Classical Mechanics Lecture 20 Today’s Concepts: A) Angular Momentum B) Precession Mechanics Lecture 20, Slide 1
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Classical Mechanics Lecture 20
Today’sConcepts:
A)AngularMomentum
B)Precession
MechanicsLecture20,Slide1
Torques and Gyroscopes
MechanicalUniverse:Episode19
Student on Stool
TherearenoexternaltorquesacCngonthestudent-stoolsystem,soangularmomentumwillbeconserved.
IniCally:Li = Ii ωi
Finally:Lf = If ωf
ω f
If
Lf
ω i
Ii
Li
MechanicsLecture20,Slide3
Clicker QuestionAstudentsitsonafreelyturningstoolandrotateswithconstantangularvelocityω1.Shepullsherarmsinandherangularvelocityincreasestoω2.
IndoingthisherkineCcenergy:A)IncreasesB)DecreasesC)Staysthesame
ω f
If
Lf
ω i
Ii
Li
MechanicsLecture20,Slide4
(usingL = Iω)
L isconserved:
Ii < If Kf > Ki K increases!
ω f
If
Lf
ω i
Ii
Li
MechanicsLecture20,Slide5
Sincethestudenthastoforceherarmstomovetowardherbody,shemustbedoingposiCvework!
Thework/kineCcenergytheoremstatesthatthiswillincreasethekineCcenergyofthesystem!
ω f
If
Lf
ω i
Ii
Li
MechanicsLecture20,Slide6
IwouldliketoseethestudentsiRnginthechairwithatopinareallifedemo.
MechanicsLecture20,Slide7
ApuckslidesinacircularpathonahorizontalfricConlesstable.ItisheldataconstantradiusbyastringthreadedthroughafricConlessholeatthecenterofthetable.Ifyoupullonthestringsuchthattheradiusdecreasesbyafactorof2,bywhatfactordoestheangularvelocityofthepuckincrease?
A)2B)4C)8
Clicker Question (like CheckPoint)
MechanicsLecture20,Slide8
=Since the string is pulled through a hole at the center of rotation, there is no torque: Angular momentum is conserved.
R
Clicker Question
MechanicsLecture20,Slide9
We just used to find
usually0,butnotnow
Food for thought (not on any test)
R
MechanicsLecture20,Slide10
But Sohowdowegetanα withoutaτ ?
NowsupposeτEXT = 0:
Sointhiscasewecanhaveanα withoutanexternaltorque!
Food for thought (not on any test)
MechanicsLecture20,Slide11
Precession
Thedirec=onofthistorqueattheinstantshownisoutofthepage(usingtherighthandrule).
Themagnitudeofthetorqueaboutthepivotisτ = Mgd.
Thechangeinangularmomentumattheinstantshownmustalsobeoutofthepage!
MechanicsLecture20,Slide13
~⌧ext
=d~Ldt
AerialView
pivot
τEXT
Precession
Ω
dφ
MechanicsLecture20,Slide14
DirecCon:TheCpofLmovesinthedirecConofτ.
Precession
Inthisexample:
MechanicsLecture20,Slide15
Adiskisspinningwithangularvelocityωonapivotedhorizontalaxleasshown.Gravityactsdown.InwhichdirecCondoesprecessioncausethedisktomove?
A)OutofthepageB)IntothepageC)UpD)Down
Torqueisoutofthepage
CheckPoint
MechanicsLecture20,Slide16
A B
InwhichdirecCondoes point?
CheckPoint
MechanicsLecture20,Slide17
InwhichdirecCondoesprecessioncausethedisktomove?
A)IntothepageB)OutofthepageC)UpD)Down
B)Gravityexertsanettorqueoutofthepage,sothediskprecessesinadirec=onoutofthepage.
D)Torqueisdownsoprecessionisdown
Torqueisoutofthepage
CheckPoint
MechanicsLecture20,Slide18
http://www.smartphysics.com/smartphysics/course/instructor/charts/stdhw_choice_histogram.aspx?qid=593054
A)Thetorqueprovidedbygravityisgoingoutofthepage.Therefore,Lmustchangeoutofthepage.However,sinceLisintheoppositedirec=onoftheaxelwhendrawnfromthepivot,precessionmustmakethediskgointothepage.
Adiskisspinningwithangularvelocityω onapivotedhorizontalaxleasshown.Ifthemassofthediskweredoubledbutitsradiusandangularvelocitywerekeptthesame:
A)TheangularmomentumofthediskdoublesB)ThetorqueaboutthepivotdoublesC)BothAandB
Clicker Question
L = Iω
MechanicsLecture20,Slide19
Adiskisspinningwithangularvelocityω onapivotedhorizontalaxleasshown.GravityactsdownandthediskhasaprecessionfrequencyΩ.Ifthemassofthediskweredoubledbutitsradiusandangularvelocitywerekeptthesame,theprecessionfrequencywould:
A)IncreaseB)DecreaseC)Staythesame
CheckPoint
MechanicsLecture20,Slide20
Ifthemassofthediskweredoubledbutitsradiusandangularvelocitywerekeptthesame,theprecessionfrequencywould
A)IncreaseB)DecreaseC)Staythesame
A)torqueduetoweightincreasesinthenumerator.
B)Whenmassisdoubled,angularmomentumdoubles,soprecessionfrequencydecreases.
C)Increasingthemassincreasesboththeangularmomentumandthetorqueequally,sotheprecessionfrequencystaysthesame
CheckPoint
MechanicsLecture20,Slide21http://www.smartphysics.com/smartphysics/course/instructor/charts/stdhw_choice_histogram.aspx?qid=593056
Adiskisspinningwithangularvelocityω onapivotedhorizontalaxleasshown.Iftheradiusofthediskweredoubledbutitsmassandangularvelocitywerekeptthesame:
A)TheangularmomentumofthediskdoublesB)ThetorqueaboutthepivotdoublesC)BothAandB
Clicker Question
L = Iω
MechanicsLecture20,Slide22
Adiskisspinningwithangularvelocityωonapivotedhorizontalaxleasshown.GravityactsdownandthediskhasaprecessionfrequencyΩ.Iftheradiusofthediskweredoubledbutitsmassandangularvelocitywerekeptthesame,theprecessionfrequencywould
A)IncreaseB)DecreaseC)Staythesame
CheckPoint
MechanicsLecture20,Slide23
Iftheradiusofthediskweredoubledbutitsmassandangularvelocitywerekeptthesame,theprecessionfrequencywould
A)IncreaseB)DecreaseC)Staythesame
A)Themomentofiner=awouldincrease,sotheangularmomentumwouldincrease,soomegawouldalsoincrease
B)Ifyouincreaseradius,youincreaseangularmomentum,andthusdecreasetheprecessionfrequency.
C)thefrequencywouldremainthesamebecausetheradiushasnoimpactonthefrequency
CheckPoint
MechanicsLecture20,Slide24
WheelsteersrightWheelsteersleY
Allofthisstuffdoesn'treallyseemprac=cal.Ifyoucan'tprovemewrongthenweshouldalljustgetA'sforthissec=onofthecourse.
http://www.youtube.com/watch?v=cquvA_IpEsA (see2:30)
Prac=calApplica=on:Keepsyoufromfallingoffyourbikewhenyouride usingnohands!
Ridingstraight(τ = 0)
LeanLeY(τ outofpage)
LeanRight(τ intopage)
Wheelsteersstraight
MechanicsLecture20,Slide25
Mg
d
MechanicsLecture20,Slide26
L (usingrighthandrule)
MechanicsLecture20,Slide27
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