Laes Tends in Applied and Theoeical Mechanics New mehod o explain and calculae he gyoscopic oque and is possible elaion o he o elecon BOJIDAR DJORDJEV Independen Reseache 968 4- Dobudja see, Ezeovo, Vana egion BULARIA bojidadj@di.bg Absac: - I a body oaes abou axis X and uns abou Y, a gyoscopic oque geneaes abou Z. Classical Mechanics explains and calculaes i by means o he mehod o he veco muliplicaion. I supposes a linea dependence on he coelaion beween he angula speeds o uning and oaion. The pape suggess a new way o examine he naue o he gyoscopic oque based on he undesanding ha i is a esul o he ineial eec o he changed diecion o he obiing masses in he plane o uning. I leads o a new omula showing ha he gyoscopic oque depends on he sine uncion o he coelaion beween he angula speeds o uning and oaion. We ind ha boh omulas calculae almos equal esuls i he angula speed o oaion is much bigge hen he one o uning. Then we ind he condiion. Exploing he sine uncion we ind ha he body does no geneae gyoscopic oque i he angula speed o oaion is / o he one o he uning. Possibly, i coesponds o he / o elecon. Key-Wods: - Classical mechanics, veco muliplicaion, gyoscopic oque, o elecon Inoducion I a lywheel (body) oaes abou axis X wih angula speed ω and in he same ime uns abou Y wih angula speed ω i geneaes a gyoscopic oque τ abou axis Z, pependicula o he is wo. Classical mechanics explains he phenomenon and calculaes is magniude by he mehod o he veco muliplicaion. The mehod can be ound in evey exbook. Fo example Feynman [], (chape 0, page 0-4) expesses he veco equaion (), whee L and J ae he vecos o angula momenum and ineia momenums o oaion. τ L ω Jω ω () Abou he diecions deemined by he mehod o he veco muliplicaion Feynman woe: [], he ac ha we say a igh-hand scew insead o a le-hand scew is a convenion, and is a pepeual eminde ha i a and b ae hones vecos in he odinay sense, he new kind o veco which we have ceaed by a x b is aiicial, a slighly dieen in is chaace om a and b, because i was made up wih a special ule. i.e. he diecion o he geneaed oque is deemined by a special ule. The magniude o he geneaed oque is equal o he ineial momen J imes he angula speed o oaion imes he one o he uning (). The ansomed in he shape () elaion () deemines a linea dependence o he magniude o he geneaed oque on he coelaion beween he angula speeds o uning ω and oaion ω. E k is he kineic enegy o oaion. ω ω τ vm Jω Ek () ω ω Poblem Fomulaion Veco muliplicaion is a model coveed by special ules as Feynman explained. I is no clea how his special ule coesponds o he undamens o mechanics. I can no explain he sense o he o he elemenay paicles and moe specially he / o elecon since i inoduces a linea dependence o he geneaed oque on he coelaion o he angula speeds o oaion and uning. In is un i leads o he undesanding ha does no coespond o Classical mechanics. 3 Poblem Soluion The Auho s eseach shows ha he gyoscopic oque is esul o he ineial eec o he change o he diecion o he obial moion/momenum in he plane o uning. We develop he idea in ew seps. ISBN: 978-960-474-377-3 55
Laes Tends in Applied and Theoeical Mechanics 3. The ineial eec o he changed diecion Fis le s deemine he ineial eec o he changed diecion, alhough he easoning is no new. Le s have a mass m, Fig., moving wih consan speed v and adius R along he ac ABC. I in he poin A he diecion o he obial velociy is along he angenial line -, and especively in he poin C he diecion is aleady along he angen -, he acual change o he diecion is wih he angle α beween he boh lines. elaion (4) can be aken as a developed om he one o he ceniugal oce. 3. The new mehod o calculae he gyoscopic oque Le s apply he connecion (4) o he case o gyoscope. Fig.. The ineial eec o he changed diecion Le s imagine ha insead along he ac, he mass m a he poin A moves along he line -. Le s apply on he mass an impulse F - mv on he way