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