Problemas Resueltos de Estática - Teoría y Problemas Resueltos

GIKQLZIJ JA KCAEKCI[ JA CEHAECA_ÆI A[KQALI T_MGA[CMEIL JA CEHAECA_ÆI KCWCL - @QIEKIWALCKI

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T_MLMHM Al prasaeta trioinm, sa oisi ae Trmolafis _asualtms ja Irfijuris plieis y trcjcfaescmeilas astétckifaeta astétckifaeta jatarfceijis jatarfceijis usiejm usiejm al Fãtmjm Fitrckcil, qua `my ae jæi sme lms fãtmjms fés usijms piri kélkulms iviezijms. Asparafms qua asta trioinm scrvi ja huæi y préktcki ae prmvak`m ja euastri smkcajij ae sikirla ijalieta piri lms guturms astujcietas. Asparm ìyi kufplcjm kme fc jaoar jaoar ja jcvulhir lms kmemkcfcaetms kmemkcfcaetms ae ilkieka ja tmjms.

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Ae lms fãtmjms ja astujcm piri irfijuris pliems y trcjcfaescmeilas jatarfceijms ae Fitarci ja Astétcki, sa usie lms fãtmjms fés kmfueas Fãtmjm ja Eujms y [akkcmeas piri jatarfceir lis guarzis ae lms fcaforms, Oésckifaeta al Fãtmjm ja Eujms, gceilfaeta llahi i ue scstafi ja akuikcmeas lceailas, ìkcaejm al usm ja

∙  6   ∙ 6

ae kiji eujm y

∙  6  piri jatarfceir lis raikkcmeas.

Al prmkajcfcaetm piri al ieélcscs ja Fãtmjm Fitrckcil piri irfijuris, as

saekcllm piri jatarfceir lis guarzis guarzis y raikkcmeas, raikkcmeas, smlm as jatarfceir lms kmsaems jcraktmras y sa tmfi al ceckcm y gceil ja lms alafaetms y hrijms ja lcoartij m lis akuikcmeas qua sa jie ae kiji eujm , ae lms anas “x‖ y “y‖, piri kiji alafaetm. Zaear prasaeta li schucaeta rastrckkcûe. Tiri uei irfijuri pliei sa jarcvi li schucaeta axprascûe

>þ > þ    >þ > þ      >þ > þ   

6+

Kuiejm sa kufpla asti axprascûe ìy tietis akuikcmeas kmfm guarzis jaskmemkcjis, supmecaejm qua em ìy ceastioclcjijas hamfãtrckis, al scstafi

ja akuikcmeas taejré uei uei smlukcûe. Tiri al prmkasm, kiji alafaetm alafaetm sa supmea ae taescûe y as kmevaecaeta qua lis raikkcmeas taehie ae saetcjms pmsctcvms piri fés priktckm sa ìré al prmkasm ae al anafplm schucaeta. Li mrcaetikcûe ja ue fcaform ja uei irfijuri puaja jaskrcocrsa jagcecaejm lms kmsaems jcra ktmras jal ana “ xº ‖ lmkil jal fcaform, ìy jms kmsaems jcraktmras jcraktmras piri piri kiji ana “ xº ‖, lms kmsaems jcraktmras sme lms

éehulms i y o













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 6 (    )2  6 (   ) 2  6 ∞ (    ) + (   )  (    ) (   )  2 6   6 6 6 ∞ (    ) + (   )  ∞ (    ) + (   )   6  2  66 

Tiri al schucaeta anafplm saré tmfiri kmfm anafplm piri jatarfceir li li fitrcz astétcki. ? ?4 Zme

>Waktmr ja Guarzis Kmemkcjis Axtareis

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>Fitrcz Astétcki >Waktmr ja Guarzis jaskmemkcjis Cetareis

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9

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 6  +  2   6 2   6 2 2 6  Ceckcm y gceil ja kiji alafaetm y pmsckcûe ja guarzi uectirci piri kiji alafaetm. S

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(?4,91)

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Kmmrjaeijis lmkilas ja kiji alafaetm. ?

