III MB0?, 20053. 5H5=8O 7040G.
9 :;0AA.
1. �5154:0
1>7=0G8< 2@5<O ?>4J5<0 >4=>9 1>G:8 ?> =0:;>==>9 ?;>A:>AB8 t0 . ">340 ;5154:0
A>25@H8B 70 MB> 2@5<O @01>BC A = �P0t0 , @02=CN 87<5=5=8O ?>B5=F80;L=>9 M=5@388 1>G:8
"U = mgh , B.5.
�P0t0 = mgh . (1)
@8 ?5@5<5I5=88 1>G:8 =0 @0AAB>O=85 L , ;5154:0 4>;6=0 =0<>B0BL =0 20; B@>A 4;8=>9
2L , ?>MB><C
2Ąrnt0 = 2L . (2)
7 MB8E C@02=5=89 =0E>48<
mgh
t0 = , (3)
�P0
�P0L
n = . (4)
Ąrmgh
2 �"@80B;>=
1>7=0G8< A:>@>ABL �10940@>G=8:0 v0 . ">340 A:>@>ABL
v0
�?;>2F0 @02=0 , 0 A:>@>ABL 153C=0 2v0 .
2
B=>H5=8O 2@5<5= 42865=8O A?>@BA<5=>2 =0E>48BAO
4>AB0B>G=> ?@>AB>
2
h2 + S
2
t1 v0 2 h2 + S
= = . (1)
t2 2h + S 4h + S
v0 2v0
@8 h = S
2
t1 2 h2 + S 2 2
= = H" 0,566 , (2)
t2 4h + S 5
B> 5ABL �10940@>G=8: ?>156405B.
!?>@BA<5=K ?@84CB : D8=8HC >4=>2@5<5==> ?@8 2K?>;=5=88 CA;>28O
2
t1 2 h2 + S
= = 1. (3)
t2 4h + S
5H0O MB> C@02=5=85, =0E>48<
4h + 16h2 + 36h2 4 + 2 13
S = = h H" 374< . (4)
3 3
'B>1K 2@5<O 42865=8O 2B>@>3> A?>@BA<5=0 1K;>
<8=8<0;L=>, >= <>65B 2K1@0BL 4@C3>9 �<0@H@CB: ?CABL
>= A=0G0;0 1568B ?> 15@53C ?> >B@57:C AD , 0 70B5<
2?;02L ?> >B@57:C DB . 'B>1K �2K1@0BL >?B8<0;L=CN
B>G:C D , 2>A?>;L7C5<AO A;54CNI8<8 @0AAC645=8O<8:
?@8 42865=88 ?> 15@53C, A:>@>ABL 53> ?@81;865=8O :
B>G:5 D8=8H0 @02=0
1
vC = 2v0 cosą . (5)
G5284=>, GB> 8<55B A<KA; 1560BL ?> 15@53C 4> B5E ?>:0, MB0 A:>@>ABL 1>;LH5, G5<
A:>@>ABL ?@81;865=8O 2 ?;02L. "0:8< >1@07><, A?>@BA<5= 4>;65= =0G0BL ?;KBL 2 B>G:5,
345 =0?@02;5=85 =0 B>G:C D8=8H0, >?@545;O5BAO A>>B=>H5=85<
v0
vC = 2v0 cosą = . (6)
2
7 =53> A;54C5B
1 15
cosą = ; siną = 1- cos2 ą = ; tgą = 15 .
4 4
h
">340 x = AD = S - H" 574< .
tgą
>;=>5 2@5<O 42865=8O ?> >?B8<0;L=><C <0@H@CBC
2
h2 + (S - x)
x
t2 = + , (7)
v0
2v0
2
 >B=>H5=85 2@5<5= 42865=8O
2
t1 h2 + S 2 37
= = H" 1,23 . (8)
t2 x + 2 h2 + - x)2
6 + 15
(S
2
3. �"5@<><5B@
3.1 !<KA; ?0@0<5B@>2, 2E>4OI8E 2 ?@82545==K5 D>@<C;K A;54CNI89:
"l 1
l0 ,V0 - 4;8=0 8 >1J5< B5;0 ?@8 t = 0�C , ą = - >B=>A8B5;L=>5 C4;8=5=85 ?@8
l0 t
"V 1
87<5=5=88 B5<?5@0BC@K =0 1 3@04CA, 0=0;>38G=> � = - >B=>A8B5;L=>5 87<5=5=85
V0 t
>1J5<0 ?@8 87<5=5=88 B5<?5@0BC@K =0 1 3@04CA.
3.2 7<5=5=85 >1J5<0 B5;0 <>65B 1KBL 2K@065=> G5@57 87<5=5=85 53> ;8=59=KE @07<5@>2
V = l10(1+ąt)l20(1+ ąt)l30(1+ ąt) H" l10l20l30(1+ 3ąt),
 MB>< 2K@065=88 <K ?@5=51@53;8 <0;K<8 A;0305<K<8 2B>@>3> 8 B@5BL53> ?>@O4:0.
!@02=820O A D>@<C;>9 4;O 87<5=5=8O >1J5<0 V = V0(1+ �t), ?>;CG05< A2O7L <564C
?0@0<5B@0<8
� = 3ą . (1)
3.3 ;O �3@04C8@>2:8 AB>;18:0 B5@<><5B@0
4>AB0B>G=> 70?8A0BL A>>B=>H5=85 4;O 87<5=5=8O
>1J5<0 @BCB8
2
Ąd0
V0 + h = V0(1+ �t). (2)
4
B:C40 A;54C5B
4V0�
h = t H" 1,15t , (3)
2
Ąd0
3@0D8:>< MB>9 DC=:F88 O2;O5BAO ?@O<0O,
?@>E>4OI0O G5@57 =0G0;> :>>@48=0B.
3.4 @8 CG5B5 87<5=5=8O @07<5@>2 AB5:;0 A>>B=>H5=85 4;O 87<5=5=8O >1J5<0 @BCB8 8<55B
284
2
2
Ąd0 (1+ 2ąt)h = V0(1+ �t).
V0(1+ 3ąt)+ (4)
1
4
7 MB>3> C@02=5=8O >?@545;O5< 2KA>BC ?>4J5<0
4V0 - 3ą 4V0� �# ą ś#
�
h1 = t H" tś#1- 3 - 2ąt ź# . (5)
2 2 ź#
Ąd0 1+ 2ąt Ąd0 ś# �
�# #
-B><C 7=0G5=8N 2KA>BK ?>4J5<0, ?> 3@04C8@>2>G=>9 7028A8<>AB8 (3) 1C45B �?@8?8A0=0
B5<?5@0BC@0
2
Ąd0 �# ą ś#
t87<. = h1 = tś#1- 3 - 2ąt ź# .
