional Materials Commission on College Physics NATTER IN NOTION

PREFACE

Physics describes the real world. The laws of physics areuseful because they describe the behavior ofactual objects in real situations. We use the language of mathematics to express thelaws, and the logic of mathematics to derive predictions from them. Butphysics is not axiomatic. The assumptions on which physical laws arebased are ex- tracted from the physical world, and the predictionsmade by the laws must be tested in laboratory experiments. This monograph draws upon selected experiences fromlife and from the laboratory to reveal clearly some of the patternsof nature. The conservation laws for linear momentum and for energydevel- oped herein are relevent to all natural phenomena, andprovide insight into a wonderful variety of events. They are notobvious from a super- ficial examination, however, but require carefultraining of the observing eye and analyzing mind to discern them. Wehope that in learn- ing to recognize this order in nature the reader willalso discover some of the beauty we see in herpatterns. From initial observations and experiments we try toformulate general laws which describe many events. Attempts tostate the laws precisely in verbal or mathematical form often lead todefinite questions which must be answered in the laboratorybefore the laws can be stated. Once a law is clearly stated, itshould make definite predictions about new situations. These predictions canthemselves be tested in controlled laboratory experiments. Ourconfidence in the general applicability of any law depends ultimately onits past success in predicting the results ofexperiments. The more times the predictions are confirmed, the more confidently we applythe law to

tested. The number and variety of experiments whichconfirm the predictions of the laws of momentum and energyconservation is so large that what we can present is limited by taste andtime rather than by any scarcity of data. Within each section of the monographwe have attempted to state clearly the principle to be developed. Thenwe have turned to experiment and description to clarify and sharpen he content of the principle.

The emphasis has been placed hereupon the general conservation laws and their identical forms in all inertial frames.The princip:a of Galilean relativity has been chosenas the expression of this invar- iance because it seems to us to be the "natural" expression of relativity for elementary mechanics.

The monograph is incomplete in that it lacks the sectionon Forces. This section, though necessary to a consistent development, would not have been large (as indicated in the block diagram of the monograph below). We hope that with a thorough understanding of these two conservation laws, together with an introduction to the study of interactions through forces and potential energies, the studentcan study further the interactions of matter in context. We hope he will use the tools of the conservation laws to study the structure of matter, the collisions of elementary particles, the nature of fields, etc., in other parts of his course. We have assumed that the student is familiar with the vector, description of motions before he reads this monograph. A suitable back- ground might be found in the first six chapters of the PSSC text ,ysics, or the monograph Motion written by Gerhart and Nussbaum at the same time as this one.

The original vision of the monograph on momentum and energyconservation looked something like the sketch below, in which the size of each area is roughly proportional to the number of pages.

Proposed

Law of Inertia . Momentum Conservation Forces energy Conservation As it turned out, the monograph is not completed. The completed sections are indicated below to roughly the same scale as above.

Done Missing - LAW OF INERTIA Galilean relativity

MOMENTUM CONSERVATION

0---Forces

ENERGY CONSERVATION

(Kinetic) Potential energies' thermal energies, etc.

The "Law of Inertia" and "Conservation of Momentum" chapters are sufficiently complete that they can probably be used in the classroom. Although some "thinking" questions are presented in the text, additional drill problems will be needed for classroom use. Even a casual reading of the text will make it clear that it is intended to be used a.,ng with classroom demonstrations as well as laboratory experiments. --1 IMMEMIMMEMMEMMOW

CONTENTS

1 THE LAW OF INERTIA 1.1 Introduction 1 1.2 Law of inertia for objects at rest 1 1.3 Law of inertia for objects in motion 2 1.4A simple principle of relativity 6 1.5The theory of Galilean relativity 8 1.5.1 The Galilean transformation 9 1.5.2 Galilean relativity and the law of inertia 12 1.6 Deviations from the law of inertia 13 1.6.1 Inertial framez, 14 1.6.2Weightlessness 17

2 THE LAW OF MOMENTUM CONSERVATION 2.1 Momentum and the conservation law 21 2.2Air track and the two-particle explosion 23 2.3Air-track experiments and results 25 2.3.1Experiment 1. Identical left- and right-side objects 25 2.3.2Experiment 2. Nonsymmetric explosion 25 2.4The concept of mass 27 2.4.1 Dependence of mass on velocity 30 2.4.2Density 31 2.5Conservation of momentum 32 2.6 Principle of relativity and the conservation law 33 2.6.1 The moving explosion 33 2.6.2The sticky collision 36 2.7The center-of-mass concept 38 2.7.1The center-of-mass point 38 2.7.2 Center of mass and the motion of the system 43 2.8Collisions 47 2.8.1 A sample collision 48 2.8.2The sample collision in thecenter-of- mass reference frame 52 2.8.3 Examples of momentum conservation described in the laboratory and in the zero-momentum frames ofreference 55

3THE CONSERVATION OF ENERGY 3.1 The principle of energy conservation 61 3.2 Isolation, external and internal 68 3.3 Energy--conservation experiments 70 3.4Kinetic energy 70 3.4.1The equal-mass elastic collision 71 3.4.2 The unequal-mass elastic collision 72 3.4.3The form of the kinetic energy 73 3.4.4The ultimate appeal 76 1 THE LAW OF INERTIA

1.1 INTRODUCTION To each object a number called its mass can be assigned to specify The law of inertia describes the this sluggishness. A way to determine tendency of matter in motion to con- quantitatively the mass of an object tinue. At the mouth of a swift river will be developed in a later chapter. ono can see the clear lake waters For now, however, it is already con- parted by the moving, muddy, river venient to introduce the word. It is water, which continues the motion it common in physics to hear or read the had overland until the drag of the words "a mass" used instead of "an ob- neighboring liquid brings it to a stop. ject" when special attention is di- An automobile at rest requires a force rected to this sluggish behavior of a to put it in motion, another force to bit of matter. turn it. An arrow once put in motion In history the difficulty in dis- by a bowstring continues until it hits cerning the behavior of a mass, i.e., something, or until the effects of air of an object, arose in separating the resistance and gravity slow it and properties of the mass itself from pull it to earth. effects peculiar to the earth's sur- The law of inertia was developed face.'It ie ,.till difficult to con- as an expression of what matter would ceive the idea of a piece of matter, do "if it were left alone." In terms an earthly object, far away from and of the last example, the law of in- independent of the surface of the ertia expresses what observations and earth. But it is not nearly so diffi- experimental tests indicate the arrow cult now as it was before children would do if there were no gravity and heard their parents and elders dis- no air resistance. The word "until" in cuss such things. On the surface of the description of the arrow's motion the earth, motions in the vertical and already implies that the motion would horizontal directions differ; there is in some way continue if there were no gravity. We must decide whether a re- outside interference. But we shall try leased object would still fall verti- to express this notion in a more pre- cally, i.e., toward the point now at cisely stated form that can be tested the earth's center, if the earth were in the laboratory. not there. To one steeped in the modern view that space itself is homogeneous and isotropic, the natural 1.2 THE LAW OF INERTIA FOR OBJECTS answer is that there would be no verti- AT REST cal direction in which to fall if the How does an object at rest behave earth were not present. if it is "left alone"? What would it If we probe the question further, do if it were placed at rest, far from rou w;"l see how an attitude toward the remainder of the universe? We ex- space influences one's answer to the pect that the object would remain at question. Suppose the object had been rest. This expectation is a generaliza- placed at rest at point A, and all on tion of the everyday experience that its own, with no outside forces, it to put an object in horizontal motion had moved to point B at the end of one at the earth's surface requires some second. It would no longer be possible sort of effort, some sort of interac- to assert that all points in space tion with another object. Matter is a are equivalent (i.e., space would not sluggish sort of thing, hard to get be homogen'eous as far as this object going. is concerned), unless the object moved

1 2 MATTER IN MOTION

always the same amount in thesame Although the notion or statement direction in the first secondno mat- that an isolated object at rest re- ter where it was placed at rest. But mains at rest is an appealing one if then that constant direction would bo wobolievo that space is homogeneous "special "; i.e., wo could notsay that and isotropic, the validity of the space is isotropic. statement does not depend upon its To one who builds his ideas from subjective appeal. The ultimate appeal common earthly sense experiences, the must be to experiment, even though a universe may not seem to be sucha direct experimental test of this sim- featureless thing with no "preferred" ple statement is very difficult. We point or direction. In buildinga sys- shall, however, be able to put this tem to explain the universe, Aristotle assumption together with others to assumed that different kinds of matter build a coherent description of grad- have natural places to which they tend ually more complicaxed physical sys- to return. In his system the earthly tems. Our trust in this and other matter of familiar solids and liquids basic notions will increase as long tends to move toward the center of as the descriptions enable us to make the universe. Thus, in his system, new verifiable predictions about these space itself has a specf.al point (the systems, and to the extent that the center of the universe) which differs predictions are indeet borne out by from others. This tendency explained experiments. to Aristotle why objects fall vertically toward the earth's center, for the earth had presumably already ar- 1.3 THE LAW OF INERTIA FOR OBJECTS rived at the center of the universe. IN MOTION Indeed, its compact spherical shape shape was explained as the result of Everyone has experienced the dif- its parts clustering as close to that ficulty of stopping or turning, as center as possible. well as starting, a very massive ob- In the modern view we take the ject in motion. In order to make a vertical motion of falling to be some- ferryboat stop just at the edge of thing peculiar to the presence of the the pier, it is not sufficient to turn earth in that direction, and take the off the engines at the instant of ar- horizontal behavior of massive objects rival. A boulder tumbling down the to be typical c- their behavior in all side of a hill does not stop right at three dimensions if the earth were not the bottom except in very unusual cir- present, or if the earth and all other cumstances; if the ground below is disturbing objects were very far away. level, it rolls on for some distance. And yet the behavior of an object in motion, isolated from all disturbing effects, is quite difficult to extract from experience. It was not until the seventeenth QUESTION century that the simple behavior of matter in motion was clearly described in the Law of Inertia: Are these two views :f space mutually consistent? In terms of what you now An isolated mass moving with a know, can you devise an experiment to given velocity will continue decide between them? An ideal experi- with the same velocity as long ment realizable in thought rather than as it remains isolated. in the college laboratory is accept- A constant velocity means both con- able here, but eventually the question stant speed and ccAstant direction, must be presented in the laboratory. of course, because velocity is a vec- THE LAW OF INERTIA 3

for quantity. "An isolated mass" re- tion of the disk is quite well iso- fers to an object alcole in space, far lated. It is free from friction away from and there-'.ore free from the against the surface on which itrides, effects of any disturbing influences. and keeping the surface hcrizontal However, when we test the law of in- makes the motion in two dimensions ertia directly 4n the laboratory, we free of the influence of gravity. must be content with systems that are When the air disk of Fig. 1.1 is approximately isolated, systems in released at rest, it remains at rest which we have reduced the disturbing on the flat horizontaltable, in ac- influences of gravity, friction, air cord with the predictions of thelaw resistance, etc., as much as possible. of inertia for isolated objectsat The motions of objects in every- rest. It may worry you a littleif the day experience are so different from table were leveled by adjusting it for the motion of an approximately iso- ro motion of the airdisk. But is it lated object that even today the law not wonderful that it is possibleto of inertia does not seem natural to find one level position such that all students of physics when they first meet it. Objects moving on the surface of the earth come eventually to a stop if they are "left alone." The reason for this is that the world around them does not really leave them un- affected. The motion of a child's wagon along a sidewalk, for example, is resisted by the bumps in the walk, the air that must be pushed aside, the crunchy dirt, and the slightly sticky oil in the wheel bearings. Snow sledding and ice skating are special fun partly because in these sports the motion persists for a long Fig. 1.1aOne kind of air disk consists of time without any effort, but even the a Lucite cylinder marked atits center with slight drag of the ice on the runners a cross on a paper circle.The disk rests eventually brings the ride to a halt. on an air table. With thisapparatus we can The success of sled runners on study almost frictionless motion. ice suggests a way to make horizontal motion at the earth's surface approx- imate the motion of an object that is really isolated. The pressure of the runners melts some of tLe iceand A provides a thin film of water as a lubricant. Introducing a fluid-between two solid surfaces (for example, 1111101M oil in bearings), is one way of reducing the frictional drag between T C them. Gases are less sticky than 4' ./ liquids and work even better aslubri- cants. Many of the most successful demonstrations of the behavior of iso,- lat.ld mechanical systems use a gas Fig. 1.1bA thin layer of air A flowing film for reducing friction. Figure from the holes H in the air table T keeps 1.1 shows some pictures of anair disk, the Lucite disk D floatingabove the solid which moves across a smoothflat sur- surface. The air is fed to the holes face with almost no friction.The mo- through channels C cut into the airtable. 4 NATTER IN NOTION

great effect, it is possible to check the law of inertia easily in three / dimensions. Laboratory experiments with neutrons or other subatomic parti- f cles usually assume the validity of the law of inertia for the velocity in three dimensions. For example, they may require an isolatedparticle to pass through a set of smallholes, lined up along a straight path, and Fig. l.lcThe motion of an air disk with artive at a target at a certain time. constant velocity on its horizontal surface. The success of such experiments indi- The disk moves from left to right. The rectly supports the validity of the light was flashed every 0.1 second to record law of inertia used in their design. the picture. This motion closely approxi- The law of inertia has been mates the ideal motion of an isolated body checked directly and indirectly in with no external forces. The disk -7es aloi4 a nearly straight line, and c er many situations. -o within the accu- nearly equal titances in equal time inter- racy expected, in every casewhere the vale. (Dhoti..; courtesy of Dr. Harohi Day, data has been carefully examined, the Seattle Physics Writing Conference.) predictions of the law of inertia have boon verified. Here is a delightful air disks placed anywhere on the flat simplicity in the motion of matter, table remain at rest? By the way, can and it applies to any isolated object: you think of another way to level it? large or small, simple or complex.I When the air disk is released The law of inertia does not tell with a certain velocity (Fig. l.lc) it everything about the motion of an continues with that same velocity isolated object - only that so long as `pled and direction), until it it is isolated it moves with constant reaches the edge of the table. Check velocity. As we now describe this mo- this for yourself from the figure. tion in the 1..nguage -of .ector mathe- With dividers you can compare the set matics we shall sharpen somewhat the of successive displacements in equal distinction between what aspects are time intervals. Can you invent a way described by the law of inertia and of specifying the maximum deviation what aspects are not. from straight-line motion? How accu- When the object itself is very rately would you say this experiment small compared with the experimental verified the law of inertia? Did the measurements of distance, we can con- speed remain constant to within ten sider the object itself as a mass percent? One percent? Were the devia- point whose description consists of tions from straight-line motion sig,- its location relative to some origin nificant? (three coordinate values). Then be- Like any other single test the cause of the relatively largescale one illustrated in Fig. 1.1verifies of our distance measurements we can the law of inertia within a certain safely ignore any internal jiggling accuracy within a limited rangeof or tumbling of the objectitself. Such conditions. Va4-!ations can and should a case is illustrated by the measure- be made. The expez:ment can be re- ment in Fig. 1.2, in which the veloc- peated with different disks in differ- ity of a tumbling cube is deta-.1:iner, ent horizontal directions. Theexperi- ments can be performed in otherlabora- 'The first part of the film Inertia by Z. tories and with other materials. Purcell (PM film #0302), could well be used For higher speeds, where appreci- here to reinforce this material in a "lamclip" able distances are covered intimes package, although a two - dimensional analysis too short for gravity to have any would be preferable. THE LAW OF INERTIA 5

from its displacement during a certain that a particle in isolation moves time interval. The velocity v, ob- with constant velocity. tained from the positions of one par- J' actual laboratory work it may ticular point on the cube at two dif- not be convenient to measure over ferent times, differs slightly from very large distances relative to the the velocity that would result from size of the object. In many cases, following the motion of a different however, we can find objects which do point. This uncertainty is not impor- not have internal motions large tant if it is less than the accuracy enough to disturb our measurements of with which we need to know v. It is in the whole body. For example, the cube this sense that we often describe the in Fig. 1.3 is not tumbling. No matter motion of a finite object in terms of what point on this cube is used for the motion of an idealized particle, measurement of v, the result is the a mass with no extension,Character- same. ized entirely by the position of a We shall see later that even for point. The law of inertia predicts objects undergoing very complicated internal motions there is one well- defined point, the center-of-mass point, that continues to move with constant velocity if the object is isolated from external influences. For this point the law of inertia strictly applies. After we have discussed the law of conservation of momentum, you will be able to calculate the location of the center of mass for any oh-

ORIGIN OF REFERENCE FRAME

Fig. 1.2The tumbling cube, free from external forces, carries the corner A from r1 at time t1 to r2t tiro- .2, through a displacement (r2 %/ The average velocity of point during this interval of time is

(r2 ri) (t2 tl). Fig. 1.3 When the cube is not tumbling, 7.1) and 71) is taken as the two displacements (r2 The displacement (r2 between times t1 and t2 are typical of the whole block because it is so (12 equal. Hence the velocity can be calculated much longer than any dimension of the cube, from the displacement of point a. or point and v is taker for the "velocity of the A, or for that matter from any other point cube." Following another point on the cube fixed on the cube: would result in a slightly different value for the velocity, but we assume that this (12 uncertainty is less than the accuracy with ;1211.261 1) (ta td which we need to know v. (t2 t1) 6 MATTER IN NOTION

ject. But you will probably not be surprised to learn now that the center of mass of an object with obvious symmetry (for example, a cube), corre-. sponds to its geometric center (see Figs. 1.4 rnd 1.5).

1.4 A SIMPLE PRINCIPLE OF RELATIVITY

It is possible to show that the Fig. 1.4. A multiple-flash photograph of a law of inertia obeys a simple princi- cylindrically symmetric air disk which ro- ple of relativity. If the law is valid tates about its center (marked with a cross) in one laboratory, then it will be while the cent( . moves with constant velocity. The law of ..nertia describes the hori- true in the same form in any other zontal motion of the center of the air disk, laboratory that moves with constant even when it is rotating. The disk enters velocity relative to the first. the picture from the left, and the interval Let us examine an experiment, between flashes is 0.1 second. (Photo cour- using very simple equipment. as it tesy of Dr. Harold Davi, Seattle Physics takes place in two different labora- Writing Conference.) tories. There is a device that one can make from a piece of string and a When the bob is put into motion weight to demonstrate the sluggisbress by moving the string's support point of matter. An elegant form of such a (Fig. 1.6b), the string must be tilted device is the plumb bob used by carpen- to change the motion of the bob from ters and surveyors to establish a rest to a finite velocity. The tilting vertical direction. When the bob or string indicates that the bob is no weight is suspended from the string longer isolated from horizontal forces. at rest, it hangs vertically beneath The tilt indicates the bob's inertia, the string support. The string is or resistance to charging its velocity. parallel to those of all other plumb If the support then continues to bobs in the neighborhood.(Fig. 1.6a) move with constant velocity, the bob The vertical string indicates that will eventually hang directly below it there iE no horizontal force on the again (after the swinging has died plumb bob at rest. This is in accord down so that it has a steady position). with the law of inertia, which pre- As long as the support moves with con- dicts that a vertical string (no hori- stant velocity, the bob can move along zontal force), should be associated at the same velocity with essentially with a plumb bob at rest with regard no tilt to the string (see Fig. 1.6c). to horizontal motion. Even the tiny tilt that remains be-

Fig. 1.5A moving wrench photographed at marks the center of mass. (From. PSSC 1/30 second intervals. The black cross Physics[D. C. Heath and Com;iny, 1960].) THE LAW OF INERTIA 7

cause of air resistance can be removed the law of inertia for Don's plumb by performing the experiment inside a bobs traveling at velocity v and for closed automobile on a straight, level his own at rest. But at the same time road. A bob suspended from a support Don in the car can verify the law of moving along inside a closed car with inertia, relative to his own reference constant velocity hangs vertically, frame, for Joe's plumb bobs traveling beneath its support. And this is what at velocity v, as well as for his own the law of inertia predicts. The verti- at rest. cal string, implying a lack of any horizontal force, is associated with Si the constant velocity of the bob. Now let us look at these results from two different viewpoints.

