Zadania z różnych lat solution

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( 5H5=8O)
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<5E0=8G5A:>9 0=0;>3859 : ?>AB@>8<
42C:@0B=>5 87>1@065=85 CG0AB:0
<564C ;C3>< 8 @5:>9. 7>1@065=8O
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G5284=>, GB> :@0BG09H89 ?CBL >B
B>G:8 4> B2 2 5ABL >B@57>: ?@O<>9
AB2 2 . -B>1@065=8N ?CB8
A>>B25BAB2C5B @50;L=0O B@05:B>@8O
<8=8<0;L=>9 4;8=K ACDB . 7 ?>AB@>5=8O A;54C5B, GB> 55 4;8=0
@02=0 4;8=5 >B@57:0 AB2 2 :
lmin = l2 + 4h2 .
2. #4>1=55 2A53> @5H0BL 7040GC, 70<5B82, GB> ?@8 ?>4>1=><
42865=88 A ?>AB>O==>9 ?> 25;8G8=5 A:>@>ABLN ?CBL, ?@>945==K9
A>10:>9, <>65B 1KBL =0945= ?> D>@S = v0� , (1)
345 v0 A:>@>ABL A>10:8, � 2@5?5H5E>40<8 4> <><5=B0 8E 2AB@5G8.
@5 =0945<, >?@545;82 A:>@>ABL A1;865=8O
?5H5E>4>2 v = vA + vB :
l0
� = . (2)
vA + vB
! CG5B>< (2) @025=AB2> (1) ?5@5?8H5BAO 2 2845:
v0
S = l0 = 11:< .
,
vA + vB
!>>B25BAB2CNI89 3@0D8: ?@82545=
( ) ( )
=0 @8AC=:5. 45AL: xA t 8 xB t
( )
7028A8<>AB8 ?5H5E>4>2 A 8 B, xC t
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=B5@5A=0 A8BC0F8O 21;878
B>G:8 2AB@5G8 ?5H5E>4>2. ><0=0O =0
3@0D8:5 1C45B 8<5BL 15A:>=5G=>5
:>;8G5AB2> 725=L52, E>BO 2@542865=8O A>10:8 :>=5G=>. >=5G=> 65, ?@8 @5H5=88 A>10:C <0B5@80;L=>9 B>G:>9, B.:. 59 >G5=L A;>6=> 1C45B
2
:@CB8BLAO <564C ?5H5E>40<8, :>340 @0AAB>O=85 <564C =8<8
AB0=5B <5=LH5 4;8=K A>10:8.
3. 7040G5 =8G53> =5 A:070=> > B><, 345 2AB@5G0NBAO
02B>1CAK ?0AA068@C: B>;L:> ?> ?CB8 8;8 =0 :>=5G=KE AB0=F8OE
B>65? -B> =0?@O A 2>?@>A><: AB>OB ;8 02B>1CAK =0
AB0=F8OE 8;8 >=8 ?@0:B8G5A:8 <3=>25==> >B?@02;ONBAO 2 ?CBL?
0715@5< >10 20@80=B0.
0@80=B 1. 2B>1CAK =5 AB>OB =0 AB0=F8OE, ?>MB>?0AA068@ =0AG8B0; 2 4>@>35 B@8 2AB@5G=KE 02B>1CA0. >AB@>8<
3@0D8: :>>@48=0BK 02B>1CA0 A =0H8< ?0AA068@>< (B>;AB0O ;8=8O
=0 @8A.). 1>7=0G5=K B>G:8, A>>B25BAB2CNI85 2AB@5G0<
02B>1CA>2., B.5. G5@57 MB8 B>G:8 ?@>E>4OB 3@0D8:8 :>>@48=0B
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>AB@>8< MB8 3@0D8:8. >;CG05< ?@>AB@0=AB25==> 2@5<5==CN
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=0 ;8=88. -B> <>6=>
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D8:A8@>20==>3> <><5=B0
2@5<5=8. 0?@8<5@, 4;O t=8
<8=; >48= 02B>1CA
=0E>48BAO 2 ?C=:B5 A,
2B>@>9 2 B, 0 5I5 420
2AB@5B8;8AL ?>A5@548=5
<564C A 8 B. B>3> 4 02B>1CA0. @5A:>@>ABL 42865=8O
8:<
v == 30:< / 80A.
16 <8=.
0@80=B 2. 0AA068@ AG8B05B 8
02B>1CAK, AB>OI85 =0 :>=5G=KE AB0=F8OE.
">340 3@0D8:8 :>>@48=0B 2 02B>1CA>2 2
7028A8<>AB8 >B 2@5<5=8 8<5NB 284,
?@54AB02;5==K9 =0 A;54CNI5< @8AC=:5.
2B>1CAK AB>OB ?> 8 <8=CB =0 :>=5G=KE ?C=:B0E (=5 <5=LH5,
8=0G5 >=8 =0G=CB G0I5 2AB@5G0BLAO). >4AG5B G8A;0 02B>1CA>2
>?OBL 405B G8A;> 4. ! CG5B>< 2@5<5=8 AB>O=:8 A@54=OO A:>@>ABL
02B>1CA>2 >:07K205BAO ?>-?@56=51CAK 54CB 8
<8=CB A> A@54=59 A:>@>ABLN 60 :OB). 2B>1CAK
<>3CB AB>OBL 8 1>;LH5, => B>340 4>;6=> 1KBL C25;8G5=> 8 G8A;>
02B>1CA>2 =0 ;8=88.
3
4. "0: :0: <564C C40@0<8 H0@8: 42865BAO ?> ?@O<>9, 0
C40@K 01A>;NB=> C?@C38, B> B@05:B>@8O H0@8:0 ?>4>1=0
B@05:B>@88 ;CG0 A25B0 <564C 75@:0;0<8. >AB@>8< <=>3>:@0B=K5
87>1@065=8O �75@:0; 4@C3 2 4@C35. -B8
87>1@065=8O 1C4CB ?@54AB02;OBL ;CG8,
2KE>4OI85 87 >4=>3> F5=B@0, C3>; <564C
:>B>@K<8 @025= ą = 10o . "> 5ABL B0:8E ;CG59
35. "5?5@L 42865=85 H0@8:0 <>6=>
@0AA<0B@820BL :0: 42865=85 2 �?@>AB@0=AB25
87>1@065=89 ?> ?@O<>9 AB. 7
B@5C3>;L=8:0 AOC ;53:> =09B8 <8=8<0;L=>5
@0AAB>O=85 OC, =0 :>B>@>5 ?@81;878BAO H0@8: : F5=B@C
rmin = h sin� = 07 <.
,
'B>1K H0@8: 8A?KB0; <0:A8<0;L=>5 G8A;> >B@065=89, 53>
=5>1E>48<> =0?@028BL ?>4 =51>;LH8< C3;>< : >A8 :>=CA0,
=0?@8<5@, 24>;L AB1. G5284=>, GB> <0:A8<0;L=>5 G8A;>
>B@065=89 18.
0 A;54CNI5<
@8AC=:5 ?>:070=0 8AB8==0O
B@05:B>@8O ABCDE , 8 55
�87>1@065=85 ABC' D' E'
4;O H0@8:0 C?@C3>
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1>;LH59 =03;O4=>AB8 87>1@065=8O).
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:0: ?>:070=> =0 @8AC=:5. 0 2@5t C3>; ?>2>@>B0 >B=>A8B5;L=>
F5=B@>2 A>>B25BAB2CNI8E
>:@C6=>AB59 @025=
� = � t.
">340 :>>@48=0BK B>G5: A 8 B 2
?@>872>;L=K9 <><5=B 2@5<5=8
xA = -R + R cos� t, xA = R = R sin� t,
ż# ż#
�# �#
yA = R sin� t, yB = R cos � t
�# �#
A;8 G:8 A, B> :>>@48=0BK B>G:8 B
>B=>A8B5;L=> 1C4CB B0:8<8
4
x0 = xB - xA = 2R + R sin� t - cos� t ,
ż#
( )
�#
y0 = yB - yA = R cos� t - sin� t .
()
�#
A;8 MB8 :>>@48=0BK A;>68BL, B> 8E AC<<0 @02=0 ?>AB>O==>9
25;8G8=5
x0 + y0 = 2R.
>>1I5 3>2>@O, MB0 D>@C@02=5=85 ?@O<>9 y0 = 2R - x0 2
A8AB5<5 >BAG5B0, A2O70==>9 A
4286CI59AO B>G:>9 A. 4=0:>,
42865=85 B>G:8 B >3@0=8G5=>.
0?@8<5@, :>>@48=0BK y ;568B 2
?@545;0E >B R 2 4> - R 2 . "0:8<
>1@07><, B>G:0 B 42865BAO
>B=>A8B5;L=> B>G:8 A 24>;L
>B@57:0 ?@O<>9 y0 = 2R - x0 .
6. 0AA<>B@8< 42865=85 >4=>9 G5@5?0E8 A. =0 42865BAO A
?>AB>O==>9 ?> <>4C;N A:>@>ABLN, =0?@02;5=85 :>B>@>9 ?>AB>O==>
<5=O5BAO, B.:. A<5I05BAO B>B >1J5:B B, =0 :>B>@K9 MB0 A:>@>ABL
=0?@02;5=0. A5 G5@5?0E8 4286CBAO >48=0:>2K< >1@07><, ?>MB>2 ;N1>9 <><5=B 2@5<5=8 >=8 1C4CB =0E>48BAO 2 25@H8=0E
?@028;L=>3> n-C3>;L=8:0, 2?8A0==>3> 2 >:@C6=>ABL <5=LH53> G5<
R @048CA0. 0945< A :0:>9 A:>@>ABLN C<5=LH05BAO @048CA R .
@>5:F8O A:>@>AB8 G5@5?0E8 =0 @048CA vr = vcosą 5ABL
25;8G8=0 ?>AB>O==0O, 2 A8;C A>E@0=5=8O A8<<5B@88 8E
@0A?>;>65=8O A> 2@5<5=5<, B.5. C3;0 ą . 0945< MB>B C3>;
Ą - �
2ą - � = Ą �! ą = ,
2
0 C3>; � 4;O ?@028;L=>3> n-C3>;L=8:0, 2?8A0==>3> 2 >:@C6=>ABL
@025=
2Ą
� = .
n
B0:
Ą Ą
ą = - ,
2 n
B>340
Ą Ą Ą
ś#
vr = vcos�# - = v sin .
ś# ź#
�# #
2 n n
5
@5AB865=8O F5=B@0 >:@C6=>AB8 B>G:8 O ?@8 @02=><5@=><
?@81;865=88 : =59
R R
t = = .
vr v sin Ą
n
-B>B 65 @57C;LB0B <>6=> ?>;CG8BL 8 4@C38< A?>A>1><. !:>@>ABL
�
A1;865=8O G5@5?0E vC = v - vcos (MB> >B=>A8B5;L=0O A:>@>ABL
2
42865=8O >4=>9 G5@5?0E8 >B=>A8B5;L=> 4@C3>9). =0O, GB>
=0G0;L=>5 @0AAB>O=85 <564C =8<8 MB> AB>@>=0 ?@028;L=>3> n -
C3>;L=8:0, 2?8A0==>3> 2 >:@C6=>ABL, <>6=> >?@545;8BL 2@5A1;865=8O. >4@>1=>AB8 @5H5=8O 8 8;;NAB@0F88 <>6=> =09B8 2
6C@=0;5 $>:CA (!1-2 70 1993 3., A. 53-55).
7. !:>@>ABL 42865=8O ;N1>9 B>G:8 =0 ?>25@E=>AB8 AD5@K
=0?@02;5=0 @0480;L=> 8 <>65B 1KBL 70?8A0=0 2 2845
r
r r1
v1 = v0 ,
R
r
345 v0 A:>@>ABL =0@0AB0=8O @048CA0 R , r1
@048CA-25:B>@ B>G:8 2 A8AB5<5 >BAG5B0,
A2O70==>9 A F5=B@>< AD5@K.
B=>A8B5;L=0O A:>@>ABL @071530=8O
r r r v0 r
vOTH = v1 - v2 = r1 - r2 .
(r )
R
"0:8< >1@07><, A:>@>ABL @071530=8O ?@>?>@F8>=0;L=0
@0AAB>O=8N <564C B>G:0<8.
r r
8..5:B>@ ?5@5<5I5=8O ;>:><>B820 S0 = AB , 25:B>@
r r
?5@5<5I5=8O ?>A;54=53> 203>=0 S1 = CC1 . /A=>, GB>
r r
ĄR
S0 = R 2 , S1 = , 0 C3>; <564C 25:B>@0<8
2
r r r
� = 45o . B=>A8B5;L=>5 ?5@5<5I5=85 "S = S0 - S1
2 2 2
"S = S0 + S1 - 2S0S1 cos� = 2R2 + Ą R2 / 4 - ĄR2
2
= R 2 + Ą / 4 - Ą
B=>A8B5;L=0O A:>@>ABL v>B= = v 2 .
6
9.�@C??8@>2:0 8AG57=5B , :>340 A0<>18;L 87
2B>@>9 3@C??K 4>3>=8B A0<>18;L 87 ?5@2>9.
0 2@5<>18;L C40;8BAO =0
,
@0AAB>O=85 S0 = v1� , B>340 �1KAB@>B@51C5BAO 2@5S0 v1�
"t = = , GB>1K 53> 4>3=0BL, ?@8 MB>< >= ?@>945B
v2 - v1 v2 - v1
v1v2
@0AAB>O=85 S = v2"� = � H" 10:< .
v2 - v1
10.1J5< @07<>B0==>9 70 2@52KA>B0, MB>B 65 >1J5< V = Ą R - r h . @8@02=820O, =0945<
( )
vd
7028A8<>ABL r(t), r = r02 - t , 0 C3;>20O A:>@>ABL 2@0I5=8O
Ą
v v
� = = .
r
vd
r02 - t
Ą
BAN40 <>6=> 2K@078BL G0AB>BC 2@0I5=8O :0BCH:8
� v
n = = .
2Ą
vd
2Ą r02 - t
Ą
11. ;O @5H5=8O 7040G8 4>?>;=8< A8AB52@0I0NI8EAO 65AB:8E AB5@6=59 B0:, :0: C:070=> =0 @8AC=:5.
>A:>;L:C ?@8 2@0I5=88 A8AB5>AB020BLAO ?0@0;;5;L=K<8 (2 A8;C 65AB:>AB8 3>@87>=B0;L=KE
AB5@6=59), B> 42865=85 B>G:8 ! 1C45B 2@0I5=85< A B>9 65 C3;>2>9
A:>@>ABLN (G0AB>B>9
2@0I5=8O) >B=>A8B5;L=>
B>G:8 O, GB> 8 B>G:8 .
A;8 AB = l , B>:
vB
� = �! vC = � �" 3l = 3vB .
l
r r
@8 MB>< 25:B>@ vC ?0@0;;5;5= 25:B>@C vB , B.5. A>AB02;O5B C3>;
Ą
- ą A 3>@87>=B><.
2
0<5B8<, GB> ?>4>1=K5 65AB:85 <5E0=8G5A:85 :>=AB@C:F88
4>AB0B>G=> G0AB> ?@8<5=ONBAO 4;O C<=>65=8O (vC = 3vB (!))
7
A:>@>AB8 8;8 ?5@5<5I5=8O 2 @07;8G=KE CAB@>9AB20E: >B 45BA:8E
83@CH5: 8 1KB>2KE ACH8;>: 4> A:;04=KE A>;=5G=KE 10B0@59
:>A<8G5A:8E 0??0@0B>2. @8 MB>< :>MDD8F85=B C<=>65=8O
>?@545;O5BAO G8A;>< 725=L52 2 A8AB5<5 (2 ?@82545==>< ?@8<5@5
n = 3 ).
12. CABL <0H8=8AB =0G0; 874020BL 3C4>: 2 <><5=B 2@5<5=8 t0 ,
:>340 ?>574 =0E>48;AO =0 @0AAB>O=88 l (2 B>G:5 C) >B ?5@55740, 0
70:>=G8; 2 B>G:5 D 2 <><5=B 2@5<5=8 t1 = t0 + � , ?@8 MB>< ?>574
=0E>48;AO =0 @0AAB>O=88 l - �� >B ?5@55740. 0G0;> 3C4:0
4>AB83;> B>G:8 A 2 <><5=B 2@5<5=8
l
t = t0 + (745AL c A:>@>ABL 72C:0), 0
AO
c
:>=5F 3C4:0 2 <><5=B 2@5<5=8
l - v�
tA1 = t0 + . !;54>20B5;L=>,
c
4;8B5;L=>ABL 3C4:0 2 B>G:5 A
v
�#1 ś#
� = tA1 - tA0 = � - = 3,24c.
ś# ź#
A
�#
c #
=0;>38G=>, 4;O B>G:8 B <>6=> ?>;CG8BL
2
l
BD BC ( - v� - l2 + h2
)
�B = � + - = � + = 325c.
,
c c c
13. 0945< A:>@>ABL 42865=8O A?CB=8:0 87 70:>=0 2A5<8@=>3>
BO3>B5=8O 8 2B>@>3> 70:>=0 LNB>=0
mv2 Mm
= G , (1)
2
R + H
R + H
( )
345 M - <0AA0 5<;8, R - 55 @048CA, m - <0AA0 A?CB=8:0, G -
3@028B0F8>==0O ?>AB>O==0O.
">340
GM v1
v = = , (2)
R + H 2
GM
345 v1 = ?5@20O :>A<8G5A:0O A:>@>ABL, 0 ?> CA;>28N
R
H = R .
"5=L A?CB=8:0 42865BAO ?> ?>25@E=>AB8
5<;8, :>340 A?CB=8: ?@>E>48B ?> 4C35 AB.
#3;>20O 25;8G8=0 MB>9 4C38 <>65B 1KBL
=0945=0 87 35><5B@8G5A:8E A>>1@065=89
8
*"
R Ą
AB = 2ą = 2 arcsin = ,
2R 3
GB> A>AB02;O5B 1/6 ?5@8>40 >1@0I5=8O A?CB=8:0 , ?@8G5<
2Ą R + H
( ) 4 2ĄR
T = = . (3)
v v1
CBL B5=8 @025= ?>;>28=5 4;8=K M:20B>@0 S = ĄR .
!;54>20B5;L=>, A@54=OO A:>@>ABL B5=8
S S 3
vcp = = = v1.
T
t
2 2
6
14. >6=> @5H8BL MBC 7040GC B@048F8>==K< A?>A>1><, 2K18@0O
>A8 Ox 24>;L =0:;>==>9 ?;>A:>AB8 8 Oy - ?5@?5=48:C;O@=> 59. K
4C<05<, GB> G8B0B5;O< ?@8 65;0=88 ?>4 A8;C ?@>45;0BL MB8
2KG8A;5=8O A0<>AB>OB5;L=>. 45AL ;55 A>2@5<5==>5
25:B>@=>5 @5H5=85 ?5@2>9 G0AB8 7040G8.
CABL B>G:0 O - =0G0;> >BAG5B0,
B>G:0 B - <5AB> ?045=8O B5;0 =0
=0:;>==CN ?;>A:>ABL. 048CA-
25:B>@ B>G:8 B =0945< 87 70:>=0
@02=>CA:>@5==>3> 42865=8O,
70?8A0==>3> 2 25:B>@=>9 D>@<5
r

gt2
r r r r
OB = rB = r0 + v0t + , r0 = 0.
2
r
2

r gt
5:B>@K v0t = OA 8 = AB
2
87>1@065=K =0 @8AC=:5.
r
1>7=0G8< <>4C;L rB G5@57 S MB> 8 5ABL 40;L=>ABL ?>;5B0 B5;0
24>;L =0:;>==>9 ?;>A:>AB8. > B5>@5<5 :>A8=CA>2 87 "OAB
8<55<
2
�# ś#
gt2 2
= v0t + S2 - 2Sv0t cos�. (1)
( )
ś# ź#
�# 2 #
! 4@C3>9 AB>@>=K, 87 ?@O<>C3>;L=KE B@5C3>;L=8:>2 AOC 8 ABC
gt2 siną
S +
OC OB + BC OB + AB siną
2
cos� = = = = .
OA OA OA v0t
>4AB028< cos� 2 (1)
9
2
2
�# ś# �# gt ś#
gt2 2
= v0t + S2 - 2S S + siną ,
( )
ś# ź# ś# ź#
�# 2 # �# 2 #
8;8
g2t4
2
- v0 - gS siną t2 + S2 = 0. (2)
()
4
#@02=5=85 (2) 5ABL 18:204@0B=>5 C@02=5=85 >B=>A8B5;L=>
=58725AB=>3> 2@5<5=8 ?>;5B0 ?@8 7040==>9 40;L=>AB8. ;O
:064>3> S < Smax 1C45B ACI5AB2>20BL 420 @07;8G=KE :>@=O t1 , t2
(425 B@05:B>@88 ?>;5B0) C@02=5=8O (2) 8 5I5 420 :>@=O A> 7=0:><
<8=CA , :>B>@K5 2 =0H5< A;CG05 A;6=K 1KBL
>B1@>H5=K. B>G:C, A>>B25BAB2CNICN <0:A8<0;L=>9 40;L=>AB8
?>;5B0, <>6=> ?>?0ABL B>;L:> 548=AB25==K< >1@07><.
!;54>20B5;L=>, 2 MB>< A;CG05 :>@=8 t1 8 t2 4>;6=K A>2?040BL. -B>
>7=0G05B, GB> 48A:@8<8=0=B C@02=5=8O (2) >B=>A8B5;L=> t2 4>;65=
>1@0I0BLAO 2 =C;L
2
g2 2
2
D = v0 - gSmax siną - 4 Smax =
()
4
4 2 2 2
= v0 - 2gSmaxv0 siną + g2Smax sin2 ą - g2Smax = 0
BAN40 ;CG05< :204@0B=>5 C@02=5=85 >B=>A8B5;L=> Smax
2 2 4
g2 cosąSmax + 2gv0 sinąSmax - v0 = 0,
@5H5=8O :>B>@>3> @02=K
2 4 4
2
- gv0 siną ą g2v0 sin2 ą + g2v0 cos2 ą v0 ą 1 - siną
Smax = = �" .
g2 cos2 ą g 1 - sin2 ą
"0: :0: A@5=L, A>>B25BAB2CNI89 7=0:C + , B> ?>A;5
A>:@0I5=8O =0 1 - siną >:>=G0B5;L=> ?>;CG05<
2
v0
Smax = .
g 1 + siną)
(
@8 ą = 0 >BAN40 A;54C5B 8725AB=K9 @57C;LB0B 4;O B5;0,
2
v0
1@>H5==>3> ?>4 C3;>< � : 3>@87>=BC: Smax = , :>B>@K9
g
4>AB8305B ?@8 1@>A:5 ?>4 C3;>< � = 45o .
