Problemas de Dinámica de Rotación

 

Jgjijl

^rlockehs

Yùcj`l

@jgèejih `k rlthijùg @jgèejih `k rlthijùg

^rlockeh 8

[g `jsil `k `k rh`jl y 8>> df ehsh,quk fjrhkmkrikg jgjijhcekgtk ilg ugh`k vkclij`h` `k>.1 86;erh`/s. Yk hpcjihg cls`karkgls ug elekgtl  E 5 -?¶t  Ge.  Ge. @ktkrejghr  ch hikckrhijùg hgfuchr kg augijùg `kc tjkepl ch vkclij`h` hgfuchr kg augijùg `kc tjkepl kc ègfucl fjrh`l kg augijùg `kc tjkepl. Kc elekgtl hgfuchr jgjijhc y kg kc jgsthgtk t 582 582 s. _kprkskgthr kc elekgtl E  kg  kg augijùg `kc tjkepl. Ileprlohr quk kc jepucsl hgfuchr ∤ > t E¶`t (èrkh) ks jfuhc h ch vhrjhijùg `k elekgtl elekgtl hgfuchr. Ch vkclij`h`, hikckrhijùg thgfkgijhc y glrehc `k ug pugtl `k ch pkrjakrjh `kc `jsil kg `jibl jgsthgtk. _kprkskgthr ksths ehfgjtu`ks.  Ylcuijùg

Elekgtl `k jgkrijh J5 8 ? 8>>¶ >.1 ? 582  dfe ? Kiuhijùg `k ch `jgèejih `k rlthijùg  J¶θ5E , θ5-t /: /: rh`/s? ch hikckrhijùg hgfuchr gl ks ilgsthgtk

Ihciuchels ch vkclij`h` hgfuchr ϊ y kc `kspchzhejkgtl `kspchzhejkgtl hgfu gfuchr chr β . ϊ∟ ϊ > 5 ∤ t > t θ¶`t   ϊ∟86;5 ∤ > t ( ∟t : )¶`t   ϊ586;∟ t ? 82  rh`/s β∟ β > 5 ∤ t > t ϊ¶`t   β5 ∤ > t ( 86;∟ t ? 82 )¶`t   β586;t∟ t 9 ;0  rh` Elekgtl hgfuchr, C5Jϊ t 5>, 5>, ϊ586;, C598;> dfe?/s t 582, 582, ϊ58;6, C5?2?1 dfe?/s

Jepucsl hgfuchr  C∟ C > 5 ∤ t > t E¶`t   C∟98;>5 ∤ > t (∟?t)¶`t  C598;>∟ t ?   df¶e ? /s Kg kc jgsthgtk t 582 582 s, C5?2?1 dfe?/s

Ch rkprkskgthijùg `kc elekgtl E  kg  kg augijùg `kc tjkepl t ks ugh rkith. Kc °rkh `kc trjègfucl `k ch ajfurh ks ∟82¶91 ? 5∟9?0 quk ks kc jepucsl hgfuchr, jfuhc h ch `jakrkgijh kgtrk kc elekgtl hgfuchr ajghc k jgjijhc ^hrh t 582 582 s Hikckrhijùg thgfkgijhc, ht 5θ¶_5(-82/:)¶>.15-8.? e/s? Hikckrhijùg glrehc, hg5ϊ?¶_58;6?¶>.158062:.1 e/s?

 

Kg ch ajfurh, sk rkprkskgth ch vkclij`h`, thgfkgtk h ch trhykitlrjh ijriuchr, ch hikckrhijùg thgfkgijhc `k sjfgl ilgtrhrjl h ch vkclij`h`, y ch hikckrhijùg glrehc `jrjfj`h bhijh kc ikgtrl. Ksths `ls ileplgkgtks `k ch hikckrhijùg gl kstèg `joumh`hs h ksihch. ^rlockeh ?