The Bonneville Salt Flats Experiment

The Bonneville Salt Flats, Utah, are chosen as the site of this experi- (a) Bobs 1, 2, 3 and their supports SI, S2, ment because they provide a large, flat S3 are all at rest. The two similar p,.umb horizontal surface on which an auto- bobs (2 and 3) as well as all others at mobile can be driven with constant rest (1) bang parallel. velocity. Joe, at rest on the ground at the es, salt flats, sets up a number of plumb bobs. To him they appear at rest and vertical. Don, inside a car driving at a constant velocity of 50 miles per hour west, also sets up a number of plumb bobs fastened inside the car. To Don his own plumb bobs appear at rest and 171 vertical. Therefore both Joe and Don (b) Bob 3 is being given a velocity to the can agree on the law of inertiafor right by moving S3 to the right from its objects at rest. But note that they position of rest. The tilting string on are able to apply the same law in two bob 3 indicates the resistance to the different laboratories, one of which change from rest to motion as the bob is moves with constant velocity relative accelerated. to the other. Joe, observing Don's plumb bobs moving at the constant velocity of 50 mpft west, and still hanging vertical, coafirms the law of inertia for objects in motion with constant velocity. Meanwhile, Don has been looking at Joe's plumb bobs. Relative to Don's reference frame in the car, Joe's i411110. plumb bobs move at the constant veloc- v3 ity 50 miles per hour east, and hang (c) Bobs 1 and 2 and their supports SI ana vertically. Dor can also verify the S2 are at rest. Bob 3 and its support S3 law of inertia for objects moving at are moving to the right with constant veloc constant velocity. ity v3. Bob 3 rides directly under its sup- the car moves with veloc- port S3, hanging parallel to bobs 1 and 2. ity 'Whenevervt Joe on the ground can verify Fig. 1.6 A plumb-bob experiment. 8 NATTER IN NOTION

It should now be clear that by fied by Don's experiment, it would be such experiments two observers could necessary to give up or somehowmodify never tell which one was moving and one of the two assumptions. which one was at rest, for the law of Notice that Don can apply the inertia takes identical forms in both same reasoning and predictfrom his laboratories. That is the essence of vertical stationary plumb bobs inside this simplest principle of relativity, the moving nar that Joe's plumb bobs, which is true for the law of inertia at rest cal the ground, will also be and for every other physical law for vertical. If the law of inertia is which it has been tested: correct for zero velocity, and if the principle of relativity is correct The laws of physics are the same for the law of inertia, then the law in two laboratories whose motions holds true for every velocity. differ only by a constant relative velocity.2 1.5 THE THEORY OF GALILEAN You will see how closely the law RELATIVITY of inertia and the principle of relativity are intertwined if we try to predict some of the results of the The theory of Galilean relativity Bonneville Salt Flats experiment from consists of two parts: the principle two assumptions: of relativity discussed in the pre- (1) The law of inertia for obvious section, and a recipe for trans- jects at rest only, and lating the description of moving ob- (2) the principle of relativity. jects from the viewpoint of one labora- Joe, at rest on the ground, verifies tory to that of another moving with as before with his plumb bobs the as- constant relative velocity. sumption (1), that objects at rest During the discussion of the and in isolation remain at rest. Bonneville Salt Flats experiment in By applying the principle of rela- the preceding section, we assume that tivity, Joe can predict that Don will the stationary observer (Joe), could obtain the same results in his labora- look into the window of the automobile, tory (moving with constant velocity). passing with constant velocity, and Therefore, Joe can predict that he ',vee" that the moving bobs were hang- (Joe) will see Don's plumb bobs hanging vertically. If Joe were to de- ing vertically while moving at con- scribe a more complicated experiment stant velocity. From these two assump- going on in the moving car, one in- tions he can predict that the law of volving velocities as well as posi- inertia for moving objects will be tions, it would require a little more verified. If it were not in fact veri- attention to detail to relate Joe's description to that of Don inside the different refer- 2The famous theory of special relativity car. Don and Joe use stmt's by Einstein early in the twentieth century ence frames, moving relative toeach includes this principle as one of its postulates. other, to describe the same phenomena. however, it also considers verycarefully the A transformation is a recipe for problems involved in describing moving systems when information can travel no fasterthan th.: translating the descriptions of phe- speed of light (another of its postulates).Very nomena from the language of onerefer- briefly, the results of these considerations are ence frame to that ofanother. The that two observers moving rela,ive to eachother Galilean transformation, in particular, will not agree on such measurements as the length of an object, the duration of a timein- translates the description of a moving terval between two events, etc. Becausethese object from the mathematicallanguage effects are small when the speeds are small com- of one reference frame to thatof an- pared with the speed of light (3 x10' m/s) we other moving at constant velocity shall not need to consider them in any of our own experiments. relative to the first. THE LAW 07 INERTIA 9

1.5.1 The Galilean Transformation. The essence of the Galilean ti-Ans- formation is this: Joe, at rest in a reference frame with origin 0, describes the motion of an object. At the time the object is at point P, described by the position vector r relative to the origin 0. Don, at rest in another reference frame withorigin 0', describes the loilntion of theob- ject at the same point P with theposition vector r', which gives the displacement from 0' to P, as in Fig. 1.7a. If 0' moves at constant velocity u relative to 0, andif we assume that Joe and Don measure time withclocks that run in perfect synchronism,then the object's velocity v relativeto 0 JOE (Joe), and v' relative to 0'(Don), are related by

=110 I.v II

All our measurements of velocity, changes of velocity, etc., arebased DON ultimately on measurements of posi- Fig. 1.7aAn object at P, described from tion and time. Figure 1.7a showshow two different reference frames0 (Joe's) the position of one object P isde- and 0' (Don's), when the two origins are separated by the displacement R,from the scribed from two different reference origin 0 to the origin 0'. Theposition r frames (in two laboratories), at the of P relative to 0 is related toits posi- same tine. We assume that oneobserver, tion r' relative to 0' by ordinaryvector Joe, always describes things fromthe addition: 0 reference frame and the otherob- server, Don, from the 0'frame. r - + r'. When the origin of the 0' frame is separated from the originof the 0 Equation (1.2) can also beobtained transforma- frame by the displacement R (Fig. directly by looking at the 1.7a), the position of an object at tion from Don's (0') viewpoint asin point P relative to the 0 frame(r) Fig. 1.7b. Joe and its position relative tothe 0' When the object is in motion, frame (e) are related by and Don separately establishits velocity by position and timemeasurements. At his time t 0, Joe finds the ob- R + r'. r ject at Po, position rorelative to 0; at time t1 he finds it atposition 1.1, This relationship shouldbe clear as in Fig. 1.8. Bythe definition of from the geometry of Fig.1.7a and average velocityv, the displacement the definition of vectoraddition. of the object from time 0 totime t1 subtracting the vector R fromboth is given by sides of Eq. (1.1) we obtainthe (1.3) equivalent statement, r ro -;t1

time ; R. (1.2) Now suppose that, during the 10 NATTER IN NOTION

JOE

Fig. 1.8 The motion of the object fromP0 to P1 and of the origin0' as seen from Joe's (0) frame of referenceduring the time interval 0 to t1. Ifthe object P moves with averagevelocity v relative to 0, DON 7:1=7:0 + ;it. Fig. 1.7bThis figure is equivalent to constant veloc- Fig. 1.7a, but it is drawnfrom Don's (0') If the origin 0' moves with viewpoint. From the geometryand the deli- ity u relative to 0, nition of vector addition 111 =104- r + the same interval 0 to ti, Don and hisrefer- object at points Po and Pi, points chosen by Joe. Fromthe geom- ence frame 0' moverelative to Joe should see that with constant velocity u. Theorigin etry of the figure you 0' will undergo a displacemental r = Ro + uti + r 1' (1.4) during this time interval, asin Fig. 1.8. From the geometrical relation- whereas 10 is given by ships of the figures, weshall be able Ko + ro'. (1.5) to describe the motionof the same ob- ro = ject in the mathematicallanguage of Don's frame of reference.Because Don The lengcii of thedisplacement moves relative to Joe,he will de- ;hi between Po and Pi in Joe's refer- scribe the positions andvelocities ence frame isobtained by substitu- of the object with numbersdifferent ting Eqs. (1.4) and(1.5) for ri and from Joe's. However, thereis a geo- ro in Eq. (1.3): metrical relationship betweentho relation- two sots of measurements, a vt, - ri rot ship which depends uponthe relative

velocity u. vti am (IL+ uti+ r1 ') In Fig. 1.9 we seeDon's two measurements of the positionsof the THE LAW OF INERTIA 11

r ro/ (1.9) t1 1

for the average velocityv' in his frame of reference. Insertingthe value of 21 - ro' fromEq. (1.8) into Eq. (1.9) gives for thq averagevelocity in the 0' system

v' (// U)tlitll. ( 1 10) We assume that theclocks of Joe and Don run in perfectsynchronism;3 i.e., that ti = t1'. Then weobtain JOE the simple vector equation,

ampb. mew Vw m V U. (1.11)

The simple velocitytransformation of DON Eq. (1.11) can be statedin words as;

DON The velocity relativeto 0' is equal to the velocityrelative to The two measurements byDon (i0' Fig. of 0' rela- and '1.9ri') of the positionsof the object at 0 minus the velocity points Po and P1 relative tohis own 0' tive to 0. frame. The picture is drawn asthough Joe The complete detailsof the geo- were standing stilland Don moving. From metrical transformationthat we have the geometry discussed, between thedescription + + r'1 of motion in onereference frame and r1 Ao utl I and that in another moving atconstant ro Ro + ro'. relative velocity, aresummarized in equations of the Galileantransforma- tion, Eqs. (1.12). Equation (1.6) can besimplified by noticing that the vectorRo disappears 0' frame in the subtraction: 0 frame

Ro = position of 0' relative to 0 at t =0 = t'; '1 vti - Utl + (r ' ro (1.7) u = constantvelocity of 0' relative to 0. Subtracting the vectoral from both r' = position vector sides of Eq. (1.7) andfactoring the 1= position vector time t1 gives a formwhich will be use- t = time t' = time ful later, v = velocity vector v' = velocity vector

ti)ti Cri (1.8) 'There is actually no universal timescale com- different velocities. Now we are ready toconsider mon to clocks moving with For relative speeds, small comparedwith the For sim- Don's velocity measurements. speed of light, two observerswill find that plicity, assume that Don agrees,with their clocks agree to within a verysmall cor- of macro- Joe, to start his clockat t = 0 when rection. For laboratory velocities scopic systems, this correction isnegligible. the object is at Po.According to comparable to the reaches P1 at If the relative speed (u) were Don's clock the object speed of light (3 x 10' meters persecond), the time t1'. By definitionof average transformation equations of thespecial theory velocity, Don's measurementgives a of relativity would have to beused instead of the Galilean transformation. value of 12 MATTER IN MOTION

t' t From this 0' frame the object "really" undergoes a displacement v'tl. Figure r' kr ut) - Ro 1.10 shows an analysis of the situation as though Don were standing still =O. ...IP v' v - u and Joe were moving. Compare Fig. 1.9. The relationship among the displacements uti, 7t1, and ;i't1in Figs. 1.9 These equations in vector form and 1.10 is illustrated in Fig. 1.11. are the equations of a general Gali- The velocity diagram in the same fig- lean transformation to the 0' frame ure illustrates the similarity of the from the 0 frame of reference. They de- vector Eq. (1.14c). pend on the constant velocity u. Equation (1.12c) applies to the aver- 1.5.2 Galilean Relativity and the Law age velocity v measured over any time of Inertia. interval. Hence the equation can also be used to relate two instantaneous The law of inertia is one of the velocities v and v's determined in the simplest examples of a law of physics limit of an arbitrarily small time in- that takes the same form in two refer- terval. ence frames whose relative motion is In some cases we shall be able to one of constant velocity. use a much simpler form of the trans- Our experiments have indicated formation. If the two origins 0 and 0' that the law of inertia holds to a coincide at t - 0 t', and if the good approximation for horizontal no- relative velocity u is directed along Po the parallel x and x' axes, sf / p t' t (1.13a)

x' x - ut, y' y, z' z (1.13b)

vx' vx -u, Vyt M Vys V zt M VZ (LIU)

The transformations we have discussed take descriptions in the mathematical language of Joe's system and translate them into the quantities JOE which describe moving objects in Don's moving system. Of course, Eq. (1.12) JOE can be solved for t, r, and v if the reverse transformation is needed:

t t', 11114 Fig. 1.10The two measurements by Don and Joe of the positions of the object at and =IP =IP =O. points Po and P1, drawn as though Don were VMVI standing still. Otherwise this picture represents the same events as Fig. 1.9, Another way of obtaining these with which it should be compared. From the same equations is to go back to the geometry beginning and redo all figures and

analysis from Don's frame of reference. ' Tto 14tz "4' 2%1 From this 0' frame the origin 0 of and Joe's frame moves with velocity -u. ro no + ro. THE LAW OF INERTIA 13

0 vt1 Fig. 1.11aThe relationship among the three displacements: uti, vtl, and v't1 of Figs. 1.9 and 1.10. The points labeled 0, 0', and P signify that al is the change in position of 0' relative to 0, vti is that of P relative to 0, and v't1 is that lof P relative to 0'. These displacements all occur during the time interval 0 to t1. 0' P.. 0 3.p v Fig. 1.12The mass m, moving with velocity Fig. 1.11bThe relationship among the v relative to the reference frame 0, moves of Figs. three velocity vectors: u, v, v with velocity v' relative to 0'. If the 1.9 and 1.10. The points labeled 0, 0', frame 0' moves with velocity u relative to and P signify that u is the velocity of 0' 0, then relative to 0, v is that of P relative to .0.. ...O m.. 0, and v' is that of P relative to 0'. VY - V - U.

If the mass m is isolated, and if 0 is an tions at the surface of the earth, inertial frame, ; is a constant velocity. relative to a reference frame fixed If u is constant, then so is ir-', and 0' is to the earth. Let us assume for the also an inertial system. purpose of the argument that we have found one reference frame (say far out in space), in which it holds exactly Because 0 is an inertial frame, v is for motions in all three dimensions. also a constant. Therefore, v' is a The law of inertia states that an constant. The isolated mass m moves isolated mass moving with velocity v with constant velocity relative to 0'. (relative to this frame), continues Thus 0' is an inertial frame, one in to move with the constant velocity v which the law of inertia is valid. as long as it remains isolated.The Every reference frame 0' moving name given to such a frame ofrefer- at constant velocity relative to an ence is an inertial frame. inertial frame 0 is also an inertial Let us imagine that an isolated frame. mass m moves with constantvelocity v relative to the inertial frame 0, as in Fig. 1.12. The Galilean transforma- 1.6 DEVIATIONS FROM THE LAW OF tion tells how the motion of the same INERTIA mass is described relative toanother frame 0', which moves with constant A careful scrutiny of real mo- velocity u relative to O. We have seen tions at the earth's surface reveals in the first part of this section that small deviations from the behavior relative to 0' the mass m moves with predicted by the law of inertia. hl- velocity given by Eq. (1.12c), though these effects are not large enough to show up in simple laboratory

...... =0. ...11. vy v u. experiments like the ones we have discussed, they are important in some By definition of the reference natural phenomena. These are all con- frame 0', is is a constant velocity. nected with the fact that a reference 14 MATTER IN MOTION .

system attached to the earth is not the observer and hds laboratory (the entirely suitable for determining the circle), with its reference frame are laws of physics in their simplest and at rest in the inertial system 0, and most general form. the observer verifies the law of inertia for the particle moving in his own laboratory. When the laboratory moves 1.6.1 Inertial Frames with constant velocity as in Fig. The earth rotates on its axis, 1.14b, his laboratory frame is still and a point fixed to the earth near an inertial frame, although he does New York City travels around a circle not measure the same constant velocity with a circumference of about 17,000 that would be measured in another in- miles every 24 hours. The 700 mph ertial frame. speed itself does not cause any diffi- If the observer's frame of refer- culties. We have seen that if the law ence moves in one direction but with of inertia is valid in one frame, it changing speed relative to an inertial is also valid in another frame moving frame, as in Fig. 1.14c, the law of at constant velocity relative to the inertia will not be valid in his labor- first. What causes deviations from the atory frame. law of inertia in the reference frame If the observer's laboratory fixed to the earth's surface is the reference frame rotates, as in the uni- fact that the velocity of the refer- form circular .emotion of Fig. 1.14d, ence frame is slowly changing with the law of inertia will not be valid time as the earth rotates. The laws of in his frame. This case illustrates motion appear in quite different forms the difficulties experienced at the in two laboratories, one of which is surface of the earth, although the ef- rotating relative to the other. A fects due to the earth's very slow glass of water resting on a rotating rotation are small. phonograph turntable behaves very dif- If, either the size or the direc- ferently from a similar glass of water tion of the velocity of the observer resting on the floor. are changing relative to an inertial Because the earth rotates very system, experiments described relative slowly (once a day), the effects of to his frame will not verify the law its rotation are not normally impor- of inertia for that frame. tant in the laboratory. They are some- Suppose that one were actually to times important in very large-scale verify the law of inertia for a mass M natural phenomena involving distances in a direct experiment. The experi.nent comparable to the earth's diameter. would have to be carried out in inter- Th3 earth's rotation is responsible stellar space, far from any disturbing for the .tact that winds in the north- influences that might be important. ro ern hemisphere do not blow radially set up a frame of reference in such a toward a low-pressure area but instead situation requires a marker to denote spiral toward and around it in a coun- the origin and some way to keep track terclockwise direction. The weather. of distances, directions, and time. map in Fig. 1.13 shows such a circula- The origin of the reference system tion around a low-pressure area in could be marked with a rock 0 suffi- the northern Great Lakes region of the ciently small that one hopes it United States, and another around an doesn't itself affect the motion of offshore low-pressure area in the the mass M. The law of inertia pre- Atlantic. dicts that the masses 0 and M will The deviations from the law of then separately move with constant inertia to be expected by an observer velocities relative to an inertial sys- whose velocity is not constant with tem. Hence it predicts that their respect to an inertial system are il- relative notion will be one of con- lustrated in Fig. 1.14. In Fig. 1.14a stant velocity. THE LAW OF INERTIA 15

Fig. 1.13 This figure is a representation heavy blue vectors. The letters H and L of the U.S. Weather Bureau weather map for identify areas of high and low pressure. July 31, 1965. The outline of the United The counterclockwise air circulations States in black is overlaid with isobars around the low-pressure areas centered in (lines connecting points of equal atmos- the northern Great Lakes and off the Atlan- pheric pressure) dashed in blue. Wind veloc- tic coast are quite well developed. ities are indicated approximately by the

Distance and time measurements QUESTIONS, alone cannot test for a constant relative velocity; some standard directions must be set up. The directions 1.1 How would you check whether the could be determined against the direc- rock 0 at the origin had any ef- tions of "fixed stars" in the distance. fect on the results of the experi- To a high degree of accuracy the stars ment? are so distant that their velocities and ours have almost no effect on the 1.2 How many directions need to be es- pattern of starry constellations dur- tablished to define a frame of ing the course of a laboratory experi- reference? If you were able to ment. Hence they provide a satisfac- measure only distances between tory set of reference directions, if reference rocks and the mass N, we make the reasonable assumptionthat and could not see the stars, how the heavens do not share some uniform many reference rocks would you (and hence undetectable in the changes need to test the prediction of a of star patterns) rotation which would constant relative velocity between make sucn a frame of reference system masses? Assume that all reference noninertial. Experiments carried out rocks could be placed at rest rela- on the earth, in which corrections are tive to .each other. made for its motions relative to the "fixed stars" indicate that they do 1.6.2 Weightlessness. indeed provide an accurate set of directions for setting up an inertial In movies and television pictures, system. millions of eyes have soon what life 16 NATTER IN NOTION

A B C D CLOCKS

9 9 1 8.. I 6.2 ti 7 .3 6 4 5

26 0 9. .1 8 / .2 7 It3

0 9. .1 8° 7 3

4's 4 5

ri 0 9. .1 8 2 7. ``N.3 3 4

0 9. .1 8. 2 7. \3 4

M snow 41 MEM IMMO

mallo Imi aimlaw=lb

Fig. 1.14The motion of a point ma 3 P (or the same motion of point P, at constant the center of mass of an extended object) velocity in the inertial frame at the lower in inertial and noninertial frames. An in- left of the figure. ertial frame is represented by the black frame in the lower left of each picture. The white circle with the cross represents The "clock" column indicates that succes- another frame of reference. Its motion is sive horizontal rows show the position of different in each column. In column A the P after successive equal time intervals. circle is stationary and its frame is in- Read vertically downward, each column shows ertial. In column B the circle's frame is THE LAN OF INERTIA 17

is like without gravity. Astronauts and the public looking over their shoulders have seen objects "floating in space" and moving in a straight line at constant velocity without falling, relative to an orbiting satellite of the earth. It is amusing to see what is special about a reference frame fixed to an orbiting satellite. Objects which would "fall" relative to a reference frame fixed on the earth "float" relative to the satellite. And yet there is a pull of the ti=0 earth's gravity where the satellite is. /11=gt We know that for certain, because that Fig. 1.15Two elevator laboratories. Joe's pull was used in the calculations that elevator is at rest. Don's is accelerating predicted the satellite's orbit. at a constant rate after the cable is cut. Briefly, the resolution of this para- dox of weightlessness in the gravita- Relative to the earth-fixed 0 tional field of the earth is this. The frame, Don's hammer also falls. But satellite and its contents aro falling Don's elevator and Don fall, too. Now together in an orbit around the earth. it is a remarkable thing about the In the relative motion of the satel- gravitational pull of the earth that lite and its contents the falling is the observer, the elevator, and the not apparent. hammer gain speed at the same rate. Later we shall discuss how the All objects falling at the same place combination of an initial horizontal under the influence of gravity gain velocity and a constant falling to- equal amounts of vertical velocity in ward the earth's center can combine equal time intervals. Thus, relative to produce a curved orbit around the to Don and his 0' frame, the hammer earth. But first let us examine what does not fall (see Fig. 1.16). other experiments tell us would happen The results of experiments with in a falling laboratory. hammers, bullets, elevators, people, Figure 1.15 shows two laborator- etc., at the earth's surface can be ies built of identical parts inside summarized as follows: If at the be- closed elevators near the earth's sur- ginning of a time interval t the veloc- face. Each has its own reference frame. ity of an object falling freely is vo, Joe is in the elevator with the 0 then at the end of that interv,1 it frame and Don is in the elevator with is the 0' frame.

