4

4.1

() , . , . Pascal, , " , , (), ( ).

, , .

(4.1)

(4.2)

p y, p0, y0 (. 4.2) g, .

, , .

, .

. . , () , . , , . , .

. , , . , ( ) (. 4.3).

1 atm V0 10- 9 mm3. ,

p p g y y= + 0 0 b g

d

d

p

yg=

95

4.2

.

4.3

.

4.1

.

V m ' .

. . ~, , .

: , , . (. 4.4). , . , ,' .

: , , , . , , , . , ,, , , , . , . .

, , . . .. , ..

: , , (). , , , (. 4.5).

. ~ (. 4.6) . ~

=

lim

V V

m

V0

96 MHXANIKH

4.4

(1)

4.5

. 4.6

, , .

, , , .

. ' . ( ) , , (. 4.7). , , ., , .

. . , . . , . , , , , , .

: , .

) ) ) ,

( ) , ,

. , Bernoulli.

: V , , , t, t.

(4.3)

S.I. m3/s L3 T -1. , ,

. (. 4.8) t , t. ,

(4.4)

( ) 4.9. 1, 1 1. Q A 2 , 2 2 . t 1 ,

= Vt

A t

t= =

V

t=

97

4.7

.

4.8

t t

t 2

V2 = 2 t = A2 2 t

:) ,

, . , 1, 2.

) . , . , 1, 2,

m1 = m2

1 1 2 t = 2 2 2 t

(4.5)

P Q

(4.6)

.

, , 1 = 2 ,

(4.7)

(4.7) (), , . .

A A 1 1 2 2=

= .

1 1 1 2 2 2 =

m V A t2 2 2 2 2 1= =

m V t1 1 1 1 1 1= =

V t A t1 1 1= =

98 MHXANIKH

4.9

.

, , , .

4-1

2,0 m2 . 3,0 m3 s-1, . 12 m.s -1. .

=

A (4.7)

= 0,25 m2

, 0,25 m2.

BERNOULLI

, (. 4.10), , 1, Q, 2.

t (. 4.10) t + t (. 4.10), t . l1 m1 = A1 1 t l2 m2 = A2 2 t.

m1 = m2 = 1 1 t = m (4.8)

H m F1 F2 (. 4.10). , , m

m Q

,

E U K m g y m Q Q Q = + = +2 221

2

E U K m g y m p p p = + = +1 121

2

= A 2 0 1 512

, ,m 2

=

AA

A =

= = 3 02 0

1 51 1,

,,m s m s

=

99

(4.9)

F1 = p1 A1 ( ) F2 = p2 A2 ( Q) ,

(4.10)

(4.11)

(4.12)

(4.10) (4.8), (4.9), (4.11) (4.12)

A tg y y A t p p A t1 1 2 1 1 1 22

12

1 2 1 11

2 + = b g e j b g

m g y y m p A t p A t2 1 22

12

1 1 1 2 1 11

2 + = b g e j

W F l p A t p A tF2 2 2 2 2 2 2 1 1= = =

W F l p A tF1 1 1 1 1 1= =

E W WF F= +1 2

Q PE m g y y m = = + 2 1 22

121

2b g e j

100 MHXANIKH

4.10

4.10 4.10 t t + t.

(4.13)

Q , ,

(4.14)

: , . , .

4.14 Bernoulli, :

,

". .

.

.

.

' , ' Bernoulli, F1 F2 () , . .

y1 = y2 (4.13)

(4.15)

, ( - ), 1 = 2 = 0

(4.16)

.

4-2

0,60 cm, 10 m. 0,15 cm, , 8,0m.s-1 : ) . ) . .

g y p g y p1 1 2 2+ = +

1

2

1

212

1 22

2 p p+ = +

pp V

V=

g ym

Vg y

U

V= =

1

2

1

22

2

m

V

V= =

1

22 g y p+ +

1

22

1 g y p+ + = .

1

2

1

212

1 1 22

2 2 g y p g y p+ + + +=

101

) (1) (2)

A1 1 = A2 2

2 = 0,50 m.s-1

) Bernoulli (1) (2)

(1) , . p1 = 1 atm = 10

5 .m-2

( p2, ;)

TORRICELLI

4.12 h . , . Bernoulli Q . y1 = h, y2 = 0, 1 = 0 2 = 4.13

p1 p2 ,

g h= 2 g h =1

22

g h p p+ = +1 2 21

2

p252 32 10= , Pap2

5 3 3 2 210 10 10 101

210 8 0 5= + + LNM

OQP,e j Pa

p p g h 2 1 12

221

2= + + e jp p g h 2 22 1 121

2

1

2+ = + +

218= (0,15)

(0,60)ms

2

2

A

2

1 1

2

=

102 MHXANIKH

4.11

4.12

To , .

, h. Torricell.

: , 4.13.

, 1 2 1 > 2,

A1 1 = A2 2 1 > 2

1 < 2 Bernoulli

1 < 2 p1 > p2

, () , (). , , , .

4-3

. h1 h2, , . h1 h2.

Torricelli, 1, (1),

1

2

1

212

1 22

2 p p+ = +

103

4.13

.

4.14

. , , ( ).

,

x1 = 1t1

, x1 = x2 ( h1) h1 = (H h2) h2 h1 + h2 = H

BERNOULLI

Bernoulli . , , . .

) : 4.15 . . , . , (1) , . (1) , , .

x H h h2 2 24= b g

x H h h1 1 14= b g

th

g1

12=h g t1 121

2=

g H h1 12= b g

104 MHXANIKH

4.15

.

4.16

. .

) : Venturi , . , . , . , . h (.4.16).

. Bernoulli ,

,

, p p .

(1) (2) p1 = p2

h .

) Pilot: , . , () (. 4.17). (), , () (). ' () , (). ~, . () () . Bernoulli

(4.17)g h

=

FHGIKJ

LNMM

OQPP

2

12

b g

1

2

1

22

22 g h

+ =

FHGIKJb g

1

2

1

22 2 p p + =

p p g h = b g

p g H p g h g H h+ = + + b g

p p g h g H h2 = + + b g

p p g 1 = +

1

2

1

22 2 p p+ = +

=

A =

105

p1 = p2. p1 = p p 2 = p + g h, ~

. ,

(4.18)

h , ' , .

) - : 4.18. . .

, 4.19.

g h= 2

1

22 p p g h+ = +

1

202 p p+ = +

106 MHXANIKH

4.17

Pitot. .

4.18

. .

4.19

.

, , . , .

) : Bernoulli . 4.20, , . , , . ( ).

, d . , 0 (. 4.21). , , 0 . 4.21. F ,

0. d .

(4.19)

. 4.19

(4.20) d

=

0

F

A

d

=

0

F A

d= 0

107

4.20

.

4.21

( ) 0 ( ).

-

. , , S.I. Pa.s ML-1 T - 1 . poise (P) CGS , 1 poise =10 -1 Pa.s.

(. 4.22). . , . F1 , F2 ., , . ' . . , , . ' , .

, : . , . , , , , , . , , , ,

(. 4.23). , . , . ,

' , .

4.24

. 20 C .1 8 10 5, Pa s

F

A=

108 MHXANIKH

4.22

4.23

m 1 (). m 2 ().

4.24

poise, 0 C.

, ' , .

, , . . , , ...

, , (. 4.25).

,

,

,

,

(4.21)

(4.22)

C1 C2 .

C1 C2 . C1 C2 , R .

. ,

Mk + L- 3 k - + + 1 - - 1 = 1 L1 T - 2

,

(4.23)

(4.21)

( Stokes) (4.24)F R = 6

C R1 6=

C R1

k 0

1

1

==

=

k 1

3k 1 1

1 2

+ = + + =

=

UV|

W|

(ML ) (ML T ) (L) (LT ) MLT3 k 1 1 1 2 =

k

R F=C F1 =

C R1 k

F C = 2 2

F C = 1

109

4.25

.

= 1, = 0, = 2

H

(4.25)

C2

(4.26)

C , , () . 4.26 C .

: , , .: .: ,

. , . , , Stokes, .

4-4

R = 10 cm m = 2,0 kg . . C2

2.

, . . . ,

F = mg

= 43 m s-1

1m s=

2

0 1

2 0 9 8

3 14 1 3,

, ,

, ,

R

m g

= 2

42 R m g2 =

C C 22

2=

C R22=

4

C R22

2

R F=[ ] [ ]C F2 2 =

C R2

110 MHXANIKH

4.26

.

