Rotational stability---an amusing physical paradox

13 Pages·2007·0.22 MB·English

IOPPUBLISHING EUROPEANJOURNALOFPHYSICS
Eur.J.Phys.28(2007)845–857 doi:10.1088/0143-0807/28/5/008
Rotational stability—an amusing
physical paradox
Carlos M Sendra1, Fabricio Della Picca1
and Salvador Gil1,2
1DepartamentodeF´ısica,JJGiambiagi,UniversidaddeBuenosAires,Argentina
2EscueladeCienciayTecnolog´ıa,UniversidadNacionaldeSanMart´ın,CampusMiguelete,
MdeIrigoyen3100,SanMart´ın(1650),BuenosAires,Argentina
E-mail:[email protected]
Received1May2007,infinalform7June2007
Published12July2007
Onlineatstacks.iop.org/EJP/28/845
Abstract
Herewepresentasimpleandamusingdevicethatdemonstratessomesurprising
resultsofthedynamicsoftherotationofasymmetricalrigidbody. Thissystem
allows for a qualitative demonstration or a quantitative study of the rotation
stabilityofasymmetrictop. Asimpleandinexpensivetechniqueisproposed
to carry out quantitative measurements to explore the theoretical predictions
ofthemodelpresentedtoexplainthemotionofthesystem. Ourresultsagree
verywellwiththeexpectationsofthetheoreticalmodel.
(Somefiguresinthisarticleareincolouronlyintheelectronicversion)
1. Introduction
It is rewarding and instructive to share with friends and the public in general the fun and
excitementthatwephysicistsoftenobtainfromexploringthewaysinwhichnaturebehaves.
Herewepresentasimpledevicethatcanbeusedtoentertainfriendsataparty,whileillustrating
thekindofproblemaphysicistenjoysworkingon. Itcanalsobeusedinaregularlaboratory
coursetocarryoutquantitativemeasurementstoexplorethephysicsoftherotationalstability
of a rigid body. The experiment consists of a rod the size of a pen or a ring attached to a
flexible wire or a fishing line of about 30 cm as illustrated in figure 1. At rest the rod or
the ring (symmetric top) hangs, as expected, with its centre of mass at its lowest position.
Ifwestartspinningthewirewithourfingers, therodortheringrotateswiththeircentreof
mass in the same position, but if we increase the rotation frequency above a critical value,
thesymmetrictopbeginstoraiseitscentreofmassapproachingahorizontalpositionasthe
frequencyincreases. Therisingofthecentreofmassisatfirstsightsurprisingandcontraryto
ourintuition. Ifwenaivelyattempttocalculatethepotentialandkineticenergyofthebarata
giveninclination,bothtermsincreasemonotonicallywiththeelevationofthecentreofmass
0143-0807/07/050845+13$30.00 (cid:1)c 2007IOPPublishingLtd PrintedintheUK 845
846 CMSendraetal
ω ω
z z
1 3 θ
p
θ θ
2
3
2 1
Figure1. Symmetrictoprotatingalongthezaxis. Left: thepositionadoptedbyabarwhen
itisbeingspunthroughthewirewithourfingers. Right: thefigureshowsthecorresponding
equilibriumpositionoftheringwhenitisrotating.Axis3representsthesymmetryaxis,whereas
axes1and2aretheotherprincipaldirectionsofthebody.
independentoftheangularfrequencyofrotation. Therefore,theobservedeffectappearstobe
paradoxical.
Thephysicsofthesymmetricaltophasimportantapplicationsinseveralareasofphysics,
such as classical mechanics, molecular, nuclear physics and astrophysics. Therefore its
understanding and inclusion in introductory and intermediate level courses is well justified
andthedevelopmentofnewexperimentstoexploreitsphysicsisalsowelcome. Thephysics
ofthefreerotatingtopistreatedinmostintroductoryandintermediateleveltextsonmechanics
[1–4]. Therearealsoanumberofveryinstructiveandamusingdemonstrationsofthemotion
of symmetric and asymmetric tops [5–8] among which the gyroscope is a classic example.
Nonetheless,theliteratureofexperimentsonthissubjectthatallowsforprecisequantitative
measurements, which can be contrasted with the corresponding theory or models, is very
scarce.
Inthisworkwedevelopasimplemodeltounderstandthisbehaviourqualitatively. Then
wedevelopamorerefinedmodelthatcanbedirectlycomparedwiththeexperimentalresults.
Themodelscanbereadilytestedusingtheresultsoftheexperiment. Weanalysethecaseof
abarandaring.
