Rotational Dynamics and the Flow of Angular Momentum

Transcription

1 Roaional Dynamics and he Flow of Angular Momenum F. Herrmann and G. Bruno Schmid, Insiu für Didakik der Physik, Universiä Karlsruhe, 7618 Karlsruhe, Germany Absrac A novel way o inroduce angular momenum in an inroducory mechanics course is presened. The presenaion is by way of a series of experimens wih wo experimens always appearing ogeher as a pair of analogies: one experimen involving elecric charge and he following analagous experimen involving angular momenum. This kind of inroducion allows he suden, from he very beginning, o become acquained wih he "subsancelike naure of angular momenum, i.e. wih he fac ha angular momenum can be hough o be disribued in and flow hrough space. In paricular, he proposed approach acquains he suden wih he physical naure of angular momenum in mechanics wih he same ease wih which he becomes acquained wih he properies of elecric charge in a radiional approach o elecriciy heory. I. Inroducion This paper presens a novel way o inroduce angular momenum L in an inroducory mechanics course. Our approach differs from more radiional demonsraions (Connolly and Connolly 1973, Klosergaard 1976, Prigo and Reading 1977, Daa 1978) of angular momenum in is srong orienaion oward he elecric charge Q. A whole series of experimens are described wih wo experimens always appearing ogeher as a pair of analogies: one of he experimens involving elecric charge and he following, analogous experimen involving angular momenum. This kind of inroducion has he advanage ha he suden becomes acquained wih he subsance-like naure of angular momenum from he very beginning of he course. Accordingly, he conservaion of angular momenum is jus as naurally expeced by he suden as is he conservaion of charge. Indeed, no paricular effor is necessary o inroduce he idea ha elecric charge is conserved during a ypical beginners course in physics whereas he usual inroducion of angular momenum conservaion via he definiion of L = r x p appears raher conrived. The far-reaching analogy beween elecric charge and angular momenum has is basis in he following facs: 1) Boh are exensive quaniies. ) Boh are subsance-like quaniies (Falk e al-in prin), i.e.a densiy and a curren densiy can be associaed wih each (which is no rue for all exensive quaniies). 3) Boh are conserved quaniies (which is no rue for all subsance-like quaniies). 4) Boh are capable of having posiive or negaive values. A furher characerisic shared by boh elecric charge and angular momenum which does no, however, play a roll in he following consideraions is he fac ha each quaniy has a universal quanum (e and h~, respecively). The analogy beween elecric charge and angular momenum is limied by he fac ha Q is a scalar whereas L is a vecor. Because of his, we will resric our consideraions of angular momenum hroughou his paper o arrangemens whereby he angular momenum vecor L and he angular momenum curren vecor I L poin along fixed direcions in space. This allows us o rea he single, non-zero componens of L and I L like scalars. We denoe hese simply by L and I L, respecively. Of course, his does no mean he pahs by which angular momenum flows in an arrangemen are one-dimensional anymore han he pahs by which elecric charge flows in a nework mus be one-dimensional.