ś# ź#
4V0� �
�# #
!;54>20B5;L=>, >B=>A8B5;L=0O ?>3@5H=>ABL 87<5@5=8O B5<?5@0BC@K @02=0
t87< - t ą
� = = -3 - 2ąt H" -5,0 �"10-2 .
t �
4. �><>65< "5E0AC
4.1 7 C@02=5=8O B5?;>2>3> 10;0=A0 A;54C5B, GB> 70 548=8FC 2@5<5=8 46>C;520 B5?;>B0
2 2
U U hb
= , 2K45;82H0OAO ?@8 ?@>E>645=88 B>:0, ?>945B =0 ?;02;5=85 m = qł
R �a
:8;>3@0<<>2 A=530, B.5.
2
U hb
= qł , (1)
�a
>B:C40 A;54C5B
2
U hb <3
q = H" 2,3�"10-2 . (2)
�ał A
4.2 A;8 70 2@5<O "t C@>25=L 2>4K ?>4=O;AO =0 "h , B> 70 MB> 2@5<O @0A?;028;>AL
"m = "habł ;L40, =0 GB> 1K;> 70B@0G5=> :>;8G5AB2> B5?;>BK, @02=>5
2 2
U U bh
"Q = "t = "t . #@02=5=85 B5?;>2>3> 10;0=A0 2 MB>< A;CG05 1C45B 8<5BL 284
R �a
2
U bh
"t = "habł , (3)
�a
87 :>B>@>3> A;54C5B, GB> A:>@>ABL 87<5=5=8O C@>2=O 2>4K 2 O<5 ?@>?>@F8>=0;L=0 2KA>B5
MB>3> C@>2=O
2
"h U
= h H" 0,18h , (4)
"t �a2ł
345 t 2 <8=CB0E, :>MDD8F85=B ?@>?>@F8>=0;L=>AB8 8<55B @07<5@=>ABL [<8=-1]
A?>;L7CO ?@82545==K5 2 CA;>288 3@0D8: <>6=> ?>:070BL, 2K?>;=5=85 MB>9 7028A8<>AB8,
4;O G53> 4>AB0B>G=> @0AA<>B@5BL <0;K5 ?@><56CB:8 2@5<5=8 (=0?@8<5@, "t = 1<8= ).
"0: :0: 87<5=5=85 2KA>BK ?@>?>@F8>=0;L=> A0<>9 2KA>B5, B> 70 >48=0:>2K5 ?@><56CB:8
2@5<5=8 2KA>B0 87<5=O5BAO 2 >4=> 8 B>65 G8A;> @07. > 3@0D8:C 284=>, GB> 70 2@5<O
t H" 4<8= 2KA>B0 C25;8G8;0AL 2 420 @070, A;54>20B5;L=>, >=0 C25;8G8BAO 2 4 @070 70 2@5<O 2
420 @070 1>;LH55, B> 5ABL 70 8 <8=CB.
3
 5H5=8O 7040G (10 :;0AA)
040G0 1. �"@0=A?>@B5@
1.1 >A;5 :>=B0:B0 45B0;8 A =0?@02;ONI59 =0 45B0;L 2 3>@87>=B0;L=>9 ?;>A:>AB8
r r
1C4CB 459AB2>20BL 425 A8;K B@5=8O: FB@1 (A> AB>@>=K ;5=BK B@0=A?>@B5@0) 8 FB@2 (A>
AB>@>=K =0?@02;ONI59), 0 B0:65 A8;0
r
@50:F88 N A> AB>@>=K =0?@02;ONI59.
C45< AG8B0BL, GB> 42865=85 45B0;59
=>A8B ?>ABC?0B5;L=K9 E0@0:B5@, B.5. >=8 =5
2@0I0NBAO ?@8 B@5=88 > =0?@02;ONICN.
;O A=OB8O (A>A:0;L7K20=8O) 45B0;8 A
B@0=A?>@B5@0 =5>1E>48<>, GB>1K ?@>5:F8O
A8;K B@5=8O FB@1 = �1mg =0 =0?@02;ONICN
8A.1
1K;0 1>;LH5 A8;K B@5=8O FB@2 = �2N >
=0?@02;ONICN (@8A. 1):
�1mg siną > �2 N .
! CG5B>< B>3>, GB> A8;0 @50:F88
N = �1mg cosą ,
?>;CG05<
tgą e" �2
!>>B25BAB25==>, 4;O <8=8<0;L=>3> C3;0 ąmin 2K18@05< A;CG09 @025=AB20
tgąmin = �2 . (1)
1@0B8B5 2=8<0=85, GB> (1) =5 7028A8B >B 7=0G5=8O :>MDD8F85=B0 B@5=8O �1,
3;02=>5  GB>1K >= 1K; >B;8G5= >B =C;O, 8=0G5 45B0;8 =5 A<>3CB 42830BLAO ?>
B@0=A?>@B5@C.
1.2 "0:8< >1@07><, ?@8 2K?>;=5=88 CA;>28O ą > ąmin 45B0;8 1C4CB A:>;L78BL 24>;L
r
=0?@02;ONI59. 0<5B8<, GB> ?> <5@5 @>AB0 A:>@>AB8 u 42865=8O 45B0;59 24>;L
r
=0?@02;ONI59 25:B>@ w A:>@>AB8 8E ?@>A:0;L7K20=8O >B=>A8B5;L=> B@0=A?>@B5@0 1C45B
?>2>@0G820BLAO, �?@868<0OAL : =>@<0;8
D . A;54AB285 MB>3> 1C45B
r
?>2>@0G820BLAO 8 25:B>@ A8;K B@5=8O FB@1
87 =0G0;L=>3> ?>;>65=8O �24>;L
B@0=A?>@B5@0 4> CAB0=>282H53>AO
?>;>65=8O, ?@8 :>B>@>< >= 1C45B
A>AB02;OBL =5:>B>@K9 C3>; � A =>@<0;LN
D : =0?@02;ONI59 (@8A.2).
8A. 2
!;54>20B5;L=>, 2 CAB0=>282H5<AO @568<5
FB@1 sin � = �2 N = �2FB@1 cos � �! tg � = �2 . (2)
! 4@C3>9 AB>@>=K 87 B@5C3>;L=8:0 A:>@>AB59 ! (A<. @8A. 2) ?> B5>@5<5 A8=CA>2
8<55<
u V
= , (3)
sin (ą - � ) sin�
Ą
345 � = + � .
2
K45;OO 87 (3) tg� A CG5B>< (2) ?>;CG05<
4
V siną - u
tg � = = �2 .
