5H5=8O 7040G.
9 :;0AA.
040=85 1. 0CB8=0
!>3;0A=> ?@82545==K< 40==K< A>?@>B82;5=8O :>;LF52KE =8B59 7=0G8B5;L=>
<5=LH5 A>?@>B82;5=8O @0480;L=KE. >MB><C A>?@>B82;5=85< :>;LF52KE =8B59 <>6=>
?@5=51@5GL. ">340 ?@82545==0O AE5<0 <>65B 1KBL 70<5=5=0 =0 M:2820;5=B=CN.
MB>9 AE5<5 A>?@>B82;5=85 :064>3> >1>7=0G5==>3> @578AB>@0 5ABL A>?@>B82;5=85 12
A>548=5==KE ?0@0;;5;L=> >B45;L=KE 725=L52 @0480;L=KE =8B59, B.5.
1 4a
R = 1 H" 3,2 "10-2 < .
12 Ąd12
U
@8 @07><:=CB>< :;NG5 ?5@2K9 0<?5@<5B@ ?>:065B 7=0G5=85 B>:0 I1 = H" 28A .
5R1
U
A;8 70<:=CBL :;NG, B> ?>:070=8O 0<?5@<5B@>2 1C4CB >48=0:>2K 8 @02=K I2 = H" 35A .
4R1
040=85 2. :20@8C<
>A;5 @07@CH5=8O AB5=>:->A=>20=89 B@0?5F88, >AB0=5BAO >B;8G=0O >B =C;O A8;0 402;5=8O
=0 1>:>2K5 AB5=:8, :>B>@0O ?> <>4C;N @02=0 @07=>AB8 A8; 402;5=8O =0 AB5=:8->A=>20=8O.
!;54>20B5;L=>, MB0 A8;0 @02=0
h
F = g " h(a - b). (1)
2
!>3;0A=> 2B>@><C 70:>=C LNB>=0, CA:>@5=85, :>B>@>5 ?@8>1@5B5B 0:20@8C<, 1C45B @02=>
F gh2(a - b).
w = =
m 2m
040=85 3. >?>B>?=K5 25AK
1>7=0G8< <0AAC ;52>3> :8@?8G0 m1 , 0 ?@02>3> m2 , 0 @0AAB>O=85 >B ;52>3> :@0O 4>A:8 4>
C?>@0 (2 ?>;>65=88 @02=>25A8O y ). 0?8H5< CA;>28O @02=>25A8O 4>A:8 2 >1>8E A;CG0OE
a a
ś# #l ś#
m1# x - ź# ś# - x - ź#
= m2
ś#
2 2
# # # #
b b
ś# #l ś#
m1# y - ź# ś# - y - ź#
= m2
ś#
2 2
# # # #
0745;82 2B>@>5 @025=AB2> =0 ?5@2>5, ?>;CG8< C@02=5=85 4;O >?@545;5=8O =58725AB=>9
25;8G8=K y :
b b
y - l - y -
2 2
= .
a a
x - l - x -
2 2
5H5=85< MB>3> C@02=5=8O O2;O5BAO 25;8G8=0
l - b l a - b
y = x - " .
l - a 2 l - a
040=85 4. !>G8=8 70:>= <0!
1. @0D8: 7028A8<>AB8 A:>@>AB8 G0AB8FK >B
2@5<5=8 ?@54AB02;O5B A>1>9 =01>@ ?@O<>;8=59=KE
>B@57:>2, :>MDD8F85=B =0:;>=0 :>B>@KE : >A8
2@5<5=8 @025= CA:>@5=8N G0AB8FK a .
028A8<>ABL :>>@48=0BK >B 2@5<5=8
87>1@0605BAO 2 2845 =01>@0 ?0@01>;, :0640O
87 :>B>@KE >?8AK205BAO DC=:F859
at2
x = x0 + . (1)
2
!@54=OO A:>@>ABL 42865=8O G0AB8FK 70
1>;LH>9 ?@><56CB>: 2@5<5=8 @02=0
A@54=59 A:>@>AB8 =0 2@5<5==>< 8=B5@20;5 @02=>CA:>@5==>3> 42865=8O, B.5.
a
VA@. = . (2)
2
2. @5<O 42865=8O G0AB8FK =0 >4=>< 8=B5@20;5 @02=>CA:>@5==>3> 42865=8O >?@545;8<
87 C@02=5=8O
2
a 2l
l = ! = . (3)
2 a
">340 A@54=OO A:>@>ABL 42865=8O 1C45B @02=0
l al
VA@ = = . (4)
2
3. 0: 8725AB=>, M;5:B@8G5A:>5 =0?@O65=85 @02=> @01>B5 M;5:B@8G5A:8E A8; ?>
?5@5<5I5=8N 548=8G=>3> 70@O40, ?>MB><C
FL eU
U = ! F = (5)
eL
4. #A:>@5=85 M;5:B@>=0 <564C AB>;:=>25=8O<8 >?@545;O5BAO ?> 2B>@><C 70:>=C LNB>=0
FeU
a = ! a = . (6)
mmL
"0: :0: 2 B5G5=85 ?@><56CB:0 2@5<5=8 M;5:B@>= 42865BAO A ?>AB>O==K< CA:>@5=85<, 0
70B5< ?>;=>ABLN B5@O5B A2>N A:>@>ABL, 53> A@54=OO A:>@>ABL @02=0 (A<. (2))
a eU
VA@ = = . (7)
2 2mL
5. !8;0 M;5:B@8G5A:>3> B>:0 @02=0 70@O4C, ?@>E>4OI5<C G5@57 ?>?5@5G=>5 A5G5=85
?@>2>4=8:0 2 548=8FC 2@5<5=8. "0: :0: :>=F5=B@0F8O M;5:B@>=>2 2 ?@>2>4=8:5
?>AB>O==0, B> A8;0 B>:0 ?@>?>@F8>=0;L=0 A@54=59 A:>@>AB8 42865=8O M;5:B@>=>2. "0: :0:
A@54=OO A:>@>ABL ?@>?>@F8>=0;L=0 ?@8;>65==><C =0?@O65=8N, B> A8;0 B>:0
?@>?>@F8>=0;L=0 ?@8;>65==><C =0?@O65=8N, GB> 8 A>>B25BAB2C5B 70:>=C <0.
6. '5@57 ?>?5@5G=>5 A5G5=85 ?@>2>4=8:0 70 2@5<O t ?@>94CB B5 M;5:B@>=K, :>B>@K5
=0E>4OBAO >B @0AA<0B@8205<>3> A5G5=8O =0 @0AAB>O=8OE <5=LH8E, G5< VA@t , B> 5ABL 2
>1J5<5 ?@>2>4=8:0 VA@tS , C<=>60O MB>B >1J5< =0 :>=F5=B@0F8N M;5:B@>=>2, ?>;CG8<
G8A;> M;5:B@>=>2 2 MB>< >1J5<5 VA@tSn , 4;O 2KG8A;5=8O AC<<0@=>3> 70@O40 =5>1E>48<>
C<=>68BL G8A;> M;5:B@>=>2 =0 70@O4 >4=>3> M;5:B@>=0. "0:8< >1@07><, 70 2@5<O t G5@57
?>?5@5G=>5 A5G5=8O ?@>2>4=8:0 ?@>B5G5B 70@O4 q = VA@tSne . !;54>20B5;L=>, A8;0 B>:0 2
F5?8
qe2 Sn
I = = VA@Sne = U . (8)
t 2mL
7. 0?8H5< 70:>= <0 2 B@048F8>==>9 D>@<5 8 2K@078< A>?@>B82;5=85 ?@>2>4=8:0
G5@57 53> @07<5@K 8 C45;L=>5 M;5:B@8G5A:>5 A>?@>B82;5=85
U US
I = = . (9)
R L
!@02=820O 40==>5 2K@065=85 A ?>;CG5==>9 @0=55 D>@<C;>9 (8), =0E>48< C45;L=>5
M;5:B@8G5A:>5 A>?@>B82;5=85 <0B5@80;0 ?@>2>4=8:0
2m
= . (8)
e2 n
10 :;0AA.
