ΚΛΑΣΙΚΗ ΜΗΧanikh
DESCRIPTION
φυσικήTRANSCRIPT
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: . , .
- :
. . . , TYPORAMA ISBN: 960-538-496-5 ISBN (SET): 960-538-495-7 , : . .
opusMAGNUM
ISBN: 978-960-538-820-1 ISBN (SET): 978-960-538-819-5 : 14/2-1 Copyright 2008 18, 26335, : 2610 314094, 314206 / : 2610 317244 . 2121/1993, , .
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:
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2008
. .., , (CERN) FERMI (FERMILAB) ... , ( ). charmonium , quarks gluons quarks, , , , , . (FERMILAB, CERN) (Northwestern Univ., Bologna Univ., Liverpool Univ.) . 1997 ( , ) CERN, (KM3NeT) (EUROCOSMICS). , . , (HELYCON). ..., , .
..................................................................................................................11 ......................................................................................................15
1
....................................................................................... 17 1.1 ...........................................18 1.1.1 .......................................................18 1.1.2 ...................................................................20 1.1.3 ...................22 1.1.4 ............................................29 1.1.5 .....................................32 1.1.6 ...................................................................36 1.1.7 .............................................................................................39 ...........................................................................................................40 .........................................................................................42 1.2 ..................................................................46 1.2.1 ................47 1.2.2 ..................48 1.2.3 , ...........53 1.2.4 ............................................................55 1.2.5 ........................................................57 1.2.6 ....................................62 1.2.7 ........................................................................................63 , , ......................................46 , , ......................................18
8
:
1.2.8 .................................................................................68 ...........................................................................................................70 ........................................................................................72 1.3 ...........................................................................75 1.3.1 ............................75 1.3.2 ......................................................................76 1.3.3 ...........................................................................80 1.3.4 ..................................................................................83 1.3.5 ..............................................................87 1.3.6 ...........................................................88 1.3.7 .......................................................................92 1.3.8 .......................................................................................................97 1.3.9 .......................................................99 .........................................................................................................101 ......................................................................................103 .....................................................................................106 , , .....................................75
2
......................................................................................109 2.1 ..........................................109 2.1.1 .............................................110 2.1.2 ...........................................................................................113 2.1.3 ..................................................117 2.1.4 ...............................................................................119 2.1.5 ......................................121 , , ...................................109
9
2.1.6 ...................................................................124 2.1.7 ......................................................127 .........................................................................................................129 .......................................................................................131 2.2 ...................................132 2.2.1 ................................................................................................133 2.2.2 1 .............................135 2.2.3 1 ........................................................................139 2.2.4 2 3 .....................................................142 2.2.5 2 3 ........................................................143 2.2.6 .....................................................................................................145 2.2.7 ............................................150 .........................................................................................................153 .......................................................................................154 2.3 ...........159 2.3.1 ......................................................159 2.3.2 ....................................163 2.3.3 .........................165 2.3.4 ..........................................168 2.3.5 ...................................172 2.3.6 ........................................175 2.3.7 ......................178 2.3.8 ............................................185 , , ....................................159 , , ....................................133
10
:
.........................................................................................................191 ......................................................................................194 .....................................................................................199 ..........................................................................207 ..............................................................221
: . , . . , , . Galileo Galilei (1564-1642) . , . Johannes Kepler (1571-1630), , . , , Isaac Newton (1642-1727), , , . , , , , , . , , . , , . . Galileo Galilei . , , ( , ) , . , . , , , , .
12
:
, : 1. . , (). . 2. , . Kepler , . , Kepler . . 19 . , (, , ). , . , . . . , . , . , , . ; , -
13
; . , . . .
. . , . , , . , , , . : , , . , . . , , , . . . . . , , . . . . . . .
16
:
, . , , . . , . . , , . . , . . . , , . , . .
. , , . , , , . : , 1.1. , , ( ), . , , , , . : (.. , , , ), ( , , , ) . , , ;
1.1
18
1:
1.1
, , . : , .
, , .
