8-5. THE MOTON OF A TOP n his secion, we shll consider he moion of n xilly symmeric body, sch s op, which hs fixed poin on is xis of symmery nd is ced pon by niform force field. The op ws chosen becse i is relively simple exmple of body whose forced moion is mrkedly ffeced by gyroscopic momens ssocied wih he spin bo is xis of symmery. The resls, however, hve pplicion in he nlysis of oher sysems, sch s gyroscopes nd spinning projeciles. Generl Eqions. Consider he roionl moion of he op shown in Fig. 8-3. is ssmed o spin wiho fricion sch h he poin O on he xis of symmery is fixed. The only exernl momen bo O is h de o he consn grviionl force mg cing hrogh is cener of mss C. Le s nlyze he moion of he op by sing Lgrnge's eqions nd choosing he Elerin ngles s coordines. We noe h his is n exmple of forced moion; hence H is no fixed in spce. So le s se he originl Eler ngle definiion of Sec. 7- in which he vecor is ssmed o poin vericlly downwrd in he direcion of he grviionl force. f we choose he fixed poin O s he reference poin, he ol kineic energy my be wrien in erms of he Eler ngle res. Noing h nd yy zz for his cse of symmery bo he x xis, we find from Eq. (8-87) h T ( & φ & sin ) (& & cos ) (8-4) where he momens, of ineri re ken bo principl xes O. Assming horizonl reference level hrogh O, he grviionl poenil energy is V mgl sin (8-4) where l is he disnce of he mss cener from. he fixed poin. Now we cn wrie he Lgrngin fncion: xx L T V (& φ& sin ) (& & cos ) mgl sin (8-4) The generlized momen re of he sme form s we obined in Eq. (8-) for he nforced cse. We see h
p p p L Ω sin & cos & L & & L Ω & φ (8-43) where we recll h he ol spin Ω is given by Ω & φ & sin. The sndrd form of Lgrnge's eqion, nmely, d d L L q& i q i is now pplied ogeher wih Eq. (8-43) o obin from which we see h boh dp dp d d φ, (8-44) p nd p φ re consn. Hence we find h Ω is consn for his cse where here is no pplied momen bo he symmery xis. Also, he precession re & cn be obined from Eq. (8-43) wih he following resl: p Ω sin &. (8-45) cos Now le s se he principle of conservion of energy o obin n inegrl of he eqion of moion. From Eqs. (8-4) nd (8-4), we see h he ol energy is E T V Ω (& & cos ) mgl sin (8-46) where we recll h Ω & φ & sin is consn. follows h he ol energy mins he kineic energy ssocied wih he ol spin Ω is lso consn. Clling his qniy E', we cn wrie E' E Ω (& & cos ) mgl sin. (8-47) Sbsiing for & from Eq. (8-45) nd solving for &, we obin & Ω sin E' p mgl sin cos. (8-48) Noe h is he only vrible on he righ-hnd side of his eqion. Ths we see from Eqs. (8-45) nd (8-48) h he precession re & nd he nion re & cn be wrien s fncions of lone for ny given cse.
n order o simplify he semen of Eq. (8-48), le s mke he sbsiion from which i follows h sin, (8-49) & & cos. (8-5) Also, le s define he consn prmeers, b, c, nd e s follows: p, Ω b, Now mliply Eq. (8-48) by The resling eqion is f we se he noion mgl c, hen we cn wrie Eq. (8-5) in he form 3 E' e. (8-5) cos nd mke he foregoing sbsiions. & ( )( e c) ( b). (8-5) f ( ) ( )( e c) ( b), (8-53) & f (). (8-54) Now le s consider he fncion f () in greer deil. Figre 8-4 shows plo of f () verss for ypicl cse. We cn ssme h he prmeer c is posiive since we cn lwys consider he disnce l from he sppor poin o he cener of mss o be posiive. Frhermore, we see from Eq. (8-53) h he cbic erm predomines for lrge bsole mgnides of. Hence we find h f () ms be negive for lrge negive vles of nd ms be posiive for lrge posiive vles of. Now f () is coninos fncion of so i ms be zero for les one rel vle of. Th f () is clly zero for hree rel vles of will now be shown. is eviden h & is zero or posiive for ll physiclly relizble siions, so f () ms be zero or posiive some poin in he inervl which is he rnge in which ms lie in he cl cse. B if we se ± in Eq. (8-53) we find h f () is zero or negive hese wo poins since ( b) ms be zero or posiive. Therefore here re wo roos in he inervl nd he hird roo 3 ms lie in he rnge 3. Smmrizing, we cn wrie. Looking gin Eq. (8-48) or (8-5), we find h he moion in sops only when or. is ppren, hen, h ms oscille beween hese vles nd will ndergo corresponding oscillion. To nlyze he moion of in ime, le s wrie Eq. (8-54) in he form Now we define 3 & f ) c( )( )( ). (8-55) ( 3