040=85 1. >2B>@8< D878:C
%>BO =0 ?5@2K9 273;O4 7040=8O :06CBAO =5 A2O70==K<8 <564C A>1>9, ?>A:>;L:C ?@54AB02;ONB @07=K5
@0745;K D878:8, >4=0:> 8E @5H5=85 >A=>20=> =0 >48=0:>2>9 <0B5<0B8:5. 5 <>6=> CA;>2=> =0720BL
C@02=5=85< =5@07@K2=>AB8, B@51CNI8< A>E@0=5=8O =5:>B>@>9 D878G5A:>9 25;8G8=K (2 >@838=0;5 @0AE>40
2>4K) ?> 2A5< ?>?5@5G=K< A5G5=8O< ?>B>:0
1S1 = 2S2 .
ż1 7 CA;>28O 7040G8 A;54C5B A>>B=>H5=85 4;O A:>@>AB59 02B><>18;O =0 @07;8G=KE CG0AB:0E 4>@>38
0 =11 =22 , (1)
@5<O 42865=8O =0E>48BAO 87 >G5284=>3> @025=AB20, 87 :>B>@>3> A;54C5B 2K@065=85 4;O =0G0;L=>9 A:>@>AB8
02B><>18;O
l l l 1 + 2
t2 - t1 = + = (1 + 2 ) ! 0 = l . (2)
0 0
0 t2 - t1
1 2
A:><K9 3@0D8: 7028A8<>AB8 t( x ) ?@54AB02;O5B A>1>9 420 CG0AB:0 ?@O<KE A @07=K<8 C3;>2K<8
:>MDD8F85=B0<8. 0 3@0=8F5 @0745;0 ?;>E8E CG0AB:>2 02B><>18;L 1C45B 2 <><5=B 2@5<5=8
l 1
t" = t1 + = t1 + t2 (3)
( - t1 .
)
V1 1 +2
ż2. ?@>F5AA5 42865=8O H091K 55 :8=5B8G5A:0O M=5@38O C1K205B ?>4 459AB285< A8;K B@5=8O. !>3;0A=>
B5>@5<5 > :8=5B8G5A:>9 M=5@388
" EK = A" , (4)
<>6=> =09B8 =0G0;L=CN M=5@38N H091K
E0 = E + m g l ( 1 + 2 ) . (5)
@0D8: 7028A8<>AB8 :8=5B8G5A:>9 M=5@388 H091K >B :>>@48=0BK x ?@54AB02;O5B A>1>9 420 CG0AB:0 ?@O<KE A
@07=K<8 C3;>2K<8 :>MDD8F85=B0<8. 0 3@0=8F5 @0745;0 H5@>E>20BKE CG0AB:>2 M=5@38O H091K 1C45B @02=0
E" = E - 2mgl . (6)
ż3. 40==>< A;CG05 4>;65= 1KBL >48=0:>2K< ?>B>: B5?;>BK 24>;L AB5@6=O, B.5.
" t1 " t2
ł = ł , (7)
1 2
l l
:@><5 B>3>, 8<55< >G5284=>5 C@02=5=85
" t1 + " t2 = t1 - t2 . (8)
5H5=85 A8AB5<K (5)-(6) ?@82>48B : @57C;LB0BC
ł
2
"t1 = ( t1 - t2 )
ł + ł
1 2
. (9)
ł
1
"t2 = ( t1 - t2 )
ł + ł
1 2
!>>B25BAB25==> 4;O ?;>B=>AB8 ?>B>:0 B5?;>BK G5@57 ?;0AB8=C ?>;CG8<
" t1 ł ł t1 - t2
1 2
q = ł = " . (10)
1
l ł + ł l
1 2
@0D8: 7028A8<>AB8 B5<?5@0BC@K ?;0AB8=K >B :>>@48=0BK x ?@54AB02;O5B A>1>9 420 CG0AB:0 ?@O<KE A
@07=K<8 C3;>2K<8 :>MDD8F85=B0<8 . "5<?5@0BC@0 =0 3@0=8F5 @0745;0 @02=0
ł
2
t" = t1 - "t1 = t1 - (t1 - t2) . (11)
ł1 +ł
2
-B> C@02=5=85 <>6=> ?@825AB8 : 284C A>2?040NI5<C A D>@<C;>9 (3)
1
ł
1
t" = t1 + (t2 - t1) . (12)
1 1
+
ł ł
1 2
ż4. 40==>< A;CG05 8<55< A8AB5<C C@02=5=89, ?>;CG5==CN 87 3@0=8G=KE CA;>289 4;O =0?@O65==>AB8
M;5:B@8G5A:>3> ?>;O 8 A2O78 <564C =0?@O65==>ABLN 8 ?>B5=F80;><
E11 = E2
2
, (13)
E1l + E2l = 1 - 2
345 E1 8 E2 8A:><K5 =0?@O65==>AB8 ?>;59 A>>B25BAB25==> 2 ?5@2>9 8 2B>@>9 ?;0AB8=0E. 5H5=85
A8AB5<K (9) ?@82>48B : @57C;LB0BC
1 - 2
2
E1 = "
l 1 +
2
. (14)
1 - 2 1
E2 = "
l 1 +
2
@0D8: 7028A8<>AB8 ?>B5=F80;0 M;5:B@8G5A:>3> ?>;O 2=CB@8 ?;0AB8=K >B :>>@48=0BK x ?@54AB02;O5B A>1>9
420 CG0AB:0 ?@O<KE A @07=K<8 C3;>2K<8 :>MDD8F85=B0<8. >B5=F80; 3@0=8FK @0745;0
1
(15)
2 1
" = 1 + (2 -1)= 1 + (2 -1).
1 1
1 + 2
1 + 2
ż5. 40==>< A;CG05 7=0G5=85 A8;K B>:0 2 >158E G0ABOE ?;0AB8=K 4>;6=> 1KBL >48=0:>2K<, ?>MB><C =0
>A=>20=88 70:>=0 <0 <>6=> 70?8A0BL
"1 "2
1 1
= . (16)
1 l 2 l
! CG5B>< >G5284=>3> @025=AB20
"1 + "2 =1 - 2 , (17)
=0945<
1
"1 = (1 - 2 )
1 + 2
. (18)
2
"2 = (1 - 2 )
1 + 2
>4AB02;OO 2K@065=8O (13) 2 @025=AB2> (11) =0E>48< ?;>B=>ABL M;5:B@8G5A:>3> B>:0 G5@57 ?;0AB8=C (A@02=8 A
(10))
1 1
1 - 2 1 " 2
j = = (1 -) . (19)
1 + 2 1 + 1
1 2
!>3;0A=> B5>@5<5 0CAA0 4;O ?>25@E=>AB=>9 ?;>B=>AB8 70@O40 =0 3@0=8F5 @0745;0 ?;0AB8= 8<55<
1 - 2
= E1 - E2 = j ( 1 - 2 ) = (1 - 2 ) . (20)
0 1 + 2
1 - 2
=0 (1 - 2 ) (21)
1 + 2
@0D8: 7028A8<>AB8 ?>B5=F80;0 M;5:B@8G5A:>3> ?>;O 2=CB@8 ?;0AB8=K >B :>>@48=0BK x ?@54AB02;O5B A>1>9
420 CG0AB:0 ?@O<KE A @07=K<8 C3;>2K<8 :>MDD8F85=B0<8. >B5=F80; 3@0=8FK @0745;0 @0AAG8BK205BAO ?>
D>@<C;5
1
" = 1 + (2 -1) (22)
1 + 2
040=85 2. #42>5=85 8 ?045=85
0AA<>B@8< A?8FC A >4=8< H0@8:>< 2 <><5=B, :>340 >=0 A>AB02;O5B =5:>B>@K9 C3>; ą A 25@B8:0;LN.
