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^ey klm^_gl•\ \bsbo gZ\qZevgbo aZdeZ^•\
Ev\•\
<b^Z\gbpl\h GZp•hgZevgh]h mg•\_jkbl_lm ³Ev\•\kvdZ ihe•l_og•dZ´
2003
;;D Y A
M>D 53 (076)
J_dhf_g^h\Zgh GZmdh\h-f_lh^bqghx jZ^hx GZp•hgZevgh]h mg•\_jkbl_lm ³Ev\•\kvdZ ihe•l_og•dZ´ yd gZ\qZevgbc ihk•[gbd
^ey klm^_gl•\ \bsbo gZ\qZevgbo aZdeZ^•\
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; : Emd•yg_pv ^hdlhj n•a -fZl gZmd ijhn_khj GZp•hgZevgh]h mg•\_jkbl_lm ³Ev\•\kvdZ ihe•l_og•dZ´;
< 1 <Zc^Zgbq ^hdlhj n•a -fZl gZmd ijhn_khj Ev\•\kvdh]h ^_j`Z\gh]h e•khl_og•qgh]h mg•\_jkbl_lm
EhiZlbgkvdbc ,/ lZ •g
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A[•jgbd aZ^Zq a n•abdb GZ\q ihk•[gbd ,/ EhiZlbgkvdbc |
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ISBN 966-553-359-2 |
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M a[•jgbdm ih^Zgh aZ^Zq• a n•abdb ^ey jha\’yaZggy gZ ijZd- |
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lbqgbo aZgyllyo • ^ey kZfhkl•cgh€ jh[hlb AZ]Zehf aZijhihgh\Zgh |
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aZ^Zq yd• ih^•e_gh gZ l_fb A dh`gh€ l_fb gZ\_^_gh |
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>ey klm^_gl•\ •g`_g_jgh-l_og•qgbo ki_p•Zevghkl_c \bsbo |
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ISBN 966-553-359-2
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1 F_oZgLdZ ...................................................................................... |
5 |
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1. |
D•g_fZlbdZ fZl_j•Zevgh€ lhqdb lZ ihklmiZevgh]h |
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jmom l•e ....................................................... |
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D•g_fZlbdZ h[_jlZevgh]h jmom l\_j^h]h l•eZ |
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3. |
>bgZf•dZ fZl_j•Zevgh€ lhqdb lZ ihklmiZevgh]h jmom l•e |
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Jh[hlZ • _g_j]•y ................................... |
16 |
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>bgZf•dZ h[_jlZevgh]h jmom l\_j^h]h l•eZ |
20 |
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6. |
F_oZgLqgL dheb\Zggy |
25 |
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7. |
=L^jh^bgZfLdZ ....................................... |
30 |
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11 Fhe_dmeyjgZ nLabdZ L l_jfh^bgZfLdZ .................................. |
33 |
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8. |
J•\gyggy klZgm •^_Zevgh]h ]Zam Jhaih^Le fhe_dme |
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33 |
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9. |
I_jrbc aZdhg l_jfh^bgZf•db L_ieh}fgLklv |
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37 |
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10. |
Y\bsZ i_j_g_k_ggy |
42 |
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11. |
Pbde DZjgh ?gljhiLy J_Zevg• ]Zab |
46 |
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51 |
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12. |
?e_dljhklZlbqg_ ihe_ m \Zdmmf• |
51 |
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13. |
?e_dljh}fg•klv Dhg^_gkZlhjb |
57 |
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IV. IhklLcgbc kljmf ...................................................................... |
