Phys 331: Ch 7,.2 More practice with Unconstrained Lagrange s Equations 1

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1 Phs 33: Ch 7 Moe ctce wth Unconstne gnge s Eutons We /3 hus / F / Mon /5 We / Genelze Vbles & Clsscl Hltonn (Recoen 79 f ou e h Phs 33) 8- -Bo Centl Foces Relte Coontes Reew E (Ch 5-7) HW7c ( ) Poject Outlne o we ll wok soe oe eles but fst we ll look t two e otnt es Conseton ws n Noethe s theoe Hlton s Euton Attel we e not gong to o uch wth these but the fst s nteestng n ts own ght n the secon gets lot of l n Quntu Conseton of Moentu n nsltonl Innce Igne ou e wtchng soethng l out I on t know boes gttonll ntectng In oe to get unttte bout oelng wht ou see the fst thng ou o s efne coonte sste hen ou cn secf the ostons n otons of the tcles elte to tht ogn n the coonte ectons ẑ ( z ( z ŷ ) ) z z ˆ Now the knetc eneg of the sste eens uon the sees n the otentl eneg of n solte sste eens on the elte seton of the tcles U ( ) ) ( hen gn ou coul he chosen ffeent locton fo ou ogn n tht wouln t chnge e uch bout ou escton of the sste s beho the two gttonll-ntectng objects woul stll be just s f t fo ech othe n the he the se eloctes the just he ffeent loctons elte to the ogn ẑ ẑ b b ( z) ( z z z ) ˆ ˆ ŷ ( z ) ( z z z )

2 So U ( ) ) ( Obousl then b Conseng slceent just n one ecton t te fo nstnce Usng the fct tht We he tht So the totl oentu s consee n fo ou nul tcles So tht sste s oentu s consee (no net etenl foce) s eulent to sng tht the lgngn s unffecte b tnsltons though sce An obous counte ele s when soethng s cte uon the gttonl foce of the Eth n we tke the Eth to be outse the sste Rotton n Angul Moentu We coul of couse ke sl guent bout ottng fo one fe to nothe he gngn s beng unffecte b ths s eulent to wht beng consee? Angul oentu o show ths ll ou ell he to o s elce wth n eo the wok boe

3 nslton though te Now s thee nthng secl f the lgngn oesn t chnge wth te? et s look t the te ete of the lgngn ( t) t But So wht we he s t he eson I wte t out ths w s so we cn ecognze the ouct ule n cton So t Now f ou coontes e Ctesn then O s the book suggests oe genell we cn tlk bout ths ete s genelze oentu Fo ele f s n ngle then s the ngul oentu So usng tht shothn we he t Rehse we he - t 3

4 hs oesn t look too nceble ll b tself but f the gngn oesn t he n elct te eenence (ecll none of the eles we e consee he h n) then the ght-hn se s zeo enng the guent ust he no (elct o othewse) te eenence t s constnt n consee untt - f t So n those cses we he consee untt n tht untt s clle H he Hltonn Clel f we e elng n Ctesn coontes the s e z fo the ffeent tcles n the s e z then H ( U) ( U) ( U) he book oes oe genell tht s long s ou genelze coonte sste s not chngng wth esect to Ctesn coontes then ou get ths U E You well wone wht s so sgnfcnt bout the Hltonn f t just totl eneg? Fo tht tte ou coul he ske wht s so sgnfcnt bout the gngn f t s just the ffeence between knetc n otentl eneg he sgnfcnce s not the thng tself but wht we cn o wth t he gngn stsfes he Hltonn stsfes H n H Soetes ths of sle eutons e ese to wok wth thn gnge s euton 4

5 Eles Ok te fo oe wok on the eles we begun lst te Pctce wth Constne sstes Now we ll contnue gettng oe ctce Fst I ll ge the set u fo hnful of obles n then I ll ge the solutons so ou cn t wokng the ouself but lso ek t the solutons Hee s el clssc: suose stng of length s connects uck of ss on fctonless tble n n object wth ss though hole (see the fgue below) s - () Wte the gngn fo the sste n tes of the uck s ol coontes (b) Fn the eutons of oton fo the sste (Book oes E 75) A sll block s slng long lne tht cn sle wthout fcton he coonte s clel not Ctesn coonte M 5

6 Eecse #5: (book sets u but oesn t o) Suose ouble enulu conssts of two g ssless os connectng two sses he sses e not eul but the lengths of the os e he enulu onl swngs n one etcl lne () Wte the gngn fo the sste n tes of the ngles shown boe (b) Fn the eutons of oton of the sste (E 9 of F&C 5 th e) Shecl enulu Fn the eutons of oton usng shecl ol coontes Note tht the length s fe hs s the se oble s we woke when the bll ws confne to the sufce of shee ecet now we efne z own n the ecton of the gttonl foce R z Ele #: (E 77 of honton) Fctonless be on snnng bolc we z c ottng wth n ngul eloct Wht s the conton on the ngul feuenc to get the be to otte t fe locton on the we? z 6

