Nine lectures for the Maths II course given to first year physics students in the Spring Term. Lecturer: Professor Peter Main

Transcription

1 Nine leces o e Ms II cose gien o is e psics sens in e Sping Tem. Lece: Poesso Pee Min Ao e cose Tis cose sows o ow o ieenie n inege ncions o seel iles. I is pesene s n eension o e clcls o le now wic els wi single ile. I is il memics o psiciss since we lie in - imensionl wol giing s les ee iles n z o escie i. Yo will me goo se o is ms in e memicl psics coses in e secon e. Te ge oo o ne is wien in e lngge o memics. Glileo Glilei p://www-ses.o.c./~pm/pmwe/ecing.m

2 Pil ieeniion So o cn ieenie ncions o one ile. Fo ncions o seel iles we m pocee s ollows: Fo cline o is n eig e olme is V = I is consn e e o cnge o V w is V π I is consn e e o cnge o V w is π V I o n e iles we cn sill se ese esls n esigne e eiies s pil eiies. Tese e wien s: I V = wee n e iles en V π n π V Te opeion we e cie o is o pil ieeniion. Te ecipe is es o ieenie pill w one ile eg ll oe iles s consns. Noion Fncions: z = z Pil eiies: z z A moe son noion is oen ope: z z z z Deiniion Fo ncion o one ile: i = en δ δ δ lim Simill we cn wie o : δ δ δ lim δ δ δ lim i.e. pil ieeniion is ciee ieeniing w one ile wile eeping e oes consn. Te genelision o ncions o n nme o iles sol e oios. Geomeicl inepeion z = gies sce i.e. e le o e ncion is ploe in e z-iecion. δ δ

3 z gies e slope o e lines consn gies e slope o e lines consn Emple I e cos Solion: e cos e cos e Cion! in n sin e cos sin I someing cn go wong i will. Hee s n emple o wee o cn esil me ig mise in pil ieeniion. Te elionsips eween Cesin n pol cooines will gie s some simple epessions o ieenie: = cos = sin cosθ θ cn cosθ cosθ I ppes eeoe cosθ!! Te po occs ecse o slc noion. Le s se e complee smol o e pil eiies: θ cosθ cosθ n ee is no eson o e n picl elionsip eween ese wo eiies since ieen iles e el consn n ieen ncions e een ieenie. Moe coecl: om = cos we oin = sec secθ n we in s epece. θ θ θ onl i e sme iles e el consn ing e ieeniion.

4 Hige oe pil eiies Js s o ncions o one ile we m eine ige oe eiies: Te is oe eiies o e n so wic cn lso e wien s wic cn lso e wien s Mie eiies e lso possile ecse e o ncions o n : wic cn lso e wien s wic cn lso e wien s Fonel we e e simpliing c poie ll e eiies eis n e coninos i.e. e opeos n comme. Oe eiies eis e.g. Emple Sow = o = e cos lso Solion: om peios emple we e e cos e cos e sin e cos sin lso e cos e sin n we in Emple e cos e sin e cos sin = Sow = sin ep- sisies onl o =. Tis is Lplce s eqion pil ieenil eqion wic is e impon in memicl psics. Yo will mee i gin in secon e coses.

5 Solion: sin ep cos ep sin ep sin ep sin ep sin ep wen = Tol ieenil Gien e mon wic e ncion cnges wen o iles cnge simlneosl is clle e ol ieenil n is eine s: Le s epess is in ems o pil eiies. Since en Simill Aing n gies Neglecing secon oe smll qniies en gies Te is em in e inl epession is me om e pil eiie wic gies e e o cnge o wi mliplie e cnge in. Te poc eeoe gies e cnge in e o e cnge in. Liewise e secon em gies e cnge in e o e cnge in. I is seen e ol cnge in e ol ieenil is e sm o e cnges e o sis in n sepel. 5

6 Emple Fin e cnge in = e. n o.5 especiel. Solion: wen = : δ δ δ wen e les o n e cnge om o e e δ e δ e δ = e + wen =..5: = e. +. =. e =.9 Te cl cnge is: =..5 =.8.78 =.9 Implici ieeniion I = is eemines s ncion o o ice es i.e. is n implici ncion o since i is no eine eplicil. Tee e wo ws o eemining / in is siion e is o wic o m e seen eoe: δ. ieenie ems in s noml: ieenie ems in s noml: ieenie ems in sing e poc le: emple: cos ieeniing gies sin n n lgeic engemen gies sin sin. since = i.e. consn en = so giing emple: cos sin sin wic is n esie clclion n e is one. Fncion o ncion Fo ncions o single ile we le now o n en