2. Theoreticalconsiderations
Webeginexaminingthecaseoftherotatingrod,whoselengthishandmassism,thathangs
fromaverylongwireoflengthL((cid:1)h). Thissystemisillustratedschematicallyinfigure2.
Thespinningangularfrequencyisω alongthezaxis(inthespace-fixedorinertialframeof
reference).
The behaviour of the bar becomes clear if we visualize our system from a body-fixed
frameofreference. Inthisframe,whichrotateswithfrequencyωaroundthezaxis,thereare
twotorquesinoppositedirections. Thetorqueduetothestringtension,τ ,tendstorestore
w
therodtotheverticaldirectionandthetorqueduetothecentrifugalforce[1,2],τ ,tendsto
c
Rotationalstability—anamusingphysicalparadox 847
Space-fixed frame
ω
Body-fixed frame
z L >>h
L T=mg
θ
F
c
F
c
mg
h
Figure2.Schematicrepresentationofarod,oflengthh,thatisbeingspunatanangularfrequency
ωalongthezaxisbyalongwireoflengthL.Left: thebarisshowninthespace-fixedframeof
reference.Right:thebarisshowninthebody-fixedframe(rotatingframe)ofreference.Herewe
assumethatL(cid:1)h.Therefore,thestringisalmostparalleltotheverticaldirection(na¨ıvemodel).
bring the rod to a horizontal position. We take the centre of mass of the rod as a reference
pointforcalculatingthetorques. Inequilibriumthesetwotorquesareequal,therefore
h h2
τ =m.g. ·sinθ =τ =m·ω2 ·sinθ ·cosθ. (1)
w c
2 12
The details of the calculation of the centrifugal torque are presented in appendix A. From
equation(1)itfollowsthattheequilibriumpositionoftherodisdeterminedby
(cid:1) (cid:2)
6g ω 2
cosθ = = c0 , (2)
hω2 ω
with
(cid:3)
6g
ω = . (3)
c0
h
Inordertoobtainasolutionofequation(1)differenttoθ =0theconditionω(cid:1)ω must
c0
besatisfied. Consequently,weexpectthatatlowrotationalfrequencies,ω<ω ,therodwill
c0
rotateintheverticalposition(θ =0),forω > ω ,theangleθ willincreaseasω increases,
c0
in accordance with equation (2). We will call the model just discussed the na¨ıve model. A
moregeneraldiscussionofthisproblemispresentedinappendicesAandB,wherethecase
ofafinitelengthofwire,andothergeometriesforthesymmetricaltop,arealsoconsidered.
ThederivationpresentedinappendixAshouldbeaccessibletobeginnerstudents,whereasthe
modeldiscussedinappendixBrequiressomeknowledgeofintermediateanalyticalmechanics
[1–4].
3. Theexperiment
Theexperimentalarrangementweusedisshowninfigure3. Itconsistedofavariablespeeddc
motor(withmechanicalreduction)poweredbyavariabledcvoltagesource,thatregulatesthe
848 CMSendraetal
Variable speed
motor
ω Photogate
toPC
Motor Regulator
Axis flag
L
h
Figure3.Schematicrepresentationofarod,oflengthh,thatisbeingspunatanangularfrequency
ωbyavariablespeedmotor.
speedofthemotor. Wetapedaplastictubetotheaxisofthemotor(axisflag)tointerruptthe
beamoflightofaphotogateoneveryrotation. Thephotogate,connectedtoacomputer,was
usedtomonitorthespeedofthemotorandmeasureitsrotationalfrequencyω. Assymmetrical
topsweusedaluminiumtubesofabout9mminanexternaldiameterwithdifferentlengths
h. WedrilledasmallholeclosetooneoftheirextremestoattachathinwireoflengthLthat
wasconnectedtotheaxisofthemotor. WeusedageneralpurposePVCinsulatedelectronic
hook-upwiregauge24andabout35cminlength. Itsmasswaslessthan0.5%ofthemassof
therodandwasneglectedinouranalysis. Wealsousedmetalringsofdiametersbetween5
and10cm.
AWebCam(GeniusGE11)wasusedtoobtainthedigitalphotographsofthetops,once
theyreachedanequilibriumposition. Aplumblinewasplacedinthebackgroundtodetermine
theangleθ ofthebarorring. AnadvantageofmanyinexpensiveWebCams,suchastheone
usedhere,isthatastheilluminationgetsdimmertheshutterspeedofthecameraisreduced.