10
15. 40==>< A;CG05 H0@8: ?>4 459AB285< A8;K 402;5=8O 25B@0
42865BAO =5 2 25@B8:0;L=>9 ?;>A:>AB8 AB , :0: MB> >1KG=>
A;CG0;>AL ?@8 ?@5=51@565=88 459AB285< 25B@0 =>@<0;L=>3>
r
25:B>@C v0 . @8 ?>4>1=>< B@5E<5@=>< 42865=88 ?045=8O A>25@H8B ?5@5<5I5=85 CB 2 =0?@02;5=88 25B@0.
"0:8< >1@07><, >= ?@875<;8BAO =0 @0AAB>O=88
AC = AB2 + BC2 (1)
>B >A=>20=8O 10H=8.
@5;5B0 >?@545;O5BAO :0: 8 2 A;CG05 A2>1>4=>3> ?045=8O:
2H
t = . 0;55:
g
2H
AB = v0t = v0 . (2)
g
2
at a
BC = = H. (3)
2 g
! CG5B>< (2) 8 (3) 2K@065=85 (1) ?@8<5B 284:
1 1
2 2
AC = 2gHv0 + a2H2 = H 2gv0 + a2H . (4)
()
g g
=B5@5A=> B0:65 >1AC48BL B@05:B>@8N H0@8:0 2 40==>< A;CG05.
0:>= 42865=8O <>6=> 70?8A0BL 2
2845
ż#
�#x = vot;
�#
2
�#y = h - gt
;
�#
2
�#
2
�# at
�#z = ;
�# 2
@>5:F88 B@05:B>@88 =0
25@B8:0;L=CN ?;>A:>ABL XY 8
3>@87>=B0;L=CN ZX O2;ONBAO
25B2O<8 ?0@01>;;K. @>5:F8O 65
=0 ?;>A:>ABL ZY ?@54AB02;O5B
>B@57>: ?@O<>9. "0:8< >1@07>< 2AO B@05:B>@8O ;568B 2 ?;>A:>AB8,
=0:;>=5==>9 ?>4=5:>B>@K< C3;>< ą : 3>@87>=BC 8 A>45@68B 2
r
A515 25:B>@ g* , >?@545;O5@=K< @025=AB2><:
r* r r
ż#
�#g = g + a .
(5)
�#tgą = g
�#
�# a
11
1KG=> 2 ?>4>1=KE A;CG0OE 53> =07K20NB MDD5:B82=K<
CA:>@5=85< A2>1>4=>3> ?045=8O : 459AB28B5;L=>, 2 ?;>A:>AB8
AA2 2 C =0 @8AC=:5 87>1@065=0 B@05:B>@8O ADC 2 2
r
A>>B25BAB2CNI0O A;CG0N 53> A2>1>4=>3> ?045=8O 2 ?>;5 g* A
r
3>@87>=B0;L=>9 =0G0;L=>9 A:>@>ABLN v0 g*Ą"v0 .
(r r )
16. 07>1L5< @5H5=88 7040G8 =0 420 MB0?0. 5@2K9 MB0? - 42865=85
1C;02:8 4> >B@K20 >B AB>;0. B>@>9 - A2>1>4=>5 ?045=85
<0B5@80;L=>9 B>G:8 A =0G0;L=>9 A:>@>ABLN A =5:>B>@>9 2KA>BK.
1 MB0? . C;02:0 ?0405B ?>4 2>7459AB285< 42CE A8;
r
- A8;K BO65AB8 8 A8;K @50:F88 >?>@K N .
!G8B05<, GB> >AB@K9 :>=5F 1C;02:8 C?8@05BAO 2
70:@C3;5=85 AB>;0 8 =5 A:>;L78B. CABL >AL
1C;02:8 >B:;>=8;0AL =0 C3>; ą >B 25@B8:0;8.
0?8H5< II 70:>= LNB>=0
r
r r
mg + N = ma
2 ?@>5:F88 =0 =0?@02;5=88 >A8 1C;02:8.
!>AB02;ONI0O CA:>@5=8O 24>;L MB>9 >A8 MB>
mv2
F5=B@>AB@5<8B5;L=>5 CA:>@5=85 , 345 R = l 4;8=0 1C;02:8.
R
">340
mv2
mg cosą - N = .
R
><5=B >B@K20 >1CA;>2;5= 8AG57=>25=85< A8;K @50:F88, B.5.
m�2
N = 0 8;8 mg cos = .
R
! 4@C3>9 AB>@>=K ?> 70:>=C A>E@0=5=8O M=5@388 3>;>2:0
1C;02:8 ?@8>1@5B05B :8=5B8G5A:CN M=5@38N 70 AG5B C<5=LH5=8O
?>B5=F80;L=>9
mv2 v2
= mgR 1 - cosą) ( .
( �! = 2g 1 - cosą)
2 R
>4AB0282 MB> 2 ?@54K4CI55 C@02=5=85 ?>;CG05<, GB>
2
g cosą = 2g 1 - cosą) �! cosą = 2 3 , v2 = gRcosą = gR.
(
3
II MB0?. !G8B0O 1C;02:C <0B5@80;L=>9 B>G:>9, 70?8H5< C@02=5=85
:>>@48=0B 4;O 3>;>2:8 1C;02:8
x = x0 + vxt ,
ayt2
y = y0 + vyt + .
2
7 @8AC=:0 284=>, GB>
12
2 2
x0 = l siną , vx = vcosą = gR ,
3 3
5 2
y0 =-l cosą , vy = v siną = gR , ay = g.
3 3
045=85 ?@>8AE>48B A 2KA>BK 2 <, A;54>20B5;L=>,
=0G0;L=K5 :>>@48=0BK x0 8 y0 25AL<0 <0;K
�# ś#
5 002
,
x = �"001 <, y0 = - < 8 8<8 <>6=> ?@5=51@5GL ?>
,
ś# ź#
3 3
�# #
A@02=5=8N, =0?@8<5@, A 2KA>B>9 ?045=8O. :>=G0B5;L=>
2 2
x = vt cosą = gRt ,
3 3
22
gt gt
y = vyt + = vt siną + .
22
<><5=B ?045=8O y = H , ?>MB>;CG05BAO :204@0B=>5
C@02=5=85 4;O 2@5<5=8 ?045=8O
2
v siną ą v2 sin2 ą + 2gH
gt
+ vt siną - H = 0, t = ,
2 g
5 2gR 10gR 10R
2
!@02=8< v2 siną = �" = 8 2gH . /A=>, GB> 2H >>
9 3 27 27
8 <>6=> 2>A?>;L7>20BLAO ?@81;865==>9 D>@9
- v siną ą 2gH
t H"
g
K18@05< 7=0: + 8 =0E>48< 2@5"5?5@L ;53:> =09B8 8 3>@87>=B0;L=>5 A<5I5=85 1C;02:8 ?@8
?045=88
x H" 01< .
,
17. A;CG05 ?5@5<5I5=8O :8@?8G0 ?> 4286CI59AO ;5=B5 A8;0
B@5=8O 1C45B =0?@02;5=0 ?@>B82
>B=>A8B5;L=>9 A:>@>AB8 :8@?8G0
?> ;5=B5. 5@5945< 2 A8AB5>BAG5B0, A2O70==CN A ;5=B>9.
!>3;0A=> ?@5>1@07>20=8O<
0;8;5O, A:>@>ABL :8@?8G0 2 MB>9
A8AB5<5 >BAG5B0
r r r
u = v1 - v0 .
r
r
">340 25:B>@ FB@ >@85=B8@>20= ?@>B82 25:B>@0 u 8 @025=
� mg = 31,4 H . !;54>20B5;L=>, A8;C, @02=CN 31,4 =5>1E>48<>
13
v1
r
?@8;>68BL 24>;L 25:B>@0 u ?>4 C3;>< ą = arctg = 2o : 25:B>@C
v0
r
(- v0 .
)
18. 0AA<>B@8< @02=>25A85 CG0AB:0 F5?>G:8 "mi , 2848<>3> 87
F5=B@0 :>;LF0 ?>4 <0;K<
C3;>< �i �i 0 . !>3;0A=>
()
CA;>28N @02=>25A8O
r r r r
r
"mg + N + T1 + T2 = O, (1)
i
r
345 N A8;0 @50:F88 A>
AB>@>=K ?>25@E=>AB8
r r
:>=CA0, T1 8 T2 A8;K
=0BO65=8O F5?>G:8 2
A>>B25BAB2CNI8E A5G5=8OE.
r
7-70 <0;>AB8 �i ?@8<5<, GB> N =0?@02;5=0 24>;L >A8 A8<<5B@88
CG0AB:0.
@>5F8@CO (1) =0 25@B8:0;L=CN 8 3>@87>=B0;L=CN >A8,
?>;CG05<:
m
ż#
N siną = "mi g = �i g,
�#
2Ą
(2)
�#
�i 1
�#
2T sin = �i 0 = 2T�i = N cosą ,
{}
�# 2 2
r r
345 T = T1 = T2 . 7 A8AB5mg
T = ctgą. (3)
2Ą
@8 ą 0 (:>=CA �?@52@0I05BAO 2 F8;8=4@) T ", 0 MB>
3>2>@8B > B><, GB> ;N10O F5?>G:0 745AL ?>@25BAO (?@8 CA;>288
>BACBAB28O B@5=8O - A@01>B05B �MDD5:B :;8=0). =K<8 A;>20<8, 2
B0:>9 A8BC0F88 =0 3;04:>< F8;8=4@5 =8:0: C45@60BLAO =5;L7O.
Ą
@8 ą = (:>=CA �?@52@0I05BAO 2 ?;>A:>ABL),
2
5AB5AB25==>, T = 0 . MB>< A;CG05 F5?>G:0 ?>:>8BAO 1C4CG8
=5=0?@O65==>9.
19. >A:>;L:C 1;>: =0E>48BAO 2 @02=>25A88, AC<<0 A8;,
459AB2CNI8E =0 =53>, @02=0 =C;N
14
2T0 - 2T = 0, T0 = T .
!8;C =0BO65=8O =8B8 A 3@C70<8 =0945<,
70?8A02 2B>@>9 70:>= LNB>=0 4;O 3@C7>2.
T - m1g =-m1a, T - m2g = m2a.
B:C40
2mm2
1
T = g.
m1 + m2
#G8BK20O, GB> m1 >> m2 , 8<55<
T0 H" 2m2 g.
=B5@5A=> >B<5B8BL, GB> MB0 A8;0 =0BO65=8O
>?@545;O5BAO 25A>< <5=LH53> 3@C70.
20. 0 @8AC=:5 >1>7=0G5=K B>;L:> A8;K, 459AB2CNI85 2
3>@7>=B0;L=>< =0?@02;5=88, ?>A:>;L:C A8;K BO65AB8 8 A8;K
@50:F88 >?>@K :>8A?>;L7CNBAO. C45< AG8B0BL
4;O >?@545;5==>AB8, GB>
F2 > F1 , B.5. 2AO A8AB5<0
42865BAO @02=>CA:>@5==>
=0?@02>, 2 AB>@>=C >A8 x.
0?8H5< ?@>5:F88 2B>@>3>
70:>=0 LNB>=0 =0 >AL x 4;O 2A5E B@5E B5;:
F2 - TB = m2a,
ż#
�#T - TB = m0a,
�#
A
�#TA - F1 = m1a.
�#
!:;04K20O MB8 C@02=5=8O 8 2K@060O CA:>@5=85, ?>;CG05<
F2 - F1
a = .
m1 + m2 + m0
>4AB0=>2:0 CA:>@5=8O 2 ?5@2>5 8 B@5BL5 C@02=5=8O A8AB5?>72>;O5B =09B8 8A:>F2 - F1 m1F2 + m0 F2 + m2 F1
TB = F2 - m2 �" = ,
m1 + m2 + m0 m1 + m2 + m0
F2 - F1 m2 F1 + m0 F1 + m1F2
TA = F1 - m1 �" = ,
m1 + m2 + m0 m1 + m2 + m0
15
>4C<09B5, GB> 87<5=88BAO, 5A;8 F1 > F2 . 5I5 >G5=L 8=B5@5A=K9
A;CG09, :>340 F1 = F2 . C4CB ;8 A8;K =0BO65=8O TA 8 TB B>65
>48=0:>2K?
21. 7>1@078< A8;K, 459AB2CNI85 2 40==>9 <5E0=8G5A:>9
A8AB5<5. 0AA<>B@8< ?5@2K9 A;CG09, :>340 : ?@02>9 =8B8
?@8:@5?;5= 1@CA>: <0AA>9 2m. 7 D0:B0 =0G0;0 42865=8O 3@C70 rm
?> 3>@87>=B0;L=>9 ?;>A:>AB8 45;05< 2K2>4, GB> A8;0 B@5=8O FB@
4>AB83;0 A2>53> ?@545;L=>3> 7=0G5=8O (B@5=85 ?>:>O A<5=8;>AL
B@5=85< A:>;L65=8O):
FB@ = �N . (1)
">340:
T1 - T2 - �mg = 0,
ż#
�#T = 2mg,
(2)
�#
1
�#
T2 = mg.
�#
7 (2) =0E>48<:
� = 1. (3)
B=>A8B5;L=> (3) 70<5B8<, GB> ?>4>1=K5 7=0G5=8O :>MDD8F85=B0
B@5=8O 4>AB0B>G=> @54:8 (=0?@8<5@, @578=0 ?> @578=5 <>65B 4020BL
8 1>;LH55 7=0G5=85 � ), >4=0:> ?@8=F8?80;L=KE >3@0=8G5=89 =0
25;8G8=C � =5 ACI5AB2C5B (2 >B;8G85 >B :>MDD8F85=B0 ?>;57=>3>
459AB28O � , =0?@8<5@).
;O 42865=8O 3@C70 2?@02> A CA:>@5=85< ?5@5?8H5<
T1 - T2 - �mg = ma,
ż#
�#
�#- T3 + m* g = m* a
(4)
�#T - mg = ma,
4
�#
�#
T2 = T4 ; T1 = T3 .
�#
345 m* 8A:><0O <0AA0 3@C70. >A;54=55 @025=AB2> A8AB5A;54C5B 87 CA;>28O =525A><>AB8 8 3;04:>AB8 =5?>4286=KE 1;>:>2 8
=8B59. 5H0O A8AB5;CG8<:
2 a + g
( )
m* = m (5)
g - a
16
7 (5) A;54C5B, GB> ?@8 a = 0 m* = 2m, B.5. 745AL A>45@68BAO A;CG09
=0G0;0 A:>;L65=8O 3@C70, >?8AK205A:>;L:C
<0AA0 =5 <>65B 1KBL >B@8F0B5;L=K9, B>:
a d" g. (6)
-B> CA;>285 4>AB0B>G=> >G5284=>: 2 ?>4>1=>9 A8AB5<5 =52>7<>6=>
4>AB8GL 1>;LH53> CA:>@5=8O . @545;L=>5 7=0G5=85 a g
@50;87C5BAO 2 A;CG05 m* ",, B.5. :>340 425H8205< �38@N :
2>74CH=>=8<0B5;L=K9 G8B0B5;L :>=5G=> 65 70<5B8;, GB> 2 CA;>288
=5 C:070=> =0?@02;5=85 CA:>@5=8O. -B> B@51C5B >B =0A @5H5=8O
r
7040G8 8 4;O 2B>@>3> A;CG0O: 25:B>@ a =0?@02;5= 2;52>. 4=0:>
4065 ?@8 m* = 0 (?@020O =8BL 2>>1I5 =5 =03@C65=0), =>2K9 3@C7 m
=5 A<>65B A428=CBL 1@CA>: A <5AB0 ?> ?@8G8=5 1>;LH>3> B@5=8O 2
A8AB5<5. "0:8< >1@07><, MB>B 20@80=B ?@8 40==KE CA;>28OE
@50;87>20BL =5 C40ABAO (?> CA;>28N 65< <5=OBL B>;L:> 3@C7
m* ).
22. 0AA<>B@8< =5:>B>@CN ?K;8=:C <0AA>9 m, =0E>4OICNAO =0
@0AAB>O=88 r >B >A8 2@0I5=8O. >?CAB8<,
GB> >=0 =5 A;5B5;0 A 48A:0, B.5. 2@0I05BAO
2<5AB5 A =8<. A=>2=>9 70:>= 48=0<8:8
70?8H5BAO 2 2845:
r r
r r
ma = mg + N + FB@ , (1)
r
r
2
345 a = a = � �" r CA:>@5=85 ?K;8=:8, N
r
A8;0 @50:F88 48A:0, FB@ A8;0 B@5=8O ?K;8=:8 > 48A:. 5:B>@0
r r
r
ma 8 FB@ ;560B 2 ?;>A:>AB8 48A:0, 25:B>@ N 59 =>@<0;5=.
r
@54AB028< 25:B>@ mg :0: AC<r r r
mg = mg'' + mgĄ" , (2)
r r
345 25:B>@K g'' = g siną 8 gĄ" = g cosą ;560B A>>B25BAB25==> 2
?;>A:>AB8 48A:0 8 =>@<0;L=K ?>A;54=59 . ! CG5B>< ?@>872545==>3>
@07;>65=8O 25:B>@=>5 C@02=5=85 (1) <>65B 1KBL ?@54AB02;5=>
A8AB5<>9:
r
r r
ż#
ma = mg'' + FB@ ,
r r (2)
�# r
O = mgĄ" + N .
�#
0AA<>B@8< 45B0;L=55 ?5@2>5 C@02=5=85 A8AB5r
4>?>;=8B5;L=K5 ?>AB@>5=8O 2 ?;>A:>AB8 48A:0: 25:B>@ mg''' 745AL
r
1C45B =0?@02;5= �25@B8:0;L=> 2=87, 25:B>@ - ma 1C45B
=5?@5@K2=> <5=OBL A2>N >@85=B0F8N, �2@0I0OAL ?> >:@C6=>AB8,
17
r
25:B>@ FB@ 4>;65=, A>>B25BAB25==>, 70<:=CBL 25:B>@=K9
B@5C3>;L=8:: @07;8G=K< ?>;>65=8O< ?K;8=:8 A>>B25BAB2CNB
@07;8G=K5 ?> =0?@02;5=8N CA:>@5=8O, => <>4C;L MB>3> 25:B>@0
?>AB>O=5=
r
2
a = � �" r.
r
G5284=>, GB> <0:A8<0;L=>5 7=0G5=85 FB@ , @50;87C5BAO B>< A;CG05 ,
r
r
:>340 25:B>@ - ma3 ?@>B82>?>;>65= 25:B>@C mg'' , B> 5ABL 2 =86=5<
?>;>65=88 ?K;8=:8. 45AL:
2
FB@ = mg siną + m� r. (3)
"0:8< >1@07><, :070;8, GB> A@K2 ?K;8=>: A 48A:0 1C45B
=0G8=0BLAO A 53> =86=59 B>G:8. #G8BK20O, GB> ?@545;L=>5 7=0G5=85
A8;K B@5=8O ?>:>O FB@ = �N = �mg cosą ?5@5?8H5< (3) 2 2845:
2
�mg cosą = mg siną + m � * r, (4)
( )
345 � * :@8B8G5A:>5 7=0G5=85 C3;>2>9 A:>@>AB8, ?@8 ?@52KH5=88
:>B>@>3> 40==0O ?K;8=:0 ?>:8=5B 48A:. 7 (4) ?>;CG8<:
g � cosą - siną).
(
�* = (5)
r
"0:8< >1@07>< ?> <5@5 @0A:@CG820=8O 48A:0 =0 =5<, =0G8=0O >B
:@0O, 1C45B @0AB8 :>;LF> >G8I5==>9 ?>25@E=>AB8. CABL 53>
2=CB@5==89 @048CA r2=.. !G8B0O, GB> >?8;:8 @0A?@545;5=K ?> 48A:C
@02=><5@=>, =0945< ?@>F5=B A;5B52H8E A =53>:
2
Ą R2 - r2=
( )
� =
ĄR2
8 @048CA r2=.:
r2= = R 1 - �. (6)
! CG5B>< (5) ?>;CG8<:
g � cosą - siną).
(
�* = (7)
R 1 - �
7 0=0;870 (7) 2848<, GB> 4;O @50;870F88 MDD5:B0 =5>1E>48<>
CA;>285:
� > tgą ,
B.5. B@5=85 4>;6=> 1KBL 4>AB0B>G=> 1>;LH8<, GB>1K 2 >BACBAB285
2@0I5=8O >?8;:8 =5 A>A:0;L7K20;8 A 48A:0. @><5 B>3>, ?@8
� = 100% 2K@065=85 (7) B5@O5B A20<8,
( )
?>;=>ABLN 48A: =5 C40ABAO >G8AB8BL =8:>340, B.:. 21;878 53> >A8
18
>?8;:8 1C4CB ?@0:B8G5A:8 ?>:>8BAO. 0:>=5F, ?@>F5AA �>G8I5=8O
g � cosą - siną)
(
=0G=5BAO ?@8 C3;>2>9 A:>@>AB8 � ** = ,
R
2KB5:0NI59 87 (7) ?@8 � = 0 (5A;8 2A5 ?@54K4CI85 CA;>28O
2K?>;=5=K).