[g oclquk `k ?>>> df kstè suspkg`j`l kg kc hjrk plr  ug ihock `k hikrl quk phsh  plr ugh plckh y hihoh kg ug tlrgl eltlrjzh`l. Kc  oclquk hsijkg`k ilg vkclij`h` ilgsthgtk `k 2 ie/s. Kc rh`jl `kc theolr  `kc tlrgl ks `k 9> ie y ch ehsh `k ch plckh ks

`ksprkijhock. ¼Iuègtl vhck kc elekgtl quk kmkrik kc ihock slork kc theolr `kc tlrgl< ¼Iuègtl vhck ch vkclij`h` hgfuchr `kc theolr `kc tlrgl<

¼Xuê pltkgijh tjkgk quk `kshrrlcchr kc eltlr s  Ylcuijùg

Qkclij`h` ilgsthgtk `kc oclquk v5>.>2 e/s Pkgsjùg `k ch iukr`h, ks kc pksl `kc oclquk, A 5?>>>¶:.258:1>> 5?>>>¶:.258:1>> df Elekgtl, E5A¶r 58:1>>¶>.95;22> 58:1>>¶>.95;22> G¶e Qkclij`h` hgfuchr, ϊ5v/r 5>.>2/>.950/8; rh`/s ^ltkgijh, ^5E¶ϊ5;22>¶0/8;58;12 V Prhohml, V5E¶β5^¶t 58;12¶8>58;12> 58;12¶8>58;12> M ^rlockeh 9

Kc sjstkeh `k ch ajfurh kstè jgjijhcekgtk kg rkplsl. Kc oclquk `k 9> df kstè h ? e `kc sukcl. Ch plckh ks ug `jsil ugjalrek `k ?> ie `k `jèektrl y ; df `k ehsh. Yk suplgk quk ch iukr`h gl rksohch slork ch  plckh. Kgilgtrhr= Ch vkclij`h` `kc oclquk `k 9> df mustl hgtks `k tlihr kc sukcl. Ch vkclij`h` hgfuchr `k ch plckh kg ksk jgsthgtk. Chs tkgsjlgks `k ch iukr`h. Kc tjkepl quk thr`h kc oclquk `k 9> df kg tlihr kc sukcl. (_kslcvkr kc prlockeh plr `jgèejih y hpcjihg`l kc ohchgik kgkrfêtjil)  Ylcuijùg

Ksirjojels chs kiuhijlgks `kc elvjejkgtl @kc elvjejkgtl ih`h ugl `k cls oclquks @kc elvjejkgtl `k rlthijùg `kc `jsil 9> ¶ :.2 ∟ P 8 5 9> ¶ h P ? ∟ ?> ¶ :.2 5 ?> ¶ h P 8 ¶ >.8 ∟ P ? ¶ >.8 5 ( 8 ? ; ¶ >.8 ? ) θ

Ch rkchijùg kgtrk ch hikckrhijùg `k cls oclquks h y ch hikckrhijùg hgfuchr θ `kc `jsil ks h5θ¶>.8

 

_kslcvjkg`l kc sjstkeh `k kiuhijlgks, h58.26 e/s? Yj kc oclquk `k 9> df ihk ? e phrtjkg`l `kc rkplsl. ?5 8 ? h t ? v5h¶t }v5?.69 e/s Ohchgik kgkrfêtjil

Kg ch ajfurh sk ilephrh ch sjtuhijùg jgjijhc y ch ajghc y hpcjihels kc prjgijpjl `k ilgskrvhijùg `k ch kgkrfåh 9>¶:.2¶?5?>¶:.2¶?+ 8 ? ?> v ? + 8 ? 9> v ? + 8 ? ( 8 ? ;¶ >.8 ? ) ϊ ? _kchijlghels ch vkclij`h` v `k cls oclquks y ch vkclij`h` hgfuchr ϊ `kc `jsil, v5ϊ¶>.8 Kc rksucth`l ks v5?.69 e/s, kc ejsel quk bkels lotkgj`l plr `jgèejih ^rlockeh 0