Therefore,byregulatingtheilluminationitispossibletoobtainaphotographofthebarorring
thataveragesoverseveralpositionsofthetopasshowninfigure4,whereweclearlyseethe
orientationoftherod. Thesedigitalpicturesallowustoobtaintheorientationangleθ witha
fewdegreesofuncertaintyforeveryfrequencyofthemotor. Naturally,thesameeffectcould
beobtainedusingadigitalcamerawithshutterspeedregulation. Anotheralternativethatcan
beusedformeasuringtheorientationangleθ istousethedigitalcamerainvideomode. The
procedurewouldinvolverecordingafewrevolutionsofthetop,andthenexaminingthevideo
frame by frame to select the one where the wire and the bar are in the plane of the picture.
Thenthepictureisusedtoproceedassuggestedinfigure4.
4. Resultsanddiscussion
In figure 5, we present the experimental results (symbols) obtained for a hollow aluminium
bar oflength h =25.5 cmand an external diameter d=0.95 cm, attached toathinwireof
L=33.4cmtoapointonthebaratδ =0.8cmfromoneofitsextremes. Inthesamefigure
Rotationalstability—anamusingphysicalparadox 849
θ
Figure4. Photographoftherotatingbar. Aslowshutterspeedwasusedsuchthateachshot
averagedoverseveralorientationsofthebar. Inthebackgroundtheimageoftheplumblineis
shownthatwasusedtodefinetheverticaldirection.Thistypeofdigitalpictureallowedustoobtain
therotationalequilibriumangleθwithanuncertaintyofabout2◦foreachrotationalfrequency.
80 Bar h=25.5 cm
s) 60
e
e
gr
de 40
(
θ0
20
0
0 1 2 3 4 5 6 7 8 9
f (Hz)
Figure5. Thesymbolsrepresenttheexperimentalresultsforabar(h=25.5cm). Thedotted
line corresponds to the prediction of the na¨ıve model given by equation (2). The dashed line
corresponds to the expectation of the model (no.2) discussed in appendix A where the finite
lengthofthewireisincluded. Theheavylineisthepredictionofthemodel(no.3)discussedin
appendixBwhereboththefinitelengthofthewireandthedistanceδofthepointofconnection
tothewirearealsoconsidered.Inthiscase,L=33.4cmandδ=0.8cm.
we also show the theoretical expectation of the na¨ıve model equation (2). The models that
includethefinitelengthofthewireandthedistanceδofthepointofattachmenttotheextreme
ofthebararealsodepictedinthesamefigure.
It is clear that for an adequate description of the experimental data model no. 3 that
includes both the finite length of the wire and the distance δ is necessary. This model is
developed in appendix B. In figure 6, we show the experimental results for another bar of a
different length and for a ring. In this figure we have also included the prediction of model
no. 3. Out of all the cases analysed in this work model no. 3 gives a very good description
oftheobservedresults. Itisinterestingtonotethatduetotheeffectofthecentrifugalforce,
athighrotationalfrequencies,thedrivensymmetricaltoptendstorotateabouttheaxisofthe
largestmomentofinertia,independentlyofwhetheritisasymmetryaxisornot.
In summary, we have devised a simple and inexpensive experiment that can be
implementedinanintroductoryorintermediatelaboratoryclassonclassicalmechanics. We
also developed a realistic model that adequately reproduces the experimentally observed
850 CMSendraetal
80 Bar h=15.6 cm
s) 60
e
e
gr
e 40
d
(
0
20
0
0 1 2 3 4 5 6 7 8 9
f(Hz)
80
Ring R = 3.1 cm
ex
s) 60
e
e
gr
e 40
d
(
0
20
0
0 1 2 3 4 5 6 7 8 9
f(Hz)
Figure6.Experimentalresults(symbols)forabarandaring.Theheavycontinuouslinesarethe
expectationofthetheoreticalmodel(no.3)presentedinappendixB.Theotherlinesinthecaseof
thebararethepredictionofthena¨ıvemodel(dotted)andmodelno.2(dashed).
equilibriumorientationsofaspinningsymmetricaltopthatcanbeusedtostudytherotational
stability of a rotating top. Our results are complementary to the analysis of supercritical
bifurcation carried out, with a different approach, for the case of the ring using a similar
experimentalsetup[9].
Acknowledgments
Wewouldliketoexpressouracknowledgmenttootherstudentswhohaveexploredthephysics
ofoursystemswithdifferentexperimentaltechniques: VBazterra,ACamjayi,MMansilla,
ASolerno´andJTiffenberg. WeexpressoursinceregratitudetoProfessorFScarlassaraforan
illuminatingdiscussionandaverycontributoryrevisionofthisarticle. Wearealsogratefulto
DrASchwintforcarefulreadingofthemanuscript.