23. @8 42865=88 B5;56:8 55 ?>B5=F80;L=0O M=5@38O
?5@5E>48B 2 :8=5B8G5A:CN, G0ABL 87 :>B>@>9 5ABL M=5@38O
?>ABC?0B5;L=>3> 42865=8O, 0 G0ABL M=5@38O 2@0I0B5;L=>3>
42865=8O :>;5A 8 >A59. >MB>;=CN :8=5B8G5A:CN
mv2
M=5@38N 2 2845 � , 345 � =58725AB=K9 157@07<5@=K9
2
:>MDD8F85=B, 7028AOI89 >B A>>B=>H5=8O <0AA :>;5A 8 ?;0BD>@@07<5@>2 :>;5A. 7 70:>=0 A>E@0=5=8O M=5@388 A;54C5B
mv2
� = mgS siną. (1)
2
"0: :0: A8;K, 459AB2CNI85 =0 B5;56:C, ?>AB>O==K, B> >=0
42865BAO @02=>CA:>@5==>. >MB>2S
v2 = 2aS, a = . (2)
2
t0
!;54>20B5;L=>,
2mS2
� = mgS siną. (3)
2
t0
>A;5 B>3>, :0: =0 B5;56:C ?>;>68;8 3@C7 (:>B>@K9 42865BAO
?>ABC?0B5;L=>), :8=5B8G5A:0O M=5@38O B5;56:8 A 3@C7>< 2K@0605BAO
2 2 2
mv1 mv1
1
D>@9 � + = � + 1 . >MB>( )mv
2 2 2
0=0;>38G=>5 (3), 8<55B 284
2
2mS
� + 1 = mgS siną. (4)
( )
2
t1
A:;NG0O 87 (3) (4) =58725AB=K9 ?0@0<5B@ � , ?>;CG8< >B25B
7040G8
�# ś#
1 2S
t1 = t0 1 + H" 23c.
,
ś# ź#
2
2 t0 g siną #
�#
24. @>I5 @5H0BL 7040GC A 8A?>;L7>20=85< 70:>=0 A>E@0=5=8O
M=5@388: :8=5B8G5A:0O M=5@38O F8;8=4@0 @0AE>4C5BAO =0
C25;8G5=85 ?>B5=F80;L=>9 M=5@388 ;5=BK, =0<0BK20NI59AO =0
19
=53>. >A:>;L:C ;5=B0 381:0O, B> 55 87381 (B.5. =0<0BK20=85) =5
B@51C5B @0AE>40 M=5@388 =0 45D>@<0F88. ">340:

E = E ,(1)

345 E :8=5B8G5A:0O M=5@38O F8;8=4@0 2 =0G0;5 =0<0BK20=8O,

E ?>B5=F80;L=0O M=5@38O ;5=BK 2 =0<>B0==>< A>AB>O=88.

@8 2KG8A;5=88 E =5 A;54C5B 701K20BL ?@> �2@0I0B5;L=CN
M=5@38N F8;8=4@0:
2 2
mv0 mv0 2

E = + = mv0 . (2)
2 2
>B5=F80;L=0O M=5@38O ;5=BK 2 :>=5G=>< A>AB>O=88:

E = MgR = 2ĄRn�gR = 2Ąn�gR2 , (3)
345 M = 2ĄRn� <0AA0 ;5=BK, g CA:>@5=85 A2>1>4=>3> ?045=8O,
R @048CA F8;8=4@0. @8 70?8A8 (3) B>=:0O ;5=B0
(A<. CA;>285), 1C4CG8 =0<>B0=0 =0 F8;8=4@, ACI5AB25==> =5
87<5=8B 2KA>BK 53> F5=B@0 BO65AB8. @><5 B>3>, A>25@H5=85
F8;8=4@>< @>2=> n >1>@>B>2 ?>72>;O5B CB25@640BL, GB> F5=B@
BO65AB8 A8AB5A8 F8;8=4@0.
! CG5B>< (1) (3), ?>;CG8<:
2Ąn�gR2
v0 = . (4)
m
A;CG05 @50;L=>3> :>2@0 1C45B A:07K20BLAO ?>B5@O M=5@388 =0
2=CB@5==55 B@5=85, =0 53> 45D>@<0F8N, =0 CB>;I5=85 F8;8=4@0.
0==K9 MDD5:B <>6=> ?@>=01;N40BL 2 >1@0B=>9
?>A;54>20B5;L=>AB8 A;53:0 ?>4B>;:=8B5 A25@=CBK9 :>25@ 8 >=,
70<5B=> CA:>@OOAL, �A0<>@0A:@CB8BAO.
0<5B8<, GB> 42865=85 :>2@0 =5 1C45B @02=>CA:>@5==K<.
>4C<09B5 B0:65, :C40 �8AG57=5B M=5@38O :>2@0 ?@8
�A0<>@0A:@CB:5?
25. >:065<, GB> >1@CG >B>@25BAO >B 3>@:8. ;O >?@545;5=8O
A:>@>AB8 F5=B@0 >1@CG0 2 B>G:5 ?5@53810 B>G:5 A>?@8:>A=>25=8O
>:@C6=>AB59 , 2>A?>;L7C5=>< A>E@0=5=8O M=5@388
2
mv0 + mgr = mv2 + mg R + r �" siną ,
( )
>B:C40
R - r
2
v2 = v0 - g�# ś#.
ś# ź#
�# #
2
20
45AL CGB5=>, GB> >1@CG CG0AB2C5B :0: 2 ?>ABC?0B5;L=><, B0: 8 2>
2@0I0B5;L=>< 42865=88, ?>MB> :8=5B8G5A:0O M=5@38O @02=0
mv2 . 7 @8AC=:0 284=>, GB>
ą = 30o siną = 1 2 . ;O B>3>, GB>1K
()
>B@K20 =5 ?@>87>H;>, =5>1E>48<>
2K?>;=5=85 CA;>28O: A8;0 @50:F88
N e" 0.
0?8H5< C@02=5=85 2B>@>3>
70:>=0 LNB>=0 4;O F5=B@0 <0AA 2
B>G:5
mv2
= mg siną - N ,
R + r
8;8
R - r
2
v0 - g
2
N = mg siną - m .
R + r
>4AB0=>2:0 G8A;5==KE 7=0G5=89 ?>:07K205B, GB> CA;>285 N e" 0 =5
2K?>;=O5BAO. "0:8< >1@07>< 2 B>G:5 >1@CG >B>@25BAO >B
?>25@E=>AB8, 8 40;LH5 53> F5=B@ 1C45B 42830BLAO ?> ?0@01>;5
?>4=O2H8AL =0 2KA>BC
v2 cos2 ą 3 R - r
�#v - g ś#
2
h == ,
ś# ź#
0
�# #
2g 8g 2
GB> A>>B25BAB2C5B 2KA>B5 =04 C@>2=5< 3>@87>=B0;L=>9 ?>25@E=>AB8
2
R + r 3v0 3 1
H = h + = - ( - r + R + r H" 43<.
R ) ( ) ,
2 8g 16 2
26. 'B>1K ?@5>4>;5BL 3>@C =5>1E>48<>, GB>1K :8=5B8G5A:0O
M=5@38O 1K;0 1>;LH5 ?>B5=F80;L=>9 M=5@388 =0 25@H8=5 3>@K
2
mv0
e" mgR
2
8;8 v0 e" 2gR . 08<5=LH0O A8;0 @50:F88
N 2 B>G:5 ?5@53810 . MB>9 B>G:5
mv2
= mg siną - N
R
2865=85 157 >B@K20 ?@8 N e" 0. A?>;L7CO 70:>= A>E@0=5=8O
M=5@388
2
mv0 mv2
= + mgR siną ,
2 2
?>;CG8<
21
v0 d" 3gR siną
3
8;8, B.:. ą = 30o , B> v0 d" gR . !;54>20B5;L=>, B0:CN 3>@:C
2
?@5>4>;5BL 157 >B@K20 H0910 =5 A<>65B =8 ?@8 :0:8E A:>@>ABOE.
27. 3>2>@5==K5 CA;>28O <0;>AB8 h x ?>72>;ONB ?@5=51@5GL
( )
@07;8G85< <564C 4;8=>9 ?@>D8;O
3>@:8 8 4;8=>9 55 >A=>20=8O l .
07>1L5< ?CBL H0@8:0 l =0 <0;K5
8=B5@20;K "xi . K@078< ?CBL
H0@8:0 2 B>G:5 A :>>@48=0B>9 x ,
8AE>4O 87 70:>=0 A>E@0=5=8O
M=5@388:
2
Cv0 = Cv2 + mgh x . (1)
( )
45AL Cv2 :8=5B8G5A:0O M=5@38O H0@8:0.
("0: :0: H0@8: 42865BAO ?>ABC?0B5;L=> 8 >4=>2@5<5==> 2@0I05BAO,
mv2
53> :8=5B8G5A:0O M=5@38O ?@52KH05B , 2 G0AB=>AB8 4;O
2
A?;>H=>3> H0@8:0
2 2
mv2 I� mv2 2 � 7
Ek = + = + mR2 = mv2 ,
2 2 2 5 2 10
>4=0:> MB>B :>MDD8F85=B =5 ACI5AB25=5= 4;O 40==>9 7040G8.)
7 (1) =0E>48<
mg
2
v x = v0 - h x . (2)
( ) ( )
C
'B>1K ?@5>4>;5BL CG0AB>: "xi , H0@8:C ?>B@51C5BAO 2@5"xi
"ti = , (3)
v xi
( )
0 2@5B>@>5 H0@8: ?@5>4>;55B 2AN 3>@:C, <>6=> =09B8,
?@>AC<<8@>202 (3) ?> 2A5< 8=B5@20;0< @07185=8O "xi
1
-
2
�#
"xi "xi "xi mg
t == = .
( )ś#
ś#1 - h xi ź#
"" "
2
v xi mg v0 Cv0 #
�#
i ( ) i
2
v0 - h xi i
( )
C
#G8BK20O <0;>ABL h, <>6=> 2>A?>;L7>20BLAO ?@81;865==>9
ą
D>@9 1 + z H" 1 + ąz , B>340
( )
22
�#
"xi mg mg
t H" = +
( )ś# l
ś#1 + 2 h xi ź# ( )
"""x 2Cv0 h xi . (4)
i
v0 2Cv0 # v0 3
�#
i i
'B>1K =09B8 2@5AB0B>G=> 2
( )
(1), 8 2 (4) 87<5=8BL 7=0: ?5@54 DC=:F859 h x
l mg
t1 = - (5)
( )
""x 2Cv0 h xi .
i
v0 3
i
!;>68< ?>G;5==> (4) 8 (5)
2l
t1 + t = ,
v0
B:C40 8A:><>5 2@52l
t1 = - t .
v0
28. 7>1@078< A8;K, 459AB2CNI85 =0 :064K9 87 1@CA:>2 . 0?8H5<
>A=>2=>9 70:>= 48=0<8:8 (II 70:>= LNB>=0) 4;O :064>3> 87
1@CA:>2 2 25:B>@=>< 2845:
r r r
r r
ż#
ma1 = mg + T1 + N + FB @.
1 1
r , (1)
�#
rr
ma2 = mg + T2
�# 2 2
r r
345 T1 8 T2 - A8;K =0BO65=8O
r
=8B59, N - A8;0 @50:F88
r
?>25@E=>AB8 AB>;0, FB@. - A8;0
B@5=8O.
0: 2848< 87 (1), 4;O 2KG8A;5=8O CA:>@5=8O 1@CA:0 m1 =0<
r r r
=5>1E>48<> 7=0BL A8;K T1 ; N ; FB@.. !?@>5F8@>202 A8AB53>@87>=B0;L=CN (X) 8 25@B8:0;L=CN (Y) >A8, 8 8A:;NG0O A8;K
B@5=8O 8 @50:F88 >?>@K, ?>;CG8<:
ma2 = mg - T2
ż#
2 2
(2)
�#ma = T1 cos� - � mg - T sin� .
()
1 1 1
�#
>A:>;L:C, ?> CA;>28N, B@5=8O 2 1;>:5 =5B 8 =8BL ;53:0O, B> A8;0 55
=0BO65=8O ?>AB>O==0. ?@545;8< :8=5<0B8G5A:CN A2O7L <564C a1
8 a2 . 0AA<0B@820O <0;K5 A<5I5=8O 1@CA:>2 A@07C ?>A;5
>B?CA:0=8O 1@CA:0 m1 , <>6=> =09B8
a2
a1 = . (3)
cos�
23
!;54C5B >B<5B8BL, GB> MB> A>>B=>H5=85 A?@0254;82>, B>;L:> 2
=0G0;L=K9 <><5=B 2@5<5=8, A:>@>AB8 1@CA:>2 @02=K =C;N, B0: :0: 2
>1I5< A;CG05 CA:>@5=85 a1 7028A8B A;>6=K< >1@07>< >B A:>@>AB8
3@C7>2.
5H0O A8AB5;CG05<:
m2 g cos� + � sin� - �m1g
()
a1 = . (4)
m1 + m2 cos� cos� + � sin�
()
B=>A8B5;L=> (4) 70<5B8<, GB> a1 =5 <>65B 1KBL >B@8F0B5;L=K<
(A;>6=> ?@54AB028BL A515 A8BC0F8N, :>340 1@CA>: m1 2B0A:8205B =0
AB>; (1) 1@CA>: m2), B.5. >B25B A;54C5B 40BL 2 A;>6=>9 D>@<5:
ż#
ż# 0,5A;8 �m1 e" m2 cos� + � sin� ;
()
�#
�#a = m2 g cos� + � sin� - �m1g
�# ()
�#
1
()
�#
�#m1 + m2 cos� cos� + � sin� ,5A;8 �m1 d" m2 cos� + � sin� . (5)
()
�# �#
�# a2 = a1 cos�
�#
>4G5@:=5< B0:65, GB> ?@8 � = 0 87 (5) ?>;CG05< E>@>H>
8725AB=K9 @57C;LB0B:
m2 g - �m1g m2 - �m1
a1 = a2 = = g .
m1 + m2 m1 + m2
Ą
=B5@5A5= A;CG09 ?@545;L=>3> ?5@5E>40 � = a2 = 0 ! , >4=0:>
( )
()
2
?@54;0305< G8B0B5;O< @07>1@0BLAO 2 =5< A0<>AB>OB5;L=>.
29. !8;0, 459AB2CNI0O =0 H0@8: 2 <><5=B C40@0, =0?@02;5=0
?5@?5=48:C;O@=> ?>25@E=>AB8 ?@871>7=0G8< 89 A8;K 70 2A5
r
2@5C40@5=8O P1 , ?> B@5BL5=C
LNB>=0 B0:>9 65 ?> 25;8G8=5 8r
?>;CG8;0 8 ?@87<0 P2 . >A:>;L:C 2@5A>C40@5=8O >1KG=> >G5=L <0;>,
8< A8;K BO65AB8 ?> A@02=5=8N A
8AB8 <>6=>
?@5=51@5GL.
7<5=5=85 25@B8:0;L=>9 A>AB02;ONI59 8>?@545;O5BAO 2B>@K< 70:>=>< LNB>=0
mv1y + mv0 = Pcosą. (1)
24
=0;>38G=>, 4;O ?@87MV = P siną. (2)
345 V A:>@>ABL ?@87A;5 C40@0.
7 C@02=5=89 A;54C5B
M
v1y =-v0 + Vctgą. (3)
m
"0: :0: 2 A8AB5<5 H0@8:-?@87<0 =5B 2=5H=8E A8;, 8<5NI8E
3>@87>=B0;L=>5 A>AB02;ONI85, B> ?@>5:F8O AC<<0@=>3> 8=0 3>@87>=B0;L=CN >AL A>E@0=O5BAO, A;54>20B5;L=>,
mv1x = MV . (4)
0?8H5< B0:65 70:>= A>E@0=5=8O <5E0=8G5A:>9 M=5@388
2 2
mv0 MV m
2
=+ v!2 + v1y . (5)
()
x
2 2 2
>4AB02;OO 2K@065=8O (3) (4) 4;O :>=5=B A:>@>AB8 (v1x 8 v1y )
H0@8:0 ?>A;5 C40@0, 8 @5H0O ?>;CG5==>5 C@02=5=85 >B=>A8B5;L=>
25;8G8=K V, ?>;CG8<
2v0 siną cosą
V = .
M
+ sin2 ą
m
B<5B8<, GB> 2 40==>9 A8BC0F88 �C3>; ?045=8O H0@8:0 =5 @025=
�C3;C >B@065=8O.
30. 2865=85 ?@874G8=O5BAO
C@02=5=8O<
Ma = F - N1 - T - �2 N2 ,
ż#
�#
Mg + T - N2 - �1 N1 = 0.
�#
;O 1@CA:0 A?@0254;82K C@02=5=8O 42865=8O (CG8BK20O, GB>
1@CA>: 42865BAO 225@E 8 2?@02>)
ma = N1 ,
ż#
�#ma = T - mg - �1 N1.
�#
!>2<5AB=>5 @5H5=85 MB8E C@02=5=89 ?@82>48B :
>:>=G0B5;L=>F = g �2 M + m 1 + �2 + a M + m 2 + �1 + �2 .
( ) )
() ( ( )
25
31.!8;K B@5=8O, 459AB2CNI85 =0 F8;8=4@ 2 =0G0;L=K9 <><5=B A>
AB>@>=K ?>;C?;>A:>AB59
mg mg
FB@1 = �1 ; FB@2 = �2 .
2 siną 2 siną
"0: :0: �1 > �2 , B> FB @.1 > FB @.2 , ?>MB>@OBLAO 2
AB>@>=C 42865=8O ?;0AB8=K , ?>:0 =5 4>AB83=5B 55 A:>@>AB8 v1 ,
?>A;5 G53> ?@>A:0;L7K20=85 ?@5:@0B8BAO. 0;L=59H55 42865=85
1C45B @02=><5@=K<. 073>= ?@>8AE>48B A CA:>@5=85<
g
a = �1 - �2
()2 siną ,
2@5=0
v 2 sinąv1
� = = .
a g �1 - �2
()
@0D8: 8<55B 284, 87>1@065==K9 =0
@8AC=:5.
32. @5=51@530O A>?@>B82;5=85< 2>74CE0, =0945<,
GB> ?@8 ?>4;5B5 : ?;8B5 8 1>;LH>9, 8 <0;5=L:89
H0@8:8 1C4CB 8<5BL >48=0:>2CN A:>@>ABL
v0 = 2gh. (1)
5@2K< 8A?KB05B C?@C3>5 A>C40@5=85 A ?;8B>9 1>;LH>9 H0@.
#G8BK20O, GB> 2@53> A>C40@5=8O 4>AB0B>G=> <0;> (>=> 8<55B
2D
?>@O4>: � = , 345 D 480<5B@ H0@0, V A:>@>ABL 72C:0 2 =5<),
V
?@8<5<, GB> <0;5=L:89 H0@8: 70 MB>B ?5@8>4 ACI5AB25==> =5
A428=C;AO. "0:8< >1@07><, A;54CNI55 A>C40@5=85
?@>8AE>48B <564C H0@>< M, 4286CI8?@56=59 ?> 25;8G8=5 A:>@>ABLN v0 , 8 H0@>< ,
?@>4>;60NI8< 42865=85 A> A:>@>ABLN v0 2=87. !G8B0O
MB> AB>;:=>25=85 C?@C38<, 70?8H5< 70:>=K A>E@0=5=8O
M=5@388 8 8r
rr r
ż#
mv0 + M v0 = mU + mv
(- )
�#
2 2 2
(2)
�#mv0 Mv0 mU Mv2 ,
+ = +
�#
�# 2 2 2 2
26
r
r
345 U 8 v A:>@>AB8, A>>B25BAB25==>, <0;>3> 8 1>;LH>3>
H0@>2 ?>A;5 2B>@>3> AB>;:=>25=8O. !B@>3> 3>2>@O, 2
25@B8:0;L=>< =0?@02;5=88 A8AB5<0 =5 70<:=CB0 ?>
?@8G8=5 459AB28O A8;K BO65AB8. 4=0:>
:@0B:>2@5<5==>ABL ?@>F5AA0 AB>;:=>25=8O 45;05B
4>?CAB8< A8;K BO65AB8 70
2@5;:=>25=8O � :
m + M g� << mu + m�.
()
?@>5:F8OE =0 25@B8:0;L=CN >AL (2) ?@8<5B 284:
Mv0 - mv0 = mU + Mv
ż#
�#mv2 + Mv0 = mU 2 + Mv2 . (3)
2
�# 0
7 MB8E C@02=5=89 =0E>48<
3 - �
�#
U = v0
�#
�#
1 + �
(4)
1 - 3�Ź#.,
�#
� = v0
�#
1 + �
�#
m
345 � = .
M
K@065=85 (4) 405B A:>@>AB8 H0@>2 ?>A;5 01A>;NB=> C?@C3>3>
A>C40@5=8O H0@>2 @07=KE <0AA. @545;L=CN 2KA>BC ?>4A:>:0
@50;87C5BAO 2 B>< A;CG05, :>340: � 0, B.5. m << M .
@8 MB><:
U 3v0
�#
(5)
v v0 Ź#.
�#
B 7=0:0 @025=AB20 2 (5) 2>745@68< 1C45B
=0@CH0BLAO 70:>= A>E@0=5=8O M=5@388.
0;55:
2
U �#
h* = 9h�#
2g �#
, (6)
v2 Ź#
�#
h** = h
�#
2g
�#
345 h* 8 h** 2KA>BK ?>4A:>:>2 ?>A;5 A>C40@5=8O, A>>B25BAB25==>,
;53:>3> (m) 8 BO65;>3> (M) H0@>2.
K@065=85 (8) =5 =0@CH05B 70:>=0 A>E@0=5=8O M=5@388, B.:. @>AB h*
(?> A@02=5=8N A h) ?@>8AE>48B 70 AG5B C<5=LH5=8O h**. =K<8
A;>20<8, G0ABL <5E0=8G5A:>9 M=5@388 1>;LH>3> H0@0 ?5@5405BAO
<0;> 27
-BC :@0A82CN 45<>=AB@0F8N <>6=> ?@>25AB8 A < 4;O
1>;LH>3> B5==8A0 8 H0@8:>< >B ?8=3-?>=30, E>BO, :>=5G=> 65,
=01;N405<0O ?@8 MB>< 2KA>B0 ?>4A:>:0 h* <5=LH5 ?@545;L=>9 9h.
?@8G8=0E MB>3> ?>4C<09B5 A0<>AB>OB5;L=>.
33.!8;C B@5=8O C4>1=> 2K@078BL 87 II 70:>=0 LNB>=0. ?@>5:F88
=0 >AL x 70?8H5<
mg siną - FB @. = ma,
>B:C40
FB@. = mg siną - ma.