nd we see h w, k, p c( 3 ), (8-56) 3 & w&. (8-57) ( )( ) Mking hese sbsiions, we cn wrie Eq. (8-55) he following form: w& p ( w )( k w ). (8-58) f we mesre he ime from he insn when is is minimm vle nd, we find h w dw p F(sin w, k). (8-59) ( w )( k w ) where we recognize he inegrl s n ellipic inegrl of he firs kind. Conversely, we cn solve for w, obining w sn p, (8-6) where, we recll, he ellipic fncion, sn, ws inrodced previosly in Sec. 3-9. Now we se he definiion of w in Eq. (8-56) o solve for ( ) sn p. (8-6) Becse he sn fncion is sqred, he period of he nion in is js hlf he period of w. Ths he period of or is K( k) 4K( k) T, (8-6) p c( 3 ) where K(k) is he complee ellipic inegrl of he firs kind. Referring o Eqs. (8-45) nd (8-5), we see h he precession re & cn be expressed s fncion of s follows: b &. (8-63) Hving solved for s fncion of ime, we cn lso obin & s fncion of ime. Also, from he definiion of he ol spin Ω given in Eq. (8-), we see h & φ Ω &, (8-64) where Ω is consn. Hence & φ is known fncion of ime. Noe h boh & nd & φ hve he sme period s. Ph of he Symmery Axis. Le s consider he ph of poin P loced on he xis of symmery ni disnce from he fixed poin O. Tking horizonl reference hrogh O, we find h 4 sin represens he heigh of P. The vles nd which & is zero correspond o posiions of
minimm nd mximm heigh nd re known s rning poins. f we represen he moion of he symmery xis by he ph of P on he srfce of ni sphere, we find h P will remin beween wo horizonl "lide" circles given by nd, corresponding o he exreme vles of. The ph of P in ny given cse cn be clssified s one of hree generl ypes. n order o simplify his clssificion, le s define p. (8-65) b p φ From Eq. (8-63) we see h & when (excep if ±, in which cse & is indeermine). Hence he moion of he symmery xis he insn when is sch h he ph of P is ngen o vericl circle of "longide." () Le s consider firs he cse where >. n oher words, lies oside he possible rnge of, nd herefore he precessionl re & does no eql zero ny ime dring he moion. This cse is shown in Fig. 8-5(), where he rrows indice he direcion of he velociy of P for posiive Ω. (b) Nex consider he cse in which < <. As shown in Fig. 8-5(b), we see h & is zero wice dring ech nion cycle, nd, for >, i clly reverses is sign compred o he verge vle of &. n generl, for cerin op wih given ol spin Ω, he ph rced by P is dependen pon he iniil vles of, &, nd &. cn be seen, for exmple, h he formion of loops s in Fig. 8-5(b) cn be ccomplished by giving he xis of symmery n iniil precession re which is opposie in direcion o is verge vle. Cspidl Moion. (c) The cse in which is known s cspidl moion nd is illsred in Fig. 8-5(c). Noe h nd -& re boh zero when, resling in csp which poins pwrd. This ype of moion 5
will occr, for exmple, if he xis of spinning op is relesed wih zero iniil velociy. The xis begins o fll vericlly, csing gyroscopic ineril momens which resl in combinion of nion nd precession sch h he xis is moionless csp reglr inervls. ncidenlly, we see from energy considerions h he xis cn be sionry only when P is is highes poin, corresponding o. n oher words, he poenil energy ms be mximm when he kineic energy is minimm. To find he limis of he nion for cspidl moion, le s se & or f () eql o zero. Consider he following wo cses: () & becse & ; () & becse cos. Cse : when &. Using he definiion of given in Eq. (8-65), we see from Eq. (8-5) h e c, (8-66) since & he csp. We cn hen wrie n expression for f () in he form f ( ) c( )( ) b ( ). (8-67) We hve lredy shown h one limi of he nionl moion occrs. Dividing o his roo, we obin or ( c ) b ( ), λ (λ ), (8-68) where λ is posiive consn given by Ω b λ. (8-69) c 4 mgl Solving for he roos of Eq. (8-68) nd inclding he one fond previosly, we cn smmrize he roos of f ( ) for his cse s follows: 3 λ λ λ λ λ λ, (8-7) where we ke he posiive sqre roo. For <, n evlion of Eq. (8-7) will show h 6 3 regrdless of he vle of λ nd in greemen wih or erlier resls. The limis of he moion in or re fond by evling from Eq. (8-66), λ from Eq. (8-69), nd hen sing Eq. (8-7) o obin. Noe gin h l 3 hs no physicl mening. Cse : & when. Consider firs h. This cse ±