!>3;0A=> 70:>=C A>E@0=5=8O <5E0=8G5A:>9 M=5@388 8<55<
2
mV
+m g l cosą = m g l cosą0 , (1)
2
345 l 4;8=0 A?8FK, m <0AA0 H0@8:0, V 53> A:>@>ABL 2 40==K9 <><5=B, ą0 C3>; =0G0;L=>3>
>B:;>=5=8O A?8FK. 7 (1), CG8BK20O, GB> V = l =0945< C3;>2CN A:>@>ABL 1 A?8FK 2 @0AA<0B@8205<K9
<><5=B 2@5<5=8
g
1 = 2 (cosą0 - cosą ) . (2)
l
@>2>4O 0=0;>38G=K5 @0AAC645=8O 4;O A;CG0O ?045=8O A?8FK A 42C<O H0@8:0<8, ?>;CG8<
2
l
2 2
( ml + m )2
l
4
= m g( l + )(cosą0 - cosą ) . (3)
2 2
K@060O 87 (3) 7=0G5=85 C3;>2>9 A:>@>AB8 2 , ?>;CG8<
6 g
2 = 2 " " (cosą0 - cosą ) . (4)
5 l
0: 2848< 87 (2) 8 (4) >B=>H5=85 C3;>2KE A:>@>AB59 2 : 1 >AB05BAO ?>AB>O==K< ?@8 ?045=88 A?8FK
2
6
= . (5)
1 5
-B> 3>2>@8B > B><, GB> B0: 65 1C4CB A>>B=>A8BLAO 8 A@54=85 C3;>2K5 A:>@>AB8 ?045=8O A?8F. !;54>20B5;L=>,
2@5<5=0 ?045=8O 1C4CB A2O70=K >1@0B=K< A>>B=>H5=85<
t2
5 5
= ! t2 = t1 . (6)
t1 6 6
040=85 3. >=45=A0B>@
>B5=F80;, A>740205<K9 =86=59 ?;0AB8=>9 2 B>G:5 =5 <5=O5BAO, B>340 :0: ?>B5=F80;, A>740205<K9
25@E=59 ?;0AB8=>9 ?@8 =587<5==>9 35><5B@88 ?@>?>@F8>=0;5=
1
1 = ą = ą ł . (1)
1 0
>A:>;L:C ?@8 ł = 1 70@O4K ?;0AB8= >48=0:>2K, B> ?>B5=F80;K, A>740205<K5 8<8 2 B>G:5 , 2 MB>< A;CG05
B0:65 @02=K. BAN40 A;54C5B, GB>
0
ą = . (2)
2
0
>A:>;L:C ?>B5=F80;K A:;04K20NBAO 0;351@08G5A:8, B> 8A:><0O 7028A8<>ABL ;8=59=0 8 8<55B 284
0
(ł ) = ( 1 + ł ) . (3)
2
! =0?@O65==>ABO<8 A8BC0F8O =5A:>;L:> =0?@O65==55. >A:>;L:C ?@8 ł = 1 @57C;LB8@CNI55 ?>;5 =5
8AG57;>, B> MB> 3>2>@8B > B><, GB> B>G:0 , =0E>48BAO 2 7>=5 B.=. :@052KE MDD5:B>2, 345 25:B>@K
r r
=0?@O65==>AB59 :064>9 87 ?;0AB8= 8<5NB =5 B>;L:> =>@<0;L=K5 (EĄ" ), => 8 B0=35=F80;L=K5 (EII )
:><?>=5=BK. 0: 8 2 ?5@2>< ?C=:B5 A;54C5B CG5ABL, GB> =0?@O65==>AB8 ?>;59 ?@>?>@F8>=0;L=K ?;>B=>ABO<
70@O4>2, B>340 2K@065=8O 4;O A>>B25BAB2CNI8E :><?>=5=B =0?@O65==>AB59 ?@8<CB A;54CNI89 284
E2
EII(ł ) = (1 + ł )
2
. (4)
E1
EĄ"(ł ) = (1 - ł )
2
!>>B25BAB25==> 8A:><0O 7028A8<>ABL E(ł ) <>65B 1KBL =0945=0 A ?><>ILN B5>@5<K 8D03>@0
2 2
E1 ( 1 - ł )2 + E2 ( 1 + ł )2
E(ł ) = (EII(ł ))2 + (EĄ"(ł ))2 = . (5)
2
@0D8:8 ?>;CG5==KE 7028A8<>AB59 ?>:070=K =0 @8AC=:5.
040=85 4. !C?5@?>78F8O
0) <><5=B <0:A8<0;L=>3> C40;5=8O >B ?>25@E=>AB8 ?;0AB8=K =0 @0AAB>O=85 h A:>@>ABL (:8=5B8G5A:0O
M=5@38O) M;5:B@>=0 >1@0B8BAO 2 =C;L. !;54>20B5;L=>, A>3;0A=> 70:>=C A>E@0=5=8O M=5@388 <>65< 70?8A0BL
BV02 B V02
= e E h ! h = , (1)
2 2e E
345 m <0AA0 M;5:B@>=0, 0 e 53> 70@O4.
1) @8 42865=88 M;5:B@>=0 ?>4 459AB285< A8;K >@5=F0 >= 1C45B >?8AK20BL >:@C6=>ABL @048CA0 R..
!>3;0A=> >A=>2=><C 70:>=C 48=0<8:8 <>65< 70?8A0BL
mV02 mV0
= e BV0 ! R = . (2)
R e B
!;54>20B5;L=>, 2 40==>< A;CG05 <0:A8<0;L=>5 C40;5=85 >B ?>25@E=>AB8 ?;0AB8=K 1C45B @02=>
mV0
R = . (3)
e B
2) A;CG05 >4=>2@5<5==>3> 459AB28O 42CE ?>;59 <03=8B=>5 ?>;5 1C45B ?>2>@0G820BL M;5:B@>=, 0
M;5:B@8G5A:>5 B>@<>78BL. 0?@028< >AL Ox 24>;L ?;0AB8=K, 0 >AL Oy ?5@?5=48:C;O@=>. A;8 M;5:B@>=
>:070;AO =0 @0AAB>O=88 h >B ?>25@E=>AB8 ?;0AB8=K, B> A>3;0A=> 70:>=C A>E@0=5=8O M=5@388
2
BV02 BV
- = e E h . (4)
2 2
B>@>9 70:>= LNB>=0 2 ?@>5:F88 =0 >AL Ox <>65B 1KBL 70?8A0= 2 2845
"VX " y
m = e B . (5)
" t " t
#<=>682 C@02=5=85 (5) =0 "t 8 ?@>AC<<8@>202 A CG5B>< =0G0;L=KE
CA;>289, ?>;CG8<
mVX = e B y . (6)
>A:>;L:C 2 <><5=B <0:A8<0;L=>3> C40;5=8O M;5:B@>=0 >B ?;0AB8=K
VY = 0 , B> 87 (6) ?>;CG8<, GB> A:>@>ABL V M;5:B@>=0 2 MB>B <><5=B
e B
V = VX = h (7)
m
>4AB02;OO 2K@065=85 (7) 2 70:>= A>E@0=5=8O M=5@388 (4), ?>;CG8< :204@0B=>5 C@02=5=85 4;O >?@545;5=8O
7=0G5=8O h . >;>68B5;L=K9 :>@5=L MB>3> C@02=5=8O 405B 8A:><>5 7=0G5=85 <0:A8<0;L=>3> C40;5=8O
M;5:B@>=0
2
2
# ś#
m E # mV0 ś# m E
ś# ź#
h = + ś# ź# - . (8)
ś# ź#
e B2 ś# e B ź# e B2
# #
# #
B@8F0B5;L=K9 :>@5=L <>6=> >B1@>A8BL, B0: :0: M;5:B@>= =0 >1@0B=>< ?CB8 =5 A<>65B ?@>A:>G8BL G5@57
?;0AB8=C. B<5B8<, GB> 2 MB>< A;CG05 B@05:B>@8O M;5:B@>=0 0@:0 M?8F8:;>84K.
