edge-up flotation behavior of the prisms is due to the fact,that they are deprived of one of their degrees of freedom:Their fourfold axis is constrained to stay parallel to theliquid surface.

After these examples it should be dalliance for the daunt-less reader to determine the equilibrium positions, as func-tions of r, of the two remaining Platonic solids: the icosahe-dron and the pentagondodecahedron.

ACKNOWLEDGMENT

The support of the Swiss National Science Foundationthrough Grant No. 20-28846.90 is gratefully acknowl-edged.

APPENDIX A

Explicit expressions for the integral I of Eq. ( 6) in thecases needed in the analytic calculation:

(1) I f a>1, b>1, and c>1:1/ — 1 ( A I )

24abc Ala2 + b2 + C2(2) I f a<1, b<1, c<1, a + b>1,b + 01, a +

I — [ ( a —1)4(12 + b 2 + C2 24abc

+ (b —1)4 + (c —1)4 —1].

All about workA. John Mallinckrodta) and Harvey S. Leib)Physics Department, California State Polytechnic University, Pomona3801 West Temple Avenue, Pomona, Cahfornia 91768

(Received 22 May 1991; accepted 4 July 1991)

A comprehensive "taxonomy of work" is developed to clarify the confusing potpourri of worklikequantities that exists in the literature. Seven types of work that can be done on a system of particlesinteracting internally and/or with its environment are identified and reviewed. Each work isdefined in terms of relevant forces and displacements; mathematical connections between theworks are delineated; work-energy relationships are derived; and the Galilean transformationproperties of the works and corresponding energy changes are obtained. The results are applied toseveral examples, illustrating subtle distinctions between the various works and showing howthey can be used to bridge the conceptual gap between the "pure" mechanics of point particles andthe thermodynamics of macroscopic matter.

I. INTRODUCTION

(A2)

It is widely appreciated that the definition of work en-countered in most introductory physics textbooks,

WnsJF.dr ( I )is well defined only when the force acts either on a pointparticle or a rigid body in pure translation. I n the realm of

(3) I fa<1, b<1, c>1, a + b>1:1 [a4 + b4 — 4(a3 +

+ b 2 + C2 2 4 a b c

+ 6(a2 + b 2) — 4(a + b) + 11. (A3)

In this case,the immersed volume isV, ( 1/6ab) [ 1 — ( 1 — a)3 — (1 — b)31.

There are five other cases, for which we do not give I. Theywere used in the preliminary numerical work ( see Sec. II).

'Part 1 of this paper. Equations of Part 1 are referenced by the prefix 1;.Dupin, "De la stabilite des corps fiottants," 1814, included in Appli-

cations de Geometrie et de Mecanique (Bachelier, Paris, 1822).'A. S. Ramsey, Hydrostatics (Cambridge UP., Cambridge, 1936), P. 82.4H. Lamb, Statics (Cambridge UP., Cambridge, 1946), 5th ed., p. 421.

Yeh and J. I, Abrams, Principles of Mechanics of Solids and Fluids(McGraw-Hill, New York, 1960), Vol. 1, p. 96.

6E. N. Gilbert, "How things float," Am. Math. Monthly 98, 201-216(1991).

"pure” mechanics—that part of mechanics dealing withnondissipative entities involving observable point parti-cles—the use of Eq. (1) is straightforward.

In contrast, when mechanics is extended into a "real-life" domain involving macroscopic objects with hiddeninternal energy modes and dissipation, Eq. ( 1) is inade-quate. Authors have invented a potpourri o f worklikequantities (hereafter referred to simply as "works") to dealwith these situations.'" These works go by a variety of

356 A m . J. Phys. 60(4), April 1992 ( ) 1992 American Association of Physics Teachers 3 5 6

names including internal work, external macroscopicwork, external microscopic work, pseudowork (also calledcenter-of-mass work), conservative work, nonconservativework, quasistatic work, thermodynamic work, and others.The existence of this veritable zoo of works raises severalquestions: How many types of works are needed? What arethe relationships between the works? What work-energyrelationships exist? Which works are frame dependent andwhich are invariant under Galilean transformations? Canthe classical mechanics of a many-particle system lead toan understanding of dissipative processes for macroscopicobjects?

Inspired by these questions and confounded by the ap-parent confusion about the definitions for and relation-ships between the various works, we present a systematicexamination of processes in which a classical, many-parti-cle system undergoes mechanical interactions both be-tween its own component particles and with its environ-ment. W e provide explicit, microscopically baseddefinitions of seven works that can, in principle, be calcu-lated for such processes. From those definitions we derivethe relationships between the works and a set of "work-energy relationships" that connect each work to a changein some form of system energy. Finally, we determine theGalilean transformation characteristics of all works andenergy changes.

Although this article is, in part, a review, its principalvalue is as a needed synthesis and generalization of pre-vious efforts. Using a consistent notation and terminology,defining all relevant quantities unambiguously, and obtain-ing the relationships between them, we provide, in essence,the fundamentals of a complete "taxonomy of work."

Our investigation is organized as follows: We begin inSec. I I with a survey of prior applications of work conceptsto macroscopic objects. Sections I I I through VI comprisethe development of our "standard taxonomy of work." InSec. VII we focus attention on a collection of engaging ap-plications chosen to illuminate subtle distinctions betweenthe works and linkages between "pure" mechanics of pointparticles and thermodynamics of macroscopic matter. Weclose with a summary of our findings and a discussion ofthe place of this study within the larger context of thermo-dynamics. Readers who wish to scan the main results, by-passing the details, are directed to Table I and Sec. VIII.

I L SURVEY OF THE LITERATUREMuch of the confusion surrounding the subject of work

arises from the lack of a standard and distinct notationsystem for quantities that are similar enough to convey theunfortunate and mistaken impression that they are identi-cal. The collection of authoritative articles referenced be-low illustrates this point. Individually, each employs nota-tion that is internally consistent and well suited to thespecific investigation being conducted. Between articles,however, one may find the same symbol representing fun-damentally different quantities and/or different symbolsrepresenting identical quantities. Our notation-free discus-sion in this section is intended to illustrate the substanceand scope of previous efforts while avoiding notationalconfusion.

