دليل المعلم للصف الثانى الثانوى 2016
TRANSCRIPT
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2016/2015 10562 / 2015
6 - 019 - 706 - 977 - 978
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36 ............................................................................................................................................................................................... 2 - 1 . 2 - 2 . ........................................................................................................................... 46 2 - 3 . ................................................................................................................................................................................................................ 52 2 - 4 . ................................................................................................................................................................................................ 56
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3 - 1 . .......................................................................................................................................... 64 3 - 2 . ....................................................................................................................................................................................................... 68 3 - 3 . .......................................................................................................................... 73 3 - 4 . ................................................................................................................................................................ 77 3 - 5 . ............................................................................................................................................................................................................... 81
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Statics 1
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Dynamics 2
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: Mechanics statics Dynamics Kinematics Kinetics Rigid bodies Elasticity Plasticity
Classical mechanics
International system of units SI
Derived quontities
Fluid mechanics QBiomechanics femtosecon Mesaure units Length Mass Time Velocity Acceleration Force
- 2
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(.
) Elasticity( Plasticity
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Classical machanics
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Quantum mechanics
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omputatuinal FluidC(
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General relativity theory :
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1 - )( :
Isaac Newton johannes Kepler Galileo Galilei
Einstein 1905 - 1916 Max plank Heyznberg
Schrodinger Dirac .
Dr. Ahmed Zewail
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Measuring Units :
...
International system of units (SI) .
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1 ........................................
2 ........................................
3 ........................................
4 ........................................
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(SI)
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Fundamental quantities )SI( :
length meter )m(
mass kilogram )kg(
time second )s(
Femtosecond 1- : ) )-15((
32 .
1990
1979
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2-
tera T1210deci d1-10 giga G910centi c2-10 mega M610milli m3-10 kilo K310micro u6-10
nano n9-10 pico p12-10 femto f15-10
-
5
: 275 . 1
635 . 2
750 . 3
1970 . 4 :
= 2750 1000 * 275 = 275 1
= 635 2 - 10 * 635 = 635 2
= 075 3 - 10 * 750 = 750 3
= 197 . 3- 10 * 1970 = 1970 4
Derived quantities :
1 .
=
: / )/(.
2 :
/ )/2(. :
: 1/ / . 1 / /. 2 1
// /2 1 / / /2 4 3
:
518 / = 1 * 100060*60
= 1/ 1
= 1000 = 10
= 10 = 10
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=24 .
= 60 . = 60 .
-
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34 *1000*10=7500 . = 34
0245 = 0245*10-3 =245*10-5
1250 =1250*10-3 =125*10-5
12 = 12*310
8 1495*10
2844*10 5
/ 8 10*2998
500 .
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460 .
4560 .
) 366 (
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385000
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2509 / = 1 * 1000 * 100
60 * 60 = 1 / 2
518 /2 = 1000
60*60 * = // 3
2509 /2 = 1000 * 10060 * 60 *
= // 4
: 1 36 // /2 1000/ / 72 / /
Force 3 )( )(
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: 1 = 510 :
: 1 1/2
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1 : 98 /2
: 1 980 /2
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:314 1
675 * 710 2
:314 = 314 * 510=314000 1
675 * 710 = 675 * 710 510= 675 2
: 2
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536 * 1250
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18 72 /=72 *
1000 * 9 = 36 /2 250
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25 * 536510 * 1250 = 67*10-4 536 * 1250 =
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Scientific calculator Graphical computer programs
. )1 - 1(:
. )1 - 2(:
)1 - 3(:
)1 - 4(:
.
Statics Force Rigid body Gravitation force acceleration of gravity Newton Dyne Kilogram weight Gram weight Line of action of the force
Resolving force force Component equilibrium of a body triangle of forces lami's rule Equilibrium of rigid body smooth plane inclined smooth plane centre of gravity
= 0 = 0
11
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Forces1 - 1 -
Scientific calculator
Graphical programs
Force Resultant Rigid body Gravitation force
Acceleration of gravity Newton Dyne Kilogram weight Gram weight
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Force :
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The resultant of two force meeting at apoint analytically :
2
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6 c135. 4
c45 .
