PHYS420 (Spring 2002) Riq Parra Homework # 2 Solutions
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1 PHYS4 Sping ) Riq Paa Hoewok # Soluions Pobles. Genealie he Galilean ansfoaion of oodinaes o oion in hee diensions b showing ha ' & '. In he deiaion of he Galilean ansfoaions ha was done in lass we assued ha he oion of he oing fae was jus in he posiie dieion i.e. ). In his poble we ae going o genealie his o an dieion i.e. ). Assuing ha a ie he wo faes ae ogehe hen a soe abia ie lae he disane beween he wo faes will be in he dieion jus like he lass deiaion) in he dieion and in he dieion. In eo noaion we wie his as d ). Fo eah dieion he onneion beween he oodinaes in he oing fae ) and he oodinaes in he saiona fae ) ae jus ' Theefoe in eo noaion ) ) ) ) ) '. In a laboao fae of efeene an obsee noes ha Newon s nd law is alid. a) Show ha i s also alid fo an obsee oing a onsan speed elaie o he laboao fae we did his in lass) & b) Show ha i is no alid in a efeene fae oing pas wih onsan aeleaion. This poble is sipl SMM Chape Pobles & obined.
2 In ode o show ha Newon s nd law Fa) is alid in a oing fae we us look a how aeleaions ansfo. a) Fo a fae oing a onsan speed in he posiie dieion) elaie o a saiona fae he Galilean oodinae ansfoaions is jus Taking a ie deiaie d/d ) and ealling ha d d and ha is a onsan we find how he eloi ansfos. This is sipl he Galilean eloi addiion law. d d d d d d u ) u d d d d d d To find he aeleaion we ake anohe ie deiaie d/d wih d d). du d du du a u ) a d d d d And so we find ha he aeleaions ae idenial. Theefoe so ae he foes F a a F b) Fo a fae oing a onsan aeleaion a in he posiie dieion) elaie o a saiona fae we anno use he sandad Galilean ansfoaion ules ha we deied in lass anoe. Assuing ha a ie he wo faes ae ogehe hen a soe abia ie lae he disane beween he wo faes will be ½ a in he dieion. In his ase he new oodinae ansfoaion is a Taking ie deiaies o find he eloi and eebeing ha d d) d d d d u a ) a u a d d d d Taking ie deiaies o find he aeleaion du d du d a u a ) a a a d d d d We see ha he wo aeleaions ae no idenial. So in his ase Newon s nd law will no hae he sae alue in he wo diffeen faes F a a a a a F a! ) 3. Wha happens o Mawell s equaions unde a Galilean ansfoaion? In a saiona efeene fae K) in fee spae he sala field ) saisfies he sala wae equaion. Show ha he fo of he wae equaion is no inaian unde Galilean ansfoaions. This quesion is eall asking whehe o no is ue unde Galilean ansfoaions. In geneal we ake use of he Chain ule o see how he deiaies ansfo. Fo he ie being and o be as geneal as
3 possible le s no assue a speifi fo fo he oodinae ansfoaion ules. We ll assue ha he oodinaes in he oing fae ae ) and ha he depend onl on ). Siilal he oodinaes in he saiona fae ae ) and he depend onl on ). The deiae opeaos an heefoe be epanded he following wa I looks painful bu i s saighfowad. Now we an assue a se of ansfoaion ules. In ou eaple we ll use he Galilean ansfoaion ules. We an now opue he oeffiiens in he deiaie epansion. This is he pa ha depends on ou hoie of ansfoaion ules. So puing eehing ogehe we see ha he deiaies ansfo he following wa: Tuning now o he sala wae equaion in he non oing oodinae fae
4 Spliing he deiaies o be lea Subsiuing he new deiaies Disibuing hough and ehanging he ode of he deiaies Subsiuing again he new deiaies Afe disibuing hough and siplifing we ge d You an see ha his is no he oiginal sala wae equaion fo; we go wo ea es on he igh side! Theefoe i is NOT inaian unde Galilean ansfoaions. 4. SMM Chape Poble 3. A -kg a oing wih a speed of /s ollides wih and siks o a 5-kg a a es a a sop sign. Show ha beause oenu is onseed in he es fae oenu is also onseed in a efeene fae oing wih a speed of /s in he dieion of he oing a. Le kg /s 5 kg & /s. Aoding o onseaion of oenu in he saiona fae final ) Soling fo final and plugging he nubes s final ) Now le s look a he sae siuaion fo a efeene fae oing wih a speed of /s. In his new fae /s
5 - /s final final /s And so we find ha oenu in his new fae is onseed ) final )) 5) ) 35).49...) Mihelson Mole epeien. Show ha we wee jusified in keeping onl he fis e of he binoial epansion when deiing he epeed finge shif. If L ou eall Shif. In ohe wods alulae wha he finge shif would be λ if ou kep he ne e and opae i o he esoluion of he epeien σ. finge). Ae we jusified? Finge The binoial epansion o he ne e is ) n n n n ) In deiing he epeed finge shif we had he following ea epession / L Appoiaing his using he binoial heoe o he ne e we find L 3 L 5 L The epeed finge is heefoe 4 L 5 L Shif λ λ λ Whee we e plugged he nubes poided in he book i.e. L 3 k/s λ 5 n). Sine his oeion is uh salle han he esoluion of he epeien we ae pefel jusified in keeping onl he fis e of he binoial epansion. 6. Snhonied loks ae saioned a egula ineals illion k apa along a saigh line. When he lok ne o ou eads noon wha ie do ou see assuing ou hae a eall poweful elesope) on he 9 h lok down he line? If he loks ae spaed illion k apa hen he 9 h lok down he line is 9 illion k fo he lok ne o e i.e. d 9 ). Ligh akes d/ 9 )/3 8 /s) 3 seonds 5 inues o ge o e. So he lok will alwas see o be 5 inues behind. In ohe wods when I see noon on he lok b side he ligh ha eahes e fo he 9 h lok down us hae lef 5 inues pio o ha. Hene he 9 h lok down will ead :55 a.
6 dp 7. Sole he non-elaiisi Newon s equaion of oion F ) in he ase of a d onsan foe in he posiie dieion F F ). As a bounda ondiion le ) and ). Ignoe oion in he & dieions. Ignoing oion in he and dieions we wan o sole he one-diensional nd ode diffeenial equaion fo ). dp d d d d F ) d d d d d Le s iniiall leae i in es of he eloi. d F d This equaion is sepaable and an be soled b inegaing wih espe o eah aiable subje o he iniial ondiions. F d d F d d ) F Diiding b o lean hings up and eplaing wih d/d F d d Sine his is again sepaable we epea he poedue subje o iniial ondiions) F d d F ) Thus we deie Newon s equaion of oion eebe ha F/ a) F ) d
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