Difficulties encountered by naively applying the work-energy theorem to nonrigid and/or rotating bodies havebeen the subject of articles by various authors.'" Erlich-son,' Penchina,3 and Sherwood'" have pointed out that

many interactions are best described i n terms o f a"pseudowork-kinetic energy" theorem. Pseudowork is cal-culated as the work that would be done by a force equal tothe net force acting on the system i f it acted along the pathfollowed by the system's center-of-mass ( c.m. ) and can beshown to equal the change in bulk translational kinetic en-ergy of the system. I t is not real work in a fundamentalsense because it generally entails forces that actually act atpoints distinct from the c.m. through displacements thatcan differ from that of the c.m.

In a recent review, "Developing the Energy Concepts inIntroductory Physics," Arons argues forcefully that "Theprincipal misconception planted in introductory physics isthat the 'work' quantity appearing in the 'work-kinetic en-ergy theorem' ob ta ined by integration of Newton's Sec-ond Law, is identical with the 'work' appearing in the gen-eral law of conservation of energy, namely the First Law ofThermodynamics." Fol lowing Penchina3 a n d Sher-wood,4'5 A rons recommends t h e use o f the t e rm"pseudowork" for the quantity connected to displacementof the c.m., "reserving the name 'work' for the quantityappearing in the First Law of Thermodynamics." He statesfurther, "...it is convenient because it does not completelysever the connection between the two quantities and be-cause it does not resort to a radically new vocabulary."

As pointed out by Arons, careless use of the work-energyequation can lead to mistakes and misconceptions. To illus-trate the kinds of difficulties that can occur, consider acompletely inelastic collision between t w o identicalmasses. Naive application of the standard work-kinetic en-ergy theorem gives the result that the work done in stop-ping each object is — mv2, where m is the mass and v theinitial speed of each object. Unfortunately, this is mislead-ing because a straightforward symmetry argument' showsthat the work done by each object on the other must beidentically zero! A proper treatment of this problem re-quires one to take into account the energy of deformation.

Bernard' and Erlichson9 have pointed out that the work-kinetic energy theorem can be applied to interactions ingeneral if we take the "work" to be that done on a system byboth external and internal forces and the "kinetic energy"to be that possessed by the system due to both its bulktranslational motion and the motions of its constituentparts relative to the cm. ( For a solid body this secondconstituent of the kinetic energy would include energy as-sociated with bulk rotation and with the vibrational kineticenergies of the molecules that make up the body. )

Canagaratna' arrived at the same conclusion but putthe result in a different form by defining an internal poten-tial energy change that is directly related to the work doneby internal forces during a change in the "relative configu-ration of the constituent parts of the body." An interestingaspect o f Canagaratna's work is the careful distinctionmade between quasistatic and nonstatic work—a conceptthat is particularly important in macroscopic mechanicswhich, in essence, is thermodynamics.

Kemp" has argued that difficulties related to the book-keeping of internal work can be circumvented entirely byappeal to the first law of thermodynamics in the formA ( mechanical energy o f system a n d surroundings)+ A ( internal energy of system) + A ( internal energy of

surroundings) O . This approach avoids explicit use ofwork and heat terms, but also conceals the mechanismsresponsible for the energy transformations. In a similar

357 A m . J. Phys., Vol. 60, No. 4, Apri l 1992 A . J. Mallinekrodt and H. S. Leff 3 5 7

vein Barrow 12 has asserted that "There is no thermody-namic role for the slippery terms 'heat' and 'work.' Weshould deal with energies and, in thermodynamics, withthe energies of the system and its thermal and mechanicalsurroundings. Then all first-law energy calculations can bedone with a good accounting system... ."

Sherwood and Bernardi' proposed use of two forms ofthe first law of thermodynamics—one frame dependentand one frame invariant. In the frame-dependent versionthe work is calculated using all external forces and the mo-tions of their points of application and the system energyincludes bulk translational kinetic energy. In the frame-invariant version the work is calculated using all externalforces and the motions of their points of application relativeto the system c.m. and the system energy does not includebulk translational kinetic energy. (Note that this secondwork is calculated in a frame that is not, in general, iner-tial.) The difference between the two works turns out to beprecisely the pseudowork, which equals the change in bulktranslational kinetic energy.

The abundance of papers devoted to energy transforma-tions and, in particular, work in macroscopic systems isimpressive and is indicative of the discomfort many physicsteachers experience in this area. I t provides evidence thatfurther clarifications are needed. The "taxonomy of work"presented in Secs. i s intended to meet this need.

ASSUMPTIONS AND ELEMENTARYDEFINITIONS

We consider a system of particles as shown in Fig. 1,which may comprise either a rigid or deformable body, andmake the following assumptions.

Assumption I : Our system consists o f a collection o f N

Origin

Fig. 1. Depiction of the AT-element system on which work is done. Elementi has mass m, , the system has total mass M, and the cm. of the system hasposition vector r, rn relative to the origin. The position vectors of element irelative to the origin, to the c.m., and to elementf, respectively, are r, , 1%,and r T h e force on element i by element] is F a n d the sum of all suchinternal forces on element i is r " . The net external force on element i ist r ' . Although this figure depicts a continuous collection of elements re-sembling a rigid body, the mathematical development assumes neithercontinuity nor rigidity.

358 A m . J. Phys., Vol. 60, No. 4, April 1992

elements, each of which behaves as a point particle. Thesignificance of this assumption is twofold: First, since ac-celeration is strictly well defined only for point particles,only point particles can strictly obey Newton's second law.Second, point particles, by definition, are devoid of internalstructure and are incapable of possessing internal modesthat store energy.

Assumption 2: The interelement forces are conservative.This enables us to define an interelement, internal potentialenergy function for the system.

Assumption 3: A classical, non relativistic analysis is ade-quate. This assumption justifies our use of the simple Gali-lean transformation and allows us to declare that all forcesand time intervals are invariant with respect to changes ofreference frame.