V cosą
:>=G0B5;L=> 4;O CAB0=>282H59AO A:>@>AB8 u ?>ABC?0B5;L=>3> 42865=8O 45B0;59
24>;L =0?@02;ONI59 ?>;CG05<
u = V (siną - �2 cosą ) . (4)
0: 2848< 87 (4) 2K@065=85 8<55B A<KA; B>;L:> ?@8 2K?>;=5=88 CA;>28O
tgą e" �2 ,
2 ?@>B82=>< A;CG05 ?>A;5 C?>@0 2 =0?@02;ONICN 45B0;8 1C4CB ?>:>8BLAO, B.5. u = 0 .
040G0 2. �8?5=85
2.1 0AA<>B@8< ?C7K@5: ?0@0 @048CA0 r (@8A. 3), >1@07>202H89AO 2=CB@8 :8?OI59
684:>AB8. C7K@5: C25;8G8205B A2>9 @048CA 2A;54AB285 8A?0@5=8O 684:>AB8 2=CB@L 53>.
!>3;0A=> CA;>28N 70 2@5<O "t A ?>25@E=>AB8 684:>AB8 S (?>25@E=>AB8
?C7K@L:0) 8A?0@8BAO >1J5< 2>4K "V :
"V = N DS"t , (1)
345 D  �MDD5:B82=K9 480<5B@ <>;5:C;K 2>4K.
!>>B25BAB25==>, >1J5< ?C7K@L:0 B0:65 4>;65= C25;8G8BLAO =0
"V . ! 4@C3>9 AB>@>=K "V <>6=> ?@54AB028BL :0: >1J5< B>=:>9
AD5@K @048CA>< r 8 B>;I8=>9 " r (A<. @8A.3):
2
"V = S" r = 4Ą r "r . (2)
7 (1) 8 (2) =0E>48< ?@8@0I5=85 @048CA0 ?C7K@L:0 ?0@0
"r 70 2@5<O "t
" r = ND"t = {� = ND}= � "t . (3)
0: A;54C5B 87 (3), ?@8@0I5=85 @048CA0 " r ?C7K@L:0 ?0@0
70 2@5<O "t =5 7028A8B >B 53> @048CA0 r , B.5. @048CA ?C7K@L:0
8A. 3
@02=><5@=> C25;8G8205BAO A> 2@5<5=5<
r = � �" t . (4)
2.2 >A:>;L:C ?;>B=>ABL ?0@0 3>@074> <5=LH5 ?;>B=>AB8 2>4K, B> A8;>9 BO65AB8
?C7K@L:0 ?0@0 ?> A@02=5=8N A 459AB2CNI59 =0 =53> A8;>9 @E8<540 <>6=> ?@5=51@5GL.
!:>@>ABL ?C7K@L:0 � 1C45B @0AB8 4> B5E ?>@, ?>:0 A8;0 @E8<540 =5
C@02=>25A8BAO A8;>9 A>?@>B82;5=8O A> AB>@>=K 2>4K (A<. @8A.3)
FA = FC ,
4 1
2 2
�0 g Ą r3 = Cx �0� Ą r ,
3 2
ż# �#
8 g 8 g 8 g
2
� = �" r = {(4)}= �" �" � t = = �" �" � = ł �" t .
�#ł Ź#
3 Cx 3 Cx �# 3 Cx �#
>4>1=K5 ?@81;865=8O, ?@8 :>B>@KE
AG8B05BAO, GB> 2 :064K9 <><5=B 2@5<5=8 A8AB5<0
=0E>48BAO 2 @02=>25A=>< (AB0F8>=0@=><)
A>AB>O=88, =07K20NBAO :2078AB0F8>=0@=K<8 (8;8
:0: <>4=> 3>2>@8BL A53>4=O �:0: 1K
AB0F8>=0@=K<8). !>>B25BAB25==> 2 =0H5< A;CG05
� (t) = ł t . (5)
@0D8: 7028A8<>AB8 (5) ?@54AB02;5= =0 @8AC=:5.
5
2.3 ;O =0E>645=8O @048CA0 ?C7K@L:0 rmax C ?>25@E=>AB8 2>4K =5>1E>48<> 7=0BL 2@5<O
53> 2A?;KB8O tmax . 0<5B8<, GB> ?@8 B>G=>< @5H5=88 =0 @8A.4 ?@8 t = tmax ?;>I04L ?>4
3@0D8:>< S (8;8 ?;>I04L :@82>;8=59=>9 B@0?5F88) 4>;6=0 1KBL @02=0 3;C18=5 A>AC40
h
S(tmax ) = h .
;O >F5=:8 MB>9 25;8G8=K <K ?>ABC?8< 4>AB0B>G=> �?@O<>;8=59=>. @8
2A?;KB88 <0:A8<0;L=0O A:>@>ABL ?C7K@L:0 �max 4>AB8305BAO C ?>25@E=>AB8
�max = ł tmax . (6)
0<5=8< :@82CN =0 3@0D8:5 7028A8<>AB8 (5) ?@O<>9, B.5. 1C45< AG8B0BL 42865=85
?C7K@L:0 @02=>CA:>@5==K<, B>340 53> A@54=OO A:>@>ABL
0 +�max 1
� = = ł tmax .
2 2
">340
3
ł
1
2
h = � tmax = ł tmax �"tmax = (tmax ) . (7)
2 2
! ?><>ILN (7) ?>;CG05< >F5=:C 4;O 2@5<5=8 2A?;KB8O tmax
2
�# ś#3
2h
ś# ź#
tmax = . (8)
ś# ź#
ł
�# #
!>>B25BAB25==> A CG5B>< (4) 8 (8) =0E>48< rmax
2
�# ś#3 3 CxD2N 2h2
2h
ś# ź#
3
rmax = � �"tmax = � = �" . (9)
ś# ź#
2 g
ł
�# #
;O @0AG5B0 ?@8<5<, GB> 480<5B@ <>;5:C;K 2>4K ?> ?>@O4:C 25;8G8=K @025=
480<5B@C 0B><0 2>4>@>40 D = 1A= 1,0 �"10-10 < , B>340 87 (9) ?>;CG05<:
rmax = 1,1�"10-3 < = 1,1<< .
040G0 3. �8M;5:B@8: 8;8 ?@>2>4=8:?
0AA<>B@8< D878G5A:CN :0@B8=C ?@>8AE>4OI8E O2;5=89. !@07C ?>A;5 70<K:0=8O
F5?8 =0 ?;0AB8=:5 2>7=8:0NB ?>;O@870F8>==K5 70@O4K (?;0AB8=:0 =0 <0;KE 2@5<5=0E
2545B A51O :0: 48M;5:B@8:), 70B5< 2 ?;0AB8:5 2 @57C;LB0B5 ?@>B5:0=8O B>:0
?@>8AE>48B ?5@5@0A?@545;5=85 70@O40, ?>:0 ?>;5 2=CB@8 ?;0AB8=:8 =5 8AG57=5B
(?;0AB8=:0 AB0=>28BAO ?@>2>4=8:><).