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2 ...................................................................................................................................................................... 1 2
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5 9 ............................................................. 4
2 3 c60 ....................................................... 5
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4 c120 16 . .
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c120 . 19
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18
45 0 6 9 8 7
3 25 + 100 + 2 * 5 * 10120 = 5 =I
1X c90
2X1X 2 + 22X + 12X = 2I a 2
5 45 = 3 2 3 * 3 * 2 + 18 + 9 = I
c26 /33 //54 = )c(X 12 =
` 169 = 225 + 64 + 2 * 15 * 8 3c120 = )c(X
12 = -
2X1X 2 + 22X + 12X = 2I 4c120 X * 8 * 2 + 2X + 64 = 2X3
X 4 - 2X - 32 =
` X = 4 X = -8
3 Xc 60
= X 2
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2
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I = 15
3 2+1 = X 9
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2 . 22 15 . . .
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c30
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2X1X 2 120 + 22X + 12X = 19
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:
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)1( )2( :
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2X1X - 34 = 19 `
15 = 2X1X
15 )3( 2X
= 1X
34 = 22X + 225 2X
25 =22X 22= 9 X `
5 = 2X `3 = 2X `
` 1X = 5 `1X = 3
22
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1
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`X= 15
3 : 2 24 5 3 = X 23
25
5 -
1 1
-
2 -
force Component
triangle of forces
centre of gravity
. .
2 - 1:
.
Resolution of a force into two components
I :)1( C
I 2 1
2 1 :
)2(: C =
) ( :
)1 + 2(2 =
11 =
2
) : ]c180 - ) 1 + 2 ( [ = ) 1 + 2(
c60 12 1 c45 .
:
12
c105 2 = c60
1 = c45
- 87846 12c105
* c45 1 = `
2
1
I
C
12
)1(
2
1
I
C
2
1 1 + 2
)2(
2I
1c60
c45
Forces resolution
- 20
2X1X
I
2X1X
.
:
. .
- - .
- - .
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- - - .
.
)24( )20(
)(.
Forces resolution
:
1X
2X
c30
3 6
1X
2X
c30
c30
12
1X
2X
c45
18
c601X2X
10
c60c60
.
- 6
-
- 107589 12c105
* c60 2 =
36 c45 c30 1
.
C 20 2
.c5
C
.
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:
:20
c170 = 2
c85 = 1
c85
: c85 c170
1=2 = 20
1=2 = 11473713- 115 .
: . c5
: 2
120.
.c48
Resolution of a force into two perpendicular components
I )(
1 2 1
I
C C :
120
C
c48
2
1
2
1
I
C
-
c90
20
Cc5
1 2
c5
c85 c85
-
21
1 1
2 =
1 =
: 90
2 =
1 = )90 - (
: 1 ) ( = I
2 ) ( = I
18 3 60
12 = 9 * 18=c60 1 =18
3 . 9 = 32
2 =18 60=18 *
2 6 3
.
Inclined Plane r2 0 > >
:
=
C
C
.
.c30 6 4 .
6
1
2 .
c601
2
18
C C
1
62
c30
c30
-
:
22
) ( )22( )21(
.
)21(
36
c75 = 2Xc30
= 1X
c45
36 * 45 - 26354 c75
= 1X
36 * 30 - 18635 c75
= 2X
)21(:
c5
.
)21(:
36
c138 = 2
X
c90 = 1
X
c48
120 * c48 - 133274 c138
=1X
120 * c90 - 179337 c138
= 2X
)22(:
2 c45 = 6 = 6
2 c45 = 6 = 6
:
.
:
.
c
.
7 -
1 1
-
.) 1 ) 1 = 6
12 = 3 * 6 = c30 6 =
) 2 )
2 = 6
3 3 = 32 * 6 = c306 =
:
.
36 4 .c60
.
)1 - 2(
:
6 1 ............................... .