;O =0E>645=8O CA:>@5=8O 2>A?>;L7C5=>< A>E@0=5=8O
M=5@388
mgh = K + KB ,
mv2
345 h = S siną 2KA>B0, =0 :>B>@CN >?CAB8;0AL B@C1:0, KK =
2
:8=5B8G5A:0O M=5@38O ?>ABC?0B5;L=>3> 42865=8O B@C1:8, KB
:8=5B8G5A:0O M=5@38O 2@0I5=8O B@C1:8. 0945< KB . @54?>;>68<,
GB> >AL B@C1:8 =5?>4286=0, 0 A0<0 B@C1:0 @0A:@CG5=0 4> :0:>9-B>
C3;>2>9 A:>@>AB8 � . -B> >7=0G05B, GB> 2A5 B>G:8 55 ?>25@E=>AB8
4286CBAO A >48=0:>2>9 ?> <>4C;N ;8=59=>9 A:>@>ABLN v . !G8B0O
B@C1:C B>=:>9, ;I8=>9. "0:8< >1@07><, 2A5
CG0AB:8 B@C1:8 8<5NB A:>@>ABL v . !;54>20B5;L=>, :8=5B8G5A:0O
mv2
M=5@38O 2@0I0NI59AO B@C1:8 @02=0 KB = . 0:>=5F, 7=0O, GB>
2
B@C1:0 A:0BK205BAO 157 ?@>A:0;L7K20=8O (MB> >7=0G05B, GB>
;8=59=0O A:>@>ABL G0AB59 B@C1:8 @02=0 A:>@>AB8 55
?>ABC?0B5;L=>3> 42865=8O), 65< 70?8A0BL
mv2 mv2
mgS siną = + = mv2 ,
2 2
345 S ?@>945==K9 B@C1:>9 ?CBL. @8 ?@O<>;8=59=><
@02=>CA:>@5==>< 42865=88, :0:>2K< 8 O2;O5BAO A:0BK20=85 B@C1:8
A =0:;>==>9 ?;>A:>AB8, CA:>@5=85, A:>@>ABL 8 ?@>945==K9 ?CBL
A2O70=K A>>B=>H5=85<
v2 = 2as, v0 = 0.
<55<
g siną
gS siną = 2aS, a =.
2
">340 A8;0 B@5=8O @02=0
mg siną mg siną
FB@. = mg siną - =.
22
28
r
34.0 :>;LF> A 459AB2CNB G5BK@5 A8;K: A8;K =0BO65=8O =8B59 T1 8
r
r r
T2 ; @50:F88 A> AB>@>=K AB5@6=O CD N ; B@5=8O FB@.. !8;>9
BO65AB8 :>;LF0 A;CG05 @02=>25A8O :>;LF0 8<55<:
r r r r r
T1 + T2 + N + FB @. = 0 . (1)
!?@>5F8@C5< (1) =0 3>@87>=B0;L=CN 8
25@B8:0;L=CN >A8:
FB@. = T1 sinł + T2 siną
�#
Ź#. (2)
N == T1 cosł + T2 cosą
�#
r r
>A:>;L:C =8BL ;53:0O, B> T1 = T2 = T..
?@545;L=>< A;CG05 (2 <><5=B =0G0;0 A:>;L65=8O :>;LF0):
FB@. = �N = �T cosł + cosą) (
= T sinł + siną)
. (3)
(
7 (3) ?>;CG05<:
sinł + siną
� = . (4)
cosł + cosą
B0: :0: <0:A8<0;L=>5 7=0G5=85 2K@065=8O (4) @50;87C5BAO ?@8
ł = � , B> ?@545;L=>5 7=0G5=85 :>MDD8F85=B0 B@5=8O <564C
:>;LF>< 8 AB5@6=5<:
sin� + siną
�* = .
cos� + cosą
"0:8< >1@07><, :>;LF> 1C45B =5?>4286=>, 5A;8
� e" �* . (5)
35. 40==>< A;CG05 ?>=OB=>, GB> ?5@2K9 �?>;C?5@8>4 :>;510=89
3@C70 =8G5< =5 1C45B >B;8G0BLAO >B A;CG0O >BACBAB28O ?;8BK
m
t1 = Ą . (1)
k
@8 42865=88 ?@C68==>3> <0OB=8:0
2=87 A?CABO =5:>B>@>5 2@5C40@8BAO > ?;8BC 8 (1C45< AG8B0BL
<3=>25==>) 87<5=8; =0?@02;5=85
A:>@>AB8 =0 ?@>B82>?>;>6=>5. 0
3@0D8:5 MB> A>>B25BAB2C5B
�8AG57=>25=8N CG0AB:0 A8=CA>84K ,
C<5=LH0NI5@>9 �?>;C?5@8>4 4> 25;8G8=K2� .
29
a 1
ś#
t2 - t1 = 2� = 2�#arcsin , (2)
ś# ź#
�#
A # �
mg
�#l ś#
345 a = h - + @0AAB>O=85 >B ?>;>65=8O @02=>25A8O 3@C70 4>
ś# ź#
�# #
k
m
?;8BK; A = v0 0;510=89; � F8:;8G5A:0O G0AB>B0
k
:>;510=89.
!>>B25BAB25==>, ?5@8>4 :>;510=89:
�# ś#
�# mg ś# Ą# mg ń#
�#l + ś# �#l ś#
h - ś# ź# ś#
ś# ź#
ó#h - ś# + k ź# Ą#ź#
�# # �# #
Ą 1 mś#
k
ź#
ó# .
T = + 2arcsinś# = Ą + 2arcsin Ą#ź#
ś#
� � ś# m ź# k m
ó# Ą#ź#
ś# v0 ź# v0
ś#
ó# Ą#ź#
�# #
�#
k k
Ł# Ś# #
G0AB=>< A;CG05, 5A;8 ?;8B0 CAB0=>2;5=0 A@07C ?>4
Ą
<0OB=8:>< a = 0 , B> (3) 405B 25@=K9 ?@545;L=K9 ?5@5E>4 T = ,
( )
�
:>340 �=86=OO G0ABL A8=CA>84K >BACBAB2C5B 8 A
�2K?@O<;5==K<8 :>;510=8O<8.
>4G5@:=5<, GB>, 5A;8 ?;8B0 AB>8B >G5=L =87:> (a > A ), B> C40@0
�# ś#
m
?@>8AE>48BL =5 1C45B T = 2Ą . A;CG05 A = a 3@C7,
ś# ź#
k
�# #
:>A=C2H8AL ?;8BK, A>E@0=8B B>B 65 ?5@8>4 :>;510=89
�# ś#
m
T = 2Ą .
ś# ź#
k
�# #
36. K 4>;6=K 2KOA=8BL, ?@8
:0:>9 0;510=89
H0@0 OI8: 1C45B =5?>42865=.
!<5AB8< H0@8: =0 @0AAB>O=85
x >B ?>;>65=8O @02=>25A8O.
0 OI8: =0G=CB 459AB2>20BL
425 >48=0:>2K5 A8;K C?@C3>AB8 ?@C68= kx . =8 459AB2CNB 2
>4=>< =0?@02;5=88 8 8E AC<<0@=>5 459AB285 =5 4>;6=>
?@52>AE>48BL A8;C B@5=8O
2kx d" FB@ = �N = �2mg.
BAN40 2K@0605< 3@0=8G=>5 7=0G5=85 0�mg m �g
xm = = �g = .
2
k k �
30
37. 0AA<>B@8< A<5I5=85 ?@02>3> :>=F0 F5?>G:8
=0 25;8G8=C x 225@E. @8 MB>< ?>B5=F80;L=0O
M=5@38O A8AB57@0AB05B =0 25;8G8=C:
m x mg
"U = 2 xg = x2 . (1)
l 2 l
0: 2848< 87 (1), "U ?@>?>@F8>=0;L=0 :204@0BC
A<5I5=8O 87 ?>;>65=8O @02=>25A8O, GB> O2;O5BAO
4>AB0B>G=K< CA;>285< 30@<>=8G=>AB8
:>;510=89.
0:>= A>E@0=5=8O M=5@388 4;O 42865=8O F5?>G:8:
mv2 mgx2
+= E (2)
2 l
A>2?0405B ?> D>@<5 A 0=0;>38G=>9 7028A8<>ABLN 4;O ?@C68==>3>
<0OB=8:0:
mv2 kx2
+ = E , (3)
2 2
345 E <5E0=8G5A:0O M=5@38O >AF8;;OB>@0. !@02=820O (2) 8 (3),
?>;CG8< >B25B:
m ml l
T = 2Ą �! T = 2Ą = 2Ą . (4)
k 2mg 2g
38. K45;8< <0;K9 A53<5=B 684:>AB8
A C3;>2K< @0AB2>@>< ą , =0E>4OI89AO
=0 @0AAB>O=88 r >B >A8, B>;I8=>9 "r 8
2KA>B>9 h .
-B0 G0ABL 684:>AB8 42865BAO ?>
>:@C6=>AB8 A C3;>2>9 A:>@>ABLN � ,
A;54>20B5;L=>, C@02=5=85 2B>@>3>
70:>=0 LNB>=0 4;O =55 ?@8<5B 284
ą
2
P + "P r + "r ah - rPąh - 2P"rh = �ąr"rh� r,
()( )
2
345 P 402;5=85 684:>AB8 =0 @0AAB>O=88 r >B >A8 2@0I5=8O,
P + "P 402;5=85 =0 @0AAB>O=88 r + "r .7 MB>3> C@02=5=8O ?>;CG8<
2
"P = �� r"r.
>;030O 402;5=85 =0 >A8 A>AC40 @02=K< =C;N, AC<<8@>20=85< 2A5E
87<5=5=89 402;5=8O "Pi =0 :064>< H035 "ri >B 0 4> R =0E>48<
402;5=85 =0 1>:>2CN AB5=:C
2
�� R2
2
P = "Pi = �� ri"ri = .
""
2
i i
31
39.1>7=0G8< 2KA>BC C@>2=O 2>4K 2 ?@>18@:5
h , 0 3;C18=C ?>3@C65=8O H . >A;54=OO <>65B
1KBL =0945=0 87 70:>=0 @E8<540
�Sh + m g = �SHg,
( )
( � ?;>B=>ABL 2>4K, S = Ąr2 ?;>I04L
?>?5@5G=>3> A5G5=8O ?@>18@:8), >B:C40
m
H = h + . (1)
�S
/A=>, GB> :>340 ?@>18@:0 ?;0205B H d" l , A;54>20B5;L=>, 2KA>B0
C@>2=O 2>4K 2 ?@>18@:5 4>;6=0 C4>2;5B2>@OBL CA;>28N
m
h < l - . (2)
�S
'B>1K ?>;>65=85 ?;020NI59 ?@>18@:8 1K;> CAB>9G82K<
=5>1E>48<>, GB>1K 55 F5=B@ BO65AB8 =0E>48;AO =865 F5=B@0
BO65AB8 2KB5A=5==>9 2>4K, :>B>@K9 =0E>48BAO =0 @0AAB>O=88 H / 2
>B 4=0 ?@>18@:8. KA>BC F5=B@0 BO65AB8 h <>6=> 2K@078BL
c
h l
�Sh + m
2 2
hc = .
�Sh + m
">340 CA;>285 CAB>9G82>AB8 8<55B 284
h2 l
� S + m
H m
2 2
< = h + . (3)
�Sh + m 2 � S
5H0O MB> =5@025=AB2>, ?>;CG8<
�# ś#
1 m
h > - (4)
ś#l ź#.
2 � S #
�#
5@025=AB20 (2) , (4) 40NB >1;0ABL 4>?CAB8:>B>@KE ?@>18@:0 ?;0205B CAB>9G82>:
�# ś#
1 m m
< h < l - .
ś#l - ź#
2 � S # � S
�#
40. @5645 G5< ?@8ABC?8BL : 70?8A8 C@02=5=89 42865=8O
?;0AB8=:8, 40209B5 :0G5AB25==> ?@>0=0;878@C5< ?@8G8=C 55
>AB0=>2:8. >A:>;L:C (?> CA;>28N) 2>40 ?>4 ?;0AB8=:C =5
?>?0405B, B> A8;0 @E8<540 (:0: AC<<0 A8;, 459AB2CNI8E =0 3@0=8
?;0AB8=:8) 87 2KB0;:820NI59
?@52@0I05BAO 2 �70B0;:820NICN,
B.5. ?@868<05B ?;0AB8=:C :> 4=C.
32
>A:>;L:C A8;0 �?@868<0 @0AB5B ?@>?>@F8>=0;L=> 402;5=8N (B.5.
3;C18=5), B>, ?> <5@5 42865=8O ?;0AB8=:8, 1C45B 2>7@0AB0BL 8 A8;0
B@5=8O, :>B>@0O 8 >AB0=>28B 40;L=59H55 42865=85 ?;0AB8=:8.
0?8H5< >A=>2=>9 70:>= 48=0<8:8 4;O 40==>3> A;CG0O:
r r r r
r r
ma = Fi = mg + N + FB @. + F , (1)
"
i
r r
r
345 mg A8;0 BO65AB8; N A8;0 @50:F88 ?>25@E=>AB8 :;8=0; FB@.
r
A8;0 B@5=8O ?;0AB8=:8 > :;8=; F - A8;0 402;5=8O 2>4K =0
r
?;0AB8=:C (�>AB0B>: A8;K @E8<540); a CA:>@5=85 ?;0AB8=:8.
?@>5:F88 =0 >AL X:
ma = mg siną - FB @. (2)
0 >AL Y:
N = mg cosą + F . (3)
r
;O 2KG8A;5=8O F = F =0?><=8< , GB> ?;0AB8=0 B>=:0O, B.5.
A8;0<8 402;5=8O =0 1>:>2K5 3@0=8 (2A;54AB285 <0;>AB8 8E
?;>I048) <>6=> ?@5=51@5GL:
F = �gha2 = �ga2x siną ,
345 x siną = h 3;C18=0 ?>3@C65=8O F5=B@0 ?;0AB8=:8. ">340:
��ga2 siną
a = g siną - �g cosą - x . (4)
m
#@02=5=85 (4) A>2?0405B A C@02=5=85< 30@<>=8G5A:8E :>;510=89 A>
A<5I5==K< ?>;>65=85< @02=>25A8O:
m
x0 = 1 . (5)
( - �ctgą)
��a2
-B> >1AB>OB5;LAB2> ;53:> ?>72>;8B =0< 2KG8A;8BL 8A:><>5 2@54> >AB0=>2:8 ?;0AB8=K :0: ?>;C?5@8>40 A>>B25BAB2CNI53>
:>;510=8O:
T Ą mm
Ą
� = = = Ą = . (6)
2 � �a2�g siną a ��g siną
!;54C5B ?>4G5@:=CBL, GB> ?@82545==>5 @5H5=85 A?@0254;82> ?@8
;N1>< <0;>< B@5=88, B.5. 2 40==>< A;CG05 (A<. CA;>285)
>B?CI5==0O 157 B>;G:0 ?;0AB8=0 =0G8=05B A:>;L78BL. A;CG05 65
1>;LH>3> B@5=8O ?;0AB8=0 <>65B 8 =5 A428=CBLAO A <5AB0 (� = 0 ).
33
41. ?>4>1=>9 A<5A8 684:>AB59 B5;> ;N1>9 ?;>B=>AB8 >?CAB8BLAO
;8HL 4> =5:>B>@>9 3;C18=K, =0 :>B>@>9 A8;0 BO65AB8 1C45B
C@02=>25H5=0 A8;>9 @E8<540. CABL F5=B@
H0@8:0 =0E>48BAO =0 3;C18=5 h .
r
KB0;:820NI0O A8;0 FA ,
459AB2CNI0O =0 H0@, G8A;5==> @02=0 25AC
684:>AB8 2 2KB5A=5==>< >1J5<5. ;O
2KG8A;5=8O 25A0 2KB5A=5==>9 684:>AB8
25@=5< 55 >1@0B=> 8 @0AA<>B@8<
420 A8<<5B@8G=KE A5G5=8O Si ,S Si = S
( )
j j
B>;I8=>9 "xi "xi << "h .
( )
0==0O ?0@0 405B 2:;04 2 25A 684:>AB8:
"Pij = �iSi"xi g + �jS "xi g =
j
(1)
= gS"x �0 + ą( - "h + �0 + ą() ( )
h0 h0 + "h = 2 �0 + ąh0 "Vi g,
)
()
345 "Vi = Si"xi . 0: 2848<, 2:;04 B>=:>9 ?0@K ?@>?>@F8>=0;5=
B>;L:> >1J520==>B>@K9 ;53:>
?@>AC<<8@>20BL
ij
P =
() ( )
""P = "2 �0 + ąh0 "Vi g = �0 + ąh0 g"2"V =
i
ij
4 4
ż#
= =
()
�#"2"V = ĄR3 �# �0 + ąh0 g 3 ĄR3 ,
Ź#
i
3
�# �#
345 AC<<8@>20=85 ?@>872>48BAO ?> 2A5< ?0@0< i, j .
"5?5@L ;53:> =0945< <8=8<0;L=CN <0AAC H0@8:0 ?@8 :>B>@>9 >=
?>;=>ABLN ?>3@C78BAO 2 684:>ABL (h0=R):
4 4
m* = �0 + ąR ĄR3 = ĄR3 �0 + ąR . (2)
() ()
3 3
"0:8< >1@07><, CA;>285 7040G8 1C45B 2K?>;=5=> ?@8:
m e" m* . (3)
?@>B82=>< A;CG05 H0@8: 1C45B ?;020BL, G0AB8G=> 2KABC?0O 87
684:>AB8.
0<5B8<, GB> ?>4>1=CN 384@>A8AB56=> A<>45;8@>20BL,
0::C@0B=> 70;82 A;>9 A>;8 2>4>9. A;54AB285 48DDC788 (G5@57 20-
30 G0A>2) :>=F5=B@0F8O A>;8 1C45B ?040BL A 2KA>B>9. A;8 2 B0:>9
@0AB2>@ >?CAB8BL :C@8=>5 O9F>, B> >=> 7028A=5B (2 A>>B25BAB288 A
@5H5=85<) �<564C =51>< 8 75<;5.
42. 0?8H5< C@02=5=85 B5?;>2>3> 10;0=A0 4;O B5?;>>1<5=0 ?@8
?5@2>< ?>3@C65=88 H0@8:0 2 2>4C.
34
C t1 - t0 = C1 tH - t1 .
( ) ( )
BAN40 =0E>48< CAB0=>282HCNAO B5CtH + Ct0 C1tH + CtH - CtH + Ct0 C
1
t1 = = = tH + t0
( - tH .
)
C1 + C C1 + C C1 + C
#@02=5=85 B5?;>2>3> 10;0=A0 ?@8 2B>@>< ?>3@C65=88
C t2 - t1 = C1 tH - t2 ,
( ) ( )
8;8
CtH + Ct1 C C2
1
t2 = = tH + t1 t0
( - tH = tH +
) ( - tH .
)
2
C1 + C C1 + C
C1 + C
()
@>45;02 0=0;>38G=K5 459AB28O ?@8 B@5BL5< ?>3@C65=88 H0@8:0 2
2>4C. ?>4B25@6405< =0;8G85 70:>=><5@=>AB8 2 2K@065=88 4;O
B52>3> @02=>25A8O
C3
t3 = tH + t0
( - tH .
)
3
C1 + C
()
B0:, ?>A;5 n ?>3@C65=89
Cn
tn = tH + t0
( - tH .
)
n
C1 + C
()
43.H81:0 2 87<5@5=88 B57=8:05B 2A;54AB285 B>3>,
GB> B5@<><5B@ 8<55B A>1AB25==CN B5?;>5<:>ABL C1 8 53> =0G0;L=0O
B54K 2 A>AC45 t0 .
!;54>20B5;L=>, :0:0O-B> G0ABL B5?;>BK ?>945B =0 =03@520=85
B5@<><5B@0, GB> ?@82545B : C<5=LH5=8N B54K. CABL
CAB0=>282H0OAO B5A;5 B>3>, :0: 2 2>4C >?CAB8<
B5@<><5B@ @02=0 t0 - "t , B>340 C@02=5=85 B5?;>2>3> 10;0=A0 8<55B
()
284
C0"t = C1 t0 - "t - t1 . (1)
()
BAN40
C1 t0 - t1
( )
"t = .
C0 + C1
-B0 25;8G8=0 8 >?@545;O5B >H81:C 87<5@5=8O
"t H" 08o C.
,
44. !G8B05<, GB> ;54 ?;0205B, B>340 <0AA0 2KB5A=5==>9 2>4K
m = 2003 . CABL <0AA0 A28=F0 2> ;L4C mA2 , <0AA0 G8AB>3> ;L40 m1 .
#@02=5=85 B5?;>2>3> 10;0=A0
35
c2 m2 - m t2 - � = m; c;t; + c2� + + mA2cA2 � - t; ,
()( ) (-
)( )
345 c2 ,c; ,cA2 - C45;L=K5 B5?;>5<:>AB8 2>4K, ;L40 8 A28=F0, t2 ,t; -
=0G0;L=K5 B54K 8 ;L40, � - CAB0=>282H0OAO
B53> C@02=5=8O =0E>48< A CG5B>< m = m1 + mA2 ,
m( -c;t; + + c2� ) - A2(m - m2 )(t2 - � )
mA2 = H" 0,018:3 .
- A;t; + AA2t; + �(A2 - AA2 )
">340 m1 = 0182:3. =0O ?;>B=>AB8 ;L40 8 A28=F0, ;53:> =09B8
,
>1J5<=>5 A>45@60=85 A28=F0 2> ;L4C
mA2 / �A2
� = H" 08%.
,
mA2 / �A2 + m1 / �2
>:068B5 A0<>AB>OB5;L=>, GB> ?@8 B0:>< A>45@60=88 A28=F0 ;54
459AB28B5;L=> 1C45B ?;020BL 8, A;54>20B5;L=>, =0H5 87=0G0;L=>5
?@54?>;>65=85 25@=>.
B<5B8<, GB> 5A;8 1K ;54 CB>=C;, B> 1K;0 1K 2KB5A=5=0
4@C30O <0AA0 2>4K, @02=0O ?@>872545=8N ?;>B=>AB8 2>4K =0 >1J5<
;L40 A> A28=F><.
45. ! B>G:8 7@5=8O B5?;>2>3> 10;0=A0 CA;>285 CAB0=>2;5=8O
B54K 2 AB0:0=5 >7=0G05B, GB> AC<<0@=K5 B5?;>2K5
?>B5@8 2 A8AB5<5 (G5@57 AB5=:8, 4=>, A2>1>4=CN ?>25@E=>ABL
684:>AB8) @02=K :>;8G5AB2C B5?;>BK, ?@>872>48<>=03@520B5;5<.