4.27

111

4-5

R = 0,50 mm. 1 = 6,0 m

.s-1. , ,) , ) 2 = 4,0 m s

-1,) . . ( = 1,8 10-5 Pa.s)

) F = C1 1 F = 6R1

F = 1,0 10-6 N) 2 , , (1 - 2). ,

F = C1 (1 - 2) F = 6 R (1 - 2)

F = 0,3 10-6 N) , () .

F = 0

= 1 = 6,0 m.s-1

' , . , () . , , F . , 4.31. F 4.31 . F ( ) F . : , 4.32. ., (1 > 2). , Bernoulli, . , ' . .

F N= 6 3 14 1 8 10 0 5 10 2 05 3, , , ,

F ( ) N= 6 3 14 1 8 10 0 5 10 6 05 3, , , , 4.28 - 30

4.31

4.32

.

112 MHXANIKH

:

) , , ( ) .

1 1 = 2 2) Bernoulli,

, ,

, y1, y2 p1, p2 o . .

Bernoulli , , Pitot ...

, d . 0 , .

= F/A d .

,

R,

.

, .

C R22=

4

C R1 6=

F C = 2 2

F C = 1

d

=

0

1

2

1

212

1 1 22

2 2 g y p g y p+ + + +=

drasthriothtesA N A

BERNOULLI1. ,

5 cm. . .

2. 5 cm 5 cm .

drasthriothtes

113

, 1 mm 2 mm . . .

3. : 250mL . . , , , . , , , , . , , . . 5 mm.

4. . , ,

. , . , , . , , ( ). , . , , , . , . . , . , .

( ;).

114 MHXANIKH

1

(). () ().

i) .

ii) .

2

;

3

. , () . . . . . () . . . . ., . () . . . . . () . . . . . .

4

. - () ().

() 1 2

5.

, .. (),

30 cm . , . 30 cm .

, Stokes .

6.

. , , . . , , , , 7 m - 80 kgf (kp).

115

1 2 ( ).

() 1 2 .

() 1 2.

5

;

6

1 2 3:1. 1, 1 2, 2 1 2.

.() 1 = 2 1 = 32() 1 = 32 2 = 31() 1 = 2 2 = 31() 2 = 31 1 = 2

7

Bernoulli () . . . . . , () . . . . . () . . . . ., () . . . . . () . . . . ..

8

. .

() Bernouli .

() , .

() m ( ) m , Bernoulli .

() .

() .

9

() ,

, . ;

() , , ;

116 MHXANIKH

10

.

11

.

; H .

12

;

13

() ().

14

h1, h2, h3 .

.

15

, , ,

() 2 , () , () 4 () / 2;

16

( , , ).

17

140 2,0 mm2 . 3,010-3m3s-1, ;

2

117

18

() A = .() 1/ 2 2 + g y + p = .,() 1/ 2 2 + p = .(1) N Bernouli .(2) Bernoulli .(3) .(4) Bernoulli, .

19

() . . , ;

() , , , ;

20

, ()

() () ;

21

, , () . . . . . () . . . . . () . . . . . (). . . . . .

22

F = C2

2.

() () () ()

23

0.

;

24

, , W .

4

3

7

9

16

3

4

=F F916

118 MHXANIKH

= 2 W () 2 W, () W, () 4 W, () 8 W C2

2.

25

D1 D2 0 - 0 ,

( ).

26

R 2R . .() C1 ,

(i) , (ii) 2 , (iii) / 4, (iv) 4 ;

() , C22,

(i) , (ii) 2, (iii) /2, (iv) 4 ;

27

, , , .(1) n (1) (2) C1 (2) kg.m

- 1 . s- 1

(3) C2 (3) kg.s

- 1

(4) C (4) kg.m

- 1

28

, () . . . . . (). . . . . . () . . . . . () . . . . ..

-

1

8000 m3s-1 44 106 m3. .

2

6,0 m s-1. A ; g = 10 m.s-2.

3

. 1 /2 = 5,0 h = 15 cm, 1. H

g = 10 m.s- 2 .

4

. 0,20 m2 0,050 m2

. 5,0 ms-1 2,0 105 N m-2 :() () . 1,0 103 kg m-3.

5

1,0 mm2 75 mm2. ,

119

, 3,5 m, 1,0 m. 10 , ; 1,0 103 kgm3 g = 9,8 m.s- 2 .

6

. , , . ;

7

0,010 m2. 2,0 10-4 m3s-1, 1,0 cm2. . . g = 10 ms-2.

8

30 cm 15 cm. 4,0 104 Pa 3,0 104 Pa, . 1,0 103 kg m - 3 .

9

H 1,75 105 Pa.

= 6 , , . (

, ). 103 kg m-3.

10

Pitot . 26,5 cm. km h-1. 0,800 103 kg m- 3 1,30 kgm-3. g = 9,80 m s- 2.

11

. h = 100 m 200 m3 s-1, . g = 10 ms-2 103 kg m- 3.

12

5,0 103 N m- 2 . m3 , 1,0 m.

13

20 m2 ( ). , 40 m.s-1, 50 m.s-1. . 1,3 kg m-3.

14

1,5 10-3 m. 1,0 103 kgm-3 g = 9,8 ms- 2. 1,3 kgm-3. = 1,8 10- 5 Pas

15

A .() ,

;() 20 %

;

16

5,0 cm 0,5 kg . . 1,3 kg m- 3 g = 9,8 m.s-2. c1 .

17

R = 40 m

.s-1. R = 2R , . :() F = C1

() F = C2 2

.

18

2,5 m. r = 2,0 m C . 80,0 kg. 1,3 kg m-3 10 m.s-2.

19

. c1 ,

20

. m R. . . g.

m g

C = +

FHGIKJ

2

1

2

120 MHXANIKH

4.2

. (), . , , . , , . .

(. . 4.33) , (

) . , , , . . , , . , , . ,

X 121

4.33

H .

.

-

, , ' , , , .

, ( ) . , () (. . 4.34), .. , ..

, ( ) . , (. . 4.35). , ..

, , , . . , , , . , , , . .

E A

. , , ' . ( CM).

, , ' , .

m1, m2, ... x, y, z. m1 (x1, y1, z1), m2 (x2, y2, z2) ...

(4.27)

(4.27)ym y m y

m m

m y

mcm

i i

i=

+ ++ +

=1 1 2 21 2

. . .

. . .

xm x m x

m m

m x

mcm

i i

i

=+ ++ +

=1 1 2 21 2

. . .

. . .

122 X

4.35

.

4.34

, - - .

(4.27)

, " " . .. , . , .

. . .. . ' .

, . . , 4.36, , y . xz. m1, m2.... m1 g y1 , m2 g y2 , ... ,'

m1 + m2 + ... = M

U = Mg ycm (4.28)

.

4-6

m1, m2 (m1 > m2), r , , . K .

: m1 = 2 m2 r = 1,2 m

U = + +m m g y1 2 .. .b g cm

m y m y m m y1 1 2 2 1 2.. . . . .+ + = + +b g cm

U = + +m y m y g1 1 2 2 .. .b g

U = + +m g y m g y1 1 2 2 . . .

zm z m z

m m

m z

mcm

i i

i

= + ++ +

=1 1 2 21 2

. . .

. ..

123

4.36

.

4.37

, , m1, m2, 1, 2 . K 1, 2, xx. .

: r1 = 0,4 m r2 = 0,8 m

, . , . t , ' , , (. . 4.38).

a , .

(4.29)

, t , .

(4.30)

1 rad/s

tt= =

lim

d

d 0 t

ta

=

rm

m mr2 1

1

1 2

= =+

r r

rm

m mr1

2

1 2

=+

r

r r

m

m m1

1 2

2

1 2+=

+

r

r

m

m1

2

2

1

=

m r m1 1 2 2= r

0 1 1 2 2

1 2

= +

+m r m r

m m

b g

xm x m x

m mcm =

++

1 1 2 2

1 2

124

4.38

~ .

, .

, . , t, t, t.

(4.31)

To a, t ,

(4.32)

1 rad/s2

.

, (). (. 4.39), .

, , , .

. -, .

, , , . r , t s (. 4.40). s

= r (4.33)

.

( ), .

,

a, (. . 4.41)

lim

lim

0 0t t

s

tr

t =

s

tr

t=

s r=

dim = T 2

= =

lim

t t t0

d

d

a =

t

dim = T 1

125

4.39

.

4.40

s s = r.

i) (arad)

(4.34)

ii) (atan ) .

(4.33)

, (4.32),

(4.35)a rtan =

atand

d= r

t

a

ttan

d

d=

a

r rrad = =

22

126

4.41

.

