Classical Mechanics Lecture 22

Today’sConcept: SimpleHarmonicMo7on:Mo#onofaPendulum

MechanicsLecture8,Slide1

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Fridayyouwilldoaprojecton! SimpleHarmonicmo7on(anywayyouwant)or! Karate! Unit14AGwillnotbegraded.Keepitforexam! Turninspreadsheets,WH14etc

DoOnlineCourseEvalua7onand

youcans7lldotheIOLabSurvey(extracredit)

Idon'tlikeradians.Whenpeoplesaypi,Isay180degrees.Orevenmorepro,Isay200gon.

thisiskindoffromthelastlecturebuthowdoyougettheequaFonofx=Asin(wt+phi)fromawordproblem?(ie.Howdoyouknowwahtphiis?andisitcosorsin?)

Who'sthegeniuswhodecidedomegashouldhavetwomeanings?DidtheyrunoutofGreekleQers?Whydon'ttheyflyoverthereandgetsomemore?Itwouldprobablyhelpboosttheireconomyatthispoint.

IamfindingitdifficulttounderstandhowthemomentofinerFaandtheradiusarebothbeingusedintheequaFon.Isn'tthemomentofinerFadependentontheradius?

IstheperiodproporFonaltoRcmthen?

talkingaboutharmonicmoFons,Letsalldance"GANGNOMstyle",itsaperfectpracFcalexample!

Forthetorsionpendulum,whatdidthelowercasekappa(κ, κ)represent?Whatcausesthatconstant?Thanks.

Howdoyouknowwhentousewhatformula?Intheprelecturetheydidn'texplainclearlyifyoucanusethesameformulaforapendulumwithamassaQachedtotheendasforapendulumwithoutamassaQachedtoit.

Your Comments

MechanicsLecture8,Slide2

“ThereisatheorywhichstatesthatifeveranybodydiscoversexactlywhattheUniverseisforandwhyitishere,it

willinstantlydisappearandbereplacedbysomethingevenmorebizarreandinexplicable.”

MechanicsLecture8,Slide3

“Thereisanothertheorywhichstatesthatthishasalreadyhappened.”

Text

DouglasAdams

Drillaholethroughtheearthandjumpin–whathappens?

Justforfun–youdon’tneedtoknowthis.

Iwanttoknowwhytheanswertolifeis42!

Youwilloscillatelikeamassonaspringwithaperiodof84minutes.Ittakes42minutestocomeouttheotherside!

Drillaholethroughtheearthandjumpin–whathappens?

k = mg/RE

MechanicsLecture8,Slide5

Iwanttoknowwhytheanswertolifeis42!

Theholedoesn’tevenhavetogothroughthemiddle–yougetthesameansweranywayaslongasthereisnofricFon.

MechanicsLecture8,Slide6

Iwanttoknowwhytheanswertolifeis42!

Youwilloscillatelikeamassonaspringwithaperiodof84minutes.Ittakes42minutestocomeouttheotherside!

Drillaholethroughtheearthandjumpin–whathappens?

Thisisalsothesameperiodofanobjectorbi7ngtheearthrightatgroundlevel.

MechanicsLecture8,Slide7Justforfun–youdon’tneedtoknowthis.

Iwanttoknowwhytheanswertolifeis42!

Panic!

“IstheresuchathingasRota7onalHarmonicMo7on?TherebeEernotbe...”

Yesthereis.

Areyouready?

I

wire

θ

τ

Torsion Pendulum

Q:IntheprelecturetheequaFonforrestoringtorqueisgivenasτ=-κθinclockwisedirecFon..soiftherestoringtorqueisincounterclockwisedirecFonsthenwouldτbeposiFve?

MechanicsLecture8,Slide8

Atorsionpendulumisusedasthe7mingelementinaclockasshown.Thespeedoftheclockisadjustedbychangingthedistanceoftwosmalldisksfromtherota7onaxisofthependulum.Ifweadjustthediskssothattheyareclosertotherota7onaxis,theclockruns:

A)FasterB)Slower

Small disks

CheckPoint

MechanicsLecture8,Slide9

Ifweadjustthediskssothattheyareclosertotherota7onaxis,theclockruns

A)FasterB)Slower

B)T=2pi*sqrt(I/MgRcm).IfRcmdecreases,Twillincrease,makingtheclockrunslower.

