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á````jƒ````fÉ````ã````dG á`````∏`````Mô`````ŸG

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∫ qhC’G Aõ÷G

ÖdÉ£dG ÜÉàc

( ) . . .

. .

áqjƒfÉãdG á MôŸG

qhC’G Aõ÷G

1437 - 14362016 - 2015

1439 - 14382018 - 2017

2013 . . . House of Education

: ©.

2014/2013 2016/2015

.

. . . .

2018/2017

á```````eó`≤e

.

.

.

.

. .

.

äÉjƒàëŸG

:

:

:

:

9

∫qhC’G Aõ÷G äÉjƒàfi

: 12

: 13

:1-1 14

:2-1 25

:3-1 29

38

39

: 42

:1-2 43

:2-2 54

:3-2 61

66

67

10

: 70

:1-3 71

:2-3 74

:3-3 78

:4-3 84

( ) :5-3 90

:6-3 95

99

101

: 104

:1-4 105

111

112

11

IóMƒdG ∫ƒ°üa

IóMƒdG ±GógCG

. .

.

. .

.

. .

. IóMƒdG ºdÉ©e

: : :

:

:

:

:

:

:

.

.

.

∂°ùØæH ∞°ûàcG

. .

. .

1 .. 2 .

.

ácô◊G

Motion¤hC’G IóMƒdG

12

) (

. . ( ) . . Parabola

. Projectile .

.

π°üØdG ¢ShQO

∫qhC’G π°üØdGäÉahò≤ŸG ácôM

Projectile Motion

13

áeÉ©dG ±GógC’G

. ( ) .

. .

. .

.

. .

.

.

1 .á¡éqàŸG äÉ« qªµdGh ájOó©dG äÉ« qªµdGScalar and Vector Quantities

. (50)kg

. kg 50 .

Arithmetic Algebra . (60)kg (40)kg

. (100)kg

.

v

y

xθ

(1 )

v

1-1 ¢SQódGá¡éqàŸG äÉ« qªµdGh ájOó©dG äÉ« qªµdG

Vector and Scalar Quantities

14

äÉHÉLEG ™e ¿ÉàdCÉ°ùe1 .

. (60)km/h . v =(60, 90∞) :

2 .

(2.5)kg. F = ((10)N, 45∞)a = (4, 45∞) :

( ) . (1 )

v AB

. AB v |AB|

. v v = (v, θ) : v

. θ

(1) ∫Éãe

. (5)N ( ) ( ) :

F = (5)N |F| = (5)N : F ( ) θ = 180∞

: . F = ((5)N, 180∞)

(1)N (1)cm ( )

:

15

A

B

45∞(2)cm

+X

(2 ) 45∞ (20)km

. (10)km (1)cm

v

(3 ) (60)km/h

. (20)km/h (1)cm

ádCÉ°ùe 60∞ (40)km . (10)km (1)cm

.

v1 v

2

(4 )

v1 = v

2

Vector Quantities á¡éqàŸG äÉ« qªµdG 1.1

. :

Displacement ( )

.

(20)km B A ) 45∞

45∞ (2)cm ((10)km (1)cm . (2)

Velocity Vector ( )

.

. (1)cm (3)

(3)cm (20)km/h. (60)km/h

2 .Properties of Vectors äÉ¡éqàŸG ¢üFÉ°üN

Equality …hÉ°ùàdG 1.2 . v2 v1

. (4 )

Transport π≤ædG 2.2 .

. : 1 . Free Vectors

. .

2 . Restricted Vectors .

16

á«°VÉjQ á©LGôe

sin = Ry

R

+y

+x

RR

y

+y

+x

R

Rx

Ry

cos = Rx

R

+y

+x

RR

y

Rx

tan = Ry

Rx

(5 ) .

(100)km/h

(120)km/h

(20)km/h

(100)km/h

(20)km/h

(80)km/h

( ) ( )

(6 ) (80)km/h (60)km/h

. (100)km/h

(1) cm = (20)km/h

(100)km/h

(60)km/h

(80)km/h

:

Addition of Vectors äÉ¡éqàŸG ™ªL 3.2

. D . v

( )

.

(100)km/h (20)km/h

. ( - 5 ) (120)km/h

. (100)km/h (U)

. ( - 5 ) v = 100 - 20 = (80)km/h

. (60)km/h (80)km/h

. ( )

. (20)km/h (1)cm (6) . (5)cm

. . (100)km/h

v2r = v2

p + v2

a :

17

äÉHÉLEG ™e πFÉ°ùe

1 .(10)N F2 F1 (15)N 60∞

.

. :

(Fr = (21.79)N, = 36.58∞)

2 . (45)m (85)m

. 30∞

. ((126)m, 10∞) :

3 . F2 F1

.O

F2 = (40)N F

1 = (30)N

(50)N :

(7 ).

B

A

R

O

180-

: v2

r = 802 + 602 = 6400 + 3600 = 10000

vr = (100)km/h

. :

tan θ = v

p

va

= 8060 = 43

θ = 53.13∞

( )

:

:( ) - O v2 v1 (7)

: 1 . O

. 2 . )

. ( 3 ..

: - :

R = A2 + B2 + 2AB cos θ

:

sin B = sin (π - )

R

: sin (π - ) = sin

sin = B sin R

18

á«FGôKEG Iô≤a

ÈàîŸG ‘ AÉjõ«ØdG

.

.

(2) ∫Éãe

(20)N F1 . O F2 F1

. 120∞ (20)N F2

1 .FeqF

2

F1

(2)cm (2)cm

(2)cmO

. 2 .. 3 ..

F

2 F1 . (10)N (1)cm . 120∞ (2)cm . ( )

. (2)cm F

eq = (2)cm ^ (10)N = (20)N :

. (20)N = 60∞ O :

: v1 v2 . v1 v2 v2 v3

. . (8)

++=v

3

v2

v1

v2

v3

v1

x

y

vR

(8 )

.

.

19

By

x

A

O

C

M(2

)cm

(9 )

(2)cm(5)cm

(3.4)c

m

30

30

60

D1

D2

DR

α

(10 )

(3) ∫Éãe

O M

. (9) O, A, B, C, M . (1500)m (1)cm

:

. ( ). ( )

:

M O ( ).

OM :

OM = 2 ^ 1500 = 3000(m). 90∞ ( )

(4) ∫Éãe

30∞ (10)km . (10 ) (4)km

( ).

( ).

1 .. : 30∞ D

1 = (10)km :

D2 = (4)km

. : 2 .:

: ( ) D

1 D2 D1 (2)km (1)cm

. (2)cm D2 (5)cm

20

60

y

= -2 RD''

R

x

= 2 DD'

(11 )

(™HÉJ) (4) ∫Éãe

θ = 150∞ 2 . (3.4)cm (6.8)km

. 43∞

: D2 D1 . R D2 D1

(6.8)km R = (3.4)cm D .