2 4 2 ............................... .
)1(: 3
2 1 I
|| = 12 I c45 c30 ||
: 1 = ............................................... 2 = ............................................... .
)2(: 4 2 1 I
c90 c45
I || = 18 : 1 = ............ 2 = ............ ||
. )1(
. )2(
.
c45
2
1
I
)1(
c30
c45
2
1
I
)2(
-
23
1 1
)3(: 5 2 1
2 6 = || 2 || 1
|| =................................................ 1 : ||
|| =................................................ . 2 ||
)4(: 6 2 c30 . 12
=.................................................. .
=............................................... .
c30 600 . 7 .c45
120 . . 8
160 c30 . 9 18 . c60 10
c30 .
42 c60. 11 .
: 130 50 390 . 12
.
: 13 C c75
E C c45
E C . 5000
.
2
1
)3(
2
1 c30 )4(
2 12
5000c45 c30C
E
-
:
24
: )23(
X X X :
.
0> > 1 0> >1.
)23(
c60 1 = 36 X
3 18 =2X= 36 c60 = 18
)2-1(:
2 4 3 878 6212
2X = 18 2 18 = 1X 4
1 = 6 .2
* 2 6 = 2X = 1X 5
. 6 2 30 = 6 12 6 2 . 2 30 = 6 12
60045 - 439231 c75
= 1X 7
600c30 -310583 c75
= 2X
2 = 120 45 = 60 8 2 = 120 45 =60
3 = 160 30 = 80 9 = 160 30 = 80
= 18 60 = 9 0
3 = 18 60 = 9
= 12 60 = 21
2 = 150
513 = 390 = 390 *
= 390
= 360 1213 * 390 =
: ) ( 24
: 5000 75
2 = 45
= 1 30
3
1 - 258819 2 - 353553
1X2X
36
c60
c60
12
513
:
- 8
-
The resultion of coplaner forces meeting at point3 -
.
:
.... 3 2 1
)1(
1 C
2 C
3 .....
. E
:
3 + ... + 2 + 1 + = I
.
23
K
1
C
I
E
)2(
2
3
K
1
I
)1(
.
.
Resultant . .
Algebraic component Unit vector .
. Scientific calculator
.
25 -
The resultant of coplanar forces meeting at a point3 - 1
.
( )
.
:
. . .
:
- - .
:
- - .
:
- - - -
.
:
- .
:
)30( )25(
)(.
9 -
-
)GeoGebra(
4 1 = 400 3 2 1
2 = 300 3 = 500
4 = 200 c30 . .
1: 100 1
C 4 . 2
3 . 3
E 5 4 .
c30 2 E E 5 .
C :
|| = 568 . C ||
= 568 * 100 = 568
c103.
The resultant of coplanar forces meeting at apoint analytically
3 ..... 2 1
N M 1 2 3 .........
3 + ..... + 2 + 1 + = I :
:
I = )1 1 1 1(
+ )2 2 2 2 (
+ ................. + ) (
M I = )1 1 + 2 2 + ....... + (
N + )1 1 + 2 2 + ....... + (
N )S S
1 = S( + M )SS
1 = S( = I
C
E
c45
c30
c10289
2
K
2
3K
1
-
:
26
SS
1 = S:
.
SS
1 = S :
.
N M + I =
2 + 2 = = I :
4 2 1 c60 5 c60 3
3 c60 . .
3 c210 c120 c60 c0 4 2 5 3
c210 3 3 + c120 5 + c60 2 + c0 = 4
2 - = 92 - 52 - 1 + 4 =
32 * 3 3 -
12 * 5 -
12 * 2 + 4 =
c2103 3 + c120 5 + c60 2 + c0 = 4 12 * 3 3 -
32 * 5 +
32 * 2 + 0 =
3 2 = 3 32 - 3 52 + 3 =
16 = 4 = 12 + 4 2 + 2 = = I 3 + 2 2 - = I `
3 - = 232- =
=
a > 0 < 0
c120 = `
c120 4
3, 40 10 20 30 1 c90 c60
c150. .