1>7=0G8< ?>25@E=>AB=CN ?;>B=>ABL 70@O40 =0
>1:;04:0E :>=45=A0B>@0 � , 0 =0 ?>25@E=>AB8 ?;0AB8=:8
0
� ' (7=0:8 70@O4>2 C:070=K =0 @8AC=:5). "0: :0:
=0?@O65=85 <564C >1:;04:0<8 ?>445@68205BAO
?>AB>O==K<, B> 2K?>;=O5BAO A>>B=>H5=85
h h
U0 = E0 + E1 , (1)
2 2
345 E0 , E1 =0?@O65==>AB8 M;5:B@8G5A:8E ?>;59 <564C
>1:;04:0<8 8 ?;0AB8=:>9 8 2=CB@8 ?;0AB8=:8,
A>>B25BAB25==>. 0?@O65==>AB8 ?>;59 A2O70=K A
?>25@E=>AB=K<8 ?;>B=>ABO<8 70@O4>2 A>>B=>H5=8O<8
� � - � '
0 0
E0 = ; E1 = . (2)
�0 �0
6
>A;5 70<K:0=8O F5?8 ?@0:B8G5A:8 <3=>25==> =0 ?;0AB8=:5 2>7=8:=CB ?>;O@870F8>==K5
70@O4K, B0:85, GB> =0?@O65==>ABL ?>;O 2=CB@8 ?;0AB8:8 1C45B 2 � @07 <5=LH5
E0
=0?@O65==>AB8 ?>;O 2=5 55 E1 = , ?>MB><C 87 A>>B=>H5=8O (1) ?>;CG8<
�
h E0 h � 2U0
�! E0 = �" . (3)
U0 = E0 +
2 � 2 � +1 h
-B> ?>;5 A>7405BAO 70@O40<8 =0 >1:;04:0E
� 2U0�0S
Q0 = � S = �0E0S = �" H" 7,9 �"10-8 ; . (4)
0
� +1 h
>4 459AB285< ?>;O 2=CB@8 ?;0AB8=:8 =0G=5B B5GL M;5:B@8G5A:89 B>:, :>B>@K9 1C45B
87<5=OBL 70@O4K, :0: =0 ?>25@E=>AB8 ?;0AB8=:8, B0: 8 =0 >1:;04:0E :>=45=A0B>@0. ">:
?@5:@0B8BAO, :>340 =0?@O65==>ABL ?>;O 2=CB@8 ?;0AB8=:8 AB0=5B @02=>9 =C;N. >A;5
7025@H5=8O MB>3> ?@>F5AA0 ?5@5@0A?@545;5=8O =0?@O65==>AB8 ?>;59 AB0=CB @02=K<8
2U0
E1 = 0; E0 = ,
h
0 70@O4 =0 >1:;04:0E
2U0�0S
Q01 = � S = �0E0S = H" 11�"10-8 ; . (5)
0
h
"0:8< >1@07><, 70@O4 :>=45=A0B>@0 1C45B 87<5=OBLAO >B Q0 H" 7,9 �"10-8 ; 4>
Q01 H" 11�"10-8 ; .
;O >?8A0=8O 48=0<8:8 87<5=5=8O MB>3> 70@O40 87 A>>B=>H5=85 (1)  (2) 2K@078<
U0�0 � '
� = +
0
h 2
, (6)
U0�0S Q'
Q0 = +
h 2
345 Q' - 70@O4 =0 ?>25@E=>AB8 ?;0AB8=:8. 7<5=5=85 MB>3> 70@O40 >1CA;>2;5=>
M;5:B@8G5A:8< B>:>< 2=CB@8 ?;0AB8=:8, A8;0 :>B>@>3> >?@545;O5BAO 70:>=>< <0
h h
E1 E1 E1 Q0 - Q'
"Q'
2 2
= I = = S = S = . (7)
h
"t R � �0 �
�
2
7102;OOAL 2 MB>< C@02=5=88 >B Q' A ?><>ILN A>>B=>H5=89 (6) ?>;CG8< C@02=5=85,
>?8AK20NI55 87<5=5=85 70@O40 :>=45=A0B>@0
"Q0 1 �#Q ś# 1
2U0�0S
I0 = = - ś# - = (8)
ź# - (Q0 - Q01).
0
"t 2�0 � h 2�0 �
�# #
7 MB>3> C@02=5=8O A;54C5B, GB> E0@0:B5@=>5 2@5<O 87<5=5=8O 70@O40 (A;54>20B5;L=>, 8
ACI5AB2>20=8O B>:0) @02=>
� = 2��0 H" 0,18c . (9)
1
0G0;L=>5 7=0G5=85 A8;K B>:0 I00 = - (Q0 - Q01) H" 1,8 �"10-7 A , :>B>@K9 70B5< C1K205B
2�0 �
4> =C;O.
7
 ;O >?@545;5=8O :>;8G5AB20 B5?;>BK, 2K45;82H59AO 2 ?;0AB8=:5, <>6=>
2>A?>;L7>20BLAO M=5@35B8G5A:8< A>>B=>H5=85<:
2
C0U0 1
- ?>A;5 70<K:0=8O F5?8 :>=45=A0B>@ >1;040; M=5@3859 W0 = = Q0U0 , 345
2 2
-1
�#�# 2�0S ś#-1 �# 2��0S ś#-1 ś#
� 2�0S
ś#ś# ź# + ś# ź# ź#
C0 = = - =0G0;L=0O 5<:>ABL :>=45=A0B>@0;
ś#�# h # �# h # ź#
� +1 h
�# #
- 70 2@5<O ?@>B5:0=8O B>:0 8AB>G=8: A>25@H8; @01>BC A = (Q01 - Q0 )U0 ;
2
C1U0 1
- ?>A;5 ?5@5@0A?@545;5=8O :>=45=A0B>@ >1;0405B M=5@3859 W1 = = Q01U0 , 345
2 2
2�0S
C = - 5<:>ABL :>=45=A0B>@0, ?>A;5 ?@5:@0I5=8O B>:0.
h
"0:8< >1@07><, M=5@35B8G5A:89 10;0=A ?@54AB02;O5BAO C@02=5=85<
~
W0 + A = W1 + Q , (10)
~
745AL Q - :>;8G5AB2> 2K45;82H59AO B5?;>BK. 7 MB>3> C@02=5=8O =0E>48<
~ 1 1
Q = A + W0 -W1 = (Q10 - Q0 )U0 - (Q10 - Q0 )U0 = (Q10 - Q0 )U0 = 7,8 �"10-5 6
2 2
040G0 4. �;>A:0O 5<;O
4.1 ! B>G:8 7@5=8O <0B5<0B8:8 70:>= 2A5<8@=>3> BO3>B5=8O LNB>=0 8 70:>= C;>=0
(>A=>2=>9 70:>= M;5:B@>AB0B8:8) O2;ONBAO AE>68<8, 5A;8 AG8B0BL <0AAC B A2>5>1@07=K<
�3@028B0F8>==K< 70@O4>< 8;8 =0>1>@>B, ?@54AB028BL M;5:B@8G5A:89 70@O4 q
�M;5:B@8G5A:>9 <0AA>9
r r
m1m2 q1q2
F = G �! F = k .