. x 43∫ : ( )

R2 = D21 + D2

2 + 2D

1D

2cos 150

R2 = 52 + 22 + 2 ^ 5 ^ 2 cos 150 = 11.67R = (3.4)cm

R = (6.8)km :

sin αD

2 = sin 150

Rsin α

2 = sin 1503.4

sin α = 0.29α = 16.85∞

α = 60 - 16.85 = 43.14∞ D2 .

3 . :

.

á«°SÉ«b á« qªµH äÉ¡éqàŸG ÜöV 4.2. (11 ) 60∞ D

R D' = 2 D .

. R D'' = -2R

. 21

3 . äÉ¡éqàŸG ÜöV

.

: 1 .. ( ) 2 ..

:

»°SÉ«≤dG Üö†dG 1.3 α B A

. (12) : B A

A . B = A ^ B cos α

B A . α .

.

(5) ∫Éãe

. x F

60∞ (50)N . (10)m

x

F

(13 )

1 .. :. 60∞ (50)N F :

. x = 10 : . :

2 .:

: W = F . x = F x (cos 60)

W = 50 ^ 10 ^ 0.5 = (250)J : 3 . :

.

óbÉædG ÒµØàdG

(85)m.

B

A

(12 )

22

A

BR

(14 )

ádCÉ°ùe

v

60∞ (10)m/s .

( ). v' = -1.5v v'

( ). v'' = -v

( ). ( ) veq = v' + v''

»gÉ qŒ’G Üö†dG 2.3 α v2 v1

. (14) B A

:R = A ^ B

:

R = A ^ B = A (B sin α)

. (14) v

(6) ∫Éãe

. (15) 120∞ (4)N F2 (5)N F1 . F1 ^ F2

F

F2

F1

(15 )

xx'120∞

1 .. : :

xíx (5)N F1 xíx 120∞ (4)N F2

. : 2 .:

: F = F1 ^ F1

: F F = F1 ^ F2 sin120 = 5 ^ 4 sin 120 = (17.32)N

. ( ) F2 F1 F

3 . :.

23

(16 )

F'

F

30∞

(10)N

(15)N

1-1 ¢SQódG á©LGôe

. -

(80)km/h -

. (80)km/h .

. 45∞ (600)km/h -

. F

1 = (3)N F2 F1 -

. F2 = (5)N

( ) ( )

25∞ (5)m/s v1 -

. . (2)m/s (1)cm v1 ( )

( ). v' = -3v

1

. v' ( ) . F2 F1 -

. Fí F (16) -

. 30∞ F' = (15)N F = (10)N

: Fíí = F + Fí ( )

F . Fí ( )F ^ Fí ( )

F2 ^ F1 -

.

24

¢SQódG

y

Ax

x

AA

y

(17 )A

.

.

.

.

1 .Vector Analysis äÉ¡éqàŸG π«∏–

.

y x A A θ (17)

. x A Ax x A

. (17) Ay y A A Ay Ax

A = Ax + A

y :

:

A = A2x + A2

y

cos θ = A

xA & A

x = A cos θ

sin θ = Ay

A & A

y = A sin θ

2-1 ¢SQódG


. .

äÉ¡éqàŸG π«∏–

Vectors Analysis

25

35∞

y

x

|v|=(120)km/h

vx

vy

(18 )

äÉHÉLEG ™e ¿ÉàdCÉ°ùe

1 . F = (50)N . x 120∞

x (25)N : y (43.3)N

. 2 .

. ay =(-4)m/s2 a

x = (3)m/s2

.

. -53∞ (5)m/s2 :

y

xB

x

By

Ay

Ax

Rx

RRy

A

B

O

(19 ). B A R

(1) ∫Éãe

v . (18 ) 35∞ (120)km/h

1 .. :θ = 35∞ v = (120)km/h :

= vy vx : 2 .:

v y x . vy vx

:

sin θ = v

yv cos θ =

vx

v:

vx= v cos θ = 120 cos 35 = (98.29)km/h

vy = v sin θ = 120 sin 35 = (68.82)km/h

3 . :

. v2 = v2

x + v2

y

v2 = (98.29)2 + (68.82)2 = 14397.11 v = (119.98)km/h

.

äÉ¡éqàŸG π«∏ëàH á∏ q°üëŸG OÉéjEG 1.1

. R B A

. R = B + A

. B A Bx Ax (19) y By Ay Rx x

. Ry R

y = Ay + By Rx = Ax + Bx

26

á«FGôKEG Iô≤a

á°VÉjôdÉH AÉjõ«ØdG •ÉÑJQG

.

1 .

(V ) (V )

. 2 .

. (V ) (V ) (V ) (V )

.

v

1 2 3

v vR

v vR

vv

(v

R)

. (v ) 3 .

.

x x y

. y :

R = R2x + R2

y

: x

tan θ = R

y

Rx

(2) ∫Éãe

. F2 F1 ( )

. . ( )

+x-x

+y

50∞40∞

(600)N (400)N

F2 F

1

1 .. :

1 = 50∞ F

1 = (400)N :

2 = 40∞ F

2 = (600)N

( ) : ( )

2 .: :

Fx = F cos θ

Fy = F sin θ

. F2 F1 27

áHÉLEG ™e ádCÉ°ùe

F

2 = (2)N F1 = (6)N

60∞ F3 = (3)N .

.

225.8∞ (4.8)N :

+x

+y

θθ

Fx

Fy

W

(20 )

+x

+y

30∞(128)N(64)N

(128)N

(21 )


FFx

Fy

F1

400 cos 50 = (257.11)N400 sin 50 = (306.41)N

F2

-600 cos 40 = (-459.62)N600 sin 40 = (385.67)N

FR

(-202.51)N(692)N

FR = F2

x + F2

y = 202.512 + 6922 = (721.02)N

tan θ = F

yF

x = 692

202.51 = 3.42

. x 106∞ x θ = 73.7∞

3 . :

.


45∞ -

: -

( ) ( )

( ) 30∞ (50)kg -

g = (10)m/s2 . y x

. (20) -

. (21)

28

(22 ).

E

DCB

A

. y x

. ( ) (22 )

.

.

3-1 ¢SQódGáØjò≤dG ácôM

Projectile Motion


. .

. . .

.

29

(24 )

.

(25 )

.

1 .áØjò≤dG ácôM QÉ°ùeThe Projectile Motion Trajectory

.

.

. .

. (23)

(23 ) ( ) ( ) :

( )( )

2 .áØjò≤dG ácôM ÉàÑ qcôeThe Components of the Projectile Motion

. . (24 )

.

. (25 ) . ( )

30

(26 )

.

á«FGôKEG Iô≤a


.

.