1.1.1
, . , ( ). . . , , .
1.1
19
( , ), . , , , . , , . . . . , , ; !!! , . ( ). ; , , . . , . . . . 10000 . , . . . , . .
20
1:
1.1.2
. , , .[1] , 1.2.
1.2
, . , (.., 1.2) (.., ). ( ) . , . , , . , , (second) s. , , . 1.2, ,
[1]
4 , , , , .
1.1
21
X, Y Z, . . , , , . .
1.3 0.5
m.
Q XYZ x = 1.0, y = 0.5, z = 0.8. XYZ, 1.3, ' 0.5m;
1.1
, tA ( , x,y,z) tB ( , x,y,z) rA rB , 1.4. r, . -
22
1:
rB rA, r = rB rA.
r =
( xB x A )
2
+ ( yB y A ) + ( zB z A ) .2 2
, ( 1.4), . , 1.5, , . , , .
1.4 . .
1.5
. , . . , , , !!! 1.1.3
. , .
1.1
23
X . . . (s), (m). 1.6 20sec . 1.1 .
1.6 20s . . .
, x, 1.7. , . . () . . () . , .
24
1:
1.2 1.7. , 22, 38 67s.
1.7 1.1
, , , 12s . 1.8. 1.9.
1.1
25
1.8 20s . . 12sec .
1.9 .
26
1:
1.6 1.8 . , , 1.7 1.9 : . ( ) . , 12s . , , . , , 22m 12sec . 20sec , , 1.10.
1.10 20s . 22m . 12s .
1.3 1.3 , , . . , , , .
1.1
27
1.3
( ) . 20sec , , 1.11. ,
1.11 20s . , .
28
1:
. . , , 1.11 1.3. ; , 1.12, . 1.7. , , ( ) , 1.12 () 1.7.
1.12 ( )
1.1
29
: ) ( ), ) ( ) ) ( ). , , , / , / .
, , : .
1.1.4
, . . , , ; . ( ) . : . . :
=
(1.1)
, , (m/s). . ,
30
1:
t1 x1 t2 x2, (1.1) := x2 x1 t2 t1
(1.2)
. , 1.1, 5 . (1.2) 5s . t1 = 0s x1= 0m t2 = 5s x2 = 3m. (30 =)3m (50 =)5s. , 5s 0.6m/s. , .. 5 1.4. . , , , .
1.4 1.2, . (1.2) .
. . .
1.5 1.3, . (1.2) .
1.1
31
1.13 . : [9s, 10s].
5s. , , , , (1.2), . , 1.13. , . , , . , 1.13 9 10 .
32
1:
. , 1.13 1.4. 1.4 , . . , . , 1.13, , , ..., 35, 36, 37, 38 39 .
( ) . t1 t2 ([t1, t2]). , , (1.2):
[t1 , t2 ] =1.1.5
x2 x1 t2 t1
(1.2)
, , (t2t1) , . :v1 = limt2 t1
x2 x1 t2 t1
(1.3)
( ) t1. , , x = x2x1 t = t2t1, (1.3) :v1 = lim x t 0 t
(1.4)
1.1
33
H . . . . ( ). (1.4) !!! . 1.14 ()
1.14 .
34
1:
. . t = 1sec. () t = 1s, [t1 = 1s, t2 = 4s]. [t1 = 1s, t2 = 4s ] = x2 x1 x = t2 t1 t
(1.5)
x1 x2 t1 t2 , x t , 1.14. (1.5) , (tan) AOE :
[t1 = 1s, t2 = 4s ] =
x = tan AOE t
(
)
(1.6)
, , -.
AOE .
[t1, t2] t1 t2 , ( ). , . 1.14, t1. (1.3) : , t = t2t1 . , , t2 t1;
1.1
35
. 1.14. t1!!! . , , t1. , t1: t1, . . , , . (1.3) (1.4) , . . . 1.15 ( ) 160 , . () , (). . (). (V) , , . (V) , .
36
1:
1.15 .
1.6 1.15.