At a particular instant Bob cuts vv -;0 + (1.15) the cable on Don's elevator. Simulta- neously Joe and Don let go of their where g is a constant vector hammers in surprise, Joe's hammer (9.8m/sec2 down). Tf non and Joe had falls to hit his toe. thrown their hammers with initial an inertial one because it moves at con- circle of column C is not an inertial sys- stant velocity relative to the inertial tem. It moves in one direction, but with a frame at the corner of the figure. The mo- smoothly varying speed. The circle of col- tion of P at constant velocity relative to umn D, which rotates at a constant rate, is the circles of columns A and B is summar- not an inertial system. The effects noted ized in the superposition of points in the in column D are typical of those observed circle at the bottom of the column. The at the surface of the rotating earth. 18 MATTER IN MOTION

C 0 7 .4-- 0 5M1 5 4\472 144 3

0 ;15 0 4 1 I 4\-7 3 fisk 2 3

)05..9-1.1 0 4,02 I 5(M1 I 4.1j2 3 3 Fig. 1.16 Two freely falling hammers ina Fig. 1.17Two hammers fall after having gravitational field. In the laboratory 0 been given an initial velocityvo to the at rest, the hammer falls to the floor. In left. In the 0 frame, at rest in thegravi- the laboratory 0', which itself falls tational field of the earth, the hammer freely, the hammer appears to float.The falls to the floor. In the 0' frame, freely clocks C and C' indicate successive equal falling with the hammer, the initial veloc- time intervals. ity vo vo appears to continue unchanged.

velocity vo, then Joe's hammer would _ButBut by substituting Eq. (1.16) have described a curved path (actually, for and (1.15) for ; into Eq. (1.17) a part of a parabola), to the floor, for v' we obtain as in Fig. 1.17. But Don's hammer, moving with vt (vo +gt)gt velocity v after a time t (Eq. 1.15), relative to the 0 frame, continuesto Vt V0. (1.18) move with velocity vo relative to the 0' frame (see Fig. 1.17). Letus see The hammer moves with constant how this comes about. Applying Eq. velocity relative to Don's freely (1.15) to the velocity U of Don's falling elevator. If Don regardsthe elevator, we find that, starting from hammer as isolated, he finds thathis rest, 0' frame is an inertial frame.A reference frame falling freely in the U it, (1.16) earth's uniform gravitational field is an inertial frame if othermasses at time t later. falling also under the influenceof If the velocity of the hammeris gravity are treated as isolated.5 v relative to the 0 frame, and the A freely falling elevator would velocity of the 0' frame isrirelative be, for a short while,a suitable to the 0 frame, then the hammer's place for demonstrating the lawof velocity relative to 0' is4 inertia in three dimensions. Thelack of weight would havea corollary bene- v° v - U. (1.17) fit in reducing friction. Objects could move freely through the thin air rather than being draggedover a rough table. Experimentswe now per- 'The Eq. (1.16) was developed insection 1.5 for two reference frames moving with constant veloc: sThe apparent absence ofgravity in a properly ity. It can also be applied when the velocity U accelerated frame of referenceis very impor- is changing if it is applied toinstantaneous tant to the theory of general relativity, which velocities. dealswith accelerated frames and gravitation. THE LAW OF INERTIA 19

Fig. 1.18An airplane flying in a para- To provide an Inertial frame the airplane bolic path, in which during.each time inter- would have to fly without changing its at- val At the velocity gains an amountAt in titude, as in the figure. In actual fact, the downward direction. For such a path the an airplane-would not fly that way very airplane provides a weightless reference well. frame in which the hammer inside "floats." form with great effort using air (1.19) above. Relative tothe airplane disks, etc., could be done easily the object has velocity with hammers, people, china cups,almost anything. ";, - IT To acquaint astronauts with the 67L + 170 + it) +it) sensations of weightlessness, an in- (1.21) genious method for producing weight- 0. lessness without the dangers of falling elevators has been devised. _.An The airplane in this special trajec- airplane with initial velocity Uo can tory is an inertial frame. The situa- be flown in a (parabolic) path such tion is illustrated in Fig. 1.18. that for 20 seconds or so the velocity A satellite in an orbit circling changes in the vertical direction in the earth has a motion similar to that the same manner as in free fall: of the airplane in a parabolic path. It has acceleration toward the center

IT lio + gt. (1.19) of the earth. Although the "weightlessness" Such a path is illustrated in Fig. effect is much more general, let us 1.18. Relative to the airplane, ob- restrict ourselves to a discussion of jects within it do not fall when re- a simple circular orbit within a few leased. The airplane provides a weight- hundred miles of the earth for which less frame. An object released attime the gravitational effect of the earth zero within the plane hasvelocity v; does not change much in size. The size relative to the plane and velocity Igi of the constant acceleration given to all masses is about the same as it is at the surface vo szt v- ; + 170 (1.20)

relative to the ground. At a laier 9.8 m/sec2. (1.22) time t its velocity relative to the ground is given by Eq. (1.15) while Of course the direction of g is at that of the plane is given by Eq. each point on the orbit given by the 20 MATTER IN MOTION

local direction to the earth's center acceleration being provided by the in- (geocentric direction). fluence of the earth's gravity. The You should recall from your study correct speed for such a circular of the description of motion that a orbit can be obtained by setting lal circular motion at constant speed in- of Eq. (1.23) equal to 11;1. Thus, volves an acceleration which is constant in size and directed toward the v2 = r I circlets center. The size of this ac- or celeration is v =y777. (1.24) v2 170= (1.23) For a radius of about 4000 miles, this gives a speed of about 18,000 mph.6 where v is the constant speed around An object isolated except for the a circle of radius r. effects of gravity and at rest rela- The gravitational influence of tive to the satellite will continue the earth near its surface gives all with the satellite in the circular or- freely falling objects, including the bit, both undergoing the same acceler- satellite and its contents, an accel- ation of size, Igl toward the earth's eration given by Eq. (1.22) in the center. geocentric direction. Thus, if the tangential velocity is exactly correct 'If the speed is not exactly that of Eq. (1.24) for the radius r, the satellite and the orbit is not circular. The satellite and objects within are still equally affected by its contents continue to move in a cir- gravity, and a situation of weightlessness pre- cular orbit with constant speed, the vails, but it is more complicated to describe. 2 THE LAW OF MOMENTUM CONSERVATION

2.1MOMENTUM AND THE CONSERVATION LAW lutely impossible depending on whether or not the conjecture agreeswith the One very important aspect of conservation law. linear motion of an object is that it As we will see, direct applica- has linear or translational momentum. tion of conservation laws can bring The momentum of an object equals the new insight to the physics inthe sys- product of the mass of the object and tem. its velocity. Since velocity is a The law of momentum conservation vector quantity and mass is a scalar, is basic and so nearly elementary that momentum is a vector quantity. The it can be considered early in the de- momentum is written as p, velopment of mechanics. Of the two alternatives - postulating the law and p = mv. then deriving and testing its conse- quences or developing the law from a Momentum and the conservation law series of experiments - we have chosen for momentum are the main topics of the latter. Some of the experiments this chapter. It is the purpose of will be thought experiments that can this chapter to develop the concepts readily be visualized, and some can be of momentum, of its conservation law, actually performed in your classroom. and of mass itself. Of course, many more than we can ac- Momentum is basic tc the struc- tually perform were needed to give mo- ture of physics. The law of conserva- mentum conservation the strong position of momentum is one of the unify- tion it holds today in physical theory, ing principles of physics. but our experiments will be representa- What do we mean when we say that tive of the important points in the something is conserved? Let us try to development of the law. define this briefly now, with the ex- The first situations we study pectation that full appreciation of will be simple ones with which we can conservation laws comes only with use easily cope. From these we shall be and understanding of the realthing. able to perceive the simplicity in If the quantity conserved is momentum, more complex situations andeventually then rie state, "For any isolated sys- to analyze some of them in detail. tcm, the total momentum of that sys- The first chapter dealt with mo- tem is constant." The conservationlaw tion of an isolated object. The object is independent of how complicatedthe was not necessarily apoint mass. Even processes may be going onwithin the though it may have consisted of system, whether they involve living several component parts, it was never- or nonlivingthings, expansions or theless visualized as one object, or contractions, fast or slow changes.If a system "as a whole." In going from the total amount of momentum ofthe one, to more than one object, we will system is known at any onetime, and find that some of the basic concepts if the system remains isolated,then of physics, such as the mass of each the total momentum will havethat same object, momentum conservation of the value at any other time. system, and force between objects, Once firmly established, such a can be developed by considering just general statement as the conservation two objects or particles interacting law can be an extremelypowerful tool. with one another. In this chapter, the A conjecture made concerningthe sys- concepts will usually be developed tem can be declared possible erabso- first in the specialized case which is 21 22 MATTER IN MOTION

(a) (b)

Fig. 2.1 The system of two spheres and a small compressed spring is released. small spring. That the system is isolated (b) The system is shown after separation is schematically indicated with an imagi- produced by release of the small spring. nary dotted line. Vectors vA and vs are the velocities of (a) The system is shown just before the A and B, respectively. easier to visualize or to reproduce in motion unequally acquired by A and B, the classroom. Subsequent to this, a what determines the division between brief general development will be A and B? given which will be more formal and Perhaps you detect the qualita- also mere abstract in that it will not tive difference between these ques- be tied to a particular given physical tions and the "obvious" question above. experiment. All concepts carry over to There are clear answers to (1) the general from the special without and (2) and from these answers, the faltering. "obvious" question as wellas (3) are Let us consider this problem: readily solved. The answers to ques- There are two spheres "floating" to- tions (1) and (2) suggest general, gether in outer space, isolated and universal conservation laws basic to far from the earth or other objects. all of physics. Two of these laws One sphere, A, is massive and other will be developed in the chapters other, B, has low mass. (Mass has been which follow. described in Chapter 1 as a sluggish- In our laboratory we will do ex- ness of matter, a property that experiments similar to the one described presses matter's inertness to changes above except that they will not be in its velocity.) A small compressed performed "floating in outer space." spring located between A and B is Instead the objects A and B will move suddenly released and forces A and B along a straight horizontal track. apart as shown in Fig. 2.1. This is a special track known as an Perhaps the obvious question "air track" or "air troagh," and the asked of the laws of physics should objects are glide blocks ("gliders"), be, "Given a knowledge of the mass that move freely on a film of air. of A, the mass of B, and complete The gliders are supported by many specification of the spring, with what small jets of air continuously coming speed will A be moving and with what out of the track. The motion of each speed will B be moving niter the glider on its cushion of air is prac- spring expansion?" That , a good tically frictionless. Although fric- question and it has an answer if all tion is an interesting and complicated the information is available. subject, we don't want to study it But consider other questions. right now! Certainly friction is evi- (1) Is there "a quantity" of this dent and useful in our lives for with- isolated system which is unchanged by out it, for example, we couldn't take (is conserved during) this spring ex- steps in walking. Some roughness be- pansion? tween our feet and the sidewalk is (2) Is there "a quantity" of mo- needed for firm footing. In mechanics tion equally acquired by A and B? experiments, however, where we would (3) If there are quantities of like to find out what controls motion THE LAW OF MOMENTUM CONSERVATION 23

A . 4 s..

40,

Fig. 2.2 (a) Photograph of an air-trough with a typical glider in position near the end of the trough. Both the glider and the (b) Photograph of a roof-top type air end post are fitted with bumper springs. track with two gliders in position on the The hoses at the end admit air to the mani- track. Both of the gliders and the end fold (shown in Fig. 2.3). This glider is post are equipped with bumper springs. The fitted with a marker to indicate its screws used to adjust the vertical height position on the scale that is mounted on of the track can be seen near the end of the side. (Photograph of Search Linear Air the track. The air input hose is at the Trough, courtesy of Macalaster Scientific back end of the track, as seen here. The Corporation.) forward end of the manifold is closed so that the only escape for the air is through of things, the friction between our the vents along the track. (Photograph of object and whatever supports it would Stull-Ealing Linear Air ?rack, courtesy of be a nuisance. Fortunately, it is al- Ealing Corporation.) most eliminated by use of an air track. low tube which can be shaped either as a trough or as a roof top. The two 2.2 AIR TRACK AND THE TWO-PARTICLE EXPLOSION usual types of tracks are shown in Figs. 2.2 and 2.3. The air track is actually a hol- When the track is in use, air

GLIDE BLOCK

AIR FLOW AIR FLOW GLIDE BLOCK

AIR AIR AIR FLOW MANIFOLD FLOW

(a) Section of trough type of air track. (b) Section of roof-top type of air track. Fig. 2.3Cut-away sections of the two vents located along the upper surface of types of air tracks. The air manifold is the manifold. This air flow, which is in- continuously supplied with air at a pres- dicated in (a) and (b), forms the air cush- sure higher than atmospheric prossure. The ion which supports the glidor. air oscapos through all of the small air 24 MATTER IN MOTION

mechanics is needed to measure velocity, v. To study motion, something is needed (as the literal motivator) A B which will get our objects, the glide MMIEZEZi blocks, into motion in a controllable 1 way. We will use a small coiled spring (a) for this purpose. Many other devices, such as a chemical explosive (a toy 0 "cap"), an electric motor, or com- r-rrrrTTTI- pressed gas, could also be used. The small spring has advantages for us at this time, and it will be used first. WiA E727,A The spring will be compressed between the left, A, and the right, B, (b) gliders shown in Fig. 2.4. After the system is quickly released, the spring Fig. 2.4 Schematic representation of a symmetric air-track explosion experiment. will expand and force the gliders (a) The glide blocks are shown in position apart. The gliders will move along the just before the spring is released. (b) The track and the velocity of each can blocks are shown in motion after the spring then be measured as it goes along. has been released. The method of spring release de- serves attention. One should not rely above atmospheric pressure is pumped on using his hands for holding the into the hollow interior and this air gliders together and for releasing escapes continuously through the small them. A person may be neither nimble jet vents (holes) that are located all nor quick enough to avoid disturbing along the track surface. For either the glider motion. A proven technique track in Fig. 2.2, the glider is large is to use a string loop which slips enough to cover several vents and it over a pin on each cart and holds rides or "hovers" over the track on them together. The string can be the cushion of air provided by the burned with a match with little or no jets. As long as the glide block does disturbance to the gliders as they fly not go too fast, it glides along fric- apart during the spring expansion. tionlessly on its air support without The track should be level. This scraping the track. is readily accomplished by adjusting We are provided with a number of the track position while using the identical glide blocks made of, for glider at rest as a level indicator. example, aluminum. Being identical With the track horizontal, we can means being made of the same "stuff" neglect all gravitational effects and and'being made the same size and with the air turned on, we very nearly shape. Let us also assume that we are simulate a "floating in space" experi- provided with meter sticks and stop ment in one dimension along the track. watches so that the velocities of our All of the spring "explosions" gliders can be measured. Other, more that we can observe on our track will sophisticated, timers such as strobe be found to have two qualitative fea- cameras, photoelectric cell timers, tures in common: etc., may be available. Such refine- (a) While the spring is expand- ments may be needed for.high-speed ing, it is always in contact with both carts, but they involve no change in gliders. Both gliders then lose con- the principle of measurement. All in- tact with the spring simultaneously. volve the determination of velocity (This means that the sprint, is small from length and time measurements. and that its motion can be neglected Note that no knowledge of the laws in the discussion that follows.) -77

THE LAW OF MOMENTUM CONSERVATION 25

(b) After the gliderslose contact with the spring, each XI X2 moves with I I its own constant velocity alongthe track. The observation (a) above is im- (a) portant in the development of mechan- XI X2 ics. The observation (b) follows di- -4---- A/mi.. WA rectly from the law of inertia discussed in Chapter 1. The free glider (b) continues to move with constant veloc- Fig. 2.5 (a) Location of interval (x2- xi) ity because it is isolatedon our near end of track. (b) Location of interval frictionless track. The constant veloc- (x2 - xl) near explosion.,Thearrows repre- ity implies that the instantaneous sent velocity vectors. velocity of a glider right after an explosion has the same valueas the average velocity we measure for any convenient length Ax of its trip along the track. It then follows that the instantaneous velocity of each glider after an "explosion" is foundas the measurement of the average velocity Fig. 2.6Two identical gliders are shown of each glider. The average velocity in position before the spring is released. The dotted line runs through the center of is easier to measure since it equals the system.

x2X Xx1 vs and vs are measured. vXave + t2 If we were to examine themeas- urements, we would find that in each and bx can be taken as a convenient explosion, the speed acquired byA displacement which allows reasonably equals the speed acquired by B. Since convenient measurements of At. Ques- vs and vs are vectors in opposite di- tion: Is the track locatioon of the in- rections, terval (x2 xi) at all critical? That is, can (x2 x1) be near the end of v vs - 0. the track as in Fig. 2.5a,or should it be near the explosion as in Fig. This is not a very surprisingre- 2.5b? sult since the experiment is perfectly symmetric about a vertical line drawn thiough the center of thespring. What 2.3 AIR -TRACK EXPERIMENTS AND is there to favor the left over the RESULTS right? If A and B were interchanged, no basic experimental change would 2.3.1 Experiment 1. Identical Left- exist since A and B are identical and Right-Side Objects. aluminum blocks. Two identical objects or gliders, A and B, are placed at the center of 2.3.2Experiment2. Nonsymmetric Ex- the air track as shown in Fig. 2.6, llosion. and "exploded" apart by the spring. The left-right symmetry isre- After the explosion, the velocity of moved by making the left glider larger A, vs and the velocity of B, vs are or smaller than the right glider. This measured. A number of small springs can first be done by simply using more of various shapes and degrees of stiff- than one glider and by connecting them noes are used, and for each explosion rigidly together possibly as a train 26 MATTER IN MOTION

and NB and for many sizes and kinds of NA Ns "explosion springs," the ratio of velocities varies in inverse linear 1 1 proportion to the ratio of the number of identical glide blocks. Or: the ratio of velocity magni- Fig. 2.7 Two trains of identical blocks, tudes is NA and NB, are shown in position on the air track before the spring is released. IVAI NB (2.1) NA 5.0 I ;)1 I Does the inverse proportionality surprise you? Take an extreme case 4.0 with NB = 10 and NA = 1, and NB/NA= 10. Perhaps you can visualize large B slowly moved by the explosion and A 3.0 scooting away rapidly from the explosion. Better yet, do the experiment! How could we study these measurements? How coUld we establish the 2.0 linear (first power). dependence of the ratio of IvAI/IvBI upon (NB/NA) and not some other "power law" depend- 1.0 ence such as, for example, (NB/NA)2 or (NB/NA)1/2? The qualitative dependence. of IvAl/IvB1 on (NB/NA) may be

Oo suggested by visual observations of 1.0 2.0 3.0 4.0 5.0 the experiment. If we were to suspect (Ne) some dependence ofIvAl/Ivill upon NA (NB/NA), we might try plotting the results as ratios on a graph as in Fig. 2.8A plot of the ratio of speeds of block trains,1;A1/1;1I, versus the in- Fig. 2.8. verse ratio of the number of blocks inthe The plot demonstrates visually trains, (Ns/NA). Each point (o) represents and quantitatively the linear depend- the result of one experiment with one set ence of IvAI/IvBI upon (NB/NA) and of NB and NA. The best fit continuous line clearly rules out the other power law through the data points is a straight line. dependences mentioned earlier. The two dotted lines are plots of The conclusion expressed by Eq. 1;1/1;1 an(N1 /Ns)2 and 1741/1;s1 (2.1) says nothing of the size of each (Ns/NA)0. individual value of Iv1. Big springs faster mo- or as a stack of glide blocks. Let NA can be observed to produce tion, and small springs produce slower and NB be the number of identical motion for both A and B. With a par- glide blocks on the left and right ticular small spring, for example, sides of the spring, respectively, as both A and B move slowly, but the in Fig. 2.7. ratio of IvAI toI;BIis still given Let us try some explosions with by the ratio of NB to NA. various values of NA and NB. The If we now rewrite the equality speeds developed by A and B are found to be no longer equal. What we find is IVAI NB that the smaller the number of blocks .... on a side, the faster that side moves. IVBI NA From a study of the measurements, we as would find that for many values of NA NBI;AI NBlvBI, (2.2) THE LAW OF MOMENTUM CONSEWATION 27

we can note that the left side of the served, remains constant, at a value Eq. (2.2) describes side A only and which for this experiment is zero. If contains no dependence on B. The right Eq. (2.4) is valid, then no matter side similarly depends on B only. This may be the nature of the explosion, equation tells us that there is still small or weak, fast or slow, the sum a left-right equality in the motion of products of Nv must be zero at the even though there are more blocks (and conclusion of the explosion. Equation thus more "stuff" on one side than the (2.4) is written to emphasize the con- other). The product of N and Ivl still servation concept and we certainly has a kind of left-right symmetry even recognize limitations in its applica- though we thought we had removed this tion at this time since it was only symmetry! developed for a particular series of In this one-dimensional experi- aluminum glide blocks on an air track! ment, the vectors which represJnt velo..71ties, vA and vB, are opposite in direction. Since the velocity is a 2.4 THE CONCEPT OF MASS vector quantity, the product Nv mcst also be a vector quantity because N To extend the experiment to other is a scalar. The direction of Nv is materials, we could make up a series given by the direction of v. of identical gliders of copper, gold, Since Ns v* and Nava have equal plastic, six., and we would always magnitudes, it follows that find a conservation principle for the sum of Nv for any one particular sub-

NAvA a Nava. (2.3) stance. Would the law hold if we mixed These vectors are shown in Fig. up, say, copper and aluminum blocks? 2.9. Equation (2.3) can also be writ- In order to do this, we need a way to ten as compare copper to aluminum in the context of this mechanics experiment. NBvA + NBA% = 0. (2.4) When is a copper block equivalent to an aluminim block? The equivalence When the conclusion is stated as can be established experimentally by in Eq. (2.4), it appears as a conserva- putting one aluminum glider on the tion principle. In words, Eq. (2.4) left and one copper glider on the says that the value of the sum of right. If the copper block acquires a products of Nv has a value after the greater speed than the aluminum block, explosion that equals the value of the then a larger copper glider will have sum of products Nv before the explo- to be chosen. If the copper block sion. The value of this sum is con. moves more slowly than the aluminum, then a little copper will have to be ift Vs's shaved from it until, with careful MMIIM)1111111111 adjustment,

NA 11Copperl = 1;eluminuml, (2.5) v,r4 and we can say that the two blocks are equivalent in this experiment. A comparison between a third ma- Nave Ns vs terial, Lucite plastic, for example, 1 and aluminum could also be established. Fig. 2.9. Since Ns is greater than NA, In a comparison of equivalent Lucite IVO is less than IvAl. The vectors -AN-A and aluminum blocks one would observe and Nes are equal in magnitude and oppo- side in direction. (2.6) IVLuciteI I-17al um inum I 28 MATTER IN NOTION

.26 GOLD COPPER ALUMINUM LUCITE PLASTIC

Fig. 2.10A collection of glide blocks ments. If the aluminum glider has a length having the same cross section. Each block of 1.00 unit, the length of the gold glider has been cut to be equivalent to the alumi- is 0.140 unit, the copper 0.303 unit, and num glider in air track explosion experi- the Lucite plastic, 2.29 units. If we have established that for track explosion experiments, it can three separate gliders, be demonstrated that equivalent blocks of the various materials can be freely Iv copperI "I :;alt..nua substituted for one another and the and conservation principle of Eq. (2.4) still sholds. 1;Lu citel I;aluminumI,

can we now assert that Ivcommr1 will NAvA + Nivs 0. equal Iv/Amite! in "air-track explosion" experimental comparison between It can also be demonstrated that them? One might expect that theclp- double blocks (twice as long) and per glider and the Lucite glider would fractional blocks can be used and the be equivalent, but this would not be conservation law still holds provided established until the gliders wereac- the meaning of N is Changed to allow tually compared in a similar "air- for multiple blocks and fractions of track explosion" experiment. Might not a block. other untested factors affect the Clearly, thenNAis a measure of equivalence? Would color, shape, and the amount of stuff in the "A" group metallic character have an effect? It and Nd is a measure of the amount of turns out that these factors do not stuff in the "B" group. From the re- matter. If it is determined by air sults of these experiments, N is also track explosions that for a certain a measure of the sluggishness or in- material, "J" ertia of the train of blocks. Trains of glider blocks that have large N I;j1 IvAI, acquire low velocities in the exoPi- ment. We have used N to label the and for material "K" "amount of stuff" and the inertia of each train in this experiment because lvAl this particular experiment deals only with blocks. If we could melt the then experiment will show that aluminum blocks of each train and form them into another shape (for ex-