A)ThemomentofinerFadecreases,sotheangularfrequencyincreases,whichmakestheperiodshorterandthustheclockfaster.

MechanicsLecture8,Slide10

CheckPoint

Pendulum

Forsmallθ

θ

XCM

RCM

Mg

θ

arc-length = RCM θ

XCM

RCM

MechanicsLecture8,Slide11

CM

pivot

θ

RCM

The Simple Pendulum

Thegeneralcase

θL

Thesimplecase

MechanicsLecture8,Slide12

Iftheclockisrunningtoofast,theweightneedstobemovedA)UpB)Down

Iftheclockisrunningtoofastthenwewanttoreduceit'speriod,T,andtodothatweneedtoincreaseomega,thefrequencyitmoveswithandtodothatweneedtheposi7onofthecenterofmasstobefurtherfromthepivot,whichisachievedbymovingtheweightdown.

MechanicsLecture8,Slide14

CheckPoint

M

pivot

θRCM

The Stick Pendulum

CM

MechanicsLecture8,Slide15

Same period

InCase1as7ckofmassmandlengthLispivotedatoneendandusedasapendulum.InCase2apointpar7cleofmassmisaEachedtothecenterofthesames7ck.Inwhichcaseistheperiodofthependulumthelongest?

A)Case1B)Case2C)Same

Case1 Case2

Cisnottherightanswer.

Letsworkthroughit

CheckPoint

m

m

m

MechanicsLecture8,Slide16

T = 2⇡

s23 Lg

T = 2⇡

s12 Lg

InCase1as7ckofmassmandlengthLispivotedatoneendandusedasapendulum.InCase2apointpar7cleofmassmisaEachedtoastringoflengthL/2?

Inwhichcaseistheperiodofthependulumlongest?

A)Case1B)Case2C)Same

Case1 Case2

m

MechanicsLecture8,Slide17

Nowsupposeyoumakeanewpendulumbyhangingthefirsttwofromthesamepivotandgluingthemtogether.

Whatistheperiodofthenewpendulum?

A)T1 B)T2C)Inbetween

m

Supposeyoustartwith2differentpendula,onehavingperiodT1andtheotherhavingperiodT2.

T1

T2

T1 > T2

MechanicsLecture8,Slide18

mm

InCase1as7ckofmassmandlengthLispivotedatoneendandusedasapendulum.InCase2apointpar7cleofmassmisaEachedtothecenterofthesames7ck.Inwhichcaseistheperiodofthependulumthelongest?

A)Case1B)Case2C)Same

Nowletsworkthroughitindetail

Case1 Case2

m

m

m

MechanicsLecture8,Slide19

Case2m

m

mCase 1

Letscompareforeachcase.

MechanicsLecture8,Slide20

(A)

(B)

(C)

Case2m

m

mCase 1

Letscompareforeachcase.

MechanicsLecture8,Slide21

Inwhichcaseistheperiodlongest?

A)Case1

B)Case2

C)Theyarethesame

m

Case1

Sowecanworkout

Case2

m

m

MechanicsLecture8,Slide22

Angle(degrees)

%differen

cebetweenθ and

sinθ

- Exact expression

The Small Angle Approximation

θ

arc-length = RCM θ

XCM

RCM

MechanicsLecture8,Slide23

Apendulumismadebyhangingathinhoola-hoopofdiameterDonasmallnail.Whatistheangularfrequencyofoscilla7onofthehoopforsmalldisplacements?(ICM=mR2forahoop)

A)

B)

C)

D

pivot(nail)

Clicker Question

MechanicsLecture8,Slide24

Theangularfrequencyofoscilla7onofthehoopforsmall

displacementswillbegivenby

R

XCM

Useparallelaxistheorem:I = ICM + mR2

m

= mR2 + mR2 = 2mR2

pivot(nail)

MechanicsLecture8,Slide25

So