11 :;0AA.
040G0 1. (0@8: =0 ?@82O78
0AA<>B@8< >1I89 A;CG09 42865=8O H0@8:0 2>:@C3 F8;8=4@0, B.5. 42865=85 A C3;>2K<
CA:>@5=85< 8 2 ?>;5 BO65AB8 mg . 5@5E>4 : G0AB=K< A;CG0O< >G52845= ;81> 0 ,
;81> mg 0 (;81> 8 B> 8 4@C3>5).
>;>65=85 H0@8:0 1C45< 704020BL A ?><>ILN
C3;0 - <564C 3>@87>=B0;LN 8 >B@57:><,
A>548=ONI8< F5=B@K 20;0 8 H0@8:0. 04048<
B0:65 C3>; ą <564C @048CA0<8 20;0,
=0?@02;5==K<8 : B>G:5 :0A0=8O 20;0 8 H0@8:0 8
B>G:5, 2 :>B>@>9 =8BL :0A05BAO ?>25@E=>AB8 20;0
(A<. @8A.). >=OB=>, GB> C3>; ą ?>;=>ABLN
>?@545;O5BAO >B=>H5=85< @048CA>2 20;0 8
r1 5
H0@8:0. "0: cosą = = , 8;8
r1 + r2 8
ą H" 51 H" 0,896 @04 .
r
r
(0@8: 42865BAO ?>4 459AB285<: A8;K BO65AB8 mg ; A8;K =0BO65=8O =8B8 T ; A8;K
r
@50:F88 N .
>1I5< A;CG05 CA:>@5=85 H0@8:0 <>6=> @07;>68BL =0 =>@<0;L=CN
2
an = (r1 + r2 ) (1)
8 B0=35=F80;L=CN A>AB02;ONI85
a = (r1 + r2 ). (2)
0?8H5< C@02=5=85 2B>@>3> 70:>=0 LNB>=0
r r
r r r
m(an + a ) = mg + N + T (3)
2 ?@>5:F8OE =0 @0480;L=>5 8 :0A0B5;L=>5 =0?@02;5=8O
2
ż#
m (r1 + r2 ) = mg sin + T siną - N
. (4)
#
(r1 )
#m + r2 = -mg cos + T cosą
@8 >B@K25 H0@8:0 >B ?>25@E=>AB8 20;0 A8;0 @50:F88 CAB@5<;O5BAO : =C;N (4> MB>3> >=0
4>;6=0 1KBL ?>;>68B5;L=>9). K@078< 87 A8AB5<K C@02=5=89 (4) 7=0G5=85 A8;K @50:F88
N :
m(r1 + r2 )+ mg cos
T =
cosą
m(r1 + r2 )siną + mg sin( +ą)
2
N = - m (r1 + r2 )
cosą
7 MB>3> A>>B=>H5=8O >?@545;8< DC=:F8N
N cosą
2
F = = (r1 + r2 )siną + g sin( +ą)- (r1 + r2 )cosą , (5)
m
7=0: :>B>@>9 >?@545;O5B CA;>285 >B@K20 (?>2B>@8<, >B@K2 ?@>87>945B, :>340
?>;>68B5;L=0O DC=:F8O C<5=LH8BAO 4> =C;O).
1. >340 20; @0A?>;>65= 25@B8:0;L=> (g = 0)8 =0G8=05B 2@0I0BLAO A ?>AB>O==>9 ( = 0)
C3;>2>9 A:>@>ABLN DC=:F8O (5) 8<55B 284
2
F = - (r1 + r2 )cosą , (6)
>=0 >B@8F0B5;L=0 ?@8 ;N1KE 7=0G5=8OE ?0@0<5B@>2 A8AB5<K. !;54>20B5;L=>, H0@8:
>B>@25BAO >B ?>25@E=>AB8 20;0 A@07C ?>A;5 =0G0;0 42865=8O, B.5. 2 MB>< A;CG05 = 0 .
2. A;8 20; @0A?>;>65= 25@B8:0;L=> (g = 0) 8 =0G8=05B 2@0I0BLAO A ?>AB>O==K< C3;>2K<
CA:>@5=85<, DC=:F8O (5) 8<55B 284
2
F = (r1 + r2 )siną - (r1 + r2 )cosą . (7)
#G8BK20O, GB> C3;>20O A:>@>ABL 7028A8B >B 2@5<5=8 ?> 70:>=C = t , 70:;NG05<, GB>
A=0G0;0 MB0 DC=:F8O ?>;>68B5;L=0, 0 70B5< >1@0I05BAO 2 =C;L. !;54>20B5;L=>, H0@8:
A=0G0;0 1C45B ?@860B : ?>25@E=>AB8 20;0, 0 70B5< >B>@25BAO >B =55, ?@8G5< MB>
?@>87>945B 2 <><5=B 2@5<5=8
2 2
F = (r1 + r2 )siną - (r1 + r2 )cosą = 0 !
2
. (8)
(r1 + r2 ) - r12
tgą
= = H" 0,65c
r1
30. A;8 20; @0A?>;>65= 3>@87>=B0;L=>, =0G8=05B 2@0I0BLAO A ?>AB>O==>9 ( = 0)
C3;>2>9 A:>@>ABLN, ?5@2>=0G0;L=> =8BL @0A?>;>65=0 25@B8:0;L=> (?@8 t = 0 = -ą ),
DC=:F8O (5) 8<55B 284
2
F = g sin( +ą)- (r1 + r2 )cosą (9)
8 2 =0G0;L=K9 <><5=B 2@5<5=8 >B@8F0B5;L=0. !;54>20B5;L=>, H0@8: A@07C ?>A;5 =0G0;0
42865=8O, B.5. ?@8 = 0 .