o make i o sop in he poin D. Then le s apply an impulse F - mv o make i o move along he line - in he way o each he velociy v in he poin C. In ac, we eplace he moion along he ac ABC wih a moion along he lines - and - wih sop in he poin D. The ineial eec o he changed diecion is equal o he veco sum o he boh impulses (3). Finally we eceive he elaion (4). The ineial eec o he changed diecion is impulse (4) acs on he poin B along he line B-D i.e. pependicula o he angen a he convexiy o he ac. The omula in no new, o example i is used o calculae he Ruheod Backscaeing [] (see elaions () and (3) om []). On he ohe hand, o vey small ime and angle α when sine o small angle is equal he angle, i ansoms o he well known omula o he ceniugal oce F c mrω. Hence, he Fig.. a/ wo masses in he plane o oaion, b/ a single mass in he plane o uning Imagine ha a single mass m o an elemenay mass m, belonging o a massive body oaes wih consan angula velociy ω aound he axis X, pependicula o he page, Fig. a. Fo evey π peiod o oaion, he mass leaves poin A and aives a poin C descibing he ac ABC in he plane o oaion. Respecively, he opposie mass m descibes he ac CDA. Theeoe, he masses move along a saigh line ABC (CDA) in he plane o page pependicula o he axis Y, Fig. b. Bu i a he same ime he axis o oaion X uns wih consan angula velociy ω aound Y Fig. b, he masses aive a poins C o A in he space insead F F + F (3) o a poins C o A, descibing a 3D acs. Tha is o say, hese masses descibe he acs ABC and CDA in he plane pependicula o he axis Y. Duing he nex π peiod o oaion, he mass m α descibes he F mvsin (4) ac C DA while (4) m descibes A BC and so on and so oh. Evey elemenay mass, belonging o he oaing body, descibes acs in he plane pependicula o he axis o uning Y. Theeoe, an ineial impulse o he changed obial diecion in he plane o uning eleases. Accoding o he subsecion 3., we can accep ha i eleases in he poins B and D pependicula o he plane o oaion when he given mass is a hese poins o space i.e. a he convexiy o he ac. Since he convexiy o A n BC n+ ac is opposiely dieced o he convexiy o he C n DA n+ one, he diecions o he impulses acing on poins B and D ae opposiely dieced. Since hey ac a he opposie ISBN: 978-960-474-377-3 56
Laes Tends in Applied and Theoeical Mechanics sides o axis Z elaive Y hey ceae he same dieced angula impulses aound Z, Fig.a. This way o explanaion ees o he Feynman s undesanding om Fig. 0-4 o [], whee he beoe posiion o he mass, m is a poin A n, he now one i is a B and he ae one i is a C n+. The beoe posiion o he mass m is a poin C n, he now is a D and he ae one is a poin A n+. We can assume ha he impulse F (4) o he changed diecion o he given mass eleases a he poins B and D peiodically, in poions (sages) o evey π (o a hal) evoluion aound X om A n o C n+ and om C n o A n+. We call his phenomenon a π-quanizaion o a hal-quanizaion. Using his aiocinaion le s calculae he gyoscopic oque τ new geneaed abou axis Z by he impulses (4) o he changed diecion o a single mass m in he plane o uning Fig.b. I o one π- quanum (a hal evoluion abou axis X) he mass m descibes ac A n BC n+, o he nex π-quanum i descibes he ac C n+ DA n+. The mass descibes a given numbe o acs ABC, C DA, A BC 3, C 3 DA 4 pe second. Fo evey ac descibed he mass eleases an impulse (4) a poins B o D pependicula o he plane o oaion when he mass is a ha poins. The numbe o he π quanum pe second N π is equal o he numbe o he descibed acs pe second, which is equal o he numbe o he impulses (4) pe second. We calculae N π dividing he angula speed o oaion ω o π (5). Dividing he angula speed o uning ω o he numbe o he π- quanum pe second N π we eceive he angle o delecion α o he obial speed a he poins B and D in he plane o uning o evey π quanum (6). The elaion (7) shows he magniude o one impulse o he changed diecion geneaed by one elemenay mass o one π- quanum. Muliplying