7 7 Tis is someimes clle e cin le. I cn e eene o e cse wee we e ncion o seel iles. Le n en: n Emple I n in cn Beoe soling e polem le s me se we cn ieenie e cn ncion. I o ememe e sn inegl C cn en ieeniing o sies gies immeiel cn Anoe w o oing i is o le = cn en = n sec cn Solion: Use is esl o sole e oiginl polem: Le = / en we e = cn n is ncion o single ile. We lso e n Teeoe cn Also cn Emple Gien in n Solion: We cnno ieenie e epession compleel ecse e ncion is no gien. Howee we cn me some pogess ows e eiies s ollows. Le = en e ncion ecomes n Ten n Emple Sow = - sisies e eqion φ φ wee is n i ieenile ncion n is consn. Solion: Using e sme memicl ies s in e peios emple e powe o clcls is emonse wen we sow e eqion is sisie n

8 8 nnown ncion. Te eqion nown s e we eqion is n impon one in memicl psics s i escies e eio o mn ins o we. Le = en n lso ecomes φ φ φ φ φ φ Cin le I we e gien n en m e epesse s ncion o lone. Te complee eiie / eeoe eiss n i cn e epesse in ems o pil eiies s ollows. Te ol ieenil o is I eeoe ollows i.e. wo ems o e om peiosl gien o ncion o single ile. Emple I e n = = - eemine / wo ieen ws. Solion: Epess s ncion o en e e e e Use e cin le e e e Cin le gin A moe genel epession o e cin le ecomes necess wen we e wee n. Usll wien s: Using ll smols: n

9 9 Dieenil opeos Yo e een sing ieenil opeos o some ime een i o en clle em nme. Fo emple o ieenie o ppl e opeo o poce e esl. Tis m e wien s. Simill. Yo cn ppl e opeo wice s in. Fo : n Moe complice opeos cn ise o wi n. Te cin le gies n is m e wien in ems o ieenil opeos s Te le-n opeo is o e se wen is in ems o n ; e ig-n opeo is o e se wen is in ems o n. Te opeos peom e sme opeion n e eeoe eqilen. Tis m e wien s Te ig-n opeo ells o o ieenie pill w n mlipl en is o e pil eiie w mliplie. Emple I = + wee = n = in sing wo ieen opeos. Solion:. Ssie o n in n se e opeo : giing. Lee in ems o n n se e opeo : Te pil eiies in e opeo e n So

10 Emple eeminion o ieenil opeos Gien = n = epess e opeos n in ems o n. Solion: Te cin le gies n Te pil eiies e: ; ; ; so e opeos ecome: Yo cn se n meo o lie o sole e eqions e ecommene meo is o se Cme s le: eeoe n Cme s le I o en seen Cme s le eoe ee i is s pplie oe. I is ecommene s i nees less lgeic mniplion n oe meos wen pplie o pi o simlneos eqions. Fo e eqions: + = + = e solion is epesse in ems o eeminns s: Te nnowns e n n e pen o eeminns is: Te ls eeminn cll i conins e o le n sie coeiciens. Te eeminn o is nnown is wi e is colmn eplce e s. Te eeminn o n nnown is wi e n colmn eplce e s. sole ese eqions o n

11 Commen Cn we oi ing o sole eqions oing e clclion in ieen w? Fo emple o wi n e cin le gies so e opeos e giing iecl wi n eqilen epession o. Howee on e oole. Te eiie oe is no e ecipocl o in e peios emple. Hee is ncion o n so e ll smol o e eiie is wees in e emple is ncion o n i.e. e eiie is. To oin e eqions = n = will e o e sole o n o epess em s ncions o n. Tis is consiel moe iicl s n soling e eqions in e emple so o cn win is w. Cnge o iles Cesin o pol Sow wee is ncion o n o e eqilen ncion o n n = cos = sin i.e. e epession is cnge om Cesin o pol cooines. Te epession isel is p o Lplce s eqion. Fom e cin le: cos sin sin cos Sole ese eqions o n o gie: cos sin n sin cos Tese epessions lso poie e opeos neee o oin e secon oe eiies: sin cos sin cos Epn e opeo: sin cos sin sin cos cos Now ieenie ememeing o se e poc le wee i pplies: sin cos sin cos sin sin cos sin cos sin cos