3 3
c60c60c30
25
4
I
)(
1 = S
.
-
27
1 1
2X 1X
) C
E (
C + = +E = :
.
.
.)Geo Gebra( :
I N M + I = 2 + 2 = I
: =
X
.
) XX = c0(
)0=c0 X )
I = = 0
=0 I=
.
) ( )27(:
c33040 + c150 3 30 + c6020 + c0 = 1012 * 40 +
32
* 3 30 - 12 * 20 + 10 = = 10 + 10 - 45 + 20 = -5
c300 40 + c1503 30 + c6020 + c0 = 10
2X
1X
I I
1XC E
:
- 20
-
3 4 C 2 8 3 C E 2 4 2 C . . C E C C C
C : c120 c90 c60 c30 c0
.
c120 4 + c90 3 2 + c60 8 + c30 3 4 + c0 = 2 ` 12 * 4 - 0 * 3 2 +
12 * 8 +
32
* 3 4 + 2 =
= 2 + 6 + 4 - 2 = 10
c60 8 + c30 3 4 + c0 = 2
c120 4 + c90 3 2 + 32
* 4 + 3 2 + 32
* 8 + 12 * 3 4 + 0 =
3 10 = 3 2 + 3 2 + 3 4 + 3 2 =
N 3 10 + M 10 = I `
3(2 = 20 10( + 2)10( 2 + 2 = = I `
3 = 3 1010
=
=
c60 = )c(X ` a < 0 < 0
EC
)1 - 3(
:
N : 3 = 6 N 2 - M 2 = M 1 = 2 1 = ......................................... = .........................................
N M - 3 C 2 = I N 8 - M 2 = 4 N 2 - M 1 = 2 2 : C = ......................................... , = .........................................
N 4 - M 6 = I N M - 3 = 4 N - M C = 2 N 2 - M 1 = 3 3 : C = ......................................... , = .........................................
C
E
3 2
3 4
8
2
4c30 c30
c30c30
I
-
:
28
: 4
)2(
c30c45
3 42 3
3 2
)3(
c30
1
2
c60
3 3
3 4
)1(
c45
2
4
2 4
)5( 6 8
C
E
7
5
10
6
8
)6(
C
E4
8
6
3 4
3 2
)4(
C
3 6
4
4
c120c30c60
3 12 c60 3 6 9 5 c90 c150. .
c30 10 20 30 6 c60 . .
3 40 10 20 30 7 c60 c30 c60
. .
C 15 20 25 8 . . C
2 = 5. 2 13 4 C E 12 9 E C . . C C C 9
-
29
1 1
32
40 - 12 * 3 30 +
32
= 0 + 20 * 6 3 5 = 3 20 - 3 15 + 3 10 =
25 + 75 = 10 = I
3 = 3 55 =
c120 =c60 -c180 = )c( X
:
11 4 3 6
3 c60 c 60 4
c 30 .
.
] : I = 8 c 60 [
)3-1(:103 C = 3 = 2
43 =
I = 5 -1 C = -1 = 1 4 = 8 = 6 = 10 3
.34
1 -1
2 = 4 - 3 = 1 = 4 + 3 - 2 = 5
-1 5 26 = 25 + 1 = I
3 = 1 + 2 - 4 = -1
3 3 - 12 * 3 4+
32
= 2 * / = I = 1
= 0 3 4 = 2
3 = 2
5 = -7 = -3 37
-1 58 = 9 + 49 = I
3 6 = 10 = 10
3 I = 20 -1
c60 = 3 + 6 5 c30 3 9-
c6012 + 32 = 3 + 3 - 9 *
32 = 6+
32
* 12 - 12 * 3 9 + 3
2 = 6 *
3 32 = 3 6 - 3
92 + 3 3 =
3 = 274 +
94 =
I
c60c60
c30
3 96
12
3
- 2
1 1
-
. . 10
c40
c30
c35
80
120
150
c35
c30
c35 100
150
200
N M + 3 = -14 N 6 + M C = 2 N 3 + M 1 = 5 11 c135 2(. C . 10( = I
2 3 : 12
.