Ylork ug pchgl blrjzlgthc kstè sjtuh`l ug iukrpl `k ;> df quk kstè ugj`l ek`jhgtk ugh iukr`h, quk phsh h trhvês `k ugh plckh `k 8; df h ltrl iukrpl `k ?>> df. Yhojkg`l quk kc ilkajijkgtk `k rlzhejkgtl kgtrk kc iukrpl `k ;> df y kc pchgl blrjzlgthc vhck >.8, ihciuchr. Ch hikckrhijùg `k cls iukrpls Chs tkgsjlgks `k ch iukr`h Ch vkclij`h` `k cls iukrpls shojkg`l quk kc `k ?>> df bh `ksikg`j`l ? e phrtjkg`l `kc rkplsl. (kepckhr `ls prlik`jejkgtls `k ièciucl phrh kstk hphrth`l)  Ylcuijùg

Kiuhijlgks `kc elvjejkgtl `k ih`h ugl `k cls trks iukrpls ?>>¶:.2∟ P ? 5?>>h P ? _∟ P 8 _5( 8 ? _ ? )θ { G5;>¶:.2 A r 5>.8¶G P 8 ∟ A r8;5;>h

_kchijùg kgtrk ch hikckrhijùg h `k cls oclquks y ch hikckrhijùg hgfuchr θ `kc `jsil h5θ¶_

Kc sjstkeh `k kiuhijlgks sk rk`uik h 8:1>∟ P ? 5?>>h P ? ∟ P 8 56.;h P 8 ∟0:5;>h }h56.0?  e/s ? Yj kc iukrpl `k ?>> df `ksijkg`k ? e phrtjkg`l `kc rkplsl ?5 8 ? h t ? v5ht }v5;.0; e/s Ohchgik kgkrfêtjil

 

Prhohml `k ch aukrzh `k rlzhejkgtl V5-A r ¶?5-0:¶?5-:2 M

V5 8 ? ;> v ? + 8 ? ?>> v ? + 8 ? ( 8 ? 8; _ ? ) ϊ ? ∟?>>¶:.2¶? _kchijùg kgtrk ch vkclij`h` v `k cls oclquks y ch vkclij`h` hgfuchr ϊ `kc `jsil v5ϊ¶_

Kc rksucth`l ks v5;.0; e/s, kc ejsel quk bkels lotkgj`l plr `jgèejih ^rlockeh ;

[gh ksakrh bukih `k ehsh E 51 51 df y rh`jl _52 ie puk`k rlthr hcrk`k`lr `k ug kmk vkrtjihc. Kg kc kmk bhy ug elekgtl `k rlzhejkgtl `k >.? G¶e. [gh iukr`h sjg ehsh ehsh kstè kgrlcch`h hcrk`k`lr `kc pchgl kiuhtlrjhc `k ch ksakrh, phsh plr ugh plckh `k elekgtl `k jgkrijh J 59¶8> 59¶8>-9 df e? y rh`jl r 5; 5; ie y kstè hth`h hc ajghc h ug oclquk `kiukr`h ehsh egl5>,1 df. Gl bhy arjiijùg kg kc kmk `k ch plckh y ch rksohch. ¼Iuèc ks ch vkclij`h` `kc oclquk iuhg`l bh `ksikg`j`l 2> ie< _kslcvkrcl `jgèejih y plr ohchgik kgkrfêtjil. J (ksakrh bukih)5?/9 E_ ?  Ylcuijùg

Kiuhijlgks `kc elvjejkgtl `k ih`h ugl `k cls iukrpls P 8 ¶>.>25( ? 9 1¶ >.>2 ? ) θ ? P ? ¶>.>;∟ P 8 ¶>.>;59¶ 8> ∟9 θ 8 >.1¶:.2∟ P ? 5>.1¶h h `kc oclquk y chs _kchijùg kgtrkhgfuchrks ch hikckrhijùg hikckrhijlgks `k ch ksakrh y `k ch plckh h5θ8¶>.>;5 θ?¶>.>2

_kslcvjkg`l kc sjstkeh `k kiuhijlgks, h58.>89 e/s?