1.1.6
. , . ; :
1.1
37
, , . t ( 1.14) t. , ( ) .
, 1.16. : () .
1.7
. , () . : . , [t1, t2] (1.2). 1.16 (1.2) -
1.16
38
1:
. . . , , . . 1.17 , . , 1.1 0, t1 t2 .
1.17
1.8 1.17 : ) t = 0 ; ) ; . , t . , .
1.1
39
1.1.7
( ) . , ( ) . !!! , , . . . 1.18 .
1.18
t1 v1 t2 v2. .
M [t1 , t2 ] =
v2 v1 t2 t1
(1.7)
40
1:
, . t1.
t1 = lim
v2 v1 t2 t1 t t 2 1
(1.8)
: ) ( ), ) ( ) ) ( ). , , , / , / . , . . x1 x2 [t1, t2], : [t1 , t2 ] = :v1 = lim x2 x1 t2 t1 t t 2 1
x2 x1 t2 t1
( ) t1. H .
1.1
41
[t1, t2]: t1 t2 . ( ). t1: t1. . . , . . () . . : [t1 , t2 ] = v2 v1 t2 t1
v2 v1 t2 t1 . : t1 = limt2 t1
v2 v1 t2 t1
42
1:
1.1 .
2000m. , Oregon State University .
1.2
1.19
. . , 1.19(). . , , ,
1.1
43
, 1.19(). , . , 1.19(). , , . 1.20 . , 1.20 :
1.20
) . ) . ) . ) , . ) . 1.19 .
1.3 1.21 , . ) ta .
44
1:
) tb; ) tc . ) ;
1.21
1.4 . (a) (t) 1.22. ) . ) . ) . ) .
1.1
45
1.22
46
1:
1.2
. .
.
1.2
47
1.2.1
. , . . , . x t :
: x = t +
(1.9)
x t = 0, 1.23. ;
1.23
(v) . (x0) .
- : x = v t + x0
(1.10)
48
1:
1.2.2 (1.10), . : . . t = 0 ( 1.1), x0, ( ) , x1. V, v. x1 > x0 V > v. , . 0 :
0 = x1 x0 1
(1.11)
x0 x1 t1. , t1 (1.10) :
x1 = V t1 + x0 t1 = x1 x0 0 = V V(1.12)
, x2. x2 (1.10) x1 x2 v t1, (1.12), :
x2 = v t1 + x1 =
v 0 + x1 V
(1.13)
1.2
49
1(=x2 x1), (1.13) :
1 = x2 x1 = 2
v 0 V
(1.14)
x1 x2 t2. . t = 0 x1 t = t2 x2. , , , t2 (1.10) :
x2 = V t2 + x11
x x t2 = 2 1 = 1 V V
(1.15)
(1.14), :t2 = v 0 V V
(1.16)
x3. x3 (1.10) x2 x3 v t2, (1.16), :
v x3 = v t2 + x2 = 0 + x2 V
2
(1.17)
2(= x3 x2), (1.17) :
2 = x3 x2 = 0 V
v
2
(1.18)
50
1:
3 x2 x3 t3. x4.
1.9 , t3 :
v t3 = 0 V V
2
(1.19)
3, :
v 3 = x4 x3 = 0 V n [2] :
3
(1.20)
v tn = V
n 1
0V
(1.21)
:
v n = 0 V .
n
(1.22)
v , , V n (1.22) n .
v lim n = lim 0 = 0 n n V
n
(1.23)
[2]
( ), (1.21) (1.22) n = 1 , n = 1, n = , n. (1.21) (1.22), 1, 2 3.
1.2
51
, . , . , , , . ; E ; . !!! , n, n . Tn = t1+ t2 + + tn. , , (1.21), n :
Tn = t1 + t2 + + tn = :
0
v + V V
0 v ++ V V
n 1
0V
(1.24)
(1.24) :
v Tn Tn V
0 v 0 = V V V
n
(1.25)
, (1.25) , :
0 v 0 0 V V = V Tn = V v v V v v 1 1 1 V V V
0
n
v n V
(1.26)
, , , .
