107

108

4-1

, . . , , .

, , - , () , , , .

-, . - .

. .

4-2

, .

. - . - .

. 4.2 . . , . (.4.2).

. 4.1 - . .

. 4.2 () - . . () . .

109

. - .

- , . - .

, r = , r .

- .

4.3 t1 1 t2=t1+dt - 2=1+d

t, .

dtd =

d 1 rad/s2.

: (- ) a .

. .. , . , ( ) . . .

4.4 . , , , cm(. 4.4) , (. 4.4). - ,

. 4.3 () d . .

110

. cm (. 4.4).

. -

. 4.2

. . . . .

(cm) , - .

. .. , .

. 4.4 () () (). - (cm) - ().

.

. 4.1 . -.

.

. 4.2 - . - .

.

111

. , . , , .

H

(. 4.5). . , dt ds, ds, d. H cm

dtds

cm = (4.1)

Rdsd = dRds =

(4.1) dtdRcm

= , =

dtd

,

Rcm =

(. 4.6). , . . cm,

Rcm = o Rdtd

dtd cm

=

Racm = cma .4-3

. 4.3 .

.

. 4.5 ds, .

. 4.6 :Racm =

112

. . (), .

.

)

zz . F .

F, l ().

Fl=

. 1 m.

. .

F -, (. 4.9)

.

, , . - .

. 4.7 H - . F .

. 4.8 F - .

. 4.9 F Fx l

113

4.10 F1 F2 . T .

221121 lFlF =+=

)

' , ( - ). - , - ( ) .

. , . ( ) - . .

.

F

Fl=

.

- F1 F2 . - . d, (. 4.12) x1 F1 x2 F2,

dFxxFxFxF 12112211 )( =+=+=

dF 1=

. , .

. 4.10 F1 F2. H - .

. 4.11 .

4.12 F1 F2 . .

114

4.1

4.13 , R1=4 cm R2=3 cm, xx. xx . , F1=6 N F2=10 . .

. 4.13:

F1 111 RF= . F2 222 RF= .

mNRFRF ==+= 06,0221121

.

4.2

4.14 . x1=5 cm x2=8 cm. F1=50 N F2=40 N. F2 = 30 . ;

:. 4.14

F1 . NmxF 5,2111 ==

F2 F2x F2y F2x=F2 30 F2y=F2 30. F2x ( F2x ), F2y

NmxF 6,1 30 222 == .

Nm9,021 =+=

H , .

115

4-4

.

. , , .

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

-

0=F 00

==

y

x

FF

0=

4.3

l= 4m w1=200N, . x= 1m w2=600 N. ;

: (w1), - w2 - 1 2 .

. 4.15

F=0 1+2-w1-w2=0 (4.2) , . . .

0= , 02 212= xwlwlT ,

Nl

xwlwT 25022 21

2 =+

=

(4.2) NT 5501 =

116

4-5

zz(.4.17). m1, m2 . , . m1,m2 , r1, r2

.

....2222

11 ++= rmrmI 1 kg m2.

4.4

, l w=400 N, . . - =30 . .

:. 4.16

.x= 30 y= 30

Fx=0 30=Fx (4.3)

Fy=0 30 + Fy=w (4.4) 0=

02

30 = lwlT (4.5)

xF Fy xT . (4.5)

wT = 30 2 NT 400= (4.6) (4.3) (4.6)

NFx 3200= (4.4) NFy 200=

F NFFF yx 400)()(22=+=

33

==

x

y

FF

30=

. 4.17 To .

117

.

- . .

cmI pI p, d -, , Steiner ().

cmI , , d o - d.

2MdII cmp +=

. 4.18 - d

118

4.5

R, .

:. 4.19

m1, m2 . m1 + m2 + =M

R, R .

2

212

22

12

222

11 ...)(...... Rmm Rm Rm rm rmI ++=++=++= 2MRI =

4.6

, m1 m2,(m1= m2=m), , l. , - ) ) m1;

:

)22

222

222

2

2

1mllmlmlmI =

=

+

=

) 221 0 mllmI =+=

. 4.20

4.7

, - R, , .

: . 4.21

2/2MRIcm = d=R

23

2

22

22 MRMRMRMdII cmp =+=+=

119

4-6

, F m= . . , ' . () ( )

I= (4.7)

, ,

( ) .

(4.7) - . , , , . .

, (4.7) , , , .

. . , - , .

- , , .

. 4.22 , , m .

120

4.8

=40 kg R=40 cm, . , F=6 N. , , . ;

221 MRI = .

. 4.23

:

FR=

I= 221 MRFR =

22 /75,0104040

622 sradmkg

NMR

F=

==

4.9

, R, , , . m. , .

