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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

........................................................................................................................................................................... 62

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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

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General relativity theory :

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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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4

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1 ........................................

2 ........................................

3 ........................................

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(SI)

:

Fundamental quantities )SI( :

length meter )m(

mass kilogram )kg(

time second )s(

Femtosecond 1- : ) )-15((

32 .

1990

1979

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tera T1210deci d1-10 giga G910centi c2-10 mega M610milli m3-10 kilo K310micro u6-10

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5

: 275 . 1

635 . 2

750 . 3

1970 . 4 :

= 2750 1000 * 275 = 275 1

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= 075 3 - 10 * 750 = 750 3

= 197 . 3- 10 * 1970 = 1970 4

Derived quantities :

1 .

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: 1 36 // /2 1000/ / 72 / /

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= 374400

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Scientific calculator Graphical computer programs

. )1 - 1(:

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)1 - 4(:

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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

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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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1 .

: :

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1 = 980 1 = 98 ) ( )2(

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The resultant of two force meeting at apoint analytically :

2

1

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1 I I

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21 = I

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21 = I a

3( + 2)3(2c45 2 3 * 3 * 2 + 2) = I `

5 3=45 = 12

* 2 18 + 18 + 9 =

12 = c45 2 * 3c45 2 3 + 3

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a =

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2X 1X2 + 22X + 12X = I `

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21 = I a

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152 =

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+ 2 1 2 22 +

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c30

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c45 .

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2 ...................................................................................................................................................................... 1 2

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3 25 + 100 + 2 * 5 * 10120 = 5 =I

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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

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= X 2

= 8

2

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3 X *X 2 + 2X3+ 2X = 2X4

2X 3 2 + 2X 4 = 2X4 ` =

` ` = 90

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I = 15

3 2+1 = X 9

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- 4

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19 . 1 2 c120 21 c60 7 . 1 2.

2 . 22 15 . . .

:

12 .. 23 6 .. .

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I

12 c30c60

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20 :

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= X `

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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

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= 1X

34 = 22X + 225 2X

25 =22X 22= 9 X `

5 = 2X `3 = 2X `

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22

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I

25

1

1X2X

2

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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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)(.

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

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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 ............................... .

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2 1 I

|| = 12 I c45 c30 ||

: 1 = ............................................... 2 = ............................................... .

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I || = 18 : 1 = ............ 2 = ............ ||

. )1(

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.

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|| =................................................ . 2 ||

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=.................................................. .

=............................................... .

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

0 = 288 + 1X 36 - 21X

1X= 12 1X 1 = 24

)1( 00 =c60 - c30 X - 3 6

= 012 -X

32

- 3 6

1 # 3 X 3 + = 12

c60 = c30 X

# 2 ) 1 2( 3 = X

9 = X 3 = 3

- 90 2 + 60 12 - 30 3 4 + X

0=c60

1 # X 2 =

3 + 60 = 0 3 30 - 12 60 + 2 4

= 4

X 1 = 2 .

- 3

1 1

2 1 I 8 1 :

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3 2 = 6 1. I 1 = 3

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: 9

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c30

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..................................................

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..................................................

N .................................................. 4 + M = 3

20 c30 .................................................

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= .................................................. C +

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c50

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c 120 = 12 = -

c30

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2R

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:

- 32

: 3

80

30

20

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1 = 30 c20 2 = 30 c70 .

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.

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80 10 c30 .

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2

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0

3 1 = 40 2 = 40

= 2 c0 + 4 c120 + 6 c 240= - 3 c 240 6 + c120 4 + c0 = 2

3 3 = - - 3 3 2 =

c210 3 2 = I

2

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E

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N = 6 c 0+ 20 )c180 +(

3 = c90 + 2 2 13 =

C 2 3 = I`

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4

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33 -

1 1

50

:

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kinematics

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Dynamics

Kincmatics

Kinetics

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. .