52
1:
, T, (1.26) n . , v . V
0 T = lim Tn = lim 0 V n n V v v 1 V 0 = 0 V V v v 1 V
v n = V
n v lim = 0 n V V v 0
(1.27)
, , (1.27) . , . 9.01m/s, 0.01m/sec 90m, (1.24) 10 . [3], . , (1.27). . , , .
x1 0 , x0 ( 0= x1x0). v V, V > v, ;
[3]
, . : . 1.1 pokemon.
1.2
53
(1.10), . , t, , x. t x, : : :
x = x1 + v tx = x0 + V t
(1.28) (1.29)
(1.28) (1.29) 1.10 (1.27). ;
1.2.3
,
(1.10) . , (1.10) x ( ) t ( ). ,
x ( t ) = v t + x0
(1.30)
, , (1.30). , , t1 x(t1) t2 x(t2), , x(t), (1.30). t1, (1.3), :
v ( t1 ) = lim
x ( t2 ) x ( t1 ) t2 t1
t2 t1
(1.31)
(1.31) ( t1). , :
v ( t ) = lim
x ( t + t ) x ( t ) t
t 0
(1.32)
54
1:
(1.32) . x(t) t.
:
v (t ) =
dx ( t ) dt
= x ( t )
(1.33)
1.11 (1.33) . : , x = k t + .
, ( , [1.30]), . .
. ( 1.16) ( ) ( ) m m m x ( t ) = 0.094 t + 0.1 2 t 2 + 0.0009 3 t 3 , s s s
.
1.12 m m m x ( t ) = 0.094 t + 0.1 2 t 2 + 0.0009 3 t 3 . s s s . 3s, 8s 50s. : , x(t).
1.2
55
1.2.4
() , (1.32), :
a ( t ) = lim
v ( t + t ) v ( t ) t
t 0
(1.34)
. :
a (t ) =
dv ( t ) dt
= v ( t )
(1.35)
(1.35) (1.33) :
a (t ) =
dv ( t ) dt
=
2 d dx ( t ) d x ( t ) = = x ( t ) dt dt dt 2
(1.36)
.
1.12 .
1.13
a(t) , , . . . , , , : ) , ) ( t = 0) V0 ) 0 t = 0. : ( (1.35)
56
1:
( ), :
v (t ) = t +
(1.37)
t. , , :
dv ( t ) dt
=
(1.38)
(1.38) , :
=a
(1.39)
(1.37) . (1.37) t = 0 . V0, :
= V0
(1.40)
(1.37), (1.39) (1.40) :
v ( t ) = a t + V0
(1.41)
: ) ) ( (1.41)), :
x (t ) =
1 a t 2 + V0 t + X 0 2
(1.42)
, (1.42) , (1.41):
dx ( t ) dt
=
d 1 2 a t + V0 t + X 0 = a t + V0 dt 2
0 (1.42) t = 0.
1.2
57
!!! , :
x (t ) =
1 a t 2 + V0 t + X 0 2 v ( t ) = a t + V0
, X 3m/s2.
1.14
1.2.5
, , ( ) . , ( g) , , . , , . , g. , ( ) .
58
1:
1.24
. , . , , 1971 David Scot, , . , Scot (Galileo Galilei). , .
1.2
59
, , (Isaac Newton) . , , V0(= 10m/s). H . , h ( 1.24), . . , , , . g= 9.81m/s2. , , , (1.41) (1.42). X ( ), ( ), 1.24. . 1.25() 3 . , , a =g = 9.81m/s2. , , X. V0 = 10m/s , (1.41), :
v ( t ) = (10 m s ) ( 9.81m s 2 ) t
(1.43)
(1.43) 1.25(). . t = 0 v = V0 = 10m/s , . .
60
1:
1.25 : ) a(t), ) v(t) ) x(t)
1.15 (1.43) , th, . th, t > th, . . : ) . (1.43) v(th) = 0. ) X.
t = 0 X0 = 0m. (1.42) :
1.2
61
1 x ( t ) = (10 m s ) t ( 9.81m s 2 ) t 2 2
(1.44)
1.25() . . . , th, 1.15. , x(th), (1.44) t = th.
