STiCM. Select / Special Topics in Classical Mechanics. STiCM Lecture 11: Unit 3 Physical Quantities scalars, vectors. P. C.

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1 STiCM Slct / Spcial Topics in Classical Mchanics P. C. Dshmukh Dpatmnt of Phsics Indian Institut of Tchnolog Madas Chnnai pcd@phsics.iitm.ac.in STiCM Lctu 11: Unit 3 Phsical Quantitis scalas, vctos. 1

2 Unit 3: Pola Coodinats Laning goals smmt Lan to us an appopiat coodinat sstm to simplif analsis. Goals: Phsical quantitis a tnsos of vaious anks. W must amin how thi componnts tansfom und th otation of a coodinat fam of fnc. 2

3 Claudius Ptolmaus (AD ) (calld Ptolm). wokd in th liba of Alandia. Th sun and th plants w considd to mov on a small cicl (calld piccl ) whos cnt would mov on a lag cicl (calld dfnt ). 3

4 Contibutions of Indian Astonoms to th Undstanding of Hliocntic Coodinat Sstm Aabhata (b. 476A.D.) - ARYABHATYA (499 A.D.) Bhaskaa I (A.D. 600) - MAHABHASKARIYA, LAGHUBHASKARIYA, ARYABHATIYA BHASHYA Bahmagupta (A.D. 591) - BRAMA SIDDHANTA Vatshwa (A.D. 880) - VATESHWARA SIDDHANTA Manjulachaa-(A.D. 932) - [ Dalt with Pcssion of quinos ] LAGHUMANASA Aabhata I I (A.D. 950) - MAHASIDDHANTA Bhaskaachaa I I (A.D. 1114) SIDDHANTA SHIROMANI [ This wok contains man fomulas fom sphical tigonomt ]..tc. 4

5 Fom Golapaada, b Aabhata, ~500 AD Nicolus Copunicus Just as a man in a boat moving ss th stationa objcts (on ith sid of th iv) as moving backwad, so a th stationa stas sn b th popl at Lanka (i.. fnc coodinat on th quato) as moving actl towad th wst. 5

6 Rn Dscats (17 th cntu, Holland) Fnch philosoph, mathmatician, scintist: "Fath of Modn Philosoph," Hliocntic sstm vs Chuch s viws Dspit admitting th advantags of th hliocntic coodinat sstm, Dscats was luctant to pomot th ctain and vidnt poof in favo of th hliocntic sstm sinc it was against th will of th chuch. 6

7 Th Tial of Galilo (fo suppoting Copnican modl) Apil 1633: Galilo is intogatd bfo th Inquisition. Jun: Galilo sntncd to pison fo an indfinit tm. Dcmb: Galilo is allowd to tun to his villa in Flonc, wh h livd und hous-ast. 1992: Catholic Chuch fomall admits that Galilo's viws on th sola sstm a coct. 7

8 Dfinition of a vcto: magnitud and diction z Is otation b 90 dgs a vcto? 8

9 EDWIN A. ABBOTT (1884) V V V Y How Y do vctos / V Y Sam vcto can also b wittn as V V This can b gnalizd into th (o N) dimnsions ' ' V X V ' V X / ' V Y / tansfom und otation of a coodinat sstm? X / X 9

10 V ' z z ' V, V, Vz ' V, V, V ' ' z' 10

11 11 / / / / z z V V V V / / / / z z V V V V / / / / z z z z z z V V V V Compact mati fom z z z z z z z z V V V V V V / / / / / / / / / / / /

12 ' R R R z ' R R R z z ' R R R z z z zz R R R z R R R z R R R z z zz 1 R ROTATION: R = + 1 PARITY / REFLECTION INVERSION ; 1 R = -1 12

13 C.N.Yang and T.D.L Z Z z z Chin-Shiung Wu X Y M I R R O R X Y 13

14 z z REFLECTION z z Lft Right Top? Bottom z z 14

15 Too much Mathmatics? IF YOU WANT TO READ THE BOOK OF THE UNIVERSE, YOU MUST KNOW ITS LANGUAGE, WHICH IS MATHEMATICS. Who said that? Gandpa of Engining? Fath of pimntal Phsics! 15

16 C AB l p angula momntum p l ighthand coss poduct l p imag p imag 16

17 C AB l p angula momntum p l ighthand coss poduct l p imag p 17 imag

18 Pola vctos and psudo- o aial-vctos und invsion,, p p but if l p, thn, und invsion, l l Aial vcto (psudo vcto) dos not tansfom lik a position vcto und flction. Its componnts a govnd b a diffnt tansfomation law with spct to otation of th coodinat sstm. 18

19 Eampls : Som al phsical quantitis Angula Momntum Vcto p Foc on a chagd paticl moving in an lctomagntic fild F q E v B Lontz Foc Th Lontz foc, lik an oth foc, is a pola vcto, sinc it includs th coss-poduct of a pola vcto v with a psudo-vcto B. 19

20 Algba of Psudo Vctos and Eampls Dot and coss poducts: Pola Pola = Aial Pola Aial = Pola Aial Aial = Aial Eampls fo aial (psudo) vctos: Toqu Angula Momntum f L p Aial. Pola = Psudoscala Magntic fild F q B mag. v Impotant: An aial vcto can nv b quatd with a pola vcto 20

21 W hav land that phsical quantitis a psntd b scalas, vctos, tnsos tc. Scalas: tnsos of ank zo Vctos: tnsos of ank on Scalas / psudo-scalas Vctos / psudo-vctos Pola vctos / Aial vctos 21

