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vt, ?@8 t d" T
Å„Å‚
x0(t) =
òÅ‚S, ?@8 t e" T . (3)
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42865=8O (0@8:0 =0 MB>< CG0AB:5
xk = u(tk -Ä ) . (5)
k -1
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u
u(tk -Ä )= v tk Ò! tk = Ä , (6)
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Ä - tk = tk -Ä .
k k -1
7 ?>A;54=8E A>>B=>H5=89 =0E>48<
u u + v
Ä = 2tk -Ä = 2 Ä -Ä = Äk -1. (7)
k k -1 k -1 k -1
u - v u - v
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u + v u + v
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Ä = Ä Ò! Ä = Ä = Ä Å" 2k (8)
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k k -1 k 0 0
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u u u + v 3
ëÅ‚ öÅ‚
tk = Ä = Ä = Ä Å" Å" 2k (9)
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k -1 0 0
u - v u - v u - v 2
íÅ‚ Å‚Å‚
8 :>>@48=0BK <5AB 2AB@5G8
k -1
u u + v 3
ëÅ‚ öÅ‚
xk = vtk = vÄ = vÄ Å" Å" 2k . (10)
ìÅ‚ ÷Å‚
0 0
u - v u - v 2
íÅ‚ Å‚Å‚
2
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4 60 3000
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S 5000
t5 = Ä + = 80 + = 113 <8= . (11)
4
u 150
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S 5000
Ä = Ä + 2 = 80 + 2 = 147 <8= (12)
5 4
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L = uÄ = 150 Å"147 = 22000< = 22 :< (13)
5
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2
x = v(t3 + "t1), ?>MB><C 4>;6=> 2K?>;=OBLAO A>>B=>H5=85
u((t3 + "t1)- (Ä + "Ä1)) = v(t3 + "t1), (13)
2
87 :>B>@>3> >?@545;O5<
(u - v)(t + "t1)-Ä = 2
"Ä1 = Å" 45 - 20 = 10 <8= . (14)
3 2
u 3
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4
(0@8: 4>;6=0 1KBL @02=0
2
x3 + x4 5250< <
< u >= = = 350 . (15)
2
Ä - t3 15<8= <8=
4
!C4O ?> 4>:C<5=B0;L=>9 E@>=8:5  B0:0O A:>@>ABL (0@8:C 4>ABC?=0!
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2
x3 2250
2 2
Ä = t3 + = 45 + = 51<8= . (16)
3
< u > 350
0: 284=>, >= 1>;55 G5< =0 10 <8=CB >?>740; 8 =0 2AB@5GC A 0B@>A:8=K<, GB>
?@825;> : 5I5 >4=><C 2>A?8B0B5;L=><C <5@>?@8OB8N!
2.3 2.4 > ?;0=C $54O 4>;65= 1K; 2E>48BL 2 45@52=N 2 <><5=B 2@5<5=8 T = 100 <8= , 0
2 @50;L=>AB8 >= 2>H5; 2 =55 2 <><5=B 2@5<5=8 Ä = 113 <8= . !;54>20B5;L=>, >= >B4KE0; 2
5
B5G5=88 ?@><56CB:0 2@5<5=8
"t2 = Ä - T = 13<8= . (17)
5
45 8<5==> $54O CAB@>8; <5AB> >B4KE0 A:070BL =52>7<>6=>. >MB><C =0 3@0D8:5 70:>=0
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2
"L = 2(x3 - x3) = 2 Å"(2250 -1500) = 1500< = 1,5:< . (18)
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C@02=5=89, A;54CNI8E 87 CA;>28O
B = B1 + B2
Å„Å‚
òÅ‚A B1 = A2B2 . (1)
ół 1
5H5=85 A8AB5<K 8<55B 284
A2
B1 = B = 0,40:3
A1 + A2
. (2)
A1
B2 = B = 0,20:3
A1 + A2
>;8G5AB2> B5?;>BK Q1 , =5>1E>48<>5 4;O ?>2KH5=8O B5<?5@0BC@K A8AB5<K =0
"t1 = 1,0°! , A:;04K205BAO 87 :>;8G5AB20 B5?;>BK Q11, 84CI59 =0 ?;02;5=85 ;L40
Q11 =  Å" m1 = 132:6 (3)
8 :>;8G5AB20 B5?;>BK Q12 , 84CI53> =0 ?>A;54CNI55 =03@520=85 2>4K <0AA>9 B = m1 + m2
=0 "t1 = 1,0°! . 0AG5B 2 40==>< A;CG05 405B
Q12 = A2(m1 + m2)"t1 = 2,52:6 . (4)
!C<<0@=>5 :>;8G5AB2> B5?;>BK ?@8 40==>9 ?@>F54C@5
Q1 = Q11 + Q12 = 135:6 . (5)
!>>B25BAB25==>, :>;8G5AB2> B5?;>BK Q2 , =5>1E>48<>5 4;O ?>=865=8O B5<?5@0BC@K
A8AB5<K =0 B>B 65 3@04CA "t1 = 1,0°! A:;04K205BAO 87 :>;8G5AB20 B5?;>BK Q21, 84CI59
=0 70<>@06820=85 2>4K
Q21 =  Å" m2 = 66,0:6
8 :>;8G5AB20 B5?;>BK Q22 , =5>1E>48<>3> 4;O ?>A;54CNI53> >E;0645=8O ;L40 <0AA>9
B = m1 + m2 =0 "t1 = 1,0°!
Q22 = A1(m1 + m2)"t1 = 1,26:6 .
!C<<0@=>5 :>;8G5AB2> B5?;>BK, =5>1E>48<>5 4;O MB>3>
Q2 = Q21 + Q22 = 67,3:6 .
5
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!1 Q1
· = = = 2,0 . (6)
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B>340 A>3;0A=> C@02=5=8N B5?;>2>3> 10;0=A0
<>65< 70?8A0BL
 "Ä = (c1m1 + c2m2)"t . (7)
7 ?>A;54=53> @025=AB20 A;54C5B, GB>
<>I=>ABL =03@520B5;O
"t
 = (c1m1 + c2m2) . (8)
"Ä
"t
5;8G8=0 ?@54AB02;O5B A>1>9 C3;>2>9 :>MDD8F85=B =0:;>=0 =0G0;L=>3>
"Ä
CG0AB:0 3@0D8:0, :>B>@K9 =5A;>6=> >?@545;8BL ?> @8AC=:C
"t 2,0°! °!
= = 0,10 . (9)
"Ä 20A A
0: 284=> 87 (9), C3;>2>9 :>MDD8F85=B (B0=35=A C3;0 =0:;>=0) ?@O<>9 2 40==><
A;CG05 8<55B «M:7>B8G5A:CN @07<5@=>ABL, >?@545;O5<CN @07<5@=>ABO<8 25;8G8=,
?@82545==KE 24>;L A>>B25BAB2CNI8E >A59 :>>@48=0B.
0AG5B ?> D>@<C;5 (8) A CG5B>< 2K@065=8O (9) 405B
"t
 = (c1m1 + c2m2) = 168B = 0,17 :B . (10)
"Ä
! B0:>9 <>I=>ABLN =03@520B5;O ;54 @0A?;028BAO 70 2@5<O
m1 Å" 
Ä1 = = 786A =13 <8= . (11)
P
!>>B25BAB25==>, 2@5<O @07>3@520 A8AB5<K 4> B5<?5@0BC@K t2 = 20°! =0945< :0:
A2(B1 + B2)t2
Ä2 = = 300c = 5,0 <8= . (12)

3. CABL 2 A>AC45 =0E>48BAO <0AA0 B2 @0AB2>@8B5;O, B>340 <0AA0 @0AB2>@5==>9 A>;8
1C45B
B1 =· Å" B2 . (13)
!>>B25BAB25==>, <0AA0 =5@0AB2>@5==>9 A>;8 2 A>AC45
B3 = B - B1 = B -· Å" B2 . (14)
;O ?>;=>9 B5?;>5<:>AB8 A8AB5<K 2 40==>< A;CG05 <>65< 70?8A0BL
! = A1B1 + A2B2 + A3B3 = A1· B2 + A2B2 + A3(B -· Å" B2) . (15)
6
5@5?8H5< ?>A;54=55 @025=AB2> 2 2845
! = A2B2 + A3B +·B2(A1 - A3) .
0: A;54C5B 87 CA;>28O 7040G8, 2 40==>< ?C=:B5 A;54C5B ?@>2>48BL G8A;5==K5
@0AG5BK, 8A?>;L7CO B@8 7=0G0I85 F8D@K. >4AB02;OO 2 (15) G8A;5==K5 7=0G5=8O, ?>;CG8<
:6
C(t) = (4,32 +1,20 Å"·(t)) . (16)
°!
@0D8: ?>;CG5==>9 7028A8<>AB8
?@54AB02;5= =0 @8AC=:5.
@8 =03@520=88 A8AB5<K >B B5<?5@0BC@K
t1 = 50,0°! 4> B5<?5@0BC@K t2 = 100°!
=5>1E>48<> ?>4AG8B0BL ?;>I04L ?>4
?@82545==K< 3@0D8:>< (?;>I04L B@0?5F88).
