Classical Mechanics Lecture 22

Today’sConcept: SimpleHarmonicMo7on:Mo#onofaPendulum

MechanicsLecture8,Slide1

Grading

Unit14and15Ac7vityGuideswillnotbegraded

Pleaseturnin:! Unit14WriIenHomeworkonMon,Dec5! TheMini-labbookonyourSHMorKarateProject,Dec12

ThelastFlipItPhysicslectureisabonus.! notonexam! useitifyoumissedanypointsearlier! AlsousetheextraTipler&Moscaques7onsonF.P.forbonusandprep

FinalExam7me:Tue.,Dec.12,3:30pm

Finalexamroom:2600

Howdoweknowwhentousethesinformulaorcosformulafortheposi7onoftheoscilla7ngsystem?Aretheybothviable?alsocanyouseethisface--> ( ͡° ͜ʖ ͡°)?

thisiskindoffromthelastlecturebuthowdoyougettheequa7onofx=Asin(ωt+φ)fromawordproblem?(ie.Howdoyouknowwahtφis?andisitcosorsin?)

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

Iamfindingitdifficulttounderstandhowthemomentofiner7aandtheradiusarebothbeingusedintheequa7on.Isn'tthemomentofiner7adependentontheradius?

talkingaboutharmonicmo7ons,Letsalldance"GANGNOMstyle",itsaperfectprac7calexample!

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

Howdoyouknowwhentousewhatformula?Intheprelecturetheydidn'texplainclearlyifyoucanusethesameformulaforapendulumwithamassaIachedtotheendasforapendulumwithoutamassaIachedtoit.

Your Comments

MechanicsLecture8,Slide2

despacito

“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–yougetthesameansweranywayaslongasthereisnofric7on.

MechanicsLecture8,Slide6

Iwanttoknowwhytheanswertolifeis42!

Youwilloscillatelikeamassonaspringwithaperiodof84minutes.Ittakes42minutestocomeouttheotherside!

Drillaholethroughtheearthandjumpin–whathappens?

Thisisalsothesameperiodofanobjectorbi7ngtheearthrightatgroundlevel.

MechanicsLecture8,Slide7Justforfun–youdon’tneedtoknowthis.

Iwanttoknowwhytheanswertolifeis42!

Panic!

“IstheresuchathingasRota7onalHarmonicMo7on?TherebeIernotbe...”

Yesthereis.

Areyouready?

I

wire

θ

τ

Torsion Pendulum

Q:Intheprelecturetheequa7onforrestoringtorqueisgivenasτ=-κθinclockwisedirec7on..soiftherestoringtorqueisincounterclockwisedirec7onsthenwouldτbeposi7ve?

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)Themomentofiner7adecreases,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.InCase2apointpar7cleofmassmisaIachedtothecenterofthesames7ck.Inwhichcaseistheperiodofthependulumthelongest?

A)Case1B)Case2C)Same

Case1 Case2

Cisnottherightanswer.

Letsworkthroughit

CheckPoint

m

m

m

MechanicsLecture8,Slide16

T = 2⇡

s23 Lg

T = 2⇡

s12 Lg

InCase1as7ckofmassmandlengthLispivotedatoneendandusedasapendulum.InCase2apointpar7cleofmassmisaIachedtoastringoflengthL/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

m

m

InCase1as7ckofmassmandlengthLispivotedatoneendandusedasapendulum.InCase2apointpar7cleofmassmisaIachedtothecenterofthesames7ck.Inwhichcaseistheperiodofthependulumthelongest?

A)Case1B)Case2C)Same

Nowletsworkthroughitindetail

Case1 Case2

m

MechanicsLecture8,Slide19

m

m

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