: 20 13 E C .
: : )( 14 45 =
2 c135 8
X .
c45c30
3 22 3
3
C
E
3 2
3 2
2
X2
X4
-
:
30
3 . -1
c15- =c 240 30 + c120 10 + 20 =M 6
3 5 - =c 240 30 +c 120 20 = N
3 10 = 75 + 225 = I
c210 =c 30 + c180 =
5- = c300 40 +c150 1 3 30 +c6020 + c010 =M 7
3 5 = c300 40 + c150 3 30 +c6020 +c 010 = N
c120 = 25 + 75 = = I
3 52 = c330 20 +c 210 15 + c90 25 =M 8
I 3 152 = c330 20 +c 210 15 +c 90 25 = N
c 60 3 5=
C = 10 9 35 )c C ( =
45 )c C ( =
2 2 c45= 14 5 + c0 3 =M
c45= 14 2 5 +c0 3 = N
c45 C 2 14= I
+ ) 3 + 6 + ( )14 - C + 5 ( = I
)1( + ) + 9 ( ) 9 - C ( =
+ 10 135 2 10 = )135 2 10( = I a
)2( 10 + = - 10 2 135
)C ` )2( )1 = -1 = 1
+ c150 3 2 + c45 2 3 + c. M = 2
c270 = 3
3 + c150 3 2 + c45 2 3 + c. = N
3 = c270
(2 = 2 + 4 2 ` )3 3 + = I `
c45 = =1 3 + 3 = I ` = 3
3 2 +c60 + c30 3 2 + c. 4 =M 3
12 -12 + 7 = c120 + c90
c90 3 2 + c60 + c30 3 2 + c. 4 = N
3
23 +
2+ 3 3 = c 120 +
E C Ia = 20
: 3 10 + 10 = I `
= 10 = 4
:
) (.
25 + 45 = N 2 - =M 4
)1( ( 25 +
45 + ) I = )2 - (
2 135 + 8 c135 2 8= I a
)2( + 8 8 =
)1( )2(
= 3 = 14 .
:
- 22
-
Equilibrium of aparticle under the action of copla-nar forces meeting at a point
4 -
Scientific calculator
.
. Triangle of forces rule
Lami`s rule . Polygon of forces .
.
. . . .
.
.
Equilibrium of a rigid body under the action of two forces
20 1-
. )1(.
-2
.
3- :
S
20 .
:1- .
2- .3- .
S = 20
= 20
= 20
= 20
20
20
)1 )
)2 )
31 -
4 - 1 Equilibrium of a particle under the action of coplanar forces meeting at a point
( )
.
:
.
.
. . .
:
- - .
:
- - -
- )Geo Gebra( .
:
.
:
-)42( )31(
.
- -
23 -
-
5 3 3 c60
5 3 :
c60 25 + 9 + 2 * 5 * 3 = I ` + 2 1 2 22 +
21 = I
49 = 7 = 15 + 9 + 25 = I `
` = 7 a )( 5 3 .
5 12 . 1
: 1 - - - . -
. 2
C . 3
)( 4 )E( .
. 5
... 2C 1C C 6 1 2 ... -
.
:
.
c60
X
3
5
CE 1 2 1C2C
-
:
32
:
) (.
.
: .
.
: :
:
.
:
.
:
.
: ) (
)32(:
.
5 12
144 + 25 = I 22X + 21X = I
I = 13 .
a )X( 5 12
` X = 13 .
)32(
.
.
32 .
:
) (
.
:
- 24
-
3 5 4 . 1 3 5 .
.
4 5 3 `
3 5 :
2X 1X2 + 22X + 21X = 2 I
5 = 2X 3 = 1X 4 = I :
30 = -18 16 = 9 + 25 + 2 * 3* 5
3-5 =
c126 5211 = c53749 - c180 = )c( `
7 8 13 . 2
Equilibrium of arigid body under the action of three coplanar forces meeting at apoint ) (
.