0AG5B A B>G=>ABLN 4> B@5E 7=0G0I8E F8D@
405B
Q = 240:6 = 0,240 6 . (17)
040G0 9- 3. !:>;L65=85.
1. !> AB>@>=K AB>;0 =0 H091C 459AB2C5B A8;0 B@5=8O @02=0O
F = µmg . (1)
01>B0 MB>9 A8;K «AJ5AB :8=5B8G5A:CN M=5@38N H091K, ?>MB><C
2 2
mv0 v0
= µmgS Ò! S = . (2)
2 2µg
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2. 0?8H5< C@02=5=85 2 70:>=0 LNB>=0 4;O H091K
ma = -bv (3)
 2>A?>;L7C5<AO >?@545;5=8O<8 CA:>@5=8O 8 A:>@>AB8
"v "x
m = -b Ò! m"v = -b"x . (4)
"t "t
-B> A>>B=>H5=85 A?@0254;82> 4;O <0;KE ?@><56CB:>2 2@5<5=8, => 5A;8 ?@>AC<<8@>20BL
?> 2A5< ?@><56CB:0< 70 2A5 2@5<O 42865=8O, B> 53> <>6=> @0AA<0B@820BL 4;O ?>;=KE
87<5=5=89 A:>@>AB8 8 :>>@48=0BK, ?>MB><C
mv0
m"v = -b"v Ò! m(0 - v0 ) = -b(S - 0) Ò! S = . (5)
b
3. "0: :0: <0AAK H091
7=0G8B5;L=> <5=LH5 <0AAK 4>A:8,
B> 42865=85 4>A:8 <>6=>
@0AA<0B@820BL =57028A8<> >B
42865=8O H091. 0 ;>A:C 459AB2C5B A8;0 B@5=8O A> AB>@>=K AB>;0 F0 = 2µmg (A8;>9
7
B@5=8O A> AB>@>=K H091 A;54C5B ?@5=51@5GL 2284C <0;>AB8 8E <0AA). !;54>20B5;L=>, 4>
?>;=>9 >AB0=>2:8 4>A:0 ?@>945B ?CBL @02=K9
2
v0
S0 = . (6)
4µg
G5284=>, GB> 2A5 H091K =0G=CB 42830BLAO >B=>A8B5;L=> 4>A:8, >13>=OO 55.
0 :064CN H091C 459AB2C5B A8;0 B@5=8O F1 = µmg (=57028A8<>9 >B A:>@>AB59
4>A:8 8 A0<>9 H091K). >MB><C 2 B>9 65 A8AB5<5 >BAG5B0, A2O70==>9 A =5?>4286=>9
?>25@E=>ABLN, :0640O H0910 <>65B ?@>9B8 (5A;8 =5 A>A:>;L7=5B A 4>A:8) 4> >AB0=>2:8
?CBL @02=K9
2
v0
S1 = . (7)
2µg
! 4>A:8 A>A:>;L7=CB 2A5 H091K, :>B>@K5 =0E>48;8AL 87=0G0;L=> =0 @0AAB>O=8OE
2
v0
<5=LH8E "S = S1 - S0 = >B ?5@54=53> :@0O 4>A:8. '8A;> B0:8E H091
4µg
2
îÅ‚ Å‚Å‚
v0
n = +1. (8)
ïÅ‚4µg l śł
ðÅ‚ ûÅ‚
45AL :204@0B=K5 A:>1:8 >1>7=0G0NB F5;CN G0ABL G8A;0.
4. 0?8H5< C@02=5=85 2 70:>=0 LNB>=0 4;O H091K 2 8=5@F80;L=>9 A8AB5<5 >BAG5B0,
A2O70==>9 A =5?>4286=>9 ?>25@E=>ABLN
"v "x "x0
öÅ‚
ma = -b(v - v0 ) Ò! m = -bëÅ‚ - ÷Å‚
, (9)
ìÅ‚
"t "t "t
íÅ‚ Å‚Å‚
345 x, x0 8 v, v0 :>>@48=0BK 8 A:>@>AB8 H091K 8 4>A:8. @8<5=OO >?5@0F8N
AC<<8@>20=8O, >?8A0==CN 2 ?C=:B5 2, ?>;CG8<
m"v = -b("x - "x0 ) Ò! m(0 - v0 ) = -b(S - S0 ) Ò!
. (10)
mv0
S - S0 =
b
!;54C5B 70<5B8BL, GB> ?CBL S - S0 , ?@>945==K9 H091>9 ?> 4>A:5 , =5 7028A8B >B 70:>=0
B>@<>65=8O A0<>9 4>A:8!
'8A;> H091, :>B>@K5 A>A:>;L7=CB A 4>A:8 2 MB>< A;CG05 @02=>
mv0
îÅ‚ Å‚Å‚
n = +1. (11)
ïÅ‚ śł
bl
ðÅ‚ ûÅ‚
5. A?>;L7CO @57C;LB0B (10), ?>;CG5==K9 2 ?@54K4CI5< ?C=:B5, =0E>48<, GB> :064K9
M;5:B@>= ?@>945B ?> ?@>2>4C ?CBL @02=K9
mRÉ
L = . (12)
²
"5 M;5:B@>=K, :>B>@K5 =0E>4OBAO =0 <5=LH8E @0AAB>O=8OE >B 30;L20=><5B@0 ?@>153CB
G5@57 =53>. E G8A;> @02=>
2
Ä„d mRÉ
N = nsL = n . (13)
4 ²
=8 =5ACB 70@O4
2
Ä„d mRÉ
q = eN = ne . (14)
4 ²
8
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r
?@8@02=820O <>4C;8 <><5=B>2 A8;K BO65AB81 mg 8
r
A8;K C?@C3>AB8 63CB0 F , >B=>A8B5;L=> H0@=8@=>3>
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mgl sinÄ… = Fl cos . (1)
2
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Ä…
x = 2l sin . (2)
2
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2K@0605BAO D>@<C;>9
Ä…
F = kx = 2kl sin . (3)
2
"0:8< >1@07><, C@02=5=8O @02=>25A8O 8<55B 284
Ä… Ä…
mgl sinÄ… = 2kl sin Å" l cos . (4)
2 2
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?@82545==>5 CA;>285
mg = kl . (5)
;O >?@545;5=8O :>MDD8F85=B0 65AB:>AB8 A;54C5B 2>A?>;L7>20BLAO @57C;LB0B0<8
87<5@5=89 4;8=K 63CB0 ?@8 8725AB=>9 <0AA5 ?>425H5==>3> 3@C70.
m0 g
m0 g = kx1 Ò! k = . (6)
x1
>4AB0=>2:0 MB>3> 7=0G5=8O 2 CA;>285 (5) 405B
m0 g
m0 g = l Ò! l = x1 (?) (7)
x1
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4=0:> C@02=5=85 (4) 8<55B :>@=8, (:>B>@K5 >:070;8AL ?>B5@O==K<8 ?@8 A>:@0I5=88)
* *
sinÄ… = 0 Ò! Ä… = 0, Ä… = Ä„ . (8)
*
@8 mg < kl CAB>9G82K< 1C45B :>@5=L Ä… = 0 , ?@8 mg > kl CAB>9G82K< 1C45B :>@5=L
*
Ä… = Ä„ . @8B8G5A:>5 7=0G5=85 <0AAK, ?@8 :>B>@>< ?@>87>945B >?@>:84K20=85 =0E>48BAO
87 8AE>4=KE 40==KE
m0 g
m*g = kl = l Ò!
x1
. (9)
l
m* = m0 = 7,0:3
x1
"@51C5<K9 3@0D8: 8<55B 284, ?>:070==K9 =0
@8AC=:5.
1
45AL <K 70?8AK205< CA;>28O @02=>25A8O 4;O ?@>872>;L=>9 <0AAK ?>425H5==>3> 3@C70
9
'0ABL 2.
#@02=5=85 @02=>25A8O 2 40==>< A;CG05 1C45B 8<5BL 284
Ä…
mg Å" l sinÄ… = k(x - x0 )Å" l cos . (10)
2
A?>;L7CO ?@82545==CN 2 CA;>288 B@83>=><5B@8G5A:CN D>@<C;C, ?>;CG8<
Ä… Ä… Ä…
mg Å" 2l sin cos = k(x - x0 )Å" l cos ,
2 2 2
Ä…
0:>=5F, CGB5<, GB> 2l sin = x , B>340 >:>=G0B5;L=> ?>;CG8< C@02=5=85 @02=>25A8O
2
Ä… Ä…
mgx cos = k(x - x0 )Å" l cos . (11)
2 2
-B> C@02=5=85 8<55B :>@5=L, A>>B25BAB2CNI89 CA;>28N
Ä…
cos = 0 Ò! Ä… = Ä„ (x = 2l). (12)
2
B>@>9 :>@5=L =0E>48BAO 87 C@02=5=8O
x0 x0
mgx = kl(x - x0 ) Ò! x = = , (11)
mg m(x1 - x0 )
1- 1-
kl m0l
@8 2K2>45 8A?>;L7>20=> 2K@065=85 4;O 65AB:>AB8 63CB0
m0 g
m0 g = k(x1 - x0 ) Ò! k = . (12)
x1 - x0
;O 0=0;870 CAB>9G82>AB8 =0945==KE B>G5: @02=>25A8O <>6=> @0AA<>B@5BL
7028A8<>ABL ?>B5=F80;L=>9 M=5@388 A8AB5<K >B 4;8=K H=C@0.