. t = 0 o. , , .

= f (t). dt , d = d t. d . t = t 0 dt.

dt . t = t 0 ().

t

t= + +0 02

= +0 t

= = =

a

t

t0

0

I

-

, , , 4.42, m1 , m2 , . . . r1 , r2 , . . . . , , ' ,

~ ,

1 = r1 , 2 = r2

, , ,

K m r =1

22 2 i ie j

K = + +12

1 12

2 22 2m r m r . . .e j

K =1

2

1

21

212

22

22m r m r+ + .. .

K = + +12

1

21 12

2 22m m . ..

127

= (t)

,

.

a = = . = 0 + at = 0 + t

2 = 20 + 2as 2 = 20 + 2

t t= +0 21

2 s t a t= +0

21

2

2 02 2= +

t = 0

= +0 21

2t t

4.42

(4.36)

, , 1kg.m2 L2 M1 T 0 I0 = L2 M 1.

, ,

(4.37)

, , , . , , , , , , 4.43. (B) , , ' (A) .

< <

, ( )

< <

( ) , .

"" "" .

4-7

( ), m = 2,0 kg , ( ) r = 0,80 m. , , = 3,0 rad/s, , :

) )

)

I m r mr

mr

12

2 2

2 2= = FHG

IKJ +FHGIKJ i i

K = I 1

22

I m r= i i 2

128

4.43

.

4.44

)

, .

- ( STEINER)

. mi (mi 0), ().

H () ri2 mi , ri

.

.

. . R, (. 4.45).

. R ,

= R2 M (4.38)

, Steiner ,

I R m R m= = 2 2

I r m= i i2

K 2 5 8= , J

K =2 221

2

1

21 3 9I J= FHGIKJ,

I 22 22 0 8 1 3= = kg m kg m, ,b g

I m r m r22 2= = i i

K =1 2 9, J

K 121

2

1

20 64 9= = FHG

IKJI , J

Im r

1

2 2

2

2

2 0 8

20 64= =

=

kg mkg m

,,

b g

129

4.45

R, .

130

P

L R R

I M R=2

52I M R=

1

22I M L=

1

122

R L

, ,

,

I MR L

=2 2

4 12+

FHG

IKJ I M

a =

2 2

12

+I M

a =

2 2

12

+

R R ,

I M R=1

22 I MR=

1

4

2 I M=1

12

2

R R R

I MR= 2 I MR=1

22 I MR=

2

3

2

z, cm, zc , z,

= cm + M d2 (4.39)

d z z c (. . 4.46)

131

4.46

z zc .

Steiner

4.47, , xcm = ycm = zcm = 0. z , (xP , yP , 0) z. mi ' xy (x i , y i , 0).

, (z)

(I)

,

(II)

xcm = 0, ycm = 0

(III)

(IV)

A d ,

(V)

(VI)

H (II) (), (), (V), (V) (VI)

I I M dp c m= + 2

m Mi =

x y dp p2 2 2+ =

m yi i = 0

m xi i = 0

I m x y x y m x m x y m yp i i i p p i p i i p i i = + + + 2 2 2 2 2 2e j e j

I m r m x x y yp i i i i p i p = = + LNMOQP

2 2 2c h c h

I m R m x ycm i i i i i = = +2 2 2e j

4.47

.

4-8

L. , ' , ,

, ' , .

, Steiner

4-9

R, , , (4.48). .

.

, , .

,

Steiner

= c + Md 2

= = =I MR M R MR

c/ 2

2 4 8 32

2 2 2b g

= =M MR

R

M/ 2

4

2

2

b g

B

m r i i2A

m r i i2

MR

m r m rA B

22 2

2= + i i i i

I m r MR02 21

2= = i i

=I M L13

2

= +I M L M L112

1

42 2

I I ML = + FHGIKJ2

2

I M L=1

122

132

4.48

4.49, (Steiner). P , . , , .

IP P.

Steiner

IP = Icm + Md2

d P.

d = cm

(4.40)

(4.40) ,

, , ,

,

.

1

22I cm

1

22M cm

K I M= +12

1

22 2

cm cm

K I M d = + 12

1

22 2

cm ( )

K I Md = +12

2 2cme j

K I P= 1

22

I MR= 1332

2

MR

I R M2

2

2

3

32= +

=I MR 2 332

= + FHGIKJI M

R M R22

32 4 2

133

4.49

, P, .

4.50 . s, . t , ., ,

, t, s,

= cm = R (4.41)

E

a = R (4.42)

.

4-10

R . , m. h . , , .

,

K M221

2=

= R

K I R 22 2 21

2

1

2= =

K m 121

2=

U m g h=

ad

dt

d

dtR

d

dt= = =cm

s

t

=

s

tcm

=

134

4.50

4.51

,

4-11

- h. , . , , ;

, ,

, , .

, , , ,

M g h M = 34

2

U = K

K M M M= + =14

1

2

3

42 2 2

R=

K M R M= +12

1

2

1

22 2 2

K I M= +12

1

22 2

U M g h=

I M R= 12

2

R R

m g h

m M= =

+ 1 2

=+

2 m g h

m M

m g h m M= +12

1

22 2

U = K + K1 2

135

> . , , "" .

(. 4.52), , . , , . , , .

, ,

r ,

=

r

r

(. 4.53) ( . 146).

= Fr sin

() r

.

.

,

r ,

.

(. 4.54).

SI 1 m ML2 T 2.

4.55 . , ,

F

F

F

F

F

F

= g h2

M g h M = 12

2

g h= 43

136

4.52

.

4.53

.

4.54

.

F l . ,

= Fl = Fr sin (4.43)

l

F., F , (. 4.56).

(. 4.57), , .

F (. 4.58), , (F / / ) (F) , .

F, , F ,

= = F l

: , , , . , 4.58, Z ( ) , (. 4.59).

F1,

F2 .., .

= 1 + 2 + 3 + ...

, . ,

4.60,

F1 1 = F1l1

F2 2 = + F2l2 .

137

4.58

F F .

4.55

l.

4.56

F , .

4.57

F .

4.59

, , .

. ,

,

F1,

F2,

F3, ... .

F (. 4.61),

. Ftan Frad. , Frad , , Ftan.

4-12

, (F1 = F2 = F) . l . () . . , , .

0 = F2 x2 F1 x1 = Fx2 Fx1, 0 = F (x2 x1) 0 = Fl. , , . .

"" . ' ' .

4-13

, , , .

F

138

4.60

.

4.61

F Ftan .

4.62

4.63

F1 = 30 , F2 = 20 N l = 2,0 m

F2 , F2x F2y.E

2 = F2x + F2y

2 = + l F2y + 0F2x

2 = + l F2 sin 30o

2 = +20 m

1 = F1 l / 2

1 = (30 2/2) Nm

1 = 30 m

= 1 + 2

= (+20 15) Nm

= 10 m

N

r, , (. 4.65).

, m. , a tan.

F = m a tan rF = m r a tan

, rF = a tan = r, .

= mr2 (4.46)

, , 4.66.

,

. , , Ftan ,

. (4.46), , .

F

F

F

F

F

139

4.64

4.65

F .

4.66

.

, (. 4.67). . (4.46)

1 = m1 r12, 2 = m2 r2

2 . . . 1 , 2 , . . . ,

,

1 + 2 + ... = m1 r12 + m2 r2

2 + ... = (m1 r12 + m2 r2

2 + ...)

(4.47)

. , . , , , (. 4.68), .

(4.47) :

, , , , , .

,

.

) : , ( 4.47), .

) : , . F = macm , . . (4.47), = cm , Icm .

.

4-14

, . , g.

=

140

4.67

.

4.68

0, .

, k

. , , ,

= ()

R ,

= F R()

,

()

cm

(IV)

() (), (), (V)

(V)

2 Newton

(V), (VI)

: ,

, , acm () =

2 / 2,

a ah

2 2 2= =cm cm

sinb g

acm sin=2

3g

M g F M asin cm = cmF M ax =

F = Macm1

2

= aRcm

I M R= 12

2

F

F

F

B

141

4.69

, 4 - 11,

4-15

m1, m2 , . R, . .

m2 ,

2 . 2 N

()

m1

()

()

, atan = R,

a , .

, ()

(IV)

(), () (IV)

(V)

Atwood o g m1 , m2 a g, (V) g.