. :
.
: .
.
4 10 6 . 3 5 7 3 5 9
C
5 3
4
1
2
3
-
33
1 1
Triangle of forces 2 1 )1(:
C
) 2 + 1 )
C .
( 2 1 + 3 )
3 . 2 , 1 , 0 3 = 2 + 1 + :
3 : 2 1
N 2 - M 3 - = 3 N 3 + M = 2 N - M 2 = 1
3 2 , 1 , )2(: .
3
= 2C
= 1C
:
: .
: .
12 130 2 50 .
.
:
)12 ( .
. C
C .
)130(2 - )50(2 = 120 C =
C : 50
= 12120
= 130
= 13 = 5
)1(C
+
1
2
1
2
3
)2(C
1
2
3
X
12
C
120 130
50
C
12
X
-
:
34
:
2X1X
C E
2X 1X
E 1 X C
2X C = E
.
) ( )33(
)13(2 = )7(2 + )8(2 + 2 * 7 * 8
c60 =
)33(: ) (
.
- 3 5 9 3 + 5 > 9
- 3 5 7 : 3 + 5< 7
- 4 6 10 : 4 + 6 =10
.
.
) (
:
1- .
2- .
0 =N 0 =M 3-
25 -
1 1
-
16 50 3
40 .
lami''s theorem
3 )1( , 2 , 1
)2(
C
2
1
3
2
3X
)2(1X2
31
3X
2X
1X)1(
:3
32 =
21 =
1 C
)180 - 3( = C
)180 - 2( =
)180 - 1(
.
60 3 c120 c90. .
:
60 :
c120 = c150
= 60c90
3 : = 30 = 30 2 3
601 = 2 =
10 4 .c40 c30
1 2 .
c150c120
60
X
c40 c30
1X
10
2X
-
35
1 1
:
.
: )(
:
.
)1S(
. E
)2S(
( . 1S E C E )
15 25 4 25 C
. .
: 15 .
)S(
)(.
C )(
)(
.)(
C
C = 25 + 25 = 50 2)25( - 2)50( : C =
3 25 =
: S25
= 15253
= 50
3 .2
= S3 : =
C
2S
1S
E
.
15
C
S
50
2515
C
S
-
:
36
3- 35:
.
16 40
= 50
= X 30
= 20 . ` X = 12 .
4- 35
:
10c110
= 2Xc120
= 1Xc130
c130 - 8152 c110
* 10 = 1X
c120 - 9216 .c110
* 10 = 2X
:
10 -1
X 75
60 .
X .
: ]X = 75 R= 125 [
15 -2
.c60
.
[ 3 15 = X 30 = R [ :
) (:
:
3X 2X 1X :
C .
2X )( 1X
I )(. `
2X I 3X ` 2X 1X a
3X I `
.
3X )(. `
C
1X
2X I
3X
:
- 26
-
: .
100 30 5 20 .
.
30 100 5 50.
.
: 30
C 1 2
.
E a
C 12 = 50 = E `
` C E
c30 = )E c(X c60 = )E Cc(X `
:
3 1 = 15 2 = 15 30
c90 2 = c120
1 = c150
: .
: 6
30 20 .
C
12
50
50
50c60
c30
E
30
.
1
2
12
C
30 20
c60
-
37
1 1
: 1 2
C C
1 = 30 2 = 20 : :
1 2
:
])c60 + ( - c180[ = 20
)c90 + c60( = 30
)c90 + (
)c60 + ) = 40 =
30
c41 /24 / /35 = )c( X = 34
)c60 + c41 /24 / /5( = 40 *
- 392107
6 600 c30 . .
Polygon of forces : :)Geo Gebra(
c120 c0 : 400 100 300 100
c240 c180 .
:
.
C . :
.
.
)c90 + ( =
)c180 - ( =
300
100100
400 C
c120
c180c240
-
:
38
` .
:
-1
.
-2
.
-3
.
: ) (
)38( )37(
.