pplies when he symmery xis psses hrogh he pper vericl posiion wih non-zero nglr velociy. We see from Eq. (8-65) h b (8-7) nd, since & > his momen, we noe from Eqs. (8-48) nd (8-5) h So we cn express f () in he form e> c. (8-7) f ( ) ( )( e c) b ( ) (8-73) nd solve for he rning poins. Dividing o he known fco (-) corresponding o, here remins ( )( e c) b ( ), which cn be wrien in he form b e b e. (8-74) c c One of he roos of his eqion corresponds o rning poin. The oher roo 3 is greer hn nd hs no physicl mening. A similr procedre cn be followed for he cse where &. n his insnce, he symmery xis psses hrogh he boom poin on he ni sphere wih non-zero velociy. We see from Eq. (8-65) h Also, since & >, we see h So we cn wrie b. (8-75) e c>. (8-76) f ( ) ( )( e c) b ( ). (8-77) Dividing o he known fcor ( ) corresponding o he roo, we obin b e b e. (8-78) c c The roos of his eqion re he pper rning poin nd he nonphysicl roo 3 which gin is lrger hn niy. Sbiliy of Moion ner he Vericl. An exmple of op moion of priclr ineres occrs if he xis poins vericlly pwrd when & is zero. This flls nder Cse of cspidl moion, nd in his insnce. From Eq. (8-7), we see h he roos re, (λ ),, (8-79), 3 where ( λ ) is designed s or 3 ccording s λ. is less hn or 7
greer hn. Now le s consider he moion for he following vles of λ : () λ <, (b) λ nd (c) λ >. () Cse : λ < or Ω < 4 mgl / 8. For his cse he csp occrs posiion of nsble eqilibrim. This cn be seen from Fig. 8-6 by noing h smll disrbnce his posiion will cse nionl moion dring which decreses nil he minimm vle corresponding o λ is reched; wherepon reversl of his moion rerns he xis o he vericl. The doble roo is illsred by he fc h he slope f '( ) is zero his poin, s my be seen from Eq. (8-67). is ineresing o noe h he period of nion cycle is infinie. To show his, we refer o Eq. (8-56) nd we find h he modls k ; hence he complee ellipic inegrl K(k) is infinie. follows from Eq. (8-6) h he period is lso infinie. (b) Cse : λ or Ω 4 mgl /. Here Here is he borderline cse in which here is nerl sbiliy he vericl. All hree roos re eql o niy nd hs here is n inflecion poin of f (). (c) Cse 3: λ > or Ω > 4 mgl /. The spin is sfficienly lrge in his cse h he vericl posiion is sble, h is, n infiniesiml disrbnce will no cse finie deviion of he symmery xis from he vericl. Agin we find h f '(), b in his cse he hird roo (λ). 3 > A op which is spinning s in Cse 3 is known s sleeping op. This nme rises becse smooh, xilly symmeric op wih is xis vericl nd λ > migh pper firs glnce o be no moving ll, nd hence "sleeping." n n cl cse here re smll fricionl momens which slowly decrese he spin nil λ becomes less hn niy. A his poin wobble or nion ppers nd grdlly increses nil he body of he op his he horizonl srfce. ncidenlly, he biliy of n cl op wih lrge spin o righ iself nd rech he sleeping condiion is de o he cion of fricionl
forces on is ronded poin. These forces hve been omied in his nlysis. Nion Freqency nd Amplide for he Cse of Lrge Spin. Now le s se Eq. (8-7), which ws obined in he nlysis of cspidl moion, o find he pproxime mplide nd freqency of he nionl moion of fs op, h is, one for which λ >>. We cn wrie he expression for in he form λ λ, λ λ or, sing he binomil expnsion nd neglecing erms of order higher hn / λ, we obin λ λ ( ). (8-8) λ λ λ λ The mplide of he nion is mgl ( ) ( λ Ω ). (8-8) Noe h he nion mplide vries inversely wih he sqre of he ol spin re Ω. becomes zero for, h is, for vericl op, he spin being mch lrger hn h reqired for sbiliy. We see from Eq. (8-7), hen, h he roos nd hve smll seprion; b he hird roo 3 is pproximed by nd hs is widely sepred from he ohers. 