-

, ,

a =

+ +

m m g

m mI

R

2 1

1 2 2

b g

T T R Ia

R2 1 =b g

T R T R I2 1 = I=

T m g m a1 1 1 =

m g T m a2 2 2 =

a g cm sin=2

3

24

3a

h

g hcm

sin=

24

3= g h

142

4.70

acm = 0 , = 0. , , . , , , , .

=M acm .

acm = 0, = 0.

= I. = 0,

= 0.

:

1. =0. , ,

2. = 0. , , .

,

x, y, =0

Fx = 0 ()

Fy = 0 () =0

(z) ( )

z = 0 () , '

(z), , . .. "" . (), (), ().

' , ' , , . , ' . , , , . , . , , . , (. 4.71). Mg , , , ' .

F

F

F

F

143

MgxK = m1 g x1 + m2 g x2 + ...

(4.48)

. . , ' . , .

, , (. 4.72).

4-16

4,0 m 100 , . , , 5,0 m. , 1,0 m , 400 . .

(). ' ;

xm x m x

MxK cm=

+ + =1 1 2 2 . ..

144

4.71

.

4.72

.

4.73

(i) 100 , , .

(ii) 400 .(iii) (

).(iv) F . '

F . (. 4.73)

Fy + T sin 400 N 100 N = 0 (I)

Fx T cos = 0 (II)

sin (4m) (100 N) (2m) (400 N) (1 m) = 0 (III)

, ,

F ( ) . ()

()

(1)

F

F

= 60

tan Fy

Fx= = =350

2001 75,

F = 400 NF Fx Fy= + = +2 2 2 2200 350 N

Fy T= = =500 35

500 2503

5350N N N N

Fx T= = =45

2504

5200N N

T = 250

T3

5150= N

T sin = 150

A = 0

Fx = 0

Fy = 0

sin cos = =1 35

2

cos

= =b gb g

4

5

145

4.74

(.

4.75). F

F .

F

Frad ()

Ftan ().

Frad . () ds F

dW = Ftan ds = Ftan Rd

Ftan R F ,

dW = d (4.49)

,

.W = (4.50)

F ( ),

P = (4.50)

(4.50) P = F , .

, , ' , .

(4.51) W I = 12

1

22 2 x

W =

PW

t

t= =d

d

d

d

146

4.75

F, .

(4.51) , .

(4.50) (4.47)

, ,

W I = 12

1

22

02

W I =

202

2

2 02 2= +

W I=

4-17

d , . , F. .

. , ,

,

= Fd

H

. , . , .

, ,

r ,

p (. 4.77).

L =r p (4.52)

)

L = rp sin = mr sin = ml (4.53)

r

p .

)

p.

) .

r

p

. (. 4.78). SI kgm2 / s L2 MT 1.

E W Fd

= =2 22

W Fd

= =2

= F d2

147

4.76

4.77

, , .

4.78

.

m, , z (. 4.79).

L = r p L = rm

L = rmr

(4.54)

4.79. , ,

(. 4.80). , , , , ,

L = L 1 + L2 + L3 + ...

(4.54) L

L = m1 r12 2 + m2 r2

2 2 + m3 r3 2 + ... = (m1 r1

2 + m2 r2 + m3 r

3 + ...)

(4.55)

.

, , (4.55)

(4.47)

(4.56)

(4.56) -

,

. - ( ) : , , .

d

d

p

tF

=

d

d

L

t=

d

d

d( )

d

d

d

L

t

I

t

tI= = =

L = I

L = m r2

148

4.79

m .

4.80

.

, (4.56)

L = .

(4.57)

. . (4.57) , , , . 1 1 , 2 2 ,

1 1 = 2 2

, , (.4.81).

4-18

( ), 4,0 kg . ; . ( ) , 3,0 kg.m2, 2,5 kg.m2. 1,0 m , 0,20 m.

, , , .

1 1 = 2 2()

- ,

1 = 3 kg m2 + 2 (4 kg) (1 m)2 = 11 kg m2

2 = 2,5 kg m2 + 2 (4 kg) (0,2 m)2 = 2,8 kg m2

f2 , ()

f1 0 5 0 5= =, ,

Hz

= .

d

d

L

t= 0

149

4.81

.

4.82

150

4.83

2 .

1 = 2 (11 kg m2 ) 3,142 (0,5 Hz)2

K1 = 54 J

K2 = 2 (2,8 kg m2 ) 3,142 (2,0 Hz)2

( ) .

4-19

m1 = 0,0200 kg

0 = 200 m/s l = 0,300 m . 1 = 6,00 rad/s . R = 0,500 m , , m2 = 2,00 kg. :

)

) .

)

L1 = m1 0 l

L1 = 0,0200 200 0,300 k

L1 = 1,2 kg m2/s ,

I = m2 R2 = 2,00 0,5002 kg m2 I = 0,500 kg m2

K2 220= J

K I v2 2 222= 2

K I v I v1 1 12

1 12

12

121

2

1

22 2= = = b g

f2 2 0= , Hz

ff

2

1 1

2

= =

11 kg m 0,5 Hz

2,8 kg m

2

2

I f I f1 1 2 2=

I f I f1 1 2 22 2 =

L2 = I1 L2 = (0,500 6,00) kg m2/s

L2 = 3,00 kg m2/s)

= mi r12 = m2 R2 + m1 R2

= (0,500 + 0,0200 0,5002 ) kg m2/s = 0,505 kg m2

A

L = L L1 + L2 = 2

(1,20 + 3,00) kg m2/s = 0,505 kg m2/s 2 2 = 8,32 rad/s

151

4.84 4.85

MA

( ) .

, .

,

,

i) = sin

,

. .

B

A

=

B

A

A B

B

A

152

ii)

,

iii) .

,

.

, .

( ).

.

.

.

, , . , .

F B

= q

B

F = sinq

A B B A

=

B

A

153

() . r

p

,

= r p

p = 0, r

F =

F 0.

()

L ,

. () . ( 4.56) . (4.56) (). z,

Li mi .

z.

A z, L, z .

Li = ri

pi

Li = ri mi i

L L L 1

= + +2 .. .

d

d

L

t

=

dL

dt

=

dL

dt p r F

= +

d

d

d

d

L

t

r

tp r

d p

dt

= +

d

d

d

d(

L

t tr p

= )

L

F

.

154

Li z = Li cos = Li sin = ri mi i sin =

= Ri mi i = Ri mi Ri = mi Ri2

+ = /2 ri sin = Ri

L z = m1 R12 + m2 R2

2 + ... = (m1 R12 + m2 R2

2 + ...)

L , z = I ()

z. , , .

()

()

() ()

(4.56) z , z.

, z = 0. d

d

t= 0

I

t z

d

d=

d

d

zo z

L

t =

d

d

L

t

=

d

d

d

d

zL

tI

t=

.

.

155

(. V),

0. ,

, . , V ( ), z, o o .

L = I

L = .

= 0

. . , . , . , . , , , , .

d

dL

t

= 0

dL

dt

0

V

, . .

V

H .

156

x

t=

d

d =

d

d

x

t

=d

d

ta

t=

d

d

I m R= i i2

F =

r

F

p = m L =

r p

2 2 = I dd

=L

tF ma= d

dF

P

t=

W = W F x=

P = P F=

x2W = 1

2

1

2

2I I xW M M= 1

2

1

22 2

= 0 p = p

= 0 L = L 1 1 = 2 2

MHXA 157

, ,

ycm y. xOz U = 0

, , , , .

,

,

cm , , p -,

d . ( Steiner)

, , .

F, ,,

=

r

F

r

F. .

= F l

, ,

, , ,

I=

K I M= +12

1

22

cm cm2

I I M dp cm= + 2

K = I 1

22

I m r= i i 2

=

t

t=

U = Mg ycm

zm z

mcm

i i

i

=

ym y

mcm

i i

i=

xm x

mcm

i i

i

=

drasthriothtesA N A

158 MHXANIKH

1. MAXWELL

Maxwell . . . , . . ;

.

F = 0 = 0

, ,

p

r

,

L =r p

,

, , .

L =I

.

: ( ) , , o.

dd

L=t

xW I= 1

22 1

22

dW d=

drasthriothtes

1

. () . . . . . . , () . . . . . . . () . . . . . . . () . . . . . ..

2

, . , .()

,

() , .

() , .

() , , .

3

. , RA = 2,0 cm RB = 6,0 cm. A / B () 3 () 1 () 1 /3 () 1 /9

4

.

R1 = 10 cm 1 R 2 = 20 cm 2. 1 /2 :() 1 /4 () 1 / 2 () 1 () 2

5

;

6

. () () ()

. .