. : (1.44) 1.15.
1.16
, t0 , X0 = 0m. (1.44), , t0. , x(t0) = 0m, :1 2 0 = (10 m s ) t0 ( 9.81m s 2 ) t0 , 2
. . x0 = 0m . . .
, t0, . :
1.17
1 2 0 = (10 m s ) t0 ( 9.81m s 2 ) t0 . 2 x = 0 m , t = 0s.
62
1:
1.18 t0. . : (1.43) 1.17.
1.19 t0 .
, , . . , , 10 , (1.44) x(t) = 10m. , , t1 t2, .
1.20 t1 t2 .
; !!! , , . ;
1.21 , . 1.25() 10m. H . . . 1.2.6
( ) .
1.2
63
2m/s2. 90m 0.01m/s. ; ; . x t. , x t, .x = 1 m 2 m 2 t x = 0.01 + 90 m 2 s2 s
.
1.22
1.2.7
. , , . . , , , . ! . , . , , . .
64
1:
1.26 . , V1 . t1 t2.
, . . . . A [4] , , 1.26. , V1 V2. . , , :
[4]
. . . , .
1.2
65
V1. V2, . . : . . . . X . X X, . , X, ( 1.26), , V1, . , X, V1 . X . t1 y1 x1, t2 y2 x2. , , :
y2 y1 t2 t1
, . :V1 = y2 y1 y2 y1 = V1 ( t2 t1 ) t2 t1
(1.45)
66
1:
, :
V2 =
x2 x1 x2 x1 = V2 ( t2 t1 ) t2 t1
(1.46)
t1 x1 , t2 x2 . , , : x2 x1 t2 t1
, . , , V2, :V2 = x2 x1 t2 t1
(1.47)
( 1.26), , x1 ( , , t1), x1 ( t1) y1 ( t1) :
x1 = x1 y1
(1.48) (1.48)
1.23
x2 = x2 y2
(1.48) (1.48) (1.47), (1.45) (1.46)
V2 =
x2 x1 ( x2 y2 ) ( x1 y1 ) ( x2 x1 ) ( y2 y1 ) = = = V2 V1 t2 t1 t2 t1 t2 t1 t2 t1V2 V1
(1.49)
1.2
67
, V1 ( ) , V2 ( ), V2 ( ), V2V1. (1.49). ( ) 20 km/h, 8 km/h. 1.26, ( x), ( ) ( x). : V1 = +20 km/h V2 = 8 km/h. (1.49), :
V2 = 8km h 20 km h = 28km h , X ( V2 ) 28 km/h. , ; : V1 = +20 km/h V2 = +8 km/h.
1.24
. : . , V2 = 0km/h. (1.49).
1.25
10 km/h X 1.26. 20 km/h X. . : V1 = +10 km/h V2 = 20 km/h. (1.49) V2.
1.26
68
1:
1.27
, ( , ) , , , 1.27. 30 km/h, 60 km/h . . X . , . (1.49), V1, V2 V2 :
V1: V2:
V2 : x :
V2 = V2 V1 +60 km h = 30 km h V1 V1 = 90 km h
1.27 .1.2.8
. . , . , . (A. Einstein).
1.2
69
1.28
. ; , . . . , , . , ( ) . . -
70
1:
. . : ( ) ; , ; 1.28. !!! ( ) ...
:
v (t ) =
dx ( t ) dt
= x ( t )
:
a (t ) =
dv ( t ) dt
= v ( t )
.
a (t ) =
dv ( t ) dt
=
2 d dx ( t ) d x ( t ) = x ( t ) = dt dt dt 2
, , , :
1.2
71
x (t ) =
1 a t 2 + V0 t + X 0 2 v ( t ) = a t + V0
, , ( ) . V1 ( ) , V2 ( ), V2 ( ), V2V1. ( )
1.5 . , . , . 500 km 20 km/h, 500 km 30 km/h 500 km 50 km/h. . , .