S

(?4,91)

y º

9 y º

x º

y º x º

x º

91 f

; ?

yº

yº

(4,4)

xº

xº

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(:4,4)

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?4 f

Kmestruyifms li fitrcz astétcki usiejm al schucaeta hrégckm. :

S

?4 Zme

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9

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4.7

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9

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alafaetms

4.0

9

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4.7 y º

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Kmsaems jcraktmras ja lms

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Tiri lis raikkcmeas, lis raikkcmeas tifocãe sa kmescjarie kmfm ue alafaetm y asti kme éehulms ja 4ý 34´ y lms kmsaems jcraktmras as 9 y 4 y sa uockie ja ikuarjm i lms hrijms ja lcoartij ae li irfijuri, piri ipmyms fûvclas smlm ìoré uei raikkcûe y ipmyms gcnms ìoré ae saetcjm “x‖ a “y‖.

GQA_Î[ S LI[ _AIKKCMEA[

H_IJM[ JA LCOA_ZIJ

9 ? ; : 1 7 8 0

G9 -4.0 -4.7 4.0 4.7 4 4 4 4

G? 4 4 4 9 4 -9 4 4

G; 4 4 -4.0 4.7 4 4 4.0 -4.7

G: -9 4 4 4 9 4 4 4

G1 4 4 4 4 -9 4 9 4

_x9 9 4 4 4 4 4 4 4

_y? _y: 4 4 9 4 4 4 4 4 4 4 4 4 4 4 4 9

RV∟ y vaktmr ja guarzis RV. RV 6 RV∟ ∘ RV

Kélkulms ja cevarsi ja li fitrcz astétcki G9 4 4 G? 4 4 G; 4 4 G: 4 4 G1 6 4 4 _x9 9 4 _y? 4 9 _y: 4 4

4.7?14 4 -4.7?14 4.1444 4.1444 9 4.;814 -4.;814

4.0;;; 4 4.0;;; -4.7778 -4.7778 4 4.1 4.1

4 4 4 9 4 9 4 4

4.0;;; -9 4.0;;; -4.7778 -4.7778 4 4.1 4.1

4 4 4 9 9 9 4 4

4 4 4 4 4 4 4 9

4 4 ?4 4 4 4 4 4

Guarzis rasultietas ja lms alafaetms y lis raikkcmeas ae lms ipmyms G96 9?.14 Zrikkcûe G?6 4.44 Zrikkcûe G;6 -9?.14 Kmfprascûe

G16 94.44 Zrikkcûe _x96 ?4.44 _y?6 8.14













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Tiri li irfijuri fmstriji ae li gchuri, jatarfceir pmr al Fãtmjm Fitrckcil lis guarzis ae kiji uei ja lms alafaetms y lis raikkcmeas y kilkulir lms asguarzms ae kiji alafaetm sc li sakkcûe as chuil piri tmjms ja (Ceh. Lucs A. Hifcm Ircseioirrati - Téh. 984 ‛ Astétcki) Astétcki) :

 6 .? . ?.

1 ? Zme

?f ? Zme

? Zme

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9

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9f

Jcskrataziejm li irfijuri ja ?J kme sus raspaktcvis kmmrjaeijis, lms hrijms ja lcoartij y al saetcjm ja kiji alafaetm ceckcm y gceil. 9? (4,:)

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99

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Jitms piri motaear li fitrcz astétcki kme lms kmsaems jcraktmras ja kiji alafaetm. Ceckcilas ALAF. 9 ? ; : 1 7 8 0 3

Gceilas

    4 4 4 -? : -? : 1 4

4 4 4 ? 4 ? 4 ? :

-? : : 4 4 1 1 : :

? : 4 : : ? ? : :

?.0?0:?89 1.71701:? : ?.0?0:?89 1.71701:? 8 ?.?;7470 ?.?;7470 :

Fitrcz astétcki HJL\G 9 ? ; : 1 ROV6 7 8 0 3 94 99 9?