( ) .

. ( )

(26) . .

. .

. a = g

∆y = 12 gt2

v = gtv2

f = 2g∆y

: ( )

. ∆x = v∆t . .

.

: .

(1) ∫Éãe

(20)m . (25)m v . v

.

1 .y vx=vi

(20)m

(25)m

. :∆y = (20)m :∆x = (25)m

= v :

31

x

y

g

O

hmax

v0

vx

vx v

x

vx

θ∞

(27 )

+x

+y

v0

v0x

v0y

(28 )


2 .:

: ∆x = v

x ∆t = vt

vy = (0)m/s

. a = g = (10)m/s2 :

∆y = 12 gt2 20 = 5t2 t = (2)s: ∆x = vt t

v = 252 = (12.5)m/s

3 . :

.

3 .ájhGõH â≤∏WoCG áØjòb ácôMMotion of a Projectile Launched with an Angle θ O m

. (27) v0

: (28) v

0x = v

0 cos θ

v0y

= v0 sin θ

m . W ( )

: ∑F = ma

mg = ma a = g ay ax a

: a

y = -g a

x = 0

: ∆x = v

0x t = v

0 cos θ t

vx = v

0x = v

0 cos θ

: ∆y = - 12 gt2 + v

0yt = - 12 gt2 + v

0 sin θ t

vy = -gt + v

0y = -gt + v

0 sin θ

32

y vx = v

xi

(29 )

á«FGôKEG Iô≤a

á°VÉjôdÉH AÉjõ«ØdG •ÉÑJQG

. . . . . . . .

.

(27 )

.

Trajectory Equation QÉ°ùŸG ádOÉ©e 1.3 Trajectory Equation

: t ∆x = v

0x t = v

0 cos θt

t = ∆xv

0 cos θ

O (0,0) t ∆y = - 12 gt2 + v

0 sin θt

y = tan θ x - g2v

02 cos2 θ x2

:

Parabola .

. . 90∞

. (29 )

Maximum Height ÉØJQG ≈°übCG 2.3 v

y 0 = -gt + v

0 sin θ :

t = v

0 sin θg

hmax

= v

02 sin2 θ

2g

: y

Range ióŸG 3.3 Range

. .

. të = 2v0 sin θg

:

:

R = v

02 sin2 θ

g33


53∞ (25)m/s .

: . g = (10)m/s2

( ) ( )

( ). ( )

(19.93)m ( ) :(60)m ( )

y = 14.96 x = 15.04 ( )v = (18.042)m/s θ = 33.5∞ ( )

60∞

15∞

30∞45∞

75∞

(31 )

. .

ádCÉ°ùe

θ

.

4 .´ÉØJQG ≈°übCGh »≤aC’G ióŸGh ¥ÓWE’G ájhGR ÚH ábÓ©dGRelation Between Angle, Range and Maximum Height

. (30)

(30 )

x

y

1

g

hmax

v1x

2

v2x

v1y

v2y

v2

(1)

. (2)

(1)

(2)

(31) . . 90∞ 60∞ (32 ) 30∞

.

(32 ). 60∞ 30∞

O x(m)

y(m

)

60

2560∞

30∞

34

(33 )

.

(50)m/s(50)m/s

(30)m/s (30)m/s

(10)m/s (10)m/s

(34 )

.

.

y

60∞

Rx

|v0|=(20)m/s

O(0,0)

hmax

(35 )

. (33 )

. .

. (34 )

. :

.

.

(2) ∫Éãe

O(0,0) 60∞ . . (35 ) v

0 = (20)m/s

. ( ). ( )

. ( ) ( )

. . ( )

1 .. :

v0 = (20)m/s :

θ = 60∞:

y = f(x) ( ) ( )

= hmax

( ) = R ( )

35


2 .

: ( )∆x = v

0x ∆t = v

0 cos θ t

∆y = - 12

gt2 + v0 sin θ t

: ∆y t = xv

0 cos θ

:

y = ( -g2v

02 cos2 θ

) x2 + tan θ x

y = -0.05 x2 + 1.73x : . vy ( )

vy = -gt + v

0 sin θ

t = v

0 sin θg = 20 sin 60

10 = (1.73)s : .

: hmax

= v

02 sin2 θ2 g ( )

hmax

= 202 sin2 602 ^ 10 = (15)m

: ( )

R = v

02 sin2 θ

gR = 202 sin(2 ^ 60)

10 = (34.64)m: ( )

vy

vx

v

θ

(36 ) v

y

t = 2 ^ 1.73 = (3.46)sv = v

x + v

y : v

: v

x = v

0 cos θ = 20 cos 60 = (10)m/s

vy = -gt + v

0 sin θ = - 10 (3.46) + 20 sin 60 = (-17.27)m/s

. (36 ) vy

: v v = v2

x + v2

y = 100 + 298.58 = (19.96)m/s

:

tan θ = v

y

vx = -17.27

10 = -1.727θ = -59.92∞

. 60∞

36


3 . :

.


. -

θ -

m

2 m

1 -

θ v0 (m

1 < m

2)

. .

-

(50)m/s v0

. (80)m :

θ ( ). (80)m

. ( ) 9∞ ( ) .

. -

(37 ) 40∞ v

0 (t = 0)

G0 .

. (x0 = 0 y

0 = (6)m) (6)m

(1)m ( ). v

0

. ( )

v0

v0x

v0y

y

x

xG0

OH

(37 )

37

∫qhC’G π°üØdG á©LGôe

º«gÉØªdG

Maximum HeightRange

Parabola Velocity Components

Scalar Quantity Trajectory Equation

Vector QuantityMagnitude

Resultant of Vectors

π°üØdG »a á°ù«FôdG QÉµaC’G

.

.

.

.

. .

.

. .v = v

1 v

2 cos α

: v = v

1 v

2 sin α

. v

.

π°üØdG º«gÉØe á£jôN

.

38

∂ª¡a øe ≥≤ëJ

: ( ) 1 .:

2 .:

3 .: (N) 60∞ (5)N

(4) (3) (2.5) (4.333) 4 .: (N) 60∞ (5)N

(4) (3) (2.5) (4.333) 5 .

: . .

. .

∂JÉeƒ∏©e øe ≥≤ëJ

: 1 . 2 . (10)km/h (1)cm

(2)cm3 . (80)km/h . (80)km/h

4 . D

2 (4)m D

1 .150∞ (6)m

∂JGQÉ¡e øe ≥≤ëJ

: 1 . ( ) vR ( )

v2 = (5)m/s v

1 = (5)m/s O

. 120∞ . ( ) vR ( )

. ( ). ( )

1 π°ü

ØdG á

©LGô

e á∏

Ä°SCG

39

2 .

(75)N

27∞ 27∞

(150)N

(75)N

.

.

3 .