22 W Will tak a Bak any qustions? pcd@phsics.iitm.ac.in Nt: vctos in Pola coodinats 22

23 STiCM Slct / Spcial Topics in Classical Mchanics P. C. Dshmukh Dpatmnt of Phsics Indian Institut of Tchnolog Madas Chnnai STiCM Lctu 12: Unit 3 Plan Pola Coodinats pcd@phsics.iitm.ac.in Clindical Pola Coodinats Sphical Pola Coodinats 23

24 Motoccl mania Th tos boths! Fist 5, thn 7 gus ac thi biks insid a 16 foot stl glob. Unblivabl! V0lLSTE1MTIN35pbjozfnE6YnJ- Ync6V0lLSTE1MTINDTDAwNzI4MDMzMDl-aW46Nn5OnJs 24

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26 cos sin sin cos cos sin cos sin sin cos tan : 0 : Y O ê ê ê ê ê coodinats (, ) (, ) X points in th diction in which th azimuthal angl incass (, ) constitut an othogonal pai of bas vctos FLATLAND spac Position vcto Plan Pola Coodinat sstm 26

27 Position vcto VELOCITY? Acclation? d Not that (, ) Poduct of two functions dt [ ] a not constant vctos. To gt acclation, w hav to do that twic! d dt d dt 27

28 Y (, ) a not constant vctos, whas (, ) a. Unit Cicl O (, ) (, ) X cos sin sin cos 28

29 (, ) (, ) How do ths unit vctos chang with th azimuthal angl? Y cos sin sin cos cos sin sin cos O X sin cos sin cos sin cos 29

30 Gomtical dtmination of ê Y 1 2 Consid (, ) at two nighboing points, infinitsimall clos to ach oth. Unit Cicl O X lim 2 1 lim

31 (, ) a not constant vctos. (, ) (, ) lim Unit Cicl 0 0 O Y ê 2 ê 1 2 ê lim cos sin sin cos 0 0 X, 31

32 If ( u) and u ( ), d thn will b a masu of th d snsitivit of to changs in : chain ul d d du d du d If ( u, v) wh u u( ), v v( ), d du dv th at at which will d u d v d chang with spct to will b givn b : If ( u, v, ) wh u u( ), v v( ), th at at which will chang with spct to will b givn b : d du dv. d u d v d 32

33 Elmntal aa in plan pola coodinats da dd Y d Position vcto & Vlocit in plan pola coodinats d ( d) d R 00 d 2 2 R dd 2 R 2 d v 2 d d dt dt d d dt dt X 0,, 0, 33

34 0,, 0, Tim-dpndnc of unit vctos d dt d and dt Motion of a paticl in plan pola coodinats chain ul v d d d d dt dt dt dt Radial vlocit and Azimuthal vlocit 34

35 v= d d dt acclation dt and instantanous vlocit d v d d a = dt dt dt =( 2 ) (2 ) a 35

36 Clindical Pola Coodinats z ê Sphical Pola Coodinats 36

37 0 0 z cos 0 2 sincos Z ê ê sinsin z tan z tan tan 1 1 Y 2 2 z 2 2 z X sin 37 z sincos sinsin cos

38 Tansfomations of th unit vctos sin cos sin sin cos cos cos cos sin sin sin cos 0 z Gt th invs mati, and wit th invs tansfomations. sin cos cos cos -sin sin sin cos sin cos cos sin 0 z 38

39 Z d ê ê d Th point is displacd to a nw point on th sph of sam adius and in th sam plan constant constant X sin Rcogniz th distanc btwn th old position and th nw position to b 39 Y d

40 d sind X Z d sin ê d ê 40 d sind Th point is displacd to a nw point on th sph and on th sufac of th invtd con Rcogniz th distanc btwn th old position and th nw position to b constant constant Y d sind

41 Volum spannd b th th displacmnts though d d sind Z θ dv ( d)( d )( sin d) 2 si n d d d φ Y X 41

42 2 1 In th limit 0, X Z d ê 42 Th point is displacd to a nw point on th sph of sam adius constant and in th sam plan constant Distanc btwn th 1 st position and 2 nd position is d 1 st position 2 nd position Y

43 Patial divativs of th unit vctos with spct to th coodinats: 0 0 If imagining complicatd gomtical th-dimnsional objcts is gtting difficult, ou can us th chain ul of taking divativs to gt th patial divativs of th unit vctos using ths tansfomation uls, as illustatd on th nt pag. =sin cos sin sin cos z =cos cos cos sin sin z sin cos 43

44 Us of chain ul to gt th patial divativs of th unit vctos using th tansfomation uls fo th unit vctos. =sin cos sin sin cos z =cos cos cos sin sin sin cos 44 z Fo ampl: cos sin cos sin cos cos cos sin sin sin sin sin cos sin cos cos Oth patial divativs can b obtaind quall asil, and lft fo ou to do as an cis!

45 45 0 sin 0 cos 0 0 cos sin

46 Motion in in sphical pola: Vlocit and acclation Infinitsimal displacmnt Position vcto d d d sind d d d d d 46

47 Motion in in sphical pola: Vlocit and acclation d d d sind v sin d v a dt ( - - sin ) 2 (2 sin cos ) (2 sin 2 cos sin ) 47

48 Gnal Rfnc on Vcto analsis : [1] Bkl Phsics Cous, Vol.1. Mchanics [2] Davis: Classical Mchanics Slightl advancd fncs: SUPPLEMENTARY OPTIONAL READING: Gnal Rfnc on Astonom : Patik Moo: Intnational Encclopdia of Astonom. Cal Sagan: Cosmos Afkn: Mathmatical Mthods fo Phsicists. Boas: Mathmatical mthods in Phsical Scincs. W Will tak a Bak any qustions? pcd@phsics.iitm.ac.in Nt, Unit 4: Dnamical Smmt of th Kpl Poblm 48