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
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)