12.7 One of the most prominent spectral lines of hydrogen is the line, a bright red line with a wavelength of 656.1

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1 .7 One of he mo prominen pera ine of hdrogen i he ine a brigh red ine wih a waeengh of m. a. wha i he epeed waeengh of he ine from a ar reeding wih a peed of km? b. he ine meaured on earh from oppoie end of he un equaor differ in waeengh b9 m. uming ha he effe i aued b roaion of he un find he period of roaion. The diameer of he un i. 6 km. Souion: a. our formua for reaiii opper effe: ν ν where i he frequen in he oure re frame i he o frequen in he oberer re frame. So he waeengh of he ine we obere wi be: o m m b. uppoe he period of he roaion of he un T i muh maer han ear. Then he earh i roaing abou he un o ow ha we oud approimae aume ha i i no roaing abou he un a a. So he reaie eoi beween he oure hdrogen aom on he un and he oberer peope on he earh i ρ where π 8 o8 o Then π R So T.775da R T

2 .8 The frequen of igh refeed from a moing mirror undergoe a opper hif beaue of he moion of he image. Find he opper hif of igh refeed dire bak from he a mirror whih i approahing he oberer wih peed and how ha i i he ame a if he image were moing oward he oberer a peed. Souion: Suppoe he oberer i a re in referene frame K; he mirror i a re in referene frame K. So in K referene frame he frequen of he inoming igh i: o where i he frequen of he inoming igh in K frame and Tha i:.. and we know a he mirror i a re in K frame he frequen of he ougoing igh houd be he ame a he frequen of he inoming igh in hi K referene frame. So in K referene frame he frequen of he ougoing igh i: o where i he frequen of he ougoing igh in K frame and Tha i:. Combining & : Suppoe he image were moing oward he oberer a peed. The frequen he oberer go woud be: I

3 .9 ab of ga moe o he righ wih peed. fah of igh i emied from and pae hrough he ga o arrie a a diane L awa. The ga ha hikne in i re frame and he peed of igh in he ga i n. How ong doe i ake he igh o go from o? Souion: Suppoe and are a re in referene frame K; he ga i a re in referene frame K. So in K he hikne of he ga: he diane he igh raeed ouide he ga: So he ime i ake for he igh o go from o : L * n L * n n Chek: L n If : L n n* * * If : L

4 . Here i he poe-auer parado. poe-auer and a farmer hae he foowing be: he poe-auer ha a poe of engh and he farmer ha a barn ong. The farmer be ha he an hu he door of he barn wih he poe ompee inide. The be being made he farmer ak he poe-auer o run ino he barn wih a peed of. In hi ae he farmer obere he poe o be Loren onraed o and he poe fi ino he barn wih eae. He am he door he inan he poe i inide and aim he be. The poe-auer diagree: he ee he barn onraed b a faor of and o he poe an poib fi inide. How woud ou ee he diagreemen? I he Loren onraion rea in hi probem? Souion: Suppoe he barn ha boh fron door and bak door and boh of hem are open o ha he poe-auer oud keep going wihou hiing he wa. efine wo een and wo referene frame. Een : he far end of he poe arrie a he bak door. Een : he poe-auer arrie a he fron door. Frame K: he referene frame where he farmer i a re. Frame K : he referene frame where he poe-auer i a re. In K: he poe i ong o In K : aording o Loren ranformaion: 5 So in K een happen afer een. The farmer ha pen of ime o hu boh door wih he poe ompee inide hen open he bak door o e he poe ome ou. In K een happen before een. So he poe-auer obere ha he far end of he poe eae he barn before he arrie a he fron door. So he poe i neer ompee inide. So i i a diagreemen abou imuanei and he order of een. The farmer aim ha he oud hu boh door a he ame ime < < wih he poe ompee inide whih i ruh in K. u he poe-auer obere ha he farmer hu he bak door wa before he hu he fron door or more ri he open he bak door o e he poe go before he hu he fron door.

5 . n quani whih i ef unhanged b he Loren ranformaion i aed a Loren inarian. Show ha i a Loren inarian where. here i he inera beween wo een and i he diane beween hem in he ame ineria em. Souion: Suppoe we hae in referene frame K. Then in referene frame K whih moe a reaie o K. ˆ So we hae: hen a where we hae ued. Now we go: Then we know ha i a Loren inarian.