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Dynamics

- 34

51

Scientific calculator Graphical calculator

Graphical programs

. )2 - 1(:

. )2 - 2(:

. )2 - 3(:

. )2 - 4(:

Rectilinear Motion Distance Vector Velocity Average Velocity Average Speed Relative Velocity Vertical Motion Universal Gravitation

Displacement Uniform Velocity Instantaneous Velocity Position Vector Uniform Acceleration Free fall Gravity

-

-

18 .

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: - - -

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. . .

Rectilinear Motion

IS

Displacement Vector Position Vector Velocity Vector Uniform motion . Average Velocity

Instantaneous Velocity

Relative Velocity

.

. . .

1 - 2:

1790 SI

International System Of Units

) (

) (.

Motion

:

.

.

Motion and its Types :

.

Rectilinear motion

- 52

mass Length displacement time

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:

. . - - . . .

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- - - -

.

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:

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Rectilimear motion

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(.

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- 36

Translational Motion

.

Distance 126

126 126

)( .

Displacement vector C

)( )C(

C

|| C ||

.

Position vector )(

N + M = S S

. N M

: Relation between position vector and displacement vector

2( )2 1( C)1 )(

C

.

0S

S : 0S ) + ( S =

) N 1 + M 1( - ) N 2 + M 2 = )

+)2 1(2)2 1(2 || = || N )12 ( + M )12 ( =

) ( || = || `

C

) (

S

S

0S

C

-

2 2

53

80 60 . . 1

C

.

= C + = 80 +60 = 140

C

:

c36 /

52 10000 = 100 , = 6080 = 12 // =

2)60( + 2)80( C = c36 .

/

52 = 100 12 //

: ) ( )

(.

H .

6 8 c 60 1 ...

: 3 2 .

S 2 = 4 N )1- 4( + M )2 + ( = )3( S N : M

N 15 + M 14 = N )1- 4 * 4( + M ) 2+ 4 * 3 ( = )4( S N - M 2 = )0( S

)0( S - )4( S = `

N 16 + M 12 = N )1 + 15( + M )2 - 14( =

, = 20 256 + 144 || = ||

C

60

80

-

:

54

.

:

:

. - . -

. -

H

.

:

53.

) (: )54(

1

36 + 64 - 2 * 6 * 8 120 =

- 12 .

= 6 + 8 = 14 .

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.

C 2

C : C = 3

0 = ) + ) -3 = 3

6

8

c120

37 -

2 2

: = 1 = 3. 3

) - (:

246810 1020304050

. 1

. 2

3 .

4 :

3

45

. 5

3 54 6 7 8 9 10

1020

30

40

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Speed

.

1 / = 518 /

185 / 1 / =

-

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55

:

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:

N )1 - 1 * 4( + M )2 + 1 * 3( = )1( S

N 3 + M 5 = )1( S

N 11 + M 11 = )3( S

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N 8 + M 6 =

64 + 36 = || ||

= 10 .

) (

.

.

3 54 6 7 8 9 10

1020304050

1 2 )(

( )

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45 9 .

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) ( :

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1 2

3 30 5 : 4

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15 / / 90 / / 4

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518 *

18018 / ... / 90 / .... / 54 / 18 / 5 * ... / 30 / ... / 20 / ... / 5 /

Velocity vector .

: : -1

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Uniform Volocity and variable Velocity :

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average speed

510 6

:

5106 = 85 / =

=

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56

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1854729010818051520253050

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39 -

2 2

Vector of the average velocity 1 2 C

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:

2 - 1 =

1S - 2S

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3 ) ( :

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.

4 / 4 = 40

10 =

= 10010 + 60 = 16 /

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.

30 18 / 4 20 15 /

.

C

. C

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1 =

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= 93 = 3 43 +

5 = 3

1S

2S

C

6050

10203040

8070

10090

65 8721 10943 )(

(

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1234

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30

20

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)(

.

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.

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2K 1K

1K - 2K

=

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.

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:

+ - 0

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)+(

)-( .

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= 10 - 20 = 30

313 = 10

3 = `

= a

C 313 / .

503 / = 20 + 30

3=

=

25 15 / 7 7 7 / . .