7 Solutons Hee s el clssc: suose stng of length s connects uck of ss on fctonless tble n n object wth ss though hole (see the fgue below) s - () Wte the gngn fo the sste n tes of the uck s ol coontes he knetc eneg fo the uck s: n fo the othe ss s: efne the gttonl otentl eneg to be zeo t the leel of the tble so: he gngn s: U g s U g s (b) Fn the eutons of oton fo the sste n g n g n constnt he secon euton s eulent to conseton of ngul oentu he fst euton cn be ewtten s: ook t soe Qultte cses: g 3 Whethe the us gows o shnks eens on whethe the obtl te s gete o less thn the gttonl te he blncng ont whee thee s no l cceleton s when 7

8 3 g It not look too fl n tht fo but ehsng t s tn gentl F l F l M look fl when the l foce eeence b s eul to / the thng eecutes ccul obt so = constnt (Book oes E 75) A sll block s slng long lne tht cn sle wthout fcton he coonte s clel not Ctesn coonte M he eloct of the block elte to the goun (netl fe) s the eloct of the lne lus the eloct of the block elte to the lne (see the g below) B the lw of cosnes the sze of the block s see sue s: cos Altentel one cn bek t own ths w n get the se esult: cos sn 8

9 he knetc eneg of the sste s: n the otentl eneg s: he gngn s: he eutons of oton e: whch ge: he fst euton ges: M cos U g sn U M cos g sn n M cos n g sn cos M cos n cos g sn cos M hese eutons cn be sole (ou cn check the lgeb) fo n to get: g sn g sn cos n cos M cos M he ght hn ses of these eutons e constnt so these e es to sole he cceleton of the block elte to the lne n the cceleton of the lne e constnt gettng these eutons usng Newton s secon lw! Eecse #5: (book sets u but oesn t o) Suose ouble enulu conssts of two g ssless os connectng two sses he sses e not eul but the lengths of the os e he enulu onl swngs n one etcl lne () Wte the gngn fo the sste n tes of the ngles shown boe 9

10 ke the to ot ont s the ogn n choose oste to the ght n oste uw he coonents of the oston of e: so the coonents of ts eloct e he coonents of the oston of e: so the coonents of ts eloct e he knetc eneg s: sn n cos cos n sn sn sn n cos cos sn sn n cos cos cos cos n sn sn he otentl eneg s: so the gngn s: = sn cos cos sn cos cos sn sn cos cos U g g U g cos g cos cos g cos g (b) Fn the eutons of oton of the sste he euton ssocte wth sn sn he euton ssocte wth s: gsn s: gsn cos cos cos sn

11 sn gsn cos sn gsn cos sn hese eutons e heousl colcte but t s not ffcult to get usng the gngn och f ou e ceful he woul be teenousl ffcult to get usng the Newtonn och! (E 9 of F&C 5 th e) Shecl enulu Fn the eutons of oton usng shecl ol coontes Note tht the length s fe hs s the se oble s we woke when the bll ws confne to the sufce of shee ecet now we efne z own n the ecton of the gttonl foce R z he oston of the bob n Ctesn coontes s: Rsn cos Rsn sn n z Rcos he esest w to get the see sue s s follows: ˆ sn ˆ ˆ So hen ˆ sn ˆ ˆ sn Wth =R constnt Rsn R he gngn s:

12 U so the eutons of oton e: gz R sn gr cos n R sn hese cn be ewtten s: cos grsn R n g sn cos sn n R sn R R sn constnt he secon euton ens tht the ngul oentu of the enulu s consee If constnt then n the enulu swngs n lne he euton of oton euces to the fl (hoefull): g sn R Ele #: (E 77 of honton) Fctonless be on snnng bolc we z c ottng wth n ngul eloct Wht s the conton on the ngul feuenc to get the be to otte t fe locton on the we? z In clncl ol coontes the eloct s: ˆ zz ˆ ˆ so: z hs sste hs one egee of feeo whee the be s long the we Use to escbe ths (coul lso use z) Snce z c the ete s z c n: c 4 he otentl eneg s U gz gc so the gngn s: he etes of the gngn e: U 4 gc c

13 he euton of oton s: 8c hs cn be ewtten s: 8c gc 8c 6c gc 8c 6c 4c 4c gc whch s colcte nonlne ( Ý s sue) ffeentl euton If the be ottes wth R constnt then so: An gc R gc 8c 3