AppendixA.Improvedmodelofarotatingrod—modelno. 2
Consider a rod forming an angle θ with the vertical. In the body-fixed frame, there is a
centrifugal force that tends to bring the rod to a horizontal position and the string tension
and weight that tend to restore the rod to the vertical position. The system is illustrated in
figure7. Throughoutallouranalysiswewillassumethatthestringorwireislongerthanthe
bar,i.e.L>h. Theequilibriumoftheverticalandhorizontalforcesimpliesthat
T cosβ =mg and T sinβ =F(cm) =mω2εsinθ, (A.1)
c
whereTisthestringtensionandF(cm)isthecentrifugalforceactingonthecentreofmassthat
c
isrevolvingaroundtheverticalthatpassesthroughthepointofsuspension,thenetcentrifugal
force. Hereε isthedistanceofthecentreofmassoftherodtothepointO(figure7)where
Rotationalstability—anamusingphysicalparadox 851
Space-fixed frame Body-fixed frame
ω
L
β L β
δ
T T.cosβ=mg
θ θ
O λ O Fc
Fc x
h ε
mg
Figure7.Schematicrepresentationofarod,oflengthh,thatisbeingspunatanangularfrequency
ωbyastringoflengthL.Left:thesystemisinthefixedorinertialframeofreference.Right:the
systemisinthebodyframeofreference(rotating). ThepointOontherodistheintersectionof
theaxisoftherodwiththeverticalthatpassesthroughthepointofsuspensionofthestring,λis
thedistancefrompointOtothepointofattachmentofthewire.
theaxisoftherodintersectstheverticalthatgoesthroughthepointofsuspension. Fromthe
geometryofoursystemwehavethat
λ·sinθ =Lsinβ, (A.2)
whereλisthedistancefromOtothepointofattachmentofthewire,therefore
λ sinθ
tanβ = · (cid:4) . (A.3)
L 1−(λ/L)2sin2θ
Fromthegeometry(figure7)wehavethatλ=h/2−δ+ε,whereδisthedistancefromthe
pointattachmentofthewiretotheextremeofthebar. From(A.1)itfollowsthat
g λ 1
ε = · (cid:4) . (A.4)
ω2L 1−(λ/L)2sin2θ
Note that ε > 0 if L is finite, since ε depends on the ratio λ/L ≈ h/2L, ε → 0 for L (cid:1)
h. Thisexpressionprovidesanimplicitdependenceofε withθ andω,sinceλalsodepends
onε. Thevaluesofεandλcanbecalculatedrecursivelyasweindicatebelow.
All torques will be calculated using the point O as a reference. The torque due to the
centrifugal force on an infinitesimal element of rod dx at a distance x from the centre of
massis
dτ =(x+ε)·cosθ ·dF =(x+ε)2·cosθ ·dm·ω2·sinθ. (A.5)
c c
Hereωistheangularvelocityoftherodanddm=(m/h)·dxisthemassoftheinfinitesimal
elementoftherod. Sinceλ≈h/2,thetotalcentrifugaltorquewillbe
(cid:5)
m h/2 m
τ ≈ cosθ ·sinθ ·ω2· (x+ε)2·dx = h2ω2cosθsinθ ·[1+12ε2/h2]. (A.6)
c
h −h/2 12
Thetorqueduetotheweightoftherodandthestringtensionis
sinβ
τ =mgλ·sinθ +mg· λcosθ −mgεsinθ, (A.7)
w
cosβ
852 CMSendraetal
or
(cid:6) (cid:7)
λ λ cosθ
τ =mg(λ−ε)·sinθ · 1+ (cid:4) . (A.8)
w (λ−ε)L 1−(λ/L)2sin2θ
Inequilibriumτ −τ ,therefore
(cid:6) w c (cid:7)
6g 1 λ λ 1 2(λ−ε)/h
ω2 = · + · (cid:4) · , for ω(cid:1)ω .
h cosθ (λ−ε)L 1−(λ/L)2sin2θ (1+12ε2/h2) crit
(A.9)
Thisrelationdescribestheconnectionbetweenθ andω. Notethatthisisanimplicitequation
betweenthesevariables. Itssolutioncanbefoundrecursively. First,basedontheconnection
between θ and ω given by the na¨ıve model, we can calculate ε(θ, ω) using equation (A.4).