1.6 1.5 s 14 (3m ). ) . ) 2.33 m/s. ) 2.33 m/s.
72
1:
1.7 () 240 km/h. . ( ) . 1.75 m.
1.8 , : . , . , . , , .
1.9 , , 120km/h. , , , . : ) , ( ), ) ( 0.5s), ) ( 100km/h 5s). , , .
1.2
73
1.10 . 343m/s 1.5 . . ( .)
1.11 10m 0.9m/s2. . ( Indiana Jones .)
1.12 15m/s 25m/s 20m. : ) ) .
1.13 10s 3m/s2. , 4m/s2. 2 , : ) ) .
1.14 . , 1.1 . .
74
1:
1.15 a1 5 . a2 = 5m/s2 10s 5m/s. a1 15s .
1.16 VT = 20m/s.
1.29
, . 15m 50m . , V, , ; .
1.3
75
1.3
, , ; ; , .
(.. ) .
1.3.1
. , , , . .
76
1:
; : , , ; , , . 2000 . , , . .
1.1 . (384322 ..) 2000 .
: !!! , 1.3.2 , . , , 1.3.2 1.3.3. 1.3.2 , . , . , . , . -
1.3
77
( ) . , . , . , , , . . , . ( ) . . . , , , , : ) ) () . . , : . ( 1.30) . , . , ; , . . , . , , , .
78
1:
1.30
1.31 . . . .
. . , . , , .
1.3
79
, 1.31. ( , , , ) . , , . , . , . . .
, , , , . ;
1.28
80
1:
1.3.3 , (GalileoGalilei). , , . , , , . , , . , , , . , . . 1.32 . : , .
1.29 ; , 1.31, . . . , .
1.3
81
, . (V) 1.33() . , V. , 1.33(). , ( 1.33[]).
82
1:
. . . , , , , . , , . . , () V.
1.3 Sir Isaac Newton (16431727) . .
(t) , y x . y x . . , , 1.33().
1.3
83
, , , . .
1.31 .
1.30
. 1.34.
1.34
1.3.4
. , , , ( , , .., ) . (Isaac Newton) Principia Mathematica Philosophiae Naturalis ( ) 1686. . .
84
1:
: . , . , . , , . .
, . : ( ). m Fx X, , ax, :[5]
Fx = m a x
(1.50)
, .. , F , , F. . , . .
[5]
O Fx =
d ( m vx ) . , dt
, , (1.50).
1.3
85
. : . . , (1.49) . , . ( Leibniz[6]). , . : . , . , dv , : a x = x dt , , () . : ( SI ) (kgr). . , . . . : . .
[6]
O 16651666. . , , . Gottfried Leibniz . .
86
1:
1.35
, . , , . : . . , , . , , . . 1.35 , F1 F2, , R, . X. , F, : F = F1+F2R. SI Newton (N).
, . , , . . , . . .
1.3
87
1.3.5
. , . : ( ). , . ( ) ( ) . . ( 1.33) . . . 1.34. , ( ). [7]. .
[7]
. , . , , , . . , , , 9.81m/s2. 4.4 103 ( ) 3.37 102m/s2 . .
88
1:
1.3.6
. m Fx X, , ax, :
Fx = m a x
(1.50)
Fx ax . , ( ) , (1.50). . . , m , , (1.50), . .
. ( ) 1kgr, . (1.50) . m0 ( 3kgr, m0). . . ( ) F , a0, a1, a2, , aN.
1.31 :
mk = m0
a0 k = 1, 2, , N ak
(1.51)
1.3
89
: , ao, , F, : F = m0 a 0 . k , ak, F: F = mk a k . .
, . , , , m0 ( 4 3kgr), . m0 . m0
;
1.32 m0 a = m0 0 k = 1, 2, , N m0 ak
: (1.51) m0 . : mk = mk
(1.50) . 1 (Newton) 1kgr 1m/s2. . / / . , m1 m2 r, , , :B =G m1 m2 r2
(1.52)
(1.52), , . G (6.67 1011Nm2kg2).