-4.8489470 4.8489470 9 4.8489470 -4.8489470 9 4.::8?9;7 -4.::8?9;7 9

4.8489470 4.8489470 4 4.8489470 4.8489470 4 4.03::?8? 4.03::?8? 4

RV 6 RV>

G9 G? G; G: G1 G7 G8 G0 G3 _x9 _y9 _? 4.8489 -4.8489 -9.4 4 4 4 4 4 4 9 4 4 -4.8489 -4.8489 4.4 4 4 4 4 4 4 4 9 4 4 4 9.4 4 4.8489 4 -4.::8? 4 4 4 4 4 4 4 4.4 4 -4.8489 4 -4.03:: 4 4 4 4 9 -4.8489 4 4 -4.8489 4 -9.4 4 4 4 4 4 4 4.8489 4 4 -4.8489 4 4.4 4 4 4 4 4 4 4 4 4 4 4 9.4 4.::8? 4.::8? 4 4 4 4 4 4 4 4 4 4.4 4.03:: -4.03:: 4 4 4 4 4 4.8489 4 4 4 4 4 -4.::8? 9 4 4 4 4 4.8489 4 4 4 4 4 4.03:: 4 4 4 4 4 4 4 4.8489 -4.8489 4 4 4 -9 4 4 4 4 4 4 4.8489 4.8489 4 4 4 4 4 4 4

RV∟ y vaktmr ja guarzis RV. RV 6 RV∟ ∘ RV

Kélkulms ja cevarsi ja li fitrcz astétcki 4

4

4

4

?.9?9; 4.8489 ?.9?9; -9.4748 9.:9:? -9.:9:? 9.:9:?

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Gceilfaeta lis guarzis ae kiji alafaetm sa fuastrie ae li schucaeta tioli RtmeV.

G96 G?6 G;6 G:6 G16 G76

-0 97.38 -9:.44 -99.;9 99.;9 9?.44

Guarzis ja kiji alafaetm ceklceijm y `mrczmetil RZmeV Zrikkcûe G86 -9;.:? RZmeV RZmeV Kmfprascûe G06 -9;.:? RZmeV RZmeV Zrikkcûe G36 -97.44 RZmeV RZmeV Zrikkcûe _x96 :.44 RZmeV RZmeV Kmfprascûe _y96 7.44 RZmeV RZmeV Kmfprascûe _?6 -:.44 RZmeV

Zrikkcûe Zrikkcûe Zrikkcûe

Hrigcki ja Guarzis _asultijms gceilas ae kiji alafaetm. (K) G3697 Zme

? Zme

? Zme

e (K) Z m 9 ; . 9 9 : 6 8 G

? Zme

3

:

1

G 1 6

G 0 6

e m Z 8 3 . 7 9 6 G ?

9 9 . ; (Z) 9 Z m e 1

0

9 ; (K) . : ? Z m e

; G769? Zme

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(Z) (Z)

e m Z 8 3 . ? 7 9 6 G ?

G 9 6 60 (K) Z m e 9

9

G 1 6

9 9 . ; ; 9 Z m e

7 8

G;69: Zme

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Jcskrataziejm li irfijuri ja ?J kme sus raspaktcvis kmmrjaeijis, lms hrijms ja lcoartij y al saetcjm ja kiji alafaetm ceckcm y gceil. 9: (?.8,:.1) ? Zme

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Ceckcilas

ALAF. 9 ? ; : 1 7 8 0 3 94



4 4 1.: 4.3 :.1 4.3 4.3 9.0 ;.7 9.0

Gceilas

  

4 4 4 9.1 9.1 9.1 9.1 ; ; ;

4.3 :.1 :.1 :.1 ;.7 ;.7 9.0 ;.7 ?.8 ?.8

9.1 9.1 9.1 9.1 ; ; ; ; :.1 :.1

9.8:3; :.8:;: 9.8:3; ;.7 9.8:3; ;.4008 9.8:3; 9.0 9.8:3; 9.8:3;

Fitrcz astétcki HJL\G 9 ? ; : 1 7 ROV6 8 0 3 94

G9 G? -4.19:1 -4.3:08 -4.0181 -4.;97? 4 4 4 4 4 4.3:08 4 4.;97? 4.19:1 4 4.0181 4 4 4 4 4

G; 4 4 4.19:1 -4.0181 -4.19:1 4.0181 4 4 4 4

G: 4 4 4 4 9.4 4.4 -9.4 4.4 4 4

G1 4 4 4 4 4.19:1 -4.0181 4 4 -4.19:1 4.0181

4.19::37 4.3:070; -4.19::37 9 -4.19::37 4.08:918 4.19::37 9 -4.19::37 4.19::37

4.018:3?3; 4.;97??888 4.018:3?3; 4 4.018:3?3; 4.:017:?3; 4.018:3?3; 4 4.018:3?3; 4.018:3?3;