:
R90
= 100 ) 180- (
= S )90 + (
)5(
R = 5 * 100 4 = S5
3 R = 100
= S
S = 75 R = 125
600 90
= R 120
= X 150
)6(
= 300 12 * 600 = X
3 300 = 3
2 * 600 = R
)Geo Gebra(
.
= . -
= . -
27 -
1 1
-
. :
c240 100 + c180 300 + c120 100 + c0 = 400
12 = * 100 - 300 - 12 * 100 - 400 =
c240 100 + c180 300 + c120 100 + c0 = 400
3 = 50 - 0 + 3 50 + 0 =
: =
=
= =
:
.
N + M 3 = 2 N 2 + M 2 = - 7 N 3 - M 1 = 5 1 3 . 2 1
3 2 + 1 + = I a
. 0 = N )1 + 2 + 3 - ( + M )2 + 7 - 5( = I `
N M + 3 = - 6 N 2 - M C - = 2 N 3 - M 1 = 4 7
C .
5 4 2 : 16 20 12 2 C C E C C E C
E. .
5 4 2 16 20 12
)i + c180( c225 c90 c0 :
c90 20 + c0 = 16 `
C
20
16
E
54
2 12
1-
2- 5
i + c180
-
39
1 1
)i + c180( 5 4 + c225 2 12 +
i 5 * 4 - 12
* 2 12 - 0 + 16 =
1 = 5
* 5 4 - 12 - 16 =
c225 2 12 + c90 20 + c0 = 16
)i + c180( 5 4 +
i 5 4 - 12
* 2 12 - 20 + 0 =
2 = 5
* 5 4 - 12 - 20 =
= =
` .
10 : 5 6 8 C E
C = 6 = 8 ,E C
C = 6. .
)1 - 4(
:
................ 1
........................... ........................... 2
N : 3 - M C = 3 N 2 - M 2 = - 7 N M + 1 = 4 3 C = .................................................... = ....................................................
3 4 = ................................... 4
5 ....................................................................................................................................................
c225i + c180 i
C
20
16
E
2 12
54
i - = )i + c180(
1-
2- 5
i + c180
C5
E
10 6
6
6
8 X
-
:
40
:
)geogebra(
)(
.
)39(:
3X + 2X + 1X = I a )7(
= 0 + )- 3 - 2 + ( )6 - C - 4( = I `
` -C - 2 = 0 C = -2- 5 + =0 = 5
:
:( 0 = I )
) 0 = (
.
:
= 0 -1 . -2
:N 0 = M = 0 = 0
) (.
)40( )8(
103
34
5
1
6
6
2 C
E
X
5
10 6
10 = 0 X + 5 - 6
0 = 1
10 * 10 * 6 - 45 * 5 + X
2 = X 0 = 6 - 4 + X35 a = 10 = 0 5 + - 6
` = 15 3
10 =
:
- 28
-
................................... 6
7 .............................................................................................................................................................
8 .................................................................................................................................................
: 9 1 = ................................ 2 = .................................
. 10 :
)1(
60
c150
X
)2(
2 1
12
c120 c150
)3( 100
40
20 X
)5(40
)4(
150
1 90 20
30
2 1
)6(
21
c30
8
3 8
. 11
.
: )Geo gebra(
.
12
6
c120
c30
c30
400
300
200 1732
-
41
1 1 C .
C
10
60 1 = 150
1 = 150
`
10 150 60
1 = 2 =
3 103 =
)4-1(:
= 0 = 0 2
5 3 3
c120 8
3 3 = 2R 1 = 3 R 9
3 40 = X 3 )1(: = 20 0
3 2R = 6 . 6 = 1R :)2(
5 X = 50 )3(: = 50
)1R :)4 = 18 2R = 24 .
2 20 = R :)5(
)30 + (
= 8c150
= 3 8
)6(: )180 - (
c( X(= c60 = 16
M = 300 + 400 c120 + 200 M 0 + c240 = 1732 - c240 200 +c120 0 + 400 = N
N-
` .