For the sake of notational clarity and precision, we col-lect here the definitions of elementary quantities that areused throughout the paper.

The element i has mass m,. The position ( i.e., displace-ment from the origin) and velocity of element i in a chosenlaboratory frame are denoted, respectively, by r, andv, d r , /dt. Each element i experiences a net external forceFr ' due to its interactions with agents outside of the sys-tem. In addition it experiences internal forces F,J due to itsinteractions with the other elementsj f i n the system. Thenet internal force on the ith element is

F:nt E

The net force on element i is +The total mass of the N-element system is MmI , m, The

position and velocity of its c.m. are denoted, respectively,by

drc rn 1y m, r, a n d vcra. = —M d t M

The net force on the system is the sum of the net externaland net internal forces; i.e.,

Ftot FE + Fint r x t + z Flnt.

However, because

= F l n t F , J ( F + F j i ) = 0

by Newton's third law, we have Ft,„ = Fe„tThe position and velocity of element i relative to the sys-

tem c.m. are denoted, respectively, by i, mr, — rc „, and_ctit/dt =v i - v c , „ Final ly, the position of the ith ele-

ment relative t o t h e j t h element i s denoted b y,J - r — r1 = if% —

IV, WORK DEFINITIONS

We now consider a process in which the system elementsinteract with each other and/or the external environmentand change their positions relative to each other and/or thelaboratory frame. The laboratory frame is inertial but oth-erwise arbitrarily chosen. The process is understood to takeplace during a well-defined "interaction time period" /-over which all integrals are carried out.

We begin with a definition of the total work done on thesystem—the sum of the works done on each of its elements:

A. J. Mallinckrodt and H. S. Leff 3 5 8

WtM=- E ( 2 )Note that the total work will, in general, include nonzerocontributions from internal forces. We may consider thework done by external and internal forces separately anddefine, respectively, the external and internal works:

AKt, ,/,‘( —1 Mv!.,„ , ( 16)2

AKmt y -1 m,E,) (17)-7' 2

We x t F r • d r i , ( 3 )

wint E Pint-dr, .

= E 17Kt•d(r + )

—J F',„t •dre., + f 17,xt•dr-,

— Wps + Wext

Wint = f Fir•d(rc.„,.

= Fint•dre.m. + P r n • d r ,

Wint

Wtot W e x t + Wint •

359 A m . J. Phys., Vol. 60, No. 4, Apri l 1992

(4)

Equations (2)—(4) define the three works, Wtot, Wextand Wm, which we characterize as "frame specific" (asopposed to "frame dependent"—a term we reserve forcharacterizing the transformation properties o f variousquantities) because each involves integrals over the pathsof the system elements in the inertial laboratory referenceframe. (We shall see that although Wm, is frame specific, itis not •frame dependent. )

A fourth worklike quantity is the pseudowork:

Wps f = Ftc.,"drc F ' , „ t •dr, m( 5)

Wps is also "frame specific" because the motion of the c.m.is relative to our chosen laboratory inertial reference frame.(Some authors prefer to call IV, "center-of-mass work," aname we avoid because of its potential for confusion withthe works defined in the next paragraph. )

We now define three quantities that are analogous toWtot W „ and Wth„ but which involve displacements rel-ative to a nonrotating frame traveling with the system cm.Note that this frame is, in general, not inertial. These "sys-tem-specific" works are defined as follows:

Low, f ( 6 )

Wext E ( 7 )

(8)

The seven works defined in Eqs. (2)—(8) ate not inde-pendent. The following four relationships are readilyderived:

W t o t — W e x t + W i n t ( 9 )

Using these four relationships all seven works can be ob-tained, for instance, from a knowledge of Wtc,„ Wps andWin,. In the next section we show that these three works—and, by extension, all the others—are directly related tochanges in distinctly different forms of system energy.

V. WORK-ENERGY RELATIONSHIPS

By definition, work is a function of a mechanical process.However, we show here that each of the "works" definedabove can be related via a "work-energy relationship" to achange in some purely state-dependent function that werefer to as an "energy." We first derive such relationshipsfor Wmt, Wps , and Wm,

Starting with Eq. (2) we find,

Wtot — I FInt-dr,

1 d v ,= . r n i - •Vdt

EI ni,v), 2

= A K t o t •

1m d ( i T h

—2 "

Here we use our assumption that system elements behaveas point particles that obey Newton's second law and intro-duce the definition,

AKto, 1 — mite) ,i 2for the change in the total kinetic energy of the systemduring the interaction period. By expressing the velocity ofelement i in terms of its velocity relative to the c.m.—i.e.,by using the relationship v, = v, + *,—we find,

AK,o, = 2 milvesm + 1 2 ),

= — 1 m, [ + + 2( vc.fits ) 1), 2

= ,6,(1 1 mi 2

+ m i ( v „ r n-r

+ A ( E ---1 111,), 2

6 , (1 ) + .6,(1 m , i 3 )\ 2 \ , 2

+ m • m , S , )

= .6,Kt„ +

(13)

(14)

(15)where we use the fact that / , m, V, = 0 (by the definition of

( 10) c m . ) and introduce the definitions,

( 11) A K r is the change in the bulk translational kinetic energyof the system—the energy that the system possesses by vir-

( 12) t u e of the motion of its c.m. relative to the observer. AK,„, is

A. J. Mallinckrodt and H. S. Leff 3 5 9

the change in the internal kinetic energy of the system—theenergy that the system possesses as a result of deviations ofthe velocities of its elements from that of the c.m. Thesedeviations can include the effects of vibration and bulkrotation.

Next, starting with Eq. ( 5) we finddv

Fr•drc.„, = f mt dt

= f mA—cit(Vcm -Fiii))•circ.,„d

1 m i d v cdt• v c . m . d t + E J mi—L•drcdt

=-- 1 M + I (-4-11,1 , ) •dr,2 d t

= A( )AK„

This is the well-knowntionship.