17J1 1- 1;11. ample, a sphere), and repeat the explosion experiments "floating in outer Figure 2.10 shows a collection of space" the same velocities would be gliders matched by experiment for measured. Some characterization or equivalence. label less restrictive than the "num- In a continuation of the air- ber of blocks" is needed for each THE LAW OF MOMENTUM CONSERVATION 29

train. There is such a physical quan- bigger the J.:.ss of the glider, the tity, and it is called mass. greater the inertia of the glider. The mass of a block or group of The larger the mass of an object, the blocks is directly proportional to the loss responsive it is tocauses that number, N. would put it into motion or change its For sets of blocks A and B, motion.' The mass isa measure of the inertia of the object. !II& NA. The mass is also measuro of (2.7) Ma Ng the amount of matter inan object. It is simply extensive in that if the We say that the particular Lucite volume of an object is increased by plastic block and the aluminum block adding the same material in thesame of Fig. 2.10 have the same mass be- state, the mass increases in direct cause they are equivalent in behavior proportion to the amount of material. in the air-track experiment. Both Mass is a positive scalar quan- blocks contain the same amount of tity. Only one number is needed to stuff or matter, although they are specify the mass of an object. The very different in other physical and connection between the scale of mass chemical characteristics. developed here and the unit of mass, The similarity in the behavior the kilogram or the gram, can be es- is similarity in one-dimensional mo- tablished by experiment. A standard tion. There is no spinning or tumbling mass is placed on one of two identical motion to the block as it slides along gliders and then the usual speed the track. The motion is purely trans- measurements following another spring lational. Certainly the blocks shown explosion are made. For example, if in Fig. 2.10 are not mechanically the empty gliders each have mass Ma, equivalent in all respects. The vari- the mass of the standard is, say, ations in lengths could be very im- 0.1000 kilogram, and the ratio of the portant in some mechanical systems speed of the empty glider to the speed involving rotations of the:'? objects,. of the_ glider c_rrying the standard is but in pure translational motion, the Ival/174.1, then, objects are equivalent. Mass is more than a bare number. Ma+ 0.1 kg Iva l Mass has physical units or "dimen- Ma I Vega I sions," and the dimension we shall and usually use is the kilogram. So far 0.1 kg we are capable of measuring ratios of N. masses only with the air-track experi- riva ment. From the definition of mass Lrisal given above, we determine the mass The mass of each glider can then be ratio of two sets of blocks from the so measured and each glider can be velocity ratio, as measured in one of labeled with its own mass in kilograms. the air-track explosions. Once these glider blocks are so cali- brated, they can be used to determine m. Ii761 (2.8) masses of other gliders in units of NB I;BI kilograms. With the definition of mass which The air-track explosion experi- has been developed, the air-track ments help in the development of the equipment could be used as a mass concept of mass as a measure of the inertia of an object or particle. In this work the mass of a block or 'Whet'r an object is "put into motion" or has its motion changed is only a question of which glider is inversely related to its frame of reference the motion is described from. final speed on the track. Or, the This is more thoroughly discussed in Section 2.6. 30 MATTER IN MOTION

measuring device. To measure an un- ship would be free of gravitational known mass, M0, the unknown could be effects and would provide an ideal placed on a pan or in a compartment of inertial reference frame in which to one glider, M1, so that the loaded work. Actually, the air-track would mass of that glider would be (M) + M0). then not be needed as a gravity iso- In an explosion experiment with an- lator! other glider of mass M2, the final There are many physical and bio- speeds Ivil and 1;21 of the loaded logical processes in which the masses block (1) and the second block (2) of the elements of the system are would be observed. Then important and all gravitational effects are unimportant. It is satisfy- (M1 + m ) 1 v2 1 ing to know that a mass concept and

112 scale can be developed independent of gravitation. and finally Mass as a measure of the inertia of matter is a concept drawn from ex- Mo = MELL M periment. The validity of this con- 2 1 cept depends upon its usefulness in understanding and analyzing more and This is admittedly an awkward more complex physics. It is a concept way of measuring mass, and it would which has withstood such tests. never be practical in a chemist's There are other concepts of mass. laboratory. The analytic balance used The famous law of universal gravita- by chemists is certainly simpler, tion discovered by Isaac Newton states faster, and more accurate. that the force of gravity exerted by The determinations of the mass one mass on another is directly pro- of a chunk of matter by the air-track portional to the product of their two method and the chemist's balance are masses and inversely proportional to in perfect agreement. Ttore is, how- the square of their separation dis- ever, a difference in the principle tance. A concept of mass developed of operation of these two methods. from this law requires some knowledge The air-track method of determining of "force." We will not discuss this mass makes a comparison of the in- further until after we develop the ertial properties of one chunk of mat- concept of "force." There is also the ter with the inertial properties of mass-energy equivalence contained in a known mass. The determination of the special theory of relativity. This mass with the analytical balance makes concept of mass will be discussed in a comparison of the gravitational at- the study of energy and energy con- traction of the earth for one chunk servation. of matter with the gravitational attraction of the earth for a known 2.4.1 Dependence of Mass on Velocity. mass. The develfrpment of a mass concept If our experiments were repeated and the measurement of mass with the with more and more powerful springs, air track or similar equipment are we would, within our ability to meas- completely independent of the earth's ure, obtain the same value for the gravitational attraction. In fact, mass of an object, no matter how fast_ the horizontal air track was used to the gliders could be separated. We isclatc the experiments on motion would conclude that the mass is in- . from the effects of gravity. The air- dependent of velocity. This conclu- track experiments could be. performed sion holds (that is, is accurate just as well in a space ship moving enough), at the low speeds within the at constant velocity, millions reach of our air tracks and springs, miles from any planet. Such a space but other experiments with micro- THE LAW OF MOMENTUM CONSERVATION 31

scopic particles (protons and elec- still very nearly one (1.0000000005). trons, for example), in the high- Microscopic (or ator0c) particles such energy physics laboratories of the as protons or electrons havebeen ac- world have demonstrated that mass is celerated in laboratories to speeds not independent of speed. Appreciable approaching c, and in such work Eq. changes in mass are not apparent un- (2.9) has been well verified. Values less the speed of the particleap- of (M/M0) of greater than 103 have proaches the speed of light. The veloc- been achieved. It is worth noting that ity dependence of mass is given by the language used to describe the Einstein's special theory of relativ- laws of mechanics at relativistic ity as (nearly equal to c) speeds uses concepts that first become familiar in Mo (2.9) the relationships and equations of v:)1' low-velocity mechanics. Our develop-. (1- mont will continuo in the nonrolativ- where Mo is the mass of the object at istic domain and the conservation laws rest. (v - 0), c is the speed of elec- so exposed will be applicableto rela- tromagnetic waves (light) in a vacuum, tivistic mechanics as well. and v is the speed of the mass. The Eq. (2.9) for the velocity 2.4.2Density. dependence of mass says that a moving The different glide blocks in mass is loss responsive to forces that Fig. 2.10 have the same mass, but they would change its motion than the same vary greatly in size. Yt is clearthen object at rest. Experiments support that size alone does not determine the the form of this equation. Equation mass of an object. The relationshipbe- (2.9) is a succinct statement of "how tween the mass of an object and its matter behaves" with no elaboration size, as measured by its volume, is of how this cones about. Substituting expressed in terms of the mass density, numbers from our own air-track experi- of that material. From daily life, we ments shows that the correction of Eq. are aware of high-densityand low- (2.9) to the one we use, density materials. A block of wood is easier for a child to move about than Ig= MOP a block of iron of the samesize. We know that if we made three is negligible within experimental aluminum glide blocks of identical error. size the three blocks would have the The speed of light is very large, same mass. The amount of massin a 3 x 106 meters/second or 186,000 given volume of aluminum is a property miles/second. With our air-track of aluminum. Copper has a different equipment, we could never make our characteristic mass in the same vol- gliders move at speeds approaching ume of copper. the speed of light. In fact, such The density of a material de- equipment could not make any macro- pends upon its composition, its tem- scopic object (that is, an object per- perature, and the pressure. For many ceivable with the human senses), move common solids and liquids, the compo- miles/ at speeds approaching 186,000 sition alone is sufficient to deter- second. For example, the mass of an mine the density to within a few per- object moving at 3meters/second cent. If a small element of volume (about 6 mph) is bigger than its rest (A vol.) of material contains an mass by a factor of1.00000000000000005 amount of mass (Am), the density of Even at speeds comparable to those of that material is defined as Garth satellites orbiting near the - (Am) earth's surface (about 20,000 mph), density the mass correction factor (MAO is vol). 32 MATTER IN MOTION

The usual units of density are sum of the products of Mv of the in- kilograms per cubic meter, kg(m-3), dividual objects. The product Mv for or grams per cubic centimeter, g(cm-3). a particle, the product of the mass of The units of g(cm-3) are most often an object times its velocity, is used for conveniet.ce. The density of called the momentum of that object. water at 4°C is 1000 kg(m-3) and the Momentum is a vector quantity. density of, for example, aluminum is Since mass is a scalar and velocity a 2700 kg(m-3). In the cgs system, these vector, the product of the two is a densities are 1.000 g(cm-3) and vector. The direction of the momentum 2.700 g(cm-3). The relative lengths is the same as the direction of the of the glide blocks shown in Fig. 2.10 velocity. The symbol p is often used can be determined from the densities for momentum: of the materials which are for gold, 19.3 g(cm-3); copper, 8.92 g(cm-3); p = mv . (2.12) aluminum, 2.70 g(cm-3); and Lucite plastic, 1.18 g(cm-3). The units of momentum are (kg m/sec) in the mks system or (g cm/sec) in the cgs system. 2.5 CONSERVATION OF MOMENTUM Equation (2.11) states then that the total momentum of this system is Return again to the two-body ex- conserved. This is our first encounter plosion as produced on the air track. with this great law of physics, the If the objects have masses MA and Ms, law of conservation of momentum. and if the system starts from rest, In words, the Eq. (2.11) states then the objects A and _.B move apart that the vector sum of the momenta of with speeds IvAI and IvAI, respec- the individual objects remains con- tively. We have seen that stant following the two-body explosion. Each object had zero momentum before the event, so that the total 117,1 MA' mor_Aum must be zero, and being con- and served, remain constant at a value of mA RA 1= mBRBI (2.10) zero. The law of conservation of momen- Again, as in Eq. (2.2), we rec- tum states that ognize a left-right or A-B symmetry in the Eq. (2.10) above. The product For any isolated system, the total MIA of object A equals the product momentum of that system is constant. MM for object B. Note also that only one number, the mass, is needed to The words "total momentum" mean characterize each object in this ex- the sum of all of the individual mo- periment. menta of all the parts of the system. Remembering that vA and vs are This sum must necessarily be a vector vectors in opposite directions, Eq. sum since the momentum of each part (2.10) can be written to contain this (or particle) of the system is a vec- information, or tor. The total momentum of an isolated -"11,V, system remains constant as time de- MA VA - I and then velops independent of any kind of chemical, mechanical, electrical, or MAvA + Mos O. (2.11) yet unnamed change that may occur within that system. The parts of the Equationi(2.11) is written as a con- system may be flying apart from one servation law, and the quantity that another, and even though they become is conserved in this explosion is the separated by vory largo distances, THE LAM OF MOMENTUM CONSERVATION 33

they still remain parts of the system. value othor than zero in any frame If the various parts of the system moving at constant velocity relative never do interact in any way with part to the frame in which the total mo- or all of any other system, then the mentum is zoro. The total momentum is system under consideration remains conserved (remains constant), in each isolated and the total momentum is frame, in accord with the principle of constant. relativity. An isolated system is one on which no external force acts. A complete discussion of what is meant by 2.6 THE PRINCIPLE OF RELATIVITY AND "isolated" will have to await further THE CONSERVATION LAW development of the concept of force. When Eq. (2.11) is written in the 2.6.1The Moving Explosion. form given, In Chapter 1 we discussed the MAVA MgVg 0, principle of relativity, which states that the laws of nature must be the the equation is expressed as a conser- same in tw., different laboratories or vation law. The total momentum of this frames of reference that move relative system is zero, and remains zero so to one another with a constant veloc- long as the system is isolated from ity. the rest of the universe. The fact If the conservation of momentum that the sum of the momenta of the law holds for an explosion in our particles of this system must be zero, frame, then it must also hold for ex- and not some other number, is a con- plosions in other frames moving at sequence of the special experiment constant velocity relative to ours. under consideration. If the system According to the relativity principle, Mocks A and B) were not at rest, but no matter in which of these frames the blocks were moving together before observers make measurements on the the explosion with the spring com- air-track explosion, all observers pressed between them, then the total would conclude that the total momentum momentum would not be zero. The total of the system is conserved. momentum would have to equal the sum It may be helpful to imagine that of the initial momenta of the system. the air track is fixed on a large cart We consider this next in section 2.6, which always moves horizontally with where we examine a moving explosion. constant speed in the +x direction. We will see in section 2.6 that The x axis is fixed to he laboratory the fact that momentum is conserved floor. There is room on the cart for at zero in one reference frame means air-track explosion equipment includ- that momentum is conserved at some ing timers and students for making

- --xD

O x

Fig. 2.11The two-particle explosion under track. The cart velocity is ;relative to investigation by two sets of observers sym- Joe, who is on the floor. The coordinate bolized by Joe and Don. Don, who is on the axes x' and x are parallel. cart, is at rest with respect to the air 34 MATTER IN MOTION ..

measurements. The student observers moves with respect to the x axis with on the cart locate block positions on velocity u, then the velocity of that the x' axis which is locatedon the block relative to the floor is, from air track, and the x' axis is parallel Eq. (1.11), to the x axis. The cart observers P. measure velocities in terms of time V =vy + U. (2.14) rate of change of the x' coordinate, and the symbol v will be used for For example, if the cart velocity their velocities. u = +3m/sec (7m/sec in the positive x For quick reference, call the set direction) and if B moves with velocity of observers on the cart (x') by the vs' = +1.5m/sec relative to the cart, name of one observer, Don, and the set then vs = +3m/sec + 1.5m/sec = +4.5 on floor (x axis) by the name ofone, m/sec relative to the x axis. Remember Joe. The cart moves with velocityu that the symbols in Eq. (2.13) carry which is directed in the +x direction. their own sign which signifies the di- The motion is indicated by the vector rection of the vector along the x' u, shown in Fig. 2.11. axis. If one of the velocities is neg- Don studies explosions, and ob- ative it appears as a negative number. serves that two given blocks of mass From Eq. (2.13), MA and Ms, acquire velocities VA' and vs' in opposite directions relativeto v' = v u. (2.14a) the track (x' axis). In Don's reference frame, the blocks are at rest be- Write Eq. (2.14a) separately for fore the explosion. Don's situationis block A and B. just exactly the experimental setup described in section 2.4. Don's obser- VA'=VA - U. vations will be the same as those of section 2.4 and his measurements show vs '= vs u. (2.15)

. Equations (2.15) make the connec- IvA'I Ms tion between what measurements Don 117.1 mA. makes (ZA', vs') and the measurements These results can be expressed in Joe makes (vA, vs). This connection or vector form, as in section 2.5, transformation is an expression of the fact that Don and Joe agree on space and time measurements. It is the MAYA' + Meg' = 0. (2.13) Galilean transformation developed in section 1.5. Conservation of total momentum of Don's velocity data fit Eq. the A-B system in Don's frame isex- pressed by Eq. (2.13). (2.13). If we substitute the right How would Joe, who has his own side of Eq. (2.15) into Eq. (2.13) for each v', then Eq. (2.13) becomes meter sticks and clocks, describe the explosion? Before the spring between blocks A and B is released, the blocks mAGA 71) ms(v. 1") o both have the same velocity, u, which or is just the velocity of the cart (or x' axis) since both blocks are at rest in that frame. Immediately after the MAYA + Mem = MAu + Msu. (2.16) completion of the explosion, A and B have different velocities relative to Equation (2.16) is our prediction Joe. The velocities measured by Joe of the equation that Joe's measure- are vA and vs. ments should obey. If a block moves with velocity Can you recognize the conserva- v' along the x' axis, and the x' axis tion of momentum law being expressed THE LAW OF MOMENTUM CONSERVATION 35

by Eq. (2.16)? The right side of Eq. Ivs +4.5 m/sec I (2.16) is the vector sum of the momenta of A and B before the explosion and the left side of Eq. (2.16) is the (3) What is the momentum in the un- vector sum of the momenta of A and B prime (Joe's) frame before the explo- after the explosion. The total momen- sion? tum remains constant, or momentum is conserved in Joe's reference frame as well as Don's.' 3TOT MAVA(before) Men (before) We know that Example: Put numbers into the above. I y vA(before) vs(before) u. Given: MA go 0.3 kg Ms.. 0.2 kg 4 P 2" MAU + MAU vA'- -1.0 m/sec u - +3 m/sec - (.3 kg) (3m/sec)

(1) Find ;11'. + (.2 kg) (3m/sec)

MAYA' + Mer 0 fum= + 1.5 kg m/sec II (.3 kg)(-1.0 m/sec) + (0.2 kg) vs' - 0 (4) What is the momentum in the un- prime frame after the explosion? vs +1.5 m/sec = MAYA +MA1711

(2) Find VA, vs = (.3 kg)(2. m/sec) m + (0.2 kg)(4.5 m/sec) for A, VA VA U -1.0 m/sec - VA - 3 m/sec 16Tor = +1.5 kg m/sec

Note that the total (vector sum) mo- 4vA - +2 m/sec mentum of the system remains constant. Momentum is conserved. vs' - vs- u How would Joe describe this explosion? 1.5 m/sec - ;'s - 3 m/sec He would say that the two blocks were initially moving together with the same speed of +3 m/sec. The spring re- 4The physics of this thought experiment is contained in the two-particle explosion in which leased and after the "explosion," A both particles are moving together before the was still moving to the right (+x di- explosion. The "system" consists of particles A rection), but more slowly at +2 m/sec and 8 in addition to a small spring the motion and B was moving more rapidly at of which is assumed to be negligible. Remember that the air track is not part of the system. +4.5 m/sec. Since the system is well isolated from the air Don says the two blocks were initially track, it is really not necessary to have the air track itself in the moving (Don's) frame. at rest and after the spring released Wu could jumt am well do the experiment in which A moved loft (-x' direction) at the air track was fixed in the x frame (Joe's) -1 m/soc and II moved right (+ x' di- and in which Don makes observations from a cart rection) at +1.5 m /sec. moving with velocity u equal to the common initial velocity of A and B. The results in the Both sets of observers' measurements form of Eqs. (2.13) and (2.16) would still obr are in accord with the law of conser- taint vation of momentum. Is it possible for Joe, with only a knowledge of his own measurements on the explosion and a knowledge of Galilean relativity, to predict what Don's measurements are? 36 MATTER IN MOTION

2.6.2The Sticky Collision. bullet imbedded in it, and by knowing the mass of the block and the mass of Lot us consider a problem that is the bullet, the original speed of the something like an inverted explosion. bullet can be found from the law of Assume that the ends of the sliding conservation of momentum applied to blocks have been prepared in such a this completely inelastic collision. way that the blocks stick together Figure 2.12 shows another example of whenever the ends touch one another. a completely inelastic collision. The ends might be coated with a glue Consider blocks E and F, of mass or mastic, or some kind of mechanical ME and Mr, respectively, which are coupling might be used. When two such projected toward one another on our blocks are projected toward one anair track. Assume that these blocks other, they collide, join together, suffer a completely inelastic colli- and move as one. Such sticky collision. It will not be necessary to un- sions are usually referred to as com- derstand the nature of the glue used pletely inelastic collisions. as a sticking agent or how the mechan- Perhaps you can think of some ical coupling works. We need only es- examples of completely inelastic col- tablish that our two-block system is lisions. In chemical reactions, one isolated in that the blocks move fric- molecule or atom might come in contact tionlessly over the track. The total with a second molecule or atom and momentum of the system of blocks E and combine with it to form the molecule F must then be conserved. of a new compound. The constituent If blocks E and F are projected parts would move together after they toward one another with equal speeds, combine, and we would call this a com- and if the blocks have equal mass, pletely inelastic collision. An open- then it is clear that the pair come to top freight car (a gondola car) coasts rest and stay at rest after they col- along under a huge sand hopper which lide. This is a symmetric collision drops a load of sand into the car as and the total momentum of the system it coasts by. The sand and the car is zero. have suffered a type of collision and, With a less symmetric collision after its completion, they move as for blocks with unequal masses and/or one. It is possible to measure the unequal speeds, the momentum conserva- velocity of a rifle bullet by shooting tion law is applied in equation form. the bullet into a large wooden block. With enough given information the If the block is large enough, the bul- equation can be solved for the unknown. let becomes imbedded in the block and For example, in Fig. 2.12, let the total mass of the first boy and his . the two then move together if the block is free to move. By measuring sled be Ml and the mass of the unsus- the velocity of the block with the pecting boy be M2. The momentum of

Vim...4111111.

(a) (b) Fig. 2.12 (a) A completely inelastic col- (b) After the collision the system moves lision in which one object (the unsuspect- with one velocity V. ing boy) is initially at rest. THE LAW OF MOMENTUM CONSERVATION 37

this system before the collision is V- -2.5 m/sec. Mivi whore viis the initial velocity of the sled. The total momentum after After the collision, the two move with the collision is (M1 + M2)11. In apply- a speed of 2 m/sec in the negative x ing the conservation of momentum law, direction. Mr had more negative momen- we write tum than Ms had positive momentum, and the total momentum in this x reference M1 Vi (M1+M2)1. frame is negative.

If M1, M2, and vi are given, then the final velocity is found to be It is interesting to note that there is an inertial reference frame .- moving with constant velocity relative - v (Ml + M2) 1. to the laboratory in which the two blocks are at rest after the collision. In the completely inelastic col- In this frame tho final momentum is lision on the air track, assume that therefore zero. It will be shown by masses Ms and My have initial veloci- explicit calculation that the momentum ties vs and vp. After the collision is conserved in this frame as well. Mx and My move together with velocity Call this frame the primed frame and put its x' axis parallel to the The total momentum before the laboratory x axis along the air track. collision is Let the constant velocity of the x frame with respect to the x frame TOT m Meg Mel, be u and since the blocks move with velocity V after the collision, and after the collision -V. r)ToT (M +Mr)V Relative to the primed frame, Momentum of this isolated system is Ms and My move before the collision conserved and the conservation is ex- with velocities vs' and Vyt which are pressed in Eq. (2.17): related to vs and vp by the transformation rule from Eq. (1.11). Msys + (MR + MF)V. (2.17)

Numerical Examelp: and (2.18) .... Vz ° Vz U. In a one-dimensional collision along the x axis, let In the primed frame the total momentum of the system before the col- +2 m/sec, Mg 0.25 kg; vs lision is ProT where, and Prar Mey + Meip'

Mr 1° 0.75 kg; -177 -4 m/sec. Mirvi- +M,(vr-u)

P.. 1° MIVic MIWz (Ms Mr)171.(2.19) Find V, the final velocity. Since u V, Eq. (2.19) becomes Mez + Mop (MR + MFR 1:16 Mzirs; +11;77 (Mg + MF)V.(2.20) (0.25 kg)(2 m/sec) + (0.75 kg)(- 4m/sec) (1.00 kg)V From Eq. (2.17) it follows that 38 MATTER IN MOTION

the right-hand side of Eq. (2.20) is initially moving in the +x' direction zero. at 4.5 m/sec and F is moving in the x' direction at 1.5 m/sec. 0. (2.21) This particular frame of refer- TOT ence is of more than usual interest, Equation (2.21) expresses the for this frame contains the center of total momentum of the E,F system be- mass of the system. fore the collision in the x' reference frame. The Eq. (2.21) was calculated from the knowledge (1) that momentum 2.7 THE CENTER-OF-MASS CONCEPT is conserved in the unprimed x frame, and (2) that the velocities of E and In Chapter 1 the law of inertia F in the primed frame could be ex- was, stated for complicated systems as pressed in terms of their velocities a law describing the motion of the in the unprimed frame in Eq. (2.18). center-of-mass point. In an isolated The result of the calculation is that system, it is this point which moves the total momentum of the system in with constant velocity during the com- the primed frame x' before the colli- plicated motions of the system's sion is found to be zero, Eq. (2.21). parts. In the last section it was We know that 14aris zero after the stated that in the inertial reference collision since the blocks are then frame that contains the center of at rest in the primed frame. This mass, the total momentum of the sys- shows that the momentum is conserved tem is zero. Even though this is one in the primed frame as well as in the of many inertial frames, the fact that unprimed frame. zero is a rather unique number leads Lot us rework the numerical ex- one to suspect that this reference ample for blocks E and F in the primed frame is particularly simple for tho frame. The x' axis moves relative to description of motion, and it is! the x axis with a velocity The center-of-mass concept is not a completely unfamiliar one. We use it V 2.5 m/sec. commonly in what seem to be "natural" Then descriptions of the internal motions vs' vs u or motions within a system. A few examples will recall the common distinc- vim' +2.0 m/sec (-2.5 m/sec) tion between motion of an object as +4.5 m/sec, a whole and the internal motions of an object. When we see a bird flying and we say that the bird "flaps his wings," but that is looking at the wing motion from the bird's viewpoint. The wing-flapping motion is an internal yr' 4 m/sec (-2.5 m/sec) motion, a motion within the (bird) sys- Vy 1.5 m/sec. tem. As seen from the ground, the motion of a wing describes a wavy line In this reference, frame E is as shown in Fig. 2.13.