Thisvalueofεthenallowsustocalculateλ. Usingequation(A.9)weobtainimprovedvalues
of θ and ω. This process can be repeated to the desired precision. In our experience, two
orthreeiterationsaresufficienttodetermineθ andωwith1–2%ofuncertainty. Thecritical
valueofω isobtainedfrom(A.9)forθ =0,i.e.,
crit (cid:6) (cid:10) (cid:11) (cid:7) (cid:8) (cid:9)
6g h 1 2δ 2ε 2 1− 2δ
ω2 = · 1+ (cid:8) (cid:9) 1− + c · (cid:8) h (cid:9), (A.10)
crit h 2L 1− 2δ h h 1+ 12εc2
h h2
sinceλ−ε =h/2−δ,tofirstorderinδ/handε/h,thislastexpressionbecomes
(cid:12) (cid:10) (cid:11)(cid:13)
6g 2δ h 4δ 4ε
ω2 ≈ · 1− + 1− + c
crit h h 2L h h
(cid:12) (cid:13)
h 2δ 2δ 2ε
≈ω2 · 1+ − − + c , (A.11)
c0 2L h L L
(cid:8) (cid:14) (cid:9)
whereε =ε(ω ,θ =0)≈ g ω2 (λ/L)isthevalueofεatthecriticalfrequency. Sincein
c crit c0
generalε andδ aresmallcomparedwithhorL,seefigure9,weneglectedthesecond-order
c
termsin(ε/h). Expression(A.11)canbefurtherapproximatedforthecaseofδ(cid:5)handε(cid:5)
h<Las
(cid:12) (cid:13)
h
ω2 ≈ω2 · 1+ . (A.12)
crit c0 2L
Consequently,theeffectofthefinitelengthofthewireistomoveω totherightofthevalue
crit
predictedbythena¨ıvemodelequation(3). ForL(cid:1)h,theexpressions(A.9)and(A.11)are
reducedtoequations(2)and(3)respectively.
Thezcoordinateofthecentreofmass,relativetoitslowestpositionis
(cid:4)
z =L+(λ−ε)− L2−λ2·sin2θ −(λ−ε)·cosθ, (A.13)
CM
and
(cid:10) (cid:10) (cid:11) (cid:11)
dz λ λcosθ
CM =(λ−ε)sinθ · 1+ · √ ·θ˙, (A.14)
dt λ−ε L2−λ2·sin2θ
where(λ−ε)isthedistanceofthecentreofmasstothehangingpointofthebar.
AppendixB.Effectivepotential—modelno. 3
Fortheanalysisoftherotationalstabilityofasymmetricaltop,subjecttoamovableconstraint
(wire)inamoregeneralmanner,itisusefultoobtaintheeffectivepotential. Tothispurpose
weusetheLagrangianformulationoftheproblem[1–4]. Letusdenotetheprincipalaxesof
Rotationalstability—anamusingphysicalparadox 853
z z=Rotation axis z
(3) L L
β
(3) β
θ
p (2) (2)
θ
θλ y λ y
O O
(1) (1)
x φ Line of x φ
nodes
Figure8.Rotatingsymmetricaltops.Weassumethatω=φ˙andψ˙ =0,sincethereisnorotation
aroundaxis3. λisthedistancefrompointOtothehangingpoint. Left: illustratesthecaseofa
ring-liketop.Right:correspondstotherod-liketop.
inertia of the top by 1, 2 and 3, axis 3 being the symmetry axis. Due to this symmetry the
momentsofinertiaalongaxes1and2areequal,i.e.I =I =I (cid:7)=I . Thezaxis(inertial
12 1 2 3
frame) is chosen to be in the vertical direction. We take z = 0 to coincide with the lowest
positionofthecentreofmassofthetop.