>B5=F80;L=0O M=5@38O A8AB5<K 2:;NG05B:
2
k(x - x0 )
- ?>B5=F80;L=CN M=5@38N @0ABO=CB>3> 63CB0 U1 = ;
2
- ?>B5=F80;L=CN M=5@38N ?>425H5==>3> 3@C70 (>B=>A8B5;L=> 25@E=59 B>G:8 :@5?;5=8O !)
U2 = -mgz .
7 B5>@5<K 8D03>@0 4;O B@5C3>;L=8:>2 "ABD 8 "BCD A;54C5B
2 2
Å„Å‚
x2
ôÅ‚d = x2 - z
Ò! z = (13)
òÅ‚
2
2 2
2l
ôÅ‚
(l
ółd = l - - z) = 2lz - z2
"0:8< >1@07><, 2K@065=85 4;O ?>;=>9 ?>B5=F80;L=>9 M=5@388 ?@8>1@5B05B 284
2
k(x - x0 ) x2 k mg k ëÅ‚ mg öÅ‚
ëÅ‚ 2 öÅ‚x2
2
U = - mg = (x - x0 ) - x2 öÅ‚ = ìÅ‚ëÅ‚1- ÷Å‚ - 2xx0 + x0 ÷Å‚ . (14)
ìÅ‚ ÷Å‚
ìÅ‚ìÅ‚ ÷Å‚
2 2l 2 kl 2 kl
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
@0D8: MB>9 7028A8<>AB8 5ABL ?0@01>;0, A 25@H8=>9 2 B>G:5
x0
x = , GB> A>>B25BAB2C5B 2B>@><C :>@=N (11) C@02=5=8O @02=>25A8O.
mg
1-
kl
@8 mg < kl 25B28 MB>9 ?0@01>;K =0?@02;5=K 225@E, ?>MB><C ?>;>65=85 @02=>25A8O (11)
O2;O5BAO CAB>9G82K<, ?@8 1>;LH8E <0AA0E mg > kl MB> ?>;>65=85 AB0=>28BAO
=5CAB>9G82K<, CAB>9G82K< AB0=>28BAO @5H5=85 x = 2l .
0:A8<0;L=>5 7=0G5=85 <0AAK <>6=> =09B8, ?>;>682 2 D>@<C;5 (11) x = 2l :
10
x0 m(x1 - x0 )
= 2l Ò! 2l - 2 = x0 Ò!
m(x1 - x0 )
m0
1-
m0l
(15)
2l - x0
m = m0 H" 8,4:3
2(x1 - x0 )
'0ABL 3.
0?8H5< 5I5 @07 CA;>285 @02=>25A8O B@C1:8:
Ä… Ä… Ä… Ä…
mgl sinÄ… = Fl cos Ò! mg Å" 2l sin cos = Fl cos . (16)
2 2 2 2
>A;5 A>:@0I5=8O =0 :>A8=CA ?>;>28==>3> C3;0 (=5 701K20O > ?>B5@O==>< ?@8 MB><
:>@=5), ?>;CG8< C@02=5=85 4;O >?@545;5=8O x
mg
x = F(x). (16)
l
-B> C@02=5=85 <>6=> @5H8BL 3@0D8G5A:8 (<>6=> 8 0=0;8B8G5A:8), 4;O MB>3> =5>1E>48<>
1
?>AB@>8BL 3@0D8: 7028A8<>AB8 25;8G8=K F(x), 4;O MB>3> 4>AB0B>G=> «?5@525@=CBL
g
3@0D8: 7028A8<>AB8 4;8=K H=C@0 >B <0AAK ?>425H5==>3> 3@C70, ?@82545==K9 2 CA;>288
m
7040G8. 0B5< =0 MB>< 3@0D8:5 A;54C5B ?@>25AB8 A5<59AB2> ?@O<KE f (x) = x 8 =09B8
l
B>G:C 8E ?@5A5G5=8O. "0:85 ?>AB@>5=8O ?>:070=K =0 @8AC=:5.
@8 <0AA0E 3@C70, ?>425H5==>3> :@>=HB59=C, <5=LH8E m1 = 6,3:3 8<55BAO >4=>
?>;>65=85 @02=>25A8O. ;8=0 H=C@0 2 MB>< A;CG05 ;8=59=> 87<5=O5BAO >B =C;O 4> 0, 4 <.
8 <0AA0E 1>;LH8E m2 = 8,75:3 B0:65 8<55BAO >4=> ?>;>65=8O @02=>25A8O.  MB><
8=B5@20;5 4;8=0 H=C@0 ;8=59=> 87<5=O5BAO >B 1 < 4> 24 < (:>340 <0AA0 3@C70 4>AB8305B
m3 = 10,5:3 ).
11
 480?07>=5 <0AA >B m1 = 6,3:3 4>
m2 = 8,75:3 5ABL B@8 ?>;>65=8O
@02=>25A8O, F5=B@0;L=>5 87 :>B>@KE
O2;O5BAO =5CAB>9G82K<. >MB><C 2
MB>9 >1;0AB8 A8AB5<0 >1;0405B
18AB018;L=>ABLN. 0:>5 87
2>7<>6=KE ?>;>65=89 @02=>25A8O
CAB0=>28BAO 7028A8B >B ?@54K4CI8E
A>AB>O=89, A;54>20B5;L=>, ?@8
C25;8G5=88 =03@C7:8 8 ?>A;54CNI5<
55 C<5=LH5=88 1C45B =01;N40BLAO
?5B;O 38AB5@578A0 (A<. @8A).
040G0 10. 2. > :0:>9 65 AB5?5=8..?
1. 0?@O65==>ABL M;5:B@>AB0B8G5A:>3> ?>;O E1, A>740205<>3> 48?>;5< =0
1>;LH8E @0AAB>O=8OE r (r >> l) 24>;L ;8=88,
A>548=ONI59 70@O4K, =0945< ?> ?@8=F8?C
AC?5@?>78F88 ?>;59. ;O MB>3> A;54C5B ?>AB@>8BL
25:B>@=CN AC<<C =0?@O65==>AB59, A>740205<KE 2
r
B>G:5 =01;N45=8O ?>;>68B5;L=K< E+ 8
r
>B@8F0B5;L=K< E- 70@O40<8 48?>;O
r r r
E = E+ + E- .
r r
>A:>;L:C 25:B>@K E+ 8 E- =0 MB>9 ?@O<>9 ?@>B82>?>;>6=K 4@C3 4@C3C, B>
<>4C;L 8E AC<<K
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
ìÅ‚
q 1 1 ÷Å‚
E1 = E+ - E- = - . (1)
4Ä„µ0 ìÅ‚ ëÅ‚r l öÅ‚2 ëÅ‚r l öÅ‚2 ÷Å‚
ìÅ‚ ÷Å‚
+
ìÅ‚ - ÷Å‚ ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
2 2
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
A?>;L7CO D>@<C;K ?@81;865==KE 2KG8A;5=89 (1+ E)Ä… H" 1+ Ä… Å" E ?@8 <0;KE E ,
?>;CG8<
Å„Å‚Ä… = -2
üÅ‚
1 1 1 ôÅ‚ ôÅ‚ 1 l
= Å" = = (1Ä… ) . (2)
òÅ‚ l żł
2
=
r2 (1m l )2 ôÅ‚x ôÅ‚ r2 r
l
ëÅ‚r m öÅ‚
ół 2r þÅ‚
ìÅ‚ ÷Å‚
2r
2
íÅ‚ Å‚Å‚
! CG5B>< (2) 2K@065=85 (1) ?@8<5B 284
q 1 l 1 l q 2l 1 2 p
ëÅ‚
E1 = (1+ ) - (1- )öÅ‚ = Å" = Å" . (3)
ìÅ‚ ÷Å‚
4Ä„µ0 íÅ‚ r r 4Ä„µ0 4Ä„µ0
r2 r2 Å‚Å‚ r3 r3
>4AB02;OO 2 (3) 2K@065=85 4;O 48?>;L=>3> <><5=B0 p = ql A8AB5<K,
>:>=G0B5;L=> ?>;CG05<
12
1 p
E1(r) = Å" . (4)
2Ä„µ0 r3
r
! CG5B>< =0?@02;5=8O 25:B>@0 48?>;L=>3> <><5=B0 p A8AB5<K (4) <>6=>
?5@5?8A0BL 2 25:B>@=>< 2845 (2 @5H5=88 =5 B@51C5BAO)
r
r
1 p
E1 = Å" .