:

. m, mg . 4.24

.

cmmaTmg = (4.8)

( ), F ( ) Mg.

g F . -

I= ITR = (4.9)

121

(4.8) T cmmamgT = (4.9) ImRamgR cm = (4.10) a

Ra = cma Racm = (4.11) (4.10) ImRmgR = 2

2mRImgR+

= (4.12)

(4.11)

2

2

mRImgRacm+

= (4.13)

(4.12) (4.9)

2mRImgR

RI

RIT

+== 2mRI

mgIT+

=

4.10

To - , . , , xx. .

221 mRI = -

g. .

: . 4.25

mg T . :

cmy maF = cmmaTmg = cmmamgT = (4.14) xx

I= ITR = (4.15)

(4.14) (4.15) ImRamgR cm =

221 mRmRamgR cm = Rag cm 2

1= (4.16)

Racm = Racm

=

(4.16) cmcm aag 21

= 3

2gacm =

122

4-7

H . .

, , , .

)

m p - r (. 4.26).

zz

rpL = rmL =

zz . 1kg m2 /s.

)

4.27 - zz . . - , , . , m1, m2 . , . , 1111 rmL = ,

2222 rmL = , . - .

... 222111 ++= rmrmL m1, m2 1, 2 11 r = , 22 r = . . .

( )... ... 222211222211 ++=++= rm rm rm r mL I rm rm =++ ... 222

211

IL = (4.17) - .

. 4.26 m . .

. 4.27 m1m2 -.

. 4.28 - zz . , .

123

2,71040 kg m2/s 5,81033 kg m2/s (=90km/h) 102 kg m2/s - (33 min) 610-3 kg m2/s 1,0510-35 kg m2/s 0,5310-34 kg m2/s

- , . , , .

- , - 0,5310-34 Js.

!21

, ! =1,0510-34 J s ( )

.

)

, - -. L1,L2, .., L

L=L1+L2+

4.17 dt d, - dIdL =

IdtdI

dtdL

==

(4.7)dtdL

= (4.18)

,

Newton.

124

O . -, , , .

. Newton ( - ). . . 4.18

dtdL

= (4.19)

L .

4-8

, . .

,

dtdL

= 0=dtdL

, , .=L -

.

, (), , , . - 24 .

.

ewton

dtdL

= .

, .

.

. 4.29 -

125

, ( ), , . :

2211 II =

- . 4.4 - , . , , - . , - - , , I . , , I , .

. , , , , . , . 4.30 , , .

1,4 2,5 , - pulsars. , , , 15-20 km.

. 4.4 .

. 4.30 - .

126

- - . 3000 . , 25 .

4-9

4.11

4.31, =4 kg. , 1=4rad/s. ; , , , -. ( ) 3,25 kgm2 2,5 kgm2.

. 4.31

1 m, 0,2 m . .

: 1I ,

, .

2211111 25,112 mkgMRIIII

=+=+=

2I , .

2222222 82,22 mkgMRIIII

=+=+=

, . :

2211 II = sradII /161

2

12 ==

127

4.32, , zz, .

,

..., 21 mm

11 r = , 22 r = , (4.20)

,

... 21

21 2

22211 ++= mmK

, (4.20)22

222

112

22

22

12

1 ...) ( 21...

21

21

++=++= rmrmrmrmK

Irmrm =++ ... 2222

11

2

21IK =

. - .

, 4.33 -.

22

21

21

IMK cm +=

M cm .

4-10

- . . .

F R, (. 4.35). d

. 4.32

. 4.33

128

dsFdW = dRds =

dRFdW =

FR . ddW = (4.21)

- ... , 21 dd . , 4.35,

=W

(4.21)

dtd

dtdW

=

dtdW / P dtd / ,

=P

. , , -

21

22 2

121

IIW = .

. 4.34 F d. - F ds=R d.

129

4.12

L=0,3 m M, . . ; -

231 ML

g=10 m/s2.

:. 4.35

. . .

2LMg

, 221I , I

.

2

21

2ILMg = 22

31

21

2MLLMg =

sradLg / 103 ==

130

ANI

4.13

R . h.H (g) .

221 MRI = .

:

. . , . ,, . , . 4.36

, Mgh .

,

221

cmM

221 I .

22

21

21

IMMgh cm += 222

21

21

21

MRMMgh cm +=

22241

21

Rgh cm += (4.22)

cm Rcm = (4.23)

(4.22) (4.23)2

22

41

21

+=R

Rgh cmcm

2241

21

cmcmgh += ghcm 34

=

131

x

dtdx

=

dtd

=

dtd =

dtd

=

F

m

F m=

I=

p=m

L=I

ewton

dtdpF =

ewton

dtdL

=

F=0 p=

=0 L=

2

21

mK =

2

21IK =

132

o . dtd

=

F, Fl=

l , - . 1 m.