60∞

F3 = (6)N

F2 = (5)N

F1 = (2)NF

4 = (4)N

30∞ 34∞

.

4 .

(120)cm

(30)cm

v (120)cm (30)cm

. . ( )

( ).

( )(g = (10)m/s2 ) .

5 . . v0 = (30)m/s O (0,0) 30∞

. . ( )

. ( ). ( )

( ).

. ( )

1 π°ü

ØdG á

©LGô

e á∏

Ä°SCG

40

6 . (18)m v

0 (2)m . (2.44)m . θ

. .

(2.44)m(2)m

(18)m

θ

v0

7 .

F2

F1

60∞

(4)N F2 (3)N F1 . 60∞

. F2 F1 ( ) F1 ◊ F2 ( )

. Fí F" F2 ◊ F1 ( )

F" Fí ( )

π°üØdG ™jQÉ°ûe

.

: .

.

1 π°ü

ØdG á

©LGô

e á∏

Ä°SCG

41

π°üØdG ¢ShQO

.

. .

.

. :

.

ájôFGódG ácô◊G

Circular MotionÊÉãdG π°üØdG

42


. .

. . .

.

C

R R

(38 ). C

. (38 )

. .

.

ájôFGódG ácô◊G ∞°Uh

Describing Circular Motion1-2 ¢SQódG

43

(39 )

(40 ) ) (

.

xxM

y

yM r

M

C

xM

= r cos y

M = r sin θ

(41 ). M y

M x

M

x

y

S

θ

M

C

(42 ). O

0 M

1 .…QGóŸG ¿GQhódGh …QƒëŸG ¿GQhódGRotation and Revolution (39) . ) . . (

. . (40 )

. .

. 24 365.25

2 .Angular Displacement ájhGõdG áMGRE’G

. . (41) M

y x M . CM

CM M r |CM | = (rθ) θ M . r . x

. y (42 ) Δθ ( ) M . r

. (43 ) θ0 = 0 rad θ

44

s

Oθ

r

(43 )

. O =

(∞)

3602π180π90π/260π/345π/430π/6

(1 )∞ (rad)

(200)m

W E

S

N

(44 )

(∞) Degree . 60 60 360∞

. . s θ

s : θ s . θ

(rad) θ s = rθ.

: 2π rad = 360∞

. (∞) (rad) (1)

(1) ∫Éãe

(200)m

.(44 )

. . ( )

( )

1 .. :r = (200)m :

θ = 90∞= π2

rad:

: ( ) = s

( )2 .

: ( )s = r θ

: s = 200 ^ 3.14

2 = (314)m

45

á«FGôKEG Iô≤a


. .

. . . . . .


( )θ = 2π

: L = r (2π)

L = 200 ^ 2 ^ 3.14= (1256)m3 . :

:

. 2πr = .

3 .ájôFGódG ácô◊G ‘ áYöùdGSpeed in Rotational Motion .

.

Linear Speed (v) á« q£ÿG áYöùdG 1.3 v . Linear Speed . . Tangential Speed .

.

46

C

(45 )

.

á«FGôKEG Iô≤a


( ) . . )

. ( .

(ω) (ájhGõdG) ájôFGódG áYöùdG 2.3

Rotational Angular Speed Rotational Speed . rad

s . ω . . . Revolution

. Per Minute

. 33.33

. (45 ) 33.33 : ω

ω = ∆∆t = t

t0 = 0 s θ

0 = 0 rad

. v = ∆x∆t

4 .ájôFGódG áYöùdGh á«°SÉªŸG áYöùdG ÚH ábÓ©dGRelation Between Rotational and Tangential Speed

.

: . ( ) ^ =

v v r ω ( )

. v = rω : ω r ω ( ) ω ( )

.

47

. ( ) ( ) . . (46 ) .

. ( ) : . ( )

. ( )

(2) ∫Éãe

45

. (4)m (2)m

. ( ). ( )

1 .. :

t = (45)s = 2π : r

1 = (2)m r

2 = (4)m

: = ω

2 = ω

1 : ( ) ( )

= v2 = v

1 : ( )

2 .

ω = t ( )ω = 2t = 245 = (0.14)rad/s

(46 )

. .

á«FGôKEG Iô≤a

É«LƒdƒæµàdÉH AÉjõ«ØdG •ÉÑJQG

.

. . . 48


1 .

. 200 ( )

. ( ) (5)cm

. T = (0.3)s ( ) :

v = (1.047)m/s ( )2 .

300 (50)cm.

( ).

( ) (10)cm M

. ( )

. M(10π)rad/s ( ) :

(10π)rad/s ( )(3.14)m/s ( )

ac

at

a

v

C

(47 )

.


:

ω1 = ω

2 = (0.14)rad/s

( ):

v = r ω:

: v

1 = r

1ω

1 = 2 ^ 0.14 = (0.28)m/s:

v2 = r

2ω

2 = 4 ^ 0.14 = (0.56)m/s

3 . :

r2 = 2r

1

. r1

.

.

5 .ájhGõdG á∏é©dGh á« q£ÿG á∏é©dGLinear and Rotational Acceleration

. . .

.

Linear Acceleration á« q£ÿG á∏é©dG 1.5

. a = ∆v∆t

:(47 ) 1 . a

t .

2 .. ac

49

Rotational Acceleration ájhGõdG á∏é©dG 2.5: ω

θ" = ∆ω∆t

. rad/s2

6 .áª¶àæŸG ájôFGódG ácô◊Gh á∏é©dGAcceleration and Uniform Circular Motion

. . . . a

c = v

2

r . r v

ω .

7 .áª¶àæŸG ájôFGódG ácô◊G ‘ …QhódG øeõdGh OqOÎdGFrequency and Period in Uniform Circular Motion . f

. f = 1T : . :

v = st

. T = 2πrv s = 2πr T

: ω T

T = 2πω

50

(3) ∫Éãe

(150)g . (60)cm

. . ( )

. ( )

1 .. :m = (150)g :

r = (0.6)m:

= v : ( ) = a

c : ( )

2 .

:ω = t ( )ω = 2 ^ 2

t = 2 ^ 21 = (12.56)rad/s

: v = r ω

: v

1 = r ω = 0.6 ^ 12.56 = (7.54)m/s

: ( )a

c = v2

r = 7.542

0.6 = (94.7)m/s2

3 . :

.

8 .á∏é©dG áª¶àæe ájôFGódG ácô◊GUniformly Accelerated Circular Motion

θ" . . .

θ" a ω v

51

θ x :

∆θ = 12 θ"t2 + ω0t

ω = θ"t + ω0

θ0 = 0 rad

. ω0 = 0 rad/s

(4) ∫Éãe

(50)cm M . (48 ) θ" = (10)rad/s2

. 10 ( )

M

(48 )

. (10) M ( )

1 .. :θ" = 10 rad/s2 : :

ω0 = (0)rad/s :

∆t = (10)s ::

= ω : ( ) = N : 10 M ( )

2 .