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14. |
Hkgh\g• aZdhgb ihkl•cgh]h kljmfm ................................... |
62 |
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9 ?e_dljhfZ]g_lbaf ................................................................... |
67 |
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FZ]gLlg_ ihe_ m \Zdmmf• |
67 |
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16. |
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71 |
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17. |
Jh[hlZ ijb i_j_fLs_gg• ijh\•^gbdZ • dhglmjm |
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18. |
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77 |
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FZ]g•lg_ ihe_ \ j_qh\bg• ?g_j]•y fZ]g•lgh]h ihey ...... |
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20. |
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83 |
91 O\bevh\Z hilbdZ .................................................................... |
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89 |
22. |
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93 |
23. |
IheyjbaZpLy k\LleZ |
97 |
9,, D\Zglh\Z ijbjh^Z \bijhf•gx\Zggy .................................. |
99 |
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L_ieh\_ \bijhfLgx\Zggy |
99 |
25. |
D\Zglh\h-hilbqg• y\bsZ |
101 |
9,11 :lhfgZ nLabdZ ........................................................................ |
105 |
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26. |
L_hjLy ;hjZ ^ey ZlhfZ \h^gx |
105 |
27. |
O\be• ^_ ;jhcey |
107 |
28. |
Ki•\\•^ghr_ggy g_\bagZq_ghkl_c ............................. |
108 |
29. |
J•\gyggy Rj_^•g]_jZ |
110 |
30. |
J_gl]_g•\kvd_ \bijhf•gx\Zggy |
112 |
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115 |
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AZdhg jZ^LhZdlb\gh]h jhaiZ^m |
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117 |
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1 F?O:G,D:
D1G?F:LBD: F:L?J1:EVGH2 LHQDB
L:IHKLMI:EVGH=H JMOM L1E
Hkgh\gL nhjfmeb
DLg_fZlbqgL jL\gyggy jmom \ dhhj^bgZlgLc nhjfL x = x(t); y = y(t); z = z(t).
DLg_fZlbqg_ jL\gyggy \ ijbjh^g•c nhjfL s = s(t),
^_ s – reyoh\Z dhhj^bgZlZ
Ijh_dpL€ r\b^dhklL gZ dhhj^bgZlgL hkL
υx = |
d x |
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υy |
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d y |
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υz |
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<_ebqbgZ fh^mev fbll}\h€ r\b^dhkl•
υ= υx2 +υy2 +υz2 .
J•\gyggy nhjfmeZ reyom
S = S(t).
Fbll}\Z r\b^dLklv
υ= ddSt = S.
Reyo ijhc^_gbc l•ehf
t2
S = ∫υ(t)dt.
t1
K_j_^gy r\b^d•klv jmom
<υ >= St .
Ijh_dpL€ ijbkdhj_ggy gZ hkL dhhj^bgZl
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ax = |
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az = ddυtz =υz = z.
<_ebqbgZ fh^mev ijbkdhj_ggy
a= ax2 + ay2 + az2 .
<_ebqbgZ fh^mev lZg]_gpLZevgh€ kdeZ^h\h€ ijbkdhj_ggy
aτ = ddυt
<_ebqbgZ fh^mev ghjfZevgh€ kdeZ^h\h€ ijbkdhj_ggy
D3 = υ5
^_ R – jZ^•mk djb\bgb ljZ}dlhj•€
1 <_ebqbgZ fh^mev ih\gh]h ijbkdhj_ggy a = aτ2 + an2 .
K_j_^g} ijbkdhj_ggy
< a >= υ2 −υ1 t2 −t1
DLg_fZlbqg_ jL\gyggy jL\ghfLjgh]h ijyfheLgLcgh]h jmom
x= xo +υx t.
DLg_fZlbqg_ jL\gyggy jL\ghafLggh]h ijyfheLgLcgh]h jmom
x= xo +υoxt + ax2t2 .
