FourierSeries

Green-underdamped

CalculusofVariations

Quiz11 Name___________________________

f {y, y '; x}dxa

b

∫ isstationarywhen ∂f∂y−ddx

∂f∂y '⎛

⎝⎜

⎞

⎠⎟= 0 .Notethatyisafunctionofx,and y ' ≡

dydx.

1.Given f {y, y '; x} = 1

2 y '2− gy ,where g isapositiveconstant.Findthedifferentialequationthat

makes f {y, y '; x}dxa

b

∫ stationaryalongapathy(x).(Itwillactuallybeaminimum.)

2.Solvethedifferentialequationbyintegration.Youwillneedtwoundeterminedconstants.

Quiz11 Name___________________________

f {y, y '; x}dxa

b

∫ isstationarywhen ∂f∂y−ddx

∂f∂y '⎛

⎝⎜

⎞

⎠⎟= 0 .Notethatyisafunctionofx,and y ' ≡

dydx.

1.Given f {y, y '; x} = 1

2 y '2− gy ,where g isapositiveconstant.Findthedifferentialequationthat

makes f {y, y '; x}dxa

b

∫ stationaryalongapathy(x).(Itwillactuallybeaminimum.)

2.Solvethedifferentialequationbyintegration.Youwillneedtwoundeterminedconstants.

HarmonicOscillator

Undamped:

Overdamped:

Critical:

Underdamped:

Driven:

Physics303MidtermExam2 Name_________________________________________1a.ThefigureshowsthepotentialenergyU=-2cos(x)(inJoules),wherex(inmeters)isplottedhorizontallyandUisplottedvertically.Amassm=0.125kgmovesinthispotentialwithoutdamping.Whatistheangularfrequencyω0 ofsmalloscillationsaboutx=0?1b.Forlargeoscillations(butwithamplitude<π),whatwillbetrue:a)themotionwillbepurelysinusoidal,becausethepotentialisb)themotionwillinclude“overtones”,bothoddandevenmultiplesofω0 c)themotionwillincludeonlyoddharmonics(motionatω0, 3ω0, 5ω0, etc.)d)themotionwillincludeonlyevenharmonics(motionat2ω0, 4ω0, 6ω0, etc.)e)themotionwillbechaotic,withnowell-definedfrequencies1c.Forlargeoscillations(butwithamplitude<π),whatelse?a)thefundamentalfrequencyω0willgetsmaller(longerperiod)b)thefundamentalfrequencyω0willgetbigger(shorterperiod)c)thefundamentalfrequencyω0willbeunchangedd)becauseofthechaoticmotion,itisnolongermeaningfultotalkaboutafundamentalfrequency2a.Amasshangsonaspring(onEarth).Whathappenstotheperiodofoscillationifthemassdoubles?a)itgetsfaster(shorter)byafactorof2b)itgetsslower(longer)byafactorof2c)itgetsfasterbyafactorof√2d)itgetsslowerbyafactorof√2e)itisunchanged2b.Whathappenstotheperiodofoscillationif,instead,gravity(G)suddenlydoubles?a)itgetsfasterbyafactorof2b)itgetsslowerbyafactorof2c)itgetsfasterbyafactorof√2d)itgetsslowerbyafactorof√2e)itisunchanged

3a.Ifthedistancetothemoonwerehalved,howwoulditaffectthegravitationalforcethemoonexertsonyou?a)itwouldbe1/8thasmuchb)itwouldbe1/4asmuchc)itwouldbehalfasmuchd)itwouldbeunchanged

e)itwouldbetwiceasbigf)itwouldbefourtimesasbigg)itwouldbeeighttimesasbig

3b.Ifthedistancetothemoonwerehalved,howwoulditaffectthetidalforcefromthemoon?a)itwouldbe1/8thasmuchb)itwouldbe1/4asmuchc)itwouldbehalfasmuchd)itwouldbeunchanged

e)itwouldbetwiceasbigf)itwouldbefourtimesasbigg)itwouldbeeighttimesasbig

4.Adynamicalsystemisdescribedbyalineardifferentialequation.Whensubjecttothetriangularpulsedrivingforceshown(solidline),whichbeginsatt=0,theresponseisthedashedline,f(t).Inadditiontotheoriginaldrivingforce,asecond“pulse”isappliedatt2,withanamplitudetwiceaslargeastheoriginal.Isitpossibletowritedowntheresponseofthesystemtothetwopulses,intermsoffandt2?(Youranswerwouldbeafunctionoft,orcourse.)Ifso,writeitdown.Ifnot,explainwhyitisnotpossible.

5a.Alinearharmonicoscillatorhasω0=2s-1andβ=1s-1.ItisdrivenwithF /m = cos2t + cos6t (inN/kg;forallt).Whichplotshowsthemotion?Thehorizontalaxisistimeinseconds,verticalxinm.Circlethecorrectplot.

5b.ThesameharmonicoscillatorisdrivenwithF/m=cos4.377t(foralltime.)Theresultingmotionisgivenby x(t) = Dcos(ω*t −δ) .Whatareω* andδ?(I’llgiveyouD=0.057m;youdon’tneeditforthispart.)5c.Instead,thisforceisturnedonatt=0.Findthepositionx(t)fort>0.

6a.FindtheGreen’sfunction(responsetoaunitimpulseattimet’)forthecriticallydampedharmonicoscillator.YourGreen’sfunctionmaycontainβ, t,t’,m.6b.Thecriticallydampedoscillatorissubjecttoaforce=αmtfor0<t<6s.m=mass,αisaconstant.Writedown(butdonotsolve)anintegralthatgivesthemotionfor0<t<6s.

7a.Findthedifferentialequationofthepaththatminimizes f (x, x ';t)dtt1

t2

∫ where

f (x, x ';t) = 12 x '

2+ x .7b.Solvethedifferentialequation.Youshouldhavetwoundeterminedconstants.7c.Solveforyourundeterminedconstants,giventhatx(0)=0andx(4)=0.