Fig. 2.13An idealized bird in flight. solid line marks the path of a white The dotted line marks the motion of the feather at the bird's wing tip. center of mass point of the bird and the THE LAW OV MOMENTUM CONSERVATION 39

When we think of a flapping wing we think of up-down motion of the wing, and the up-down motion is the wing motion in the reference frame of the bird. It is the motion in the center of mass reference frame. As a helicopter ascends, we think of the whole helicopter rising with its great whirling propeller spinning above it. We tend to think of the propeller blades spinning relative to the body of the helicopter. Actually the path of a point on the tip of that great propeller in the air is a helix, or spilt', as shown in Fig. 2.14. In the study of mechanics and in the study of the conseyvation laws it is often useful ana illumieating to separate internal motion from motion of the object as a whole. The most useful point of reference of a system Fig. 2.14 A helicopter in vertical take- about whicn to describe the motion off. The dotted line narks the path of the within that system is the center-of- center of mass of the helicopter and the mass point. solid line marks the path of one tip of one of the propeller blades.

2.7.1The Center-of-Mass Point. center of mass of a system or object,

and let ri g We would expect that this point T2 T3 ri re be the position vectors of each of would move along with the system and the N elements of mass which comprise be somewhere in the "middle" of it. the system. Let mi, m2, m3, . . . One might first think that the point ml, . . s be the masses of the re- could he located at the average of spective mass elements. positions of all of the parts of the Then by the definition, system. This thought is partly right. In averaging the positions of each part of the system, the parts with mlr1 i 1112T2 +..+ iri + + mars re "" more mass are counted more. For this 1111 +m2 1111 kind of averaging, the average position is called the center-of-mass Or point. It may not be at the geometric center of the system. If some parts have greater mass, the center-of-mass (2.22) re will be shifted toward the massive parts away from the geometric center. The formal definition of the location of the center-of-mass point can be specified in relation to any where N is the total nusoer of mass convenient coordinate system. The posi- elements in the system. tion of the center of mass is the weighted average of the positions of (The symbol, E, is shorthand for all the elements of mass composing "the sum of." Each term of the sumr that system, with the Mass of each nation is represented by the sub- element used as a weighing factoi. script "i," where "i" is a running Let re be the position of the index covering a range of values 40 NATTER IN NOTION

position, think of an object shaped like a dumbbell made of two homogeneous metal spheres, A and B, having masses, MA and Ns, respectively, ;:on- A nected by a thin plastic rod. Assume

CI that the mass of the plastic rod can be neglected for the purposes of this discussion. The system is shown in x axis Fig. 2.15. X A XC xs There are two mass elements in Fig. 2.15 Two metal spheres A and B con- this system, MA and Ms, and their po- nected by a thin plastic rod. The x axis is sitions are identified as the posi- parallel to the rod. tions of the centers of the spheres. from one (1) through N. The range of (This means that MA and Ms are as- values of "i" over which the sum is sumed to be point masses located at made is indicated by the subscript the center of the spheres. This as- "il" below E and the stiperscript sumption is justified later in this

"N" above Z.) . section.) If MA > Ms, then the center-of- The denominator of Eq. (2.22) mass point will be located closer to equals the total mass of the system. A than to B. Let the center-to-center Equation (2.22) is then distance between A and B be represented by L. This is a one-dimensional system so that, for convenience, one (2.23) coordinate axis (x) is established parallel to the, rod connecting A and where B. Then N V " Esti. 1.4 xa -x1 =L,

and the center-of-mass coordinate xc Before studying some examples of is- the application of this definition, it will be useful to have Eq. (2.23) expressed in terns of individual x, y,. MAxA + M321 XGa° and z coordinates of Cartesian co- ONA + Ms) emdinate system. Thest are Suppose M. 2Mis; then

2MIXA MIXX x c (211, + MA)

2 1 N xC --xA3 +3 . 22m4Y1 (2.24) 1-1 Since Ye -

then xc x + L/3.

INI a The center-of-mass point, C, is located between A and B and one third As an introductory specific ex- of the spacing, L, away from A. Tho ample, in obtaining the center-of-mass ratio of tho distancos from C to B and THE LAW OF MOMENTUM CONSERVATION 41

A is two tono and this, clearly, re- fleets the mass ratio.

XA Xc MA 2. xc - xA Ms 1

In general, one can find a coordinate system that will have its origin at the center of mass even though the location of that point may not Fig. 2.16The origin (0') of the x' axis yet be known. This is done by requir- is established at C. With MA 2. 2N1, ing that rc equal zero in this coor- Ks/ +2L/3 and xA' m. 113 where L is the dinate system. In the example just con- center-to-center spacing of A and B. sidered, let x' be the coordinate axis which has its origin at C. (2.23). If the system has some ele- Then ment or elements of symmetry, the symmetry can be used to help locate Max A' + the center of mass. If the system has xc' (2.25) (MA + MA) a plane of symmetry, the center of mass will lie in the plane. Now, since we have set xe - 0, For example, a tennis racket is and if we choose the system shown in symmetric about two planes that are Fig. 2.16 with MA - 2MA, Eq. (2.25) parallel to the handle and perpendicu- becomes lar to one another. Since the center of mass must be on both of these 0 2x A' + xst. (2.26) planes, it is on the line of intersec- tion of these planes. The location is The origin of the x' axis must be shown in Fig. 2.17. located in such a way that Eq. (2.26) A frying pan has only one plane is obeyed. Also, since the spacing be- of symmetry. Can you describe it? tween'B and A is equal L, The symmetry of the water molecule can be used to good advantage in locating its center-of-mass point. Xst XAt e (2.27) The water molecule is approximat.ad as It follows from Eqs. (2.26) and (2.27) three points in Fig. 2.18. Its center that, of mass lies along the bisector of the 105' angle shownas a dotted line in XAt - -L/3 Ffg. 2.18. Prove to yourself that the center xat +2L/3. of mass is located on the bisector of the angle MOH in Fig. 2.18 and at a The x' axis with points xe and xs distance OC - (R/9) cos 52.5° where marked is shown in Fig. 2.16. The thin R is the 0- H spacing in the mole- plastic mod connecting A and B in our cule. example is actizslly unnecessary, for Sometimes there are lumps or if it were removed the center-of-mars groupings of matter in the system point would still be loted at C. It that are more or less distinguishable is not necessary that there actually from one another. All of the mass ele- be some material at the mater-of-mass Lents of such a group can be repre- point. Its position is elc.7-rly speci- sented by a point mass, equal to the fied without a marker being there. group mass, located at the center-of- The location of the center of mass point in the group. We did this mass of more complicated objects is in locating t'ae center of mass of thu determined by application of Eq. dumbbell- shaped object in an earlier 42 NATTER IN NOTION

Fig. 2.17The xy and xz planes are the mass element, ml, is shown. The use of the mutually perpendicular planes of symmetry symmetry has established the y and z coor- of a tennis racket. The line of intersec- dinates of C and only the x component is tion containing C, the center of mass left to be determined. point, is taken as the x axis. A typical

Fig. 2.19, which is a first approxima- discussion. Each sphere, M and Ma, tion to a hockey stick. was represented as a point mass lo- The centers of mass of the handle cated at the center of each sphere. A and the blade B are at their respec-A This trick is especially useful when tive geometric centers. To find the the ccmposite parts of the system have center of mass of the hockey stick, easily determined centers of mass, but replace the handle with a point mass the whole system may not be very MA located at its center of mass CA, symmetric. and replace the blade with a point Suppose we want to find the cen- mass 116 located at its center of mass ter of mass of the object shown in Ca. The center of mass of the whole hockey stick is then located by find- H ing the center of mass of the MA and Ms point masses. This procedure is exact and the formal definition of center of mass

MEM. =MD

Cs Fig. 2.18Schematic representation of a Fig. 2.19A hockey stick without rounded water molecule. Point C is the center-of- edges and corners. The center of mass of mass point. The mass of the oxygen atom, 0, part A is located at CA and the center of is 16 times the mese of one of the hydrogen mass of part B is located at Cs. The center atoms, N. The ON distance is R, and of mass of the hockey stick is located at OC (1MR cos 52.5'. point C on the line connecting CA and Cs. THE LAW OF MOMENTUM CONSERVATION 43

can be used to demonstrate its validity. The center-of-mass position is defined by Eq. (2.23),

Mrc

Divide the summation into two parts involving in one only mass elements of A and in the other only mass elements of B. Then Eq. (2.23) can be written as

Mrc - Emirs +Emiri A B Fig. 2.20 In (a) of this figure, the photograph of subsequent positions of thedumb- where the A and B mean "over part A bell is made from a fixed camera while the only" and "over part B only" respec- dumbbell moves past it, rotating "end over tively. The term f mire is just the end" as it goes. Each set of position indi- cators on the picture provides asimulta- summation we would have if we were neous location of both massesand the cen- finding the center of mass of part A ter of mass. only. Or, we can write In (b), the camera moves in thereference frame of the center of mass while thedumb- in (a). Mar* EKiri (2.28) bell repeats the motion shown A (Photos courtesy Film Studio, Educational and Services, Incorporated) gars 2:miro (2.29) erty is that the internal motionsof B the system do not contribute to the total linear momentum of the system. where and re are position vectors under- of the center-of-mass points of A and These characteristics can be stood from a description of the cen- B, respectively. ter-of-mass motion of the system. Each individual summation can be Even though some or all of theele- replaced by the product of a point ments of mass in the system may he mass and the position vector of its moving, the instantaneous positionof center of mass. the center-of-mass point can be established from the instantaneous lo- Finally, cations of the mass elements. Figure 2.20 gives an excellent example of lirc Mari + Nara. a* this. Fig. 2.20 is a stroboscopic The argument is readily extended ("time history"), photograph of the mo- to systems with more than two parts. tion of a dumbbell-shaped objectsimi- lar to the one discussed above.One 2.7.2Center of Mass and the Motion mass, indicated by an opencircle (o) of the System. has twice the mass of theother, shown A useful property of the motion as a cross (x) inthe photograph. The of complex systems is that the total center of mass is located at a dis- momentum of the system equals the tance equal to one third of the two product of the total mass of the sys- mass center-to-centerspacing away tem and the velocity of the center-of- from the larger mass. A closedcircle mass point. A corollaryto this prop- indicates the center-of-mass position. Marna IN NoTIoN

'A;

Fig. 2.21aThis figure shows the initial Fig. 2.21b Instantaneous position of dumb- instantaneous position of the dumbbell- bell shownat a time at later than Fig. shaped object in the (0) reference frame. 2.21a. Thedisplacements of MI, 112, and lass 1, equals 214. The center-of-mass point C inthe (0) reference frame are in- point is labeled C. dicated byvectors Ari, Art, and Arc, respectively The motion in Fig. 2.20a looks a bit complicated, but the suncessive There is an appealing simplicity positions of the dumbbell-shaped ob- in the separation of motion into ject can be located easily. For every translation and rotation for this sys- instantaneous position of the center tem. The rotational mot-on is the in- of mass, the simultaneous positions of ternal motion of this object. The 112 and 12 can be located on a straight point at which the division between line passing through the center of the two motions is made is the center- mass. For each subsequent position of. of-mass point in the system. If we the center of mass, the angular posi- look again at the defining equation tion of this line is changed. It ap- for the center-of-mass position rc and pears from Fig. 2.20a that the object see how rc varies with time, we can is rotating in some way as it moves see how this separation is conven- along. The rotational motion is more iently made. apparent in Fig. 2.20b. Let rc be the instantaneous posi- The first thing that strikes tion vector to the center-of-mass one's eye in Fig. 2.20b is the cir- point C, and let ri and r2 be the cular motion of the x and o syrSols. simultaneous position vectors to Clearly, the dumbbell is simply rota- masses (1) and (2), respectively, of ting at constant angular velocity in the dumbbell used in Fig. 2.21. The this frame of reference. To make the location of an origin, (0), used in photograph of Fig. 2.20b, the experi- specifying these vectors, as shown in menter had to move his camera at con- Fig. 2.21 is in an inertial frame of stant velocity in order to "stop" the reference. The position of each mass center-of-mass point. The velocity element relative to the center of needed for his camera could be ob- mass C is given by vectors in the tained from Fig. 2.20a in which this center-of-mass coordinate system (0') velocity could be determined from the having its origin at point These equal spacing of the center-of-mass vectors are labeled r19 and r2° for points. masses (1) and (2) respectively. We conclude then that the motion For each mass element in the figure is one of rotation about the center-of-mass point superimposed r1 r (2.30)( upon translation of the center of and mass. ra re + 2.21. (2.31) ThE LAW OF MOMENTUM CONSERVATION 45

rc' 0.

(2.35) [ r 1 +M2r2'] 0,

and Eq. (2.34) becomes

Mira + 112r2 - (N1 + Mdre. (2.36)

In a short time, At, theposition vectors will, in general, changeby a shown Fig. 2.21cTwo instantaneous positions of small amount. These changes are the dumbbell are shown in the(0') refer- in Fig. 2.21. Both;1 and 71' change ence frame. The origin ofthe (0') frame is during the interval At, but thechange The located on the center-of-mass point C. in does not equal the change inrat "internal displacements" measured relative since these vectors are in twodiffer- and Ar2'. to (0') are indicated bya., ent reference frames in relative mo- Note thattwitandAir,'are antiparallel in this two -mass system. What is theratio of tion. the magnitude of Ars' to the magnitudeto During the time interval At, ers'? 2.1 becomes671 + A;1),

Multiplying Eq. (2.30) and Eq. r2 becomes(i.2+A;;), (2.31) by M1 and M2, respectively, these equations become rc becomesGC+A-4).

M1r1 - Mrc + (2.32) Equation (2.36) is rewritten for this later time as and M2rc + M2r2- '. (2.33) M2r2 1111 +4111:1 M2 G2 + 4:17:2 igc). (2.37) Thesumof Fogs. (2.32) and (2.33) = (M1 + M2) )(re + is By subtracting Eq. (2.36) fromEq. MAri + 112r2 (MI + (2.37); we find

+[Ma +M2r2'], (2.34) MIA;1 + M1Ar2 -(N1+M2)ac. (2.38) and from this sum we will be ableto Divide Eq. (2.38) by At to obtain find the total momentum,(M171 + Nava). Before doing so, it should be AlF2 noted that the sum of terms inthe mgit mr+2 At (m1 bracket I] of Eq. (2.34) is zero. This must be true, because the bracket or, in the limit asAt C, sum is the weighted(by mass) sum of the positions of the elementsof'mass Ara /At - va Are /At and composing this system where these positions (the primedvectors) are vc. specified relative to C. From the 111 + M2r2 + M2)"..vc definition of center of mass, it must be true that Therefore, (2.39) 111r1 + M2r2 (ME + 112);ct Pim' (M1 +M2);c,

Since the origin of the(0') coordi- which is our expression for the total nate system is located at C, the cen- momentum of tile two-particle(mass) system shown in Fig. (2.21). ter ofmass, 46 MATTER IN NOTION !MI

Equation (2.39) follows from Eq. mass point. You recall from previous (2.36), and Eq. (2.39) is valid for study of motion that the displacement any reference frame used to descrilv of a point, or the velocity of a the position and motion of the center- point, is the same in all reference of-mass point. frames which differ from one another In the (0') reference frame, the by only a fixed constant displacement velocity of the center of mass is between their origi:is. The set of zero, reference frames in which the center of mass is at rest is often referred to as the set of "zero momentum" refer- vc 0, ence frames because the total momen- so that in this frame tum of the system is zero in each of these frames. 166- o. Several interesting things are expressed by Eq. (2.39). The total momentum in the center- of-mass frame is zero. During the time (1) The total momentum of the interval At considered above the dis- system is independent of the internal placements of M1 and M2 are All and motions of the system. Ar21 in the center-of-mass frame. By the phrase "internal motion" These displacements are shown in Fig. we mean motion relative to the frame 2.21c. The velocities of M1 and M2 in of reference in which the center of the (0') frame are, in the limit of mass is at rest. This frame is the At 0, (0') frame in our discussion. Internal motion involves time rate of change of r1' and r2' and these do not appear v1' 6;11/At in Eq. (2.39), because of the special ;2' Ai21/At. property of the center of .pass, Eq. (2.35). The displacements Ail anda2t There is internal motion in this are oppositely directed and they obey, system because the individual 0r2 ' the relation and Die' displacement vectors are not zero in the (0') reference frame. The internal motions are always of such a mi Arei 1126.72t9 combinaticn that the internal momenta because these displacements do not Mori' and M2v2' exactly cancel and displace the center of mass in the make no contribution to the total mo- (0') frame. Therefore,- mentum of the system. When we say "total momentum" we mean the momentum of the system as a v1 m2v2 . whole, and it may then not be surpris- The momenta of the two objects ing that the internal-motion veloci- exactly cancel one another in the (0') ties do not appear in Eq. (2.39). That they dr, not appear is the case only frame and P4.01. ... 0. In the above discussion the (0') when toe internal motion is defired 1.'rame was defined with its origin relative to the center-of-mass (or, located at the center-of-mass point. zero-momentum) frame of reference. However, all of the above statements concerning displacements and veloci- (2) The total momentum of the ties in the (0') frame are also valid system equals the product of the total in other reference frames in which the mass and the velocity of the centerof center of mass is at rest, vet 0, mass. even though the origins of suchframes While the internal motions of the may not be located on thecenter-of- system may be involved and complicated, THE LAW OF MOMENTUM CONSERVATION 47

the motion of the center-of-mass point Divide Eq. (2.41) by At. is representative of the motion of the system as a whole. The observations (1) and (2) above are based upon 11(aciAt) Emi(A1/At). mathematics and they express, with better precision, our earlier division or separation of motion into motion of In the limit as At . 0, the system as a whole and motion within the system in the discussion (Arc /At) Vc of Fig. 2.13 and Fig. 2.14. (eri/At) ;1. (3) If the system is isolated Then from the world outside itself, the center-of-mass point obeys the law of 1Gc - Emivi inertia. If the system is isolated, then the total momentum of the system is conserved. Since the total mass does Mve EPit not change, the velocity of the cen- 1-1 ter of mass must be constant according to Eq. (2.39). Whatever compli- where pi - mvi is the momentum of the cated internal motions the system has, i'th mass element. its center of mass will continue to The total linear momentum is move with constant velocity. Thiel last statement is the law of inertia, and N it is thus contained within the law PTA- EPl. of conservation of momentum. Conclusions (1), (2), and (3) Therefore, above are general and can be derived formally and quickly from the general pm"'MVlc (2.42) definition of the center-of-mass vector of Eq. (2.23), Conclusions (1), (2), and (3) hold for Eq. (2.42).

Misc (.2.23) i -i 2.8 COLLISIONS During a time At, each value of rj changes so that Momentum is a vector quantity. The vector character of momentum may ri becomes (ri + not have been fully apparent in the and one-dimensional explosions and colli- r becomes (r+ ) sions discussed earlier. In systems c,. free to move in two or three dimen- Then, sions, the vector character of the conservation of momentum law is clear. N In this section we extend this Mao + Emi(ri+ bi";1). (2.40) discussion to more than one dimension, tie]. and we will also consider collisions Subtract Eq. (2.23) from Eq. in which the colliding objects re- (2.40) and obtain bound and do not stick together after they collide. The completely inelastic N collision discussed earlier is not an Mao Elai(ai) (2.41) uncommon occurrence in nature, but it 1-1 is a special case. Collisions in which MATTga IM MOTION 48 11.1=MIN

during the col- the collision is not enough time objects move apart after world" to nature. In the study lision for the "outside are abundant in the system by processes, change the motion of of physical or biological in- interaction of one very much.Look at one particle we often study the and examine the interac- volved in the collision particle with another or during the system, with its change in motion tion of one object, or there is a interactions collision impact. If another. Often these world, We say that a coupling with the outside occur as collisions. particle will collision occurs when twoobjects, the momentum of this be changed partly bythe interac- which were initiallyseparated, come tions with the outsideworld as together to interactwith one another well as by theinteractions with and then, in most cases,separate the other particle(s)in the sys- again. tem during the collision.The sum An understandingcf collisions of these two changesin momentum is helpful, and there rresome things equals the resultantchange of that the laws ofphysics can say in momentum of thatparticle, Ap. general about them.One of these is Then that if the collisionoccurs in such a way thatthe participantsof the Pow collision are isolated(or temporarily a - dicou + "outside world," isolated) from the change in momentum of the system where acou equals the then the total momentum due tointrasystem interac- of participants mustbe conserved. particle(s) of that there is a tions with the other We will assume equals the before and after the system and Apo, period of time both due to the inter- during which eachof change in momentum the collision (with the "out- particles moves with a system interactions the colliding assumption of momentum. The total side world"). The constant definable used in study- before or afterthe "temporary isolation" momentum either ing much of collisionphysics can collision will equalthe vector sum be stated in termsof these momen- of the collidingparti- of the momenta tum changes as, cles. that the time We will also assume awn )1> APow duration of the actualcollision or in- the collidingparti- teraction between so thatAL, can be neglected. cles is small andnegligible. The image of thecollision is thatof a brief impact duringwhich each parti- 2.8.1A Sample Collision. less suddenly cle involved more or Consider the collisionshown in This condition changes its motion. Figs. 2.22a and2.22b. Figure 2.22a is upon the timeduration of the colli- photograph of two hard if the system a stroboscopic sion is not necessary steel spheres of unequalmass which involved is wellisolated. If the sys- move toward oneanother, collide, and tem is notisolated, we can assume separate again. that there is a"temporary isolation" drawing world Figure 2.22b is a scale of the systemfrom the outside of Fig. 2.22a. providing that made from the photograph during the collision between successive time duration isshort. The time intervals the collision positions of eachsphere (1 and 2), are equal.The displacementbetween The analysis ofsuch a collision successive positions ofeach sphere will be onlyapproximately correct; is then proportionalto the velocity often an approximationis good of the sphere. Velocitymeasurements enough. To say thatthe collision can be takendirectly from the draw,- time is short isto say thatthere THZ LAW OF MOMENTUM COMBRVATION 49

O o O 00 Qb b Qc CO d SPHERE 1 0 da

SPHERE 20 eC> 0°

CENTER-OF-. fC Of MASS POINT 90 09 hO Oh ;Cs 0 io 0 i O 0 O O Fig. 2.22b A scale drawingfrom the photograph of Fig. 2.22a. Thecenter-of-mass point of the two spheres has beendeter- mined for each instantaneousposition shown in Fig. 2.22a. The constancyof the velocity of the center-of-masspoint is a measure of the constancyof the total momentum.

most horizontal when the spheres are released. The speed of the spheres increases as they swing down, but Fig. 2.22aA multiple-flash photograph which shows equal-time intervals inthe they move at almost constant velocity collision between two spheres ofunequal in the region of the "bottom"of the masses. The mass of the largesphere is swing where the collision occurs.We 201 grams (0.201 kg) and the massof the assume that the systemconsisting small sphere is 85 grams (0.085kg). Both of these two spheres is isolatedfrom the photo- spheres enter from the top of the outside world insofar ashorizon- graph. (From PBSC Physics[D. C. Heath and tal motion is concerned, eventhough Company, 1960J.) each sphere is connected to theout- ing proportional to the displacement side world by its string. Id the re- is between successive positions ofeach gion of the collision the string almost vertical and has only a small sphere. The spheres shown in Fig.2.22a effect on the horizontal motion. The are made of hardsteel and are sus- total momentum of the system is the dis- pended from long strings, likependu- vector sum of pi and p2. In this be lum bobs. The two bobs arepulled up cussion, the subscript (i) will of and away from theirequilibrium posi- used to designate "initial" values tions and released. They arepulled momentum or velocity before thecol- (f) back so farthat,the strings are al- lision. Similarly, the subscript 50 MATTER IN MOTION

p t I p 0 0.25 0.5 0 1 2 3 4 0 0.25 0.5 v(m/sec) kg Woe kgmbec

Pit PT T m, +m?