90
1:
. , . , , m h . , 1.36, . , , .
1.36
( , , ) , . R, [8]:
B =G
M m
( R + h)
2
(1.53)
6.37 ( 5.98 1024kg), , h , (1.53) [9]:[8]
. (1.53), : ) , . ) . , , . . () . h10m (1.54) 99.9999%.
[9]
1.3
91
B G
M m
( R)
2
(1.54)
. , , . (1.54), :
B = ma M m M g a =G 2 M m ma = G 2 R R B =G R2
(1.55)
, (1.55), , , G. . , , .
( (1.54)). . ; ( 6.37 106 . .) : , a, B = M a : M m B = G R2 1 , x = 1mm = a t 2 , t. 2 , , ( ) ( ) ( ) .
1.33
B = mg
(1.56)
92
1:
1.34 ( ) . : (1.54). 7.36 1022kg 1.74 106m .
1.37
, . 1.3.7
1.37, , . , ( ) . .
1.3
93
, . 1.37; . , .. , F , , F. . 1.37 ( 100) . , . , a1 a2 , .
; a1 a2 50kg 2000kg. , 100. : . (1.50) ( m1 a1) ( m2 a2).
1.35
t , . . .
94
1:
1.36 50kg , 2000kg. : . , t 1.35. , t (.. 2s) . , .
. . 1.38 . . . 1.38() 1.38() . , , . , , . . , , F. , , F. , . , , .
1.3
95
( , .) . , , B F. , . 1.38() . ( ). .
: . , . . . N .
F . : , . S = B T. . , , S, . F , , , .
1.37
96
1:
a, 34. . 33. a.
1.39 a
() 1.39 . , a , , . , :
T B = ma m . (B = mg) , :
T = m (g + a ) > m g , , .
1.3
97
= F , F , , , . F . , : ( ) , mg , , .
, a, (a = g/2); : : T B = m a T = m (g a ) < m g
1.38
1.3.8
. , 1.40 : F Fk . , , , (), .
98
1:
1.40
. , 1.40 (.. ), . , , , , . ( [1.40]) . , . :
T = Fk
(1.57)
, , . . .
1.3
99
1.3.9
, , . . . , . , . , , , . , . mA = 10 mB = 20kg, , , 1.41. . t = 0 h = 1m . . 1.41 : , , , BB = ( 20 kgr ) ( 9.81m s 2 ) = 196.2 N. , , , BA = (10 kgr ) ( 9.81m s 2 ) = 98.1N. , , FB. ,
FA.
100
1:
FA FB , , . , , .
FA = FB = F
(1.58)
( , FA FB ). : :
BA FA = mA a ABB FB = mB a B
(1.59) (1.60)
a a . , . , , .
1.39 : ( ) , .
, , , . ( [1.41]):
a B = a a A = a
(1.61)
(1.58) (1.61) (1.59) (1.60), :
BB BA = a ( mB + mA ) a =g mB mA = 3.27 m s 2 mB + mA
101
, , , 3.27m/s2. , , . , , , , , .
. : 0.5m 3.27m/s, .
1.40
, . : ( ). m Fx x, , ax, :
Fx = m a x , .. , F , , F. . . , . .
102
1:
. N . m1 m2 r, :
B =G
m1 m2 r2
, g, , M, , R , G. .
g =G
M R2
103
1.17 12kg. , () 30m/s2;
1.18 80km/h , . 5 45km/h. 100N ( ), .
1.19 , 1.42.
1.42
F . . . .
104
1:
1.20 1.43 , m1, m2 m3 . . a1, a2 a3 : a1+a2+2a3 = 0. m2 = 2m1 m1 , : , 2 3, 3.
1.43
1.21 m1. m2 m1, 1/5 . m1/m2.
105
1.22 , r. r/2, , 1.44. 4r . .
1.44
1.23 m1 = 2200kg , m2 = 310kg m3 = 260kg , 1.45. 1.9m/s2, : ) , Fth, , ) , 1, ) , 2, .