Yj kc iukrpl `k >.1 df `ksijkg`k >.2 e phrtjkg`l `kc rkplsl >.25 8 ? h t ? v5ht }v58.?69 e/s Ohchgik kgkrfêtjil

 

^rjgijpjl `k ilgskrvhijùg `k ch kgkrfåh >.1¶:.2¶>.25 8 ? >.1¶ v ? + 8 ? 9¶ 8> ∟9 ϊ 8 ? + 8 ? ( 8 ? 1¶ >.>2 ? ) ϊ ? ? _kchijùg kgtrk ch vkclij`h` v `kc oclquk y chs vkclij`h`ks hgfuchrks `k ch ksakrh y `k ch plckh v5ϊ8¶>.>;5 ϊ?¶>.>2

Kc rksucth`l ks v58.?69 e/s, kc ejsel quk bkels lotkgj`l plr `jgèejih ^rlockeh 1

@ls iukrpls `k 9 y ; df kstèg ugj`ls plr ugh iukr`h quk  phsh plr ugh plckh kg alreh `k `jsil `k ? df `k ehsh y ?> ie `k rh`jl. Heols `kscjzhg slork pchgls jgicjgh`ls `k 9>½ y 0;½. Cls ilkajijkgtks `k rlzhejkgtl kgtrk cls iukrpls y cls pchgls jgicjgh`ls slg >.9 y >.8 rkspkitjvhekgtk. Ihciuchr= Ch hikckrhijùg `kc sjstkeh, Chs tkgsjlgks `k ch iukr`h, Ch vkclij`h` quk h`qujkrkg cls oclquks iuhg`l sk `kspchzhg ; e h cl chrfl `k cls pchgls jgicjgh`ls rkspkitjvls,  phrtjkg`l `kc rkplsl. (Kepckhr `ls prlik`jejkgtls `k ièciucl phrh kstk hphrth`l, ileprlohg`l quk sk lotjkgkg cls ejsels rksucth`ls).  Ylcuijùg

Kiuhijlgks `kc elvjejkgtl `k ih`h ugl `k cls iukrpls

{ G ? 59¶:.2¶ils9> 59¶:.2¶ils9> A ? 5 γ ? G ? 5>.9¶9¶:.2¶ils9> 5>.9¶9¶:.2¶ils9> P ? ∟ A ? ∟9¶:.2¶sjg9>59h ∟9¶:.2¶sjg9>59h { G 8 5;¶:.2¶ils0; 5;¶:.2¶ils0; A 8 5 γ 8  G 8 5>.8¶;¶:.2¶ils0; 5>.8¶;¶:.2¶ils0; ;¶:.2¶sjg0;∟ ;¶:.2¶sjg0;∟ P 8 ∟ A 8 5;h P 8 r∟ P ? r5( 8 ? ? r ? )θ _kchijùg kgtrk ch hikckrhijùg h `k cls oclquks y ch hikckrhijùg hgfuchr θ `k ch plckh h5θ¶r 

Kc sjstkeh `k kiuhijlgks sk rk`uik h 98.2∟ P 8 5;h P ? ∟??.9059h P 8 ∟ P ? 5h }h5>.:29  e/s ? Yj kc iukrpl `k ; df `kscjzh ; e phrtjkg`l `kc rkplsl ;5 8 ? h t ? v5ht }v59.89 e/s Ohchgik kgkrfêtjil

 

Prhohml `k ch aukrzhs `k rlzhejkgtl V5-A 8¶;-A ?¶;5-;;.;? M

V5 8 ? 9 v ? + 8 ? ; v ? + 8 ? ( 8 ? ? r ? ) ϊ ? +9¶:.2¶;¶sjg9>∟;¶:.2¶;¶sjg0; +9¶:.2¶;¶sjg 9>∟;¶:.2¶;¶sjg0; _kchijùg kgtrk ch vkclij`h` v `k cls oclquks y ch vkclij`h` hgfuchr ϊ `k ch plckh v5ϊ¶r 