2 2
r r
r
 ?@>4>;65=85 MB>9 0=0;>388 <>65< A:070BL, GB> CA:>@5=85 A2>1>4=>3> ?045=8O g 2
B5>@88 3@028B0F88 O2;O5BAO �A8;>2K< 0=0;>3>< 25:B>@0 =0?@O65==>AB8 M;5:B@8G5A:>3>
r
?>;O 
r r r
r
F = m g �! F = q E .
8
"0:8< >1@07><, <>65< A>AB028BL =51>;LHCN B01;8G:C �3@028B0F8>==>-M;5:B@8G5A:8E
0=0;>389.
! @028B0F8O -;5:B@>AB0B8:0
q
1. m
r r
g
2.
E
1
k =
3. G
4Ą �0
1
4. �0
4Ą G
0?@O65==>ABL M;5:B@8G5A:>3> ?>;O E ?;0AB8=K B>;I8=>9 h , 70@O65==>9 A
?>AB>O==>9 >1J5<=>9 ?;>B=>ABLN � :
�h
E = .
2�0
!;54>20B5;L=>, <5B>4>< 0=0;>389 =0E>48<, GB> CA:>@5=85 A2>1>4=>3> ?045=8O =0
A525@=>< ?>;NA5 (240;8 >B :@052 ?;0AB8=K)
� h
g = 4Ą G = 2Ą G � h . (1)
2
! ?><>ILN (1) =0E>48< 8A:><CN B>;I8=C 5<=>3> 48A:0
g
h = = 4,3�"106 < = 4,3�"103 :< . (2)
2Ą G�
4.2 @8 42865=88 :0<5H:0 2=CB@8 H0EBK =0 =53>
1C45B 459AB2>20BL ?5@5<5==0O A8;0 BO65AB8 F(x) ,
3@0D8: 7028A8<>AB8 :>B>@>9 >B @0AAB>O=8O x 4>
h
�=C;52>3> A5G5=8O ?;0=5BK (=0 3;C18=5 )
2
?@54AB02;5= =0 @8A. 5.
01>B0 A8;K BO65AB8  @02=0
70HB@8E>20==>9 ?;>I048 ?>4 3@0D8:>< S
(?;>I048 ?@O<>C3>;L=>3> B@5C3>;L=8:0)
8A. 5
1 h
A = mg ,
2 2
345 h 405BAO 2K@065=85< (2). !:>@>ABL :0<5H:0 1C45B <0:A8<0;L=>9 8<5==> 2 =C;52><
A5G5=88, ?>A:>;L:C ?@8 40;L=59H5< 42865=88 3@028B0F8>==K5 A8;K =0G=CB 53>
?@8B>@<06820BL.
!>3;0A=> B5>@5<5 > :8=5B8G5A:>9 M=5@388 2 MB>< A;CG05 ?>;CG05<
2
m�max h
= A = mg , (3)
2 4
345 m  <0AA0 :0<5H:0. 7 (3) =0E>48<
g h g 1 < :<
�max = = = 4,6 �"103 = 4,6 . (4)
2 2 Ą G� A A
0: 2848< 87 (4) 4065 ?;>A:0O 5<;O @073>=8B :0<5H5: 4> �?>;C:>A<8G5A:>9 A:>@>AB8.
4.3 0) @0AA<>B@8< 2KAB@5; 87 AC?5@:0B0?C;LBK 24>;L <5@8480=0 >B F5=B@0 ?;0=5BK.
2Ą
D8=K 2@0I0NBAO 2>:@C3 F5=B@0 5<;8 A ;8=59=>9 A:>@>ABLN � = �r , 345 � = 
T
9
C3;>20O A:>@>ABL 2@0I5=8O ?;0=5BK, " = 86 400A  ?5@8>4 >1@0I5=8O 5<;8 2>:@C3
A2>59 >A8.
!>>B25BAB25==>, MBC 65 A:>@>ABL ?@8>1@5B05B A=0@O4
2 078<CB0;L=>< =0?@02;5=88 >B=>A8B5;L=>3> 01A>;NB=>3>
=01;N40B5;O, =0E>4OI53>AO 2 F5=B@5 5<;8 (@8A. 6). 4=0:>
;8=59=0O A:>@>ABL 2@0I5=8O B>G5: 2 @09>=5 ?@875<;5=8O
1>;LH5 (>=8 40;LH5!) =0 25;8G8=C "�
"� = �(r + " r) -� r = � " r , (5)
2 2
�0 �0
345 " r = sin 2ą =  40;L=>ABL ?>;5B0 A=0@O40 =0
g g
=5?>4286=>9 5<;5. !>>B25BAB25==> 70 2@5<O ?>;5B0
8A. 6
2�0 siną
A=0@O40 t = MB8 B>G:8 �C94CB 2?5@54 =0 25;8G8=C
g
3
2�0 siną 4Ą �0
" S = "� t = � "r = siną = 1,1< . (6)
2
g T g
>A:>;L:C <5@8480= C945B 2?5@54 ?> =0?@02;5=8N 2@0I5=8O 5<;8, B> 4;O
=01;N40B5;O A=0@O4 >B:;>=8BAO =0 70?04 >B <5@8480=0 =0 25;8G8=C " S = 1,1< .
53:> 4>3040BLAO, GB> ?@8 ?@81;865=88 A=0@O40 : F5=B@C 5<;8 ;8=59=0O A:>@>ABL
2@0I5=8O B>G5: 55 ?>25@E=>AB8 1C45B C<5=LH0BLAO, B.5.
B5?5@L, 2 >B;8G85 >B ?@54K4CI53> A;CG0O, A=0@O4 1C45B 8E
�>13>=OBL. !>>B25BAB25==>, <5@8480= 2 MB>< A;CG05
�>BAB0=5B =0 BC 65 25;8G8=C " S = 1,1< , => 2 4@C3><
=0?@02;5=88. "0:8< >1@07><, 4;O 75<=>3> =01;N40B5;O ?@8
2KAB@5;5 87 :0B0?C;LBK 24>;L <5@8480=0 ?> =0?@02;5=8N :
F5=B@C 5<;8 A=0@O4 >B:;>=8BAO C65 =0 2>AB>:, => =0 BC 65
25;8G8=C " S = 1,1< .