7

d1 d2 d1 > d2 . 1 2 , () 1 > 2() 1 < 2()

. .

8

.

MHXA 159

, . () () () .

9

, , .

() ()

() ()

10

. , :

() . . . . . . () . . . . . . () . . . . . ..

11

, ()

:

()

,

,

() :

( ) .

12

, . m R,

. ;

() .

() .

() .

13

, . . .() ,

, .()

, .()

.()

.()

.

14

;

15

; .

2

52m R

K I= 12

2K I = +12

1

22 2

cm cm

K I= 12

2

K I = +12

1

22 2

cm cm

K I = +12

1

22 2

cm cm

K I m= +12

1

22 2

cm cm

K

K1

2

2=KK

1

2

1=

K

K1

2

1

4=K

K1

2

1

2=

160 MHXANIKH

16

z.

. ; .

17

F1,

F2,

F3, . (), () ()

. () ().

()

F1,

F2 , .()

F1,

F3 , , .

()

F2,

F3 , , .

()

F1

F3 .

18

. , () . . . . . . () . . . . . ..

19

.

mI a x

F

20

, , = 5,0 rad/s. , , I = 50 kg . m2. , 10 s:

() 100 N.m () 25 N.m

() 1 N.m () 250 N.m

21

. F ;() ()() ()() .

22

. , . .

MHXA 161

23

.

() ()

()

;()

.

() , , , ;

24

;

25

;

26

. 2, ;() () () ()

27

, ;

28

: , () . . . . . . . (). . . . . . .

29

kg m/s kg m2

J kg m2/s m

30

, . ;

31

, . ;

()

() .

()

32

,

. .

162 MHXANIKH

() ()

()

, ()

33

O , ,

, . ;()

()

()

()

,

34

. ( ) ,

. () () () .

35

m . , .

r. . () () () ()

MHXA 163

-

1.

xOy. , , m1 = 1,0 kg, m2 = 2,0 kg,m3 = 1,0 kg. (xA = 2 cm, yA = 3 cm) (xB = 1 cm, yB = 2 cm) (x = 4 cm, y = 2 cm) . .

2

100 cm .

. . : 2,70 g/cm3, : 7,80 g/cm3

3

9 cm . . .

4

, . 6,4 103 km

5

1,5cm.

6

R = 0,50 m ( ) = 6,0 rad/s. .

7

, , = 3,0 m mA = 1,0 kg, mB = 2,0 kg, m = 3,0 kg. , ()

()

()

.

8

m = 1,0 kg R = 0,10 m = 4,0 rad/s, , . .

9

m l = 2,0 m . . ( ). :

, g = 10 m/s 2

10

m = 2,0 kg R .

m1 = 6,0 kg m2 = 3,0 kg . . m1 6,0 m.

, g = 10 m/s2. -

.

11

6,0 km/h 42 km/h 5,0 s. 40 cm, ; .

12

r = 1,0 cm

R = 8,0 cm.

.

, ,

g = 10 m/s2. .

13

R = 0,2 m ().

I m r= 25

2

1

22m R

I m l= 13

2

164 MHXANIKH

MHXA 165

. F = 10 N(), . , = 2 10 2 kg.m2 (). 2 s.

14

m R.

. . :()

.()

h = 0,30 m. , ,

. g = 10 m/s2

15

, R.

( ) ,

.

m = 2 M . ( ). g = 10 m/s2

16

m1 = 1,0 kg m2 = 2,0 kg , M = 2,0 kg.

.

m1 . ( ). g = 10 m/s2

17

R m, . (.

1

22M R

1

22M R

I m R= 12

2

). F. , . (W = F S, S ) . = 1/2 mR2.

18

. , . , . l = 1,0 m, m = 4,0 kg, g = 10 m/s2 . ,

19

. , . , = mR 2/ 2.

20

A , , 0 . , . :() ,

= 5 0 / 7.

() 12 0

2 / 49 g, , , g . = (2/5) mR 2, R .

21

, 100 ( )

F, 60 , . N F.

22

d, 1, 2. F1 1 F2 2, F1 / F2 .

23

100 cm = 20

I m= 13

2l

166 MHXANIKH

MHXA 167

F, .

F.

24

4,0 m 900 , () = 2,5 m. 7560 .

x .

25

5m 200 , . 3,0m 495. . 70 , , ;

26

, = 40,0 , . 100,0 . .

27

1m ()

1200 0,20 m.

F 450 , . .

28

105 cm. , , 40 . , , 30 . .

29

. 30 cm, 10 cm. .

30

30

12 cm, , . .

31

Alfa Romeo 156. 1.6l 118 (hp) 6200 145 N.m 4190 . 1 hp = 746 W, () 6200 .() 4190 .

32

800 kg 1,0 m. t0 = 0, 180 , , , . 5 . , () ()

.()

t0 = 0 t1 = 3 s.

2 = 10

33

= 12 .m , , , = 2,5 kg.m2. t0 = 0. :() ,

t1 = 4,50 s.() t1 = 4,50 s.

34

, , 2,0 Hz. , , I = 2,0 10-4 kg.m2. m = 20 g 0,10 m . ;

35

. , . . 6,0rad/s. . . , 0,50 kg.m2, ( ) 3,0 kg.m2.

36

, . (r) (r) .

, :

37

0

60 . h = R R .

0 ,

. R g0 .

r

r

=

168 MHXANIKH

MHXA 169

38

l = 1,2 m M = 2,0 kg. . m = 0,020 kg 0

d = 0,90 m . 0 /2.

90. :()

() = 1/3 Ml2 g = 10 m/s2.

39

m1 = m2 = 60 kg 6,0 m/s . , .

. 22 kg m2 0,60 m.

40

m1 = m2 = m (). m1 R0 O

0 R0

.

m2 , R 0 .

41

(OA) h = 1,8 . , O

. . g = 10 m/s2 1/3 ml2, m .

gR0 02=

170 MHXANIKH

4.3 EI

. .

, (), () , () , . O ,

, , . O , , , .

171

4.86

.

(), ,.. ( , , ), , . , , ( . 4.87), . , . 4.88 .

172 MHXANIKH

4.88

( ) .

4.87

() ( ).

( 12 , 21)

12 = 21

12 = 12 ti = 21 ti = 21 ( 4.89).

12 + 21 = 0

12 = 1 1

21 = 2 2

1 + 2 = 1 + 2

, , : , , , t, , , (, ..) . , , , .

p

p

p

p

p

p

I

p

p

I

I

I

I

F

F

I

F

F

F

F

173

4.89

12 21 .

F

F

, .

, , . . :

) K1 + K2 = 1 + 2 , , , . . , . () , ( ). , , , , . , . , .

) K1 + K2 > K1 + 2 ( K1 + K2 < K1 + 2 ) . , .

, . . , ., . . . .

, , , . () , 4.90. , , . , , , . , , . , , ,

174 MHXANIKH

4.90

.

. , . () . . .

. m1 m2

, ,

1

2. ( )

( )

1

2 (. 4.91).

,

m1 1 + m2 2 = m1 1 + m2 2 (4.58)

(4.59)

(4.58) (4.59)

m1 (1 1) = m2 (2 2 ) (4.60)

m1 (12 12 ) = m2 ( 22 22) (4.61)

1 1 2 2 (4.60) (4.61)

1 + 1 = 2 + 2 (4.62)

1 2 = (1 2) (4.63)

(4.62) (4.58)

(4.64)

(4.65)

: (4.63) (

, ) , , , , .

(e) , , ,

=+

+ +

m

m m

m m

m m2

1

1 21

2 1

1 22

2

=+

++

m m

m m

m

m m1

1 2

1 21

2

1 22

2

1

2

1

2

1

2

1

21 1

22 2

21 1

22 2

2m m m m + = +

175

4.91

( ).

(4.66)

1 , 2, 1 2 . ,

100 %.

: , e, ,

0 e 1. : .

.

.

. ) m1 = m2 = m.

(4.64) (4.65) -

.

) m2 : 2 = 0 2 = 0 (4.64) (4.65)

(4.67)

(4.68)

1) () m1 = m2 = m

m1

.

(1) , . ( 107 m/s) 103 m/s. 235 238U, 235U .

2 = 1

1 = 0

=+

m

m m2

1

1 21

2

=+

m m

m m1

1 2

1 21

2 = 11 = 2

e = 0

0 < e < 1

e = 1

e =

=

1 2

1 2

1

e=

=

x

x

1 2

1 2

176 MHXANIKH

(), . , , (. ).