3 λ, (8-8) To find he nion freqency of fs op, le s refer o Eq. (8-6) which gve he generl solion for in erms of ellipic fncions. However, becse 3 is widely sepred from nd, we see from Eq. (8-56) h he modls k is very smll; hence we cn pproxime he ellipic fncion sn p by he rigonomeric fncion sin p. Ths we obin ( ) sin p, (8-83) where he ime is mesred from he momen when. Using Eq. (8-8) nd rigonomeric ideniies, we cn rewrie his resl in he form: mgl ( )( cos p ). (8-84) Ω From Eqs. (8-5), (8-56), (8-69), nd (8-8), we' see h he circlr freqency of he nion is Ω p c( 3 ) cλ. (8-85) An expression for he precession re & cn now be obined, for we 9
noe from Eqs. (8-63) nd (8-65) h b &. Sbsiing for b nd ( ) from Eqs. (8-5) nd (8-84), we obin B we see from Eq. (8-84) h or ( mgl & ( cos p). (8-86) Ω ) where we neglec erms of order of fs op h he precession re is ( mgl ) ( cos p) Ω ( cos p), (8-87) λ / λ or higher. So we obin for his cse mgl & ( cos p) ; (8-88) Ω he pproximion being vlid even for cspidl moion ner he vericl, so long s λ >>. Precession wih No Nion. Precession wih no nion is indiced by hving, h is, by he presence of doble roo of f () poin oher hn ±. The condiions for doble roo re h f () nd f '( ) be zero for he sme vle of, s shown in Fig. 8-7. Ths, sing Eq. (8-53) we cn wrie Also, f ( ) ( )( e c) ( b). (8-89) f '( ) ( e c) c( ) b( b). (8-9) From Eqs. (8-89) nd (8-9), we hve ( b) c( ) b( b) e c,
from which we obin he following qdric eqion in ( b) : ( b) b( )( b) c( ). (8-9) Solving for ( b), we obin b b± b c. (8-9) B we recll from Eq. (8-63) h so we find h he precession re is or, in erms of, b & ( ) & cos, (8-93) b c & ±, (8-94) b b csin & ±. (8-95) sin b is ineresing o noe h wo sedy precession res re possible, provided h he vles of nd b Ω / re sch h he sqre roo in Eq. (8-95) is rel. This ls reqiremen is similr o reqiring sfficien spin for sbiliy in vericl op. Ths we see h he condiion on he ol spin Ω in order h sedy precession be possible given vle of is h Consider now he cse in which 4 mgl Ω > sin. (8-96) Ω >> 4 mgl sin / ; h is, he spin is lrge enogh so h he second erm in he sqre roo of Eq. (8-95) is smll compred o niy. Then we cn pproxime (8-95) by b csin & ±, sin b from which we obin he following possible res of niform precession: ( & ) ( & ) c mgl b Ω. (8-97) b Ω sin sin Noe h he slow precession re (& ) is independen of. This is he precession re which is slly observed in fs op or gyroscope nd is lso eql o he men vle of & fond in Eq. (8-88). On he oher hnd, he fs precession re (& ) is independen of he ccelerion of grviy nd, in fc, is idenicl wih he free precession re obined previosly in Eq. (8-4).
Exmple 8-7. A op wih ol spin Ω nd velociy v is sliding on smooh horizonl floor wih is symmery xis vericl (Fig. 8-8). Sddenly,, he poin srikes crck O nd is prevened from moving frher lhogh he nglr moion is nhindered. f is he ngle beween he xis of symmery nd he floor, find. The nglr velociy & ( ) js fer he verex is sopped. b. The liner implse exered on he op. c. min in he ensing moion, ssming h Ω v / l g / l nd (/ 4) ml / 5, where he momens of ineri re ken bo xes hrogh he verex. Or pproch will be o noe h he nglr momenm bo he verex O is conserved dring he iniil insn becse he recion forces of he floor pss hrogh his poin. Hence he nglr implse pplied is zero. Eqing expressions for he horizonl componen of nglr momenm before nd fer he impc, we obin H mvl & ( ), (8-98) h where his componen is direced ino he pge. Ths we find h & mvl (). (8-99) The liner implse is fond by clcling he chnge in he liner momenm dring impc. Noing h we obin v ( ) & ()l, ~ ml F mv( ) mv mv( ), (8-) where he implse is posiive when direced o he righ. n order o clcle he minimm vle of, we noe firs h & sin. Therefore we hve he cspidl moion which ws described previosly nder Cse on pge 47. Evling he consn