12 Collec lie ems n p em in logicl oe: sin sin sin sin cos A simil nlsis gies e epession o : θ θ sin cos cos sin sin Aing em ogee gies e eqie esl: Tlo seies o ncion o wo iles Fo ncion o single ile we e: n n n!! '''! '' ' o leniel: n n n!! '''! ' ' ' Tese wo epessions e e sme wi = +. Fncions o seel iles cn e ee in simil w. Fo wo iles eine e ieenil opeo D. Te Tlo epnsion o o e poin is en:!!! D n D D D n wee D n mens ppl e ieenil opeo n imes o n ele e esl e poin. Le s loo D : Tis is e sme memicl om s inomil epnsion. Wiing o ll e ems in e Tlo epnsion p o oe gies:!! Te lenie om is oine ping = + = + :!!

13 Emple Epn = ep sin s powe seies o e poin / o ems o e secon egee. Hence oin n ppoime le o. /. = ep sin / = e = sin ep sin / = e = cos ep sin / = = sin ep sin / = e = cos ep sin + cos sin ep sin / = = - sin ep sin + cos ep sin / = -e Te Tlo seies o / is: Ping in e les o e eiies gies: ep sin e e e e e e Fo =. n = / e ppoime le o. /.. Usel pope Fo ncions o single ile: e Te coec le o. / ep.sin. I cn e wien s PQ en Tlo epnsion o = [Tlo epnsion o P] [Tlo epnsion o Q] Fo ncion o wo iles: I cn e wien s PQ en Tlo epnsion o = [Tlo epnsion o P] [Tlo epnsion o Q]

14 Inegion Recll e e ne ce is gien inegion: Elemen o e = lim Tol e In e limi s ol e Dole inegls Le s een is ie o gie n epession o e olme ne sce. Te ncion z = gies e sce n e olme eween is n gien egion in e -plne is e olme o e clcle. Te olme ssocie wi ec elemen o e in e -plne is n e sm o ese gies e esie olme. Howee is is one in e ssemic mnne. Le e olme e one : Top z = Boom -plne Sies ce ABCD Le e ce ABC e = Le e ce ADC e = Now e n elemen slice o consn. A D δ = φ C Elemen o olme = Teeoe olme o elemen slice lim δ B = φ lim So ol olme ne sce In is ole inegl noe e is inegion is oe wee is ee s consn. Dole inegls e no nomll wien in is w e moe commonl epesse wio ces s: o In e oe nlsis i is possile o inecnge e oles o n o oin secon ole inegl eql in le o e is. Ting elemen slices o consn les o e ole inegl:

15 Volme ne sce = c wee e olme is one on e le = ψ n on e ig = ψ. We cn eeoe eqe e wo ole inegls: c n we e cnge e oe o inegion. On e le we inege oe is oling consn en inege oe. On e ig inege oe is oling consn en inege oe. Noe e limis on e wo ole inegls e e ieen. o escie e sme iel o inegion. Cnging e oe o inegion Howee e Yo cn onl sel ece e limis on ole inegl wing e iel o inegion. Te e ollowing inegl s n emple. Te limis on e inegion oe sow e iel o inegion is one e lines = n =. Teeoe e iel o inegion is s sown in e igm wi going om o / s gien e limis on e inegion oe. On ineging oe is e igm sows goes om e line o. Tese eeoe om e limis on e inegl n e nge o is seen o e < < /. We eeoe e: In is secon emple e ole inegl s o e spli ino e sm o wo inegls o ccommoe e ieen lowe limis wen ineging oe is. c = ψ A = D B = = ψ C wen < 5