Finally, starting with Eq. (4) we find,

Wint 1 1 1 , n t • d r i = F u • d r iI

- E ( J F6-dr, + F i r - d r i )all pairs

w,o, — AKintin which we introduce the following definitions:

(24)

A tf AK , „ , + A4), (25 )A E = A K t r + A U . (26 )

- 1 Ia F i f d ( r i — ri )all pa i rs

- E f F•ci i ;all pairs r i r i

- E f d W#all pairs

( 18 )pseudowork-kinetic energy rela-

= ( — Actly ) = —My, ( 19 )all pa i r s

where we use Newton's third law (Fr = — F„, ) and ourassumption that all internal forces are conservative, andintroduce the definitions,

and

dWiJF• 4•di 0, 60:1) ( 1 1 3 7 , : i

A4) / 1 4 ) r . , ( 2 0 )all pairs

A4) is the change in the internal potential energy of thesystem—the energy that the system possesses by virtue ofthe conservative interaction forces between its elements. Inthe case of a rigid body these would be the "binding forces"and changes in 4) would result from temporary or perma-nent deformations. In the usual fashion it is the assumptionof conservative internal forces that allows us to assert thateach work integral is path independent and to define thepotential energy functions 4),J for each interacting pair ofelements.

Equations ( 13 ), ( 18 ), and ( 19) are work-energy rela-tionships for three of the seven works defined in the pre-vious section. With the help of the four relationships be-tween the works in Eqs. (9)—( 12) we derive the followingfour additional work-energy relationships:

360 A m . J. Phys., Vol. 60, No. 4, April 1992

W e x t

West = A U ,

win, = — Ad),

A U is the change in internal energy of the system, whichincludes both kinetic and potential energy contributions.AE is the change in total energy of the system, which in-cludes changes in both the internal energy and the bulktranslational kinetic energy. Note carefully that there is noterm representing the potential energy of the system withrespect to its surroundings since the contributions of allexternal forces—conservative or not—are accounted foron the "work side" of our work-energy relationships.

The seven works and their six associated energy changesare summarized in Table I, which also gives the defmitionof each quantity, the auxiliary relationships between it andthe others, an interpretation of each work-energy relation,and whether i t is frame dependent or frame invariant.(Transformation properties are discussed i n the nextsection. )

We reiterate that among the seven works and six energychanges there are only three independent quantities. Al -though many sets of "basis" quantities might be chosen,perhaps the most appropriate candidates are: ( 1) A K „ ,which is simply associated with the choice of laboratoryframe; ( 2 ) A K w h i c h characterizes the change in theinternal state of motion of the system; and (3) A4), whichcharacterizes the change in the system's configurationalenergy. Taking these three quantities as our basis we maysummarize the relationships between the quantities in ma-trix form:

W t o t

wex,

w„W W I

Wext

W in t

A K t o t

A UAE

11

1

1to1

11

11

11

(21)( 22 )( 23 )

14

• ( A K „AKim ( 27 )A4)

VI. TRANSFORMATION CHARACTERISTICS

In this section we investigate the transformation charac-teristics o f the previously defined works and energychanges. As indicated by Eq. (27), we need only determinethe transformation characteristics of AKt, , AK , „ „ and Ail)because all other quantities can be obtained in terms o fthese. We determine the values of these energies in a "uframe" traveling with velocity u with respect to the labora-tory frame. The interelement forces are independent of ref-erence frame by assumption and the Galilean transforma-tion to the u frame yields,

A. J. Mallinekrodt and H. S. Leff 3 6 0

Table!. Seven works with definitions, equivalent energy changes, interrelationships, interpretations, and frame-dependence properties. Each of the sevenworks and six energies is precisely defined in terms of the masses, trajectories, and velocities of the particles that make up the system and the forces that acton them during the interaction process.

W,„,

= Wnn + Wird A K „ + AK,„,

E l Fr-dr, A E A K „ + A U

+ w , = . A K „ + AK,n, + Acl)

E I cn•dr,

= W i n t

Work

wr„, E I F I " • di,

Wext + W i r d

we„, E A U A K + Art)

—

= ves,„• — u, $ „ a n d f ; . / =Therefore,

ac ( m Iv c.„ - 1,112)AKtr — u•AP,

Associated energychange

AK,„, --- A( E —1 t n j O, 2

— Eall p a i r s

(28)

(29)where PEEMvc „, , the linear momentum of the system. Ofcourse, AP equals the impulse I = fFext dt on the systemduring the interaction period. The impulse and, therefore,the momentum change AP are manifestly invariant withrespect to Galilean changes of reference frame. Equation(29) makes i t clear that A K „ = AK t', unless u•AP = O.Proceeding:

A K , = A( y m , b:2) A K ( 3 0 )'7' 2and

= — 1 ' u - d e 1 = A4). ( 3 1 )all pairs

Interpretation andframe dependence

The frame-specific total work is equal tothe change in the total kinetic energy.

Frame dependent.

The frame-specific external work is equalto the change in the total energy.

Frame dependent.

The frame-specific internal work is equaland opposite to the change in the internalpotential energy.

Frame invariant.

The pseudowork is equal to the change inthe bulk translational kinetic energy.

Frame dependent.

The system-specific total work is equal tothe change in the internal kinetic energy.

Frame invariant.

The system-specific external work is equalto the change in the total internal energy.

Frame invariant.

The system-specific internal work is equaland opposite to the change in the internalpotential energy.

Frame invariant.

Thus AK a n d A43 are frame invariant while AK„ is framedependent.

Equations (29)—(31) demonstrate that, o f our threechosen basis quantities, only AK„ is frame dependent. As aresult, only those works and energy changes that show adependence o n A K „ i n E q . ( 2 7 ) - - i . e . ,Wtot Wext Wps AKtot a n d SE—wil l themselves beframe dependent. Furthermore, because of their simple ad-ditive dependence on AK„, each will transform in a man-ner identical to Eq. (29).