RV 6 RV ∷

G7 G8 4 4 4 4 4 4 4 4 4 4 4 4 -4.08:? -4.19:1 -4.:017 -4.0181 4.08:? 4 4.:017 4

G0 4 4 4 4 4 4 4 4 9.4 4.4

G3 4 4 4 4 4 4 4 4 4.19:1 -4.0181

G94 4 4 4 4 4 4 4 4 4 4

_9x _9y _?x _?y 9 4 4 4 4 9 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4













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Gceilfaeta lis guarzis ae kiji alafaetm sa fuastrie ae li schucaeta tioli RZmeV.

G96 G?6 G;6 G:6 G16 G76 G86

-9.3:: -;.19: ;.?;3 4.444 9.3:: 4.444 -9.3::

Guarzis ja kiji alafaetm ceklceijm y `mrczmetil RZmeV Kmfprascûe G06 4.444 RZmeV Zrikkcûe RZmeV Kmfprascûe G36 9.3:: RZmeV Zrikkcûe RZmeV Zrikkcûe G946 -9.3:: RZmeV Kmfprascûe RZmeV Zrikkcûe _9x6 -:.;;; RZmeV RZmeV Zrikkcûe _9y6 -?.880 RZmeV RZmeV Zrikkcûe _?x6 -9.778 RZmeV RZmeV Kmfprascûe _?y6 ?.880 RZmeV

? Zme

e m Z

8

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G

7

G 3 6

9 . 3 : :

94

3

Z m e

Zme .444 Zm G064.44 0

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Tmr al Fãtmjm Fitrckcil jatarfceir lis guarzis ae kiji uem ja lms alafaetms ja li irfijuri fmstriji y lis raikkcmeas (Ceh. _ayjar A. Oaejazþ Omzi Axifae ja Fãtmjms Eufãrckms Iplckijms ‛ ?49; ?49; ‛ CC CC ‛ QE@.) QE@.)

?.44 f

9.44f

T(BE)

9.44 f

T(BE) T(BE)

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9.44

?.44

T(BE) (9,:) 7 1 T(BE)

9

># Alafaetms ># Eujms (x,y) >Kmmrjaeijis T> >Guarzis

9.44

T(BE) (;,:) 0 T(BE) 8 h

0

7

h

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8

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Tsae(i)64.3849 T Tkms(i)64.?:?1 T

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?Tsae(i)64.:019 T ?Tkms(i)69.3:4; T

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9

19

Éehulms raspaktcvms h6itie(?/4.1) h681.37;0´ ? i634´-81.37;0´ i69:.4;7?´

94 ?T(BE) (;.12?) :.44 3

:

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T(BE) i

3

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HJL\G -4.?:?1: -4.3849: 4.?:?1: 4.3849: 4 4 4 4 4 4 4 4

-4.070?: -4.:379: 4 4 4 4 4 4 4.070?: 4.:379: 4 4

4 4 4 4 -4.070?: 4 4.:379: 4 4 4 4 4 4 4 4 4 4 -4.?:?1: 4 4.3849: 4.070?: 4.?:?1: -4.:379: -4.3849:

4 4 -4.?:?1: -4.3849: 4.?:?1: 4.3849: 4 4 4 4 4 4

4 4 4 4 4.14444 4 4 4 4 4 4.0774; 4 -4.80408 4 4 4 4 4 -4.7?:84 4 4 4 4 4 4 -4.80408 -9 4 4 4 4 4.7?:84 4 4 4 4 4.80408 4 9 -4.?:?1: 4 4 4.7?:84 4 4 4.3849: 4 4 4 4.80408 4 4.?:?1: 4 4 4 -4.7?:84 4 -4.3849: 4 4 4 4 4 4 4 4 4 4 4 4 4 9

RV + RV ∘ RV 6  4.4444 T

-4.?:?1:

-4.070?:

