Chem 373- Lecture 1: Classical Mechanics and the Schrödinger Equation
Classical Mechanics Lecture 22 - Simon Fraser University · 2016. 12. 1. · Classical Mechanics...
27
Classical Mechanics Lecture 22 Today’s Concept: Simple Harmonic Mo7on: Mo#on of a Pendulum Mechanics Lecture 8, Slide 1
Transcript of Classical Mechanics Lecture 22 - Simon Fraser University · 2016. 12. 1. · Classical Mechanics...
Classical Mechanics Lecture 22
Today’sConcept: SimpleHarmonicMo7on:Mo#onofaPendulum
MechanicsLecture8,Slide1
Grading
Unit14and15Ac7vityGuideswillnotbegraded
Pleaseturnin:! Unit14WriEenHomeworkonMon,Dec5! TheMini-labbookonyourSHMorKarateProject,Dec9
ThelastFlipItPhysicslectureisabonus.! notonexam! useitifyoumissedanypointsearlier
FinalExam7me:Fri.,Dec.9,12pm
Finalexamroom:2600
Announcements
PizzaPartynextMondayandReview! D200studentscancomeat1oreatcoldpizzaat2:30.
Midterm2solu7onswillbepublishedWednesday.! Turninyourrewritesbeforethenifyouwant.(karmapoints)
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