III and final momenta of m1 and m2.(III) The Fig. 2.22cThe velocity and momentum vec- (vector) sum of the initial momenta and tors found from Fig. 2.22b.(I) .the initial have been velocity vectors of m1 and m2, v22and v22, the sum of the final momenta found separately. The value of vcobtained and the final velocity vectors vatand v212 from pToT divided by the total massis also are shown. The vector veis the velocity of shown. the center-of-mass point.(II) The initial ity expressed in Eq.(2.43) can be will be used for "final"values after checked. the collision. There is another way of express- The initial velocities are seen ing the momentum conservationin this from the Fig. 2.22 to be almostequal. collision that will be usefulin un- The exact magnitude of one ofthe ini- derstanding Newton's third lawof tial velocities is taken as the spa- mechanics when we encounter it.This cing of subsequent positions ofthe way of expressingmomentum conserva- spheres before the collisionand the velocity direction of each sphere is tion is along the line of images. The product + . 0, (2.44) of mass and velocity of each disk Cpl equals the momentum of each; the where a, and Ape are themomentum initial momenta are shown drawn to changes of particles 1 and2. scale in Fig. 2.22c, for masses The Eq. (2.44) is obtaineddi- m1 85 g, m2 .= 201 g. rectly from Eq. (2.43),' Since the two objects, 1 and 2, form an isolated system, the final (2.43) total momentum (after tice collision) PTOT(1) Pirar(f) equals the value of the initial total which is, in the formof the particle momentum. Both objects experience momentum changes due to the collision, momenta,

but the sum of their momenta remains .110 unchanged. Because momentum is a vec- Pli P21 - Pit P2 tor, both magnitude and direction of the system's total momentum must re- Then main constant. Let Pror equal the to- Poi ) - 0.(2.45) tal momentum of this system. Then (13a Gat

Since (2.46) (2.43) APl um (P1 f ) PTar(i) " PTca(t) (2.47) The vector sum of pif and P2f is and 11132 (7)2f 711 ) shown in Fig. 2.22b. By comparing their sum, puma:, in Fig. 2.22c with then, from Eqs.(2.45), (2.46), and (2.44) pw,T(1) in the samefigure, the equal- (2.47) we obtain Eq. 52 MATTIta IN MOTION

P11

a 0 b 0 0 0 0 0 0 0 0 Pie f ghi j P11°

Pee1 Fig. 2.24The collision as it appears in The magnitude of each final momentum is the center-of-nabs (0') of reference. about 12% less than the magnitude of each The pairs of letters a-a, b-b, etc., cor- initial momentum in this collision. In comr respond to these pairs of letters exactly paring the changes of momentum of each as they appear in Fig. 2.22b. This ff..gure sphere in this reference frame, found from is drawn to the same scale as Fig. 2.22b. pt' and 'pi', with the changes of momentum The vectors representing the momenta have shown in Fig. 2.23, keep in mind that there been drawn at twice the length they would is a factor of two difference in scale of have if the scale of Fig. 2.22c were used. these two figures.

in the laboratory frame as shown in the collision is not "head on," the Fig. 2.23. initial trajectories do not necessarily lie along one straight line even though they are parallel.) The 2.8.2The Sample Collision in the sum + p29 is zero both before Center-of-Mass Reference Fi awe. and after the collision. The description of collisions is The diagrams for vector addition often simpler in the center-of-mass in this reference frame are certainly reference frame. Choose the moving simpler than in the frame of Fig. frame (0') so that it moves with the 2.22b. The momentum of particle 1 is center of mass. We can place the ori- related to the momentum of particle gin of the (0') frame at the center 2 by a simple change in sign. There of mass of the two particle system. are other features of this representa- Therefore choose tion that make it attractive is understanding and analysis. Before getting U Vc, to them, let us see how to transform the description of this collision from wheivic is the velocity of the center the laboratory frame to the center-of- of mass point of our two particles in mass frame. Fig. 2.22. How is the velocity of this frame, Figure 2.24 shows how this colli- Irc, determined from a knowledge of sion looks in the center-of-mass frame piiandp"? The most direct way to The system has a total momentum equal obtain vc is to make use of the fact to zero in this reference frame. No- developed in section 2.7 that the to- tice that the momentum of particle 1 tal momentum of the system equals the is antiparallel tohe momentum of product of the total mass of the sys- particle 2 in both the initial and tem and the velocity of the center-of- final states of this! system. (Since mass polut. Then, 52 NATTER IS NOTION

oreolfs. PIII

0 0 c 41111 000oo o o Pt, 4fgbil PHs

P2os Fig. 2.24The collision as it appears in The magnitude of each final momentum is the center-of-mass (0') of reference. about 12% less than the magnitude of each The pairs of letters a-a, b-b, etc., cor- initial momentum in this collision. In com- respond to these pairs of letters exactly paring the changes of momentum of each as they appear in Fig. 2.22b. This figure sphere in this reference frame, found from is drawn to the same scale as Fig. 2.22b. pp=' and pe, with the changes of momentum The vectors representing the momenta have shown in Fig. 2.23, keep in mind that there been drawn at twice the length they would is a factor of two difference in scale of have if the scale of Fig. 2.22c were used. these two figures.

in the laboratory frame as shown in the collision is not "head on," the Fig. 2.23. initial trajectories do not necessarily lie along one straight line even though they are parallel.) The 2.8.2The Sample Collision in the sum(p1 + p21) is zero both before Center-of-Mass Reference Flame. and after the collision. The description of collisions is The diagrams for vector addition often simpler in the center-of-mass in this reference frame are certainly reference frame. Choose the moving simpler than in the frame of Fig. frame (0') so that it moves with the 2.22b. The momentum of particle 1 is center of mass. We can place the ori- related to the momentum of particle gin of the (0') frame at the center 2 by a simple change in sign. There of mass of the two particle system. are other features of this representa- Therefore choose tion that make it attractive is understanding and analysis. Before getting .110 U Vc, to them, let us see how to transform the description of this collision from where 14 is the velocity of the center the laboratory frame to the center-of- of mass point of our two particles in mass frame. Fig. 2.22. How is the velocity of this frame, Figure 2.24 shows how this colli- determined from a knowledge of sion looks in the center-of-mass frame. pitand 21? The most direct way to The system has a total momentum equal obtain ve is to make use of the fact to zero in this reference frame. No- developed in section 2.7 that the to- tice that the momentum of particle 1 tal momentum of the system equals the is antiparallel to rye momentum of product of the total mass of the sys- particle 2 in both the initial and tem and the velocity of the center-of- final states of this system. (Since mass polut. Then, 'III LAW OF SOSINTUS CONSIRVATION 53

P2. OIL 112);3 ;al + ....hi and Ai 4' 161 V (2.51) 6 + s2)° where ml and a2 are the masses of particles 1 and 2, respectively. Equation (2.51) could also be written in terms of the final particle momenta of the system since ploy is constant.

Pit P2f v (2.52. 0 (2111 +

.A1. When Tic is obtained from the to- P2f tal momentum of the system as in Eq. (2.51.) or Eq. (2.52), it is often referred to as the velocity of the "center -of- momentum" or the "zero- momentum" reference frame as well as "the center-of-mass" reference frame. (-81 c) We will use the phrases "center-of- mass frame," "center-of-momentum P2f frame," and "zero-momentum frame" in- terchangeably. (The center-of-momentum concept is especially useful for example if one of the particles is a photon which has momentum even though Fig. 2.25 Scale drawings representingthe determination of the momentum of each parti- its rest mass is zero.) frame We already have the total momen- cle in the zero - momentum reference according tc Sq. (2.50) of the text. The tum of particles 1 and 2 defined in values of the momenta in the laboratory Fig. 2.22c. When pTer is divided by frame and the value of ;sic are taken from the total mass, Oh f 212), the result Fig. 2.22c. A convenient way of finding the is the center-of-mass vector, veloc- length of the lines representing the vec- ity vc, shown in Fig. 2.22c(111). tors my& is to draw them as theappropri- For convenience, the instanta- ate fractional length of vector hero!Fig. pmr/(mi + mi)9 mlvc neous position of thecenter-of-mass 2.22c. Since vet + mid and mere equals point, C, of particles 1 and 2 have equals immr(mi/mi proy(a2/m1 + mi). The scale, of this drawing been indicated on Fig. 2.22b. The dis- is the same as that of Pig. 2.24. placement between subsequent positions of C in Fig. 2.22b is a measure of irsro which can readily be compared with By applying the momentum trans- ire in Fig. 2.22c(1). Note also that formation rule of Eq. (2.53) to each the spacings between subsequent posi- particle, the momentum of each parti- tions of C are equal, or,is is con- cle in the (0') frame is foundfrom stant. each particle's momentum in the(0) The momentum of a particle in the frame. The application of the trans- "serormomentum" frame is obtained formation rule is represented by the Iron (2.49) with u set equal to vectors, drawn to scale, in Fig.2.25. vs: The direction of the vectors drawn in Pig. 2.25 are identical to

- 31/476- (2.53) the directions of the same vectors 34 MATTIS IN LOTION

cally. Collisions in a plane are ml .71/ 4 defined by two-dimensional vectors. b Two numbers are required to define m2 b = IMPACT PARAMETER v2I / a two-dimensional vector:its direction (an angle) and its magnitude. Fig. 2.26 The trajectories of two particles, mi and ma, are shown by dotted lines If, for example, both final momenta as they would move if the particles were are unknown, there are fourunknown to continue their initial motions without numbers, two for each momentum deflecting one another. Tho "impact parame- vector. The law of conservation of ter," b, is shown. momentum provides one vector equation that connects these four un- as they appeared ;n Fig. 2.k2. The knowns with the total momentum vectors pia10 Nil, Nil and p2f' ap- which is assumed known. Since this pear in Fig, 2.24 as they are deter- is a vector equation, it must place mined in Fig. 2.25. two conditions on our unknowns. In The reader can find the vector algebraic form, the vector equation representing the change in momentum becomes two algebraic equations; of particle 1 and of particle 2 in for example, as one equation for the 0' frame from the defining rela- the x components and one for the v tions, component:. These two equations dpi' lm P11 Pli reduce the number of unknowns from AZ n n four to two. Even a knowledge of -r2" r2f x'2i what happens to the kinetic ener- Draw or sketch these vector diagrams gies of the particles (discussed showing al' and4;2'. Does in Chapter 3) would add only one Ap2°, that is, are the momen- more equation and thus reduce the number of unknowns to one. A unique tum changes equal and opposite in this reference frame? CompareAii° with Ail solution cannot be found from the of Fig. 2.23. Compare fps' withAi; of conservation laws alone. Fig. 2.23. Let us say that the momentum of The missing information has to one particle is known in the zero- do with the details 'of the collision. momentum frame after the collision. In principle, the final.momenta can We can quickly find the momentum of be predicted fully if (a) the separa- the other particle at that same time tion between the two initial trajecto- by simply multiplying the first parti- ries and (b) the nature of theinter- cle's momentum by minus one. action between particles are known. Notice that the law of conservation of momentum does not provide enough information to allow the unique Sometimes the two objects are prediction of both momenta after the spherizal; i.e., the interaction collision even though the initial mo- between them depends only on the menta of both particles may be known. distance separating them and not There are many (an iL:inite num- upon the direction of one from ber of) pairs of vectors representing another. Then all that one needs to i;2 and Tqf that add up to zero total know about their initial trajecto- momentum for the system. The question ries is the "impact parameter" il- of which of these pairs is the cor- lustrated in Fig. 2.26. The impact rect one appearing in Fig. 2.24 can- parameter is the distance between not be answered on the basis of mo- the two trajectories that would be mentum conservation alone. obtained if there were no interaction. It would be the distance of This limitation on predictabil- closest approach between centers ity oan be demonstrated matbemati- if the two objects continued undo- 1'

MX LAW OF MOMENTUM.0.11 CONSERVATION 55

fleeted. Head-on collisions have Look at each of these photographs, zero impact parameter. with these.questions in mind. We shall not be concerned here (1) In the (a) photographs, can with prediction of trajectories of you locate the center of mass for each all particles in a collision. subsequent position of the dry-ice pucks? (2) Can you predict what the (b) photograph should look like from the 2.8.3Examples of Momentum Conserva- (a) photograph and a knowledge of the tion Described in the Labora- mass ratio? tory and in the Zero-Momentum (3) Is the velocity of the center Frames of Reference. of mass in (a) constant? Should it be? Each of the pairs of photographs (4) Is the momentum of this 2- used in these examples was made by two particle system conserved? cameras simultaneously. One camera was (5) Is the change of momentum of fixed in the laboratory and the photo- one particle equal and opposite to graphs made by it are labeled (a). The the change of momentum of the other other camera was mounted in a moving particle in each photograph? frame which traveled at the same velocity as the center of mass of the Example 1. Collision Between TwoOb- system being photographed. The photo- j of Equal Mass. graphs made by this camera are labeled (b). A similar camera set-up was used The collision between two mag- in making Fig. 2.20. netic dry-ice pucks of equal mass is The colliding objects used in mak- shown in Fig. 2.2 ?. In this collision ing these photographs ara d:y-ice disks one dry-ice puck (1) is initially in or pucks similar to those tescribed in motion and the second (2) is initially Section 1 of Chapter 1 with just one at rest in the laboratory frame of exception. Each of the dry-ice pucks reference. The Fig. 2.27a is from the used here contains a nine-inch-long laboratory camera and the Fig. 2.27b bar magnet mounted vertically along shows the collision in the zero- its axis. The magnets are mounted so momentum frame. that their "north" ends are down and In Fig. 2.27a, mass (1) initially their "south" ends are up. When the approaches mass (2) from the left. dry-ice pucks are near one another During the collision (1) forces (2) on the horizontal surface on which away and (1) reacts by beingdeflected they glide, the magnets repel one an- from its original trajectory. After other. Their "interaction" is a mag- the collision they both move to the netic one. These dry-ice pucks collide right while (1) also moves toward the without actually "touching" each other. lower edge of the picture and (2) The interaction between the magnets moves toward the upper edge. causes the momentum of each dry-lee In Fig. 2.27b, in the center-of- puck to change during a collision or mass reference frame, the masses move an explosion." toward one another with (1) coming The center of mass of the dry-ice from the left and (2) coming from the pucks is marked; one is marked by a right. After the collision, (1) is cross (x) and the other is marked by moving toward the bottou of the photo- an open circle (o). The collisions can graph and (2) is moving toward the be considered to be collisions be- top. tween point particles located at the Figure 2.28a and b show the vec- marked positions. The photographs tors representing the momenta of (1) show instantaneous positions of the and (2) of Fie. 2.27a and b. Since dry-ice pucks taken with successive (2) is initially at rest in the labora- light flashes equally spaced in time. tory frame its momentum in that frame MATTER INMfil MOTION

Fig. 2.27bA multiflush photograph of the Fig. 2.27aMultiflash photograph of colli- same collision shown in Fig. 2.27a. This sion between two magnetic drrice pucks of photograph was made simultaneously with equal mass. One mass, marked (x) is ini- Fig. 2.27a by a camera moving in the zero- tially at rest and the other, (o) is ini- momentum (center-of-mass) frame of refer- tially in motion from the left. The photo- ence. There is one-to-one correspondence be- graph was taken in the laboratory frame of tween the positions of the pucks In this reference. (Courtesy Film Studio, Educa- figure and in Fig. 2.27a. (Courtesy Film tional Services, Incorporated.) Studio, Educational Services, Incorporated.) is zero. Let Pia po in the lab frame. The initial and final momenta of Then the total momentum in this frame particles 1 and 2 are shown in Fig. is 2.28b in the zero-momentum frame. The total momentum can be seen to be zero PTOT " Pi i + P2 1 in this frame. Prove to yourself that the value of pli and 1;4, should be PIMP PO (50/2) and (-40/2), respectively, in this reference frame. The vectors plf and pot shown in Fig. 2.28a are drawn from velocity Example 2. Collision Between Two Ob- measurements made from Fig. 2.27a. jects of Unequal Mass Mass Ratio Two The vector sum of pif and prs equals p0 withina two percent accuracy and to One. momentum is conserved. The collision becween two magni,tie dry-ice pucks of unequal mass is shown in Fig. 2.29. In Fig. 2.29a 0 1 2 3 4 5 0 1 2 3 4 5 i.i.1.1.i one dry-ice puck (1), mass MI, in I 1 1 approaches a second puck (2)

p11 =PO =1.=1 44 2f 17111 P211 Fig. 2.28aThe momentum vectors representing initial and final momenta of pucks (o), pi, and (x); ps, of the collision ofFig. 2.27a. The wAentum of each puck is shown Fig. 2.28b The momentum vectors of the for the laberaicr7 frame of reference. Com- pucks of Fig. 2.27b before and after the parison of the resultant pit + pot and pi' collision in the zero - momentum frame. The is used to confirm the conservation of mo- sum of the momenta is zero both before and mentum. after the collision. TO LAW OF VONSMTUM COMMATION 37

"!:, ,f( J*

.1111 photograph of the Fig. 2.29a Multiflashphotograph of a col- Fig. 2.29b A multiflash lision of two magnetic dry-icepucks of un- same collisionshown in Fig. 2.29a. This equal mass. One mass, marked(x) is ini- photograph was made simultaneouslywith tially at rest. The otherdry-ice puck, (o), the one used in Fig.2.29a by a camera mov- is initially in motion from theleft. The ing in the zero - momentum(center-of-mass) (0) puck has twice the mass cfthe (x) frame of reference. Each puckposition in puck position puck. The photograph was takenin the labor- this figure corresponds to a initial motion is atory frame of reference.(Courtesy Film shown in Fig. 2.29a. The (Courtesy Film Studio, Educational Services,Incorporated.) from the left and the right. Studio, Educational Services,Incorporated.) which is at rest. The massof (1), M, equals twice the massof (2), M2. so that The general directionsof motion pToT 2po of the magnetic dry-icepucks in Fig. 2.29 are similar to thedirections of The vector's pitand p2f shown in motion in Fig. 2.27. The exactdiffer- Fig. 2.30a are drawnfrom velocf'.y ences in theirtrajectories are due measurements taken fromFig. 2.29a, ratio. to the different massratio of the and a knowledge of the mass pucks in the Fig. 2.29. Compare the vector sum(iiIf + p2f) Figure 2.30a and b showthe vec- with 2p0 to see that themomentum is tors representing themomenta of (1) coraerved. Note that neitherthe vec- and (2) of Fig. 2.29aand b, respec- tor sum of vela -;ties nor analgebraic tively. Since (2) isinitially at rest sum of momenta isconserved. Only the in tilt:, lab frame(Fig. 2.29a) the tom vector sum of momentum isconserved. tal momentum of the systemequals the Momentum is a vectorquantity! initial momentum of (2), 1)21.Let It Fig. 2.30b theinitial and fin.:; momenta of (1) and(2) are piigm 2p0 sh,iln for the zero-momentumframe. Prove to youraelf that theinitial 8 10 02 4 6 8 10 0,;.1A. values, and p2it, in this refer- I I ence.frame should be(00/3) and

I I (-2;0/3), respectively. 1 I

I I PTOT f I

Pil=21); Fif66, Pp Ps' Fig. 2.30a The momentum vectorsrepresent- ing the initial and finalmomenta of pucks (a), pi and (x), ps, of thecollision in The momentum vectors of the Fig. 2.29a. The momentum ofeach puck is Fig. 2.30b after the of reference. pucks of Fig. 2.29b before and shown in the laboratory.frame frame. The The resultant of plf + p,1 canbe compared collision in the zero-momentum sum of the momenta is zeroboth before and with p11 to confirm theconservation of after the collision. momentum. MATTER IN MOTION