F Fl= l , 1 m, .

0=F 00

==

y

x

FF

0=

-

...2222

11 ++= rmrmI

I=

- -

prL = rmL = . 1kg m2 /s. - - .

IL =

.

Newton dtdL

=

221IK =

=W

=P

To - 21

22 2

121

IIW =

133

1. .

. . , . .

.

2. .

. 4.38, . . () - (. 4.38). . . . . .

.

0=mgdMgx 0 = mgMg Mm =

. 4.37

. 4.38

134

, - m.

3.

, . .

, :

, . . ;

4.1 . 4.40 , , , ;

4.2 . ;

4.3 . 4.41 ;)

t1-t2 .

. 4.39

. 4.40

. 4.41

135

) t4 t1.

) t1 t4.

) t3 t1.

136

4.4 . - . ) ) ) ) , ;

4.5 , , - ;

4.6 . - ; .

4.7 : , . . .. ..

4.8 -. ;

4.9 4.42 . -. - .

4.10 4.43 - . F1. F2 .) F2; ) F1 F2.

. 4.42

. 4.43

137

4.11 4.44 - . F1 F2 . . - F1 2/3 . F2 ) F1/2 ) 2F1/3 ) F1/3 ) 2F1 ) 3F1/2 ) 3F1 .

4.12 4.45 m . , ) .) .) .) .

4.13 , , . ;

.

4.14 4.46 zz. . zz.

4.15 . ;

4.16 1, 2, 3 4 1,2, 3 4 (.4.47)

. 4.44

. 4.45

. 4.46

. 4.47

138

4.17 - )

.) .) .) . .

4.18 4.48 -. . 1 2. - . - ; - .

4.19 4.49 . F . F 4.49

;) .) t1 ,

F .) .) F .

4.20 (. 4.50). , . ) .) .) .

. 4.48

. 4.49

. 4.50

139

-

4.21 , . ;

4.22 4.51 . , , , . t.

4.23 )

.) .) .)

. .

4.24 . - . ;)

.) ) .) .

4.25 , . ; .

4.26 . . . . , ; - , , - -. ;

4.27 1 (. 4.52) , . 2 3 1. - :) ) 2 ) 3 ) /2 ) /3 .

. 4.51

. 4.52

140

4.28 . . . 1. .2.

.

4.29 - F :) 1 rad/s 3 rad/s.) 4 rad/s 6 rad/s.) -2 rad/s 5 rad/s.) -3 rad/s 4 rad/s. ;

4.30 , . - . .

-

120 J 40 J 20 J 80 J

4.31 :

1 SI rad/s

rad/s2

mL

1 .

141

4.32 , 4.53. ; 20 rad/s;[: 0,25 rad/s2 , 72 s ]

4.33 20 m/s. 40 cm. .[: 50 rad/s ]

4.34 , cmr 40= , 2 m/s2. ;[: 5 rad/s2 ]

4.35 8cm . 5 m/s. :) .) .[: 10 m/s , 9,9 z ]

4.36 o , R=20 cm, , =8 m/s. x=20 m. ;[A: 8rad/s2 ]

4.37 , 20cm. 200. ; ;[ : 40 m ]

4.38 4.54 R= 0,5 m - . F1=20N F2=30N. - ;[ : 5 m ]

. 4.53

. 4.54.

142

4.39 H 4.55 ' . F1=20N, F2=2N F3=10N. . : x=2 m =30.[ : 16 m ]

4.40 R1=20 cm, , R2=0,5 m. , ( ) 150 . .[ : 60 ]

4.41 , l=2 m, - w1=200 w2=300 N (. 4.56). ;[ : 1,2m ]

4.42 4.57 l=4 m w1=150 N. H 1 m, . w2=700 N. ;[ : 79 cm ]

4.43 l w1=100 N - (. 4.58). . =30 . w2=40 N. .[ : =180 , F=163,7 N, =0,32 ]

4.44 (.4.59) . 6m 100kg. . L,

, 2121 MLI =

[: 4800 kg m2 ]

. 4.55

. 4.56

. 4.57

. 4.58

. 4.59

143

4.45 , =2 kg R=0,5 m, =100 rad/s F, . 5s. :) () .) F.