Two corollaries are immediately evident.Corollary /: For any interaction process with AP = 0, all

works and energy changes are frame invariant andLKtr = a I t then also follows t ha t Wps = 0, tha tWtot = wtot = LiKtot A K , „ „ a n d t h a t Wext — Wext= AE = A U.

Corollary 2 : Fo r any interaction process i n whichAP =0, we can find families of reference frames in whichany given frame-dependent quantity takes on any desiredvalue including, for instance, zero. These families are de-

361 A m . J. Phys., Vol. 60, No. 4, Apri l 1992 A . J. Mallinckrodt and H. S. Leff 3 6 1

termined by the requirement that their frame velocities rel-ative to the lab frame obey u • AP ( value of the quantityin the lab frame) ( desired value). This relationship im-plies that the relative velocity between any two frames insuch a family is perpendicular to AP.

VII. APPLICATIONS

In this section we consider four examples illustrating theapplications of our results.

A. Example 1: Head-on collision

Here we consider a one-dimensional, completely inelas-tic collision between two objects, each of mass m .7 In thecam frame, their velocities prior to the collision are + vand v ; after the collision they are both at rest. We takeour system to comprise both objects. In any frame we findAK„ = 0 because the c.m. velocity is constant. Further-more, because there are no external forces acting on thesystem We, = 0 again in any frame. The second row ofTable I then shows that A U A K , „ , + Acl) = O. Noticethat Km, includes the translational kinetic energies of bothobjects as well as the system's total microscopic kinetice n e r g y ; i . e . , = K t r i + A K t r , 2 + K i n t microscopic. I tis straightforward to show that AK,,, + AK tr,2 = — mv2in any frame. Therefore, AK1„,,,,,,„0,cop,c + 4143 = mti2 ( inany frame) and we conclude that the effect of the head-oncollision is that an amount of energy my' ( which was ma-croscopically observable before the collision in the form ofthe bulk kinetic energy of its two important components) isconverted into macroscopically unobservable internal po-tential energy and microscopic kinetic energy. Notice thatthere is no heat transfer involved in this process—althoughthere might be as a result of the process. The energy trans-formations we are discussing here are calculable, at least inprinciple, by a purely mechanical theory.

I f we define our system to be one of the two objects, wefind highly frame-dependent values for We,„ and AK„ andit becomes more convenient to consider the system-specificworks. For instance, since the contact surface with the oth-er object wil l , predominantly, move toward and exertforces toward the system c.m., we find that we„, = A U> 0;i.e., the internal energy increases due to the compressiveforces of the collision. On the other hand, without furtherdetails we cannot determine the sign of w,„, and, therefore,of M . Although we might anticipate increased internalpotential energy, the collision may just as well act as thecatalyst for its release.

B. Example 2: Free expansion of a gas

Next we consider a gas confined to the left half of anenclosure with the right half entirely empty. The systemelements here are gas molecules. The enclosure is assumedto be "perfectly insulating" so that energy transfer betweenthe gas molecules and the environment is negligible. Wedenote the internal energy of the gas by UO, view the systemin the rest frame of the enclosure, and assume the initialcm. velocity of the gas is zero. Now suppose the partitionseparating the left and right halves of the container spon-taneously collapses allowing gas molecules to fill the entireenclosure.

With the collapse of the partition, the leftward force pre-viously constraining the molecules to the left half becomes

zero. The force from the left wall is still nonzero, so there isa net force directed to the right. Although this force does noreal physical work, it does generate positive pseudowork( because the rightward force acts on the gas while its c.m.moves to the right) that is related to the increased bulktranslational kinetic energy—i.e., Wp, = AK„ > O. A t alltimes, however, we have We„, = 0, so Table I shows thatA U = — AK„ < O. That is, the bulk kinetic energy is de-rived from the internal energy of the gas. Furthermore, i fthe gas is "ideal" in the sense that the intermolecular poten-tial energy cl) = 0 ( or is at least negligibly small) thenA U = K 0 , and we see that the translational kinetic ener-gy is, in a sense, nothing more than "redirected" internalkinetic energy. The nonzero bulk translational energy ofthe system is only temporary, however. As the gas expandsto fill the enclosure, the new right wall begins to exert aforce greater than that from the left wall. This generatesnegative pseudowork (because a net leftward force nowacts on the gas while its cm. is still moving to the right) anddecreases the bulk translational kinetic energy. After beingbrought to rest the cm. may reaccelerate to the left and theprocess may repeat as the gas "sloshes" back and forthuntil the amplitude of the c.m. motion decreases to the levelof statistical fluctuations.

It is instructive to look at the process from the stand-point of the system—specific works also. Just after the parti-tion collapses, the left wall moves to the left ( from thestandpoint of a frame moving with the system cm.) whileexerting a force toward the right and, therefore, performsnegative system-specific external work. Table I shows thattv,„, = A U = AK,„, < 0; i.e., the internal energy of the sys-tem decreases in agreement with the analysis above. Later,when the right wall begins to exert its large leftward forcewhile moving to the left ( again as seen from the systemcm.), positive system-specific external work is done andthe internal energy of the system increases again.

Once equilibrium is reached and the net force is againzero, the gas has its initial internal energy U = U0. I f thegas is "ideal" this means that Km, H o w e v e r , thefinal configuration leaves the N molecules farther apart onaverage and, i f the gas molecules interact according to atypical intermolecular potential energy, this configurationis expected to have an increased potential energy (1). Theconstancy of U then means that the internal kinetic energyhas decreased. We learn in thermodynamics that the "freeexpansion" o f an ideal gas leaves the temperature un-changed while that of an interacting gas leads to a tempera-ture decrease. Thus we find a correlation between tempera-ture and kinetic energy for a classical gas. Although thisconnection is not necessarily valid for other systems," it iscommonly encountered in both kinetic theory and statisti-cal mechanics. The fact that it is suggested by a classicalmechanics argument is noteworthy.