Kc rksucth`l ks v59.89 e/s, kc ejsel quk bkels lotkgj`l plr `jgèejih ^rlockeh 6

[g `jsil `k >.? df y `k 8> ie `k rh`jl sk bhik fjrhr ek`jhgtk ugh iukr`h quk phsh h trhvês `k ugh plckh `k >.; df y `k 6 ie `k rh`jl. @k ch iukr`h iukcfh ug oclquk `k 9 df, thc ilel sk eukstrh kg ch ajfurh. Kc `jsil fjrh hcrk`k`lr `k ug kmk vkrtjihc kg iuyl kxtrkel bhy ugh vhrjcch `k >.6; df ehsh y `k ?> ie `k clgfjtu` pkrpkg`jiuchr hc kmk y kg iuyls kxtrkels sk bhg ajmh`l `ls ksakrhs jfuhcks `k ? df `k ehsh y ; ie `k rh`jl. Yk sukcth kc  oclquk y kc `jsplsjtjvl ilejkgzh h fjrhr. Ihciuchr= Kc elekgtl `k jgkrijh `kc `jsplsjtjvl. Ch hikckrhijùg `kc oclquk. Ch vkclij`h` `kc oclquk iuhg`l bh `ksikg`j`l ? e phrtjkg`l `kc rkplsl (rkslcvkr kstk hphrth`l plr kgkrfåhs).  Ylcuijùg

Elekgtl `k jgkrijh `kc `jsplsjtjvl J5 8 ? >.?¶ >.8 ? + 8 8? >.6;¶ >.? ? +?( ? ; ?¶ >.>; ? +?¶ >.8 ? )5>.>06;  df¶e ?

Kiuhijlgks `kc elvjejkgtl `k ih`h ugl `k cls iukrpls P ? ¶>.85J θ ? P 8 ¶>.>6∟ P ? ¶>.>65( 8 ? >.;¶ >.>6 ? ) θ 8 9¶:.2∟ P 8 59¶h _kchijùg kgtrk ch hikckrhijùg h `kc oclquk y chs hikckrhijlgks hgfuchrks `kc `jsil y `k ch plckh h5θ8¶>.>65 θ?¶>.8

Kc sjstkeh `k kiuhijlgks sk rk`uik h P ? 50.6;h P 8 ∟ P ? 5>.?;h ?:.0∟ P 8 59h }h59.16;  e/s ? Yj kc iukrpl `k 9 df `ksijkg`k ? e phrtjkg`l `kc rkplsl ?5 8 ? h t ? v5ht }v59.29 e/s Ohchgik kgkrfêtjil

 

^rjgijpjl `k ilgskrvhijùg `k ch kgkrfåh 9¶:.2¶?5 8 ? 9¶ v ? + 8 ? J ϊ ? ? + 8 ? ( 8 ? >.;¶ >.>6 ? ) ϊ 8 ? _kchijùg kgtrk ch vkclij`h` v `kc oclquk y chs vkclij`h`ks hgfuchrks `kc `jsil y `k ch plckh v5ϊ8¶>.>65 ϊ?¶>.8

Kc rksucth`l ks v59.29 e/s, kc ejsel quk bkels lotkgj`l plr `jgèejih ^rlockeh 2

Kc sjstkeh `k ch ajfurh ilgsth `k ugh plckh alreh`h plr `ls `jsils ilhxjhcks slc`h`ls `k ehshs ;;> y 9>> f y rh`jls 2 y 1 ie, rkspkitjvhekgtk. @ls ehshs `k 1>> y ;>> f iukcfhg `kc olr`k `k ih`h `jsil. Ihciuchr= ¼Kg quê skgtj`l fjrh< Ch tkgsjùg `k ih`h iukr`h Ch hikckrhijùg `k ih`h ehsh Ch vkclij`h` `k ih`h iukrpl iuhg`l ugl `k kccls (¼iuèc.;¶:.2¶>.>17>.1¶:.2¶>.>1, fjrh kg kc skgtj`l `k chs hfumhs `kc rkclm.