:
:
.
:
.
10 .
:
1S2S
C
10
c60 c60
c30c30
29 -
1 1
-
28 60 12 14
.
200 60 80 13 100. .
200 14 c30 100 .
.
800 = 06 15 .
)( c30 16 36 .
.
3 17 c30. .
50 20 . 18 30 40 .
2 5 2 6 4 19 . .
5 4 3 7 20 c60. .
: 6 21 30
3 2
.
c30 .
32
6
S
-
:
42
60
c90 =
c150 = X
c120 2
3 30 = X :
= 30 .
1R
)c180-( = 1
R
)c180-( 3
200 c90
=
= 160 .45 * 200= 1R
= 120 . 35 * 200= 2R
R)c180- (
= 100 c150
4
200
)c30- ( =
200 = 200)c30- (
=
` )c 30+ ( = 90
3 = c60 = 100 3
X
)c180 - ( =
S
c90 5
800
)c90 + ( =
800 =
X = S
12 =
X800 =
X = 600 . S = 1000
36c150
= S
c90 =
S
c120 6
3 = 72 36 = S
2S 1S 7
3 c120
= 2S
c90 =
S
c120
3 2 = 6 3 = 1S
c3014
28X
60
2R
200
1R
100
c30
200
100R
3
4
5
S
X
800
c30c30
S
20
c90 = 2
R
)c90+ (= 1
R
)c90+ ( 8
:
1R =12 2R =16
2 5 + c135 2 4 + c90 6 + c0 X 90 = c270 + c225
= 9 . 0 = 5 - 4 - X
c225 2 5 + c135 2 4 + c90 0 + 6 X
0 = c270 +
= 5 6 + 4 - 5 - = 0
5 c0 +4 c60+ c120 + 3 c180 + 200 = c300 7 + c240
)1( : + = 15
+ c180 3 + c120 +c 60 4 + c0 5
0 =c 300 7 + c 240
)2( : X - = 3
)1( )2( = 9 = 6
2R1R
2525C
2540
30
:
- 30
-
) (
:
4 [r 0] = ..... . 1
5 8 = ........................................... 2 = .................................. .
)( 3 ..................................................
6 8 ..................... . 4
N M + 3 = 9 N 4 + M 2 = - 3 N 6 - M C = 1 5 C = .............................. = ..............................
. : 6
= ........................
= ........................
X
6
)1(
= ........................
= ........................
6
8
12
)3(
X
= ........................
= ........................
c135
)2(
X
2 4
C . : 7
4
2
C
8 6
20
C
S
30
20 20
E2
1
39 C
65
65
50
120
1 = ........................... . = ........................... . 1 = ........................... .
1 = ........................... . S = ............................... . 2 = ........................... .
45 -
1 1
:
2 :
S)c120 + (
= 6)c90 - (
= 3 2c150
`
c30 = ` 3
2` =
3 2 = 12 * 3 = 4
4 * 2 X + 4)22 - 1( =
`
42X + 42 a =
` X = 4 8 2 = X + 8 2 - 4
13 3 3 2
= -6 = 2 10 5 4
3 )X :)1 = 3 = 3 6 16 = X 20 = R :)3( = 4 = X :)2(
125 = 2R
165 = 1R :)1( 7
)2(: = 50
E C9 .
2030 =
2R25 =
1R25
503 = 2R = 1R
3965 =
2R25 =
1R60 )3(:
1R = 36 2R = 15
64 + 225 + 2 * 8 * 15 90 = 17 . = I 8
c30 = 602 = 3 I = 2 * 6030 = 60
12 6 = -3 = -
6 3 + 6 90 =
.c120 = )c(X
c120 36 + 9 + 2 * 6 * 3 = I
3 3 = I 3
2 = - 6
+ 6 3 3 = c90
c150 =c 30 - c180 = )c(X
c150 3 * 6 3 * 2 + 36 + 27 = I
I = 3
150 3 12 * X * 2 + 432 + 21X = 144
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- 44
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