C. Example 3: Slow expansions of an ideal gasHere we consider slow expansions o f an ideal gas

( = 0) under two different special circumstances. I nboth cases the N-element gas is assumed to be in a horizon-tal cylinder, the right vertical wall of which is a movable,frictionless piston. The expansions, whereby the pistonmoves through a distance d, are assumed to be sufficientlyslow that the piston's force Fpwon (leftward) on the gas isequal to the opposing external force Ffixed (rightward) at

362 A m . J. Phys., Vol. 60, No. 4, April 1992 A . J. Mallinekrodt and H. S. Leff 3 6 2

the fixed, left container wall. This infinite slowness meansthat the interaction time r becomes arbitrarily large; suchprocesses are called quasistatic. Of course, Fpiston and F fixedare both expected to decrease in magnitude as the gas ex-pands but this does not affect our analysis.

For the first expansion type we assume that the onlyenergy interactions with the gas are via work done by thepiston. Because the molecules of an ideal gas do zero workon each other,

Wtot = W e t = I - d r < Fpiston a

The inequality holds because the piston force is directedleft while its displacement is directed right. Referring torows I and 2 of Table I ( which are identical for an idealgas), this implies We,„ = AK„ + A K < O. Row 4 tells usthat Wps = 0 because the net force on the system is identi-cally zero at all times. This implies 6,1C„ = 0, as expectedbecause of the slowness of the piston action. Using the lat-ter result in row 1, we have lK„t A K , „ , < 0; i.e., the workon the gas is negative and equals the change in its internalenergy. I n other words, the gas does positive work( — Wext ) on the piston at the expense of its internal ener-gy. The piston's energy does not change because it movesarbitrarily slowly. Of course some other agent, which isunspecified here, must receive the energy given up by thegas. In thermodynamics an expansion where the only ener-gy transfer is mechanical is referred to as an adiabaticexpansion.

For the second expansion type we assume that (some-how) the internal energy of the gas does not change; i.e.,AK„, O . The zero-net-force constraint assures, as in thelatter case, that AK „ O . Thus row 1 of Table I implies

Wtot = W e x t = F m s t „ „ • d r = A K , „ t O .

This contradicts the obvious fact that the integral musthave a nonzero ( in fact, negative) value. ' We concludethat :f it is possible to maintain the constancy ofK du r i nga slow expansion this must come about via nonmechanicalmeans. Referring to the suggestion in Example 2 that theconstancy of Km, can be related to temperature constancywe are tempted to envision the possibility of a slow, nona-diabatic, isothermal expansion during which a nonme-chanical energy transfer occurs. Such expansions are com-monplace in thermodynamics where the nonmechanicalenergy transfer is called heat. Again, classical mechanicshas led to an important thermodynamic concept.

D. Example 4: Conveyor-belt variant

The standard conveyor-belt problem envisions the con-tinuous flow of mass onto a constant-velocity horizontalbelt. Despite its simple description, this problem containssubtleties that have engendered a rather interesting litera-ture." We consider a variant here in which a crate of massm is placed, with zero velocity in the lab frame, onto ahorizontal belt moving with velocity v. As the crate is accel-erated to velocity v in timer, a force Fe is applied to the beltin order to maintain constant speed. Row 2 of Table Ishows that, during the acceleration, the external work doneon the crate is W„„,, = imv2 + A U,. Similarly, the exter-nal work done on the belt

363 A m . J. Phys., Vol. 60, No. 4, April 1992

Wext,b = F e d x + ( — Wext ,c = A K t r , b + A U , = A U ,

because t h e constant-speed constraint insures t h a tAKtr,b =

We assume that the friction force on the crate (and thuson the belt) is constant throughout (0,r) , whence theabove integral reduces to Fed = Fein- and we find fromabove that Four — mu2 = A U, + AUb. The force Fe gen-erates impulse Fer that must equal the change in momen-tum mu of the crate. Thus Feur = mv2 and we have theresult AU + A Ub = -11111)2. The interpretation here is thatthe crate cannot be accelerated without the existence offriction and such friction cannot act without being accom-panied by energy dissipation. We might account for theenergy transformations by saying that half of the workdone by Fe is dissipated into internal energy and the otherhalf goes into the bulk translational kinetic energy of thecrate. We will see, however, that this view of the overallenergy accounting is highly frame dependent.

Consider an analysis of the same situation from the view-point o f the (inertial) frame o f the belt. I n this frame

= — Imv2 + AUG. The force Fe does zero work inthis frame because we assume that the displacement of itspoint of application is zero. Thus

W;xt , ty = + ( — W Lt,c = AK t , + A U, = A U,Combining t h e l a t t e r t w o equa t ions g i v e sA U, + A Ub = imv2, in agreement with our result in thelab frame. It is also easy to verify that the works on the crateand o n t h e b e l t t r a n s f o r m p r o p e r l y ; i . e . ,We'xt,c = Wext ,c — U . A P c = Wext ,c — "11'2 a n d W e ' x t , b— Wext,b — = Wext,b • However, in contrast with theoverall energy accounting given in the lab frame, the sourceof the dissipated energy ( which is frame invariant) nowappears to be the initial kinetic energy of the crate. Theseresults serve to emphasize the slipperiness of trying to per-form this type of energy accounting.

Because the belt's momentum does not change (i.e.,AP, 0 ) , Corollary I ( i n the previous section) showsthat all works on and energy changes of the belt are frameinvariant. On the other hand, since AP, 0 , we may applyCorollary 2 to the crate. Consider a frame moving rightwith velocity v/2 relative to the lab frame. In this frame

= 0 because the crate begins with velocity — v/2just after it is placed on the belt and ends with velocity+ v/2 at full speed. The total work done on the crate in this

f r a m e i s — We'x t ,c + W :nt,c A K i n t , c w h i c h i sequivalent to saying that W ect,c = A U,.

We may locate ye t another frame w i t h velocityv/2 + A K , / m v relative to the lab frame in which thetotal work on the crate vanishes; i.e., W = O. I t isstraightforward t o v e r i f y t h a t , i n t h i s f r ame ,AK = — AKint,c • This example makes it clear that thetotal work can be made to vanish simply by choosing ourinertial frame of observation such that the bulk transla-tional kinetic energy change is equal and opposite to the( frame invariant) internal kinetic energy change. A framein which the external work vanishes can be found in a simi-lar manner.