ω = θ"t ( ):

ω = 10 (10) = (100)rad/s: ∆θ = 12 θ"t2 ( )

θ = 12 (10) (100) = (500)rad:

θ = 2πN N = θ2π = 500

2 ^ 3.14 = (79.61)rev 3 . :

. 10 (100)rad/s

52

r=(10)m(2)m

C

(49 )

(50 )


. -

-

-

( ) -

: (49) . (10)m. ( )

. ( ). (600) -

. ( ) v ( )

. (40)cm ( ) (2)kg -

. (1)m (5)rad/s. ( )

. ( ) (240)cm -

. (50 ) 30 . ( )

. ( ). ( )

-

. θ" = (2)rad/s2

5 ω ( ). θ

0 = 0

rad

. ( ). ( )

53


. .

(51 ).

.

. .

1 .ájõcôŸG áHPÉ÷G Iqƒ≤dGDefinition of the Centripetal Force (51 )

.

.

2-2 ¢SQódGájõcôŸG áHPÉ÷G Iqƒ≤dG

Centripetal Force

54

(52 ) ( )

. ( )

.

m

F

mgF

h

c

Fv

(53 )

.

2 .ájõcôŸG áHPÉ÷G Iqƒ≤dG ´GƒfCGTypes of Centripetal Force

. . .

. (52 )

3 .ájõcôŸG áHPÉ÷G Iqƒ≤dG QGó≤eMagnitude of the Centripetal Force

. . .

.

. F

. F (53 )F = Fv + F

h

Fv . F

h . Fc

: ∑F = ma

Fc = ma

c

ac = v2

r a :

. Fc = mv2

r:

.

55

(54 )

.

. (54 )

.

. . ( )

.

(1) ∫Éãe

. (50)m (1.5)tons . (314)s

(1)ton = (1000)kg

1 .. : m = (1.5)tons = (1500)kg : :

r = (50)m : N = 5

Δt = t = (314)s : :

= Fc :

2 .

ω :

ω = t = 2πNt

ω = 2 ^ 3.14 ^ 5314 = (0.1)rad/s

v = r ω : v = 50 ^ 0.1 = (5)m/s :

Fc = mv2

r

Fc = 1500 ^ 25

50 = (750)N : 3 . :

. (1500)kg 56


1 . (50)m/s (360)m

. (20000)N .

(1440)kg :2 .

v = (10)m/s .

(80)kg (350)N

. r = (22.85)m :

(2) ∫Éãe

. (188.5)m (56.6)m/s

. (1.89 ^ 104)N

1 .. :r = (188.5)m : :v = (56.6)m/s :

Fc = (1.89 ^ 104)N :

: = m :

2 .

Fc = mv2

r :

m = F

c r

v2 = 1.89 ^ 104 ^ 188.5(56.6)2 = (1112.09)kg :

3 . :.

4 .ájõcôŸG áHPÉ÷G Iqƒ≤dG ∫GhROmission of the Centripetal Force

. .

.

. . (55 )

(55 ).

57

F

(56 )

N

mg

F

(57 )

Fr

(58 )

5 .á«∏ª©dG IÉ«◊G ‘ ájõcôŸG áHPÉ÷G I qƒ≤dG ∫ƒM äÉ≤«Ñ£JApplications of Centripetal Force in Practical Life

á«≤aC’G äÉØ£©æŸG ≈∏Y ¥’õf’G 1.5 . . : (100)m (1000)kg . (14)m/s

= 0.66 :

. = 0.25 :

. N f

. = fN

f

. (58 57 ):

F = mv2

r:

F = 1000 ^ 142

50 = (3920)N

: : f

1 f

1 =

1 ^ mg = 0.6 ^ 1000 ^ 10 = (6000)N

. f

2 :

f2 =

2 ^ mg = 0.25 ^ 1000 ^ 10 = (2500)N

.

R

58

mg

xy

θ

θN sin θ

N cos θ N

(59 )

á∏FÉŸG äÉØ£©æŸG 2.5

.

. (59 ) .

. N sin θ = mv2

r

. Design Speed

(3) ∫Éãe (50)m

(50)km/h .

1 .. :

r = (50)m : :v = (50)km/h = (13.88)m/s :

: = θ :

2 .

. N

N sin θ = mv2

r : :

. N = mgcos θ N cos θ = mg

: N sin θ = mv2

r tan θ = v2

rg = (13.88)2

50 ^ 10 = 0.385

θ = 21.07∞3 . :

. (50)km/h

59


-

(1000)kg -

. (60 ) (32.5)m . (2500)N

(1.1)m (25)kg -

. (1.25)m/s . ( )

. ( ) -

(1500)kg (70)m

0.8 -

(50)m/s (4000)kg (360)m

. (61 ) -

25∞ (1500)kg. (50)m

(50)km/h (1350)kg -

. (400)m . ( )

. ( ) ( )

(60 )

(61 )

60

(62 ) )

. (


. .

.

. . Centrifugal Force

.

.

1 .ájõcôŸG IOQÉ£dG Iqƒ≤dGh ájõcôŸG áHPÉ÷G Iqƒ≤dGCentripetal and Centrifugal Forces

. (62 ) .

. . . . .

.

ájõcôŸG IOQÉ£dG Iqƒ≤dG

Centrifugal Force3-2 ¢SQódG

61

á«FGôKEG Iô≤a

AÉjõ«ØdG ∞«XƒJ

1884 . (100)m . (150)km/h . . .

.

.

. (63 )

(63 ) ( ) .

.

. (64 ) . . . .

.

(64 ).

62

(65 )

.

.

á«FGôKEG Iô≤a


. . . .

.

2 .QGqhO »©Lôe QÉWEG ‘ ájõcôŸG IOQÉ£dG Iqƒ≤dGCentrifugal Force in Rotating Reference

. Frame of Reference

.

. . (65 ) .

. .

.

. . . .

.

63

á«FGôKEG Iô≤a

. (66 )

. .

. . . . . .

. .

. g . (2)km

. . .

.

(66 ) .

.

(67 ) . ( ) .( )

.

(68 )

.

.

. ª 64

(69 )

.

(70 )

g . (0.5)g . (0)g (g) . (0.5)g . (0.2)g

. .

.


. -

-

-

-

(70)

65

ÊÉãdG π°üØdG á©LGôe

º«gÉØªdG

Rotation Tangential Speed

Revolution Centripetal Force

( ) Rotational Speed Centrifugal Force

Linear SpeedAxis


. .

.

. .

.

.

.

.


.