1) ?@8 2KAB@5;5 24>;L ?0@0;;5;8 =0 2>AB>: (70?04) (@8A.
7) A=0@O4 B0:65 4>102;O5B (B5@O5B) 078<CB0;L=K9 :><?>=5=B
8A. 7
A:>@>AB8, GB> <5=O5B 53> 40;L=>ABL ?>;5B0. @><5 B>3>, :0:
A;54C5B 87 @8A.7, 70 2@5<O ?>;5B0 A=0@O40 B>G:8 ?0@0;;5;8
�?>25@=CB >B=>A8B5;L=> ?;>A:>AB8 53> ?>;5B0, 2A;54AB285
G53> B>G:0 ?045=8O A=0@O40 >:065BAO =0 =5:>B>@>< @0AAB>O=88 " S >B ?0@0;;5;8.
"0:8< >1@07><, 40;L=>ABL ?>;5B0 A=0@O40 " r ?@8 2KAB@5;5 =0 2>AB>: (?>
=0?@02;5=8N 2@0I5=8O 48A:0)
2�0 siną
" r = (�0 cosą + �r) . (7)
g
@8 2KAB@5;5 =0 70?04 (?@>B82 =0?@02;5=8O 2@0I5=8O 75<=>3> 48A:0)
2�0 siną
" r = (�0 cosą - �r) . (8)
g
0<5B8<, GB> 2 MB>< A;CG05 (A<. @8A.7) A<5I5=85 A=0@O40 " S ?@>8AE>48B 2A5340 =0
N3, B.:. >= 2 ;N1>< A;CG05 C40;O5BAO >B F5=B@0 5<;8 (?0@0;;5;8). @8 MB><, A>3;0A=>
B5>@5<5 8D03>@0,
2
2 2
(r + " S) = r + " r . (9)
>A:>;L:C " S << r , B> 2 (9) <>6=> ?@5=51@5GL A;0305<K< ("S)2 , GB> ?@82>48B :
@57C;LB0BC
2 2
2 2
" r (�0 cosą ą �r) (2�0 siną)2 2 (�0 cosą ą �r) �0 sin2 ą
" S = = = . (10)
2 2
2r 2r g r g
10
>4G5@:=5<, GB> ?@8 @0AG5B0E 2 (10) A;54C5B 1@0BL 7=0: �+ 2 A;CG05 2KAB@5;0 =0
2>AB>:
" S = 3,4 < ,
8 7=0: �- 2 A;CG05 2KAB@5;0 =0 70?04
" S = 1,3 < .
"0:8< >1@07><, 4;O 75<=>3> =01;N40B5;O ?@8 2KAB@5;5 87 AC?5@:0B0?C;LBK 24>;L
?0@0;;5;8 A=0@O4 >B:;>=8BAO =0 N3 2 ;N1>< A;CG05, => 25;8G8=K A<5I5=89 2 MB>< A;CG05
1C4CB @07=K<8: " S = 3,4 < ?@8 2KAB@5;5 =0 2>AB>: 8 " S = 1,3 < ?@8 2KAB@5;5 =0 70?04.
5H5=8O 11 :;0AA.
040G0 1. �%>@>H> ;8 K 7=05B5 A8;C B@5=8O?
1.0 ;O 2K2>40 40==>3> A>>B=>H5=8O (C@02=5=8O 48=0<8:8 2@0I0B5;L=>3> 42865=8O)
4>AB0B>G=> @0AA<>B@5BL C@02=5=85 2B>@>3> 70:>=0 4;O =51>;LH>3> CG0AB:0 F8;8=4@0 8
70B5< ?@>AC<<8@>20BL 53> ?> 2A5< CG0AB:0< A CG5B>< B@5BL53> 70:>=0 LNB>=0.
"0: 4;O i -B>3> CG0AB:0 <>6=> 70?8A0BL
r r r
r
"miai = Fi + Ti,i+1 + Ti,i-1 ,
r r
345Ti,i+1,Ti,i-1 - A8;K, 459AB2CNI85 =0 i-BK9 CG0AB>: A>
AB>@>=K A>A5459.  ?@>5:F88 =0 B0=35=F80;L=>5
r "�
=0?@02;5=85 4;O 2A5E CG0AB:>2 ai = R , ?@8
"t
AC<<8@>20=88 2A5 2=CB@5==85 A8;K �8AG57=CB, >AB0=CBAO B>;L:> B0=35=F80;L=K5
A>AB02;ONI85 2=5H=8E A8;.
1.1 !8;0 B@5=8O, 459AB2CNI0O =0 B@C1:C, A>>1I05B 59
;8=59=>5 CA:>@5=85, ?@82>4OI55 : C25;8G5=8N A:>@>AB8
F5=B@0 B@C1:8
ma = FB@ . (1)
!:>@>ABL 1C45B 87<5=OBLAO ?> 70:>=C
FB@
V = t . (2)
m
-B0 65 A8;0 1C45B B>@<>78BL 2@0I5=85 B@C1:8, A>3;0A=>
C@02=5=8N 48=0<8:8 2@0I0B5;L=>3> 42865=8O
"�
mR = -FB@ . (3)
"t
>MB><C C3;>20O A:>@>ABL 1C45B C<5=LH0BLAO ?> 70:>=C
FB@
� = �0 - t . (4)
Rm
@>A:0;L7K20=85 (A;54>20B5;L=>, 8 87<5=5=85 A:>@>AB59) ?@5:@0B8BAO ?@8 2K?>;=5=88
CA;>28O V = �R . 7 70?8A0==KE C@02=5=89 A;54C5B, GB> A:>@>ABL CAB0=>282H53>AO
42865=8O
�0R
VCAB = . (5)
2
1.2 A;8 ?@8;>65==0O A8;0 F ?@52KH05B <0:A8<0;L=CN A8;C B@5=8O ?>:>O Fmax = �mg ,
B> CA:>@5=85 1@CA:0 =0E>48BAO 87 C@02=5=8O 2B>@>3> 70:>=0 LNB>=0
F
a = - �g , (6)
m
2 ?@>B82=>< A;CG05 CA:>@5=85 1@CA:0 @02=> =C;N.