boh sides by he adius R (he disance O-B o O-D) we deemine he magniude o one angula impulse (8). To deemine he magniude o he oque τ new geneaed o one second we muliply boh sides by he numbe o he π- quanum pe second N π (9). τ new πω F mvsin (7) ω πω FR mvrsin (8) ω π ω FRNπ mvrω sin (9) π ω We eplace he obial speed o he elemenay mass v by he angula speed o oaion imes he adius o oaion vω R (0). Then eplacing he momen o ineia o a poin (elemenay) mass m a a disance R by JmR we ge an equaion (). The equaion () expesses he ole o he kineic enegy o he oaing E k. We assume ha using he new mehod o analysis, we deemine he magniude, he axis and he diecion abou he axis o he geneaed gyo oque τ new wihou he help o he veco muliplicaion. τ new τ τ π ω mr ω sin (0) π ω new 3.3 Fis Check π ω Jω sin () π ω new E k 4 π ω sin () π ω ω N π (5) π ω πω α (6) N ω π Table. I compaes he esuls calculaed by he τ vm and τ new omulas o dieen coelaions ino angula speeds o oaion ond uning ISBN: 978-960-474-377-3 57
Laes Tends in Applied and Theoeical Mechanics We have on hand wo ival omulas, he one o he veco muliplicaion () and he new one () o/and (). Le us y o ind how hey wok o dieen coelaions beween he angula speeds o uning and oaion given in he is column a he Table. We ake ha he momen o ineia o he given body is equal o one (J). The unis o ω, ω, τ vm and τ new ae no given because he inal pupose is he calculaion o he pecenage o elaive dieence d, displayed in he las column. Compaing he esuls, we ind ha boh omulas calculae esuls wih negligibly small dieence, i he speed o oaion is much geae han he one o uning. 3.4 Second Check Le us y o ind he mah condiion unde which boh omulas () and () calculae equal esuls. Equaing boh omulas, we eceive (3) and (4). Then muliplying boh sides o (4) by (π/ω ) we eceive (5). Replacing he connecion o he angle o delecion α (6) we eceive (6). τ vm τ new (3) ω π ω π Jω sin ; Jω (4) ω π ω ω π ω π ω sin (5) ω ω α α sin (6) The igonomeic condiion (6) shows ha boh mehods calculae equal esuls i he sine o hal o he angle o delecion om Fig. b is equal o he hal o he angle. Tha is o say, he boh mehods calculae equal esuls i he angle o delecion o he obial momenum in he plane pependicula o he axis o uning o evey π- quanum o he oaion is small enough o saisy condiion (6). The equaion (6) shows ha he condiion (6) is saisied i he angula speed o oaion is much geae han he one o uning, ω >>ω, especively i ω <<ω. We call i a Classical yoscopic condiion because i deemines he join ange o he boh mehods. Bu wha i he angula speed o oaion is jus geae han he one o uning (ω >ω )? Wha i boh angula speeds ae equal (ω ω )? O wha i he angula speed o oaion is lowe han he one o uning (ω <ω )? Obviously, hese cases say ou o he ange o he mehod o he veco muliplicaion because i i calculaes a coec esul only i he Classical yo condiion (6) is saisied, he uncion sine o a given angle calculaes an always-coec esul independenly, i he angle is small enough o saisy condiion (6), no small enough, big o vey big. We can assume ha he Classical gyo condiion (6) does no show he limi o he gyoscopic popeies. I demonsaes he limi o he popeies o he veco muliplicaion as o explain gyoscope. 3.5 Sine uncion analysis We can pesen he new elaion o geneaed gyo oque () as a muliplied consan and acos (agumens). Le us denoe a yoscopic Consan K g /π (7) and hen he yoscopic Faco (8) as a coelaion beween he angula speeds o uning and oaion. Then we pesen he Qualiaive yoscopic Faco q g (9) as a sine uncion o imes / K g. The kineic enegy o oaion E k plays he ole o he Quaniaive yoscopic Faco Q g (0). The equaion () shows ha he magniude o he geneaed oque is equal o wo imes he gyoscopic consan K g imes he Quaniaive yoscopic Faco Q g imes he Qualiaive yoscopic Faco q g. q g new K g (7) π ω (8) ω sin (9) K g Q g E k (0) τ K Q q () Obviously, he behavio o he Qualiaive yo Faco q g as a sine uncion o he coelaion beween he angula speeds o uning and oaion deemines he qualiaive behavio o he geneaed gyoscopic oque. Fig.3 shows he change o he q g as a sine uncion o. in he ange om -4 o 4. We can see ha q g 0 and heeoe he geneaed g g g ISBN: 978-960-474-377-3 58