66


: ( ) 1 . 30∞ (25)m

: (m) (13) (7.5) (1.2) (750)

2 . (100)m : (157)m

1.57∞ 60∞ 90∞ 30∞

3 . (300)m (1000)kg : (s) (25)m/s

(1.04) (37.68) (25.12) (18.84)

4 .: (N) (83.3) (830)

(3802) (4166.6) 5 .:

. .

. .


: 1 .

. 2 . .

. 3 . 4 .

2 π°ü

ØdG á

©LGô

e á∏

Ä°SCG

67


:

1 . . :

( ) ( )

( )

2 . (2)m (1)kg : . (2)m/s

. ( ) . (1.8)N ( )

3 . . (90)km/h (2)m (200)tons .

4 .. (22)m (70)cm

5 . (2)rad/s2 ( ). (10)s (4)m

. (10)s ( ). (10)s ( )

6 . (50)m (1000)kg . 20∞

.


.

.

2 π°ü

ØdG á

©LGô

e á∏

Ä°SCG

68

. :

.

.

.

. .

2 π°ü

ØdG á

©LGô

e á∏

Ä°SCG

69

. . . .

.

π°üØdG ¢ShQO

ådÉãdG π°üØdGπ≤ãdG õcôe

Center of Gravity

70


. .

(Baseball) . . . (71 )

: .

.

.

(71 ).

1 .π≤ãdG õcôe ∞jô©JDefinition of the Center of Gravity

.

ª .

.

1-3 ¢SQódGπ≤ãdG õcôe

Center of Gravity

71

(72 )

.

hhh/4h/3

(73 ).

(74 ).

(75 )

.

.

. ª . . . (72 ) . h

. (73 ) h ) . ( . (74 ) .

.

2 .º°ù÷G π≤K õcôe QÉ°ùePath of the Center of Gravity of a Body (75) . ) ( . .

.

72

á«FGôKEG Iô≤a


. .

.

.

.

( ) . (76 ) . (77) .

.

(76 )

(77 ).


. -

-

-

-

. . -

73

(78 )

.


. .

.

. (Point Mass ) .

.

. .

.

1 .á∏àµdG õcôe ∞jô©JDefinition of Center of Mass

. (78 )

á∏àµdG õcôe

Center of Mass2-3 ¢SQódG

74

(79 )

.

(80 )

.

(81 )

.

2 .π≤ãdG õcôeh á∏àµdG õcôe ÚH ¥ôØdGDifference Between Center of Mass and Center of Gravity

. . (1)mm (541)m 2013 .

.

. . (79 ) . . (80 )

. .

.

. .

. (81 )

75

.

. (82 )

(82 )

.

3 .ΩƒéædG íLQCÉJh á∏àµdG õcôeCenter of Mass and Swinging Stars 800

.(83 ) 1.5

.

.

(83 ) .

.

76


. -

-

-

.

. . -

. -

.

77

(84 )

.


. .

. .

.

.

.

.

.

1 .º°ù÷G ¿RGƒJh π≤ãdG õcôeCenter of Gravity and Equilibrium of the Body

. . (84) . .

.

π≤ãdG õcôe hCG á∏àµdG õcôe ™°Vƒe ójó–

Determining the Position of the Center of Massor Center of Gravity

3-3 ¢SQódG

78

(86 )

.

( )( )

(87 ) ( )

. ( )

.

2 .πµ°ûdG áª¶àæe ΩÉ°ùLC’G π≤K õcôeCenter of Gravity of Regular-Shaped Bodies

.

.

. (85)

.

GGGG

(85 )

3 .πµ°ûdG áª¶àæe ÒZ ΩÉ°ùLC’G π≤K õcôeCenter of Gravity of Irregular-Shaped Bodies

.

. ) . . (

. (86 ) ( )

. .

. ( -87) . .

. ( -87) 79

x

y

x1

x2

m2

m1

xc.m.

(89 )

.

.

. . (88) .

.

CG

CG

CG

CG( )

(88 ).

4 .Ú«£≤f Úª°ùL á∏àc õcôe ™bƒe ÜÉ°ùMCalculating the Position of Center of Mass of Two Point Objects m

2 m

1 m

2 m

1

. (89 ) x2

x1

:

xc.m.

= m

1x

1 + m

2x

2

m1 + m

2

(1) ∫Éãe

m2 = (8)kg m

1 = (2)kg

. (6)cm .

1 .. :

m1 = (2)kg :

m2 = (8)kg

80

m1

m2

m3

mn

c.m.

r3

rn

r1

r2

rc.m

(90 )


x1 = 0 O(0,0) m

1

. x2 = 6 cm

: = x

c.m. :

2 . :

xc.m.

= m

1x

1 + m

2x

2m

1 + m

2

:

xc.m.

= 2(0) + 8(6)10 = (4.8)cm

3 . : (4.8,0)

.

5 .óMGh iƒà°ùe ‘ IOƒLƒe πàc I qóY á∏àc õcôeCenter of Mass of Several Bodies on the Same Plane

Ö m3 m

2 m

1

Ö r3 r

2 r

1

:

Rc.m.

= m

1r

1 + m

2r

2 + ...

m1 + m

2 + ...

(Oy) (Ox) :

xc.m.

= 1M

N! m

i x

ii=1

yc.m.

= 1M

N! m

i y

ii=1

.

.

81

x

y

m2m

1(0,0) (10,0)

m3

(5,53 )

(91 )

r3

rn

m1

m2

m3

mn

x

y

z

c.m.r1

r2

rc.m

(92 )

äÉHÉLEG ™e πFÉ°ùe

1 . (50)cm .

. :

2 . m

2 = (300) m

1 = (100)g

A g . AB = (40)cm B

. A A (30)cm :

3 . L O .

. O (L4 L4) :

(2) ∫Éãe

m2 = (2)kg m

1 = (1)kg

m3 = (3)kg

. (91 ) (10)cm

1 .. :m

1 = (1)kg :

m2 = (2)kg

m3 = (3)kg

L = (10)cm : :

yc.m

= ? = xc.m.

:

2 .

(91) (Oy) (Ox) (5,53 ) (0,10) (0,0)

. m1

:

xc.m.

= 1(0) + 2(10) + 3(5)(1 + 2 + 3)

= (5.8)cm

yc.m.

= 1(0) + 2(0) + 3(53)(1 + 2 + 3)

= (4.3)cm

3 . :.

6 .ÆGôØdG ‘ IOƒLƒe á«£≤f πàc I qóY á∏àc õcôeCenter of Mass of Several Point Objects in Space Ö m

3 m

2 m

1

Ö (92 ) r3 r

2 r

1

:

Rc.m.

= m

1r

1 + m

2r

2 + ...

m1 + m

2 + ...