2Ä„µ0 r3
"0:8< >1@07><, :0: A;54C5B 87 (4), <>4C;L =0?@O65==>AB8 ?>;O E1, A>740205<>3>
48?>;5< 24>;L 53> >A8 =0 1>;LH8E @0AAB>O=8OE (r >> l) C1K205B >1@0B=>
?@>?>@F8>=0;L=> :C1C @0AAB>O=8O r 4> 48?>;O
1
E1 ~ . (5)
r3
0<5B8<, GB> 2K@065=85 (4) <>6=> ?>;CG8BL 8 B@048F8>==K< A?>A>1><, 157
8A?>;L7>20=8O ?@82545==>3> 2 CA;>288 ?@81;865==>3> @025=AB20. 59AB28B5;L=>, ?@8=O2
>1>7=0G5=8O
l
r1 = r -
2
,
l
r2 = r +
2
?5@5?8H5< (1) 2 2845
2
ëÅ‚ öÅ‚
q 1 1 q r2 - r12
ìÅ‚ ÷Å‚
E1 = E+ - E- = - = .
4Ä„µ0 ìÅ‚ 2 ÷Å‚ 4Ä„µ0 2
r12 r2 Å‚Å‚ r2 r12
íÅ‚
2
! CG5B>< B>3>, GB> r2 - r12 = 2rl , 8 2 @0<:0E ?@8=OBKE ?@81;865=89 <>6=>
2
AG8B0BL, GB> r2 Å" r12 H" r4 , ?>;CG8<
q 2rl 1 2 p
E1 = Å" = Å" .
4Ä„µ0 r4 4Ä„µ0 r3
=B5@5A=>, GB> 7028A8<>ABL, ?>4>1=0O (5), A>E@0=O5BAO 8 ?@8
=0E>645=88 =0?@O65==>AB8 ?>;O 2 «?5@?5=48:C;O@=> 48?>;N,
B.5. =0 ?@O<>9, ?5@?5=48:C;O@=>9 >A8 48?>;O 8 ?@>E>4OI59 G5@57
53> F5=B@.  MB>< A;CG05 87<5=O5BAO B>;L:> 157@07<5@=K9
:>MDD8F85=B
1 p
E2 = Å" .
4Ä„µ0 r3
1I0O :0@B8=0 A8;>2KE ;8=89 48?>;O =0 1>;LH><
@0AAB>O=88 >B =53> 40=0 =0 @8AC=:5 A?@020. 0A?>;>65=85
?>;>68B5;L=>3> 8 >B@8F0B5;L=>3> 70@O4>2 48?>;O A>>B25BAB2C5B
@0A?>;>65=8N, ?@82545==><C 2 CA;>288 7040G8.
2. !>3;0A=> 70:>=C C;>=0 ?@>1=K9 70@O4 q0 (>= O2;O5BAO
13
?>;>68B5;L=K< B>G5G=K< 70@O4><) 1C45B ?@8BO3820BLAO : B>G5G=><C 70@O4C - q1 2
20:CC<5 A A8;>9
1 q0q1
F1 = Å" . (6)
4Ä„µ0 r2
!>3;0A=> B@5BL5<C 70:>=C LNB>=0 A B0:>9
65 ?> <>4C;N A8;>9 F1 70@O4 - q1 1C45B
?@8BO3820BLAO : 70@O4C q0 .
!;54>20B5;L=>, 2 40==>< A;CG05
A?@0254;82> CB25@645=85
1
F1 ~ .
r2
"0:8< >1@07><, 2 40==>< ?C=:B5 7040G8
n =2 . (7)
r
3. @8 2708<=>< @0A?>;>65=88 48?>;O p 8 ?@>1=>3> 70@O40 q0 :0: =0 @8AC=:5
2708<>459AB285 1C45B =>A8BL E0@0:B5@ >BB0;:820=8O, ?>A:>;L:C >4=>8<5==K9
(?>;>68B5;L=K9) 70@O4 48?>;O =0E>48BAO
1;865 : ?@>1=><C (?>;>68B5;L=><C) 70@O4C,
G5< @07=>8<5==K9 (>B@8F0B5;L=K9).  B0:><
A;CG05 A8;K >BB0;:820=8O ?@5>1;040NB =04
r
A8;0<8 ?@8BO65=8O, GB> ?@82>48B : 2>7=8:=>25=8N @57C;LB8@CNI59 A8;K F2 ,
=0?@02;5==>9 2?@02> (A<. @8A).
!;54>20B5;L=>, 2 40==>< A;CG05 48?>;L 1C45B >BB0;:820BL ?@>1=K9 70@O4 24>;L
A2>59 >A8 A A8;>9
q0 p
F2 = q0E1(r) = {(4)}= Å" . (8)
2Ä„µ0 r3
r
0<5B8<, GB> ?@8 87<5=5=88 >@85=B0F88 48?>;O ( - p ), E0@0:B5@ 2708<>459AB28O
48?>;O A ?@>1=K< 70@O4>< 87<5=8BAO A >BB0;:820=8O =0 ?@8BO65=85.
!;54>20B5;L=>, 2 40==>< A;CG05 A?@0254;82> CB25@645=85
1
F2 ~ .
r3
"0:8< >1@07><, 2 40==>< ?C=:B5 7040G8
n =3 . (9)
r
4. ;O >?@545;5==>AB8 1C45< AG8B0BL, GB> ;52K9 48?>;L p A>7405B =0?@O65==>ABL
?>;O 2 ?@>AB@0=AB25, :>B>@>5 459AB2C5B
r
=0 ?@02K9 48?>;L - p .
>A:>;L:C >4=>8<5==K5 70@O4K
48?>;59 =0E>4OBAO =0 <5=LH8E
14
@0AAB>O=8OE, G5< @07=>8<5==K5, B> A>3;0A=> 70:>=C C;>=0 A8;0 >BB0;:820=8O 2 A8AB5<5
1C45B 1>;LH5 A8;K ?@8BO65=8O.
r
-B> ?@82545B : 2>7=8:=>25=8N @57C;LB8@CNI59 A8;K F3 >BB0;:820=8O <564C
48?>;O<8, =0?@02;5==>9 2?@02> (A<. @8A.).
A?>;L7CO (4), 4;O @0AA<0B@8205<>9 A8AB5<K <>65< 70?8A0BL
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
l l q p 1 1
ìÅ‚ ÷Å‚
F3 = qëÅ‚ E1(r - ) - E1(r + )öÅ‚ = - . (10)
ìÅ‚ ÷Å‚
2 2 2Ä„µ0 ìÅ‚ (r - l )3 (r + l )3 ÷Å‚
íÅ‚ Å‚Å‚
ìÅ‚ ÷Å‚
íÅ‚ 2 2 Å‚Å‚
! ?><>ILN D>@<C;K ?@81;865==>3> 2KG8A;5=8O (1+ E)Ä… H" 1+ Ä… Å" E (?@8 E 0 )
2 40==>< A;CG05 =0E>48<
Å„Å‚Ä… = -3
üÅ‚
1 1 1 ôÅ‚ ôÅ‚ 1 3 l
= Å" = = (1Ä… ) .
òÅ‚ l żł
3
=
r3 (1m l )3 ôÅ‚x ôÅ‚ r3 2 r
l
ëÅ‚r m öÅ‚
ół 2r þÅ‚
ìÅ‚ ÷Å‚
2r
2
íÅ‚ Å‚Å‚
!>>B25BAB25==>, @07=>ABL 2 2K@065=88 (10) ?@8<5B 284
1 1 3l
- H" .
l l
(r - )3 (r + )3 r4
2 2
! CG5B>< B>3>, GB> p = ql , >:>=G0B5;L=> ?>;CG8<
3p2 1
F3 = Å" . (11)
2Ä„µ0 r4
!;54>20B5;L=>, 2 40==>< A;CG05 A?@0254;82> CB25@645=85
1
F3 ~ .
r4
"0:8< >1@07><, 2 40==>< ?C=:B5 7040G8
n =4 . (12)
0<5B8<, GB> 2K@065=85 (11) ?@>I5 ?>;CG8BL, 8A?>;L7CO ?@>872>4=CN >B (4) ?>
@0AAB>O=8N r , ?>A:>;L:C 8A:><0O A8;0 F3 2KG8A;O5BAO :0:
"E
F3 = - p ,
"r
>4=0:> ?>4>1=K9 ?>4E>4 2KE>48B 70 @0<:8 459AB2CNI59 H:>;L=>9 ?@>3@0<<K.
15
5. C45< AG8B0BL, GB> =0?@O65==>ABL E (r) M;5:B@>AB0B8G5A:>3> ?>;O,
A>740205<>3> B>G5G=K< 70@O4>< q0 2 >1;0AB8 =0E>645=8O <>;5:C;K,
<5=O5BAO =57=0G8B5;L=> (2 A8;C <0;>AB8 @07<5@>2 <>;5:C;K) 8 @02=0
1 q0
E (r) = Å" .
4Ä„µ0 r2
">340 8=4CF8@>20==K9 48?>;L=K9 <><5=B <>;5:C;K ?@8<5B 7=0G5=85
1 q0 Ä… q0
pi = Ä…µ0E(r) = Ä…µ0 Å" Å" = Å" .
4Ä„µ0 r2 4Ä„
r2
 ?C=:B5 3 7040G8 <K 2KG8A;8;8 A8;C 2708<>459AB28O 48?>;O 8 B>G5G=>3> 70@O40,
r
?@0240 2 @0AA<0B@8205<>< A;CG05 8=4CF8@>20==K9 (=02545==K9) 48?>;L=K9 <><5=B pi
8<55B ?@>B82>?>;>6=>5 =0?@02;5=85 («>B 70@O40), GB> ?@82545B : 2>7=8:=>25=8N 2
40==>9 A8AB5<5 A8;K ?@8BO65=8O.