Fig. 2.31aMultiflash photograph of an Fig. 2.31bA multiflash photograph of the explosion of two equal mass magnetic dry same explosion shown in Fig.2.31a. This ice pucks. The "explosion" istriggered by photograph was taken simultaneously with the breaking of a solder wireconnection the one used in Fig. 2.31a by a camera between the two pucks. Thr, solderwire is moving in the zero-momentum (center-of- broken by melting it in a match flame.The mass) frame of reference. In this frame of wiggly white line is due to the match reference the pucks are initially at rest flame. The coupled pair are initially mov- and they move away from each other along ing toward the right. The motionis shown a straight lineafter the explosion. (Cour- In- in the laboratory frameof reference. (Cour- tesy Film Studio, Educational services, tesy Film Studio, EducationalServices, In- corporated.) corporated.) trigger for the explosion is a negli- Example 3. Moving Explosion of Two gible perturbation on the system and Objects Having Equal Mass. the system can be considered to be The previous moving explosion con- isolated. sidered earlier was for one dimen- The explosion of two equal mass sional motion. In this example, the pucks is shown in Fig. 2.31a and b. line along which the two-particle ex- The wiggly white line in the photo- plosion occurs is not parallel to the graphs is due to motion of the match initial motion of the system in the flame used. In Fig. 2.31a, the coupled laboratory reference frame. pair of dry-ice pucks initially move The explosion is caused by the toward the right. In the center-of- repulsive force between two magnetic mass referenceframe, shown in Fig. dry-ice pucks. Initially the two ob- 2.31b, the dry-ice pucks are ini0.ally jects are held together by a thin at rest and move away from one another solder Acta]. wire. This metal melts along a straight line after the ex- at a relatively low temperatureand plosion. it will melt in the flame from a Let the initial momentum of each match. In performing the experiment, dry-ice puck in the pair equal po. the two magnetic pucks are firsttied The total momentum of this systemin together by a loop of solder wire. the laboratory frame is then The pair is given its initialvelocity 111. along the table, and then theexperi- Purr." Pli P21 to menter brings a flame from a match Ptarga 2Po. the wire solder. When the soldermelts, and the "explosion" occurs in thatthe two Fig. 2.32a shows the initial pucks of magnetic pucks fora,:' each other apart. final momenta of the dry-ice measurements The me of the match flame asthe Fig. 2.31a as drawn from THE LAW OF MOMENTUM CONSERVATION 59

1 1 2 3 4 5 6 7 01.1.1.1.1.t.INI2 3 4 5 6 7 01.1.1.1.1.1.1.1 I I I I I I FIf' I I I 1 1 I I I PTOTf i Fit +itte'2P-O PI, P2f Fig. 2.32bThe momentum vectors of the Fig. 2.32aThe momentum vectors represent- pucks of Fig. 2.31b after the explosion in ing the initial and final momenta of the the zero-momentum frame of reference. The pucks (o), pl, and (x), p2, in the moving pucks are at rest before the explosion in explosion of Fig. 2.31a. The momentum of this reference frame. each puck is shown for the laboratory reference frame. netic dry-ice pucks move in from the left and explode apart near the center of velocity taken from Fig. 2.31. The of the photograph. Clearly the more vector sum of final momenta shows the massive dry-ice puck acquires a lower conservation of momentum at a value of velocity perpendicular to the original 2p0, to within experimental accuracy. direction of motion than does the less Since the initial momentum of massive one. We let ml equal 2m2 and each dry-ice puck is zero in the zero draw the momentum vectors for this ex- momentum frame, only the final momenta plosion from velocity measurements on the vector appear in Fig. 2.32b. That the photographs. Figure 2.33b shows sum of the final momenta of the two the explosion in the center-of-mass pucks foil=' and 1/2f1 is zero can be frame of reference. seen in Fig 2.32b. In Fig. 2.34a, the initial and final momenta are drawn to scale for Example 4. Moving Explosion of Two the explosion in the laboratory frame. Objects Having Unequal Mass, Mass Ratio Two to'Oce. Figure 2.33a shows the two meg-

.4741014:

Fig. 2.33bA multiflash photograph of the same explosion shown in Fig.2.33a. This photograph was taken simultanecusly with Fig. 2.33a Nultiflash photograph of two dry-ice pucks of unequal mass. The mass the one used in Fig. 2.33a by a camera mov- ratio is two. The coupled pair ofpucks is ing in the zero-momentum (center-of-mass) initially moving toward the right inthe frame of reference. In this frame of refer- and photograph. The puck marked (o) hattwice ence the pucks are initially at rent the mass of the puck marked (x).The motion they move away from each other after tt.e explosion in a straight line. (Courtesy is shown in the laboratory frameof reference. (Courtesy Film Studio,Educational Film Studio, Educational Services, Incorpo- Services, Incorporated.) rated.) SATTER IN NOTION

0 1 2 34 5 6 7 89 1011 I .6 I .1 I I I. 1 I.

)01

F s++ -1;21 3 ic

0 1 2 61 4 5 67 8 9 10 11 .I .1.1.1.1.1.1.1.1.1.1

Fig. 2.34bThe momentum vectors of the pucks of Fig. 2.33b after the explosion in the zero- momentum frame of reference. The pucks are at rest before the explosion in this reference frame.

Fig. 2.34aThe momentum vectors represent- as a vector quantity. The rules we ing the initial and final momenta of the follow in finding sums and differences pucks (o) and (x) in the moving explosion are the rules of vector algebra. We of Fig. 2.33a. Te momentum vectors are see that physical systems demonstrate shown for the laboratory frame of reference. these rules. There is something marvel- ous about this unity of mathematics The total momentum is and physics. If you don't begin to see this unity and understand some PTolrm 3m2 vo 3;06 examples of it, then you will rot have seen physics. When physics problems where v0 is the initial velocity ob- get solved they are first expressed tained from the photograph. The vector in a mathematical form. The transforma- sum of the final momenta equals 3p2. tion of physics problems into mathe- In the center-of-mass frame, the matical problems is often the toughest final velocities are in ratio two to part of physics problem solving. Once one so that the momenta are equal and this is properly done, the solution to opposite in this reference frame. The the ensuing mathematical problem, vectors representing plf' and in found by the logic of the mathematics, the center-of-mass frame are shown in will be the answer to the physics Fig. 2.34b. problem! If you were given pa and pm As we study physics further, we for this explosion could you find the will see furthe7 physics examples and momentum in the center of mass frame, problems. Without becoming expert in pit'? problem solving, it is hoped that These examples have demonstrated there Will be sufficient exposure to that the law of conservation of mo- grant some insight into the unity of mentum works only if we treat' momentum physics and mathematics.

6 3 THE CONSERVATION OF ENERGY

3.1 THE PRINCIPLE OF ENERGY CON- forms of energy as well. The useful- SERVATION ness of the concept of energy stems from the fact that we can define each Matter in motion has a price! The of these forms of energy in a measur- inertia of mass at rest requires a able way, such that the total amount force to put that mass in motion, and of energy in the universe never momentum still does not tell the whole changes. Whenever an experiment hat, story. In the experiments discussed in been performed with careful bookkeep- this book, a variety of agents were ing of all the different forms of en- used to put matter in motion: a com- ergy, it has been found that the total pressed spring, a chemical explosive, amount of energy remains constant. En- another piece of moving matter, and so ergy ttat is removed from one form ap- forth. These agents possess a potenti- pears in others, and energy that ap- ality for putting matter in motion pears in one form is always balanced that differs somehow from the ability by energy that disappears !mothers. It of passive &Tents to change the direc- is in this sense that physicists asso- tion of the vector momentum. Energy, ciate their formulas for many different the "price" of matter in motion, dif- forms of energy. A compressed spring, fers qualitatively from momentum. An a nuclear reactor; a gasoline-air mix- ice skater works very hard to build up ture waiting to burn do not share a speed to perform his tricks, but then common appearance or feel or sound. he gliues around intricate curves and They do not appear at all alike to the loops with little further effort, senses, and the different forms of en- ,/'.41e his vector momentum may change ergy are very elusive, but there is rapidly and even reverse. The effort always a strict correspondence in the required to make a moving railway amount of one kind of energy that is train turn a corner and redirect its exchanged for a given amount of another. constant speed is qualitatively dif- The great variety of forms in ferent from the ranting and snorting which energy exists is suggested by of the engine that accompanies putting the number of methods one might use it into motion from rest, giving it to put a mass in notion. The mass energy of motion. There is something could be struck with another moving called energy released in the burning mass, or it could be accelerated by a of the fuel that is r_;eded to put the stretched elastic band. It ceild be train or the skater in motion, but put into motion by tying to It one end which is not required to change their of a rope that is wound at the other directions. Energy is a measure of the end around the shaft of an electric price of mass in motion. motor, or accelerated from rest simply A moving mass itself can, in a by the pull of the earth. A rush of collision, transfer motion to another hot gases from the burning of gasoline, mass while leaving the rest of the the steam± from heated water, or a hur- world viaffected, and one form of en- ricane born in the tropical oceans ergy is present in the motion itself. can start the mass moving. In many This form of energy is called kinetic casPa it is fairly easy to trace the energy. Because there, are many sys- history of the energy given the mov- tems which do not involve obvious mo- ing mass even farther back. The elec- tion, but which contain the potential- tric motor that accelerates the mess ity for producing it, a description of may te connected by wires to a genera- the physical world must include other tor driven by a nuclear reactor many 81 62 MATTIR IN MOTION

miles away. The living things from bonds between atoms are slightly dis- which the gasoline was ultimately de- torted, as in stretching a spring, rived tx.ok energy from the brilliant the change in electrical potential rays of the sun. Fortunately many energy is sometimes called elastic kinds of energy in this bewildering potential energy. When new electrical array have features in common, and bonds between atoms are created and the different forms of energy can be old ones destroyed, as in the burning grouped into classes and subclasses of gasoline, the change in electrical that simplify the over-all picture. potential energy is included as chem- The most convenient classifica- ical energy. tion of energies for our purpose is The gravitational potential en- in terms of the kinds of quantities ergy associated with a rzlck separated needed to calculate them. In some from the surface of the earth can be cases these classifications will be converted to kinetic energy by letting rough and will overlap a little, but rock and earth come together. If the they are still useful. The kinetic rock falls only a few hundred feet, energy of matter in motion depends the kinetic energy is most apparent only on the mass involved and the in the motion of the rock; but if it speed with which it is moving. If a falls for thousands of miles, much of mass moves with velocity v, its the energy will beamed in heating kinetic energy KZ depends on and v and vaporizing the rock as a white- (the size of v) according to the equa- hot shooting star flashes across the tion, sky. Any two objects have a gravitational potential energy that depends KB imvs. (3.1) only on their masses and their separa- In the mks system of units (meter- tion. This statement applies not only kilogram-second) this equation gives to the earth and another object, but the energy in units called Joules. to two asteroids, two dust particles, Other kinds of energy, which can or two atoms isolated in space. At be converted to kinetic energy, de- the surface of the earth gravitational pend only upon a position. The name interactions are very important. Be- potential energy for this form prob- tween two dust particles the gravita- ably arose because it has the poten- tional potential energy is very small, tiality of being converted to the more but then so are the dust particles obvious kinetic energy. The elastic themselves. The gravitational poten- potential energy of a stretched or tial energy has an important effect compressed spring depends up,_ its on the future of a cloud of countless length, the position of one end rela- billions of atoms and dust particles tive to the other. Opposite charges as it collapses into a hot (energetic) on the top and bottom of a thutler- ball of fire called a star. However, cloud have a great attraction for each in assembling a proton and an elec- other. The electrical potential en- tron into one hydrogen atom, the ergy which depends upon their separa- change of gravitational potential tion is converted to a rushing kinetic energy is so small compared with the energy when the charges flow together changes in electrical energy that it in a lightning stroke, and then to can always be neglected. For masses light energy, heat energy, sound en- typical of elementary particles like ergy, and chemical energy in the s r. the electron and proton, and for the sequent collisions with intervening amount of electric charge on these air molecules. Because the forces be- particles (the smallest nonzero elec- tween atoms themselves are basically tric charge), the electrical energy electrical in nature, many forms of changes completely dominate the gravi- potential energy are really kinds of tational energy. Only for 'electrically electrical potential energy. When the neutral matter (with no net electrical THS COMMATION 07 =MGT 63

charge) is gravitational potential energy significant. The motit,ns of macroscopic ob- 4-Jects can be described in terms of gravitational or electrical forms of potential energy. Even the interactions of electrons and atomic nuclei as they interact to form atoms and molecules involve potential energies that are basically electrical. Of course, describing the wealth of var- Fig. 3.1aA man walking. This is a very ied phenomena that make up the motions complicated motion which involves more than just one mass moving with a velocity. The of the stars, the planets, their in- next two parts of this figure illustrate habitants, and the atoms of which they just a little of this complexity. are composed into changes of kinetic and two kinds of potential energy is a terribly incomplete description. But it does reveal a thread that runs through the whole tapestry of natural phenomena. In laboratory experiments which probe the nucleus of the atom itself, in events which involve distances --4 only one ten-thousandth of an atomic V V diameter, new forms of potential en- Fig. 3.1b A simple model that can "walk." ergy become important. Nuclear po- Here it is sliding. The main mass H and tential energies describe the effects the two feet share the common velocity V, of forces which act only over the very which is therefore the velocity of the center of mass. The kinetic energy is short distances found between parti- i(11 + 2807/2. cles inside the atomic nucleus (less than 10r14 meter). They are important to the macroscopic world in that they determine the constituents from which it is made. In this submicroscopic realm the description of nature in terms of forces is very complicated, and the use of the energy concept has 2V been a necessary part of whatever 0 progress has been made in understanding it. Fig. 3.1cThe model "walking." If one foot To build a conservation law for is at rest on the ground and the other mov- energy, the kinetic and potential en- ing forward at velocity 2V, the over-all center of mass still moves with the mean ergies of an object treated as a parti- velocity, V. The total kinetic energy ex- cle (i.e., specified by the position ceeds that in Fig. 3.1b: of a point) are not enough. The -motions of the parts must also be taken ICE - 411V2 + im(2V)2 into account, if energy can be trans- - i(11 + 2m)V' + mV2. ferred to or from these motions. A walking man has more kinetic energy haps the next step in complexity from than just the kinetic energy of his a single mass moving with one velocity. center-of-mass motion. Figure 3.1 does As well as internal kinetic en- not do justice to the full complexity ergies associated with their veloci- of a man in motion, but it illustrates ties, real objects commonly have in- the point with a model which is per- ternal potential energies associated MATTIS IN MOTION

with the relative positions of their parts. The moving vibrating dumbbell of Fig. 3.2 is made of two equal masses m connected by a spring of v negligible mass. The motions of the (a) Two equal masses connected bya spring masses give this object kinetic energy, of negligible mass. In the inertialframe and the spring stores a potentialen- in which the center of mass P is atrest, ergy which depends on its length. the two masses have equal and opposite when the dumbbell participates in velocities. A particular situation atone processes in which energy is trans- instant of time is illustrated here. ferred only to or from the motion of the whole (as for example, when it is released to fall under the influence of gravity), then the whole dumbbell can be treated as one particle with a rogrr- ,,,111110. total mass 2m, as in Fig. 3.2c. The we. v v v=u+vt energy of the internal motions need (b) Viewed from another inertialframe, not be taken into account when it which happens to be our laboratory frame, remains constant. Only when the dumb- the center of mass moves with thevelocity bell is involved in more complicated u. The masses move in this frame with situations, where energy can be trans- velocities (II i172) and (u + 71) atthe ferred to the motion of onemass rela- moment illustrated. tive to the other, is it necessary to treat the two masses separately. The total energy in the more detailed analysis includes the kinetic energy of a particle of mass 2m moving with the center-of-mass velocity plus the kinetic energies of the twomasses in the center-of-mass reference frame (c) A single mass, 2m, moving with the cen- plus the potential energy of the ter-of -mass velocity W. compressed or stretched spring. If the Fig. 3.2The total energy of a vibrating center of mass of the system moves dumbbell from three viewpoints. Theenergy with velocity u relative to the labor- of the dumbbell seen in the laboratoryas tory, and if two masses move with in (b) is velocities +v and v relative to the center of mass, the total energy I of I im(u v')2 + im(u + v')2+ VS of spring the moving, vibrating dumbbell in the mu2 + mv" + PS of spring. laboratory rcEerence frame is (see Fig. 3.2), This quantity differs from tha kineticenergy of a single mass 2s moving with the I i(20u2 + imv" + inn,"+ PE of center-of-mass velocity u, as in (c). The spring, kinetic energy in (c) is Or KS 1(2012 mu2. E mu' + my" + PE of spring. we shall see later that it ismore than That the separation into center-of- coincidence that the internal energy (dif- mass energy (Mu') plus internal energy ference between IC for (b) and SS of (c) is (mv2+ PE of spring) takes place so Just the total energy calculated 'rem the naturally will be shown later to be viewpoint of (a), in the rest frame of the center of mass: more than a coincidence. When a part of each mass m can change its motion Internal energy im + Inv" separately from the rest, aneven more + pi of spring - Res + PS of spring. detailed analysis is required. THE CONSERVATION OF ENERGY 65

Objects of a size we can handle of internal energy associated with are all made of smaller pieces called the myriad random motions that go on atoms, and, in spite of our best ef- among the atoms of the hot brake lin- forts, the energy of a large mass in ing or any other material substance. motion has a tendency to become dis- The heat energy associated with sipated in the random motions of the the motions of atoms in a solid brake atoms that comprise it and its envi- lining or molecules whizzing about in ronment. The air pushed aside as an a gas is too complicated and involves automobile passes, the viscous oil in too many particles to keep track of the bearings, the squashing of the in retail. Fortunately, the total in- tires, all transfer energy from the ternal energy of such a system is of- macroscopic motion of the car to the ten found to be equally shared, on random motions of atoms. The conver- the average, among all the kinetic and sion of the over-all directed kinetic potential energies of its parts. When energy of center-of-mass motion to the many particles in a complicated the energy of random atomic motions system inter act with each other we call heat can even be useful. The quickly, so that the particles share kinetic energy of an automobile in their energies in mutual interactions motion must sometimes be dumped in a many times before and during the time hurry. The automobile's brakes use it takes to make a measurement on the friction to convert its kinetic energy system, the system can be thought of to heat. A typical 1500-kilogram auto- as "homogenized," In such systems each mobile moving at 30 meters per second particle has its average share of en- (about 60 mph), has a kinetic energy ergy, just as each drop in a quart of of about 7 x105 Joules according to homogenized milk has its share of Eq. (3.1). After its brakes have cream. Then the total energy of the brought it to a halt, there is no more system can be simply related to the kinetic energy associated with the average energy of each particle. The car's center-of-mass motion. In search- average particle energy suffices to ing for a clue as to where the energy predict many of the gross properties went, even an initially uniformed ob- of the whole system, for example, the server would be impressed by the tem- pressure of a gas in a container or perature rise of the car's brakes. the length of a solid. A useful meas- For the car above, the approximately ure of the average energy of the parti- 30 kilograms of steel in the brake cles in these complicated but homo- drums and shoes could be warmed by geneous systems is the temperature. more than 40°C. in one stop, becoming Two identical objects (say, two too hot to touch. Further experimenta- brake shoes) can be placed in physical tion would reveal that the tempera- contact so that the atoms of one can ture rise of the brakes increases with bump against the atoms of another, ex- the kinetic energy of the car's motion. changing energy. If, on the average, More quantitative and careful experi- heat energy flows in these collisions ments on a variety of moving objects from object A to object A', we say would show that a definite amount of that the temperature of A is higher kinetic energy is always required to than that of A'. The temperatures of produce the same thermal effect. (A A and A' are equal when, on the aver- common unit of thermal energy, the age, no energy flow takes place in calorie, is the amount of heat re- either direction, as demonstrated by quired to raise one gram of water one watching some macroscopic property degree centrigrade.) This consistency that depends on the average particle in the amount of heat obtained from energy. When the two objects A and A' each unit amount of kinetic energy are identically constructed, it comes leads to the conclusion that heat is a as no great surprise that energy flows form of enema. Heat is a variety of from the system with more energy to MATTER IN MOTION

meters per second). If amass mo at rest is given a kinetic energy KE, associated with the speed v by Eq. (3.1), the corresponding increase in mass is given by substitution of KE from Eq. (3.1) for AE in Eq. (3.2):

move (3.3) c2

The fractional mass change (Am/mo) is Fig. 3.3Brake shoe A and thermometer B in obtained by dividing both sides of Eq. physical contact. The brake shoe and the (3-3) by in: Thermometer are ac the same temperature when there is no net energy flow into or Am 2 out of the thermometer, as demonstrated by m (3.4) the fact that the length of the liquid in MO the capillary tube does not eunge with time. The speed c is so extremely large, compared with any reasonable labora- that with less; i.e., that a higher tory speed that the fractional in- temperature for the same system of crease in mass is negligible. A speed particles implies a higher total heat of 1000 miles per hour is equal to energy. The temperature provides a about 500 meters per second in mks measure of this tendency of heat en- units. For this speed the frac..ional ergy to flow that can also be applied increase in mass is only tc iifferent objects. Two different Am (5 x 102)2 objects A and B have the same temper- m = 1 X 10-12. (3.5) ature if, when they are placed in MO 3 X 108 physical contact (for example a brake shoe placed against a thermometer as It is only with microscopic particles, in Fig. 3.3), no energy flows on the where speeds can be made comparable average in either direction. with the speed of light, that the frac- Energy appears in still another tional mass increase with kinetic en- wonderful disguise as mass itsele. ergy is significant. The kinetic and potential energies of For speeds v comparable to c, an object, from both its over-all mo- Eq. (3.1) is no longer adequate, and tion and its internal motions, affect must be replaced by as equation for its mass. The conversion factor from the total energy, energy to mass is so small, however, E mc2 that the mass changes associated with (3.6) the kinetic energies we find in E. in which the mass m is not themass macroscopic object are insignificant of the object at rest, but themass of fractions of the mass of the object at the object now, in the situation where rest. The kinetic energy of a mass we want to know its energy. For an ob- moving at typical laboratory speeds ject with mass mo at rest, itsmass is given Eq. (3.1). The mass change when moving with speed v is Am associated with any energy change E in a system is given by MO (3.7) AE Am m # (3.2) c$ This gives it a total energy when in where c is the speed of light (3 x 10$ motion of THE CONSERVATION OF ENERGY 67

E I.------rT2114( (3.8) C2

at rest, its total energy, according to Eq. (3.6) was only

E0 m2c2. (3.9) M The kinetic energy, which had to be (a) Two H2 'Aolecules and ope 02 molecule at supplied to set it in motion, is rest. The total mass is M. air- E0 V2 C2

When (v2/c2) is very small compared A) one, it is a very good approximation to write

2 4111111. M 1 (3.11) Et2 1 4. 2 V -4; 1 (b) After the reaction the two moving jiggling H2O molecules still have mass II if they retain the energy Q released by the When tbLs approximation is valid (and reaction as energy of internal motion. v2/c2 is certainly small in Eq. (3.5)) the expression (3.10) for the kinetic energy can be written

m0c2[1 (1+i5)]

v imov2. (3.12)

The error in tha approximation in Eq. M Q/c2 (3.12) is of the size of me4/c2, (C) If the energy Q released by the reac- which is smaller by a fraction some- tion is removed, the mass of the products thing like v2 /c2 than the value im0v2. at rest is smaller by the amount Q/c2 than Recall Eq. (3.5) for "typical" numbers. the mass M of the reactants. Not only kinetic energy changes In the reaction 2H2 + 02.1. 2H20 but also other kinds of energy changes Fig. 3.4 the mass decrease is about 2 x 10-36 kg, affect the mass of an object. The out of a total mass M of 6 x 10-2' kg, a kinetic and potential energy changes fractional decrease of 3 x 2n-11. in the chemical reaction

2H2 +102 2H20, two water molecules, the total mass stays exactly constant during the re- are such that the internal energyof action. When, however, the energy es- the two water molecules is less than capes and the total mass of two H20 that of the oxygen and two hydrogen molecules at rest is compared with the molecules by about 2 x 10-1 Joules. total mass of one 02 and two H2 mole- If this energy remains in the system cules, there should be a very small as internal or kinetic energy of the difference (see Fig. 3.4). Because the

1 68 NATTER IN NOTION

so MD fa MI MID OD ..... IND MEIIIMM/ MEI MO MEP MIMIMil MO MIR MO MI MEP MEP11111111 Om.