82

ádCÉ°ùe

:

(1,1,0) m1 = (1)kg

(0,0,1) m2 = (0.5)kg

(-1,2,2) m3 = (2)kg

G1 G

2

(93 )

(Oz) (Oy) (Ox) :

xc.m.

= 1M

N! m

i x

ii=1

yc.m.

= 1M

N! m

i y

ii=1

zc.m.

= 1M

N! m

i z

ii=1

7 .á∏°üqàe ΩÉ°ùLCG I qóY á∏àc õcôeCenter of Mass of Several Attached Bodies

. (93)

.

.

(3) ∫Éãe

m1 = (2) (93 )

. (60)cm m2 = (1)kg (20)cm kg

1 .. :

m1 = (2)kg :

m2 = (1)kg

: = x

c.m. :

2 .

. (Ox)

. (50, 0) . (0,0) :

xc.m.

= 2(0) + 1(50)1 + 2 = 50

3 = (16.66)cmy

c.m. = (0)cm

. (16.66,0) 3 . :

.

83

O

(94 )

B

C

A

D

(20)cm

(95 )

r = (20)cmR = (40)cm

O1

O2

(96 )


-

. . -

-

-

. (94) O

. (10)cm : -

mD = (4)kg m

C = (3)kg m

B = (2)kg m

A = (1)kg

(20)cm . (95)

(40)cm (500)g -

(20)cm (200)g . . (96)

84

(98 )


. .

. .

.

(97 )

.

1 .Toppling ΩÉ°ùLC’G ÜÓ≤fG

. . (98) . :

.

ΩÉ°ùLC’G ÜÓ≤fG

Toppling4-3 ¢SQódG

85

(99 ).

(100 )

.

(101 )

.

CG

CG

(102 )

.

. . (99 ) 28∞ . .

. . (100 ) . .

.

(101) . . .

.

2 .á∏eÉ◊G áMÉ°ùŸG øe π≤ãdG õcôe ÜôbCloseness of the Center of Gravity to the Supporting Area . :

:

(102 ).

86

(103 ) Formula 1

.

bV1

aV

3

V2

V8

V7

V6

V5

V4

c

(104 )

CG

hCG

cV

1 V

2

(105 )

.

rh

CG

c2

CGCG

b2

r

(106 ). >

c

.

. .

. . (103 )

.

3 .Critical Angle of Toppling áj qó◊G ÜÓ≤f’G ájhGR

b a c

. (104) V

2 V

1

θ . .

. ( ) θ

C

. (105 ) V1V

2

θC

θ . V1V

2

. (106 )

. θ

C

b α .

87

b θ . (106 )

θ = θc

:

tan α = h

CG

(b/2) tan α = 2h

CG

b

: θc α

θc = 90 - α

: α

θc = 90 - tan-1 (

2hCG

b )

hCG

90∞ b .

. b h

CG

. .

(1) ∫Éãe

a = (5)cm : c = (20)cm b = (5)cm

. c

.

1 .. :c = (20)cm a = b = (5)cm : :

: = θ

c :

88

á«FGôKEG Iô≤a

á©«Ñ£dÉH AÉjõ«ØdG •ÉÑJQG

. . . . .

.


2 . h

CG = (10)cm

tan α = 2h

CG

b :

tan α = 2 ^ 105 = 4 α = 76∫

θc = 90 - 76 = 14∫

3 . :

.


. -

-

-

. -

(101)

-

(10)cm -

. .

89

(107 ).


. ( ) ( )

. ( ) .

.

. . (107 )

. .

1 .Definition of Stability ¿GõqJ’G ∞jô©J. ( ) :

.

.

.

(äÉÑãdG) ¿GõqJ’G

Stability5-3 ¢SQódG

90

CG

(108 )

(109 )

.

(110 )

.

(111 )

.

(112 )

.

2 .Cases of Static Stability Êƒµ°ùdG ¿GõqJ’G ä’ÉM

. .

. (108) .

.

. (109) .

.

. (110)

. . .

. . . (111 ) ( )

. . (112)

.

91

(113 ) .

CG

(115 )

.

(116 )

.

.

3 .π≤ãdG õcôeh ΩÉ°ùLC’G QGô≤°SG ÚH ábÓ©dGRelation Between Stability of Bodies and Center of Gravity .

. . . (113)

. ( -114) . ( )

. ( -114)

( ) ( )

(114 ). ( )

. ( )

. . (115) .

(116)

. 92

(118 ) ( )

. ( )

. (117) .

.

(117 ). ( )

. ( ).

. . (118 )

. . ) (

. . . .

93


-

. (115) -

: -

. -

-

-

94


(119 ) ( ) .

.

. .

.

. . ) . (

.

¿É°ùfE’G º°ùL π≤K õcôe

Center of Gravity of People6-3 ¢SQódG

95

(120 )

. (2)

O

(121 )

1 .¿É°ùf’G ‘ π≤ãdG õcôe ™°VGƒeLocations of Center of Gravity in the Human Body

. 3 2

.

. 5%

2 .¿É°ùfEG º°ùL ‘ Év«°VÉjQ π≤ãdG õcôe ™°Vƒe ÜÉ°ùMMathematical Calculation of Center of Gravity in Human Body

. (2) . (120 ) ì î

.

93.56.9 71.146.1

71.76.6 55.34.2

43.11.7 42.521.5

18.29.61.83.4

(2 )ì î

. 58%

. (121) (1.7)m

: .

96

(122 )

.

. (2) . 1.7

100

O : O

: x1

x1 = 42.5 ^ 1.7 = (72.25)cm

: x2

x2 = 18.2 ^ 1.7 = (30.94)cm

: x3

x3 = 1.8 ^ 1.7 = (3.06)cm

:

xCG

= (x

1 ^ m

1) + (x

2 ^ m

2) + (x

3 ^ m

3)

m1 + m

2 + m

3

: x

CG = 21.5 (72.25) + 9.6 (30.94) + 3.4 (3.06)

21.5 + 9.6 + 3.4 = (53.93)cm

(53.93)cm (121) . O

3 .á«FÉjõ«ØdG ÉæJÉ£°ûfCG ‘ π≤ãdG õcôe ™°Vƒe ÒKCÉJInfluence of the Position of the Center of Gravity on Our Physical Activities

.

. .

. . (122) . .

.

97

(123 ) .

y

0

9.6cm

23.6cm

28.5cm 1.8cm18.2cm

x

(125 )

4 .»°VÉjôdG AGOC’Gh π≤ãdG õcôe ™°VƒeLocation of the Center of Gravity and Athletic Performance "u" . 8 5 . "c" (123)

. (124) . .

.

(124 )

.


-

. "c" "u" -

-

. -

3.4% (125) 9.6% 21.5%

. Oy Ox

98

ådÉãdG π°üØdG á©LGôe

º«gÉØªdG

Toppling Non Uniform Shape

Static Stability Center of Gravity

( ) Unstable Equilibrium Center of Mass

Neutral Equilibrium Supporting Area

Stable Equilibrium Uniform Shape

Weight System of Particles

Critical Angle


.