TheLagrangianofthissystemis
L=T +T +T(cm)−V , (B.1)
tras rot rot grav
where T is the translational kinetic energy associated with the raising or lowering of the
tras
centreofmassofthetop. Accordingto(A.10)wehave
(cid:10) (cid:11) (cid:10) (cid:10) (cid:11) (cid:11)
1 dz 2 1 λ2 cosθ 2
T = m· CM = m (λ−ε)sinθ · 1+ √ ·θ˙ ,
tras 2 dt 2 (λ−ε) L2−λ2·sin2θ
(B.2)
where(λ−ε)isthedistancefromthecentreofmasstothehangingpointofthetop. Inthe
caseofthebar,(λ−ε)≈h/2andforaring(λ−ε)≈radius. ThetermT(cm) indicatesthe
rot
kineticrotationalenergyofthecentreofmass,i.e.,
(cid:10) (cid:11)
1 1 g2 λ 2 sin2θ
T(cm) = mω2ε2sin2θ = m . (B.3)
rot 2 2 ω2 L 1−(λ/L)2sin2θ
The rotational kinetic energy, T , written in terms of the Euler angles [1–4], for a
rot
symmetricaltopis
(cid:15)
1 1
T = I ω2 = [I (θ˙2+φ˙2sin2θ)+I (ψ˙ +φ˙cosθ)2], (B.4)
rot 2 i i 2 12 3
i
whereθ,φ andψ aretheEuleranglesthatareusedinourproblemasgeneralizedcoordinates
as indicated in figure 8. As usual, ψ describes the rotation of the top around axis 3 [1, 2].
In our case there is no rotation around this axis, since the wire does not twist, we have that
ψ = constantandψ˙ = 0. Theconstraintimposedbytherotatingwireimpliesthatφ˙ = ω.
Therefore,accordingtoourmodelθ becomestheonlydegreeoffreedomforoursystemand
854 CMSendraetal
weuseitasthegeneralizedcoordinatetodescribeitsstate. Expression(B.4)isdifferentfor
thecasesofaring-likeandtherod-liketopsasillustratedinfigure8. Inthecaseofarod-like
topwehave
T = 1[I (θ˙2+ω2sin2θ)+I ω2cos2θ]. (B.5)
rot 2 12 3
Inthecaseofaring-liketopwehave
(cid:16) (cid:17)
T = 1 I (θ˙2 +ω2sin2θ )+I ω2cos2θ , (B.6)
rot 2 12 p p 3 p
withθ =π/2−θ andθ˙ =−θ˙. Ifweintroduceanewvariable,χ sothat
p p
(cid:18)
π/2 forthecaseof thering-liketop
χ = (B.7)
0 forthecaseof therod-liketop.
Thenexpressions(B.5)and(B.6)canbecastinasingleconvenientform
T = 1[I (θ˙2+ω2sin2(χ −θ))+I ω2cos2(χ −θ)], (B.8)
rot 2 12 3
Forthegravitationalpotentialenergyaccordingtoequation(A.13)wehave
(cid:4)
V =mg·z=mg·[L+(λ−ε)(1−cosθ)− L2−λ2·sin2θ]. (B.9)
grav
Consequently,theLagrangianofoursystemis
(cid:10) (cid:11)
1 λ2 cosθ 2
L= m(λ−ε)2sin2θ · 1+ √ ·θ˙2
2 (λ−ε) L2−λ2·sin2θ
1
+ [I (θ˙2+ω2sin2(χ −θ))+I ω2cos2(χ −θ)]
12 3
2
(cid:10) (cid:11)
1 g2 λ 2 sin2θ
+ m
2 ω2 L 1−(λ/L)2sin2θ
(cid:4)
−mg·[L+(λ−ε)(1−cosθ)− L2−λ2·sin2θ] (B.10)
andtheconjugatedmomentumofθ is
(cid:6) (cid:10) (cid:11) (cid:7)
∂L λ2 cosθ 2
p = = m(λ−ε)2sin2θ 1+ √ +I ·θ˙. (B.11)
θ ∂θ˙ (λ−ε) L2−λ2sin2θ 12
TheHamiltonian[1–4]ofoursystemis
p2
H =p ·θ˙−L= (cid:16) (cid:8) θ (cid:9) (cid:17) +V (θ), (B.12)
θ 2 m(λ−ε)2sin2θ 1+ λ2 √ cosθ 2+I eff
(λ−ε) L2−λ2sin2θ 12
whereV (θ)istheeffectivepotentialgivenby
eff
(cid:10) (cid:11)
1 1 g2 λ 2 sin2θ
V (θ)=− ω2(I sin2(χ −θ)+I cos2(χ −θ))− m
eff 2 12 3 2 ω2 L 1−(λ/L)2sin2θ
(cid:4)
+mg·(L+(λ−ε)(1−cosθ)− L2−λ2sin2θ) (B.13)
and
∂V (θ)
eff =−ω2sinθcosθ|I −I |+mgsinθ
12 3
∂θ (cid:19) (cid:10) (cid:11) (cid:20)
λ2cosθ g λ 2 cosθ
· (λ−ε)+ √ − (cid:8) (cid:9) . (B.14)
L2−λ2sin2θ ω2 L 1−(λ/L)2sin2θ 2