0 �O7K:5 <>I=>AB59 MB> <>6=> 70?8A0BL A;54CNI8< >1@07><:
P= = P? , (1)
345 P= <>I=>ABL =03@520B5;O, P? <>I=>ABL ?>B5@L.
>I=>ABL ?>B5@L P? 7028A8B >B @07<5@>2 AB0:0=0, 53>
B5?;>?@>2>4=KE A2>9AB2, @07=>AB8 B574CE0,
B.5. 2>7@0AB05B ?> <5@5 =03@520=8O 2>4K. =K<8 A;>20<8, 4;O
:064>3> :>=:@5B=>3> A;CG0O ACI5AB2C5B <8=8<0;L=>5 7=0G5=85
<>I=>AB8 =03@520B5;O, 4>2>4OI53> 684:>ABL 2 A>AC45 4> :8?5=8O.
>A;5 2K:;NG5=8O =03@520B5;O ( P= = 0 ) P? A>E@0=O5B A2>5
7=0G5=85 (AG8B05<, GB> B5
87<5=8BLAO, B.5. 4;O 8A:><>3> 2@5<5=8 � A?@0254;82> @025=AB2>:
cm �" "T cm"T
P? �"� = cm"T �! � = == 17c, (2)
P? P=
36
6
345 A = 422 �"103 C45;L=0O B5?;>5<:>ABL 2>4K.
,
:3 �" :
@8 ?>;CG5=88 (2) 5<:>ABLN A0<>3>
=03@520B5;O, AG8B0O, GB> >=0 3>@074> <5=LH5 B5?;>5<:>AB8 2>4K.
70:;NG5=85 70<5B8<, GB> 25;8G8=0 <>65B 1KBL
>?@545;5=0 87 3@0D8:0 7028A8<>AB8 B54K >B 2@5<5=8,
B.:. >=0 ?@>?>@F8>=0;L=0 B0=35=AC C3;0 =0:;>=0 :0A0B5;L=>9 : >A8
01AF8AA 2 40==>9 B59 B>G:5 (A<.7040GC 9-3,
8B51A:,1990).
46. 0: C:070=> 2 CA;>288, 1@>H5==K9 :CA>G5: ;L40 AB0=>28BAO
F5=B@>< :@8AB0;;870F88, AB8F5AA
>1@07>20=8O ;L40. @8 MB>< 2K45;ONI0OAO B5?;>B0 845B =0
=03@520=85 >AB02H59AO 2>4K 4> B51@07><,
A>AB028< C@02=5=85 B5?;>2>3> 10;0=A0
m1 = m2c"t ,
345 m1 - <0AA0 >1@07>202H53>AO ;L40 (<0AAC 1@>H5==>3>
:@8AB0;;8:0 =5 CG8BK205<), m2 - <0AA0 >AB02H59AO 2>4K. BAN40,
CG8BK20O , GB> ?5@2>=0G0;L=0O <0AA0 2>4K 2 A>AC45 m = m1 + m2 ,
?>;CG05<
m2 c"t
= E" 006 .
,
m c"t +
47. @8 @5H5=88 7040G8 1C45< AG8B0BL, GB> @0AA58205<0O <>I=>ABL
> >
?@0:B8G5A:8 =5 <5=O5BAO ?@8 >ABK20=88 >B 95 ! 4> 90 !. ">340
:>;8G5AB2> B5?;>BK, ?>;CG5==>5 2>4>9 ?@8 =03@520=88:
P0
( - P1 �1 = Q, (1)
)
345 P0 ?>;57=0O <>I=>ABL =03@520B5;O; P1 <>I=>ABL ?>B5@L ?@8
40==KE CA;>28OE; �1 2@5ABK20=88:
P1�2 = Q, (2)
345 � 2@5ABK20=8O.
2
">340
P1 �1
= = 01 = 10%.
,
P0 �1 + �2
48. >A:>;L:C A>AC4 B5?;>87>;8@>20= 8 35@<5B8G5=, B> ?@8 3>@5=88
C3;5@>40 (8;8 ;N1>< 4@C3>< ?@>F5AA5) 1C45B A?@0254;82 70:>=
A>E@0=5=8O M=5@388 8 <0AAK. >;L7COAL C@02=5=85< A>AB>O=8O
8450;L=>3> 3070 2 D>@<5 5=45;5520 ;09?5@>=0, =0945< <0AAC
:8A;>@>40 2 A>AC45:
37
PV�O
0
2
mO = ,
2
RT0
345 �O <>;O@=0O <0AA0 :8A;>@>40, P0 8 T0 7=0G5=8O 402;5=8O 8
2
B5@<0;L=KE CA;>28OE.
mO = 14:3 .
,
2
#G8BK20O C@02=5=85 E8<8G5A:>9 @50:F88 3>@5=8O C3;5@>40,
>?@545;8< <0AAC A3>@52H53> C3;5@>40:
�C
C + O2 = CO2 �! mC = mO
2
�O
2
PV
345 �C = 123 / <>;L <>;O@=0O <0AA0 C3;5@>40 mC = �C 0 = 053:3 ,
,
RT0
B.5. 2K3>@8B =5 25AL ?>@>H>:, => 2 E>45 3>@5=8O @0AE>4C5BAO 25AL
:8A;>@>4.
!>3;0A=> 70:>=C A>E@0=5=8O M=5@388 <>65< ?@8@02=OBL 8AE>4=CN
2=CB@5==NN M=5@38N :8A;>@>40 8 >A2>1>645==CN ?@8 3>@5=88
B5?;>BC : 2=CB@5==59 M=5@388 >1@07>202H53>AO C3;5:8A;>3> 3070:
5 5 5 �C�0V
�RT = �RT0 + qmC = �RT0 + q . (1)
2 2 2 RT0
0<5B8<, GB> 2 >158E G0ABOE (1) AB>8B @02=>5 :>;8G5AB2> 25I5AB20,
B.:. 270<5= � <>;59 :8A;>@>40 >1@07>20;>AL � <>;59 C3;5:8A;>3>
3070 (A<. C@02=5=85 E8<8G5A:>9 @50:F88)
PV
0
�C = �O = .
2
RT0
:>=G0B5;L=> ?>;CG8<:
2 q�C
T = T0 + H" 19 �" 104 . (2)
,
5 R
!;54C5B 70<5B8BL, GB> 5<:>ABLN A>AC40 8
>AB02H53>AO 2 =5< 3@0D8B0. #G5B MB8E D0:B>@>2 =57=0G8B5;L=>
C<5=LH8B 7=0G5=85 (2).
49. @8 AB>?@>F5=B=>9 2;06=>AB8 G8A;> <>;5:C;, 2K;5B0NI8E 87
?>25@E=>AB8 2>4K, @02=> G8A;C <>;5:C; 2>4K, ?>?040NI8E 87
2>74CE0. ;06=>ABL 2>74CE0 ?@>?>@F8>=0;L=0 ?0@F80;L=>402;5=8N 2>4O=KE ?0@>2, ?>MB>;8G5AB2> <>;5:C;,
?>?040NI8E 87 2>74CE0 2 2>4C, ?@>?>@F8>=0;L=> 2;06=>AB8 � . "0:
:0: :>;8G5AB2> <>;5:C;, ?>?040NI8E =0 ?>25@E=>ABL 2>4K 7028A8B
B>;L:> >B B5 A:>@>ABL 8A?0@5=8O (@07=>ABL <564C
G8A;>< 2K;5B52H8E 8 G8A;>< �25@=C2H8EAO <>;5:C;)
?@>?>@F8>=0;L=0 25;8G8=5 1 - � , A;54>20B5;L=> 2@5( )
38
1
� ,
~
1 - �
;8
�1 1 - �0
= , (1)
�0 1 - �1
345 �0 , �1 2@5<5=0 2KAKE0=8O ?@8 2;06=>AB8 �0 , �1,
A>>B25BAB25==>. 7 (1) =0E>48<
1 - �0
�1 = �0 = 4 G0A0.
1 - �1
50. !B5?5=L 48AA>F80F88 ą = aT , B>340 G8A;> G0AB8F ?@8
T < 900K @02=> N = (1 + ą )�N , � = 2 <>;O. >=F5=B@0F8O G0AB8F
A
1 + �N
( ą) ,
A
n =
V
0 402;5=85
1 + �RT 1 + aT �RT
( ą)= ( )
p = nkT =
V V
?@8 T < 900K ,
�RT
p = 2 ?@8 T > 900K .
V
!E5<0B8G5A:89 3@0D8: 7028A8<>AB8 402;5=8O 2
7028A8<>AB8 >B B5@8AC=:5. 1@0B8B5 2=8<0=85 =0 87;>< 3@0D8:0
2 B>G:5 T = 900K , >1JOA=8B5 53> ?>O2;5=85.
51. !@54=55 G8A;> C40@>2 <>;5:C; > 2=CB@5==NN ?>25@E=>ABL
10;;>=0 70 2@5 G8A;C <>;5:C;, :>B>@K5 4>;5BOB 4>
?>25@E=>AB8 10;;>=0 70 MB> 2@5, 2 A2>N >G5@54L, @02=>
G8A;C <>;5:C; 2 >1J5<5 v"tS . 0 548=8FC 2@5<5=8 G8A;> C40@>2
1C45B @02=>
1 1 N 3 N
� = nvS = v4ĄR2 = v, (1)
4
4 4 4 R
ĄR3
3
N
345 n = :>=F5=B@0F8O <>;5:C;, N >1I55 G8A;> <>;5:C; 2
V
4
A>AC45, V = ĄR3 >1J5< A>AC40, S = 4ĄR2 ?;>I04L 53>
3
?>25@E=>AB8, � - A@54=OO A:>@>ABL B5?;>2>3> 42865=8O <>;5:C;.
>MDD8F85=B 1/4 CG8BK205B 4>;N <>;5:C; 2 2K45;5==>< >1J5<5,
39
:>B>@K5 ;5BOB ?> =0?@02;5=8N : AB5=:5. <5AB> 1/4 <>6=>
8A?>;L7>20BL 1>;55 ?@82KG=K9 2 H:>;L=KE :C@A0E :>MDD8F85=B
1/6, GB> ?@8=F8?80;L=K< >1@07>< =5 <5=O5B @5H5=85 7040G8.
!@54=55 G8A;> C40@>2 >4=>9 <>;5:C;K
� 3 v
�1 = = ,
N 4 R
!;54>20B5;L=>, A@54=55 2@51 4R
� = = .
�1 3v
B<5B8<, GB> MB0 25;8G8=0 =5 7028A8B >B G8A;0 <>;5:C; 2 A>AC45 8,
?> ?>@O4:C 25;8G8=K, @02=0 2@5<5=8 A2>1>4=>3> ?@>;5B0 >B >4=>9
2R
B>G:8 ?>25@E=>AB8 4> 480<5B@0;L=> ?@>B82>?>;>6=>9 .
v
>=5G=>, C25;8G5=85 ?;>B=>AB8 3070 2 A>AC45 ?@82>48B :
C25;8G5=8N G8A;0 AB>;:=>25=89 <>;5:C; <564C A>1>9, >4=0:> MB>
>1AB>OB5;LAB2> =5 A:07K205BAO =0 A@54=5< G8A;5 C40@>2 2K1@0==>9
<>;5:C;K > AB5=:C. ".:. B5 <>;5:C;K, :>B>@K5 87=0G0;L=> 1K;8
@0A?>;>65=K 1;865 : AB5=:0<, 1C4CB C40@OBAO G0I5 B5E, :>B>@K5
=0E>48;8AL 1;865 : F5=B@C, 2 A@54=5< 65 @57C;LB0B >B G8A;0
<>;5:C; =5 7028A8B.
52. '8A;> <>;5:C;, 2K;5B0NI8E 87 >B25@AB8O 70 2@5
1 N
"N = vS� (A<. @5H5=85 7040G8 51) . "0: :0:
4 V
"p "N vS�
= = � = 001, B> � = .
,
p N 4V
4V� 4V� Ą �
BAN40 � = = H" 90c .
Sv S 8 RT
53.>A:>;L:C <5<1@0=0 �?@>7@0G=0 4;O 35;8O, B> ?>A;5
CAB0=>2;5=8O @02=>25A8O >= 709<5B 25AL A>AC4, ?@8G5< 53>
P0
?0@F8>=0;L=>5 402;5=85 C?045B 242>5 8 AB0=5B @02=K< .
2
!>3;0A=> 70:>=C 0;LB>=0 2 >4=>9 ?>;>28=5 A>AC40 (345
P0
A>45@68BAO B>;L:> 35;89 He) 402;5=85 1C45B , 0 2 4@C3>9
2
3P0
(He+N2) . ">340 8A:><0O @07=>ABL
2
"P = P0 .
40
>6=> 1K;> @0AAC640BL 8 8=0G5: 2 :>=5G=>< A>AB>O=88 459AB285
35;8O <>6=> =5 CG8BK20BL, B.:. >= 709<5B 25AL A>AC4 8 1C45B 4028BL
=0 <5<1@0=C A >158E AB>@>=. 2>B 07>B 1C45B ?>-?@56=5=0 <5<1@0=C: "P = P0 .
54.>@H5=L 1C45B 42830BLAO A=0G0;0 CA:>@5==>, 0 70B5<
70<54;5==>. !:>@>ABL 1C45B <0:A8<0;L=0, :>340 CA:>@5=85
>1@0B8BAO 2 =C;L, B.5. :>340 ?>@H5=L ?@>945B G5@57 ?>;>65=85
@02=>25A8O. -B> CA;>285 5ABL CA;>285 @025=AB20 A8; 402;5=8O 8
A8;K BO65AB8.
pS = mg. (1)
>A:>;L:C ?@>F5AA ?>4J5<0 ?>@H=O 4> ?>;>65=8O @02=>25A8O
?@>E>48B 4>AB0B>G=> 1KAB@>, 65< ?@8 @5H5=88 7040G8 =5
CG8BK20BL B5?;>>1<5= <564C 307>< 8 >:@C60NI8< ?@>AB@0=AB2><.
0?8H5< 70:>= A>E@0=5=8O M=5@388
K + P + U =U0
345 K 8 P :8=5B8G5A:0O 8 ?>B5=F80;L=0O M=5@388 ?>@H=O, U 8 U0
:>=5G=0O 8 =0G0;L=0O 2=CB@5==85 M=5@388 3070. ;8
mv2 3 3
+ mgh + �RT = �RT0 . (2)
2 2 2
A?>;L7C5< C@02=5=85 5=45;5520 ;0?59@>=0 4;O =0G0;L=>3> 8
:>=5G=>3> A>AB>O=89 3070
PV0 = P0Sl = �RT0 , PV = PS h + l = �RT. (3)
( )
0
>4AB02;O5< (1) 8 (3) 2 C@02=5=85 (2)
mv2 3 3
+ mgh + mg h + l = P0Sl.
( )
2 2 2
BAN40 =0E>48< 8A:>@>ABL
3PSl
0
v =- g 5h + 3l .
()
m
41
55.!45;05< AE5<0B8G5A:89 @8AC=>: =0H59 CAB0=>2:8. #AB0=>2:0
A>25@H05B AC<<0@=CN @01>BC 70 F8:; A = A1 + A2 , 345 A1 @01>B0
70 F8:; @01>G53> B5;0 ?5@2>9 <0H8=K, 0 A2
@01>B0 70 F8:; @01>G53> B5;0 2B>@>9 <0H8=K. >
>?@545;5=8N .
A A1 + A2 A1 A2
� = = = + .
QH QH QH QH
111 1
;O ?5@2>9 <0H8=K @025=
A1
�1 = .
QH
1
;O 2B>@>9 A>>B25BAB25==>
A2 A2 A2
�2 = = = .
QH QX 1 - �1 QH
( )
1 1 1
BAN40
A2
= �2 1 - �1 ..
( )
QH
1
">340 >:>=G0B5;L=>5 2K@065=85 4;O 2A59
A8AB5� = �1 + �2 - �1�2 ,
GB> 3>2>@8B > ?>2KH5=88 MDD5:B82=>AB8 8A?>;L7>20=8O
42830B5;59 2 A>?@O65==>9 A8AB5<5. @8G5< >B ?>@O4:0 A;54>20=8O
42830B5;59 MB>B =5 7028A8B.
56.!>3;0A=> >?@545;5=8N :
A
� = , (1)
Q1
345 A @01>B0, A>25@H5==0O 8450;L=K<
307>< 70 F8:;, Q1 :>;8G5AB2> B5?;>BK,
?>;CG5==>5 @01>G8< B5;>< >B =03@520B5;O.
40==>< A;CG05 B>G:8 1 8 3 ;560B =0
>4=>9 ?@O<>9, B.5.:
P1 = ąV1
ż#
P = ąV �!
�#P = P3 = ą(2V1 = 2ąV1 = 2P1 ,
)
2
�#
B.5. 2 B5G5=85 ?@>F5AA0 402;5=8O 2>7@0AB05B B0:65 2 2 @070.
01>B0 3070 70 F8:; @02=0 ?;>I048 70HB@8E>20==>3>
?@O<>C3>;L=8:0:
A = V2 -V1 P2 - P1 = P1V1. (2)
()()
42
7 3@0D8:0 2848<, GB> =03@520B5;L ?>4:;NG8;AO 2 B>G:5 1 F8:;0, 0
>B:;NG8;AO 2 B>G:5 3. 0 >AB0;L=KE CG0AB:0E F8:;0 @01>G55 B5;>
:>=B0:B8@>20;> A E>;>48;L=8:><. "0:8< >1@07><:
Q1 = Q12 + Q23 . (3)
!>3;0A=> ?5@2>48=0<8:8:
3 3
Q12 = "U12 + A12 = � R T2 - T1 + 0 = � R T2 - T1 , (4)
() ()
2 2
345 � :>;8G5AB2> 25I5AB20 8450;L=>3> 3070. =0;>38G=>:
3
Q23 = � R T3 - T2 + P2 V3 -V2 . (5)
() ()
2
>;L7COAL (3) (5) ?>;CG05<:
PV1 = �RT1 3
3 ż# �#
1
Q1 = � R T3 - T1 + 2P1V1 = = PV2 - PV1 + 2PV1 = PV1.
() ()13
�#PV = �RT3 Ź#
2 1 1 1
2 2 2
�# 2 2 �#
:>=G0B5;L=>:
PV1 2
1
� == = 15%. (6)
13
PV1 13
1
2
0: 2848< 87 (6) � 4>AB0B>G=> <0; 8 ?@8<5@=> A>>B25BAB2C5B
7=0G5=8N ?0@>2KE <0H8=. %>B5;>AL 1K >1@0B8BL 2=8<0=85 =0
<5B>48:C ?>8A:0 CG0AB:>2 F8:;0 A @01>B0NI8< =03@520B5;5<.
A;CG05 �?@O<>C3>;L=>3> F8:;0 MB> =5A;>6=>: B>G:0 T3
A>>B25BAB2C5B <0:A8<0;L=>9 B5< A>AB>O=88
>B:;NG05BAO =03@520B5;L. A;8 65 F8:; 1>;55 A;>65=, =0?@8<5@,
�:@C3>2>9, B> 4;O >BKA:0=8O A>>B25BAB2CNI8E B>G5: >B:;NG5=8O
=03@520B5;O =5>1E>48<> 8A:0BL B>G:8 :0A0=8O F8:;0 A 048010B0<8
ł
PV = const , GB> ?@54AB02;O5BAO 1>;55 A;>6=K< 8 3@><>74:8<.
()
57. 725AB=>, GB> @01>B0 @02=0 ?;>I048 F8:;0, 87>1@065==>3> 2
,
:>>@48=0B0E ( PV ), B.5. ?;>I048
B@5C3>;L=8:0 2 =0H5< A;CG05
1
A = P2 - P1 V3 -V1 .
()()
2
>A?>;L7C5 B@5C3>;L=8:
@02=>154@5==K9, 0 B>G:8 2 8 3 ;560B =0
>4=>9 87>B5@<5.
V1 + V3 RT2 2R�T0
V2 = ,T2 = T3 = �T0 �! P2 = = .
2 V2 V1 + V3
>4AB028< P2 2 2K@065=85 4;O @01>BK
43
ś#
�# ( )ź#.V3
1 RT0 ś# �# P1 V1 +V3 -V1
A = - P1ź# V3 -V1 = �RT0 - .
ś#2� () ś#
2 V1 +V3 # �# #
�# 2 V3 +V1
7 C@02=5=8O A>AB>O=8O 8450;L=>3> 3070
PV1 = RT0
ż#
1
( ) ( )
�#PV = RT3 = �RT0 �! V3 = �V1 ,V3 - V1 = � - 1 V1 ,V3 + V1 = � + 1 V1.
�# 1 3
">340
2
�# PV1 1 + � ś# �# RT0 1 + � ś# �
( )ź# - 1 ( )ź# - 1 ( - 1
� � )
1
A = �RT0 - = �RT0 - = RT0 .
ś# ś#
2 � + 1 2 � + 1 2 � + 1
�# # �# # ( )
58. "0: :0: ?@8 048010B=>< ?@>F5AA5
B5?;>>1<5= >BACBAB2C5B, B> =0 CG0AB:5 2
2=0G0;5 307 ?>;CG05B B5?;>BC, 0 70B5< 55
>B405B. 0945< 7028A8<>ABL :>;8G5AB20
?>;CG5==>9 307>< B5?;>BK Q >B 53>
>1J5<0 V , ?@8 C25;8G5=88 ?>A;54=53> >B
V0 4> 2V0 . > ?5@2>=C
B5@<>48=0<8:8
3 P0 + P
Q = "U + A = R"T + V
( -V0 , (1)
)
22
3
345 "U = R"T 87<5=5=85 2=CB@5==59 M=5@388 3070, A @01>B0
2
A>25@H5==0O 307><. "0: :0: 402;5=85 87<5=O5BAO ?> ;8=59=>70:>=C, B> @01>B0 ;53:> 2KG8A;O5BAO :0: ?;>I04L A>>B25BAB2CNI59
B@0?5F88 ?>4 3@0D8:>< DC=:F88 P(V ) . #@02=5=85 ?@O<>9 AB
<>65B 1KBL 70?8A0=> 2 2845
P0 - P1
P = P0 - ( -V0 . (2)
V
)
V0
#G8BK20O B0: 65 C@02=5=85 A>AB>O=8O 8450;L=>3> 3070 PV = RT,
25;8G8=C Q(V ) <>6=> 2K@078BL:
�# P0 - P1 ś#
ś#2P0 + 3P1 - 4 "V "V
ź#
V0 #
Q "V = , (3)
( )�#
2
44
345 "V =V - V0 . -B0 DC=:F8O 8<55B <0:A82P0 + 3P1
*
"V = V0 .