1.1.>\Z Z\lhfh[•e• jmoZxlvky ih ^\ho ijyfhe•g•cgbo • \aZ}fgh i_ji_g^bdmeyjgbo ^hjh]Zo m gZijyfdZo ^h i_j_oj_kly a• klZebfb
df ]h^ • v2 df ]h^ I_j_^ ihqZldhf jmom i_jrbc Z\lhfh[•ev i_j_[m\Z\ gZ \•^klZg• S1 = df \•^ i_j_oj_kly ^jm]bc gZ \•^klZg• S2 df Q_j_a ydbc qZk i•key ihqZldm jmom \•^klZgv f•` Z\lhfh[•eyfb [m^_ f•g•-
fZevghx" ]h^)
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1.2.Jmob ^\ho fZl_j•Zevgbo lhqhd aZ^Zxlvky j•\gyggyfb x1 = A1 +
+ B1t + C1t2, x2 = A2 + B2t + C2t2 ^_ A1 f, B1 f k,
C1 = – f k2, A2 f, B2 f k, C2 f k2 < ydbc fhf_gl qZkm t r\b^dhkl• pbo lhqhd [m^mlv h^gZdh\bfb" <bagZqblb
r\b^dhkl• v1 • v2 lZ ijbkdhj_ggy .1 • .2 lhqhd m p_c fhf_gl
k f k f k2 f k2)
1.3.>\• fZl_j•Zevg• lhqdb jmoZxlvky a]•^gh a j•\gyggyfb x1 = A1 +
+ C1t2 + D1t3, x2 = B2t + C2t2 + D2t3 ^_ A1 f, C1 f k2,
D1 = – f k3, B2 f k, C2 2, D2 f k3 < ydbc fhf_gl qZkm ijbkdhj_ggy pbo lhqhd [m^mlv h^gZdh\bfb" <bagZqblb r\b^dhkl• lhqhd m p_c fhf_gl k f k f k
1.4.H^ghqZkgh a h^gh]h • lh]h kZfh]h imgdlm \ h^ghfm gZijyfdm
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ihqbgZxlv jmoZlbkv ijyfhe•g•cgh ^\Z Z\lhfh[•e• AZe_`g•klv |
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ijhc^_gh]h reyom \•^ |
qZkm hibkm}lvky j•\gyggyfb S1 = C1t2, |
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S2 = C2t2 + D2t3 ^_ C1 |
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f k2, C2 |
f k2, D2 |
f k3 AgZclb |
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\•^ghkgm r\b^d•klv Z\lhfh[•e•\ m fhf_gl qZkm W |
k. f k |
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1.5. |
AZe_`g•klv ijhc^_gh]h |
l•ehf |
reyom |
\•^ |
qZkm |
hibkm} |
j•\gyggy |
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S = Ct2 + Ht4, ^_ K |
– f k2, G f k4 <bagZqblb _dklj_- |
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fZevg_ agZq_ggy r\b^dhkl• l•eZ f k |
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1.6. |
JZd_lZ \klZgh\e_gZ \ |
fhf_gl klZjlm gZ \bkhl• G0 |
f gZ^ |
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ih\_jog_x A_fe• jmoZ}lvky \]hjm a ijbkdhj_ggyf yd_ a qZkhf |
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f
1.7.AZe_`g•klv r\b^dhkl• l•eZ \•^ qZkm aZ^Z}lvky j•\gyggyf v = A +
f k3 Ydbc reyo
k ^h t2 k" YdZ k_j_^gy r\b^d•klv <v> l•eZ • k_j_^g} ijbkdhj_ggy . ! aZ p_c ijhf•`hd qZkm" f f k f k2)