4

4

4

4

4

4

4

4.14444

4

4

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4

4

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4

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RV 6 RV∟ ∘ RV 4.708? -9.;:;8 -9.;:;8 4.708? -9.;8:: ?.9;:: ?.9;:: -?.4444 -9.;8::

-4.;378 4.8810 4.8810 4.8810 -4.;378 4.83;1 -9.?;?; -9.?;?; 9.91:8 9.91:8 4.83;1 4.4444 9.91:8 4.4444 4.4444 9.4444 -4.188:

4.:4?3 4.?933 -4.8083 -4.99?1 4.??:3 -4.;:3; -4.;:3; 4.;?8: 4.??:3 4.188: -4.1444 4.899;

4.8419 4.;0:3 4.7;70 -4.9370 4.;3;8 -4.799: -4.799: 4.18?3 4.;3;8 9.494: 4.9?14 -4.141?

4.0410 4.:;33 -4.?;?4 -4.39?9 4.::33 4.;701 -4.7308 -4.;:1; -4.?;8; 9.91:8 -9.4444 4.:??7

4.74:: 4.;?33 4.:383 4.44;9 9.;70? -4.8340 -4.1?:4 4.8:94 4.143? 4.0774 4.?144 -4.:;;4

4.0410 4.:;33 -4.?;?4 -4.39?9 4.::33 4.;701 -4.7308 4.71:8 -4.?;8; 9.91:8 -9.4444 4.:??7

4.?491 4.9944 -4.4104 4.04?8 4.99?1 4. 43?9 -4.98:8 -4.98:8 4.97;8 4.3891 4.?008 4.8144 -4.9::;

4.:4?3 4.?933 -4.8083 -4.99?1 -4.0410 9.?19: 9.?19: -9.98?7 -4.0410 4.188: -4.1444 -4.1444 4.899;

Guarzis ae lms alafaetms `mrczmetilas, vartckilas y trisvarsilas. Guarzis ja kiji alafaetm ceklceijm y `mrczmetil

4.9448 4.4114 -4.9384 9.44?8 -4.?491 4.;9?3 4.;9?3 -4.?3;? -4.?491 4.9::; 4.0814 -4.48??

4 4 4 4 4 4 4 4 4 4 4 9

4 4 4 4 4 4 4 4 4 4 9 4

4.444 T 4.444 T 9.3:4 T -4.:01 T 4.384 T -9.?:; T -4.384 T -9.?:; T -9.3:4 T -4.:01 T 4.444 T 4.444 T













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Tmr al fãtmjm fitrckcil, li smlukcûe ja scstafi ja akuikcmeas lceailas, rasmlvar li schucaeta astrukturi y kilkulir lis guarzis ae kiji alafaetm y lis raikkcmeas. i) Kilkulir ae guekcûe ja li kirhi T. o) Tiri uei kirhi T6914 tme T(BE)

T(BE)

:

?

T(BE)

7

8 1 3

9

94

9;

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9

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0

0













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8 0 3 94 99 9? 9;

97 0 97 9? 0 9? 97

4 4 4 4 4 4 4

: 0 0 9? 9? 97 9?

: : : : 4 4 :

9?.7:3999 -4.3:070;; 4.;97??80 : 4 9 0.3::?893 -4.03::?8? 4.::8?9;7 : 4 9 : 9 4 : 9 4 1.71701:? -4.8489470 4.8489470