Elekgtl `k jgkrijh `k cls `jsils slc`h`ls J 5 8 ? >.;; ¶ >.>2 ? + 8 ? >.9 ¶ >.>1 ? 5 >.>>?9   df¶e ? Kiuhijlgks `kc elvjejkgtl `k ih`h ugl `k cls iukrpls >.; ¶ :.2 ∟ P ? 5 >.; ¶ h ? P ? ¶ >.>2 ∟ P 8 ¶ >.>1 5 J θ P 8 ∟ >.1 ¶ :.2 5 >.1 ¶ h 8 _kchijùg kgtrk chs hikckrhijlgks `k cls oclquks y ch hikckrhijùg hgfuchr `k cls `jsils h85θ¶>.>1 h?5 θ¶>.>2

_kslcvkels kc sjstkeh `k kiuhijlgks, θ5;.8? rh`/s?, h 5>.9>6 8 e/s?, h?5>.0>: e/s? Ohchgik kgkrfêtjil

 

Yj kc iukrpl `k >.; df `ksijkg`k 9 e phrtjkg`l `kc rkplsl 95 8 ? h ? t ? v ? 5 h ? t } v ? 58.;16 e/s Iuhg`l kc iukrpl `k >.; df `ksijkg`k b?59 e kc iukrpl `k >.1 df hsijkg`k b8 β5 b 8 >.>1 5 b ? >.>2    b 8 5?.?; e Ilgskrvhijùg `k ch kgkrfåh >.;¶:.2¶95 8 ? >.;¶ v ? ? + 8 ? J ϊ ? + 8 ? >.1¶ v 8 ? _kchijùg kgtrk chs vkclij`h`ks `k cls oclquks y ch vkclij`h` hgfuchr `k cls `jsils v85ϊ¶>.>1 v?5 ϊ¶>.>2

Kc rksucth`l ks v?58.;16 e/s, kc ejsel quk bkels lotkgj`l plr `jgèejih ^rlockeh :

Ylork ug pchgl blrjzlgthc y quk prkskgth ugh rksjstkgijh hc `kscjzhejkgtl `k ilkajijkgtk γ5>.?, `kscjzh ug oclquk `k 9 df `k ehsh ugj`l h ugh iukr`h quk sk kgrlcch kg ch pkrjakrjh `k ugh plckh alreh`h plr ug `jsil ; df y >.9 e `k rh`jl quk tjkgk ugh bkg`j`urh `k >.8 e thc ilel sk vk kg ch ajfurh. @k ch iukr`h kgrlcch`h kg ch bkg`j`urh pkg`k ug oclquk `k 8> df `k pksl. Ihciuchr= Chs tkgsjlgks `k chs iukr`hs Ch hikckrhijùg `k ih`h iukrpl Kc oclquk `k 8> df `ksijkg`k ? e phrtjkg`l `kc rkplsl, ihciuchr ch vkclij`h` `k ih`h ugl `k cls oclquks (rkslcvkr kstk hphrth`l rkchijlgh`l trhohmls y kgkrfåhs).  Ylcuijùg

Kiuhijlgks `kc elvjejkgtl `k ih`h ugl `k cls iukrpls { P 8 ∟ A r 59 h 8 G59¶:.2 A r 5>.?¶G5;.22 P ? ¶>.8∟ P 8 ¶>.95( 8 ? ;¶ >.9 ? )θ 8>¶:.2∟ P ? 58>¶ h ? _kchijùg kgtrk chs hikckrhijlgks `k cls oclquks y ch hikckrhijùg hgfuchr `k cls `jsils h85θ¶>.9

 

h?5 θ¶>.8

_kslcvkels kc sjstkeh `k kiuhijlgks, θ589.;8 rh`/s?, h850.>; e/s?, h?58.9; e/s? Yj kc iukrpl `k 8> df `ksijkg`k ? e phrtjkg`l `kc rkplsl ?5 8 ? h ? t ? v ? 5 h ? t } v ? 5?.9? e/s Ohchgik kgkrfêtjil

Iuhg`l kc iukrpl `k 8> df `ksijkg`k  x?5? e kc iukrpl `k 9 df sk `kspchzh  x8 β5 x 8 >.9 5 x ? >.8    x 8 51 e Prhohml `k ch aukrzh `k rlzhejkgtl V5-A r ¶x85-;.22¶15-9;.?2 M