16 I is impon e iel o inegion sol e e sme eoe n e cnging e oe o inegion. Emples eling ole inegls. Ele oe e e one = n =. Vei e sme esl is oine wen e oe o inegion is eese. Solion: Te limis on e inegls e o e oine om igm o e iel o inegion. Ineging oe is: Ineging oe is: Ele sin Solion: Tee is no es w o peoming e inegion oe so consie cnging e oe o inegion. Tis cn onl e one ploing o e iel o inegion is. Te inegl ecomes sin Noe sin is consn in e inegion oe so i cn e en o o inegl. Te inegion oe is now iil: sin sin sin n e inegion oe now ecomes possile ecse e inegn s cnge. Use e ssiion en n e inegl ecomes:

17 sin cos Sepion o iles In e specil cse wee e limis o inegion e consn no ncions o e iles n e inegn is o e om FG we e: F G c F G Tis is e poc o wo single inegls cn e ele inepenenl o ec oe. c Dole inegls in pol cooines = cos; = sin Uni ecos ˆ n θˆ e eine o lie in e iecions o incesing n especiel. Cnging om Cesins o pols in ole inegl is n emple o ssiing o o iles simlneosl. θˆ ˆ is n elemen o e in e -plne. Te eqilen elemen o e in e -plne is: = i.e. elemen o e = = Noe is no eplce e imensions e wong! A A Cnge o oe o inegion ccos cos ccos = cos Te eqion = cos gies cicle o is cene on. Ce line in cicle: inege oe consn. Sig line in cicle: inege oe consn. Emple Ele wee R is e egion one + = R 7

18 8 Solion: Coneing o pol cooines will simpli o e inegn n e escipion o e iel o inegion. = cos = sin = Emple Ele 5 Solion: Tnsom o pol cooines o simpli e inegn: = cos = sin = cos cos Tee is sepion o iles so is is poc o wo single inegls. Use e ieni cos = cos n le + = en = 5 5 cos Tiple inegls Te memicl ies gie s ole inegls cn esil e eene o iple inegls: Inegn = z Fiel o inegion = olme in D spce Elemen o olme = z = V Repee inegl: z z In oe o el wi iple inegls we nee o loo D cooine ssems. Speicl pol cooines A poin P z in Cesin cooines is lso in speicl pols wee = sin cos = sin sinz = cos + = = ˆ θˆ φˆ

19 A P ni ecos ˆ θˆ φˆ e eine o lie in e iecions o incesing n especiel. Tese ecos om ig-ne oogonl cooine ssem poin. A sce o consn is spee A sce o consn is cone A sce o consn is semi-ininie plne Tese sces inesec e poin Te elemen o olme is mos esil oine geomeicll: Te o in e igm s sies o sin n Tis mes e elemen o olme = V = z = sin = sin Clinicl pol cooines A poin P z in Cesin cooines is lso z in clinicl pols wee = cos = sin z = z A P ni ecos ˆ θˆ zˆ in e iecions o incesing z om n oogonl ig-ne ssem P. A sce o consn is cline A sce o consn is semi-ininie plne A sce o consn z is n ininie plne Tese sces inesec e poin z. Te elemen o olme is oine geomeicll: Te o in e igm s sies o n z Tis mes e elemen o olme = V = z = z = z ẑ ˆ θˆ 9

20 Soli ngle Now we e ino D cooine ssems le s loo e mesemen o ngles in ee-imensionl spce. I will e es o s wi ngles o e mili wi in wo-imensionl spce: Te igm sows n c o cicle o is. Te leng o e c is. Te ngle is e io o e leng o e c o e is i.e. leng o c is ins. I e c is lengene o complee e cicle e ngle ecomes cicmeence o cicle ins. is In D ngles e clle soli ngles e e eine in simil w o ngles in D. Te igm sows p o spee o is wic sens soli ngle is cene. e o speicl sce Te ngle is eine s is o spee seins. I e sce is eene o complee spee e soli ngle e cene ecomes sce e o e spee seins. is o e spee Soli ngles e se in psics o emple o escie e D ngle ino wic soce o iion m ie. Emple o iple inegl A simple illsion o e se o iple inegl is o clcle e olme o spee o is. Solion: Use speicl pol cooines n inege e elemen o olme oe e spee. V spee V θ sin θ sin θ θ.. Me se o nesn e ssignmen o e limis: Te inegion oe is long line om e oigin o e sce o e spee. Te inegion oe oes is line o e oigin o sweep o e e o semicicle. Te inegion oe oes e semicicle o e z-is o sweep o e olme o e spee. Te limis on e ollowing iple inegl lso gie e iel o inegion s spee: c o cicle cp o spee sin θ θ..