2 The ideas presened in his paper are direced oward firs semeser universiy sudens. However, he overall approach is so elemenary ha i can be adaped wihou difficuly for presenaion o high school sudens as well. Applicaions of our ideas o elemenary and high school physics eaching have already been published elsewhere ( Falk and Herrmann 1981, Schmid 198). II. Descripion of he Course In his secion we presen an ouline of he course. Many deails are lef ou of discussion where we feel hese are obvious and simple enough o be filled in by he reader himself, for example, figures o simple experimens and all quesions concerning unis. In addiion, only hose feaures which we consider novel (and no he enire discussion of angular momenum acually presened in he course such as consideraions of he momen of ineria, he parallel-axis heorem, ec.) are presened here. Mos of he maerial necessary for he experimens presened in his paper should be available in he sock of any well-equipped college. For he mechanical experimens, wo flywheels are needed which can be eiher held by hand or fasened o a able op. The flywheels can be spun up eiher direcly by hand or indirecly wih, say, he edge of a disc sander fied o an elecric drill. For some demonsraions, he flywheels mus be conneced via a fricion coupling. To his end, a shor piece of garden hose can be afixed o one flywheel concenric o is axis and a bole cork (which can be insered ino he hose) o he oher flywheel (Figure la). For one experimen (Secion II.11), he flywheels mus be conneced via a orsion spring. A clock spring serves paricularly well for his purpose. Such a spring can be oufied so ha i auomaically decouples when i is no under ension (Figure lb). The arrangemen for he experimen in Secion II.3 can be easily buil wih he help of a consrucion ki. For he elecrical experimens, he mos frequenly used elemens are capaciors. A circui is needed which lighs a lamp upon he discharging of a capacior. Thus, he capaciance should be of he order of several mf. Accordingly, i will be sufficien o charge he capacior up o abou 10 vols. The inducance of he coil used in he experimen of Secion II.11 should be on he order of a few H so ha he oscillaions are slow enough o be followed easily on a volmeer. In order ha he capaciors are no significanly discharged upon connecing a volmeer across hem, he volmeer mus have a sufficienly high inernal resisance, for example, a volmeer wih a preamp could be used. Fig. 1 a) Two flywheels which can be axially conneced across a fricion coupling by pushing hem ogeher. b) Two flywheels which can be axially conneced across a orsion spring. The flywheels decouple as as soon as he spring is relaxed. II.1 Angular Momenum as Spin A sphere which has been charged wih he help of, say, a van-de-graaf generaor has elecriciy. If he sphere is well insulaed, i keeps his elecriciy. If we carry he sphere around he room, we are also carrying elecriciy around he room. A physicis calls elecriciy 'elecric charge'. A flywheel which has been spun up wih he help of, say, a moor has spin. If he axle of he flywheel is fied wih good bearings, i keeps his spin. If we carry he flywheel around he room, we are also carrying spin around he room. A physicis calls spin 'angular momenum'.

3 II. Angular Momenum Currens If we ouch an elecrically charged sphere o he ground, he sphere will discharge: Elecric charge flows ino he earh. If we ouch a spinning flywheel o he ground, he flywheel will sop spinning: Angular momenum flows ino he earh. If we connec an elecrically charged sphere wih a wire across a resisor o an uncharged sphere, he amoun of elecric charge possessed by he one sphere decreases while ha of he oher increases. We conclude from his ha an elecric charge curren flows from he one sphere hrough he wire and resisor ino he oher sphere. To aid in demonsraing his flow of elecric charge, i is helpful o use an objec which can hold much more elecric charge han a sphere, namely, a commercial capacior. We can repea he above experimen wih each sphere now replaced by a capacior. To make eviden he flow of elecric charge hrough he wire joining he capaciors, he wire can be conneced across an elecric ligh bulb (Figure a). The flow of elecric charge hrough he wire joining he capaciors is evidenced by he fac ha he bulb lighs up. Fig. a) Two capaciors conneced across a ligh bulb. Iniially, he one capacior is charged und he oher uncharged. Upon he flow of elecric charge from he one capacior o he oher, he ligh bulb lighs up. b) Two flywheels conneced across a fricion coupling. Iniially, he one flywheel is spinning and he oher is a res. Upon he flow of angular momenum from he one flywheel o he oher, he fricion coupling heas up. If we connec a spinning flywheel wih a shaf across a fricion coupling o anoher flywheel a res, he amoun of angular momenum possessed by he one flywheel decreases while ha of he oher increases. We conclude from his ha an angular momenum curren flows from he one flywheel hrough he shaf and fricion coupling ino he oher flywheel (Figure b). The flow of angular momenum hrough he shaf joining he flywheels is evidenced by he fac ha he fricion coupling heas up. Consider wo idenical capaciors. Le hem be equally bu opposiely charged and connec hem wih a wire across a resisor. I is easy o deermine wih he help of a volmeer ha, afer waiing a shor while, boh capaciors are compleely discharged. We see ha elecric charge can ake posiive as well as negaive values such ha equal amouns posiive and negaive charge add o zero. Consider nex wo idenical flywheels. Le hem be equally bu opposiely spun and connec hem wih a shaf across a fricion coupling. I can be direcly seen ha, afer waiing a shor while, boh flywheels are compleely a res. We see ha angular momenum can ake upon posiive as well as negaive values such ha equal amouns of posiive and negaive angular momenum add o zero. II.3 Angular Momenum Conducance Consider charging a capacior wih a volage sabilized power supply in each of several differen neworks. Le a differen elemen be buil ino each respecive nework in addiion o he capacior and charging device and follow he charging process on a volmeer. The ime required o charge up he capacior o some arbirary poenial in each case depends upon he naure of he exra elemen buil ino he respecive nework. For example, if his exra elemen is a piece of copper wire he charging process goes quickly; if he exra elemen is a sal bah (Figure 3a), more ime is required; if his exra elemen is simply an air gap, i.e. if he circui is open, he capacior does no charge up a all. From his we conclude ha copper is a good conducor of elecric charge, sal waer is no such a good conducor and air is a nonconducor of elecric charge. Wheher an objec is a good or a poor conducor of elecric charge depends (among oher hings) upon is maerial composiion. The abiliy of a maerial o conduc elecric charge is described in erms of he maerial's conduciviy.