prmeers b, c, nd e from Eq. (8-5), we hve Ω g b 4 l mgl 8g c 5l E' mvl e mgl 56g 5l. (8-) The rning poin is fond by sbsiing hese vles ino Eq. (8-74) wih he resl:.4 8.6. The roos of his eqion re nd 3, he hird roo of f ( ) being. We obin Ths we find h 3.958. 9.494 sin (.958) 64.9. min Exmple 8-8. A niform circlr disk of mss m, rdis, rolls on horizonl srfce in sch mnner h is plne is inclined wih he vericl consn ngle nd is cener of mss describes circlr ph of rdis R - Fig. 8-9(). Solve for he precession re &. Firs Mehod: Alhogh he disk is n xilly symmeric body sbjec o grviionl momens, he geomery nd he consrins re differen from hose enconered in he nlysis of op. So le s sr gin wih he bsic roionl eqion M H &, where we shll choose he fixed poin O s reference. Le s se Elerin ngles o describe he orienion of he disk. The precession re & cn be obined in erms of he ol spin Ω by 3
eqing wo expressions for he rnslionl velociy of he cener of mss. from which we hve v R& Ω, & Ω. (8-) R cn be seen h & ms be consn. A chnging & wold imply chnging kineic energy. B, since is consn nd no slipping occrs, here re no working forces cing on he disk; hence, by he principle of work nd kineic -energy, & cnno chnge. From Eq. (8-), we see h Ω is lso consn. Consider now he xil nd rnsverse componens of he nglr momenm bo he cener of mss. From Eqs. (8-6) nd (8-), we cn wrie H H Ω ( R / ) &. & cos To find he ol nglr momenm bo O, we ms dd he nglr momenm de o he rnslionl velociy of he cener of mss bo O. Tking horizonl nd vericl componens of his porion, we obin H H mr & mr& cos As he disk rvels in is circlr ph, he sysem of for componen vecors roes wih nglr velociy &, s shown in Fig. 8-9(b). Ths we see h he ol nglr momenm vecor H is of consn mgnide b precesses bo vericl xis he sme re s he disk. Now we cn evle H & by noing h H & & H (8-3) for his cse in which he mgnide H is consn. We find h H & & H H cos H sin ) e (8-4) ( h or, sbsiing he preceding expressions for where H h, H nd H, we obin H & R & cos mr sin e, (8-5) e is ni vecor direced o of he pge. The pplied momen bo O is independen of where we noe h F h nd is given by M mg sine, (8-6) F v mg since he mss cener hs no vericl moion. Eqing he expressions on he righ hnd sides of Eqs. (8-5) nd (8-6), we obin 4
or R & cos mr sin mgsin, mg n &. (8-7) R mr sin F or hin niform disk, we recll h m, m. 4 Sbsiing hese vles ino Eq. (8-7), we obin 4g n &. (8-8) 6R sin Second Mehod: Now le s consider his problem from he Lgrngin viewpoin. Using Eq. (8-98) for he roionl kineic energy nd dding he rnslionl kineic energy, we obin he ol kineic energy T m( R & & ) (& φ& sin ) (& & cos ). (8-9) n order o se sndrd form of Lgrnge's eqion, he coordines ms be independen. B we sw in Eq. (8-) h he condiion of rolling imposes he consrin h Ω & R φ& sin &, So he coordines,, φ re no independen s hey snd. f, however, we wrie he consrin eqion in he form & R' sin φ & sin &, (8-) nd sbsie ino he kineic energy expression of Eq. (8-9), we obin he resl T m ( R' sin ) & ( ' & & cos ), (8-) where we hve le ' m nd R R' sin, (8-) in ccordnce wih Fig. 8-9(). Now φ hs been elimined s generlized coordine nd he remining coordines nd re independen in he kinemic sense. Also, we hve inrodced new consn, R, becse R is no longer consn if is considered o be vrible. cn be seen h he sysem is conservive nd he poenil energy is 5
6 given by mg cos V. (8-3) The generlized forces re sin mg V M V M. (8-4) Le s se Lgrnge's eqion in he form of Eq. (6-73), nmely, i i i Q q T q T d d &. Firs we obin he generlized momen: & & & & T p R m T p ' cos ) sin ' (. (8-5) Also, we evle he following erms: cos sin ) sin ' ( & R m T T. (8-6) From Eqs. (8-4), (8-5), nd (8-6), we cn now wrie he eqion in he form d dp, (8-7) indicing h p is consn. For his priclr cse in which is consn, i follows from Eq. (8-5) h he precession re & is consn. n similr fshion we find h he eqion is sin cos sin ) sin ' ( ' mg R m & & &. (8-8) Seing & & nd solving for &, we obin sin n R mr mg & in greemen wih or erlier resl.