VII I . SUMMARY AND CONCLUSIONS

Table I summarizes the main findings of this paper. Wereturn briefly to the questions raised in Sec. I. First, how

A. J. Mallinekrodt and H. S. Leff 3 6 3

many types of works are needed? The answer is that threeworks are sufficient to generate the set of seven works thathave been identified. Second, which works are independentof the others? Because of the four relationships that existbetween the seven works ( indicated after the definitions ofWtot Wext Wint and tot°, in the first column of Table I),there are only three independent works. Third, what work-energy relationships exist? These are shown by the asso-ciated energy changes in the second column of Table I,which also gives the elementary definitions of the energychanges and their interrelationships. Fourth, which worksare frame dependent and which are invariant under Gali-lean transformations? These are indicated by a comment inthe third column of Table I and comprise all works andenergy changes that show a dependence on the value ofA K , in Eq. (27).

Finally, can the classical mechanics of a many-particlesystem lead to an understanding of dissipative processes formacroscopic objects? The answer here is a conditional"yes." Classical mechanics enables us to associate dissipa-tion with systems that have hidden energy modes—namelythe modes associated with the large numbers of moleculesthat comprise a macroscopic system. It illustrates that ma-croscopically observable "bulk" energy can be convertedinto internal energy and makes plausible the experientialfact that the reverse process is less easily achieved becausewe have no practical way to control the numerous energytransformations among the molecules that make up an ob-ject. After all, the reversal of a head-on, totally inelasticcollision would require arranging energy transfers among— l e molecules such that they spontaneously exhibitedbulk translational motion. This would be quite remarkable!

On the other hand, if the head-on collision is only slight-ly inelastic then the two colliding objects do recover a frac-tion of their bulk motion as they bounce off each other. Inthis sense elasticity can be associated with memory. Elasticmaterials "remember" their initial configurations—at leastpartially—and facilitate the ordered energy transforma-tions needed to produce bulk motion. Just as elastic pro-cesses exhibit memory effects, inelastic processes do thereverse—erasing memory. For example, if a ball is droppedfrom a height of 1 m, bounces a few times and then comes torest, there is no way to uniquely reconstruct its initial con-figuration from an analysis of the internal energy gains ofthe ball and ground. Loss of information regarding the ini-tial condition can be thought of in terms of a gain of missinginformation. This has nothing to do with a particular ob-server; the information is missing for any observer whosepower of measurement is limited to typical macroscopicresolution levels. The importance of memory effects inthermodynamics is exemplified by Bennett's innovativeanalysis of the Maxwell's demon puzzle, whereby the de-mon is prevented from violating the second law because itsmemory must be erased." Of course, classical mechanicsdoes not predict this loss of information. That requires theapparatus of thermodynamics and/or statistical physics todevelop the second law of thermodynamics.

The results of this paper have led us indirectly and in-completely to the concept of heat. Roughly, conductivebeat transfer can be viewed as external work performed atthe system interface via "collisions" with the elements ofneighboring systems. Though appealing, this explanationfails to make a clear distinction between heat transfer andmechanical work at a surface. That distinction can indeed

be a difficult one. 19 In thermodynamics, i t is sometimesbased upon observable macroscopic work processes andsometimes upon heat transfers between objects with mea-surable temperature differences. While the spirit of classi-cal mechanics is to describe all the elements of a system, thespirit of thermodynamics is to deal only with macroscopi-cally observable quantities. Attempts to bridge the gap be-tween these two disciplines easily lead to the simultaneousconsideration of both macroscopic and microscopic phe-nomena. This is conceptually pleasing but, as in the classiccase of Maxwell's demon, we must be mindful of whatquantities are measurable macroscopically and what quan-tities are useful only from a conceptual viewpoint.'

a) AJMALLINCKRO@CSUPOMONA.EDIJ.HSLEFF@ CSUPOMONA.ED1.1.

ID. Halliday and R. Resnick, Fundamentals o f Physics (Wiley, NewYork, 1988), 3rd ed., pp. 159-162, 193-195.

'H. Erlichson, "Work arid kinetic energy for an automobile coming to astop," Am. J. Phys. 45, 769 (1977).

'C. M. Penchina, "Pseudowork-energy principle," Am. J. Phys. 46, 295–296 (1978).

'B. A. Sherwood, Notes on Classical Mechanics (Stipes, Champaign, IL,1982), Chap. 5. A detailed review of this book was given by J. T. Cush-ing, Am. J. Phys. 51, 958 (1983).

'B. A. Sherwood, "Pseudowork and real work," Am. J. Phys. 51, 597–602 (1983).

6A. B. Arons, "Developing the energy concepts in introductory physics,"Phys. Teacher 27, 506-517 (1989). See also A. B. Arons, A Guide toIntroductory Physics Teaching (Wiley, New York, 1990), Chap. 5.

71I. S. Leff and A. J. Mallinckrodt, "Stopping objects with zero work:Mechanics meets thermodynamics!," in preparation.

8W. H. Bernard, "Internal work: A misinterpretation," Am. J. Phys. 52,253-254 (1984).

911. Erlichson, "Internal energy in the first law of thermodynamics," Am.J. Phys. 52, 623-625 (1984).

'S. G. Canagaratna, "Critique of the treatment of work," Am. J. Phys.46, 1241-1244 (1978).

"H. R. Kemp, "Internal work: A thermodynamic treatment," Am. J.Phys. 53, 1008 (1985).

"G. M. Barrow, "Thermodynamics should be built on energy—not onheat and work," J. Chem, Ed. 65, 122-125 (1988).

141. A. Sherwood and W. H. Bernard, "Work and heat transfer in thepresence of sliding friction," Am. J. Phys. 52, 1001-1007 (1984).