3 7 C )5 2( )9 10( 5 .

:

) 1S ( C

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1S - 2S = :

= )9 10( - )5 2(

= )4 8(

2 - 1

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5 = 2)2( + 2)1( || = ||

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26 2 6 //

3 8 C)7 2( )4 6(

.

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10

6 1082 4

C

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a -7 .

. = 32 + 7 = 25

83 =

77 +

25 = 15

= 12 3283

= ` K

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12/.

8)25+7( * 3 = 12/. =

:

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(

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.

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.

:

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N 45 + M

35 - =

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925

= || ||

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1 -

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52 //

: 11

9 K 9

0 !K9

9 K 9

=

:

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3 4 5 61 2 )(

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Instantaneous Velocity

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Relative velocity

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.

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.

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C-

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59

41 -

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.

.

C

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:

= C

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= - 30 - 80 50 = C

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30 / .

:

80 = C =- 50

C - = C a

= - 130 - 80 50 - = C

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130 / .

72 / . 9 28 / . :

.

.

100 7 / 300 / 250

.

C

80 /

50 /

C

80 / 50 /

-

:

60

C

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)

(.

300 - = C

250 = C `

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= - 50

50 / .

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= 2

40 / 10 120 /

)2 - 1(

:90 / = ...................... / 20 / = ...................... / 2 1

72 / = ...................... . 3

...................... = C

M = 22 M 15 = C 4

= ...................... 50 = C C = 65 5

/ 15 C 6 12 / C ...................... / .

: 75 / 20 : 7

30 25 20 15

C

250 /

-

2 2

61

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:

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.

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. -4

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:

C 10-

C .

C - = C -

= - 44 - 72 28 = C

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= - 100 - 72 28 - = C a

100/

= C = -11

M 120 - = C M 40 = C

C + C = a

M 80 - = N 120 - M = 40 `

- 42

20 / 180 : 3 212

2 - 1 12

: M 35 = C M C = 15 9

M 50 M 20 M 20 - M 50 -

)( : 10

S : 2 S = )2 2 + 3(

11 8 6 4

83 : 11 1494 *1110 .

12 70/ 84 / .

49

150 13 15

90 / .

30 15 / 10 14 10 / .

800 9 / 15 45 / .

C 120 C 16 88 / C

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3 18

7 4

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5 .

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21

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30:9

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C)4 3( )12 9( . 3 8 23 .

: C 38 C 24 C 60/

C

.

10

6050

20

70

30

80

40

90

3 5 74 6 81 2

1

65

2

7

3

8

4

9

30 50 7040 60 8010 20

-

2 2

63

1K = 42 2K = 35 = 7 12

= 225 5 * 9018

150 + = 15

13

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10 = 110

= 2K 2 = 3015

= 1K 14

203

=

10 - 303

=

8 = 45

= 2K 163

= 1K 15

= 16

= 1600 16 = 100 /

K 72 + K 88 = 120 K = 45 16

= 66

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.

)17( )18(

.

= - 90 60 = C 17

) - )- 90 60 = - C C = `

= - 30 - 90 C = 60 `

1 # 60 = C - 18

2 # 0 = C - 2

1 2 ` C = 60/ = 120/.

.

.

- 44

2 - 2

Acceleration

Uniform variable motion

Uniform acceleration

Uniform deceleration

. - . . - . - . - .

Rectilinear motion with Uniform accelerated

:

.

:)Rectilinem variable motion- non variable motion(

)( :

- )( =

= / 2 / 2 / 2

1 :

.

Velocity-Time curve ) - ( :

( ) )(

)1(.

:

)( )2(.

: .

20406080

6 82 4

/

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20406080

6 82 4

/

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- 64

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) N 6 + M 8( 13-8

=

N 65

+ M 85

=

(2 = 2 65 ( + 2)

85 ( =

34

-1

:

.

24

60 /

C C/

38

38

28

C = .

C )

(.38

38

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' C C

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=

=

68

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45 / 30/ 19

90/ 3

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:

.

.

.

.