2 ) , m2 >> m1

(4.67) (4.68) :

, .

2 ) , m2 > m2 3) m1 > m2 .

(1) m12

0 +

0

1 02b g,

m

m

m

m

=

+FHG

IKJ

4

1

2

1

2

1

2

m

m2

1

0

= 1

m m

m m=

+

4 1 2

1 22b g

m m

m m=

+FHG

IKJ1

1 2

1 2

2

=+

m m

m m1

1 2

1 21

=

=

=

=

1 11

1

1

12 1 1

2

12 1 1

21

2

12

1 1 1m

m

2 211 1

2 01 1

177

4.92

3) m1 >m2) 1 1 (m1

M, m . :

y

1 = 3 2 = m

.

, ( ) . () , , .

, , , , , .. (). , , , ,

= 9h

1

23

2m m g Hb g =

m

M 0

=+

=

+

m

M m

m

Mm

M

23

1 3

1

=+

=

+

m

M m

m

Mm

M

13

3

1

+ = = +

m

1 2

1 2

b g b g

+ =

=

m

m

1 2

21

22

2 2

b g b ge j e j

+ = +

+ = +

UV|W|

m m

m m

1 2

2 21

22

21

2

1

2

1

2

1

2

179

. , . .

. ( )

m1 m2 1 2 1 2 ( 4.94).

, m11 + m22 = m11 + m22

() .

. ( )

m1 + m2

( . 4.95).

m11 + m22 = (m1 + m2)

.

4-22

m1 m2 (). .

m11 + m22 = (m1 + m2)

m m

m m=

++

1 1 2 2

1 2

1

2

1

2

1

21 1

22 2

21 2m m m m E+ = +b g 2 +

m m

m m= +

+1 1 2 2

1 2

1

2

1

2

1

2

1

21 1

22 2

21 1

22 2

2m m m m E+ = + +

180 MHXANIKH

4.94

.

4.95

.

1 2 =

:

, .

4-23

m1 = 1200 kg m2 = 1500 kg . , . , 16 m 30 . 100 km/h;

g = 10 m/s2 = 0,80.

1 1 1

1 2 m m= +

FHG

IKJ

m m

m m1 2

1 2+=

x2E

m m

m m=

+1 2

1 22b g

Em m

m m =

+1 2

1 21 2

2

2b g b g

E m m m mm m

m m= + +

+

+

1

2

1

2

1

21 1

22 2

21 2

1 1 2 22

1 22

b g b gb g

1

2

1

2

1

21 1

22 2

21 2

2m m m m E+ = + +b g

181

4.96

= 16 m/s (1)

x, y

p(x) = p(x) m11 = (m1 + m2) cos 30o (2)

p(y) = p(y) m22 = (m1 + m2) sin 30o (3)

(2)

(1)

1 112 km/h

1 52 km/h

.

, , . ( ) , x y.

(.4.97) ( m1 m2) m2 . b ( . 4.97) ( b = 0 ). , m1 x ( ) m2 x ().

1

2700 163

21200

= m

s

m m

m1

1 2

1

=+b g cos30 o

= 2 0 8 10 16, m s g s= 2

01

21 2

21 2 + = +m m m m g sb g b g

182 MHXANIKH

x y ,

x

p (x) = p (x)

0 = m22 sin m1 1 sin (4.69)

y

p (y) = p (y)

0 = m2 2 sin m11 sin (4.70)

(4.71)

:) ,

, (4.71).) (m1, m2

1),

(1, 2, ) , . (..).

4-24

, , .

1

2

1

2

1

21 1

21 1

22 2

2m m m = +

183

4.98

.

4.97

() () .

p =

p

p1 =

p 1 +

p 2 (1)

1 + 0 = 1 + 2

, p 1 p2 .

, , : , , .

. . , . , ( , ). , ()

( ) , . : , , . . , : . , , , .

, , ( = .) , . ;

F m Fd p

dt

= =

p p p12

12

22= +

p

m

p

m

p

m12

12

22

2 2 2=

+

184 MHXANIKH

= . (

= 0)

. .. , () 4,4 10 -3 m/s2 () , 3,37 10-2 m/s2. . , , .

, ,

p

. , , , .. , ..

, , ,.. , , , . , . , , , . , . , , , , , ( )

. = m

, .

, 1 2 , . ( ). , , . ( ) , ( )

, F

F

F

F

F m Fd p

dt

= =

185

, = m , .

. : , .

, ,

, ,

0 ,

.

=

+ 0

, = 0.

= m

= m + m 0

(4.72)

, ,

,

m 0 () , . DAlembert . . , .

, , , . .

1 :

~

,

m ( . 4.99, 4.100). ~ , ,

. ~ ~ , . , .

F

F m a m a

= 0

F

F

F

186 MHXANIKH

. , , , , .

2

~ m . , , , 4.101, 4.102. ( ~), .

187

() ,

,

. 2 , ,

mg .

, .

mg . ~

, , . ~ (),

,

= m

mg , :

( )

(~) . , . , , . , , :

mg,

m , .

~ (~), -

mg , m :

( ~)

() .

m g T m a + = 0

T

T

tan a

g=

F m g T m a

= + =

T

F

T

F

F

T

4.99 4.100

188 MHXANIKH

()

,

,

r . ( ).

:

( )

(~)

~, . , , , ., , m , . ~ ~,

m , :

( ~)

).

T m T m

r = =

0 02

T

T m a T m

r

= = 2

T

r =

2

4.101 4.102

, , , , , .

1) . (.. ). , 6 104 , , 4 105 .

2) (.. Houston ..) . , 15 m , 24 , , , 10 .

M .

~ (.. , , , ..). (xyz) . (x, y, z) t. , (. 4.103).

( ). , , . ~. , , x ~x~ ( . 4.104) y ~y~ Oz ~z~. t~ = t = 0, , ~ , u () ~ , (~ ) =

ut. ,

, , (4.104), , (x~, y~, z~, t~) (x, y, z, t).

(4.73)

. , , t = t~. , .

(x , y , z )

. ~ u (u, 0,

0). ~

. t (x, y, z) t + t (x + x, y + y,z + z). ~ ( ) t ~ = t (x~ = x ut, y~ = y, z~ = z) t ~ + t ~ = t + t [x ~ + x ~ = x + x u (t + t),y~ + y ~ = y + y, z~ + z~ = z + z].

x~ = x uty~ = yz~ = z t~ = t

~x = x u

=

x

t

x

tu

x = x ut, y = y, z = z, t = t

189

4.103

.

4.104

P. , . u.

t 0

(4.74)

() .

.

u x .

(4.75)

, ~ u mu.

A .E ,

( ) t, t + t (= t ~ + t ~ )

x + x = x + x uxy + y = y + y uyz + z = z + z uz

t + t = t + t

(4.76)

~

.

. , , ,

=

= = =

t

tz z

zz

= = =

t

ty y

yy

= = =

t

tx x

xx

= = =

u

u

u

x x

y y y

z z z

p =

p + m

u

p =

p m u

=

~ +

u

~ =

u

=

=

=

UV|

W|

u

x x

y y

z z

z

= = =

z

t

tz z

= = =y

t

y

ty y

190 MHXANIKH

( ),

~ = . , m ~ = ~ m

= (o m

, ). .

, = m

m = m

~ = ~.

(4.77)

4-25

25 C 346 m/s. , 25 m/s

) ~ .) ~ .) ~ .) ~ ,

.

.

~ xy z ,

. ~x~y~z~ u = 25 m/s, = 346 m/s ~ ~x~y~z~, ~.

~ =

u

Fp p

F

= = =

lim

lim

t tt t0 0

F

F

F

lim

t t

=

0

pFlim

t t

=0

pF

F

F

F

F

191

4.105 4.106

1) () ~ = u = 321 m/s2) () ~ = ( u) = 371 m/s,

x.3) ()

u

~ =

u

~

x~ 94.

4) , () ~ z~,

~

u

~ =

u

=

~ +

u

2 = 2 u2

, , x

86

, , , . , , ( ), . , Coulomb. . .

tan

u= = 13 8,

= u2 2 345 ms

tan

u= = 13 8,

= + = u2 2 347 ms

192 MHXANIKH

4.108 4.107

, (CM), E

, N () m1, m2, ... ~ , ,

, , (4.78)

, (4.78)

, c ,

~

(4.79)

= mi (4.79)

(4.80)

mii = M c

.

(4.81)

, , . , .

, .