Laes Tends in Applied and Theoeical Mechanics gyo oque is equal o zeo, i ω 0. This is he socalled Cenal o Classical zeo. The igh side owad he Cenal zeo is he zone o he posiive whee he coelaion beween angula speeds o uning and oaion is posiive. The le side is he zone o he negaive. The hin sip occupying boh sides close o he Cenal zeo is he Classical yo zone whee he Classical yoscopic condiion (6) is saisied. The zone o a Squae gyoscope is deemined on he condiion ha he angula speed o uning is equal o he one o oaion i.e. is equal o plus/minus one. A Supe gyoscope occupies posiive and negaive zones (in blue) beween a Squae and a Classical gyoscope whee he coelaion beween boh angula speeds is less han one bu no low enough o saisy he Classical yo Condiion (6). Obviously, he Hype gyoscope (in geen) esponds o he condiion >. Fig.3. Qualiaive gyo aco q g as a uncion o yo aco As we can see, he sine uncion acceps is maximal value o one (q g +/-; whee +/-) when he angula speed o oaion is equal o he angula speed o uning. This is he Squae gyoscope. The physical explanaion is ha o evey hal evoluion abou axis X (π-quanum) he plane o oaion complees hal a evoluion abou axis Y, pependicula o he page, Fig. b. Tha is o say ha evey mass m (o m ) leaving poin A n (o C n ) in he space aives a he same poin A n (o C n ) compleing he ac A n BA n (C n DC n ) wih maximum possible angle o delecion equal o π i.e. a closed 3D cuvaue. Beyond his poin o coelaion begins a coundown whee he eal angle o delecion deceases. We can noe ha he Qualiaive yo Faco is also equal o one i he angula speed o uning is +/-3, +/-5, +/-7 imes bigge han he one o oaion i.e. i acceps posiive o negaive odd whole numbes (ineges). As we have menioned, evey ime he sine uncion cosses he abscissa he geneaed oque (gyo couples) becomes zeo. Tha is o say, ha he gyoscope is sable because i does no lose enegy o geneae a gyo oque i.e. i is in a poenial well. The uncion acceps zeo i 0, +/-, +/-4, i.e. i acceps zeo and posiive o negaive even ineges. Howeve, hee ae some dieences. I ω 0 a he Cenal (Classical) zeo gyoscope conseves he oienaion o is plane o oaion in space. This is well-known popey used in gyocompasses. Bu i ω ω, ω 4ω and so on, he gyoscope does no geneae oque alhough is plane o oaion uns abou he axis o uning. The physical explanaion o ha phenomenon is ha i he angula speed o uning is equal o zeo, in he ame o evey π- quanum, evey elemenay mass moves along he line A n BC n (o C n DA n ) in he plane in uning, Fig. b. Theeoe, he angle o delecion is equal o zeo i.e. he geneaed gyo oque is equal o zeo. Bu when he angula speed o uning is wo (ou, six ) imes geae han he one o oaion, o evey π-quanum he enie body complees a whole evoluion (o, o moe) abou he axis o uning. I makes so ha inally he mass om poin A n aives a poin C n (om C n o A n ) he same as when he angula speed o uning is zeo i.e. i he angle o delecion is equal o zeo. Theeoe, a he Hype zeos, hee is no geneaion o gyoscopic oque abou axis Z, i.e. he degees o eedom o he gyoscope ae disconneced (isolaed), despie ha he body oaes abou X and simulaneously uns abou Y. On he ohe hand, disubance coming om he ouside can geneae oscillaions abou he zeos. Fo example, we know nuaion as an oscillaion abou he Cenal (Classical) zeo. Similaly, oscillaions can be povoked aound he Hype zeos. Bu i an oscillaion abou he Classical zeo is expessed as an oscillaion o he plane o oaion, he oscillaion abou evey Hype zeo is expessed as an oscillaion o he coelaion beween he angula speed o uning and oaion. In ac, evey divesion o he coelaion in posiive and negaive diecion connecs he degees o eedom posiively o negaively. 3.6 Possible connecion o he o elecon The exising Quanum heoy as o example inoduced in [3] and [4] saes ha he chaged paicles, (emions, lepons including elecon...) ISBN: 978-960-474-377-3 59