1.8.AZe_`g•klv r\b^dhkl• l•eZ \•^ qZkm hibkm} j•\gyggy v = A + Bt ^_ : f k, < f k2 Ydbc reyo ijhoh^blv l•eh aZ ijhf•`hd qZkm
\•^ t1 k ^h t2 k" <bagZqblb k_j_^gx r\b^d•klv <v> l•eZ aZ p_c ijhf•`hd qZkm f f k
7
1.9.F¶yq dbgmeb a i•^\bs_ggy \ ]hjbahglZevghfm gZijyfdm a•
r\b^d•klx v0 |
f k AgZclb r\b^d•klv v |
lZg]_gp•Zevg_ .2 • |
ghjfZevg_ .n ijbkdhj_ggy f¶yqZ q_j_a qZk W |
k i•key ihqZldm |
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jmom f k f k2; 4,45 f k2) |
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1.10. LhqdZ jmoZ}lvky ih dhem a• r\b^d•klx v = Bt |
^_ < f k2. |
I•key ihqZldm jmom \hgZ ijhoh^blv reyo ydbc ^hj•\gx} 0,2
^h\`bgb dheZ <bagZqblb ih\g_ ijbkdhj_ggy . lhqdb m p_c fhf_gl qZkm f k2)
1.11.GhjfZevg_ ijbkdhj_ggy lhqdb sh jmoZ}lvky ih dhem jZ^•mkhf
5 f, af•gx}lvky aZ aZdhghf .n = A + Bt + Ct2 ^_ : f k2, < f k3, K f k4 AgZclb lZg]_gp•Zevg_ ijbkdhj_ggy .2 lhqdb
reyo ydbc \hgZ ijhcreZ aZ qZk t1 k i•key ihqZldm jmom ih\g_ ijbkdhj_ggy . m fhf_gl qZkm t2
1.12. Jmo fZl_j•Zevgh€ lhqdb \ iehsbg• xy hibkm}lvky j•\gyggyf x = At, y = At + Bt2 ^_ : f k, < – f k2 HljbfZlb j•\gyggy
ljZ}dlhj•€ fZl_j•Zevgh€ lhqdb y(x) r\b^d•klv v lhqdb \ fhf_gl
qZkm t1 k ijbkdhj_ggy . lhqdb fhf_gl qZkm t2 m ydbc dml f•` r\b^d•klx • ijbkdhj_ggyf . Œ . (y = 0,5 + 4x2 f k
f k2 k
1.13.J•\gyggy jmom fZl_j•Zevgh€ lhqdb fZxlv \b]ey^ x = A1 + B1t,
y = B2t + C2t2, z = 0 ^_ :1 = – f, <1 f k, <2 f k,
K2 = – f k2 <bagZqblb r\b^d•klv v ijbkdhj_ggy . lZ dml f•` \_dlhjZfb pbo \_ebqbg m fhf_gl qZkm dheb dhhj^bgZlZ y gZ[m\Z} _dklj_fZevg_ agZq_ggy f k f k2; 900)
1.14. J•\gyggy jmom fZl_j•Zevgh€ lhqdb fZxlv \b]ey^ x = B1t + D1t3, y = B2t + C2t2, z = 0 ^_ <1 f k, D1 = – f k3, B2 f k,
C2 = – f k2 <bagZqblb lZg]_gp•Zevg_ .2 , ghjfZevg_ .n ijbkdh- j_ggy • jZ^•mk djb\bgb R ljZ}dlhj•€ \ fhf_gl qZkm t1 dheb dhhj-
^bgZlZ y gZ[m\Z} _dklj_fZevg_ agZq_ggy • \ fhf_gl t2 dheb lZd_ agZq_ggy gZ[m\Z} dhhj^bgZlZ x. ( f k2 f k2, 1 f f k2;
f k2 f)
1.15. Jmo fZl_j•Zevgh€ lhqdb ih dhem jZ^•mkhf 5 f hibkm} j•\gyggy S = A + Ct2 ^_ : f, K – f k2 <bagZqblb fhf_gl qZkm t dheb
8
ghjfZevg_ ijbkdhj_ggy lhqdb ^hj•\gx} .n f k2 AgZclb
r\b^d•klv v lZg]_gp•Zevg_ .2 • ih\g_ . ijbkdhj_ggy lhqdb \ p_c fhf_gl qZkm k f k f k2 f k2)
1.16. AZe_`g•klv reyom \•^ qZkm ^ey lhqdb sh jmoZ}lvky ih dhem jZ^•mkhf 5 f hibkm} j•\gyggy S = Dt3 ^_ ' f k3. <bagZqblb fhf_gl qZkm t dheb e•g•cgZ r\b^d•klv lhqdb v f k.