. 1 11 110...ii../ I I /.1.:

1 1 11 I IO. 1 .1.f I 01/ 1 1 1 % I j,1 .1 % I 111 eel/1,10 INSIDE z. SYSTEM BOUNDARY OUTSIDE Fig. 3.5The selection of a system sepa- imaginary boundary. rated from the rest of the universe by an two H2O molecules have 2 x 10-18 so large compared with the mass of the Joules less internal energy, they particles involved (a total of about should have a mass smaller by 4 x 10-85 kilograms), that the fractional change is significant (0.001 or AE 2 x 1010 -18 0.1%). Of course, the missing mass Am = 2 x 10-88 kg: (3 x 8)8 was not "lost"; it was the mass of the escaped energy. In all reactions where The fractional difference in the there are energy changes the principle 6 x 10 -26 kg total mass of four hydro- of energy conservation guarantees that gen and two oxygen atoms is only about mass will be conserved as well, if the 3 x 10-11. Even if many water mole- mass of each form of energy is prop- cules are used to boost the total mass erly accounted for. In chemical reac- change Am by a large factor, such a tions and most ot:er laboratory experi- small fractional effect is undetect- ments the energy charges are suffi- able with present laboratory tech- ciently small compared to the masses niques. The mass changes in reactions involved that the mass of the particles are significant fractions of the total appears to be conserved within experi- mass only when nuclear potential en- mental error even if the mass-equiva- ergies are involved. A slow neutron lent of the energy involved is ignored. (almost zero kinetic energy), can be absorbed by the nucleus ofU23 (an isotope of the element uranium that 3.2 ISOLATION, EXTERNAL AND INTERNAL has, in addition to the 92 protons of any uranium atom, the particular num- The principle of energy conserva- ber of 143 neutrons in its nucleus). tion states that the total energy in The resulting nucleus of 236 particles the universe remains constant, but in is unstable and splits roughly in half. practice the principle is seldom ap- The result of the reaction is the re- plied to tte whole universe. A labora- lease of two lighter atomic nuclei, tory experiment or a theoretical calcu' some excess neutrons, and an energy lation usually is confined to one of about 3 x 10-11 Joules. If the en- small but interesting part of the uni- ergy escapes and the mass of the prod- verse. It is common to call this piece ucts at rest is compared with the of the universe a system, and to mass of the reactingpaiticles at imagine completely surrounded by a rest, the lower internal energy of the boundary surface through which pass products is reflected in a mass de- only things and influences that we crease of 4 x10"" kilograms. In this know, understand, and can measure nuclear reaction the energy release is (see Fig. 3.5). THE CONSERVATION OF ENERGY 69

The total energy of the universe term E1 merely adds complication with- can be written as the sum of the en- out adding anything of interest. ergy inside and the energy outside If the brakes are applied to the the boundary. If no energy crosses the trait, of Fig. 3.5, the total energy boundary, then we can conclude that must contain another term E2 for the the energy of the system (inside) and heat energy of the brakes. Even in the energy of the rest of the universe braking situations, the energy (outside) are separately constant. It is always implicit in such an argu- Et imv2 + gravitational PE + E2 ment that we know of all forces that might act across the boundary so that remain. constant. If the engine burns we can be really certain that no en- fuel in air, this energy is included ergy is transferred. When this is the as well. Calling the chemical energy case, the system is called an isolated of the fuel-air mixture E3, the con- system. stant quantity is Even for a system which is not isolated, some useful statement of energy conservation is possible if we E" - imv2 + gravitational PE + E2 Es. can identify and measure all the forms in which energy crosses the boundary. Always the idea is to have as few In such circumstances any change AE terms as possible, consistent with all in the energy of the system during a the energy changes that actually occur. time interval At is equal to the net The total energy of any system energy that flows inward across the includes other internal energies in boundary during that time: addition to the terms included in a typical description of its "relevant" AE (energy flow in) energy. The watch ticking in the engineer's pocket, the electrons within - (energy flow out). atoms, the protons within nuclei, etc., With a system like the train of all involve energies that are usually Fig. 3.5, the principle of energy con- neglected in writing down the total servation is most useful when it can energy of a train. These energies are be written in terms of a few simple neglected because they do not change quantities. If the train merely coasts separately in the course of the up and down hills, the total energy E train's motion. In laboratory mechan- can be written as ics experiments it is a great simpli- fication to reduce friction to such a E - imv2 + gravitational PE. small effect that macroscopic energies are not mixed in with random atomic Internal energies like those of the energies, for then the total energy parts of the engineer's watch remain of the system can be written without separately constant during the coast- including heat energy. Fortunately, ing; and hence add nothing of inter- the internal energies associated with est to the conservation of total en- motions of nucleons within nuclei and ergy. If we add to E the energy, with the structures of elementary particles themselves are quite well in- El - energy of watch - constant, sulated from the macroscopic world. These energies do not change in the sum E + E1 is a constant but is macroscopic experiments and need not of no more interest than E itself. It be included in their description. The is the flow of energy back and forth only place where the total of all the from the kinetic energy term (imv2) energy in an object is important, to the potential energy term that is whether it changes or not, is in de- interesting. Adding another constant termining its mass. 70 MATTER IN MOTION

3.3 ENERGY-CONSERVATION EXPERIMENTS (4) If we can obtain from Eq. (3.14) a formula for E4 (like The historical development of the KE a imv2) in terms of all the vari- law of conservation of energy depended ables that describe the system (length, upon the discoveries that various temperature, etc.), and if the same kinds of energy wore interchangeable formula describes all such experiments, according to unique rules. For a fixed then we have confirmed the conserva- amount of one kind of energy, a fixed tion law, Eq. (3.13), limited to these amount of another can be obtained. But four kinds of energy. to make such a rule apparent, one must The confirmation of Eq. (3.13) have a formula for the amount of each supports indirectly our faith in the kind of energy. assumptions (1), (2), and (3) used to We shall somewhat parallel this obtain it. Of course, each such experi- historical development in form, ment also helps confirm that we have though much more efficiently, indeed, the correct formulas for E1, E2, and if we examine some selected experi- E3 ments to determine the formulas for The fact that we can for so many different kinds of energy. Briefly, different cases find a conservation the analysis of each exxaiment will law of the form of Eq. (3.13), summing go something like this: many and various kinds of energy to a (1) We assume that there is a constant total, leads us to believe law of conservation of energy.9 that there is a quantity corresponding (2) We isolate a system; that is, to what we call energy. And it does we attempt to prevent it from exchang- seem to be conserved. Whenever we keep ing energy with the rest of the uni- careful account, none is found to be verse. Together with assumption (1), created or destroyed. this means that we assume the total We shall see some examples of energy of our isolated system to be a this kind of analysis in the follow- constant. ing sections, beginning with kinetic (3) Furthermore, we chodse a energy. system in which our observations lead us to assume that only a few forms of energy are changing, including among 3.4 KINETIC ENERPY them only one kind of energy whose formula we do not know. If El, E2, In this section we shall start E3 represent energies whose formulas with the assumption that a mass m mov- we already know, and whose changes we ing with speed v has a kinetic energy can recognize, and if we assume for determined by these two numbers. By our system considering cases in which kinetic energy seems to be conserved, we shall E1+ E2 + E3 + E4 - constant, (3.13) show that if it is conserved in any collisions (called elastic collisions), then we can calculate changes in E4: it must have the form ' AE4 m (AEI + AE2 + AE2). (3.14) KE (constant) 7 v2.

'Historically, the Erand conservation law en- The development wi7.1 reveal con- compassing all the energy forms did not come un- nections among kinetic energy, momen- til the late nineteenth century. But it was al- tum conservation, and Galilean rela- ready known around 1700 that there are some simple systems in which the kinetic energy is tivity. Let two observers, moving with conserved. Early studies of heat energy (then constant relative velocity, watch the interpreted as "caloric") separate from Mechan- same collision. To show that they ical energy invoked a separate conservation law. agree on whether or not energy is con- WO now know that the two laws are spacial oases of the energy oonmorvation law. served in the collision, we shall THE CONSERVATION OF ENERGY 71 41.1111

need the law of conservationof mo- TRIAL 3 mentum. NO. 1 2 Eventually Eq. (3.1) must meet r-BEFORE 0.115 0.573 1.25 additional tests. In experiments in 7A I which KE from Eq. (3.1) is conserved, /A I *zero itzero *0.01 does the rest of the energy also re- AFTER 7s t 0.114 0.565 1.24 main constant, as predicted by the law of conservation of energy? Inad- Units an rnisee ditional experiments involving other Table 3.1Elastic collisions for data for forms of energy as well, does this formula permit writ:.ng an energy con- oAma- 0.400kg servation law that is confirmed by V 0 measurements? A mass mA, moving with, velocity initial and final velocities arepresented strikes a body of mass ms, which for three trials. had initially a velocity vii. This is the start of a collision. We shall periment would be well describld by show that there are some such colli- the equations sions in which kinetic energy is "obviously" present in equal amounts be- VAIr 0, (3.15a) fore and after the collision, even to Vp- VAi. (3.15b) one who does not know Eq.(3.1). There are indeed such collisions. They occur The before and after pictures of very frequently in the microscopic such an elastic collision between realms of atomic and nuclear physics. equal masses would resemble Fig. 3.6. A macroscopic collision in which With the assumption that the kinetic energy is "obviously" the same kinetic energy of a moving mass de- afterwards as before can be demon- pends upon m and v only, Eqs.(3.15) strated in the class room on the air tell us that kinetic energy is con- track. Collisions which do not change the kinetic energy are called elastic NrAt collisions. 1.31111111. A ber-1:3b bar-mb 3.4.1 The Equal -Mass Elastic Colli- sion. (a) Two identical gliders A and Bbefore the collision. Glider B isat rest while Two identical gliders A and B glider A moves with velocity vAi.The glid- are placed on the air track,B ini- ers are equipped withelastic steel bumpers tially at rest and A with initial ve- b. The kinetic energy is containedin the locity vAl. The gliders have equal motion of mass mA at speed:vAll. mass,

mA ms mt

and are equipped with spring-steel (b) After the collision A is found at rest bumpers on the ends. These bumpers can and B moves with velocity vilf- val.Be- be distorted in a collision and returned cause the two gliders areidentical, it is to their original shapes with very here "obvious" that B carries off the same little transfer of energy to the motion kinetic energy originally brought to the of the atoms within the springs. The collision by A. A collision in whichki- measurements will show that this is so. netic energy is conserved is called an Some of the data from a typical elastic collision. experiment might look like that in Fig. 3.6 An elastic collision between Table 3.1. The results of such an ex- equal masses. 72 NATTER IN MOTION

nets will collide elastically,as did the gliders with spring-steel bumpers. If they have equal masses andone in motion is sent against another at rest, Fig. 3.7aThe unequal masses A and Bon they will exchange velocities in the the air track before the collision. Ini- magnetic collision. Theone initially tially in motion will stop, and the other vA ;91 acquire the initial velocity. A proton and a neutron have V9 0. nearly equal masses. Quite often when

"*. a neutron in motion collides "head-on" VM Vhf with a proton at rest ("head-on"as- A sures straight-line motion), it comes cr-i=1 to rest, and the proton goeson ahead _J with the neutron's total kineticen- Fig. 3.7b The unequal masser, A and B ergy; i.e., with the same mass and separating after the collision. Finally the same speed.

VA - ;Ai 3.4.2The Unequal-Mass Elastic Colli- ;O - ;Of' sion. served in this experiment. Initially The same elastic springs employed a mass mA - m travels with speed IvAil, in our first experiment can be used in and an equal mass sits at rest. another experiment with unequalmasses. Finally, the data describea situation We shall find the conservation ofen- in which mass ms - m travels with ergy in the slightly more complicated speed IvAil, and an equalmass sits at collision just as convincing (after rest. The kinetic energy is unchanged some analysis) from the data we re- in this collision. cord. As a reward for carrying through This experiment serves to demon- the more complex analysis, we shall strate that we can find a set of "elas- be able to determine the actual form, tic" springs. But an experimentwith Eq. (3.1), for the kineticenergy. equal masses will not demonstrate that Th:3 second experiment begins the kinetic energy must have the with mass ms at rest on the air unique form of Eq. (3.1). Just for track.1° The initial velocity ofmass fun, try calculating from thedata the mA is Vett. The two gliders have dif- initial and final kinetic energies ferent masses, but theyuse the same on the following assumptions: spring bumpers used so successfully before. Try: KE'oc(mA + ms I-170) (3.16a) A few of the data from suchan Try: KE'cc(mA riAla+ ma 1;1'12) (3.16b) experiment, pictured in Fig. 3.7, might look like those in Table3.2. Try: KE'oc(sA IVAI3+ ma 1781 3) (3.16c) A quick inspection of the data in Table 3.2 is not sufficient1,o see They all are conserved. Canyou see why this particular experiment will not distinguish among them? "We choose ms initially at rest forsimplicity. Not all collisions are elastic, It will turn out that there is nothing to be but one very simple collision experi- gained but complication by choosing anonzero vs" unless we could set ment, in which kinetic energy is "obviously" conserved, has been described here. Equivalent experiments can be U9vat, performed in other ways. Two air to make the laboratory the zero-momentum (center- gliders with mutually repellingmar of-mass) frame. TER CONSERVATION OF ENERGY 73

whether kinetic energy is the same be- TRIAL I

fore and after the collision.11 But NO. 1 2 1 3 4 look at the experiment from the center- of-mass viewpoint! Table 3.3 shows IIEFORE 7A, 0247 0.553 1.07 1.54 the same glider motions from the cen- if -0.104 -0.235 -0.450 -0.651 ter-of-mass frame. Look at the con- AFTER stancy of kinetic energy! In the V,, 0.140 0.314 i 0.601 0.870 center-of-mass frame, each mass leaves Units are in/sec the collision with the sane speed Table 3.2Velocity data from an air-track that it entered; only its direction is collision. The collisions between unewial changed. masses use spring steel bumpers that Are If the kinetic energy depends supposed to make them elastic. The data only on the mass and the speed, it will tell if this is so. The initial conditions are must surely be unchanged before and a after these collisions described in Table 3.3. The data of this table can m. - 0.200 kg, ira vai be described as far as energy is con- mini 0.500 kg, 0. cerned by

, TRIAL

Ivolll - Ivoill (3.17a) NO. 1 2 3 4

Iva?' - Ivaill. (3.1713) -;As ' 0.176 0.395 0.765 1.10 IEFORE 178,1 -0.0706-0.158-0.306 -0.440 For collisions which obey Eqs. (3.17), it is "obvious" to an observer in the 7A,' -0.175 -0.393-0.756 -1.09 center-of-mass frame that kinetic en- AFTER 1' 0.6694 I 0.156 0.295 0.430 ergy i- the same after the collision as before. Uftitsammbee The question we now want to an- Table 3.3The velocities of Table 3.2 from swer is "Shia quantity remains con- the center-of-mass frame. The center of stant in the laboratory trare before mass moves in the laboratory with velocity and after collisions which are 'obviously' elastic in the center-of- (114) 2- mass frame?" ; (a + moia " We cannot try all possible func- tions of and v in the time allotted. Table 3.4 shows that in these But we might compare the three assump- collisions for which kinetic energy is tions of Eqs. (3.16): conserved in the center-of-mass system the quantity (mAlvAl2 + mol;O12) is (a) Is the quantity(salvid .+ alvo') conserved in tho laboratory system. constant? (Table 3.4a) This is the first justification for the form of Eq. (3.1) (b) Is the quantity(salvo's + salvo's) constant? (Table 3.4b) KE intArvii! 2 + 1111.!Zal 2 (c) Is the quantity(salvo's + salvo's) The factor 4 arises from the relation constant? (Table 3.4c) between force and changes of kinetic energy. Any other multiple of mv2 would also be conserved in an elastic collision. "Actuallya very clever observer might notice a clue to the elasticity is Table 3.2 in that the size of the relative velocity between ma and 3.4.3 The Form of the Kinetic Rivera, ms is the same after the collision as before. But the center-of-mane frame gives a better van- It is possible to show mathemati- tage point for viewing the elastic collision. cally that a collision for which 74 PATTER IN LOTION

TRIALWX 1 2 3 4

BEFORE mal vA,1 0.0494 0.111 0.214 0.308

AFTER malvat I+ms1vsfI 0.0908 0.204 0.390 0.595

Table 3.4a Is the quantity (KARA,+ ne1;111) the sane before and after the collision?

TRIALFAX i 2 3 4

I AFORE makia 0.0122 0.0612 0.230 0.474

AFTER mAllirm12 +ine Iva 0.011* 0A605 0.222 0.463

Table 3.4b Is the quantity (nAlvAl2+ n,lval2) the sane before and After the collision?

TRIAILNCI. 1 2 3 4

MAX( mANIV 0.00302 0.0338 0.246 0.684

AFTER maival 13 + osivilf q3 0.00160 0.0181 0.127 0.384

Table 3.4c Is the quantity (RAlvals+ alvals) the same before and after the collision?

VIsI"' 11 (3.18a) in the laboratory frame, we need to V f -if Al (3.18b) do a little geometry. Figure 3.8 shows that the length IV' + of the vec- ma in the center-of-mass frame will not tor, v + U, is either thesum of the change the quantity lengths if the vectors art parallel, or the difference of the lengths if mAl;Al 2 + mil Val 2 they are opposite.

in any other inertial frame. 1-1;11 + I uI if aLd r1 are The center-of-mass frame isan parallel inertial frame in which the center-of- al if 17, and II mass point moves with the constant 1;2 + i4 1;11 are velocity zero. opposite and if 1101 >IUI The center of mass moves with con- rdI 1;21 if v; and Uare stant velocity U with respect toan- opposite and if IUI >1;21. other.inertial frame. Ta this frame, the initial and final velocities for (3.20) the elastic collision described by Hence Eqs. (3.18) are

vAl - 4.1a: ;:,- r; 17212 + 12;11lir'+lul2 if v and U are parallel (3.19a) Ivl +u1'- `al +14 ;it ;'st+ U. 1y'12 21,I if v' and U are opposite. (3.19b) (3.21) In order to write the quantity To allow for these two cases with 464ill 110At + U12 one simple notation, we introduce the T811 CONSSIVATION OF AMOY 75

"scalar product" or "dot product" of .111.vt two one-dimensional vectors:12 MITXOW

+ 1711ail for ;7° + U parallel to ri 1;2 V- V UM 1;11IV!for ;;° {opposite to U. (3.22) 4411F-s

This notation permits us to write V' +u

1;212 (3.23) and 1; + "612-17112.4. 2;1 + (3.24) .111111&

Notice how much simpler it is to write Sq. (3.24) instead of Eq. (3.21). The scalar product has a pair of useful Fig. 3.8.. Theformulafor the length of the properties which can be verified from vector (v' + U) depends on whether71 and the definition, Eq. (3.22): V are parallel or antiparallel multiplication by a scalar, iv, + -1;g1 a17' 6- (111) (46). 1;1 + -1;1 (2.35) 1;g - and addition

6°A+ U if/a U. Collecting appropriate terms together, we find that (3.26) ae1;s112 nia1A112 Before the collision isal;a113 +as1;si 12 + 2(514;°,1 sial;a112 + Rands 141;lal + TI12 + a,;',1) V + (aa + an) Fiji 2* lial;lej + V12 (3.28) - m-4.-1A1.2i --A-'Al9m7, + 21012 +1121;231 12 After the collision, the same expansion and collection of terms gives 2ajI111 + anirlli (3.27) 1111;Ati +1121;8112 litARtA f 2 +Ma/2W + 2(14V /At +112;25f) U + 01A + Irv. (3.29)

12The same notatioo will be used later for the three-dimensional generalization, where the For elastic collisions defined by tern "scalar product" is especially appropriate. Eq. (3.18), tho two expressions in For one-dimensional vectors expressed by posi- Eqs. L3.28) and (3.29) are equal. Be- tive and negative numbers it is just likethe first ordinary arithmetical product. cause of Eq..(3.16), each of the 76 NAM* IN NOTION

two terms on the right side of Eq. equation for kinetic energy alone in (3.28) is separately equal to each of elastic collisions, the formula for the first two terms on the right side kinetic energy must be of Eq. (3.29): KE o (constant) mv2. atAl 2 *IVtai2 (3.30a) 11111;t111 I2 141;t1)1 2 (3.30b) 3.4.4The Ultimate Appeal. Our few experiments and their Because momentum is conserved, analyses only indicate but do not prove that kinetic energy is propor Re Al + Re 111AvtAt Re of tional to mv2. Nor is it at all justi- (3.31) fied to jump to any grand conclusion yet that ther is such a thing as en- Bence, the third term on the right ergy which is conserved. side of Eq. (3.28) is equal to the But there are a few things that third term on the right side of Eq. we can do to check our hypothesis that (3.29): kinetic energy is the same after an elastic collision as before. We can 2(11;;tai+ ajts1) 6 check the other properties (besides velocity) of the isolated air-track 2(147t.,+ jtsf) U. gliders-before and after the experi- (3.32) ment to see if they are the same. If other properties (temperature, shape, etc.) of the gliders stay the same The last term on the right-hand side in such collisions, perhaps that in- of Eq. (3.28) is identical with the dicates that the other energies, in- last term on the right hand side of ternal to the glider, remain constant Eq. (3.29). The next step in the development Let us summarize what we have of energy conservation is to go back just shown. If there are in fact elas- to the laboratory, to put the equation tic collisions described in the center- of-mass frame by merely a reversal of velocities Eq. (3.18), and if momentum KE imv2 is conserved Eq. (3.31), then the quantity malval2 + milvij2 in any other to an experimental test. This can be inertial frame is the same after the done by studying experiments in which collision as it was before it. kinetic energy is exchanged for an- Hence, if there is a kinetic en- other form. The laboratory provides ergy associated with matter in motion, the ultimate test for physical theor- and if we can write z conservation ies.