4 Fig. 3 a) An elecrical nework used o charge up a capacior. The bah of sal waer buil ino he circui as a conducor delays he charging process. b) A mechanical nework used o spin up a flywheel. The bah of oil buil ino he circui as a fricion coupling delays he spinning process. Consider spinning a flywheel from res wih a crank, i.e. charging a flywheel wih angular momenum by means of a crank. The crank is roaed a a fixed angular rae hroughou he process. Le any one of several differen elemens be buil ino he shaf of he crank and observe he rae by which he flywheel can be spun up from res in each case. The ime required o spin he flywheel up (o some arbirary angular velociy) in each case depends upon he naure of he exra elemen buil ino he shaf. For example, if his exra elemen is a solid objec, he spinning process goes quickly; if he exra elemen is a fricion coupling, say, an oil bah (Figure 3b), more ime is required; if he shaf is open, he flywheel does no spin up a all. From his we conclude ha a solid rod is a good conducor of angular momenum, hick oil is no such a good conducor and air is a nonconducor of angular momenum. Wheher an objec is a good or a poor conducor of angular momenum depends (among oher hings) upon is maerial composiion. The abiliy of a maerial o conduc angular momenum is described in erms of he maerial's viscosiy. II.4 Measuremen of he Angular Momenum Curren If we send an elecric curren hrough a coil, a magneic field is creaed in he coil. This magneic field can be used o measure he srengh of he elecric curren flowing hrough he wire of he coil (Figure 4a). If we send an angular momenum curren hrough an elasic shaf (or bar), he shaf will be wised. This wis, namely, he angle hrough which he shaf is wised, is a measure of he srengh of he angular momenum curren flowing hrough i (Figure 4b). The (negaive) angular momenum curren is also called 'orque'. Fig. 4 a) Measuremen of he elecric curren srengh via he magneic field produced in a coil. b) Measuremen of he angular momenum curren srengh via he wis produced in a ribbon of seel. II.5 The Coninuiy Equaion for Angular Momenum If he amoun of elecric charge Q conained in a region of space R is changing, a ne elecric charge curren I Q mus be flowing hrough he boundary surface of R (Figure 5a)

5 Consider now wo idenical flywheels. Le one of hese be spun up unil is angular velociy is ω and is corresponding angular momenum is L. Then bring his flywheel ino equilibrium wih he oher, iniially resing flywheel. Because of he symmery of he arrangemen, he angular momenum of each flywheel a equilibrium will be L/. The corresponding angular velociy of each flywheel can be measured and will be found o be ω/. The angular momenum L could jus as well be disribued over an arbirary number of idenical flywheels and he angular velociy of each could be measured. Regardless of he number of flywheels used, he measured angular velociy of each flywheel is proporional o he angular momenum of he flywheel: L ω. Differenly consruced flywheels, however, will no have he same angular momenum a equilibrium, i.e. a he same angular velociy. The proporionaliy facor J beween L and ω is caldq d + I Q = 0 (1) Equaion (1) is he coninuiy equaion in inegral form for elecric charge. If he amoun of angular momenum L conained in a region of space R is changing, a ne angular momenum curren I L mus be flowing hrough he boundary surface of R (Figure 5b) dl d + I L = 0 () Equaion () is he (one-dimensional) coninuiy equaion in inegral form for angular momenum. Fig. 5 a) Illusraion of he coninuiy equaion for elecric charge (Equaion (1) of ex). b) Illusraion of he coninuiy equaion for angular momenum (Equaion () of ex). II.6 Angular Momenum Equilibrium Consider wo differen capaciors wih differen poenial differences across each. Le hem be conneced ogeher via a resisor (like Figure a) and measure he poenial drop across each capacior as a funcion of ime. An elecric curren coninues o flow hrough he resisor unil he volages across boh capaciors are equal. Afer he elecric curren ceases o flow, we say he wo capaciors are in elecrical equilibrium. Now consider wo differen flywheels spinning a differen angular velociies. Le hem be conneced ogeher via a fricion coupling (Figure b) and observe he angular velociies of he wo flywheels as a funcion of ime. An angular momenum curren coninues o flow hrough he fricion coupling unil he angular velociies of boh flywheels are equal. Afer he angular momenum curren ceases o flow, we say he wo flywheels are in roaional equilibrium. II.7 Angular Momenum Capaciy Consider wo idenical capaciors. Le one of hese be charged up unil he poenial difference across i is U and he corresponding amoun of charge on each plae is Q. Then bring his capacior ino equilibrium wih he oher, iniially uncharged capacior. Because of he symmery of he nework, he charge of each capacior a equilibrium will be Q/. The corresponding poenial drop across each capacior can be measured and will be found o be U/. The charge Q could jus as well be disribued over an arbirary number of idenical capaciors and he poenial drop across each could be measured. Regardless of he number of capaciors used, he measured poenial drop across each capacior is proporional o he charge of he capacior: U Q. Differenly consruced capaciors, however, will no have he same charge a equilibrium, i.e. a he same poenial drop. The proporionaliy facor C beween Q and U is called he 'capaciance' of he capacior. The capaciance is a measure of he charge of a capacior for a given poenial drop across i.