8 P0 - P1
()
!;54>20B5;L=>, 4> B0:>3> 7=0G5=8O ?@8@0I5=8O >1J5<0 307
?>;CG05B B5?;>BC, 0 70B5< 55 >B405B. "0:8< >1@07><, :>;8G5AB2>
?>;CG5==>9 B5?;>BK
2P0 + 3P1 2
()
*
Q1 = Q "V = V0 . (4)
( )
32 P0 - P1
()
07=>ABL <564C ?>;CG5==>9 8 >B40==>9 B5?;>B0<8 5ABL @01>B0,
A>25@H5==0O 70 F8:;, :>B>@CN <>6=> ?>4AG8B0BL, ?>;>682 2 (3)
"V = 2V0 - V0 = V0
7 P1 - 2P0
A = V0 .
2
">340 =0E>48< :>MDD8F85=B ?>;57=>3> 459AB28O
7 P1
( - 2P0 P0 - P1
)( )
A
� = = 16 H" 026 .
,
2
Q1
2P0 + 3P1
()
59. (0@8: 1 =0E>48BAO 2 @02=>25A88 ?>4 459AB285< 42CE A8;:
r r r
T1 + Fk 1 = 0 , (1)
r r
345 T1 A8;0 =0BO65=8O =8B8, Fk1 @02=>459AB2CNI0O
:C;>=>2A:8E A8;, 459AB2CNI0O =0 H0@8: 1 A> AB>@>=K >AB0;L=KE.
!>3;0A=> ?@8=F8?C AC?5@?>78F88 ?>;59:
1 q2 1 1 49 1 q2
�#1 + + ś#
T1 = = . (2)
ś# ź#
4Ą�0 l2 �# 4 9 # 144 Ą�0 l2
=0;>38G=> 70?8AK20O CA;>285 @02=>25A8O H0@0 2, ?>;CG05<:
ż# q2 1
�#T3 - T2 =
(3)
�# 4Ą�0 4l2 .
�#
T2 = T1
�#
58 q2
7 (3) =0945< T3 = .
144 Ą�0l2
45
T3 58
"0:8< >1@07><, = .
T1 49
60. >A:>;L:C ?;>A:>ABL 3;04:0O 8 70@O4K 2A5 >4=>3> 7=0:0, B> ?>4
459AB285< :C;>=>2A:8E A8; >=8 =0G=CB @07J5760BLAO �2 @07=K5
AB>@>=K. "0: :0: 70@O4K 2708<>459AB2CNB B>;L:> <564C A>1>9 (2
?;>A:>AB8 42865=8O), B> ?>;>65=85 F5=B@0 <0AA A8AB565B
87<5=8BLAO.
>AB@>5=85< >?@545;8<
?>;>65=85 F5=B@0 <0AA A8AB5B>G:8 . @8 MB>< (?> >?@545;5=8N
F5=B@0 <0AA 42CE B>G5:):
AB 2 AO 1
= ; = .
AC 1 OD 1
>2B>@OO 0=0;>38G=K5
?>AB@>5=8O 2 ?@56=8E ?@>?>@F8OE
4;O 8725AB=KE B>G5: B 8 C ;53:>
=0945< <5AB>?>;>65=85 3@C70 3m D . @8 MB><:
A' B' 2 AO 1
'
= ; = .
A' C' 1 OD' 1
=B5@5A=>, GB> >B25B =5 7028A8B >B 25;8G8= 70@O4>2 3;02=>5,
GB>1K >=8 1K;8 >4=>3> 7=0:0. MB>< A;CG05 B@5C3>;L=8: 1C45B
�@0AB8. #1548B5AL A0<>AB>OB5;L=>, GB> B@5C3>;L=8:8 ! 8
C B D ?>4>1=K 8 @071530=85 70@O4>2 @50;87C5BAO B0:8< >1@07><,
GB> A>>B25BAB2CNI85 AB>@>=K 2KH5C:070==KE B@5C3>;L=8:>2
>AB0NBAO ?0@0;;5;L=K<8.
61. A;54AB285 O2;5=8O M;5:B@>AB0B8G5A:>9 8=4C:F88 =0 1;86=5<
H0@8:5 1 ?>O28BAO 70@O4 ?@>B82>?>;>6=>3> 7=0:0 - q' , 0 =0
40;L=5< H0@8:5 2 70@O4 B>3> 65 7=0:0 + q' . @8G5< A>3;0A=>
70:>=C A>E@0=5=8O 70@O40 2
87>;8@>20==>9 A8AB5<5 q' = - q' ,
B0: :0: 87=0G0;L=> H0@8:8 1K;8
=5 70@O65=K.
5@5@0A?@545;5=85 70@O4>2 (B. 5. M;5:B@8G5A:89 B>:) ?@5:@0B8BAO,
:>340 8AG57=5B =0?@O65=85 (B. 5. @07=>ABL ?>B5=F80;>2) <564C
H0@8:0<8:
�1 = �2 (1).
A>>B25BAB288 A ?@8=F8?>< AC?5@?>78F88 :
46
ż# 1 q q' q'
�# ś#
�1 =- + ,
ś# ź#
�#
�# #
�# 4Ą�0 l r a
(2)
�#
�#�2 = 1 �# q + q' q' ś#.
-
ś# ź#
�# #
�#
4Ą�0 l + a r a
�#
7 (1)-(2) ?>;CG05<:
q a2r q ar
q' = H" r << a H" . (3)
{ }( )
2 l + a l a - r 2 l l + a
( ) ( )
0: 2848< 87 (3), q' 0 ?@8 l " , B. 5. ?@8 C40;5=88 70@O40 q
O2;5=85 M;5:B@8G5A:>9 8=4C:F88 8AG5705B. =0;>38G=> q' 0 ?@8
A1;865=88 H0@8:>2 , B0: :0: ?@8 MB>< 2K@02=8205BAO 8E
�2=5H=89 ?>B5=F80;.
@8 @5H5=88 7040G8 5<:>ABL
B>=:>9 ?@>2>4OI59 =8B8, =0 :>B>@>9 B>65 >:06CBAO =5:>B>@K5
8=4CF8@>20==K5 70@O4K.
62. 5>1E>48AB0B>G=K< CA;>285< 2>7<>6=>AB8 :>;510=89
2 A8AB5<5 O2;O5BAO =0;8G85 ?>;>65=8O CAB>9G82>3> @02=>25A8O.
CABL 1CA8=:0 =0E>48BLAO =0 @0AAB>O=88 x >B
F5=B@0 :>;LF0. !8;C M;5:B@8G5A:>3>
2708<>459AB28O 1CA8=:8 A :>;LF>< <>6=>
=09B8 A ?><>ILN ?@8=F8?0 AC?5@?>78F88,
@071820O :>;LF> =0 <0;5=L:85 70@O65==K5
CG0AB:8 8 AC<<8@CO 70B5< ?@>5:F88 =0
=0?@02;5=85 >A8 :>;LF0 A8; 2708<>459AB28O
1CA8=:8 A> 2A5<8 CG0AB:0<8 :>;LF0
qQx
F = . (1)
3/ 2
4Ą�0 x2 + R2
()
">G:0< @02=>25A8O A>>B25BAB2C5B CA;>285
qQx
mg = F = . (2)
3/ 2
4Ą�0 x2 + R2
()
5?>A@54AB25==>5 @5H5=85 MB>3> C@02=5=8O 70B@C4=8B5;L=>,
>4=0:> >=> =0< 8 =5 B@51C5BAO, 4>AB0B>G=> 2KOA=8BL ?@8 :0:8E
7=0G5=8OE ?0@0<5B@>2 >=> ACI5AB2C5B. 5@5?8H5< (2) 2 2845
47
4Ą�0mg x
a == f ( x ) = . (3)
3/ 2
qQ
x2 + R2
()
$C=:F8O f ( x ) =5?@5@K2=0,
=5G5B=0, ?@8 x ą" ,
f ( x ) ą0 , :@><5 B>3> ?@8
x = 0, f ( x ) = 0 .
!E5<0B8G5A:89 3@0D8: MB>9
DC=:F88 87>1@065= =0
@8AC=:5. 0945G:8
M:AB@52 x * 87 CA;>28O f ' ( x*) = 0
R2 - 2x2
f 2 ( x ) = . (4)
5/ 2
x2 + R2
()
B:C40 =0E>48<
R
x" =ą ,
2
2
8 7=0G5=8O f ( x" ) = ą .
3 3R2
"0:8< >1@07>< C@02=5=85 (3) 1C45B 8<5BL @5H5=8O (?@8G5<
420 :>@=O!) ?@8
4Ą�0mg 2
< f ( x" ) =,
qQ
3 3R2
B> 5ABL ?@8
6 3Ą�0 R2mg
q > . (5)
Q
KOA=8<, 5ABL ;8 A@548 MB8E 42CE ?>;>65=89 @02=>25A8O E>BO 1K
>4=> CAB>9G82>5.
CABL 70@O4K q 8 Q 8<5NB >48=0:>2K5 7=0:8. ">340
@5H5=8O C@02=5=8O (3) x1 8 x2 ?>;>68B5;L=K. 0<5B8<, GB> A
B>G=>ABLN 4> ?>AB>O==>3> ?>;>68B5;L=>3> <=>68B5;O f ( x ) 5ABL
A8;0, 459AB2CNI0O =0 H0@8: 225@E, a A8;0, 459AB2CNI0O 2=87.
0AA<0B@820O A8;K ?@8 <0;>< >B:;>=5=88 >B B>G:8 x1 70<5G05<,
GB> ?@8 x > x1 M;5:B@8G5A:0O A8;0 ?5@525H8205B A8;C BO65AB8, 0 ?@8
x < x1 A8BC0F8O >1@0B=0O. !;54>20B5;L=>, B>G:0 x1 ?>;>65=85
=5CAB>9G82>3> @02=>25A8O. =0;>38G=>5 @0AA<>B@5=85 A8; 21;878
B>G:8 x2 ?@82>48B : 2K2>4C, GB> MB> ?>;>65=85 @02=>25A8O
CAB>9G82>.
48
@8 @07=KE 7=0:0E 70@O4>2 CAB>9G82K< >:07K205BAO
?>;>65=85 @02=>25A8O 2 B>G:5 x1.
"0:8< >1@07><, ?@8 2K?>;=5=88 CA;>28O (5) 2 A8AB5<5
ACI5AB2C5B CAB>9G82>5 ?>;>65=85 @02=>25A8O 8, A;54>20B5;L=>,
2>7<>6=K :>;510=8O 1CA8=:8.
63. >A:>;L:C @048CA H0@0 1>;LH5 @048CA0 :>;LF0, B> 4;O
2K?>;=5=8O CA;>28O 7040G8 >= 4>;65= @0ABO=CBL :>;LF> A8;0<8
:C;>=>2A:>3> 2708<>459AB28O. CABL 2 @57C;LB0B5 =5>1E>48<>3>
2708<>459AB28O :>;LF> @0ABO=C;>AL 4> @048CA0 R1 , B>340 =0
548=8FC 53> 4;8=K ?@8E>48BLAO 70@O4:
Q
= ,
2ĄR1
345 Q 70@O4 :>;LF0.
0AA<>B@8< <0;K9 M;5<5=B :>;LF0 "l ( "l << R1 )A C3;>2>9
25;8G8=>9 2ą . 0 =53> 459AB2CNB
r
:C;>=>2A:85 A8;K Fi , :>B>@K5
C@02=>25H820NBAO A8;0<8 C?@C3>AB8 2
@0ABO=CB>< :>;LF5: (2 ?@>5:F88 =0
r
25@B8:0;L=CN >AL): = T2 = T
{T r }
1
1 �" "l �" QH
2T siną = . (1)
2
4Ą�0 R1
r
0?8AK20O (1), 2 A8;C <0;>AB8 "l , 2A5 25:B>@0 Fi
?0@0;;5;L=K >A8 X . @><5 B>3>, 745AL ?>;5 H0@0 A
70@O4>< QH M:2820;5=B=> ?>;N B>G5G=>3> 70@O40, @0A?>;>65==>3>
2 B>G:5 O .
!>3;0A=> 70:>=C C:0:
T = k "l = k 2Ą( R1 - R2 ), (2)
345 k :>MDD8F85=B C?@C3>AB8 :>;LF0.
@><5 B>3>, 2 A8;C <0;>AB8 "l :
"l
siną H" ą = . (3)
2R1
! CG5B>< (2) 8 (3) 2K@065=85 (1) ?5@5?8H5BAO 2 2845:
2k2Ą R1 - R2 =
()2"l 1 Q "lQH
2
R1 4Ą�0 2ĄR1 R1
. (4)
3
16Ą �0 R1 - R2 R12
()
QH = k
Q
49
>4G5@:=5<, GB> A0< <5E0=87< ?@>E>645=8O H0@0 A:2>7L
:>;LF> 1AC6405<. K@065=85 (4) 405B ;8HL =5>1E>48<>5
CA;>285 MB>3> ?@>F5AA0.
64. 2865=85 1@CA:0 ?@>8AE>48B ?>4 459AB285< G5BK@5E A8;:
r rr r r
ma = mg + qE + N + FB @ , (1)
r r
345 N A8;0 @50:F88 >?>@K (?@87
r r
AB>@>=K ?@8720B5;L=>, 25:B>@=0O AC<<0 A8; - N + FB@
()
?@8;>65=0 : ?@87<5 8 2K7K205B 55 42865=85 A CA:>@5=85<,
>?@545;O5<>< C@02=5=85<:
r r r
r r
Ma0 = - N + FB @ + N0 + Mg,
()
r
345 N0 - A8;0 @50:F88 >?>@K, 0
r
Mg - A8;0 BO65AB8 ?@87#G8BK20O, GB> 42865=85 1@CA:0
A;>6=>5 (2<5AB5 A ?@87<>9 8 24>;L
?> =59), ?@54AB028< 53> CA:>@5=85
2 2845 25:B>@=>9 AC<r r r
a = a0 + a1 , (2)
r r
345 a0 CA:>@5=85 ?@87CA:>@5=85 24>;L ?@87@>5F8@CO C@02=5=85 (1) =0 >AL
Y , ?>;CG8<
ma0 siną = mg cosą + qE siną - N . (3)
;O ?@875:F88 =0 3>@87>=B0;L=CN >AL):
Ma0 = N siną + F cosą . (4)
"
@><5 B>3>, CGB5<, GB>: FB@. = �N .
7 (3) ?>;CG8<:
N = mg cosą + qE siną - ma0 siną . (5)
">340 A CG5B>< (4)
qE siną + mg cosą)(
siną + � cosą)
(
a0 = . (6)
M + msiną(
siną + � cosą)
A;CG05 ?@545;L=>3> ?5@5E>40 (ą 0 ) 87 (6) ?>;CG05< E>@>H>
8725AB=K9 @57C;LB0B:
�mg
a0 = ,
N
50
0 ?@8 ą Ą / 2 ?>;CG05< B0:65 2?>;=5 @50;L=K9 ?@545;L=K9
?5@5E>4:
qE
a0 = ,
M + m
B0: :0: 2 MB>< A;CG05 4286CI59 A8;>9 O2;O5BAO =5 A8;0 B@5=8O, 0
A8;0 M;5:B@>AB0B8G5A:>3> 459AB28O.
r
65. (0@8: 42865BAO 2 ?>;5 42CE ?>AB>O==KE A8;: BO65AB8 mg 8
r r
"
M;5:B@>AB0B8G5A:>9 qE . E AC<<0 F B0:65 ?>AB>O==0 8 @02=0
22
"
F = mg + qE
( ) ( )
qE
0?@02;5=0 MB0 A8;0 ?>4 C3;>< ą : 25@B8:0;8, tgą = . ;O
mg
<8=8<0;L=>9 A8;K =0BO65=8O =8B8 2
B>G:5 , GB>1K H0@8: <>3 42830BLAO
?> >:@C6=>AB8 4>;6=> 2K?>;=OBLAO
mv2
"
CA;>285 = F , 345 v -
l
<8=8<0;L=> 2>7<>6=0O A:>@>ABL 2
B>G:5 . ;O B>3>, GB>1K =09B8
A:>@>ABL 2 B>G:5 , 2>A?>;L7C570:>=>< A>E@0=5=8O M=5@388
mv2 mv2
"
A
= + F l 1 - cosą)
.
(
2 2
7 70?8A0==KE C@02=5=89 ?>;CG8<
l 22
vA = 3 mg + qE - 2gl .
( ) ( )
m
66. -;5:B@>=K, ?@>94O CA:>@ONICN @07=>ABL ?>B5=F80;>2,
?>?040NB =0 C548=5==K9 H0@8:, :>B>@K9 ?@8>1@5B05B
>B@8F0B5;L=K9 70@O4. -B>B 70@O4 A>7405B M;5:B@8G5A:>5 ?>;5,
>BB0;:820NI55 =>2K5 =0;5B0NI85 M;5:B@>=K. '0AB8FK 1C4CB
4>;5B0BL 4> ?>25@E=>AB8 H0@8:0, ?>:0 :8=5B8G5A:0O M=5@38O,
?@8>1@5B05<0O 8<8 2 ?@>F5AA5 @073>=0, =5 AB0=5B @02=>9
?>B5=F80;L=>9 M=5@388 M;5<5=B0@=>3> 70@O40 =0 ?>25@E=>AB8
H0@8:0. "0:8< >1@07>< H0@8: 70@O48BLAO 4> ?>B5=F80;0 - U
G8A;5==> @02=>3> CA:>@ONI5B5=F80;C. "5?5@L =5B@C4=>, 7=0O
5<:>ABL 548=5==>3> H0@0
C = 4Ą�0 R ,
51
>?@545;8BL 8 70@O4 H0@8:0
Q = -UC = -4Ą�0 RU .
67. 0?5;L:0 @07>@25BAO =0 G0AB8, 5A;8 402;5=85 M;5:B@8G5A:>3>
?>;O =0 ?>25@E=>AB=K5 70@O4K ?@52KA8B ;0?;0A>2A:>5 402;5=85.
0: 8725AB=>, 402;5=85 M;5:B@8G5A:>3> ?>;O Pe =0 ?>25@E=>AB=K5
70@O4K =0 ?@>2>4=8:5 @02=> ?;>B=>AB8 M=5@388 ?>;O
2
2
�0 E �0 �# ś#
Q
Pe == , (1)
ś# ź#
2 2 4Ą�0 R2 #
�#
0 ;0?;0A>2A:>5 402;5=85 ?>4 AD5@8G5A:>9 ?>25@E=>ABLN
2�
P1 = . (2)
R
@8@02=820O (1) 8 (2), =0945< <8=8<0;L=K9 70@O4 :0?;8, ?@8
:>B>@>< ?>A;54=OO @07>@25BAO
�
Q = 8Ą�0 R2 .
�0 R
68. 0: 8725AB=>, =0?@O65==>ABL ?>;O 2 48M;5:B@8:5 E 2 � @07
<5=LH5 =0?@O65==>AB8 ?>;O 2 20:CC<5 E0 (5A;8 A8;>2K5 ;8=88
?5@?5=48:C;O@=K 3@0=8F5 @0745;0, GB> 2K?>;=O5BAO 2 40==><
A;CG05)
E0
E = . (1)
�
-B> C<5=LH5=85 =0?@O65==>AB8 ?>;O >1CA;>2;5=>
?>;O@870F8>==K<8 70@O40<8, 8=4CF8@>20==K<8 =0 ?>25@E=>AB8
48M;5:B@8:0. CABL ?>25@E=>AB=0O ?;>B=>ABL 8=4CF8@>20==KE
70@O4>2 �' , B>340 =0?@O65==>ABL ?>;O 2 48M;5:B@8:5 <>6=>
?@54AB028BL 2 2845
�'
E = E0 + , (2)
�0
345 �'/ �0 =0?@O65==>ABL ?>;O A>740205<>ć� ?>;O@870F8>==K<8
70@O40<8. 07=>ABL ?>B5=F80;>2 U <564C >1:;04:0<8
:>=45=A0B>@0 <>6=> ?@54AB028BL 2 2845
d d
U = E + E0 . (3)
2 2
5H0O A>2<5AB=> (1)-(3), ?>;CG8<
52
� - 1 2�0U
�' = - �" .
� + 1 d
=0: <8=CA C:07K205B, GB> ?>;5 8=4CF8@>20==KE 70@O4>2
=0?@02;5=> ?@>B82>?>;>6=> ?>@>482H5 ?>;N (A<. B0:65
7040GC 10-4, 840,1995).
69. 0==0O 7040G0 ?>GB8 ?>;=>ABLN ?>2B>@O5B 7040GC 10-5
5A?C1;8:0=A:>9 >;840, >M4>< A;CG05
>3@0=8G8B25B0
1 + 2 a
R = 2 � H" 037 R1.
,
S
5 + 3 2
70.">: G5@57 30;L20=><5B@ >BACBAB2C5B, 5A;8 ?>B5=F80; B>G:8 !
@025= ?>B5=F80;C B>G:8 <564C @578AB>@0<8 R1 8 R2 . -B> CA;>285
2K?>;=O5BAO, 5A;8
R1 l1
= . (1)
R2 l2
@><5 B>3>, 2 CA;>288
>3>2>@5=>, GB> AB5@65=L
A;01>?@>2>4OI89, ?>MB><>6=> ?@5=51@5GL B>:>< G5@57 AB5@65=L ?> A@02=5=8N A B>:><
G5@57 @578AB>@K. >MB>I( R1 + R2 ) = U (2),
345 I 8 U - ?>:070=8O 0;LB<5B@0 A>>B25BAB25==>. 7
C@02=5=89 (1) 8 (2) =0E>48?@>B82;5=85 F5?8 2 ?5@2>< A;CG05
R1 = R + R / 2 = 3 / 2R ,
<>I=>ABL
2 2
P1 = U / R1 = 2 / 3U / R .
>A;5 ?5@53>@0=8O A?8@0;8 3 A>>B25BAB25==>
2
R2 = 2R 8 P2 = U / 2R .
">340 87<5=5=85 <>I=>AB8
2
"P = P2 - P1 = -U / 6 R ,
53
0 55 >B=>A8B5;L=>5 87<5=5=85 "P / P1 = -1 / 4, B.5. <>I=>ABL
=03@520B5;L=>9 10B0@58 C<5=LH8BAO =0 25%.