Ą
31. A;8 65 ?5@2>=0G0;L=> =8BL 3>@87>=B0;L=0 (?@8 t = 0 = -ą ), B> 7=0G5=85
2
DC=:F88 (9) ?@8 t = 0
2
F = g sin( + ą)- (r1 + r2 )cosą =
(10)
2
= (g - (r1 + r2 )cosą)H" 9.4 > 0
>MB><C ?5@2>=0G0;L=> H0@8: 1C45B ?@860B : ?>25@E=>AB8 20;0. B@K2 ?@>87>945B,
:>340 MB0 DC=:F8O >1@0B8BAO 2 =C;L (?@8G5< A;54C5B 2K1@0BL <8=8<0;L=K9
?>;>68B5;L=K9 :>@5=L MB>9 DC=:F88), B.5
2
# ś#
r1
ś# - arcsin -ą
ź#
ś#Ą ź#
g
2 # #
F = g sin(t + ą)- (r1 + r2 )cosą = 0 ! t = H" 0,73c
4. 0:>=5F, =0< =5>1E>48<> @0AA<0B@820BL ?>;=CN DC=:F8N (5). 0; @0A?>;>65=
3>@87>=B0;L=>, =0G8=05B 2@0I0BLAO A ?>AB>O==K< C3;>2K< CA:>@5=85<, ?5@2>=0G0;L=>
2
t
=8BL @0A?>;>65=0 25@B8:0;L=> ( = t , = -ą ); MB0 DC=:F8O 8<55B 284
2
2
F = (r1 + r2 )siną + g sin( + ą)- (r1 + r2 )cosą . (11)
=0G0;L=K9 <><5=B 2@5<5=8 MB0 DC=:F8O ?>;>68B5;L=0, A;54>20B5;L=>, H0@8: ?@860B :
?>25@E=>AB8 20;0, 4;O >?@545;5=8O <><5=B0 >B@K20 =0< =5>1E>48<> =09B8 <8=8<0;L=K9
?>;>68B5;L=K9 :>@5=L MB>9 DC=:F88. =0;8B8G5A:>5 @5H5=85 C@02=5=8O F = 0
=52>7<>6=>, =5>1E>48<> 8A?>;L7>20BL ?@81;865==K5 G8A;5==K5 <5B>4K. ;O 0=0;870
2
t
@5H5=8O >1>7=0G8< +ą = = , B>340 = t = 2 8 ?5@5?8H5< C@02=5=85
2
F = 0 2 2845
2(r1 + r2 )cosą (r1 + r2 )siną
sin = - , (12)
g g
8;8 >1>7=0G0O ?>AB>O==K5 :>MDD8F85=BK
2(r1 + r2 )cosą (r1 + r2 )siną
k = H" 3,06 "10-2; b = H" 1,91"10-2 , ?>;CG8< C@02=5=85,
g g
:>B>@>5 2K3;O48B 4>AB0B>G=> ?@>ABK<,
sin = k - b . (13)
>AB@>8< AE5<0B8G5A:85 3@0D8:8 DC=:F89, AB>OI8E 2 ?@02>9 8 ;52>9 G0ABOE C@02=5=8O
(13)
84 3@0D8:>2 ?>:07K205B, GB> C@02=5=85 (13) 8<55B ?>;>68B5;L=K9 :>@5=L ?@8 ;N1KE
?0@0<5B@0E A8AB5<K. @8 7040==KE 2 CA;>288 G8A;>2KE 40==KE =0:;>= 3@0D8:0 ;8=59=>9
"
7028A8<>AB8 <0;, ?>MB><C A;54C5B >6840BL, GB> :>@5=L C@02=5=8O 1;87>: : Ą , ?>MB><C
MB> G8A;> <>6=> 27OBL 2 :0G5AB25 =0G0;L=>3> ?@81;865=8O :>@=O, :>B>@>5 ?>B>< <>6=>
CB>G=8BL ;N1K< <5B>4><. @57C;LB0B5 ?>;CG05BAO A;54CNI55 7=0G5=85 :>@=O
"
2
" "
H" 3,067 , A>>B25BAB2CNI55 7=0G5=85 2@5<5=8 >B@K20 H" 3,067 = H" 1,4c .
0<5B8<, GB> H0@8: >B>@25BAO, A>25@H82 GCBL <5=LH5, G5< ?>; >1>@>B0.
5. 'B>1K H0@8: 4> >B@K20 A>25@H8; ?>;=K9
>1>@>B =5>1E>48<>, ?5@2K9 ?>;>68B5;L=K9
:>@5=L C@02=5=8O 1K; @025= 2Ą . 7>1@078<
3@0D8G5A:8 B0:CN A8BC0F8N. 0: A;54C5B 87
MB>3> @8AC=:0, 4;O @50;870F88 >?8A0==>9
A8BC0F88 4>;6=K 2K?>;=OBLAO A;54CNI85
A>>B=>H5=8O
b
= 2Ą , (14)
k
2 MB>< A;CG05 2Ą 1C45B O2;OBLAO :>@=5<
C@02=5=8O;
k e" 1, (15)
?@8 2K?>;=5=88 MB>3> CA;>28O 1C4CB
>BACBAB2>20BL <5=LH85 :>@=8.
A?>;L7CO 2K@065=8O 4;O :>MDD8F85=B>2,
CA;>285 (14) 8 53> @5H5=85 ?@8>1@5BCB 284
2
(r1 + r2 ) - r12
tgą
2
= 2Ą ! = 4Ą ! r2 = ( 16Ą +1 -1)r1 H" 58A<
2 r1
7 CA;>28O (15) =0E>48<
2(r1 + r2 )cosą 2r1 g
k = = e" 1 ! e" H" 98c-2 .
g g 2r1
040G0 2. ;8:8 =0 4=5
G5284=>, GB> 1;8:8 =0 4=5 2>7=8:0NB 1;03>40@O ?@5;><;5=8N =0 87>3=CB>9
?>25@E=>AB8 2>4K, ?@8 MB>< 3>@1K A;C60B A2>5>1@07=K<8 A>18@0NI8<8 ;8=70<8, 0
2?048=K @0AA5820NI8<8 ;8=70<8. !;54>20B5;L=>, 4;O >?8A0=8O 1;8:>2 =5>1E>48<>
@0AA<>B@5BL E>4 ;CG59 ?>A;5 ?@5;><;5=8O =0 ?>25@E=>AB8. 0<5B8<, GB> 2>;=0 42865BAO
?5@?5=48:C;O@=> ?;>A:>AB8 ?045=8O ;CG59, ?>MB><C <>6=> 87CG0BL E>4 ;CG59 2 >4=>9
25@B8:0;L=>9 ?;>A:>AB8, A>2?040NI59 A =0?@02;5=85< @0A?@>AB@0=5=8O 2>;=K. @><5
B>3>, <>6=> AG8B0BL, GB> ;CG8 ?040NB =0 ?>25@E=>ABL 25@B8:0;L=> (B>G=55, <>6=>
?@8=8<0BL 2> 2=8<0=85 B>;L:> 25@B8:0;L=K5 A>AB02;ONI85 A>;=5G=KE ;CG59).
1. 04048< C@02=5=85 ?@>D8;O 2>;=K (@8A.1):
2Ą
y = a cos# xś# . (1)
ś# ź#
# #
2. CABL 2 B>G:5 2>;=K A :>>@48=0B>9 x
r
25:B>@ =>@<0;8 n : ?>25@E=>AB8 2>;=K
>1@07C5B A 25@B8:0;LN C3>; (@8A. 2). 0945<
7028A8<>ABL MB>3> C3;0 >B :>>@48=0BK.
>=OB=>, GB> MB>B C3>; @025= C3;C, :>B>@K9
>1@07C5B :0A0B5;L=0O A 3>@87>=B0;LN. 0:
8725AB=>, B0=35=A C3;0 =0:;>=0 :0A0B5;L=>9
3@0D8:C DC=:F88 : >A8 0@3C<5=B0 @025=
?5@2>9 ?@>872>4=>9 >B 87CG05<>9 DC=:F88.
"0: :0: 2 @0AA<0B@8205<>9 7040G5 4;8=0 2>;=K 7=0G8B5;L=> 1>;LH5 55 0<?;8BC4K, C3>;
<>6=> AG8B0BL <0;K<, A;54>20B5;L=>, 745AL 8 40;55 B0=35=A 8 A8=CA MB>3> C3;0 <>6=>
?@8=8<0BL @02=K<8 A0<><C C3;C. "0:8< >1@07><, 8A:><0O 7028A8<>ABL 8<55B 284
2Ą 2Ą
2
(x) = - y = a sin# xś# . (2)
ś# ź#
# #
2Ą
0<5B8<, GB> <0:A8<0;L=>5 7=0G5=85 MB>3> C3;0 max = a H" 0,3. @8 MB>< ?>3@5H=>AB8
70<5= A8=CA>2 8 B0=35=A>2 =0 7=0G5=85 A0<>3> C3;0 =5 ?@52KH05B 4% .