A?>;L7CO (8) 8 B@5B89 70:>=
LNB>=0, =0945< A8;C ?@8BO65=8O
2K=C645==> ?>;O@87>20==>9 <>;5:C;K :
B>G5G=><C 70@O4C
2
q0 pi q0 1 Ä… q0 Ä… q0 1
F4 = Å" = Å" Å" = Å" (13)
2
2Ä„µ0 2Ä„µ0 4Ä„
r3 r3 r2 8Ä„ µ0 r5
!;54>20B5;L=>, 2 40==>< A;CG05 A?@0254;82> CB25@645=85
1
F4 ~ .
r5
"0:8< >1@07><, 2 40==>< ?C=:B5 7040G8
n =5 . (14)
6. C45< AG8B0BL, GB> =0?@O65==>ABL
M;5:B@>AB0B8G5A:>3> ?>;O, A>740205<>3>
r
48?>;5< p 2 >1;0AB8 =0E>645=8O
<>;5:C;K, <5=O5BAO =57=0G8B5;L=>. ">340
A>3;0A=> (4) <>65< 70?8A0BL
1 p
E(r) = Å" .
2Ä„µ0 r3
!>>B25BAB25==>, 8=4CF8@>20==K9 48?>;L=K9 <><5=B <>;5:C;K 2 MB>< A;CG05
?@8<5B 7=0G5=85
1 p Ä… p
pi = Ä…µ0 Å" Å" = Å" . (15)
2Ä„µ0 r3 2Ä„
r3
 ?C=:B5 4 7040G8 <K 2KG8A;8;8 A8;C >BB0;:820=8O <564C 48?>;O<8,
>@85=B8@>20==K<8 «=02AB@5GC 4@C3 4@C3C.  40==>< A;CG05 48?>;8 >@85=B8@>20=K 2
16
>4=>< =0?@02;5=88, ?>A:>;L:C =02545==K9 48?>;L=K9 <><5=B 2A5340 >@85=B8@>20= «?>
?>;N. -B> >1AB>OB5;LAB2> ?@82545B : 2>7=8:=>25=8N A8;K ?@8BO65=8O <564C 48?>;O<8
2 @0AA<0B@8205<>9 A8AB5<5.
!G8B0O 2708<>459AB285 48?>;59 ?> D>@<C;5 (11), ?>;CG8< 25;8G8=C A8;K
?@8BO65=8O <564C =8<8 4;O @0AA<0B@8205<>3> A;CG0O
3p Å" pi 1 3 p Ä… p 1 3Ä… p2 1
F5 = Å" = Å" Å" = Å" . (16)
2
2Ä„µ0 2Ä„µ0 2Ä„
r4 r3 r4 4Ä„ µ0 r7
!;54>20B5;L=>, 2 40==>< A;CG05 A?@0254;82> CB25@645=85
1
F5 ~ .
r7
"0:8< >1@07><, 2 40==>< ?C=:B5 7040G8
n =7 . (17)
=B5@5A=>, GB> 2 ?C=:B0E 3 8 4 7040G8 8<55B <5AB> A8;0 >BB0;:820=8O <564C
>1J5:B0<8, 0 2 ?C=:B0E 5 8 6, A2O70==KE A
8=4CF8@>20==K< 48?>;L=K< <><5=B><, 2A5340 8<55B
<5AB> A8;0 ?@8BO65=8O.
-B> A2O70=> A B5<, GB> =0?@02;5=85
=0?@O65==>AB8 2=5H=53> M;5:B@8G5A:>3> ?>;O
>4=>7=0G=> >?@545;O5B =0?@02;5=85 2>7=8:0NI53>
r
48?>;L=>3> <><5=B0 p ?>;O@87C5<>3> >1J5:B0  >=
2A5340 >@85=B8@>20= «?> A8;>2>9 ;8=88 2=5H=53>
(8=4CF8@CNI53>) ?>;O.
 7025@H5=85 7040G8 =0 @8AC=:5 A?@020
?@82545=0 4>AB0B>G=> 70102=0O (=> 25@=0O!)
7028A8<>ABL ?>:070B5;O AB5?5=8 n 2 2K@065=88 4;O
A8;K 2708<>459AB28O Fi >B =><5@0 N ?C=:B0 7040G8.
0: 3>2>@8BAO, :><<5=B0@88 87;8H=8, => B0: 8 =5 ?>=OB=>, ?>G5<C ?>A;54=OO
B>G:0 =5 ;>68BAO «=0 ?@O<CN. K;> 1K B0: :@0A82>&
17
040G0 10-3 «0@>0B<>AD5@=0O <0H8=0 LN:><5=0
1>7=0G8< 402;5=85 ?0@0 2 @01>G5< F8;8=4@5
P1, 0 402;5=85 2>4K =0 ?>@H5=L F8;8=4@0 =0A>A0 P2 .
"0: :0: ?>@H=8 4286CBAO @02=><5@=>, B> AC<<0 A8;,
459AB2CNI8E =0 =8E @02=0 =C;N. @><5 B>3>, B0: :0:
:>@><KA;> C@02=>25H5=>, A8;K =0BO65=89 B@>A>2,
?@8:@5?;5==KE : ?>@H=O< @02=K. A;8 ?>@H5=L
@01>G53> F8;8=4@0 =0E>48BAO =0 2KA>B5 x , B> ?>@H5=L
=0A>A0 =0E>48BAO =0 2KA>B5 h - x . K@078< MB8
CA;>28O 2 C@02=5=8OE
F = (P0 - P2 )S2
Å„Å‚ S2
(P0 )
òÅ‚F = - P1)S1 Ò! P1 = P0 - - P2 . (1)
(P0 S1
ół
G5284=>, GB>
P0 - P2 = Ág(h - x) (2)
@8 >?CA:0=88 ?>@H=O =0A>A0, 8
P0 - P2 = Ág(H + h - x) (3)
@8 53> ?>4J5<5.
"0:8< >1@07><, 7028A8<>AB8 402;5=8O >B ?>;>65=8O ?>@H=O @01>G53> =0A>A0 8<5NB 284:
- ?@8 ?>4J5<5 ?>@H=O @01>G53> F8;8=4@0
P1 Ágh S2
ëÅ‚1- x
öÅ‚
2
P1 = P0 - Ág(h - x)S Ò! = 1- ; (4)
ìÅ‚ ÷Å‚
S1 P0 P0 S1 íÅ‚ h
Å‚Å‚
- ?@8 53> >?CA:0=88
S2 P1 Ágh S2 H
ëÅ‚1+ - x
öÅ‚
P1 = P0 - Ág(H + h - x) Ò! = 1- . (5)
ìÅ‚ ÷Å‚
S1 P0 P0 S1 íÅ‚ h h
Å‚Å‚
1>7=0G8< ?0@0<5B@ A8AB5<K (8 2KG8A;8< 53> G8A;5==>5 7=0G5=85):
2
S2
Ágh 1,0 Å"103 Å"10 Å"1 20
ëÅ‚ öÅ‚
b = = = 1,11Å"10-2 . (6)
ìÅ‚ ÷Å‚
P0 S1 1,0 Å"105 60
íÅ‚ Å‚Å‚
"0:8< >1@07><, 2 «157@07<5@=KE ?0@0<5B@0E 7028A8<>AB8 402;5=8O ?0@0 >B 2KA>BK
?>4J5<0 8<5NB 284:
P1 x
Å„Å‚
= (1- b)+ b
ôÅ‚
P0 h
ôÅ‚
. (7)
òÅ‚
P1 ëÅ‚ H öÅ‚ x
öÅ‚÷Å‚
ôÅ‚
= ìÅ‚1- bëÅ‚1+ + b
ìÅ‚ ÷Å‚÷Å‚
ôÅ‚
P0 ìÅ‚ íÅ‚ h h
Å‚Å‚
íÅ‚ Å‚Å‚
ół
"5?5@L =5 ?@54AB02;O5B B@C40 ?>AB@>8BL
4803@0<<C F8:;8G5A:>3> ?@>F5AA0.
7 @0AA<>B@5=8O ?@>F5AA0 A;54C5B, GB>
<0:A8<0;L=0O B5<?5@0BC@0 ?0@0 2 F8:;5
@02=0 tmax = 100°C . 8=8<0;L=0O
B5<?5@0BC@0 @02=0 B5<?5@0BC@5 :8?5=8O
?@8 <8=8<0;L=>< 402;5=88 2 F8:;5, B.5.
?@8
ëÅ‚ H öÅ‚
öÅ‚÷Å‚
Pmin = P0ìÅ‚1- bëÅ‚1+ = 0,92Å"105 0 . (7)
ìÅ‚ ÷Å‚÷Å‚
ìÅ‚
h
íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
18
;O >?@545;5=8O A>>B25BAB2CNI59 B5<?5@0BC@K =5>1E>48<> 2>A?>;L7>20BLAO
?@54;0305<>9 2 CA;>288 ;8=50@870F859 7028A8<>AB8 B5<?5@0BC@K :8?5=8O >B 402;5=8O.