(4.81)

F

=c( )

V

t

Fp

=

t

p V c

=

V

V

=c i im

M

V

=c ii

i

m

m

Vm

mz

zc

i i

i

=V

m

my

yc

i i

i

=

Vm

mx

xc

i i

i

=V

X

t

mx

tm

xcc

ii

i

= =

V

Zm z

mc

i i

i

=Y m y

mc

i i

i

=X m x

mc

i i

i

=

193

(4.82)

.

,

F. ( , - , ).

(4.82) : ( ), . , , , , .

, () , , .. ,

, c . , .

, 1,

2, ... ,

(.. ), , ( ). , ,

p = .

(4.81) ,

(4.83)

. (

), .

( c = 0) CM. , , .

, , . .

V

Vp

=c M

V

V

=cc

t

F a

= cM

FV

=c

M

t

194 MHXANIKH

( ) m1,m2, ...,

1,

2, ..., (

) ,

(4.84)

(4.85)

(4.81) ( ),

, c , ,

(!)

, , !

.

(4.84) ,

c . .

1 =

1 c ,

2 =

2 c , ... ,

=

c (4.86)

1 =

1 + c ,

2 =

2 + c , ... ,

=

+ c (4.87)

(4.87) (4.85)

(4.88)

mi i =

p i =

p = 0, .

K m M V K M V= + = +1

2

1

2

1

22 i i c2 . c2

K m m m V V m = + = + + 1

2

1

2

1

22 2 i i c i i 2 i c c i i( ) ( ) V

V

V

V

V

V

V

V

1

22M Vc

K M V= 12

2c

V

K m = 12

2 i i

K m m = + +12

1

21 1

22 2

2 . ..

195

, , (.. ) :

. -

c ( ,

, c). , . H .

, (). .

1

.

(4.89)

2

(.. ), , ,

(4.90)

-

3 . .. . ( . 4.109).

,

F F

= 121

21

2m m t t

1 2

F

=212m

t

2F

=121m

t

1

F m

t

=21 2

2F

=12 1mt

1

K K M V= +c c21

2K m M V= +1

2

1

22 i i c

2

p c = mi

i = 0

K K M V= + c21

2

V

V

1

22M Vc

196 MHXANIKH

4.109

m1 , m2 .

F12 =

F21

~ 1

2 =

12 m1 m2

12 m1 m2.

, ,

(4.91)

(4.92)

: , ,

, , . , , , , ,

F12 = 12

.~

.

:1)

.. m2 >> m1 , .

( )

. .

2) m1 = m2

m m

m m

m=+

=2

m

m1

2

0m m

m m

mm

m

m=+

=+

1 21 2

1

1

2

1

1

F12 = 12

m m

m m=

+1 2

1 2

1 1 1

1 2 m m= +

12

=t

12

F

+FHG

IKJ =

12

1 2

1 1

m m t

( )

1 2

197

, mp mn.

4-26

~ m1 , 1

m2 = 3m1 . , (. . 4.110).

) ( ).

:

1) m1 1 m2.

2) m2 3) m2

(4.79)

:1) m1

m2 .

= 1 11

2 =

+

m m

m m1

1 2

1 21

V c =1

41V

m

m mc = +

1 1

1 2

198 MHXANIKH

4.110

() m1 3 m, .

4.111

() m 3 m, .

2) m2

m1.3) m2 ,

.

) .

: :

1) m1 m2

2) m2

3) . :

pc

pc = 0

. pc~ = 0 m1 m2 . , 4.111 . , , . , , , . .

:)

(1)K m L = +1

201 1

2

p m m m m c = = 1 1 2 1 1 1 1 13

4

1

4

3

43

1

4

V 2 101

4c c= =

V 1 1 13

4c c= =

= =V Vc c14

= +

Vm m

mc

11

2 1 11

2 1

1

3

4

e j

= + +

Vm m

m mc

1 1 2 2

1 2

= 2 11

2 =

+

m

m m2

2

1 21

2

199

)

(2)

(3)

(1) (3) , , . (1) (2)

~ (

) .

: (2) Vc

,

, , , , , () .

K c =1

212

Km m

m mc = +

1

21 2

1 212

Vm

m mc = +

1 1

1 2

1

24 1

2m Vc

K m c =3

81 1

2

K K m VL c c2= + 1

24 1

K K m m m VL c c2 = = +1

8

1

21 1

21 2b g

K m c =3

81 1

2

K m m

c =FHGIKJ +

FHGIKJ

1

2

3

4

1

2 41 1

2

21

2

K m m Vc c2= FHG

IKJ +

1

2

1

4

1

21 1 1

2

2

K m V m Vc c c= + 1

2

1

201 1

22

2b g b g

200 MHXANIKH

4-27

(), , . , .

:

. ;

DOPPLER

( ) , (), , , . ~ ( ) ( ) , , . , - Doppler ( Doppler).

Doppler , x~x,

u

u .

t = 0 0 t t , (0 t ) u t.

, t

0 t

K m m

m mc = = +

1

2

1

212 1 2

1 212

201

4.112

m 3 m , .

4.113

m 3 m , .

(t ) = t + u t = ( + u) t (4.93)

f , t = f t (4.94)

, (t ).

, (4.93) (4.94)

(4.95)

(t )

(4.96)

() xx~

u ,

() = u( 4.114).

( (+) u

()

, () xt t x).

() ) x~t

(4.95) (4.97)

(4.98) =+

f u

uf

=

f

x

=

u

f

=+

u

f =

+

u t

f t

b g

=

( )t

202 MHXANIKH

4.114

, u u (t )

(t ) (). (t ) > t < , .

) t x

(4.96) (4.97)

(4.99)

(4.98) (4.99)

(4.100)

, f ~, (),

u, ,

u ,

, (4.100).

(. ),

,

: = [(+)

() ]. , , .

. ~ u, ,

(4.101)

.

: 1) u > , (4.101) f ~~ < 0, .

2) , , (), , ().

. ~ , u, , :

= f f u

=+

f f u

=

f u

uf

=

f u

uf

=

f

x

203

(4.102)

: 1) u > , (4.102) f ~ < 0, , .

2) , , (), , ().

. ~ . , . . , (4.100), u u ,

, (. 4.115). ,

(4.103)

. ~ .

. , (4.100) .

. , ,

, , , . , , .

=++

f f u

u

cos

cos

=+

f f

u

=

f f

u

204 MHXANIKH

4.115

, u

u .

DOPPLER

Doppler , . . , . , ..

1) radar . radar ( f = 9 GHz).

2) . .

3) , , , .

4) Doppler . ~ , . , .

: Doppler , , (.). , .

4-28

. , t. 600 Hz 25 s. :

) .) t .

, 340 m/s.

)

u

=

f f

u

205

4.116

~ u = 40 m/s u = 144 km/h.) ~ ,

, , .

, , .

4-29

K 2,00 z , , , . 160 z. 1500 m/s. .

~ . f . ~ , ,

, ,

f = f ~~ f .

A

, , = 6 10 2 m/s.

f

f f

=+2

=+

f f

=

f f

=+

f f

fN

t

fN

t

tf t

ft

=

=

UV|

W|= = s s680 25

60028

um

s

m

s=

=

340 680 600

68040

b g

u f f

f =

b g

206 MHXANIKH

207

( ) , , .

() , , . ( -.)

p1 +

p2 =

p 1 +

p 2

, , , () , .

:

: .

: - .

( ) .

,

e = 1: 0 < e < 1: e = 0:

()

: 1 2:

-.

.

x = x ut, y = y, z = z, t = t

( )

1 2 -.

p

= c

V

Ci imV

M

=

= +

+++

m e m

m m

e m

m m2

2 1

1 22

1

1 22

1b g

= +

+++

m e m

m m

e m

m m1

1 2

1 21

2

1 21

1b g

e

= =

x

x

1 2

1 2

drasthriothtesA N A

208 MHXANIKH

1H - ( ), , .

2 , .. (L), , ,

() , ,

Doppler ( ) , , , . Doppler ( Doppler). f ,

: u u

=

f f u

u

m m

m m=

+1 2

1 2

K K MVL c c= +1

22

K m MVL i i c= +12

1

2

2 2

= =

p c im i 0

() , ( ). , . . () t = 0,02 s . , .

drasthriothtes

209

() m1 = 201,1 g m2 = 85,4 g. t = 0,033 s.

. .

. , . . m1

m2 . .

h g t s t= =12

2

210 MHXANIKH

1

K ()

,()

,()

,

() .

2

4.86, ,

- ( ) .

3

, , () .() .() .() .