' W e have deliberately avoided introducing potential energy terms to ac-count for the work done by conservative external forces. Although, as apractical matter, using external potential energy can simplify the solu-tion of many problems, its introduction here is at best a needless compli-cation. A t worst, i t raises fundamental questions with respect to the"location" of potential energy. Because potential energy always arises inthe context of an interacting pair (or set) of objects, assigning it com-pletely to one or the other of those objects lacks a rational basis.

"R. Baierlein, "The meaning of temperature," Phys. Teacher 28, 94(1990).

'Another impractical possibility would be to invoke the supervision of aMaxwell's demon to permit piston motions only between the impacts itsuffers from the gas molecules. In this case the expansion is mathemat-ically identical to the free expansion of example 2.

"H. S. Leff, "Conveyor-belt work, displacement, and dissipation," Phys.Teacher 28, 172 (1990).

'Sc. H. Bennett, "Demons, engines and the second law," Sci. Am. 257(5), 108 (1987). Bennett uses the result that memory erasure is a neces-sarily dissipative act, shown first by R. Landauer, "Irreversibility andHeat Generation in the Computing Process," IBM J. Res. Dev. 5, 183(1961). A bibliography on Maxwell's demon may be found in H. S. Leffand A. F. Rex, "Resource Letter MD-1: Maxwell's demon," Am. J.Phys. 58, 201 (1990); and a detailed explanation of memory erasure is in

364 A m . J. Phys., Vol. 60, No. 4, Apri l 1992 A . J. Mallinckrodt and H. S. Leff 3 6 4

H. S. Leff and A. F. Rex, Maxwell's Demon: Entropy, Information,Computing (Princeton U.P., Princeton, NJ, 1990 and Adam Hager,Bristol, Great Britain, 1990).

19G. Laufer, "Work and heat in the light of ( thermal and laser) l ight,"Am. J. Phys. 51, 42-43 (1983). Here it is argued that energy transfer of

Using small, rare-earth magnets to study the susceptibilityof feebly magnetic metals

R. S. Davisa)Bureau International des Folds et Mesures, F-92312 Sevres Cedex, France

(Received 28 August 1991; accepted 23 October 1991)

Small rare-earth magnets have remarkable properties that can be exploited in the construction ofsimple measuring devices. Examples are a torsion pendulum that measures the earth's horizontalmagnetic field and a gravimetric pendulum that responds to the small susceptibilities of various"nonmagnetic" materials. These preliminary demonstrations lead to the design of a practicaldevice for measuring the susceptibilities of feebly magnetic metals in low magnetic fields. Ourlaboratory has used this last method to determine relatively small magnetic susceptibilities, downto 0.0003, to an uncertainty of less than 20%. A convenient feature of our apparatus is thatcalibration by a susceptibility standard is not required.

I. INTRODUCHON

Small but powerful rare-earth magnets are readily avail-able from commercial sources. Cylindrical magnets withdiameter twice their length are so stable that they areshipped already magnetized and no "keeper" is required.The building blocks of the devices to be discussed below arecylindrical magnets of samarium—cobalt' having a diame-ter of 5 mm and a height of 2.5 mm. They have a magnetiza-tion M of about 900 kA/m, aligned along the geometricaxis of symmetry. Although this magnetization ( an inten-sive quantity) is typical of the most powerful permanentmagnets of any size, the magnetic dipole moment m ( anextensive quantity) is conveniently small.

The small magnetic moment and compact shape of thesemagnets lead to the following desirable properties: the di-pole approximation becomes appropriate at about 10 mmfrom the geometric center of a magnet;' at 10 cm from themagnet, the fields are already far too weak to destroy mag-netic coding on credit cards and other tape media; there islittle chance of physical injury to the experimenter as at-tracting magnets are placed together. The last two proper-ties are important safety features, as anyone who hasworked with powerful magnets will agree.

Section IV describes a simple device, based on a magnet-ic dipole, for low-field measurements of the weak magneticsusceptibility of materials such as AISI 304 stainless steel.The two preliminary experiments of Sec. I I I provide in-sight into the magnetic dipole and its interaction with fee-bly magnetic materials. The work presented below was mo-tivated by our need to select metals that are sufficientlynonmagnetic for constructing precision mass balances.This application is also discussed in Sec. IV.

A basic acquaintance with magnetostatics is required inwhat follows although expertise in the rather specialized

365 A m . J. Phys. 60 (4), April 1992

laser light should properly be considered work rather than heat.2°11. Erlichson, "Are microscopic pictures part of macroscopic thermody-

namics?," Am. J. Phys. 54, 665 (1986); B. A. Sherwood, "Reply to 'Aremicroscopic pictures part of macroscopic thermodynamics?,' " Am. J.Phys. 54, 666 (1986).

discipline of permanent magnets will not be necessary. Wehave chosen Duffin's Electricity and Magnetism' as ourprincipal reference in part because it uses SI units through-out. The magnetic dipole plays an important role in ourtheoretical analysis so we begin with a brief review of itsmathematical description.

IL THE MAGNETIC DIPOLE

When a permanent magnet is placed in a uniform B field,it is subject to a torque r,

r = mxtt ( 1 )Equation (1), in Duffin's treatment,' defines the mag-

netic dipole moment m. The direction of m is that of the Bfield when the freely suspended magnet is in equilibrium.The magnitude of m is the torque experienced by the mag-net in a B field of 1 T when m is perpendicular to B. A l -though the units of m are evidently N m T - 1, it is custom-ary to use the equivalent units A m2. Duffin defines themagnetization M as the magnetic dipole moment per unitvolume:5

dm = M ( 2 )where dr is a small volume with magnetic moment dm.

With the magnetic dipole moment defined, we restatethe familiar relations in spherical coordinates for the Bfield in vacuum at distances large2 compared to the magnetdimensions:

2m cosB, = Ito4777 3

; Be =tio m sin 04irr3 Bg, =0,

( 3)where r is the distance from the dipole, 8 is the angleformed by r and the dipole axis, and ito is the vacuum per-

© 1992 American Association of Physics Teachers 3 6 5