11
1.3 0 @8AC=:5 87>1@065=K A8;K,
459AB2CNI85 =0 B5;0 (2A5 >1>7=0G5=88
B@048F8>==K).  ?@>5:F88 =0 3>@87>=B0;L=>5
=0?@02;5=85 C@02=5=8O 2B>@>3> 70:>=0
LNB>=0 8<5NB 284
m1a = F1 - T
. (1)
m2a = -F2 + T
K@078< 459AB2CNI85 A8;K B@5=8O (2A5 >=8
O2;ONBAO A8;0<8 A:>;L65=8O):
- <564C F8;8=4@>< 8 1@CA:><
F0 = �T , (2)
- <564C F8;8=4@>< 8 3>@87>=B0;L=>9 ?>25@E=>ABLN
F1 = �N1 = �(m1g - F0 ) = �(m1g - �T ), (3)
- <564C 1@CA:>< 8 ?>25@E=>ABLN
F2 = �N2 = �(m2 g + F0 ) = �(m2 g + �T ). (4)
5H0O MBC A8AB5<C C@02=5=89, ?>;CG8< CA:>@5=85 B5; (5A;8 >=8 4286CBAO)
m1 m2
-
2 2
1+ � 1- �
a = �g . (5)
m1 m2
+
2 2
1+ � 1- �
"0:8< >1@07><, 1@CA>: =0G=5B 42830BLAO ?@8
2
1- �
m2 < m1 2 H" 0,92:3 .
1+ �
<
@8 7040==>< 7=0G5=88 <0AA CA:>@5=85 A8AB5<K @02=> a H" 1,6 .
A2
1.4 0A?@545;5=85 A8; 2 40==>< A;CG05 0=0;>38G=> @0AA<>B@5==><C @0=55. >MB><C 5A;8
?@54?>;>68BL, GB> 2A5 A8;K O2;ONBAO A8;0<8 B@5=8O A:>;L65=8O, =04> ?@87=0BL
A?@0254;82>9 D>@<C;C (5) 4;O CA:>@5=89. 4=0:>, ?@8 m1 = m2 87 =55 A;54C5B, GB>
CA:>@5=85 4>;6=> 1KBL >B@8F0B5;L=K<. !;54>20B5;L=>, A8;0 B@5=8O F2 O2;O5BAO B@5=85<
?>:>O. >MB><C A8AB5<0 C@02=5=89 4;O CA:>@5=89 8<55B 284
ma = -T + F1 = -T + �(mg - �T ),
(6)
ma = T - F2
: :>B>@K< =5>1E>48<> 4>1028BL C@02=5=85 48=0<8:8 2@0I0B5;L=>3> 42865=8O 4;O
"� a
2B>@>3> F8;8=4@0 (A CG5B>< = )
"t R
a
mR = F2 - F0 = F2 - �T . (7)
R
7 MB>9 A8AB5<K =0E>48<
F2 = ma + �T
2ma
T = ma + F2 = 2ma + �T �! T = .
1- �
�g <
2
ma = �mg - T(1+ � ) �! a = H" 0,54
2
1+ � A2
1+ 2
1- �
040G0 2. �%>@>H> ;8 K 7=05B5 70:>= ?@5;><;5=8O A25B0?
12
2.1 >74CE =04 4>@>3>9 ?>4 459AB285<
A>;=5G=KE ;CG59 @07>3@5205BAO, ?>MB><C
53> ?>:070B5;L ?@5;><;5=8O <5=LH5, G5<
2 1>;55 2KA>:8E 8 E>;>4=KE A;>OE. CG8,
84CI85 >B =510, <>3CB 8A?KBK20BL
?>;=>5 >B@065=85 >B @07>3@5B>3> A;>O, GB> 2>A?@8=8<05BAO :0: ?>O2;5=85 =0 0AD0;LB5
�;C6. ;O >?8A0=8O 8E ?>;>65=8O >B=>A8B5;L=> =01;N40B5;O A;54C5B @0AA<>B@5BL
CA;>28O ?>;=>3> >B@065=8O.
7 70:>=0 ?@5;><;5=8O A;54C5B CA;>285 ?>;=>3> >B@065=8O
n(T0)cosą = n(T ) (2.1)
#G8BK20O <0;>ABL C3;0 ą (8;8, GB> @02=>A8;L=> h << S ), <>6=> ?>;>68BL
1
-
�# ś#
S h2 2 h2
cosą = = ś# ź# H" 1- .
2
ś#1+ S ź# 2
2
2S
S + h2 �# #
0;55 ?@5>1@07C5< 2K@065=85 4;O >B=>H5=8O ?>:070B5;59 ?@5;><;5=8O (?@8 @07;>65=88
a
1C45< CG8BK20BL, GB> >B:;>=5=85 ?>:070B5;O ?@5;><;5=8O >B 548=8FK <0;> << 1, 0
T
@07=>ABL B5<?5@0BC@ A@02=8<0 A> A@54=59 B5<?5@0BC@>9 2>74CE0) 
a
1+
n(T )= n(T0 + "T ) T0 + "T a a a"T
= H" 1+ - = 1- .
a
n(T0 ) n(T0 ) T0 + "T T0 T0(T0 + "T )
1+
T0
>4AB02;OO MB8 2K@065=8O 2 C@02=5=85 (2.1),
a"T h2
=
2
T0(T0 + "T ) 2S
?>;CG8< 8A:><CN 7028A8<>ABL
T0(T0 + "T )
S = h
2a"T
@0D8: MB>9 7028A8<>AB8 ?>:070= =0 @8AC=:5.
2.2 "0: :0: ?@54<5B (=8BL) =0E>48BAO =0
@0AAB>O=88 2F >B ;8=7K, B> 8 M:@0= A;54C5B
@0A?>;>68BL =0 @0AAB>O=88 b = 2F = 40A< 70
;8=7>9.
2.3 !>3;0A=> 8725AB=>9 7040G5 > :06CI59AO 3;C18=5 2>4>5<0, 87>1@065=85 =8B8 ?>A;5
?@5;><;5=8O =0 704=59 3@0=8 1@CA:0 =0E>48BAO =0 @0AAB>O=88
l
l'= >B =55. !;54>20B5;L=>, MB> 87>1@065=85 =0E>48BAO =0 @0AAB>O=88
n
n -1
a'= a - l + l'= a - l H" 33,3A<
n
>B ;8=7K. A?>;L7CO D>@<C;C ;8=7K, ?>;CG8< @0AAB>O=85 b', =0 :>B>@>< A;54C5B
@0A?>;>68BL M:@0=
-1
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2>A?>;L7>20BLAO ?@8=F8?>< N935=A0. 1;878 25@E=59
3@0=8 1@CA:0 A25B>20O 2>;=0 4>AB83=5B :@09=59 B>G:8 70
2@5<O
n1l (n0 + �n)l
t1 = = ,
c c
n0l
0 21;878 =86=59 70 2@5<O t0 = .
c
0 @07=>ABL MB8E 2@5<5= 70 704=59 3@0=LN, A25B, 8A?CI5==K9 2 =86=59 B>G:5, ?@>945B
?CBL r = c(t1 - t0 ) = �nl . !;54>20B5;L=>, D@>=B 2>;=K (8 ?5@?5=48:C;O@=K5 5<C ;CG8
�nl
?>25@=CBAO =0 <0;K9 C3>; ą H" = 1,0 �"10-3 . !>>B25BAB25==>, 87>1@065=85 =0 M:@0=5
d
A<5AB8BAO 225@E =0 25;8G8=C �z = ą �" F = 2,0 �"10-2 A< .