Laes Tends in Applied and Theoeical Mechanics ae wih /. The unchaged paicles (bosons) ae wih. Femions wih ohe s including 3/ and 5/ and bosons wih ohe s as 0,, and 3 ae no known o exis, even i heoeically pediced. In 03, he Higgs boson wih 0 has been poven o exis. All o his educes he possible numbes o vey ew: 0, / and. Wha is? Fo example, Shanka, [3], ch.4 woe: I ollows ha elecon has ininsic angula momenum no associaed wih is obial moion. This angula momenum is called, o i was imagined in he ealy days ha i he elecon has angula momenum wihou moving hough space, hen i mus be ning like a op. We adop his nomenclaue bu no he mechanical model ha goes wih i, o a consisence mechanical model doesn exis. Nobuly, [4] ch.0.4 saes: As nicely explained his angula momenum is ininnsic o he elecon and does no aise om obi eecs. Can we elae he popeies o he / o he popeies o he Hype zeos om Fig. 3 and he dependence ()? I we accep ha o elecon is a physical oaion o he elecon abou is axis wih an angula speed / o he speed o uning abou a pependicula axis, an elecon does no geneae gyo oque abou he axis pependicula o he is wo. Tha is o say ha an elecon acs as a Hype gyoscope wih a yoscopic aco +/-. This elecon exiss in he poenial wells because he degees o eedom ae disconneced and an elecon o acually an elecon-nuclei sysem does no lose kineic enegy. In ac, he o an elecon e is ecipocal o he yoscopic aco a he Hype zeos + and - (). e ± ω ± ± ω ± () Suppoed by he analysis om he above we eun o he oiginal undesanding ha elecon oaes like a op. 3.6.. The ininsic (pinciple, pimay) axis poblem As we acceped, expesses a single physical oaion abou an axis. On he ohe hand a gyoscope woks i hee ae wo oaions (o oaion and uning accoding o he Auho s eminology) aound a pependicula axes leading o geneaed oque abou he hid axis. Evey oaion needs an axis o deemine i, so an ininsic equies a clealy deemined ininsic axis. The unchaged paicles oae () abou hei axis eely. They eceive uning abou a pependicula axis causing gyo eecs in special cases like o example collusion and diacion. Unlike hem, he chaged paicles exis nomally in a sysem conneced o opposie chaged paicles because o he aacive oces. They oae () eely only when due o ceain easons hey lose he sysem. Hence, exising in a sysem wih nuclei, an elecon eceives a second moion belonging o he sysem in addiion o is ininsic. On he ohe hand, i a he +/- Hype zeos he speed o uning is wo imes geae han he one o oaion, he kineic enegy o uning is ou imes geae han ha o oaion (o spheical body). Theeoe, an elecon cans easily swich ove he axes o and uning. This canno happen only i an elecon s axis o s is clealy deemined by an ininsic popey o he elecon. Fig.4. Elecon-nuclei sysem The abovemenioned consideaions/equiemens can be saisied i we suppose ha he cenes o chage and mass o he elecon ae divided a some disance p, Fig. 4. The cenes o chage and mass o he nuclei ae also divided, bu since he nuclei o dieen chemical elemens and isoopes consis o dieen numbes o poons and neuons, he cenes ae divided a dieen disances. Obviously, he ininsic axis o an elecon is he line connecing he elecon s cenes o mass and chage. The Coulomb s aacive oce diecs he elecon s cene o chage o he nuclei s cene o chage while he acing on he elecon s cene o mass ceniugal oce diecs i opposiely. Theeoe, he oces make so ha he hee cenes ae nomally in line. Since he elecon s cene o chage is dieced o he nuclei, he elecon complees one un abou he axis Y pependicula o he obial plane evey ime i complees one ISBN: 978-960-474-377-3 60