72."0: :0: <0:A8<0;L=K9 B> : G5@57 ?@>2>;>:C 2 n = I / I0 = 5 @07
<5=LH5 @0AG5B=>9 A8;K B>:0 I , =0H @578AB>@ 1C45B A>AB>OBL 87 n
?0@0;;5;L=KE ?@>2>4>2. E AC<<0@=>5 A>?@>B82;5=85 @02=> Rt 8
@0AAG8BK205BAO ?> D>@1 n
= .
Rt R1
BAN40 R1 = Rtn, 8 65< @0AAG8B0BL B@51C5?@>2>;>: 4;O 873>B>2;5=8O @578AB>@0
l1 = R1 / r = Rtn / r .
!C<<0@=0O 4;8=0 ?@>2>;>:8 2 n @07 1>;LH5
L = nl1 = n2 Rt / r = I / I0 2 Rt / r = 250<
( )
73.!>?@>B82;5=85 <8;;80G5=L <0;> ?> A@02=5=8N A
A>?@>B82;5=85< @578AB>@>2 R1, R2 8
2>;LB<5B@0 RV , ?>MB>:>2 2
B0:8E F5?OE 8< <>6=> ?@5=51@5GL. =0O
=0?@O65=85 =0 2>;LB<5B@5 (0 7=0G8B 8 =0 R1)
<>6=> @0AAG8B0BL ?>;=K9 B>: I1
U1 R1RV U1( R1 + RV )
I1 = ,R1v =�! I1 = .
R1V R1 + RV R1RV
0?@O65=85 =0 :;5<<0E U0 ?>;CG05< ?> 70:>=C <0
U0 = I1( R1V + R2 ).
A;8 2>;LB<5B@ ?>4A>548=8BL :> 2B>@>@C, B> A8;0 B>:0 2
F5?8
R1V + R2 U1 R1V + R2 R2RV
I2 = U0 / ( R2V + R1 ) = I1 = ,R2V = .
R2V + R1 R1V R2V + R1 R2 + RV
74. 0==0O AE5<0 =5 <>65B 1KBL @0AAG8B0=0 ?> AB0=40@B=K<
<5B>48:0< 4;O :>78F88 ?>A;54>20B5;L=>3> 8 ?0@0;;5;L=KE
A>548=5=89, ?>MB>A?>;L7C54>< M:2820;5=B=KE
?>B5=F80;>2.
59AB28B5;L=>, ;53:> 70<5B8BL GB> >AL O2;O5BAO >ALN
A8<<5B@88 40==>9
AE5A8<<5B@8G=KE B>G5: 8
54
, ! 8 D 1C4CB 8<5BL >48=0:>2K5 ?>B5=F80;K
�B = �A ;�C = �D
0==K9 D0:B ?>72>;O5B =0< 157 CI5@10 4;O AE5548=8BL
C:070==K5 B>G:8 8 ?@825AB8 F5?L : M:2820;5=B=>
?>440NI5R0 = R / 2 + R / 4 + R / 2 = 5 / 4R = 1,25R ,
345 R0 - 8A:><>5 A>?@>B82;5=85 F5?8, R - A>?@>B82;5=85 >4=>3>
725=0.
75. !>3;0A=> 70:>=C <0 ?@8 @07><:=CB>< :;NG5 8<55:0 2 F5?8, r -
A>?@>B82;5=85 0A;54C5B
U
r = - R1 E" 02<.
,
I
"0:8< >1@07>< r << R1 , r << R2 8 ?@8 70:070=8O
020<8,
HC=B8@>20BL 01E>48<> <0;K<8 A>?@>B82;5=8O<8,
A@02=8 25;8G8=5 A r .
76. '8A;> ;0G5: <>6=> =09B8 :0: N = U0 / U , 345 U -
N N
=><8=0;L=>5 =0?@O65=85 =0 ;0G:5, U0 - =0?@O65=85 A5B8.
N = 37 ( A >:@C3;5=85< 2 1>;LHCN AB>@>=C ). ">: 2 F5?8
U0
I == 0198 < 02 A.
,,
U
N
N
I
N
0@0;;5;L=>5 ?>4:;NG5=85 4>?>;=8B5;L=KE ;0G5: =5 C25;8G8B
A@>: @01>BK 38@;O=4K: B>: G5@57 :064CN ;0G:C >AB0=5BAO
?@56=8<, 0 5A;8 ?5@53>@8B >4=0 87 ;0G5:, B> A8;0 B>:0 G5@57
2B>@CN A@07C 2>7@0AB5B ?@81;878B5;L=> 2 420 @070. >MB>;0G:0 1KAB@> ?5@53>@8B.
55
77. 0 @8AC=:5 C:070=K ?@54?>;03057=0:8 70@O4>2 =0 >1:;04:0E :>=45=A0B>@>2.
> ?@028;C 8@E3>D0
IR1 = U1 - U3 ,
ż#
�#IR = U2 + U3 ,
�# 2
U0
345 I = .
R1 + R2
> 70:>=C A>E@0=5=8O 70@O40 q2 = q1 + q3 8;8 U2 = U1 + U3 .
5H5=85 A8AB5I 1 2 1 2
U3 = R2 - R1 , U2 = IR1 + IR2 , U1 = IR2 + IR1 .
()
3 3 3 3 3
-=5@38O :>=45=A0B>@>2, :>B>@0O 2K45;8BAO 2 F5?8 2 2845 B5?;>BK
?@8 >B:;NG5=88 8AB>G=8:0
2
C CU0
2 2 2 2 2
Q = U1 + U2 + U3 = R1 + R2 + R1R2 .
() ()
2
3 R1 + R2 2
()
78. 5@2>=0G0;L=> :;NG 70<:=CB , 8
B>: G5@57 @578AB>@ R2 B5G5B. 045=85
=0?@O65=8O =0 @578AB>@5 R1
�
U1 = R1
r + R1
@02=> C42>5==>=45=A0B>@5 C . -=5@38O
70@O65==KE :>=45=A0B>@>2
2 2
CU CU12 C� R12
C
W = 2 = C(U1 / 2 )2 = =
2 4 4(r + R1 )2
?@8 @07?@>B82;5=8OE
R1 8 R2. ?@>F5AA5 @07@O4:8 B>:8, B5:CI85 G5@57 @578AB>@K R1 8 R2
>48=0:>2K, ?>MB>B=>H5=85 B5?;>B Q1 8 Q2 , 2K45;82H8EAO =0
R1 8 R2 , @02=> >B=>H5=8N A>?@>B82;5=89
Q1 / Q2 = R1 / R2 .
! 4@C3>9 AB>@>=K ?> 70:>=C A>E@0=5=8O M=5@388:
Q1 + Q2 = W .
5H0O A>2<5AB=> MB8 C@02=5=8O ?>;CG05< 8A:>Q1 = R1W / ( R1 + R2 ), Q2 = R2W / ( R1 + R2 ),
8;8
56
2 2
R1 C� R12 R2 C� R12
Q1 = , Q2 = .
R1 + R2 4(r + R1 )2 R1 + R2 4( R1 + r )
79.>=45=A0B>@ 70@O605BAO 4> =0?@O65=8O 706830=8O ;070B5< >G5=L 1KAB@> @07@O605BAO G5@57 ;0> U1 (345
U0 - =0?@O65=85 8AB>G=8:0; U1 - =0?@O65=85 706830=8O ;0
A8;C B>:0 70@O4:8 I1 = U0 / R <>6=> AG8B0BL ?>AB>O==>9. ">340
2@5=45=A0B>@5
?@8 =0?@O65=88 U1 . ;O 2@5<5=8 70@O4:8 ?>;CG05<
� = RCU1 / U0 .
@5=51@530O 2@5<5=5< @07@O4:8 , ?>;CG8< G0AB>BC 2A?KH5:
U0 1
� = 1 / � = H" 10:F .
,
U1 RC
80. "0: :0: ?@>2>4=8:8 A>548=5=K ?>A;54>20B5;L=>, B> A8;0 B>:0
>48=0:>20 2 >1>8E ?@>2>4=8:0E. > 70:>=C <0 ?;>B=>ABL B>:0 j
A2O70=0 A =0?@O65==>ABLN M;5:B@8G5A:>3> ?>;O E A>>B=>H5=85<
1
j = E , (1)
�
87 :>B>@>3> A;54C5B, GB> =0?@O65==>AB8 ?>;59 2 @07=KE
?@>2>4=8:0E @07;8G=K. 0@O4, 2>7=8:0NI89 =0 3@0=8F5, A>7405B
B0:>5 4>?>;=8B5;L=>5 M;5:B@8G5A:>5 ?>;5, :>B>@>5 :>@07;8G85 2 ?@>2>48<>ABOE <0B5@80;>2 8 >15A?5G8205B ?>AB>O=AB2>
A8;K B>:0. CABL ?>25@E=>AB=0O ?;>B=>ABL 70@O40 �, B>340 2 ?5@2><
?@>2>4=8:5 =0?@O65==>ABL AC<<0@=>3> ?>;O
�
E1 = E0 + , (2)
2�0
0 2 4@C3><
�
E2 = E0 - , (3)
2�0
345 E0 - =0?@O65==>ABL ?>;O, A>740205<>3> 8AB>G=8:>:0, 0
�
E' = - =0?@O65==>ABL ?>;O 70@O4>2 =0 3@0=8F5. 7 (1)-(3) ;53:>
2�0
=09B8
� = j�0( �1 - �2 ). (4)
#<=>60O (4) =0 ?;>I04L ?>?5@5G=>3> A5G5=8O, ?>;CG8< 8A:>70@O4 =0 3@0=8F5 Q = I�0( �1 - �2 ) H" 15 �" 10-17 ;.
,
57
81. @0I0NI85AO 70@O4K, =0E>4OI85AO =0 ?>25@E=>AB8
?@>2>4=8:0, ?@54AB02;ONB A>1>9 M;5:B@8G5A:89 B>:, ?>4>1=K9 B>:C
A>;5=>840. 0: 8725AB=>, 8=4C:F8O <03=8B=>3> ?>;O 2 F5=B@5
A>;5=>840 2KG8A;O5BAO ?> D>@B = ��0nI ,
345 n - G8A;> 28B:>2 =0 548=8FC 4;8=K A>;5=>840. @>872545=85
nI 5ABL 70@O4, :>B>@K9 ?@>B5:05B 70 548=8FC 2@5<5=8 G5@57
548=8FC 4;8=K @07@570 =0 ?>25@E=>AB8 A>;5=>840 (?>25@E=>AB=0O
?;>B=>ABL B>:0). ;O 2@0I0NI53>AO F8;8=4@0 MB0 25;8G8=0 @02=0
�v = ��r . >MB>3> ?>;O 2=CB@8 F8;8=4@0
B = ��0�r� .
82. >340 AB5@65=L @0A:@CG8205BAO, M;5:B@>=K 2=CB@8 AB5@6=O
=0G8=0NB A<5I0BLAO : 53> 40;L=5=FC, A>74020O B5< A0871KB>G=K5 70@O4K 8 M;5:B@8G5A:>5 ?>;5 2=CB@8 AB5@6=O.
CAB0=>282H5 ?>;5 1C45B A>>1I0BL M;5:B@>=0<
F5=B@>AB@5<8B5;L=>5 CA:>@5=85. ;O M;5:B@>=0, =0E>4OI53>AO =0
@0AAB>O=88 x >B >A8 2@0I5=8O, A>3;0A=> 2B>@>=C LNB>=0
<>6=> 70?8A0BL:
2
m� x = eE , (1)
345 m - <0AA0 M;5:B@>=0.
B:C40 A;54C5B
2
m�
E = x . (2)
e
0?@O65==>ABL M;5:B@8G5A:>3> ?>;O 87<5=O5BAO ?> ;8=59=>70:>=C, ?>MB>AB8 ?>B5=F80;>2 <>6=>
8A?>;L7>20BL 55 A@54=55 7=0G5=85 (2 A5@548=5 AB5@6=O ?@8
x = l + L / 2 ).
">340
2
m�
"� = Eav L = (l + L / 2 )L . (3)
e
@8 =0;8G88 <03=8B=>3> ?>;O 2 C@02=5=88 (1) ?>O2;O5BAO =>2>5
A;0305<>5 - A8;0 >@5=F0
2
m� x = eE ą evB . (4)
=0: MB>9 A8;K >?@545;O5BAO =0?@02;5=85< 25:B>@0 8=4C:F88
<03=8B=>3> ?>;O. 'B>1K =0?@O65==>ABL M;5:B@8G5A:>3> ?>;O
r
2>7@>A;0, =5>1E>48<> GB>1K 25:B>@ B 1K; =0?@02;5= 225@E. ">340
87 (4) A;54C5B (A CG5B>< v = �x )
2
m�
E = ( + �B )x . (5)
e
58
03=8B=>5 ?>;5, ?@8 :>B>@>< @07=>ABL ?>B5=F80;>2 (0
A;54>20B5;L=> 8 =0?@O65==>ABL) 2>7@0AB05B 2 420 @070,
C4>2;5B2>@O5B CA;>28N
2
m�
�B =
e
8;8
B = m� / l . (6)
0<5B8<, GB> 87-70 <0;>9 <0AAK M;5:B@>=0 25;8G8=K 2>7=8:0NI59
@07=>AB8 ?>B5=F80;>2 :@09=5 <0;K (A<. 7040GC 11-2, @5AB,1993).
83."0: :0: @0<:0 ?@8 42865=88 8A?KBK205B A>?@>B82;5=85 2>74CE0,
B> ?> ?@>H5AB288 =5:>B>@>3> ?@><56CB:0 2@5<5=8 >=0 >AB0=>28BAO
2 ?>;>65=88 @02=>25A8O. CABL MB>
?>;>65=85 >?@545;O5BAO C3;>< ą. #A;>285
@02=>25A8O 8<55B 284
a
FAa cosą = mb ga siną + 2ma g siną ,
2
345 ma , mb <0AAK AB>@>= @0<:8 A
4;8=0<8 a 8 b , FA = IBl A8;0 459AB2CNI0O =0 =86=NN AB>@>=C @0<:8.
"0: :0: @0<:0 O2;O5BAO A25@E?@>2>4OI59,
B> M.4.A. 8=4C:F88, 2>7=8:0NI0O ?@8
42865=88 @0<:8, @02=0 M.4.A. A0<>8=4C:F88
"I "Ś
L =
"t "t
8;8
"Ś Bab
I = = 1 - siną)
.
(
L L
45AL "Ś 87<5=5=85 2=5H=53> <03=8B=>3> ?>B>:0 G5@57 @0<:C.
#G8BK20O, GB>
m m
ma = a,mb = b,
2 a + b 2 a + b
( ) ( )
CA;>285 @02=>25A8O ?@5>1@07C5BAO 2 284C
B2a2b2
1 cosą = siną.
( - siną)mga
L 2
02=>25A=>5 7=0G5=85 C3;0 <>65B 1KBL =0945=> :0: @5H5=85 MB>3>
C@02=5=8O. >A?>;L7C5ABLN <03=8B=>3> ?>;O (<0;>ABLN
C3;0 ą ), B>340 siną H" ą , cosą H" 1 8
59
2B2ab2
1 H"
( - ą) ą ,
mL
>B:C40
2B2ab2
ą H" .
mL
84. CABL :>=BC@ A<5AB8;AO =0 @0AAB>O=85 "x . !8;C B>:0 2 :>=BC@5
<>6=> =09B8 87 CA;>28O
"I "Ś
L = ,
"t "t
>B:C40
"Ś B0ąa2
I = = "x .
L L
45AL B>:0 "Ś = S"Bz = a2ąB0"x . !C<<0@=0O
A8;0, 459AB2CNI0O =0 @0<:C,
2 2
B0 ą a4
F = Ia( B( x + "x ) - B( x + "x + a )) = - "x .
L
#@02=5=85 42865=8O ( w - CA:>@5=85 )
2 2
B0 ą a4
mw = - "x
L
B0ąa2
5ABL C@02=5=85 30@<>=8G5A:8E :>;510=89 A G0AB>B>9 �0 = .
Lm
! CG5B>< =0G0;L=KE CA;>289, 70:>= 42865=8O 8<55B 284
�#
Lm B0ąa2 ś#
"x = v0 sinś# tź# .
B0ąa2 �# Lm #
85."0: :0: :>;LF> 70@O65=>, B> =0 ;N1>9 CG0AB>: :>;LF0 459AB2C5B
A8;0 A> AB>@>=K M;5:B@8G5A:>3> ?>;O. @8
2@0I5=88 :>;LF0 : MB>9 A8;5 4>1028BAO
A8;0 >@5=F0, =0?@02;5==0O @0480;L=>, B0:
65 :0: 8 M;5:B@>AB0B8G5A:0O A8;0. >
CA;>28N 7040G8 =5>1E>48<> >?@545;8BL =0
A:>;L:> C25;8G8BAO A8;0 =0BO65=8O :>;LF0,
?>MB>6=> =5 CG8BK20BL
M;5:B@>AB0B8G5A:>5 2708<>459AB285.
0AA<>B@8< =51>;LH>9 CG0AB>: :>;LF0, 2848
F5=B@0 ?>4 <0;K< C3;>< ą .
!8;0 >@5=F0, 459AB2CNI0O =0 =53> :
60
Q
FL = 2ą�RB (1)
2Ą
(745AL Q2ą / 2Ą - 70@O4 CG0AB:0, �R - 53> A:>@>ABL)
C@02=>25H8205BAO 4>?>;=8B5;L=K< =0BO65=85< "T , ?@>5:F8O
:>B>@>3> =0 @0480;L=>5 =0?@02;5=85 "Tą . 7 CA;>28O @02=>25A8O
:
Q
2"Tą = 2ą�RB . (2)
2Ą
0E>48< 4>?>;=8B5;L=>5 =0BO65=85 :>;LF0
Q
"T = �RB . (3)
2Ą
0<5B8<, GB> 2 7028A8<>AB8 >B =0?@02;5=8O ?>;O B, 25;8G8=0 "T
<>65B 1KBL :0: ?>;>68B5;L=>9 B0: 8 >B@8F0B5;L=>9.
86. 0AA<>B@8< @0<:C 2 <><5=B, :>340 25:B>@ 8=4C:F88 <03=8B=>3>
r
?>;O B1(t ) =>@<0;5= : @0<:5 (B.5.
>@85=B8@>20= 24>;L 25:B>@0 =>@<0;8 :
r
@0<:5 n1. 0 2@5?>25@=5BAO =0 C3>;
ą = 2ĄN"t .
@><5 B>3>, =0 MB>B 65 C3>; , => 2
r
4@C3>< =0?@02;5=88 ?>25@=5BAO 25:B>@ B . "0:8< >1@07><, C3>;
r
r
<564C =>2>9 =>@<0;LN n2 8 =>2K< 25:B>@>< 8=4C:F88 B2
n2 ,B2 = 2ą = 4ĄN"t .
(r r )
">340 7=0G5=85 -! 8=4C:F88 2 @0<:5:
"Ś
� = - = -( S cos(4ĄNt ))' = 4ĄNS sin(4Ąnt ),
"t
0 55 459AB2CNI55 7=0G5=85
�eff = 2 2ĄNS
(A@02=8B5 A �'eff = 2ĄNS ?@8 2@0I5=88 B>;L:> @0<:8 ). 0==>5
O2;5=85 (7028A8<>ABL -! >B >B=>A8B5;L=>3> 2@0I5=8O 8=4C:F88
?>;O 8 @0<:8) ?>;>65=> 2 >A=>2C @01>BK M;5:B@8G5A:8E
0A8=E@>==KE 42830B5;59, C :>B>@KE C3;>2K5 A:>@>AB8 @0<:8 8 ?>;O
=5 A>2?040NB.
87. !8;0 >@5=F0, 459AB2CNI0O =0 G0AB8FC 2 <03=8B=>< ?>;5, =5
<>65B 87<5=8BL 25;8G8=C 55 A:>@>AB8. -B> ?>72>;O5B A@07C
61
70:;NG8BL, GB> 42865=85 G0AB8FK 1C45B @02=><5@=K<, B.5. 8A:>?CBL
S = vt (1),
345 v - A:>@>ABL G0AB8FK, t - 2@5B25B0 =0
2B>@>9 2>?@>A 7040G8 2A?><=8<, GB> 2 >4=>@>4=>< <03=8B=>< ?>;5
B@05:B>@8O 42865=8O 70@O65==>9 G0AB8FK >:@C6=>ABL @048CA><
mv P
R = = (2)
QB QB
?@8G5< ?5@8>4 >1@0I5=8O
2ĄR 2Ą m
� = = ,
v B Q
345 m - <0AA0 G0AB8FK, Q - 55 70@O4, B - 25;8G8=0 8=4C:F88
<03=8B=>3> ?>;O. -B> 7=0G8B, GB> ?5@5:;NG5=85 ?>;O <>6=>
?@>872>48BL ?>A;5 F5;>3> G8A;0 >1>@>B>2 G0AB8FK, B>340 55
B@05:B>@8O (2 2845 2>AL<5@:8) 8 1C45B 70<:=CB0. B0:,
2Ą m
T = i , i = 12,3...
,
B Q
0AA<>B@8< :0G5AB25==> : :0:8< ?>A;54AB28O< <>65B ?@825AB8
CG5B 8=4CF8@>20==>3> 28E@52>3> M;5:B@8G5A:>3> ?>;O. !>3;0A=>
70:>=C M;5:B@><03=8B=>9 8=4C:F88 28E@52>5 M;5:B@8G5A:>5 ?>;5
r
E 2>7=8:05B 2> 2@53> ?>;O, ?@8G5<
r
25:B>@ =0?@O65==>AB8 E ?5@?5=48:C;O@5= 25:B>@C B , B> 5ABL ;568B
2 ?;>A:>AB8 B@05:B>@88 G0AB8FK. 5;8G8=0 =0?@O65==>AB8 E
?@>?>@F8>=0;L=0 A:>@>AB8 87<5=5=8O <03=8B=>3> ?>;O
dB
E = C ,
dt
345 C - :>MDD8F85=B, 7028AOI89 >B :>>@48=0B B>G:8 8 >B
35><5B@88 <03=8B=>3> ?>;O. A;8 70 2@572@0I05BAO
2 8AE>4=CN B>G:C (A>25@H82 F5;>5 G8A;> >1>@>B>2), B> MB>B
:>MDD8F85=B >AB05BAO =587<5==K<. "0: :0: <03=8B=>5 ?>;5
87<5=O5BAO 1KAB@> , B> <>6=> ?@5=51@5GL A<5I5=85< G0AB8FK 70
2@520=8O M;5:B@8G5A:>3> ?>;O. ">340 2B>@>9 70:>=
LNB>=0 4;O G0AB8FK 8<55B 284
dv dB
m = C
dt dt
>B:C40 A;54C5B
2BC
"v =
m
"0:8< >1@07><, AB@>3> 3>2>@O, A:>@>ABL G0AB8FK 87<5=O5BAO :0: ?>
25;8G8=5, B0: 8 ?> =0?@02;5=8N. 4=0:>, ?5@8>4 >1@0I5=8O >B
A:>@>AB8 =5 7028A8B, ?>MB>285 70<:=CB>AB8 B@05:B>@88
62
>AB0=5BAO =587<5==K<, 0 ?CBL, ?@>945==K9 G0AB8F59, 1C45B
7028A5BL >B 35><5B@88 ?>;O 8 <5AB>?>;>65=8O G0AB8FK.