040G0 3. �'B> K 7=05B5 > !>;=F5?
3.1 > 70:>=C !B5D0=0->;LF<0=0 70 2@5<O "t !>;=F5 87;CG05B 2 ?@>AB@0=AB2> B5?;>2CN
M=5@38N
4 4
W = � T �"S �" "t = � T 4Ą R2"t , (1)
345 R  @048CA !>;=F0, T  01A>;NB=0O B5<?5@0BC@0 53> ?>25@E=>AB8. 0A?@>AB@0=OOAL
157 ?>B5@L 2 :>A<8G5A:>< ?@>AB@0=AB25, MB0 M=5@38O 4>AB8305B 5<;8, �>15A?5G820O =0A
A25B>< 8 B5?;><. K@078< W G5@57 A>;=5G=CN ?>AB>O==CN ł . !>3;0A=> 70:>=C
A>E@0=5=8O M=5@388 70 ?@><56CB>: 2@5<5=8 "t G5@57 2>>1@0605<CN AD5@C @048CA>< L , 2
F5=B@5 :>B>@>9 =0E>48BAO !>;=F5, 4>;6=0 ?@>9B8 B0 65 M=5@38O
W = ł 4Ą L2 "t . (2)
7 (1)  (2) =0E>48< 8A:><CN 01A>;NB=CN B5<?5@0BC@C T ?>25@E=>AB8 !>;=F0
ł L2 ł L2
4
4
T = �! T = �" . (3)
� R2 � R2
>A:>;L:C @048CA !>;=F0 R =0< =5 8725AB5=, B> 2K@078< 53> G5@57 2848<K9
C3;>2>9 480<5B@ ą 2574K
L �"ą
D = 2R = L �"ą �! R = . (4)
2
! ?><>ILN (3)  (4) =0E>48<
4ł
T = 4 = 5,8�"103  . (5)
2
�ą
>4G5@:=5<, GB> 4;O @0AG5B0 2 (5) A;54C5B ?>4AB02;OBL C3;>2>9 480<5B@ 72574K,
2K@065==K9 2 @0480=0E
Ą
ą ( @04) = ą � .
180 �
3.2 A;54AB285 =5?@5@K2=>3> 87;CG5=8O M=5@388 !>;=F5, A>3;0A=> D>@<C;5
-9=HB59=0, �EC455B, C<5=LH0O A2>N <0AAC. 0 ?@><56CB>: 2@5<5=8 "t 2 B5@<>O45@=>9
�A>;=5G=>9 ?5G8 A3>@05B ?>@F8O B>?;820 " m
E = W = ł 4Ą L2 "t = " mc2 .
" m
!>>B25BAB25==> 4>;O � = = 10 % A>;=5G=>9 <0AAK 2K3>@8B 70 2@5<O t ,
m
C4>2;5B2>@ONI55 CA;>28N
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� mc2
ł 4Ą L2 t =� mc2 �! t = . (6)
4Ą ł L2
4=0:> ?@>872>48BL @0AG5B A ?><>ILN (6) <K =5 <>65<, B0: :0: 2 CA;>288 =5
C:070=0 <0AA0 !>;=F0. K@078< 55, @0AA<0B@820O II->9 70:>= LNB>=0 (>A=>2=>9 70:>=
48=0<8:8) 4;O 42865=8O 5<;8 2>:@C3 !>;=F0
2 2
m7 m � L3 4Ą L3
2
m7� L = G �! m = = �" . (7)
2
L2 G T G
! CG5B>< (7) 2K@065=85 (6) ?@8<5B 284
2
� c2 4Ą L3 Ą�Lc2
t = �" �" = , (8)
2 2
4Ą ł L2 T G ł GT
345 T = 3,15�"107 c  ?5@8>4 >1@0I5=8O 5<;8 2>:@C3 !>;=F0. 0AG5B ?> D>@<C;5
(8) 405B
t = 4,6 �"1019 A .
!@02=820O ?>;CG5==K9 @57C;LB0B A 2>7@0AB>< A5;5==>9
(t* H" 4,5 <;@4 ;5B = 1,4 �"1017 A ) 2848<, GB> ?@>1;5<0 �?>EC40=8O !>;=F0 =5
?@54AB02;O5BAO AB>;L >AB@>9  2 @0<:0E <=>38E �75<=KE 7040G 53> <0AAC A =5?;>E>9
B>G=>ABLN <>6=> AG8B0BL ?>AB>O==>9.
3.3 ;O >F5=:8 MDD5:B82=>9 B>;I8=K h A>;=5G=>9 0B<>AD5@K 2>A?>;L7C5<AO
10@><5B@8G5A:>9 D>@<C;>9
Mgh
p (h) = p0 exp(- ) . (1)
RT
1I5?@8=OBK< AG8B05BAO A;54CNI89 ?>4E>4  70 3@0=8FC 0B<>AD5@K
?@8=8<05BAO 2KA>B0, =0 :>B>@>9 402;5=85 C<5=LH05BAO 2 5 = 2,718281... ( 5  >A=>20=85
=0BC@0;L=KE ;>30@8D<>2) @07.
! CG5B>< (1) ?>;CG05<
p0 Mgh RT
p (h) = = p0 �" e-1 = p0 exp(- ) �! h = , (2)
e RT M g
:3
345 M = 1,0 �"10-3  <>;O@=0O <0AA0 0B><0@=>3> (48AA>F88@>202H53>) 2>4>@>40,
<>;L
m <
g = G = 270  CA:>@5=85 A2>1>4=>3> ?045=8O =0 ?>25@E=>AB8 !>;=F0.
R2 A2
0AG5B 405B
h = 1,8�"105 < = 180:< .
0: 2848< ?@8 B0:>9 1>;LH>9 B5<?5@0BC@5 ?>25@E=>AB8 !>;=F0 MDD5:B82=0O
2KA>B0 0B<>AD5@K >:070;0AL 4065 <5=LH5 75<=>9 ( h H" 300:< ), GB> <>6=> ?>=OBL,
?@8=8<0O 2> 2=8<0=85 25;8G8=C A8;K 2A5<8@=>3> BO3>B5=8O =0 ?>25@E=>AB8 =0H53>
A25B8;0.
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