4

, , m 1

m 2 . , , m2 / m1 ) 1 ) >> 1 ) 3 ) > 1()

7

2 3 , , m .

1 , m, 0 , . , :() m > M() m M

()

()

()

211

()

()

() m = M

8

m1 m2 m2 m1 1 = 10 m/s()

m1 m1 = m2

() :

() m1 [20, +10] m/s

() m2 10 m/s

9

. 3,0 s 0,45 m, 3,0 s 0,36 m()

() ()

().

10

() o

.()

.()

.()

.

11

. ()

()

()

()

12

. , :() 60

() 120

() 90

() 180

() 0

13

, :

() () 0 < e < 1() () e = 0() () e > 1

() e = 1

14

N , , m1 m2 1 2, , .() m1 = m2 2 = 0 () 1 = 0 2 = 1() m1 >> m2 1 = 0 () 1 1 2 21() m1 >> m2 2 = 0 () 1 2 2 0() m1

212 MHXANIKH

, m2 , , , , () 3 m1 = m2 () m2

() m1 = m2 () m2

() m1 >> m2 () m2

() m1

213

() .()

.() .

22

. . () ()

()

23

, , . . () .() .() .

24

. . , :()

.()

.()

.()

.

25

. , , , , . ;() ;() ;()

.

26

, . :()

.() ().() ().

27

, . , , , , .()

.() .() .

28

, , . , .

, ,

:()

.()

.()

. . . =1,29 kg/m

3, =0,178 kg/m3,

g=9,81 m/s.

29

, = 30, :() 5,7 m/s2 () 8,7 m/s2 () 5,0 m/s2

214 MHXANIKH

30

, . ;

31

() () ( ) , .

(blackout), ( ) ( ). () .() .

32

8,00 m , . . - , ,:() 8,00 m

.() 7,70 m

.()

, .

3500 kg 140 kg. . - .

33

m1 , m2 ,

1

2

(.. ), :

, Vc (cm) 1 ,2 m1 m2 cm., , :

x , .

34

, m1 , m2 .()

( , , );

() ;

35

, , , . (). ;

36

e, , e2,:

= e2 .

37

(), , , ,

K MV = +12

1

22 2

c x

K MV m m c = + + = + 1

2

1

2

1

22

1 12

2 22

c

215

. () .() .() .

38

m1 , 1 , m2 .()

;()

;()

; (), (), () -

Vc;() ,

, ;

39

, , m1 m2 1 2 .()

, ,

() , , ;

40

. ,

f0 , t0 . ;

41

( -) , 1500 Hz. , , . .() () 1500 Hz

: () >1500 Hz(B) H ()

216 MHXANIKH

1

m1 = 0,30 kg m2 = 0,50 kg 1 = 20 m/s 2 = 10 m/s, . , . ) ;) ; .

2

3m 1 = 10 m/s 2 m m, .()

, .

() 2m 0,20 kg,

-

43

, , . ()

.()

.()

.()

.

44

f. . , , () () f .() f .()

.

45

, ,

() .()

.() .

46

, . , 1000 Hz, :() .() f > 1000 Hz.() f > 1000 z.() f < 1000 z.() f < 1000 Hz.

47

() () ( ). , :() () ()

217

0,010 s.

3

m R = 0,050 m.

9m . . (g = 10 m/s2).

4

, (). : m = 1,0 kg , k = 50 N/m, = 2,0 m/s.

5

To . , m = 0,5 kg

, -

= 2,0 kg, , = 60 . k = 200 N/m , (g = 10 m/s2).

6

h . e :()

.()

.() ,

.

7

. 36 % , ;

8

4,00 m. 2,25 m. ;

9

m . 60 . 30 . ;

10

, m1 , m2 , ... m

= 150 ms

218 MHXANIKH

. ( ) 1, . , ;

11

m1 , m2 , 0,10 kg 0,30 kg m1 h = 0,20 m,

, :

() e = 1,0 , () e = 0,50 , () e = 0,0

12

, 0,80 kg, . , ( ).

0 = 0,50 m/s . 0,30 m/s , :() .()

( , ).

13

. , ,

m, ( >> m).

14

, , .

0 = 7,0 m/s (), . N , .

15

. 5,0 m/s, .

( ) ; .

16

m

. . .

17

~ m1 = 900 kg m2 = 1200 kg.

3m

s

219

40 . . , , 14,0 m/s. 17,4 m 0,850. , 60 km/h. : sin 40o 0,642.

18

m R, ,

, . (, ) , , b (). .

19

: ) - ) - . , .: me = 9,1 10-31 kg, mp = 1,672 10-27 kg,mn = 1,674 10 -27 kg.

20

~ m1 m2 , 1 2, . , .

21

, ,

h , . , - , :

m2 .

22

m . K = Q.()

.() , ,

.

23

, c , , :

.

~ : .

24

45 kg, , 640 m/s. 32 kg 13 kg. 450 m/s 1500 m/s.

m2

2

2c

c=

m1

1

2c

c=

2Q

m

m Q

2

m g h

= 2

2

m

g h= 2 2

220 MHXANIKH

, . ;

25

1500 kg 3500 kg . 80 km/h 50 km/h. . :() ,

, .

() ;

26

m1 m2 1 2 , , :()

:

.

() - , K,

: ,

e: :

.()

i) ii) ().

27

~

, -

. 43,2 km/h. :()

, .

() ,

, .

28

, .

29

, , 4,0 m/s 30, . , , , 2,0 m/s , (sin19o 0,327).

30

, 3,0 m/s . , 80,0 m, . , 5,0 m/s , , . .

31

( ) .

240 3 m

12 3m

s

e2 x2 = 1

21e j

=+

+FHG

IKJ

m

m

m

m2

1

2

21

2

1

11

eeb g

=+

+FHG

IKJ

m

m

m

m1

2

1

12

1

2

11

eeb g

221

.

32

~ 15 kg , .

, , 18 .() .()

;()

, , , .

33

-

, . ~ . , . . 0,40 4,0 m, , .

34

-

.()

;() 44,0 kg

6,00 kg, ;

() , , .

g = 10,0 m/s2 ,

35

m = 2m l = 10,0 m, . , . :

3 1 73= ,

222 MHXANIKH

() ( )() ,

h = 5,0 m .

36

1 ,

1, m. 1 2 , , 2 . . , , 2 , 2 , 1 , 1.

37

, ,

, . , .

38

() ;

() ~ = 600 : i) 72 km/h, 10min , ii) 3,14 s F2 = 600 , iii) 10 s F3 = 120 iv) 1,57 s F4 = 8000 v) 5 s F5 , . . .

() F5 ;

39

680 m . , .() .() ~ ,

, , 5 :4. ;

340 m/s.

40

~ 2,00 rad/s R = 25,0 m. , . 850 Hz 340 m/s.

223

41

~ 40 m/s.~ 30,0 m/s . 400 Hz 340 m/s, ;

42

360 km/h 900 m. , 1 2 , 1350 m, , . f f~1, f~2 , f1 / f f2 / f. 340 m/s.

43

800 Hz. 340 m/s. , , , 20 000 Hz;

44

500 Hz, , x = 10,0 m/s2.()

33,0 s .

() . 330 m/s.

45

435 Hz , 80 m . , : ()2,0 s () 2,0 s . 340 m/s.

46

() = 589 m , 927 Hz, (), 72,0 km/h. 900 Hz.() () () .

47

1000 m, 1500 m . , , 360 Hz, , , 340 Hz. 355 Hz. ( = 340 m/s).

48

~ , 16 Hz 16 kHz, . ~E 15,5 Hz. , , . 340 m/s.

224 MHXANIKH

4.4

1865 Maxwell, . , , , . , , () , , . . , .

. . c. , c. K , Maxwell . :

1. Maxwell .

2. () , , Maxwell .

3. Maxwell , , .

1905 3. . . Lorentz , , ( Maxwell) . Lorentz .

1899 1900 1904 Poincare ( Michelson Morley)

X 225

, . . () .

1905 , :

(1)

(2) .

. Lorentz , . . .

., .

Lorentz , . , , ( ) .

, .

(.. , , ..) () .

( ) , . ,

226 MHXANIKH

() . () . . , , . . . . . , , 4.117. () , , u = ux x. () (x, y, z, t) ~ ( )

227

4.117

P, x, y, z, x , y , z . ux = u Ox , Ox . t = t = 0.

(x , y , z , t ). ~ 4.117

. , . . ( , ). ~ . , ( ) .

t1 (x1 , 0, 0) t2 (x2 , 0, 0). t 1 (x 1 , 0, 0)