3. CABL 25@B8:0;L=K9 ;CG ?>?0405B =0 ?>25@E=>ABL 2>4K
2 B>G:5 A A :>>@48=0B>9 x . MB>9 B>G:5 25:B>@ =>@<0;8
=0?@02;5= ?>4 C3;>< : 25@B8:0;8, A;54>20B5;L=>, >= 8
O2;O5BAO C3;>< ?045=8O. 0945< C3>; ą <564C ;CG><
?@5;><;5==K< 8 25@B8:0;LN (@8A.3). #G8BK20O, GB> 2A5
C3;K <0;K 8 8A?>;L7CO 70:>= ?@5;><;5=8O, =0945< C3>;
?@5;><;5=8O ł = , B>340 8A:><K9 C3>; <>6=>
n
@0AAG8B0BL ?> D>@<C;5
n -1
ą = -ł = . (3)
n
4. #G8BK20O, GB> @0AA<0B@8205<K5 3;C18=K 7=0G8B5;L=> ?@52KH0NB 0<?;8BC4C 2>;=K,
?>A;54=59 <>6=> ?@5=51@5GL ?@8 >?@545;5=88 :>>@48=0BK B>G:8 ?045=8O ;CG0 =0 4=>.
0: ;53:> 70<5B8BL (@8A.3)
x1 = x - h " tgą H" x - hą . (4)
5. 0?8H5< DC=:F8N 7028A8<>AB8 x1 >B x 2 O2=>< 2845
2Ąa n -1 2Ą
x1 = x - h " sin# xś# . (5)
ś# ź#
n
# #
!E5<0B8G5A:85 3@0D8:8
MB>9 DC=:F88 ?@8 @07;8G=KE
7=0G5=8OE 3;C18=K ?>:070=K =0
@8A.4 (=><5@0 3@0D8:>2 40=K 2
?>@O4:5 2>7@0AB0=8O 3;C18=K).
6. !0<>9 ACI5AB25==>9 >A>15==>ABLN 40==KE 7028A8<>AB59 O2;O5BAO ?5@5E>4 >B
<>=>B>==>3> 2>7@0AB0=8O : ?>O2;5=8N <0:A8<C<>< 8 <8=8<C<>2.
59AB28B5;L=>, @0AA<>B@8< <0;K9 8=B5@20; (?>;>A:C) =0 4=5 H8@8=>9 "x1, =0 =55
?>?04CB ;CG8, :>B>@K5 ?5@5A5:;8 ?>25@E=>ABL 2>4K 2 A>>B25BAB2CNI5< 8=B5@20;5 "x .
>=OB=>, GB> G5< 1>;LH5 "x , B5< 1>;LH5 A25B0 ?>?045B =0 ?>;>A:C "x1 (@8A. 5). >MB><C
"x
A@54=OO >A25I5==>ABL ?>;>A:8 4=0 2>7;5 B>G:8 x1 ?@>?>@F8>=0;L=0 >B=>H5=8N .
"x1
>MB><C G5< 1>;55 ?>;>30 7028A8<>ABL x1(x) 2 40==>9 B>G:5, B5< 55 >A25I5==>ABL
1>;LH5. 1;878 <0:A8<C<0 >=0 AB0=>28BAO >G5=L 1>;LH>9, 0 21;878 B>G:8 ?5@53810 5I5
1>;LH5. >MB><C <0:A8<C< 8=B5=A82=>AB8 1C45B 4>AB830BLAO =0 B>9 3;C18=5, 345 3@0D8:
7028A8<>AB8 x1(x) 1C45B 8<5BL B>G:8 ?5@53810.
AB0B8, =0 1>;LH8E 3;C18=0E 4;O @0AG5B0 8=B5=A82=>AB8 2 =5:>B>@KE B>G:0E 4=0
B@51C5BAO ?@>25AB8 AC<<8@>20=85 ?> B@5<, ?OB8 8 1>;LH5<C G8A;C 8=B5@20;>2 =0
?>25@E=>AB8. 5 1C45< 70=8<0BLAO MB>9 @01>B>9. >AB0B>G=>, GB> <K ?>;CG8;8 <5B>4
@0AG5B0 3;C18=K <0:A8<0;L=>9 D>:CA8@>2:8 - 2 7028A8<>AB8 x1(x) 4>;6=K ?>O2;OBLAO
B>G:8 ?5@53810. ?@81;865=88 <0;KE C3;>2 MB0 7028A8<>ABL 8<55B 284 (5)
@>872>4=0O >B MB>9 DC=:F88 @02=0
2
2Ą n -1 2Ą
2
x1 = 1- ha# ś# " cos# xś# . (6)
ś# ź# ś# ź#
n
# # # #
@8 <0;KE 3;C18=0E h DC=:F8O (5) M:AB@5<C<>2 =5 8<55B (@8A. 5), ?@8 h 1>;LH8E
=5:>B>@>3> :@8B8G5A:>3> 7=0G5=8O ?>O2;ONBAO B>G:8 <8=8<C<>2 8 <0:A8<C<>2. -B>
?@>87>945B B>340, :>340 ?@>872>4=0O DC=:F8O (6) 1C45B ?@8=8<0BL 2 =5:>B>@KE B>G:0E
2
=C;52K5 7=0G5=8O. =K<8 A;>20<8, :>340 C@02=5=85 x1 = 0 8<55B :>@=8, 2 7028A8<>AB8
x1(x)?>O2;ONBAO M:AB@5<C<K - 3@0=8G=>5 7=0G5=85 <564C MB8<8 42C<O A;CG0O<8 - B>G:8
?5@53810. B0:, CA;>285 D>:CA8@>2:8 ?>;CG05B 01AB@0:B=CN <0B5<0B8G5A:CN
D>@<C;8@>2:C - C DC=:F88 (6) ?>O2;ONBAO =C;8!
0: A;54C5B 87 0=0;870 DC=:F88 (5) 5A;8 ?5@20O ?@>872>4=0O @02=0 =C;N ?@8
x = x1 = 0, MB0 B>G:0 1C45B B>G:>9 ?5@53810, B> 5ABL 8<5==> 2 MB>< A;CG05 O@:>ABL ?>;>A>:
1C45B <0:A8<0;L=0. ;O >?@545;5=8O MB>9 >?B8<0;L=>9 3;C18=K =5>1E>48<> @5H8BL
?@>AB59H55 C@02=5=85
2
2Ą n -1
1- ha# ś# " = 0 , (7)
ś# ź#
n
# #
87 :>B>@>3> =0E>48< 3;C18=C D>:CA8@>2:8
2
n
# ś#
h0 = " H" 2,0< . (8)
ś# ź#
2Ą a(n -1)
# #
;LB5@=0B82=K9 A?>A>1.
0AA<>B@8< D>:CA8@>2:C ;CG59, ?>?040NI8E 2 <0;CN >1;0ABL 21;878 <0:A8<C<0 2>;=K. ;O MB>3>
1C45< AG8B0BL 2 DC=:F88 (5) x <0;K<, B>340 MB0 DC=:F8O C?@>I05BAO
2
#
2Ąa n -1 2Ą
ź#x .
ś#1- ś# 2Ą ź# n -1ś#
x1 = x - h " sin# xś# H" ha# ś# " (9)
ś# ź#
ś# ź#
n n
# # # #
# #
0;55 70<5G05<, GB> 5A;8 :>MDD8F85=B 2 A:>1:0E >1@0I05BAO 2 =C;L, B> 2A5 ?040NI85 ;CG8 A>18@0NBAO 2
B>G:5 x1 = 0 , B> 5ABL D>:CA8@CNBAO. 7 MB>3> =5?>A@54AB25==> A;54C5B 2K@065=85 (8).