88.CABL 2 ?5@28G=>9 >1<>B:5 ?5@2>3> B@0=AD>@<0B>@0 2>7=8:05B
M.4.A. �1 , 0 2> 2B>@8G=>9 >1<>B:5 4@C3>3> �2 , B>340 �1 + �2 = U0 .
2 2
#G8BK20O, GB> �1 " n1 , 0 �2 " n2 , 345 n - G8A;> 28B:>2 2 >1<>B:5, B>
�1 2
= k . ! CG5B>< MB>3>
�2
2
1 k
�1 = U0 2 ;�2 = U0 2 .
1 + k 1 + k
!>>B25BAB25==> =0?@O65=8O =0 2B>@8G=KE >1<>B:0E:
1 k k
U1 = �1 = U0 ;U2 = k�2 = U0 .
2 2
k 1 + k 1 + k
>4AB0=>2:0 G8A;5==KE 7=0G5=89 405B @57C;LB0B 103,5 .
89.7 M;5<5=B0@=KE ?>AB@>5=89 E>40 ;CG59 A;54C5B, GB> =8
@07<5@K �709G8:0, =8 53> ?>;>65=85 87<5=OBLAO =5 1C4CB.
59AB28B5;L=>, 709G8: :0: 1K D>@<8@C5BAO <=8G=8:><
S' . 07<5@K 709G8:0 >?@545;8<
A ?><>ILN 35><5B@8G5A:8E
?>AB@>5=89. CABL 53> 480<5B@
@025= x . ">340
L
x = 2a tgą = 2a �" = 2L ,
a
B.5. =5 7028AOB >B @0AAB>O=8O a
>B 8AB>G=8:0 S 4> 75@:0;0.
90. 2545< A8AB5>@48=0B :0:
?>:070=> =0 @8AC=:5. >>@48=0B0
D>=0@O 87<5=O5BAO ?> 70:>=C
y = H - vt . (1)
7 ?>4>18O B@5C3>;L=8:>2 <>6=>
70?8A0BL
x - l / 2 l / 2
= , (2)
h y - h
345 x - :>>@48=0B0 B5=8. 7 (1)-(2) =0E>48<
l hl / 2
x = + . (3)
2 H - h - vt
!:>@>ABL 42865=8O B5=8 <>6=> =09B8 :0: ?@>872>4=CN >B
:>>@48=0BK x ?> 2@5<5=8
63
hl / 2
vx = x' = v (4)
( H - h - vt )2
8;8 ?>4AB02;OO G8A;5==K5 7=0G5=8O 2 A8AB5<5 !, ?>;CG8<
45
vx = . (5)
(9 - 0,2t )2
!>>B=>H5=8O (4)-(5) A?@0254;82K
4> B5E ?>@, ?>:0 B5=L =5 4>AB83;0
AB5=K, B.5. ?@8 y < 2h , >B:C40
t < ( H - 2h ) / v0 = 30c. @8 t > 30c
, >G5284=>, A:>@>ABL B5=8 AB0=5B
@02=>9 A:>@>AB8 >?CA:0=8O D>=0@O.
:>=G0B5;L=>, B@51C58<55B 284, 87>1@065==K9 =0
@8AC=:5.
91. ;O >?@545;5=8O 345 >AB@>2, 0 345 53> 87>1@065=85 =0 ?>4>1=KE
D>B>3@0D8OE, 40209B5 2>AAB0=>28< >?B8G5A:CN AE5>1J5:B82 =0E>48BAO =0 =5:>B>@>9 2KA>B5 h =04 ?>25@E=>ABLN
2>4K. -B> >1AB>OB5;LAB2> ?@82>48B : =5:>B>@K< @07;8G8O< 2 C3;0E
8 @0AAB>O=8OE =0 AE5<5. A5 C3;>2K5 @07<5@K ?@54<5B>2
(A;54>20B5;L=>, 8 8E @07<5@K =0
D>B>3@0D88) =5A:>;L:> 1>;LH5, G5< 8E
87>1@065=89 2 2>45. 7 @8AC=:0 284=>,
GB> C3>; ACO =5A:>;L:> 1>;LH5 C3;0
A CO.
@0240, >B<5G5==K5 35><5B@8G5A:85
MDD5:BK 8AG570NB ?> <5@5 C40;5=8O >B
>AB@>20 8 =5 <>3CB ?><>GL ?@8
845=B8D8:0F88 :0@B8=K 8 55 87>1@065=8O, =0?@8<5@, ?@8 AJ5<:5
70:0B0 !>;=F0. MB>< A;CG05 =5>1E>48<> 0=0;878@>20BL O@:>ABL,
:>=B@0AB=>ABL 87>1@065=8O, GB> 4>2>;L=> BO65;> A45;0BL �=0 3;07
(G0ABL ;CG59 ?@8 >B@065=88 ?@5;><;O5BAO 8 ?>3;>I05BAO 2 2>45).
64
92. 1>7=0G8< 2KA>BC C@>2=O 2>4K 2 75@:0;5
G5@57 l . ;O ?>AB@>5=8O 87>1@065=8O
2>A?>;L7C5=0?@02;5= ?5@?5=48:C;O@=> ?>25@E=>AB8
75@:0;0 8 2>4K; 2B>@>9 SA - 2KE>48B 87
8AB>G=8:0 S ?>4 <0;K< C3;>< ą : =>@<0;8.
0945< 3>@87>=B0;L=>5 A<5I5=85 2B>@>3> ;CG0
AC = (h - l )ą , (1)
ą
AB = 2l� = 2l . (2)
n
@8 2K2>45 MB8E A>>B=>H5=89 CGB5=0 <0;>ABL
C3;>2 ą 8 � 8 8A?>;L7>20= 70:>= ?@5;><;5=8O 2 2845
ą
� H" .
n
"5?5@L <>6=> =09B8 ?>;>65=85 87>1@065=8O S
BC
n - 1
SS' = SC + = (h - l ) + (h - l ) + 2l / n = 2h - 2l . (3)
ą n
A;8 2KA>B0 C@>2=O 2>4K 87<5=O5BAO ?> 70:>=C l = vt , B> A:>@>ABL
42865=8O 87>1@065=8O v1 , :0: A;54C5B 87 (3), @02=0
n - 1
v1 = -2v . (4)
n
=0: �<8=CA C:07K205B, GB> ?@8 C25;8G5=88 l 87>1@065=85
?@81;8605BAO : 75@:0;C.
A;8 2KA>B0 C@>2=O 2>4K ?@52KH05B @0AAB>O=85 4>
8AB>G=8:0 (l > h ), B> @5H5=85 7040G8 87<5=8BAO.
MB>< A;CG05 E>4 ;CG0, 2KE>4OI53> ?>4 C3;>< ą :
=>@<0;8, 8<55B 8=>9 284. !<5I5=85 ;CG0 �#CB�#2
MB>< A;CG05
CB = hą + lą , (5)
0 @0AAB>O=85 4> 87>1@065=8O
CB
SS' = CS' - CS = - (l - h ). (6)
�
! CG5B>< 70:>=0 ?@5;><;5=8O � = ąn ?>;CG8<
h + l n + 1 n - 1
SS' = - l + h = h - l . (7)
n n n
!:>@>ABL 42865=8O 87>1@065=8O
n - 1
v1 =-v , (8)
n
GB> 2 420 @070 <5=LH5, G5< 2 ?5@2>< A;CG05 (l < h ).
65
93. #A;>285 7040G8 1C4CB 2K?>;=5=K, 5A;8 ;CG8 ?>?040NI85 =0
3@0=L 12 2K94CB G5@57 3@0=L 45 (0=0;>38G=>, ;CG8, ?@>E>4OI85
G5@57 3@0=L 16, 2K94CB G5@57 3@0=L 34). 20 ?@545;L=KE A;CG0O
B0:>3> E>40 ;CG59 ?>:070=K =0 @8AC=:0E. ?5@2>< A;CG05
siną
?>:070B5;L ?@5;><;5=8O n = <>65B 1KBL =0945= 87
sin �
35><5B@8G5A:8E A>>B=>H5=89. >A:>;L:C 2 A5G5=88 ?@87?@028;L=K9 H5AB8C3>;L=8:, B> 53> AB>@>=0 @02=0 @048CAC
>?8A0==>9 >:@C6=>AB8, B.5. 12 = R , 14 =2R . 53:> ?>:070BL,
GB> C3>; ?045=8O ą @025= Ą / 6 .
">340 87 70:>=0 ?@5;><;5=8O
siną 1
= n �! n = .
sin� 2 sin�
#3>; � <>6=> >?@545;8BL 87
B5>@52 4;O
B@5C3>;L=8:0 1!4 , GB>
1
?@82>4OB : @57C;LB0BC tg � = .
2 3
:>=G0B5;L=>
13
n = H" 18 .
,
2
B>@>9 ?@545;L=K9 A;CG09
@50;87C5BAO ?@8 n " . !;54>20B5;L=>, >B25B 7040G8 n > 18 .
,
94. ! ?><>ILN ?5@2>3> ?>AB@>5=8O >?@545;8< ?>;>65=85 D>:CA>2
;8=7K. 7 CA;>28O OA=>, GB> S - MB> <=8<>5 87>1@065=85. @>2545<
G5@57 B>G:C S =5:>B>@CN ?@OB;8G=CN >B 3;02=>9 >?B8G5A:>9
>A8. ">G:0 B - MB> B>G:0 ?5@5A5G5=8O ?@O<>9 A ?;>A:>ABLN ;8=7K.
!25B>2>9 ;CG ?>A;5 ?@5;><;5=8O 2 B>G:5 B ?>945B B0:8< >1@07><,
GB> 53> ?@>4>;65=85 4>;6=> ?@>9B8 G5@57 87>1@065=85 S . 0;55,
G5@57 >?B8G5A:89 F5=B@ ;8=7K O ?@>2>48< ?@OSB. 5@5A5G5=85 MB>9 ?@O<>9 8 ?@O<>9 S B 405B ?>;>65=85
?>1>G=>3> D>:CA0 F .
66
"5?5@L ?@>2545< 2B>@>5 ?>AB@>5=85, @5H0NI55 7040GC, 0
8<5==>: AB@>8< ?@OA;5 ?@5;><;5=8O
2 B>G:5 B1 ;CG 4>;65= ?@>9B8 G5@57 ?>1>G=K9 D>:CA F . 3>
?5@5A5G5=85 A 3;02=>9 >?B8G5A:>9 >ALN (B>G:0 S1 ) 8 5ABL
87>1@065=85 8AB>G=8:0 S1. B<5B8<, GB> 87 @025=AB20 (?> @8AC=:C)
�#OS�#=�#SS �#A;54C5B, GB> ?>;>65=85 B>G:8 S A>2?0405B A
?>;>65=85< ?5@54=53> D>:CA0 ;8=7K. MB>< <>6=>
=5?>A@54AB25==> C1548BLAO ?>4AB0=>2:>9 @0AAB>O=89 2 D>@B>=:>9 ;8=7K.
95. >A;5 H;8D>2:8 425 ?>;>28=K <>6=> @0AA<0B@820BL, :0: 425
;8=7K A> A428=CBK<8 >?B8G5A:8<8 >AO<8 8 A> A428=CBK<8
>?B8G5A:8<8 F5=B@0<8. !4283 :064>3> F5=B@0 >B=>A8B5;L=>
8AE>4=>9 >?B8G5A:>9 >A8
>1>7=0G8< h / 2 ( h -
B>;I8=0 A;>O, A=OB>3> ?@8
H;8D>2:5). 7 @8AC=:0
A;54C5B
h a
= ,
d d + f
>B:C40
ad
h = .
d + f
0<5B8<, GB> 25@E=55 87>1@065=85 S D>@<8@C5B =86=OO G0ABL
;8=7K 8 =0>1>@>B.
96. !=0G0;0 ?>AB@>5=85< =0E>48< ?>;>65=85 D>:0;L=KE
?;>A:>AB59 >158E ;8=7 (@8A.0 8 @8A.1). "0: :0: 459AB28O 0=0;>38G=K
>?8H5< ?>AB@>5=85 B>;L:> 4;O ?5@2>9 ;8=7K.
@>2>48< ?>1>G=CN >?B8G5A:CN >AL O1 O1 ?0@0;;5;L=CN
?040NI5<;5==K9 ;CG B1F1 4>;65= ?@>E>48BL
G5@57 ?>1>G=K9 D>:CA, B0: 65 :0: 8 ?>1>G=0O >AL O1 O1 .
!;54>20B5;L=>, 8E B>G:0 ?5@5A5G5=8O 8 5ABL ?>1>G=K9 D>:CA F1 .
>AAB0=>282 ?5@?5=48:C;O@ 87 B>G:8 F1 , =0E>48< 3;02=K9
?5@54=89 D>:CA F1. ;O @0AA5820NI59 ;8=7K ?>4>1=K<
67
?>AB@>5=85< :CA F2.
!>2<5AB8< ;8=7K, AG8B0O ?>;>65=85 8E D>:CA>2
8725AB=K<8. >2B>@O5< E>4 ;CG59 A1B1 8 B1F1 . CG B1F1 ?0405B =0
@0AA5820NICN ;8=7C. !B@>8< ?>1>G=CN >?B8G5A:CN >AL O2 O2. 5
?5@5A5G5=85 A 704=59 D>:0;L=>9 ?;>A:>ABLN F2F2 ?>72>;O5B
>?@545;8BL =0?@02;5=85 ;CG0 B1C, ?@5;><;5==>3> A8AB5<>9 42CE
;8=7 (@8A.2).
97. !@07C 65 >B<5B8<, GB> 87>1@065=85 :>;LF0 1C45B @07>@20==K<,
B.:. 2 A8AB5<5 ?@8ACBAB2CNB B>G:8, ?@8=04;560I85 D>:0;L=>9
?;>A:>AB8. 048< �0;351@08G5A:>5 @5H5=85 40==>9 7040G8.
254< A8AB52KE :>>@48=0B, B0:CN, GB> >AL X A>2?0405B
A >?B8G5A:>9 >ALN ;8=7K, 0 >AL Y ;568B 2 ?;>A:>AB8 ;8=7K.
1>7=0G8< :>>@48=0BK B>G:8 A - ( x0 ,y0 ) (=0 =0H5< @8AC=:5
x0 < 0 ). 0945< :>>@48=0BK B>G:8-87>1@065=8O A' - ( x,y ). -B0
B>G:0 O2;O5BAO B>G:>9 ?5@5A5G5=8O ?@OA>>B25BA288 A ?@028;0<8 ?>AB@>5=8O 87>1@065=89 2 B>=:>9 ;8=75).
0?8I5< C@02=5=8O MB8E ?@O 22545==>9 A8AB5<5 :>>@48=0B.
y x y0
BFA' : + = 1; AOA' : y = x . (1)
y0 F x0
5H0O MBC A8AB548< :>>@48=0BK B>G:8 ?5@5A5G5=8O -
x0 F
x = ;
x0 + F
(2)
y0 F
y = .
x0 + F
K@078< :>>@48=0BK B>G:8 G5@57 C3>; � , >?@545;ONI89 55
?>;>65=85 =0 7040==>9 2 CA;>288 7040G8 >:@C6=>AB8
F F
x0 = cos� - F; y0 = sin�, 8 ?>4AB028< 2 C@02=5=8O (2). >;CG8<
22
68
2
x = F(1 - );
cos�
(3)
sin�
y = 2F
cos�
-B8 C@02=5=8O 7040NB ;8=8N - 87>1@065=85 2 ?0@0<5B@8G5A:>9
Ą 3Ą
D>@<5. -B0 :@820O O2;O5BAO 38?5@1>;>9. @8 � ; 87
2 2
C@02=5=89 (3) <>6=> ?>;CG8BL y H" ą( x - 1), C@02=5=8O :>B>@K5
>?@545;ONB 0AA8BK
40==>9 38?5@1>;K.
@54>AB02;O5< G8B0B5;N
A0<>AB>OB5;L=> =09B8
A>>B25BAB285 <564C
B>G:0<8 >:@C6=>AB8 8 8E
87>1@065=89 =0
38?5@1>;5.
98. 7>1@065=85G:8 2 ?5@2>9 ;8=75 O2;O5BAO B>G:0 !, 0
87>1@065=85G:8 - B>G:0 , B> 5ABL 87>1@065=85 >B@57:0
5ABL >B@57>: ! . 7>1@065=85 MB>3> >B@57:0 2> 2B>@>9 ;8=75
@07>1L5BAO =0 420 15A:>=5G=KE ;CG0 (?>4C<09B5 ?>G5B@57>:
F 1 40AB <=8<>5 87>1@065=85 O F 1C 8 40;55 4> 15A:>=5G=>AB8;
>B@57>: CF 1 40AB 459AB28B5;L=>5 87>1@065=85 >B B>G:8 A (O A =
2 F) 4> 15A:>=5G=>AB8.
99. "0: :0: 48A: =0E>48BAO 2 2>74CE5, B> <>I=>ABL B5?;>>B40G8 W'
?@>?>@F8>=0;L=0 @07=>AB8 B525@E=>AB8 48A:0 8
69
2>74CE0 "T . CABL ?;>B=>ABL ?>B>:0 M=5@388 ?040NI53> A25B0
@02=0 w0 . ">340 <>I=>ABL A25B>2>9 M=5@388, ?>?040NI59 =0
?5@54=NN 3@0=L 48A:0 @02=0
W1 = Ąr2w0 (1)
CAB0=>282H5I=>ABL @02=0 <>I=>AB8
B5?;>>B40G8
W1' = "T , (2)
345 - :>MDD8F85=B B5?;>>B40G8.
7 (1)-(2) A;54C5B
2
Ąr w0
"T1 = . (3)

#G8BK20O, GB> D>:CA=>5 @0AAB>O=85
AD5@8G5A:>3> 75@:0;0 @02=> R / 2 , ;53:>
=09B8, GB> =0 704=NN 3@0=L 48A:0 ?>?0405B
M=5@38O, ?5@5=>A8<0O A25B>2K< ?>B>:><
G5@57 :>;LF> A 2=CB@5==8< @048CA>< r 8
2=5H=8< 2r .
2
W2 = 2r - Ąr2 = 3Ąr2w0 . (4)
( )
(Ą )w0
A>AB>O=88 B5@<>48=0<8G5A:>3> @02=>25A8O >=0 @02=0 <>I=>AB8
B5?;>>B40G8
W2' = "T2 . (5)
!;54>20B5;L=>,
2
3Ąr w0
"T2 = . (6)

:>=G0B5;L=>
"T1 1
= .
"T2 3
100. 0@O4 H0@8:0 ?5@5AB0=5B 87<5=OBLAO, 5A;8 M=5@38O
2K;5B52H53> A ?>25@E=>AB8 M;5:B@>=0 1C45B <5=LH5 M=5@388,
=5>1E>48<>9 4;O B>3>, GB>1K M;5:B@>= C40;8;AO >B H0@8:0 =0
15A:>=5G=>ABL, B> 5ABL ?@8 2K?>;=5=88 CA;>28O
h� = A2KE + 5� ,
345 � = c / - G0AB>B0 ?040NI53> A25B0, A2KE = hc / 0 - @01>B0
2KE>40 M;5:B@>=0, 0 - 4;8=0 2>;=K :@0A=>9 3@0=8FK D>B>MDD5:B0,
q
� = - ?>B5=F80; H0@8:0, q - 70@O4 H0@8:0,h - ?>AB>O==0O
4Ą�0 R
70
;0=:0, e - 70@O4 M;5:B@>=0, c - A:>@>ABL A25B0. 7 MB8E
A>>B=>H5=89 =0E>48<
q = 4Ą�0 Rhc(1 / - 1 / 0 ) .
101. '8A;> D>B>=>2, ?040NI8E =0
1>:>2CN ?>25@E=>ABL :>=CA0 >AB05BAO
?>AB>O==K<. >MB>B=>H5=85 A8;
402;5=89 1C45B @02=> >B=>H5=8N
82, ?5@540==KE ?>25@E=>AB8
:>=CA0 D>B>=0<8 ?>3;>I5==K<8 8
>B@065==K<8 ?>25@E=>ABLN. CABL
84=>3> D>B>=0 @025= p0 , B0:>9
65 8;CG8B :>=CA ?@8 ?>3;>I5=88 MB>3> D>B>=0. A;8
D>B>= >B@0605BAO, B> 3>@87>=B0;L=0O A>AB02;ONI0O 87<5=5=8O 53>
8"px = p0 (1 - cos2� ) .
AB0;L=K5 ?@>5:F88 87<5=5=8O 86=> =5 CG8BK20BL, B0:
:0: 87 A8<<5B@88 7040G8 A;54C5B, GB> 8E AC<<0 4;O 2A53>
?040NI53> ?>B>:0 1C45B @02=0 =C;N. "0:8< >1@07>< >B=>H5=85
A8; A25B>2>3> 402;5=8O 4;O 75@:0;L=>9 8 ?>3;>I0NI59
?>25@E=>AB59 1C45B @02=>
P75 @: "px
� == = 1 - cos2�
P?>3; p0
B<5B8<, GB> ?@8 � > 450 A8;0 402;5=8O =0 75@:0;L=CN
?>25@E=>ABL 1>;LH5, G5< =0 ?>3;>I0NICN, 0 ?@8� < 450 1>;LH5
A8;0 402;5=8O =0 ?>3;>I0NICN ?>25@E=>ABL.
71

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