"0: 87<5=5=85 B5<?5@0BC@K ?@8 548=8G=>< 87<5=5=88 402;5=8O @025=
"t 110 -100 
= = 25 Å"10-5 . (8)
"P (1,4 -1,0) Å"105 0
!;54>20B5;L=>, 8A:><0O B5<?5@0BC@0 1C45B @02=0
"t
ëÅ‚ öÅ‚
(P0
tmin = t0 - ìÅ‚ ÷Å‚ - Pmin ) = 100°C - 25 Å"10-5 Å" 0,08 Å"105 = 98°C . (9)
"P
íÅ‚ Å‚Å‚
>;57=CN @01>BC, D0:B8G5A:8, A>25@H05B 0B<>AD5@=>5 402;5=85! 01>B0, A>25@H5==0O 70
F8:;, >B<5G5=0 =0 4803@0<<5 70;82:>9. '8A;5==>5 7=0G5=85 MB>9 @01>BK
d12
A = P0 Å" (1- 0,93) Å"Ä„ h = 2,0 Å"103 6 . (10)
4
"5?;>B0 >B =03@520B5;O ?>ABC?05B =0 8A?0@5=85 2>4K (?@0:B8G5A:8 ?@8 0B<>AD5@=><
402;5=88), 4;O 70?>;=5=8O :>B;0. 0:A8<0;L=0O <0AA0 ?0@0 2 @01>G5< F8;8=4@5
=0E>48BAO 87 C@02=5=8O A>AB>O=8O
m MPV 18 Å"10-3 Å"105 Å"Ä„ Å" 0,32 Å"1
PV = RT Ò! m = = H" 1,6 Å"10-1:3 .
M RT 8,31Å" 373
(11)
>;8G5AB2> B5?;>BK, B@51C5<>5 =0 8A?0@5=85,
Q = Lm = 3,7 Å"105 6 . (12)
"5?5@L <>6=> @0AAG8B0BL  <0H8=K
A
· = = 5,4 Å"10-3 . (13)
Q
0 =03@520=85 :>B;0 B@51C5BAO :>;8G5AB2> B5?;>BK
Q1 = c1m1"t = 0,46 Å"103 Å" 200 Å" 2 = 1,8Å"105 6 . (14)
!;54>20B5;L=>, A CG5B>< B5?;>5<:>AB8 :>B;0,  ?>=8605BAO 4>
A
2
· = = 3,6 Å"10-3 . (15)
2
Q + Q
0:>=5F,  <0H8=K, @01>B0NI59 ?> F8:;C 0@=> 2 B>< 65 480?07>=5 B5<?5@0BC@,
@025=
"T 2
·0 = = H" 5,4 Å"10-3 . (16)
T0 373
GB>, :0: MB> =5 C4828B5;L=>, A>2?0405B A @0=55 ?>;CG5==K< @57C;LB0B><.
19
040G0 11-1. >?;02>:
1. !8;0 BO65AB8, ?@8;>65=0 : F5=B@C <0AA AB5@6=O C , :>B>@K9
l
=0E>48BAO 2 53> A5@548=5 =0 @0AAB>O=88 >B 53> :>=F0. !8;0
2
@E8<540 ?@8;>65=0 : F5=B@C <0AA 2KB5A=5==>9 2>4K B>G:5 P ,
x
:>B>@0O =0E>48BAO =0 @0AAB>O=88 >B 53> :>=F0 ( x - 3;C18=0
2
?>3@C65==>9 G0AB8 AB5@6=O). "0: :0: ?>?;02>: ?;0205B, B> x < l ,
?>MB><C B>G:0 ?@8;>65=8O A8;K BO65AB8 =0E>48BAO 2KH5 B>G:5
?@8;>65=8O A8;K @E8<540. >MB><C 25@B8:0;L=>5 ?>;>65=85
AB5@6=O =5CAB>9G82>  ?@8 <0;59H5< A;CG09=>< >B:;>=5=88 >B
25@B8:0;8 2>7=8:05B <><5=B A8;, AB@5<OI89AO >?@>:8=CBL
?>?;02>: (A<. @8A.)
2-4. 0 ?>?;02>: 459AB2CNB:
r
- A8;0 BO65AB8 m1g ;
r
r m1 r
x
- A8;0 @E8<540 FA1 = -Á0Vx g = -Á0 g ;
Á1 l
r
=0 3@C78;> B0:65 459AB2CN A8;0 BO65AB8 m2 g 8 A8;0 @E8<540
r
r m2 r
FA1 = -Á0V2g = -Á0 g . "0: :0: 25:B>@=0O AC<<0 MB8E A8; C =0A 1C45B
Á2
G0AB> 2AB@5G0BLAO 2 40;L=59H5<, B> >1>7=0G8< 55
r
r r Á2 - Á0 r
µg = m2 g + FA2 = m2 g
Á2
#A;>285 @02=>25A8O ?>?;02:0 70?8AK205BAO 2 2845 A;54CNI53>
C@02=5=8O (AC<<0 A8; @02=0 =C;N):
m1 x
µg + m1g = Á0 g . (1)
Á1 l
7 MB>3> C@02=5=8O =0E>48< 4;8=C ?>3@C65==>9 G0AB8 ?>?;02:0
ëÅ‚ öÅ‚
Á1 ìÅ‚ µ
÷Å‚
x = l Å" . (2)
Á0 ìÅ‚1+ m1 ÷Å‚
íÅ‚ Å‚Å‚
0:A8<0;L=>9 <0AA5 3@C78;0 A>>B25BAB2C5B ?>3@C65=85 =0 2AN 4;8=C ?>?;02:0.  MB><
A;CG05 8E D>@<C;K (2) A;54C5B
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
Á1 ìÅ‚ µ Á0 Á2 ìÅ‚ Á0
÷Å‚ ìÅ‚ ÷Å‚ ÷Å‚
= 1 Ò! µ = m1ìÅ‚ -1÷Å‚ Ò! m2 max = m1 -1÷Å‚ . (3)
Á0 ìÅ‚1+ m1 ÷Å‚ Á1 Á2 - Á0 ìÅ‚ Á1 Å‚Å‚
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚ íÅ‚
20
5. 0 @8AC=:5 87>1@065=K A8;K, 459AB2CNI85 =0 ?>?;02>: A
3@C78;><, 2 B>< A;CG05. >340 ?>?;02>: >B:;>=8;AO =0 C3>; Ä… .
B>@>5 CA;>285 @02=>25A8O (@025=AB2> =C;N AC<<K <><5=B>2 A8;)
<>6=> 70?8A0BL >B=>A8B5;L=> B>G:8 C - F5=B@0 <0AA ?>?;02:0
l l x
öÅ‚sinÄ… . (3)
µg sinÄ… = FA1ëÅ‚ - ÷Å‚
ìÅ‚
2 2 2
íÅ‚ Å‚Å‚
-B> CA;>285 >G5284=> 2K?>;=O5BAO ?@8 Ä… = 0 , B.5. ?@8
25@B8:0;L=>< ?>;>65=88 ?>?;02:0. AA;54C5< 2>7<>6=>AB8
F = (µ + m1 )g
4@C38E @5H5=89. ! CG5B>< B>3>, 8 ?>A;5
A1
A>:@0I5=8O =0 A8=CA C3;0 >B:;>=5=8O, ?>;CG8<
x
öÅ‚
. (4)
µ = (µ + m1)ëÅ‚1- ÷Å‚
ìÅ‚
l
íÅ‚ Å‚Å‚
7 MB>3> C@02=5=8O ?>;CG8< 2K@065=85 4;O 4;8=K ?>3@C65==>9
G0AB8:
x x m1
öÅ‚
µ = (µ + m1)ëÅ‚1- ÷Å‚
Ò! = , (5)
ìÅ‚
l l µ + m1
íÅ‚ Å‚Å‚
:>B>@>5 2 >1I5< A;CG05 =5 A>2?0405B A 2K@065=85< (2), ?>;CG5==K< 87 CA;>28O
@02=>25A8O A8;.
!;54>20B5;L=>, 4@C38E ?>;>65=89 @02=>25A8O, :@><5 25@B8:0;L=>3> 8;8 3>@87>=B0;L=>3>
=5B!
48=AB25==K9, 8A:;NG8B5;L=K9 20@80=B ?@8 :>B>@>< ?>?;02>: =0E>48BAO 2 @02=>25A88
?@8 ;N1>< C3;5 =0:;>=0 <>6=> =09B8, ?@8@02=820O @5H5=8O (2) 8 (5):
2
ëÅ‚
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
Á1 ìÅ‚ µ m1 µ Á0 Á0 öÅ‚
÷Å‚ ìÅ‚ ÷Å‚
= Ò! = Ò! µ = m1ìÅ‚ -1÷Å‚ .