AgZclb ghjfZevg_ .n lZg]_gp•Zevg_ .2 • ih\g_ . ijbkdhj_ggy lhqdb \ p_c fhf_gl qZkm k f k2 f k2 f k2)
1.17.FZl_j•ZevgZ lhqdZ ihqbgZ} jmoZlbky aZ ]h^bggbdh\hx klj•edhx ih dhem jZ^•mkhf 5 f •a klZebf lZg]_gp•Zevgbf ijbkdh- j_ggyf .2 = 0 f k2 Q_j_a ydbc qZk t \_dlhj ijbkdhj_ggy . ml\hjblv a \_dlhjhf r\b^dhkl• v dml . = 600" Ydbc reyo ijhc^_ aZ p_c qZk jmohfZ lhqdZ" k f
1.18. FZl_j•ZevgZ lhqdZ jmoZ}lvky ih dhem jZ^•mkhf 5 f M fhf_gl qZkm dheb \_dlhjb ih\gh]h . • ghjfZevgh]h .n ijbkdhj_gv ml\h-
jxxlv dml . = 600 ghjfZevg_ ijbkdhj_ggy lhqdb .n |
f k2. |
<bagZqblb r\b^d•klv v • lZg]_gp•Zevg_ .2 ijbkdhj_ggy |
lhqdb |
f k f k
2.D1G?F:LBD: H;?JL:EVGH=H JMOM L<?J>H=H L1E:
Hkgh\gL nhjfmeb
DLg_fZlbqg_ jL\gyggy h[_jlZevgh]h jmom
ϕ = ϕ(t).
Fbll}\Z dmlh\Z r\b^dLklv
ω = ddϕt .
K_j_^gy dmlh\Z r\b^dLklv
<ω >= ϕ2 −ϕ1 . t2 −t1
9
<_ebqbgZ fh^mev dmlh\h]h ijbkdhj_ggy
ε= ddωt = ω = ϕ .
DLg_fZlbqg_ jL\gyggy jL\ghfLjgh]h h[_jlZevgh]h jmom
ϕ= ϕ0 + ωt.
I_jLh^ h[_jlZggy
T = 2ωπ .
QZklhlZ h[_jlZggy
n = T1 , n = 2ωπ .
ELgLcgZ r\b^dLklv
υ= ωR.
LZg]_gpLZevg_ ijbkdhj_ggy
aτ = εR.
GhjfZevg_ ijbkdhj_ggy
an = ω2 R.
2.1. |
Rd•\ jZ^•mkhf R1 |
kf a¶}^gZgbc j_f•gghx i_j_^Zq_x a• rd•- |
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jZ^•mkhf |
R2 |
kf QZklhlZ h[_jlZggy |
i_jrh]h |
rd•\Z |
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h[ k |
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2.2. |
FZoh\bd h[_jlZxqbkv j•\ghijbkdhj_gh a[•evrb\ aZ qZk W |
k |
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qZklhlm h[_jlZggy \•^ n1 = 4 h[ k ^h n2 |
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h[ k <bagZqblb |
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dmlh\_ ijbkdhj_ggy ε fZoh\bdZ • d•evd•klv h[_jl•\ N yd• \•g |
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a^•ckgb\ aZ qZk t. jZ^ k²; 18 h[_jl•\ |
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2.3. |
FZoh\bd ydbc h[_jlZ\ky a qZklhlhx n1 h[ k i•^ qZk ]Zevfm- |
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\Zggy ihqZ\ h[_jlZlbky j•\ghkih\•evg_gh • \bdhgZ\ N = 314 h[_j- |
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l•\ Dheb ]Zevfm\Zggy ijbibgbehkv qZklhlZ h[_jlZggy fZoh\bdZ |
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klZgh\beZ n2 |
h[ k <bagZqblb dmlh\_ ijbkdhj_ggy ε fZoh\bdZ • |
qZk t mijh^h\` ydh]h a^•ckgx\Zehkv ]Zevfm\Zggy jZ^ k2;
k
10