21 Cn e olme o e spee ell e zeo? In speicl pols e elemen o olme V is sin In e is inegl e nge o is om o so V is lws posiie. Howee in e secon inegl V is negie oe l is nge cncelling o e posiie coniion o e inegl n ening p wi zeo. I o inen o e negie olme en e secon inegl is peecl coec o nee o e we o w o e oing. Dimensions in inegls emples o pplicions in psics I epesens e olme ne sce w oes z z epesen? Te ole inegl onl epesens olme i n ll e e imensions o leng. Te imension o e inegn is en leng wic is olme. We e le se iple inegl o eemine e olme o spee go c n cec e imensions e coec. No eeing is mese in mees n we nee o consie e imensions o e qniies in e inegl wen ppling i o psics. Te ing o noe is e smols ec. e no js lels emining o o e iles oe wic o peom e inegion e e lso psicl qniies wi imensions. Le s loo some emples o ow o consc ios inegls o pplicions in psics.. A egion o spce conins n elecic cge ensi o z coloms/m. W is e ol cge in picl olme V? Cge in elemen o olme = z z Teeoe ol cge in olme V = V. Volme o iel o inegion = V ρ z z V m. coloms.. Te spee o picle cnges wi ime s. Fin e isnce elle in ime T. Disnce elle in ime = Teeoe ol isnce elle in ime T = mees.. Te ensi o in see o meil ies s g/m. Fin is ol mss. Mss o elemen o e = Teeoe ol mss o see = M = R T g wee R eines e spe o e see. 5. Te men ensi o e see in e peios emple is oine iiing e ol mss e e i.e. men ensi = g/m wee A is e A e o e see. Simil inegls will eemine e ege le o ncion in D o D spce: Aege o eween = n = is R

22 Aege o z wiin e olme V = V V z z. Anoe impon se o inegls is o eemine e cene o mss o n ojec. Using e sme in see is in emple e mss is g so is momen o e -is is g m. Te ol momen o e -is is eeoe n is ms e eql o M wee M is e ol mss R n is e -cooine o e cene o mss. We eeoe e o e cooines o e cene o mss: n M M R 7. A sligl moe enos se o mliple inegl is in e clclion o momen o inei. Te momen o inei o poin mss m isnce om e is o oion is m. Fo o o ensi z g/m e poin mss z is z z I z is e oion is e isnce om e is o e poin mss is Te momen o inei o e poin mss is eeoe Teeoe e momen o inei o e complee o = Emple R V z z z z Fin e posiion o e cenoi o niom soli cone o eig n se is R. Solion: Te cenoi is e cene o mss o o o niom ensi. B smme e cenoi lies on e cone is so onl e z-cooine is eqie. Using e inegls in e peios emples e z- cooine o e cenoi is gien : z V cone z V Tee is coice o cooine ssem clinicl pols will e e esies. Elemen o olme = V = z Te ce sce o e cone is gien e iple inegl is R z so z R V z R z R z R R z z z z z z z R B e olme o e cone is gien V R R so z R

23 Volme o cone I o ogoen o nee nown e oml o e olme o cone se in e peios emple we cn eie i ee. We e le ece e limis on iple inegl escie cone so we cn immeiel se em ee: V cone V Emple cone z z R z z R R z R z Deemine e ol mss n men ensi o o occping e posiie ocn wee n z e ll posiie one + + z = n wose ensi is z = z g/m. W e e psicl imensions o e consn? Using is esl conim e psicl imensions o o nswes o p e coec. Solion Mss o elemen o olme = z V g Teeoe ol mss = V z V g Use speicl pol cooines: = sin cos; = sin sin; z = cosv = sin ol mss sin θ cosθ sinφ cosφ sin θ θ φ 5 sin θ cosθ θ sinφ cosφ φ le = sin en = cos lso se e ieni sin cos = sin cos φ sinφ φ 8 g Men ensi = ol mss olme 8 g m Te imensions o n z e ll mees. Since z is ensi is imensions ms e g m - wien s [z] = g m - Tis mes [] m = g m - so [] = g m - Te mss o e ojec is 8 g. Fom e oe we e [ ] = g m - m = g Te men ensi o e ojec is 8 g m - n [ ] = g m - m = g m -.

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