6 led he 'momen of ineria' of he flywheel. The momen of ineria is a measure of he angular momenum of a flywheel for a given angular rae of roaion. II.8 Energy ransfer hrough roaing shafs Energy can be ransferred from a baery o a moor (Figure 6a) wih he help of a wire. In he process, an elecric charge curren flows in a closed loop: from he baery hrough he wire o he moor and back again o he baery hrough he earh. The energy curren (or power ) I E depends boh upon he poenial difference U beween he wire and he earh as well as upon he elecric charge curren I Q according o he relaion I E = U I Q. If he moor is shored ou of he circui, U = 0. Accordingly, I E = 0 even hough an elecric charge curren I Q coninues o flow hrough he nework. If, on he oher hand, he circui is opened a he moor so ha I Q = 0, I E will vanish even hough U 0. Fig. 6 Illusraion of he he ransfer of energy in (a) an elecrical or (b) a roaional sysem. a) Energy is being ransferred from a baery o a moor wih he help of a wire. b) Energy is being ransferred from a moor o a coffee grinder wih he help of a shaf. Energy can be ransferred from a moor o a coffee grinder (Figure 6b) wih he help of a shaf. In he process, an angular momenum curren flows in a closed loop: from he moor hrough he shaf o he coffee grinder and back again o he moor hrough he earh. The energy curren I E depends boh upon he difference ω in angular velociy beween he shaf and he earh as well as upon he angular momenum curren I L according o he relaion I E = ω I L. If he coffee grinder is shored ou, i.e. if he end of he shaf is rigidly conneced o he earh, hen ω = 0. Accordingly, I E = 0 even hough an angular momenum curren I L coninues o flow hrough he moor, shaf and earh. (Tradiionally, one speaks of a orque acing in he moor, shaf and earh in his case). If, on he oher hand, he shaf is allowed o roae freely so ha I L = 0, i.e. so ha he orque in he shaf vanishes, I E will vanish even hough ω 0. In Figure 6a, one commonly says ha energy is being ransferred in he form of elecric energy from he baery o he moor. However, we prefer o say in his case ha energy is being carried (Falk e al-in prin, Schmid 198) by elecric charge from he baery o he moor since he erm elecric energy suggess here is more han one physical quaniy: energy. In Figure 6b, one commonly says he moor is doing work on he coffee grinder. However, we prefer o say in his case ha energy is being carried by angular momenum from he moor o he coffee grinder since he erm work convers up he fac ha he occurrence being deal wih is an energy ransfer process, i.e. ha he relevan physical concep is ha of an energy flow. Furhermore, speaking abou work insead of an energy curren does no bring ou he analogy beween his example and he one illusraed in Figure 6a (as well as several oher analogies which are radiionally hidden under differen names). In Figure 6a, he elecric poenial is a measure of how much energy he energy carrier curren I Q is loaded wih. In he same way, in Figure 6b, he angular velociy is a measure of how much energy he energy carrier curren I L is loaded wih.