ìÅ‚
Á0 ìÅ‚1+ m1 ÷Å‚ µ + m1 ìÅ‚1+ m1 ÷Å‚ Á1 Á1 ÷Å‚
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
#G8BK20O >?@545;5=85 µ , ?>;CG8<
Á2 - Á0 ëÅ‚ Á0 öÅ‚ Á2 ëÅ‚ Á0 öÅ‚
" "
ìÅ‚
m2 = m1ìÅ‚ -1÷Å‚ Ò! m2 = m1 -1÷Å‚ . (6)
ìÅ‚
Á2 Á1 ÷Å‚ Á2 - Á0 ìÅ‚ Á1 ÷Å‚
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
>?;02>: 709<5B 25@B8:0;L=>5 ?>;>65=85 ?@8
l l x
öÅ‚
. (7)
µg > FA1ëÅ‚ - ÷Å‚
ìÅ‚
2 2 2
íÅ‚ Å‚Å‚
*
-B> =5@025=AB2> 2K?>;=O5BAO ?@8 m2 > m2
6. @8 25@B8:0;L=>< ?>;>65=88 ?>?;02:0 A8;0 BO65AB8
C@02=>25H8205BAO A8;>9 @E8<540, ?>MB><C
FA1 = µg + m1g . (8)
7 MB>3> C@02=5=8O ;53:> =09B8 4;8=C ?>3@C65==>9 G0AB8 ?>?;02:0
FA1 = µg + m1g Ò! Ä„R2 x0Á0 g = µg + m1g Ò!
(9)
µ + m1
x0 =
Ä„R2Á0
7. !8;K, 459AB2CNI85 =0 ?>?;02>: 2 =0:;>==>< ?>;>65=88,
87>1@065=K =0 @8AC=:5.
21
"0: :0: 25;8G8=0 x0 =5 87<5=O5BAO, B>, 2>-?5@2KE, >=0 >?@545;O5BAO D>@<C;>9 (9), 2>-
2B>@KE, 87 35><5B@88 A8AB5<K A;54C5B, GB>
x0 = x + R tgÄ… . (10)
0AAG8B05< <>4C;8 A8; 8 8E <><5=BK:
!8;0 BO65AB8 ?>?;02:0:
F1 = m1g
l
öÅ‚sinÄ… (11)
M1 = m1gëÅ‚ x0 - ÷Å‚
ìÅ‚
2
íÅ‚ Å‚Å‚
!C<<0 A8;, 459AB2CNI8E =0 3@C78;>:
F2 = µg
(12)
M = µg Å" x0 sinÄ…
2
!8;0 @E8<540, 459AB2CNI0O =0 =86=NN F8;8=4@8G5A:CN G0ABL AB5@6=O
FA1 = Ä„R2Á0gx = Ä„R2Á0g(x0 - R Å"tgÄ…)
x
öÅ‚sinÄ… = (12)
M = Ä„R2Á0g(x0 - R Å"tgÄ…)Å"ëÅ‚ + R Å"tgÄ… ÷Å‚
ìÅ‚
A1
2
íÅ‚ Å‚Å‚
1
2
= Ä„R2Á0g (x0 - R2 Å"tg2Ä…)sinÄ…
2
!8;0 @E8<540, 459AB2CNI0O =0 «A@570==CN G0ABL
2
FA = Ä„R3Á0g Å"tgÄ…
R4 3 1 3
öÅ‚ öÅ‚sinÄ… = (13)
2 2
M = FA Å" d0 = Ä„ Á0g Å"tgÄ… Å"ëÅ‚5cosÄ… - ÷Å‚ ìÅ‚
= Ä„R2Á0g R2ëÅ‚5 - ÷Å‚
ìÅ‚
A
8 cosÄ… 8 cos2 Ä…
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
1
= Ä„R2Á0g R2(2 - 3tg2Ä…)sinÄ…
8
#A;>28O @02=>25A8O 70?8AK20NBAO 2 8455 42CE C@02=5=89:
!C<<0 A8; @02=0 =C;N:
µg + m1g = Ä„R2 xÁ0 g + Ä„R3Á0 g Å" tgÄ… . (14)
-B> CA;>285 2K?>;=O5BAO ?@8 2K?>;=5=88 A>>B=>H5=8O (10).
2
!C<<0 <><5=B>2 A8; @02=0 =C;N (>B=>A8B5;L=> B>G:8 O )
2
M1 + M + M - M = 0 (15)
2 A A1
>4AB028< 2 MB> C@02=5=85 =0945==K5 <><5=BK A8;.
"0: :0: 2A5 >=8 ?@>?>@F8>=0;L=K A8=CAC C3;0 =0:;>=0, B> =0 =53> A@07C A>:@0B8<,
=5 701K20O > :>@=5 Ä…0" = 0 C@02=5=8O (15). B0:, ?>A;5 A>:@0I5=8O ?>;CG05<
l 1 1
öÅ‚
2 2 2
m1ëÅ‚ x0 - ÷Å‚
+ µ Å" x0 + Ä„R2Á0 R2(2 - 3tg Ä…)- Ä„R2Á0 (x0 - R2 Å"tg Ä…)= 0 (16)
ìÅ‚
2 8 2
íÅ‚ Å‚Å‚
-B> C@02=5=85 C4>1=> @0745;8BL =0 25;8G8=C Ä„R2Á0 , B>340 ?>A;5 M;5<5=B0@=KE
?@5>1@07>20=89 ?>;CG8< C@02=5=85
22
1 1 Á1 2 1 1
2 2
x0 - l + R2 + R2 Å" tg Ä… = 0. (17)
2 2 Á0 4 8
m1 Á1
 ?@5>1@07>20=8O 8A?>;L7>20=> A>>B=>H5=85 = l .
Ä„R2Á0 Á0
#@02=5=85 (17) 8<55B @5H5=8O (0 A;54>20B5;L=> ?>?;02>: 4@C385 ?>;>65=8O @02=>25A8O)
?@8 2K?>;=5=88 CA;>28O
ëÅ‚ öÅ‚
Á1 2 2 ÷Å‚
ìÅ‚
2ìÅ‚ l - x0 ÷Å‚ - R2 > 0 (18)
Á0
íÅ‚ Å‚Å‚
 MB> @5H5=85 5ABL
Á1 2 2
l - x0
Á0
"
tg Ä… = 4 - 2R2 . (19)
R2
>6=> ?>:070BL, GB> ?@8 =0;8G88 @5H5=8O (19), MB> ?>;>65=85 @02=>25A8O CAB>9G82>, 0
?@8 25@B8:0;L=>5 =5CAB>9G82>. A;8 CA;>285 (18) =5 2K?>;=O5BAO, B> 548=AB25==K< 8
CAB>9G82K< ?>;>65=85< @02=>25A8O
O2;O5BAO 25@B8:0;L=>5.
>:070B5;LAB2> MB>3> CB25@645=8O
<>65B 1KBL ?@>2545=> @07;8G=K<8
A?>A>10<8. 0?@8<5@, ?> 0=0;87C
?>B5=F80;L=>9 :@82>9. "0:, =5
A;>6=> ?>:070BL, GB> ?>B5=F80;L=0O
M=5@38O A8AB5<K ?@>?>@F8>=0;L=0
A;54CNI59 DC=:F88 C3;0 =0:;>=0
2
U " (tg Ä… + b)cosÄ… (20)
ëÅ‚ öÅ‚
Á1 2 2
ìÅ‚ ÷Å‚
4ìÅ‚ l - x0 ÷Å‚ - 2R2
Á0
íÅ‚ Å‚Å‚
345 >1>7=0G5=> b = . 0 @8AC=:5 ?>:070= 284 MB>9 DC=:F88 ?@8
R2
?>A;54>20B5;L=>< C25;8G5=88 ?0@0<5B@0 b >B -2 4> 6. @8 >B@8F0B5;L=KE 7=0G5=8OE
MB>3> ?0@0<5B@0 ?>B5=F80;L=0O :@820O 8<55B 548=AB25==K9 <8=8<C< (2 =C;5), :>340 65
MB>B ?0@0<5B@ AB0=>28BAO ?>;>68B5;L=K<, M:AB@5<C< 2 =C;5 AB0=>28BAO <0:A8<C<><, =>
?>O2;ONBAO 420 <8=8<C<0 21;878 3>@87>=B0;L=>3> ?>;>65=8O.
040G0 11-2 $>B>M;5<5=B.
'0ABL 1. 450;L=K9 D>B>M;5<5=B
1.1 ">: 2 =03@C7:5 @025= @07=>AB8 D>B>B>:0 8 B>:0 B5:CI53> G5@57 48>4:
(1).
0?@O65=85 =0 =03@C7:5 B0:>5 65, :0: 8 =0 48>45. >4AB02;OO 8 ,
?>;CG8< :204@0B=>5 C@02=5=85:
(2).
5H5=85 8<55B 284:
(3).
">: 2 =03@C7:5:
23
 (4).
1.2 >4AB02;OO 2 (1) , ?>;CG8< A2O7L <564C B>:>< 8 =0?@O65=85<:
(5).
1.3 @8 =C;52>< A>?@>B82;5=88, =0?@O65=85 =0 =03@C7:5 @02=> =C;N.  ?@8 1>;LH><
A>?@>B82;5=88 =03@C7:8  B>: @025= =C;N. AE>4O 87 (5):
(6),
(7).