. .

.

.

.

. . . θ

c

. .

.

99

π°üØdG »a á«°VÉjôdG ä’OÉ©ªdG

Rc.m.

= m

1 r

1 + m

2 r

2 + ...

m1 + m

2 + ...

xc.m.

= 1M

N! m

i x

ii=1

yc.m.

= 1M

N! m

i y

ii=1

θc = 90 - tan 1(

2hcg

b )


.

100


: ( ) 1 . . (30)cm m

2 = (100)g m

1 = (500)g

: . m

1 m

2 m

1

. m2

m1

. m2 m

2 m

1

. m1

2 . mA > m

B L m

B m

A

: A L m

AxCG

= mB

L m

BxCG

= mA

L mAx

CG = m

A + m

B

L mBx

CG = m

A + m

B

3 .: . .

. . 4 .:

. .

. .

5 .: . .

. .


: 1 . .

2 .

.

3 π°ü

ØdG á

©LGô

e á∏

Ä°SCG

101

3 . 4 ..

( )( )

5 ..

( )( )( )

6 .. ∂JGQÉ¡e øe ≥≤ëJ

:

1 . m2 = (400)g m

1 = (200)g

. (50)cm 2 .: . m

3 = (30)g m

2 = (20)g m

1 = (10)g

. (126) (50)cm ( )

m1

(50)cm

m2

m3

(50)cm

(126 )

m3

m2m

1

L

LL

A B

C

x

y

(127 )

( )m

2 A m

1 L

A C m3

B Ay Ax

. (127 )

3 π°ü

ØdG á

©LGô

e á∏

Ä°SCG

102

3 .

(128 )

(2)L(2)L

(4)L

(4)LO

O (128) ) .

(.

4 .

ab

c

: a = (5)cm c = (40)cm b = (5)cm

. c ( )

. ( )

b c .

b

c a

( )


. .

.

:

3 π°ü

ØdG á

©LGô

e á∏

Ä°SCG

103

π°üØdG ¢ShQO

.

. .

. :

.

™HGôdG π°üØdGá«YÉæ°üdG QÉªbC’G ácôM

Satellite Motion

104


. .

. .

. .

1 .Shapes of Orbits äGQÉ°ùŸG ∫Éµ°TCG . B

. OB v0 v0

(11.2)km/s ve v

0

(v0 > v

e)

v

0 = v

e Hyperbolic

. ( ) Parabolic

v0 < v

e

(8)km/s (129 ) ( ).

:.

2 .Circular Orbits ájôFGódG äGQÉ°ùŸG . . (142 )

á«YÉæ°üdG QÉªbC’G äGQÉ°ùe

Orbits1-4 ¢SQódG

. (8)km/s

. (8)km/s

. (11.2)km/s

. (11.2)km/s

(129 )105

S

dF

(131 )

.

90∞

90∞

(130 ) ( )

. . ( )

.

. . . . ( )

.

»YÉæ°U ôª≤d á« q£ÿG áYöùdG ÜÉ°ùM 1.2

Calculating the Linear Speed of a Satellite:

F = G Mmd2

(1)

M G = (6.67 ^ 10-11)N.m2/kg2 G d m

. . a

c = v

2

d : F = m.a

c

F = m v2

d (2)

: (2) (1)

v = GMd

106


. d

d :

. T = (125)mind = (1.9 ^ 106)m :

¬æeRh »YÉæ°U ôª≤d (ájhGõdG) ájôFGódG áYöùdG ÜÉ°ùM 2.2…QhódG

Calculating the Rotational Speed and the Period of a Satellite

: ( )

ω = vd = GM

dd

ω = GMd3

ω T = 2π T :

T = 2π d3

GM

T2 = 4π2 d3

GM

(1) ∫Éãe

3 . R = (6400)km : M = (6 ^ 1024)kg :

1 .. :

M = (6 ^ 1024)kg : :R = (6400)km :

: = d :

2 .

T = 2π (R + d)3

GM ( )

:

3600 ^ 3 = 2π (6400 ^ 103 + d)3

6.67 ^ 10-11 ^ 6 ^ 1024

d = (4.17 ^ 106)m = (4.17 ^ 103)km3 . :

. 3

107

á«FGôKEG Iô≤a


. .

.

(133 ).

F F

F

F

PE + KE

PE + KE PE + KE

PE + KE

(134 ) .

.

3 .Elliptical Orbits ¢übÉædG ™£≤dG äGQÉ°ùe v

e = (11.2)km/s (8)km/s

. (132 ) . Ellipse .

(132 ) (8)km/s .

. .

.

Drawing an Elliptical Orbit ¢übÉf ™£b º°SQ 1.3 . (133) .

4 .á«YÉæ°üdG QÉªbC’G ácôMh ábÉ£dG ßØMEnergy Conservation and Satellite Motion (KE) . (PE) .

. (134 ) . .

.

108

KE + PE

KE + PE

KE + PE KE + PE

(135 ) .

(136 )

:

. (8)km/s

Apogee . . Perigee . (135 ) .

5 .Escape Velocity äÓaE’G áYöS . (8)km/s . (8)km/s

. : . (11.2)km/s .

. . . .

.

109

(137 ) 1972 10

. 1984

Conservative Force

. v

e m

:

12 m v

e2 - GMm

r = 0 + 0

Kinetic Energy = Potential Energy :

ve = 2GM

r = (11.2)km/s

r M G . .


. -

: (6370)km (6.0 ^ 1024)kg

T = (23)h (55)min (40)s . G = (6.67 ^ 10-11)N.m2/kg2

-

-

.

. v = (30)km/h -

. (8)km

. ( ) . ( )

110

™HGôdG π°üØdG á©LGôe

º«gÉØªdG

Universal Gravitation Energy Conservation

Escape Velocity Circular Orbit

Elliptical Orbit


. (11.2)km/s (8)km/s (8)km/s

. (11.2)km/s

v = GMd

: . d M G

:

T = 2π d3

GM

F = G m

1 m

2

d2

d m2

m1

: . G


.

(8)km/s

(8)km/s

111

112

4 π°ü

ØdG á

©LGô

e á∏

Ä°SCG


: ( ) 1 .: (8)km/s

. . . .

2 .: M v = GMd

. . . .

3 . (T) : .

. 2T . . T

2 2 . T2


: 1 . . m/s

. (150 ^ 106)km 2 .

. 3 . M

. (G = (6.67 ^ 10-11)N.m2/kg2) (r)


.

. .

.

112

ÖdÉ```£dg ÜÉ``à`c§لصف...a _ o. a = = 4 a a 2 1. . . _ a + b = =

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