7 II.9 The Gear Box: An Angular Momenum Transformer An energy curren flows hrough an elecric ransformer simulaneously wih he flow of an energy carrier curren, he elecric charge curren I Q. A he one side of he ransformer, he carrier curren I Q is large and he energy load facor U is small. A he oher side of he ransformer, he carrier curren is small and he energy load facor is large (Figure 7a). Fig. 7 a) Elecric ransformer. b) Gear Box, an angular momenum ransformer. An energy curren flows hrough a gear box simulaneously wih he flow of an energy carrier curren, he angular momenum curren I L. A he one side of he gear box, he carrier curren I L is large and he energy load facor ω is small. A he oher side of he gear box, he carrier curren is small and he energy load facor is large (Figure 7b). II.10 Flywheels and Torsion Springs as Energy Conainers While an elecric charge curren is flowing ono a capacior plae, an energy curren is flowing ino he capacior. Energy is being sored in he capacior. The amoun E of sored energy can be calculaed by inegraing he ne inflowing energy curren I E over ime: E = I d UI d C UdU C E = Q = = U U Here use has been made of he relaion I Q = C U/ (from Q = C U) beween he charge curren and he ime-rae-of-change U/ of he volage across he capacior. C is he capaciance of he capacior. While a volage is being applied across a coil, an energy curren is flowing ino he coil. Energy is being sored in he coil. The amoun E of sored energy can be calculaed according o Q L E = IEd = UIQd = L IQdIQ = I I Q Here use has been made of he relaion U = L I Q / beween he poenial difference U across he coil and he ime-rae-of-change I Q / of he elecric charge curren I Q flowing hrough he coil. L is he inducance of he coil. While an angular momenum curren is flowing ino a flywheel, an energy curren is flowing ino he flywheel. Energy is being sored in he flywheel. The amoun E of sored energy can be calculaed by inegraing he ne inflowing energy curren over ime:

8 J E = IEd = ωild = J ωdω = ω ω Here use has been made of he relaion = J ω/ (from L = J ω) beween he angular momenum curren I L and he ime-rae-of-change ω/ of he angular velociy. J is he momen of ineria of he flywheel. While an angular velociy difference is applied o he ends of a orsion spring, an energy curren is flowing ino he spring. Energy is being sored in he orsion spring. The amoun E of sored energy can be calculaed according o IL E L E= Id= ωid= D IdI = L L D I L Here use has been made of he relaion ω = D I L / (from α = D I L, where α is he angular displacemen) beween he angular velociy difference ω across he spring and he ime-rae-of-change I L / of he angular momenum curren I L flowing hrough he spring. D is he inverse roaional compliance of he orsion spring. II.11 Oscillaions: The Torsion Pendulum Consider wo capaciors wih he same capaciance conneced across a swich hrough a coil. Le he charge of each capacior be moniored wih wo separae volmeers (Figure 8a). Assume an iniial sae wih he swich open: one capacior, say, capacior 1 is charged up and he oher capacior (labelled ) is uncharged. Afer he swich is closed, oscillaions can be observed in he readings of boh volmeers. These readings indicae harmonic oscillaions in he elecric charge of he plaes of each capacior, in he srengh of he elecric curren flowing beween he capaciors, in he elecric poenial difference across each capacior, in he energy sored wihin each capacior and in he flow of energy beween he capaciors. Consider now wo flywheels wih he same momen of ineria. The flywheels can be conneced along heir axes hrough a orsion spring (Figure 8b). Assume an iniial sae wih he flywheels disconneced: one flywheel, say, flywheel 1 is spinning and he oher flywheel (labelled ) is a res. Afer he flywheels are conneced, harmonic oscillaions can be observed in he angular momenum of each flywheel, in he srengh of he angular momenum curren (orque) flowing beween he flywheels, in he angular velociy difference beween each flywheel and he earh, in he energy sored wihin each flywheel and in he flow of energy beween he flywheels. Fig. 8 a) Two capaciors wih he same capaciance conneced across a swich hrough a coil. The charge of each capacior can be moniored wih wo separae volmeers. b) Two flywheels wih he same momen of ineria conneced along heir axes hrough a spring. If a recifier is buil ino he circui of Figure 8a, he oscillaions in he nework las for only half a period. A he end of his ime, capacior is charged, capacior 1 is discharged. If he flywheels of Figure 8b are coupled along heir axes so ha hey auomaically decouple whenever he angular momenum curren changes direcion, he oscillaions in he arrangemen las for only half a period. A he end of his ime, flywheel is spinning, flywheel 1 is a res. (This process is someimes called an elasic collision. The approach o equilibrium menioned above in Secion II.6 is called an inelasic collision ).