Laes Tends in Applied and Theoeical Mechanics evoluion aound he nuclei. We can assume ha i a he same ime an elecon complees hal a evoluion () abou is ininsic axis ha elecon acs as a Hype gyoscope in one o he poenial wells o he Hype zeos + o -. The degees o eedom o he elecon-nuclei sysem ae disconneced and heeoe i does no lose kineic enegy. The axis o canno be changed by anohe one because i is ievesibly deemined by he ininsic popey o elecon. Spin hee is no measued in adians pe second bu as a sic one-hal coelaion () beween angula speed o abou he ininsic axis dieced o he nuclei and angula speed o uning abou an axis pependicula o he plane o obiing, equal o he obial angula speed. e up sae e doun sae ω ω le igh obi obi obi ω + obi obi obi obi ω + obi Table. Spin-up and -down saes like a coelaion ω /ω obi e up sae e doun sae + le obi obi + + igh Table 3. Spin-up and -down saes inepeed by he signs o ω I he elecons om he elecon couple ake posiive and negaive Hype zeos a poins + and - om Fig. 4, we can ind ha hee ae ou possible combinaions beween he diecions o obi obi + and obial momenums. Accoding o adiion, we can classiy he possible -up and -down saes om Fig. 4 also as le and igh ones, Table. Table 3 inepes he elecon s -up and down saes wih hei signs. 3.6.. Some basics on he elecon s dynamics I o example, because o some eason, he obi o he elecon educes by a adius R, i inceases is angula speed aound he nuclei i.e. he angula speed o uning abou axis Y by ω, inceasing he obial momenum/angula momenum. The yoscopic aco becomes geae han wo ( >). Hence, he om Tables and 3 becomes less han one-hal o he obiing. The elecon comes ou o he given Hype zeo (poenial well) and sas o geneae Hype gyo oque accoding o he elaions () o (). I he elecon comes ou o he + Hype zeo i geneaes a negaive oque, i i comes ou o he - Hype zeo i geneaes a posiive one. The Hype oque acs aound an axis Z pependicula o he is wo passing hough he elecon s cene o mass, Fig.4. I shis he elecon s cene o chage o he line connecing he elecon s cene o mass and he nuclei s cene o chage. I makes he elecon s cene o chage o do peiodical moions dependen on he π- quanizaion. Pobably i leads o emission o elecomagneic wave. On he ohe hand, evey piece o elecomagneic wave aacking he elecon om ouside pobably makes is cene o chage o do simila peiodical moions. We can call he phenomenon o -obi ineial ineacion a -obi ineial coupling. On he ohe hand, he obial and magneic momens sepaaely ineac wih any applied magneic ield (he Zeeman eec) and also wih each ohe i.e. hee is a -obi magneic coupling. In ac, and obi ae coupled in wo ways - an ineial and an elecomagneic one. All ogehe wok in sysem. In ac, hee ae many deails. Biely, we know ha evey sysem aken ou o is equilibium ends o ecove i. The only way he ineial eleconnuclei sysem can ecove is equilibium is o bing he coelaion beween he angula speed o and he obial one o he naual value o one-hal. How do he ineial and he elecomagneic couplings wok ogehe o ecove he equilibium o he elecon-nuclei sysem? Can we ely ha easoning o povoke a uhe developmen? ISBN: 978-960-474-377-3 6
Laes Tends in Applied and Theoeical Mechanics 4 Conclusion The new mehod o calculae he gyoscopic oque coesponds o he Newonian Laws o Dynamics, he veco muliplicaion and he o elemenay paicles. I pomises a oppouniy o nex developmens. Reeences: [] X. Feynman, R. P., The Feynman s Lecues on Physics, nd pining, Calionia Insiue o Technology, Novembe 964, Volume, Chape 0, page 0-4. [] Lindholm A. and ohes, Ruheod Backscaeing, Laboaoy manual, Uppsala Univesiy, Depamen o Nuclea and Paicle Physics; hp://www.sl.uu.se/~od/kp/ruheod.pd [3] Shanka, R. Pinciples o Quanum Mechanics, nd ed. Yale Univeciy, New Haven, Connecicu, Kluwe Academic/Plenum Publishe, New Yok, 994, Chap. 4 [4] Nobily, J. Quanum Mechanics, Physics Depamen, Univeciy o Wisconsin- Milwaukee, Novembe, 0, 000, Chap. 0-4 ISBN: 978-960-474-377-3 6