8. ;C18=0 2>4>5<0 h = 0,5< O2;O5BAO <0;>9 ?> A@02=5=8N A =0945==>9 3;C18=>9
D>:CA8@>2:8. >MB><C 2 MB>< A;CG05 DC=:F8O (5) M:AB@5<C<>2 =5 8<55B, ?>MB><C, :0:
"x 1
1K;> >B<5G5=> @0=55, >A25I5==>ABL 4=0 ?@>?>@F8>=0;L=0 = , 8 <>65B 1KBL
2
"x1 x1
70?8A0=0 2 2845
I0
I = . (10)
2
2Ą n -1 2Ą
1- ha# ś# " cos# xś#
ś# ź# ś# ź#
n
# # # #
!B@>3> 3>2>@O, 2 MB> 2K@065=85 A;54C5B
?>4AB028BL x :0: DC=:F8N x1, 2K@065==CN 87
DC=:F88 (5). 4=0:>, CG8BK20O <0;>ABL C3;>2
>B:;>=5=8O ;CG59 >B 25@B8:0;8, <0;>ABL
3;C18=K, 4;O ?>AB@>5=8O AE5<0B8G5A:>3>
3@0D8:0 <>6=> ?>;>68BL x1 H" x . "@51C5<K9
3@0D8: ?>:070= =0 @8A. 6.
9. @8 3;C18=5 >75@0 h = 2,5< DC=:F8O
(5) 8<55B 4>AB0B>G=> 1;87:>
@0A?>;>65==K5 <0:A8<C<K 8
<8=8<C<K. !;54>20B5;L=>, =0 MB>9
3;C18=5 :0640O O@:0O ?>;>A0
@074208205BAO, B0: :0: :064><C
<8=8<C<C 8 <0:A8<C<C A>>B25BAB2C5B
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?>:070=0 =0 @8A.7.
;O >?@545;5=8O ?>;>65=8O <0:A8<C<0 >A25I5==>AB8 x1 =0 4=5 =5>1E>48<> @5H8BL
A8AB5<C C@02=5=89 (5) (6). 7 CA;>28O >1@0I5=8O ?@>872>4=>9 (6) 2 =C;L =0E>48<
2
2Ą n -1 2Ą h0
1- ha# ś# " cos# xś# = 0 ! x = ą arccos H" ą0,10< ,
ś# ź# ś# ź#
n 2Ą h
# # # #
>4AB02;OO MB> 7=0G5=8O 2 DC=:F8N (5), =0E>48<
2Ąa n -1 2Ą
x1 = x - h " sin# xś# H" m1,7A< .
ś# ź#
n
# #
:>=G0B5;L=>, H8@8=0 ?>;>A:8 @02=0 "x1 = 2x1 H" 3,5A< .
040G0 3. >153 8 ?>3>=O
1. 0AG5B <0AAK >G52845= m = łV = ĄR2hł .
2. 1AC48< D878G5A:85 ?@8G8=K, ?@82>4OI85 2 42865=85 <>=5BC. @8 2:;NG5=88 B>:0
2>7=8:05B ?5@5<5==>5 <03=8B=>5 ?>;5, :>B>@>5 8=4CF8@C5B 2 <>=5B5 M;5:B@8G5A:85 B>:8,
:>B>@K5 2 A2>N >G5@54L, 2708<>459AB2CNB A <03=8B=K< ?>;5<. -B> 2708<>459AB285
?@82545B : ?>O2;5=8N A8;K <?5@0, A>>1I0NI59 <>=5B5 CA:>@5=85. "0: :0: ?> CA;>28N
B>:, 0, A;54>20B5;L=>, 8 <03=8B=>5 ?>;5 2>7@0AB0NB 1KAB@>, B> <>6=> ?@5=51@5GL
A<5I5=85< <>=5BK 70 2@5<O 2>7@0AB0=8O B>:0 2 :015;5. "> 5ABL, 4>AB0B>G=> @0AAG8B0BL
8<?C;LA, :>B>@K9 ?@8>1@5B5B <>=5B0. 0 >A=>20=88 ?@028;0 5=F0 <>6=> CB25@640BL,
GB> 8<?C;LA <>=5BK 1C45B =0?@02;5= >B :015;O (GB>1K C<5=LH8BL 2>7@0AB0=85
<03=8B=>3> ?>B>:0), ?>MB><C 40;L=59H85 @0AG5BK 1C45< ?@>2>48BL 157 CG5B0 7=0:>2, B>
5ABL ?> <>4C;N. @8ABC?8< : @0AG5BC A:>@>AB8 <>=5BK.
!B@>3> 3>2>@O, <03=8B=>5 ?>;5 2 ?@545;0E
<>;5:C;K O2;O5BAO =5>4=>@>4=K<. 4=0:>,
?@8=8<0O 2> 2=8<0=85 <0;K5 @07<5@K <>=5BK, ?>
A@02=5=8N A @0AAB>O=85< 4> :015;O, ?@8 @0AG5B5
@0A?@545;5=8O B>:>2 2 <>=5B5 =5>4=>@>4=>ABLN
?>;O ?@5=51@565<. #G8BK20O >A52CN A8<<5B@8N
<>=5BK, A;54C5B AG8B0BL, GB> @0A?@545;5=85 B>:>2
2 <>=5B5 O2;O5BAO :@C3>2K<, ?@8G5< 25;8G8=0
?;>B=>AB8 B>:0 7028A8B @0AAB>O=8O 4> F5=B@0
<>=5BK. K45;8< 2 <>=5B5 B>=:>5 :>;LF> @048CA0
r , B>;I8=>9 dr 8 =0945< A8;C B>:0 2 MB><
:>;LF5. 03=8B=K9 ?>B>: G5@57 ?;>I04L :>;LF0 @025=
0I 0I
2 2 2
Ś = Ąr B0 = Ąr = r , (1)
2Ąl 2l
0I
745AL B0 = - 8=4C:F8O <03=8B=>3> ?>;O 2 F5=B@5 <>=5BK, @0AAG8BK205<0O ?>
2Ąl
8725AB=>9 D>@<C;5 8=4C:F88 ?>;O ?@O<>3> B>:0.
@8 87<5=5=88 A8;K B>:0 2 :015;5, 2 @0AA<0B@8205<>< :>;LF5 2>7=8:=5B -! 8=4C:F88,
@02=0O ?> 70:>=C $0@045O
2
dŚ 0I
2
ind = = r , (2)
dt 2l
dI
2
745AL I = - A:>@>ABL 87<5=5=8O B>:0 2 :015;5, :@><5 B>3>, A>3;0A=> A45;0==>9 @0=55
dt
>3>2>@:5 >?CI5= 7=0: <8=CA.
-;5:B@8G5A:0O ?@>2>48<>ABL 2K45;5==>3> :>;LF0 @0AAG8BK205BAO ?> 8725AB=>9 D>@<C;5
1 hdr
g = = , (3)
RM;. 2Ąr
l
745AL RM;. = - M;5:B@8G5A:>5 A>?@>B82;5=85, =5 ?CB0BL A @048CA>< <>=5BK 1C:2 =5
S
E20B05B.
> 70:>=C <0, A8;0 B>:0 2 2K45;5==>< :>;LF5
2 2
hdr 0I 0I h
2
di = gind = " r = rdr . (4)
2Ąr 2l 4Ąl
0<5B8<, GB> 2 >4=>@>4=>< <03=8B=>< ?>;5, A8;0, 459AB2CNI0O =0 :>=BC@ A B>:><,
>@85=B8@>20==K9 B0:, :0: =0H0 <>=5B0, @02=0 =C;N.