Ohchgik kgkrfêtjil V5 8 ? 8>¶ v ? ? + 8 ? J ϊ ? + 8 ? 9¶ v 8 ? ∟8>¶:.2¶? _kchijùg kgtrk chs vkclij`h`ks `k cls oclquks y ch vkclij`h` hgfuchr `k cls `jsils v85ϊ¶>.9 v?5 ϊ¶>.8

Kc rksucth`l ks ϊ5?9.?0 rh`/s, v851.:6 e/s, v?5?.9? e/s, kc ejsel quk bkels lotkgj`l plr `jgèejih ^rlockeh 8>

Ylork ug pchgl jgicjgh`l 9>½ y quk larkik ugh rksjstkgijh hc `kscjzhejkgtl `k ilkajijkgtk γ5>.?, `kscjzh ug oclquk `k 9 df `k ehsh ugj`l h ugh iukr`h quk sk kgrlcch kg ch pkrjakrjh `k ugh plckh alreh`h  plr `ls `jsils hilpch`ls `k 8 df y >.; df y `k rh`jls >.9 e y >.8 e rkspkitjvhekgtk. @k ch iukr`h kgrlcch`h hc `jsil pkqukøl pkg`k ug oclquk `k 8> df `k pksl. Ihciuchr= Chs tkgsjlgks `k chs iukr`hs Ch hikckrhijùg `k ih`h iukrpl Ch vkclij`h` `k ih`h iukrpl sj kc oclquk `k 8> df `ksijkg`k ? e phrtjkg`l `kc rkplsl (kepckhr `ls prlik`jejkgtls `jstjgtls phrh kstk hphrth`l).  Ylcuijùg

 

Elekgtl `k jgkrijh `k cls `jsils slc`h`ls J5 8 ? 8¶ >.9 ? + 8 ? >.;¶ >.8 ? 5>.>06;  df¶e ? Kiuhijlgks `kc elvjejkgtl `k ih`h ugl `k cls iukrpls { P 8 ∟ A r ∟9¶:.2¶sjg9>59 ∟9¶:.2¶sjg9>59 h 8 G59¶:.2¶ils9> G59¶:.2¶ils9> A r 5>.?¶G P ? ¶>.8∟ P 8 ¶>.95Jθ 8>¶:.2∟ P ? 58>¶ h ? _kchijùg kgtrk chs hikckrhijlgks `k cls oclquks y ch hikckrhijùg hgfuchr `k cls `jsils h85θ¶>.9 h?5 θ¶>.8

_kslcvkels kc sjstkeh `k kiuhijlgks= θ5:.?; rh`/s?, h85?.66 e/s?, h?5>.:? e/s? Yj kc iukrpl `k 8> df `ksijkg`k ? e phrtjkg`l `kc rkplsl ?5 8 ? h ? t ? v ? 5 h ? t } v ? 58.:? e/s Ohchgik kgkrfêtjil

Iuhg`l kc iukrpl `k 8> df `ksijkg`k  x?5? e kc iukrpl `k 9 df sk `kspchzh  x8 β5 x 8 >.9 5 x ? >.8    x 8 51 e Prhohml `k ch aukrzh `k rlzhejkgtl V5-A r ¶x85->.?¶9¶:.2¶ils9>¶15-9>.;;

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V5 8 ? 8>¶ v ? ? + 8 ? J ϊ ? + 8 ? 9¶ v 8 ? +9¶:.2¶1¶sjg9>∟8>¶:.2¶? +9¶:.2¶1¶sjg 9>∟8>¶:.2¶? _kchijùg kgtrk chs vkclij`h`ks `k cls oclquks y ch vkclij`h` hgfuchr `k cls `jsils v85ϊ¶>.9 v?5 ϊ¶>.8

Kc rksucth`l ks ϊ58:.?0 rh`/s, v85;.66 e/s, v?58.:? e/s, kc ejsel quk bkels lotkgj`l plr `jgèejih

Aug`hekgtls Aåsjils © Ègfkc Arhgil Fhriåh