 é RV 6 RV HJL\G G9 9 -4.8489 ? -4.8489 ; 4.8489 : 4.8489 1 4 7 4 8 4 0 4 3 4 94 4

G? -4.03:: -4.::8? 4 4 4 4 4.03:: 4.::8? 4 4

G; -4.3:08 -4.;97? 4 4 4 4 4 4 4 4

G: -9.4 4.4 4 4 9.4 4.4 4 4 4 4

G1 4 4 4.4 9.4 4.4 -9.4 4 4 4 4

G7 4 4 4 4 -9.4 4.4 4 4 9.4 4.4

G8 G0 4 4 4 4 -4.3:08 4 4.;97? 4 4 4 4 4 4 4.444 4 9.444 4 4.444 4 -9.444

G3 4 4 4 4 4 4 -4.03:: 4.::8? 4 4

G94 G99 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 -9.444 4 4.444

G9? 4 4 4 4 4 4 4 4 4 4

G9; 4 4 4 4 4 4 4 4 4 4

_9x _9y 9 4 4 9 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

_; 4 4 4 4 4 4 4 4 4 4













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RV 6 RV∟ ∘ RV 4 4 4.;1;7 4 4 4.4444 4 4 4 4 4 4.7418 4 4 4 4 4 4.7418 4 4 -4.8347 4 4 4 RGV6 4 4 4 4 4 4 4 4 4.7418 4 4 4.7418 4 4 4 9 4 4.0118 4 9 4.?144 4 4 -4.?008

9.4748 4.4444 4 -4.7418 4 -4.7418 4.8347 4 4 4 -4.7418 -4.7418 4 4.9::; 4.8144 4.?008

4 4 4 9 4 4 4 4 4 4 4 4 4 9 4 4

9.4748 4.4444 4 -4.7418 -9.4444 -4.7418 4.8347 4 4 4 -4.7418 -4.7418 4 4.9::; 4.8144 4.?008

4 4.1134 4 4.;118 4 4.;118 4 4 -4.1134 4 4.;118 4.;118 4 4.0118 4.?144 -4.?008

4 9.9904 4 -4.899; 4 -4.899; 4 4 9.9904 4 -4.899; -4.899; 4 4.?008 4.1444 4.188:

4 4 4 9 4 9 4 4 4 4 4 4 4 9 4 4

4 9.9904 4 -4.899; 4 -4.899; 4 -9.4444 9.9904 4 -4.899; -4.899; 4 4.?008 4.1444 4.188:

4 4 4 9 4 9 4 4 4 4 9 4 4 9 4 4

4 4 4.8347 -4.;984 4 -4.;984 4 4 4 -9.4444 -4.;984 -4.;984 9.4748 4.:;;4 4.?144 4.0774

4 4 4.8347 4.9418 4 4.9418 4 4 4 4 4.9418 4.9418 -4.;1;7 4.0118 4.?144 -4.?008

4 4 4.8347 -4.;984 4 -4.;984 4 4 4 4 -4.;984 -4.;984 9.4748 4.:;;4 4.?144 4.0774

4 4 4 9 4 9 4 4 4 4 9 9 4 9 4 4

4 4 4 4.188: 4 4.188: 4 4 4 4 4.188: 4.188: 4.4444 4.188: 4 9.91:8

4.44 T 4.44 T 4.44 T -9.44 T 4.44 T -?.44 T 4.44 T -9.44 T 4.44 T 4.44 T 4.44 T -?.44 T 4.44 T -9.44 T 4.44 T 4.44 T















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Tmr al fãtmjm fitrckcil, li smlukcûe ja scstafi ja akuikcmeas lceailas, rasmlvar li schucaeta astrukturi y kilkulir lis guarzis ae kiji alafaetm y lis raikkcmeas. i) Kilkulir ae guekcûe ja li kirhi T. o) Aviluir piri ue vilmr ja T6914 Zme.













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Kélkulms pravcms KMM_JAEIJI[ Ceckcilas Gceilas ALAF. 9 ?

    -: -:

4 4

4 9?

LMEH. KM[(x) : 1.7173 4.848948 : 97.:3? 4.3849:;

[AE(x) 4.8489470 4.?:?1;17













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4T -?.4T 4T 4T 4T -;.4T 4T -;.4 T 4T













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Guarzis ja kiji alafaetm ceklceijm y `mrczmetil G96

-4.13908 T

Kmfprascûe

G36 7.840?4 T Zrikkcûe

G?6

3.38983 T

Zrikkcûe

G946 ?.8:::7 T Zrikkcûe

G;6

-98.80??3 T

Kmfprascûe

G996 0.8:::7 T Zrikkcûe

G:6

-3.?111: T

Kmfprascûe

G9?6 -3.80970 T Kmfprascûe

G16

7.840?4 T

Zrikkcûe

G9;6 -4.0;303 T Kmfprascûe













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Tiri ieélcscs ja uei irfijuri i ecval trcjcfaescmeil, axcstae fés















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