4=0:> <>6=> ?>:070BL, GB> 2 =5>4=>@>4=>< ?>;5 ?>O2;O5BAO A8;0,
:>B>@0O 4;O <0;>3> :>=BC@0 <>65B 1KBL 70?8A0=0 2 2845
dB
F = IS , (5)
dx
dB
745AL I, S - A8;0 B>:0 2 :>=BC@5 8 53> ?;>I04L, - ?@>872>4=0O
dx
>B 25;8G8=K 8=4C:F88 ?>;O ?> :>>@48=0B5, 2KG8A;5==0O 2 F5=B@5 :>=BC@0. =0H5<
A;CG05
0I 0I
dB d # ś#
= = (6)
ś# ź# - .
2
dx dx 2Ąx 2Ąl
# #
"0:8< >1@07><, A8;0, 459AB2CNI0O =0 2K45;5==>5 :>;LF>, @02=0
2
2 2
0I 0I h 0I 0 II h
2 2
dF = di "(Ąr )" = rdr "Ąr " = r3dr . (7)
2
2Ąl 4Ąl 2Ąx2 8Ąl3
@>8=B53@8@>202 ?> 2A5<C :>;LFC, ?>;CG8< 2K@065=85, 4;O ?>;=>9 A8;K, 459AB2CNI59
=0 <>=5BC
R
2 2
2 2 2
0 II h 0 II h R4 2
0 II hR4
F = r3dr = " = (8)
3
+"
8Ąl3 8Ąl 4 32Ąl3
0
> 2B>@><C 70:>=C LNB>=0 87<5=5=85 8<?C;LA0 B5;0, @02=> 8<?C;LAC A8;K, :>B>@K9 2
40==>< A;CG05 <>6=> 2KG8A;8BL, 4065 =5 7=0O O2=>3> 2840 7028A8<>AB8 A8;K B>:0 2
:015;5 >B 2@5<5=8
2 2 2 2 2 2 2
0 hR4 0 hR4 0 hR4 0 hR4 I0 0 hR4I0
mV = = " = . (9)
+"Fdt 32Ąl3 +"II 2 dt = +"I dI dt = +"IdI =
32Ąl3 dt 32Ąl3 32Ąl3 4 128Ąl3
0:>=5F, 8A?>;L7CO 2K@065=85 4;O <0AAK, ?>;CG8< 2K@065=85 4;O A:>@>AB8 <>=5BK
2 2 2 2
0 hR4I0 1 0 R2I0
V0 = " = . (10)
2
128Ąl3 ĄR2hł 128Ą l3ł
3. @8 42865=88 <>=5BK 2 AB0F8>=0@=>< ?>;5 B0:65 1C45B 2>7=8:0BL -! 8=4C:F88 8
:0: A;54AB285 M;5:B@8G5A:85 B>:8, 2 MB>< A;CG05 70 AG5B =5>4=>@>4=>AB8 <03=8B=>3> ?>;O.
!;54>20B5;L=>, =0 :0BOICNAO <>=5BC 1C45B 459AB2>20BL B>@<>7OI0O A8;0, G0AB>
=07K205<0O <03=8B=>9 2O7:>ABLN. 0?@02;5=85 MB>9 A8;K, ?@>B82>?>;>6=>5 A:>@>AB8
B0:65 <>6=> >1>A=>20BL ?@028;>< 5=F0. ;O @0AG5B0 MB>9 A8;K ?@8<5=8<
M=5@35B8G5A:89 ?>4E>4 - @01>B0 (8 <>I=>ABL) B>@<>7OI59 A8;K @02=0 :>;8G5AB2C
(<>I=>AB8) 2K45;O5<>9 2 <>=5B5 46>C;52>9 B5?;>BK. "0:8< >1@07><, =0< =5>1E>48<>
@0AAG8B0BL <>I=>ABL 2K45;ONI59AO B5?;>BK. I5 @07 >B<5B8<, GB> -! 8=4C:F88
2>7=8:05B 2A;54AB285 42865=8O <>=5BK 2 =5>4=>@>4=>< ?>;5, 2 >AB0;L=>< 65 @0AG5B
B>:>2 >AB05BAO ?@56=8<, ?>MB><C ?@>2545< 53> 157 ?>4@>1=KE :><<5=B0@852.
03=8B=K9 ?>B>: G5@57 2K45;5==>5 :>;LF>
0I0 0I0 2
2 2
Ś = Ąr B(x) = Ąr = r . (11)
2Ąx 2x
-! 8=4C:F88 2 :>;LF5
2 2 2
# ś#
dŚ d 0I0r 0I0r dx 0I0r
ś# ź# -
ind = = = = - V . (12)
ś# ź#
dt dt 2x 2x2 dt 2x2
# #
>I=>ABL 46>C;52>9 B5?;>BK, 2K45;ONI59AO 2 :>;LF5
2
2 2 2
# ś#
hdr 0I0r 0 I0 h
2 2
ś# ź#
dP = gind = " V = V r3dr . (13)
ś#
2Ąr 2x2 ź# 8Ąx4
# #
>A;5 8=B53@8@>20=8O ?> 2A59 ?;>A:>AB8 <>=5BK ?>;CG8< ?>;=CN <>I=>ABL
R
2 2 2 2
0 I0 h 0 I0 hR4
2 2
P = V r3dr = V , (14)
+"
8Ąx4 32Ąx4
0
:>B>@CN ?@8@02=O5< : <>I=>AB8 A8;K P = FV , 8 =0945< O2=>5 2K@065=85 4;O MB>9 A8;K
2 2
0 I0 hR4
F = V . (15)
32Ąx4
0: A;54C5B 87 MB>9 D>@<C;K, B>@<>7OI0O A8;0 ?@>?>@F8>=0;L=0 A:>@>AB8 8 7028A8B >B
:>>@48=0BK :>;LF0. 0?8H5< C@02=5=85 2B>@>3> 70:>=0 LNB>=0 4;O 4286CI53>AO :>;LF0
dV C
m = - V , (16)
dt x4
2 2
0 I0 hR4
345 >1>7=0G5=> C = - ?>AB>O==K9 :>MDD8F85=B. "0: :0: =0A 8=B5@5AC5B B>;L:>
32Ą
:>=5G=>5 ?>;>65=85 :>;LF0 (:>340 53> A:>@>ABL AB0=5B @02=>9 =C;N), <>65<
?@5>1@07>20BL C@02=5=85 (16), 8710282H8AL 2 =5< >B 2@5<5=8:
dV dV dx dV C dV C
m = m " = m V = - V ! m = - . (17)
dt dx dt dx x4 dx x4
>A;54=55 65 C@02=5=85 <>65B 1KBL ;53:> ?@>8=B53@8@>20=> A CG5B>< =0G0;L=KE 8
:>=5G=KE CA;>289
L
dx dx 1 1
ś#
. (18)
mdV = -C ! - mV0 = -C = -C# - ź#
ś#
+"
x3 x4 l3 L3
# #
l
>4AB02;OO 7=0G5=8O :>=AB0=BK 8 =0G0;L=>9 A:>@>AB8, 87 C@02=5=8O (18) =0E>48<
4
3
:>=5G=>5 ?>;>65=85 <>=5BK L = " l 8 ?CBL ?@>945==K9 5N
3
# ś#
4
3
ś#
"x = L - l = -1ź#l (19)
ś# ź#
3
# #
>4AB0=>2:0 G8A;5==>3> 7=0G5=8O =0G0;L=>3> @0AAB>O=8O ?@82>48B : @57C;LB0BC
"x H" 5,0A<.
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