1.4 @0D8: 7028A8<>AB8  25B2L ?0@01>;K. 5@H8=0 8<55B :>>@48=0BK ,
B>G:0 ?5@5A5G5=8O A >ALN 01AF8AA  .
1.5 #<=>68< >15 G0AB8 @025=AB20 (5) =0 :
(8).
@8@02=O2 : =C;N ?@>872>4=CN ?> =0?@O65=8N, ?>;CG8<:
(9).
!>>B25BAB25==>:
(10),
(11),
(12).
1.6 A?>;L7CO G8A;5==K5 7=0G5=8O, ?>;CG8<:
(13),
(14),
(15),
(16),
(17),
(18).
'0ABL 2. >B5@8 M=5@388 2 D>B>M;5<5=B5
2.1 $>B>B>: @0745;O5BAO =0 B@8 G0AB8: B>: 48>40 , B>: G5@57 ?0@0;;5;L=>5 A>548=5=85
8 B>: G5@57 ?>A;54>20B5;L=>5 A>?@>B82;5=85 8 =03@C7:C :
(19).
0?@O65=85 =0 48>45 8 ?0@0;;5;L=>< A>?@>B82;5=88, <>6=> 2K@078BL G5@57 B>: 2
=03@C7:5:
(20).
">340:
(21),
(22).
>4AB02;OO MB8 2K@065=8O 2 (19) ?>;CG8< :204@0B=>5 C@02=5=85 >B=>A8B5;L=> :
(23).
5H5=85 C@02=5=8O:
24
 (24).
0?@O65=85 =0 =03@C7:5 .
2.2 <5AB> C@02=5=8O (20) 70?8H5<:
(25).
>4AB0282 2 (19) ?>;CG8< 8A:><CN A2O7L:
(26).
2.3 >4AB0282 2 2K@065=85 (24) 7=0G5=85 , ?>;CG8< B>: :>@>B:>3> 70<K:0=8O:
(27).
=0;>38G=>5 @5H5=85 <>6=> ?>;CG8BL, ?>;030O 2 C@02=5=88 (26) .
0?@O65=85 E>;>AB>3> E>40 =0945<, ?@8=O2 2 (26) .
(28).
2.4. '8A;5==K5 7=0G5=8O B>:0 :>@>B:>3> 70<K:0=8O 8 =0?@O65=8O E>;>AB>3> E>40:
(29),
(30).
2.5 >4AB028< G8A;5==K5 7=0G5=8O 2 (26) 8 C<=>68< >15 G0AB8 =0 103:
(31).
0A:@K2 A:>1:8, ?>;CG8<:
(32),
".:. B>: 2A5340 <5=LH5 , B> A;0305<K< <>6=> ?@5=51@5GL.
0?8H5< C@02=5=85 2 A;54CNI5< 2845:
(33).
5;8G8=0 87<5=O5BAO =57=0G8B5;L=>  >B 0 4> 312. >;>68 55 @02=>9 150.
:>=G0B5;L=> ?>;CG8<:
(34).
0 @8AC=:5 87>1@065=K 3@0D8:8 7028A8<>AB8 .
0<5B8<, GB> >B;8G8O >B B>G=>3> ?>AB@>5=8O
=525;8:8 8 ?@>O2;ONBAO ?@8 <0;KE =0?@O65=8OE
2.6 A?>;L7CO C40G=>5 ?@81;865=85 87 ?@54K4CI53>
?C=:B0, <>65< 70?8A0BL:
(35).
@8@02=O2 : =C;N ?@>872>4=CN, ?>;CG8<:
(36);
(37).
25
040G0 11 - 3. «20 35=5@0B>@0
'0ABL 1. @C3;K9 35=5@0B>@.
5@5<5==>5 <03=8B=>5 ?>;5 ?>@>6405B 28E@52>5
M;5:B@8G5A:>5 ?>;5.  40==>< A;CG05 A8;>2K<8 ;8=8O<8
M;5:B@8G5A:>3> ?>;O 1C4CB :>=F5=B@8G5A:85 >:@C6=>AB8,
?;>A:>ABL :>B>@KE ?5@?5=48:C;O@=0 ;8=8O< 8=4C:F88
<03=8B=>3> ?>;O. 0AA<>B@8< :>;LF> @048CA0 ak . !>3;0A=>
70:>=C M;5:B@><03=8B=>9 8=4C:F88 $0@045O, ?5@5<5==>9
<03=8B=>5 ?>;5 ?>@>6405B 2 =5< -! 8=4C:F88 @02=CN
"Åšk
µk = - , (1)
"t
2
345 Åšk = Ä„ ak B0 cosÉ t - <03=8B=K9 ?>B>: G5@57 MB> :>;LF>. !;54>20B5;L=>, -!
8=4C:F88 MB>3> :>;LF0 @02=0
"Åšk 2
µk = - = Ä„ ak B0É sinÉ t . (2)
"t
-;5:B@8G5A:>5 A>?@>B82;5=85 :>;LF0 @02=>
2Ä„ak ak
rk = Á = 8Á . (3)
2 2
(Ä„d ) 4 d
A;8 ?@5=51@5GL A>?@>B82;5=85< 0<?5@<5B@0, B> A8;0 B>:0 G5@57 =53> 1C45B @02=0
2 2 2
10
sinÉ
k
i0 = = =
"ik "µ "Ä„ ak B0Éak t = Ä„d a0 B0É sinÉ t"k = 55 Ä„dÁa0 B0É sinÉ t
rk k 8Á 8Á 8
k k k =1
2
d
59AB2CNI55 7=0G5=85 A8;K B>:0, A>>B25BAB25==>, 2 MB>< A;CG05 @02=>
2
55 Ä„d a0
I0 = B0É . (4)
Á
8 2
A;8 CG5ABL A>?@>B82;5=85 0<?5@<5B@0, B> 4;O :064>3> :>;LF0
<>6=> 70?8A0BL
µk = ikrk + i0R . (5)
0745;8< MB8 C@02=5=8O =0 rk 8 ?@>AC<<8@C5< ?> 2A5< :>;LF0<, 2
8B>35 ?>;CG8<
k
"µ
1 rk
k
k
i0 = = - i0R Ò! i0 = . (6)
"ik "µ "
1
rk k rk
k k
1+ R
"
rk
k
!C<<0 2 G8A;8B5;5 ?>4AG8B0=0. KG8A;8< AC<<C ?@>2>48<>AB59
2 2 2
10
1 d 1 d d
= H" Å" 2,930 H" 0,366 (7)
" "
rk 8Áa0 k =1 k 8Áa0 Áa0
k
"5?5@L <>6=> 70?8A0BL >:>=G0B5;L=>5 2K@065=85 4;O 459AB2CNI53> 7=0G5=8O A8;K B>:0
2 MB>< A;CG05
2
55 Ä„d a0
B0É
2 2
15,27 Å" d a0 B0É
Á
8 2
I = = . (8)
2 2
d Áa0 + 0,366Rd
1 + 0,366R
Áa0
26
'0ABL 2. @O<>C3>;L=K9 35=5@0B>@.
0 :064K9 M;5:B@>= 2
4286CI5<AO ?@>2>4=8:5 2 >1;0AB8
<03=8B=>3> ?>;O 459AB2C5B A8;0
>@5=F0
F = eBv .
(1)
-B0 A8;0 A>25@H05B @01>BC ?>
?5@5<5I5=8N M;5:B@>=0 ?>?5@5:
4286CI59AO ;5=BK, B> 5ABL A>7405B -!, @02=CN
µ = Bva . (2)
-;5:B@8G5A:>5 A>?@>B82;5=85 CG0AB:0 ;5=BK, =0E>4OI59AO 2 M;5:B@8G5A:>< ?>;5, @02=>
a
r = Á . (3)
bh
> 70:>=C <0 A8;0 B>:0 2 :>=BC@5 >?@545;O5BAO 2K@065=85<
µ Bva
I = =
(4)
a
r + R
R + Á
bh
!> AB>@>=K <03=8B=>3> ?>;O =0 MB>B B>: 459AB2C5B A8;0 <?5@0, =0?@02;5==0O 2 AB>@>=C
?@>B82>?>;>6=CN 25:B>@C A:>@>AB8 ;5=BK 8 @02=0O
B2a2
F = IBa = v . (5)
a
R + Á
bh
'B>1K ;5=B0 42830;0AL @02=><5@=>, =5>1E>48<> 55 BO=CBL A B0:>9 65 ?> <>4C;N A8;>9.
-B0 A8;0 @0728205B <>I=>ABL
B2a2
P0 = Fv = v2 . (6)
a
R + Á
bh
>I=>ABL, 2K45;ONI0OAO =0 @578AB>@5 <>65B 1KBL @0AAG8B0=0 ?> 70:>=C 6>C;O-5=F0
B2v2a2
2
P = I R = R . (7)
2
a
ëÅ‚ öÅ‚
R + Á
ìÅ‚ ÷Å‚
bh
íÅ‚ Å‚Å‚
0:>=5F,  35=5@0B>@0 @02=>
P R
· = = , (8)
P0 R + Á a
bh
GB> @02=> B@048F8>==><C >B=>H5=8N A>?@>B82;5=8O =03@C7:8 : ?>;=><C A>?@>B82;5=8N
F5?8.
27