9 II.1 Charged and Uncharged Conducors of Angular Momenum When an elecric curren is flowing hrough a wire, he wire is almos elecrically neural, i.e. he posiive and negaive charge wihin he wire compensae one anoher. However, here are also elecric currens wih a large non-zero charge densiy, for example, he flow of elecric charge wih a curren of elecrons in a elevision ube. When an angular momenum curren is flowing hrough a rod, he angular momenum of he rod is almos zero. However, here are also angular momenum currens wih a large non-zero angular momenum, for example, he flow of angular momenum wih a curren of polarized neurons in an acceleraor arrangemen. A curren of polarized elecrons represens boh an elecric charge curren and an angular momenum curren (plus currens of several oher subsance-like quaniies as well). III. The Choice of Analogy An analogy of he ype presened above in Secion II can be jus as well carried ou beween elecric charge and linear momenum p. This migh already have been expeced, since he analogy beween angular momenum and linear momenum is well-known. Indeed, we have inroduced linear momenum in exacly his way in a beginners mechanics course a he Universiy of Karlsruhe. Of course, one difficuly wihin his presenaion is inheren o he fac ha, unlike elecric charge or angular momenum, a body charged wih linear momenum canno be simply held in one's hands and carried around he lecure hall: The naure of a body conaining linear momenum is o move across he room on is own! In a cerain sense, i would be more physically saisfying o choose he following analogy: elecric charge linear momenum and elecric dipole momen angular momenum. A comparison of he elecric dipole momen wih angular momenum is, on he one hand, very insighful especially concerning discussions of he angular momenum densiy. Indeed, he dipole momen densiy is an esablished quaniy wellknown under he name of elecric polarizaion. On he oher hand, his comparison is no so far-reaching since he elecric dipole momen is no a conserved quaniy like angular momenum. Finally, we would like o make a few commens abou oher more familiar analogies beween mechanics and he heory of elecriciy as hese are usually referred o or sysemaically presened in manyex books (Olson 1958, MacFarlane 1964). These do no, as we do, compare Q wih L (or Q wih p) and U wih ω (or U wih he linear velociy v). However, because of he logical symmeries wihin mechanics and elecrodynamics hemselves, hese analogies are, from a formal sandpoin, no less jusifiable han he one presened in his paper. Neverheless, he usual analogies suffer from a major concepual drawback: The conserved subsance-like quaniy Q does no correspond o he conserved subsance-like quaniy L bu, raher, o he angle α o which here is neiher a curren (dα/d is no angle curren) nor a densiy, nor a conservaion law. IV. Conclusions The analogy beween elecric charge and angular momenum presened in his paper is advanageous: 1) For he economy of hough i inroduces ino one's undersanding of how angular momenum and elecric charge can be reaed in mechanics and elecrodynamics. Economy of hough has always been an imporan aspec o every physical analogy. ) For he concepual clariy i inroduces ino one's undersanding ino he physical naure of angular momenum. The radiional inroducion of angular momenum L via he equaion L = r x p does no bring ou he wo main poins emphasized by he analogy presened in his paper, namely, (i) ha angular momenum is a physical quaniy of is own, which in general is independen of he linear momenum p and (ii) ha angular momenum is jus as subsance-like in naure as elecric charge (or as mass or as amoun of subsance or as enropy). Emphasizing hese wo poins upon he inroducion of angular momenum as a physical quaniy enables one o have jus as firm an inuiive grasp upon angular momenum as one radiionally has upon elecric charge.

10 References Connolly W C and Connolly H C 1973 Am.J.Phys. 41(1) Daa Am.J.Phys (11) Falk G and Herrmann F 1981 Neue Physik: Das Energiebuch (SchroedelVerlag Hannover) Falk G, Herrmann F and Schmid G B Energy Forms or EnergyCarriers? Am. J. Phys. (in prin) Klosergaard H 1976 Am. J. Phys. 44 (1) 1 MacFarlane A G J 1964 Engineering Sysems Analysis (George G. Harrap & Co. Ld., London) Olson H F 1958, Dynamical Analogies (D. Van Nosrand Co., Inc., Princeon, New Jersey) Prigo R B and Reading M 1977 Am. J. Phys. 45 (7) Schmid G B198 Phys.Educ. 17, 1-18