221 MRI = .

[ : 2/20 srad = NF 10= ]

4.46 , L=1 m, (.4.60). , , , ;

2

31 MLI = g=10 m/s2.

[ : 15 rad/s2 ]

-

4.47 , m=100 g , 4.61. - =16 rad/s, zz. . l=0,8 m.[: 5,12 kg m2/s ]

4.48 =2 kg R=0,4 m, =10 rad/s . .[ : 3,2 kg m2 ]

4.49 20 cm 1 kg 2 Hz . 100gr, 10 cm . .

221 MRI =

[ : 1,9 Hz ]

. 4.60

. 4.61

144

-

4.50 =3 kg L=40 cm =10 rad/s . .

2121 MLIcm = .

[ : 8 J ]

4.51 =8 kg R . =5 m/s. - .

221 MRI = .

[ : 150 J ]

4.52 4 m 50 z. ; [ : 400 W ]

4.53 m=40 kg R=20 cm, 5 z . ) ; ) 5s.

221 mRI = 102 .

[: 400J , 80W ]

4.54 H 4.62 L=2 m = 3kg, . F=10 N . F . :) F, .)

.)

( ) .

231 MLI = .

[ : W=40 J, =7,9 rad/s, P=158 W ]

4.55 H , l=30 cm, (. 4.63). .

. 4.62

. 4.63

145

, .

12

2mlI = g=10 m/s2.

[ : 3 m/s ]

4.56 l w=100 N 4.64. . =60.

[ : 3/32 ,750 ,350 === NFNT ]

4.57 4.65 h R w. F .

[ : F >hR

hRhw

)2(]

4.58 400 (. 4.66) 30. . .

[ : 2/3 ]

4.59 R=0,33 m 1 kg. 100 - . 5 s. -, 0,69. . ( ). [: 1 ]

4.60 4.67 . L=1 m =0,6 kg. r=0,2 m m=0,1 kg , . f1=10 z. , , . .

2121 MLI = .

. 4.64

. 4.65

. 4.66

. 4.67

146

[ : 5,8 z ]4.61

5 m/s. 30 . . . ,

, 252 mR . g=10 m/s2.

[ : 3,5 m ]

4.62 30. ,

, 252 mRI =

g=10 m/s2. . [ : 25/7 m/s2 ]

4.63 4.68 m=2 kg R. 1 2 m1=3 kg, m2=1 kg. .

221 mRI = g=10m/s2.

.: - . [ : = 4 m/s2 ]

4.64 . 4.69 200 gr 30 cm 40 rad/s. . 15 cm; ( ).[ : 43,2 J ]

4.65 O 4.70 0,18 kg m2 =25 rad/s . . .[: 4,5 2 kg m2/s ]

. 4.68

. 4.69

. 4.70

147

4.66 , 20 cm, - . 20 - . f=4 Hz. .

221 mRI = . [: 50 kg

]

4.67 . ;

2

21 mRI = .

[: > 3/ ]4.68 m=120

g R=1,5 cm, (. 4.72). , . )

, )

30cm. .

221 mRI = . g=10 m/s2.

[: ) 610-3 kg m2/s ) 2 m/s ]

4.69 m r , , 4.73. , h ; R=20 cm.

252 mrI = .

R. [: 54cm ]

4.70 1 2 , d. 12 , 4.74. , .

. 4.71

. 4.72

. 4.73

. 4.74

148

[: g

d

2

2 ]

149

4.71 m= 2kg (. 4.75). - . . =8,4m/s. . =11,52 kgm2 R=1,2 m.[: 1,75 rad/s ]

. . ( ) ,

ABC = .

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

( ), C.

C, - .

. 4.74

. 4.76

150

x, Ay, Az , - Bx, By, z, ,

zyx

zyx

BBBAAAkji

BAC ==

i, j k x, y z, .To

ABBA =

F

Fr =(4.24)

r F. r F .To

rF= (. 4.77 ) dr = d, F, Fd= (4.25) - F r.

(4.24) (4.25) - , (4.24) .

,

rprL == m(4.26)

r . 4-7. (4.26) .

, , . - . .

. 4.77

. 4.78

151

(4.24) (4.26) . ,

, . ( Lorentz) :

F = q

, - . - . - .

()

. - , -

,() .

- , - .

- , , - (4000 /min), - 450 km/h. - .

.

- ,- , ,

1 .

1 .

. 4.79

152

.

. - ( ) (. 4.80). , ( -) .

, 1 1 - (. 4.80). 1 ( )1R

1 ( )1ER . -

1

1

1

1

=

RRE

1

1. 1 1 1 (. 4.81).

, - . , , - . .

( )lF = . .

. 4.80

153

- .

, 2 2 (. 4.80).

12

2

2=

ERR

. 2

2 .

2/3 , - . - . , , - , .. .

, . - (. 4.80).

.

.

. .

.4.81

154

(. 4.82 ) . .

, - .

. 4.82