1.45
106
1:
1.1 110km/h. 0.14s . 60kgr .
: 13100
1.2' X 108m/s. 1018s. Y . H 9.11 1031kg.
: 9 105
1.3 . , . , 532. 60kg ( ), . ;
: 8.87m/s2
1.4 50kg 40km/h. .
: 490
1.5 450 . 32kg. .
: 4.26m/s2
107
1.6 74kg . : ) , ) 2.4m/s, ) 2.1m/s, ) 1.7m/s2 ) 1.7m/s2;
: ), ), ): 725, ): 851, ): 599
1.7 1, 2 3kg, , , 1.46. 12 . . : 10.
1.46
1.8 (, 3kg).
: 6
1.9 11kg 1.8kg. 2.3m/s2. (1) (2); ; .
: 1 = 29.4 2 = 25.3
108
1:
1.47
1.10 , m, 1.48, . , a. .
1.48
1.11 0.42, ( ) . ( ) 60km/h 1.5m.
: 4.12m/s2, 86.8m/s2
. . . , , , , , . , . , , . ( ) ( ). 1 . , , .
2.1
1 ( ) ( ).
.
110
2:
2.1.1
. 2.1, . , ( ) . [1], 2.2.
2.1
2.2
[1]
M , .
2.1
111
. . , 1. . , , 2.3. . , , 30 . 2.1 2.2 , . , 2.1 2.2 . . , , .
2.3
112
2:
. , .
2.1
113
, . , . , , . , , ( ), . ( 2.4) 2.935951m 2.629045m. {2.935951,2.629045}[2] . , R:
R = 2.935951 i + 2.629045 j
(2.1)
i j [3] .
2.1.2 . . R (2.1) , , : R=+ : (2.2)
= 2.935951 i = 2.629045 j
(2.3)
[2] [3]
H X . i j .
114
2:
(i j) R . R , , , 2.4. R (2.2) ( , R) :
R = 2 + 2
(2.4)
2.1 , (2.2), (2.3) (2.4), 2.4.
2.4 K
2.1
115
2.4 R :
= R cos = R sin cos = =R sin =
+ 2 2
(2.5)
R
=
2 + 2
R .
(2.5). (2.2) , , :
2.2
R = R cos i + R sin j
(2.6)
2.4 ( [2.3]) 2.3 (2.6). , , . . . A B. 2.5() :
A = x i + y j B = x i + y j
(2.7)
, , , , :
= A + B = ( x + x ) i + ( y + y ) j
(2.8)
116
2:
, :
= B A = ( x x ) i + ( y y ) j
(2.9)
, , 2.5(). 2.5() 2.5().
2.4 A B . .
2.5
2.1
117
2.1.3 R , 2.6.
2.6
:
R = k + m k = ai R = ai + b j m = bj
(2.10)
i j , k m R . , , :
k =a m=b R = a 2 + b2 , , ( 2.6). R . (2.11)
118
2:
R . k m , (2.6) (2.11).k = k cos k sin m = m sin + m cos
(2.12)[4]
(2.10) (2.12) (2.11), :
R = k + m = ( a cos + b sin ) + ( b cos a sin )
(2.13)
( 2.7) , R. , . , tx ty, :
T = tx i + t y j
(2.14)
2.7
[4]
k Y.
2.1
119
, , , , . R ( 2.7). R R T.
2.5(2.15)
R = R T = ( a t x ) i + ( b t y ) j
R . , , (2.13) (2.15), :
R = ( a t x ) cos + ( b t y ) sin + ( b t y ) cos ( a t x ) sin (2.16)
2.1.4
2.1.1: . 2.8() . 2.8() ( ), 2.8() ( 2.1 2.1.1). , . ( 2.8[] 2.8[]) . 2.8() 2.8() x(t) y(t), .
120
2:
2.8 : () , () , ()
, , x(t) y(t) ( : m m m x ( t ) = 1.736 t y ( t ) = 9.848 t 4.905 2 t 2 ), s s s (2.2). , , :
= x (t ) R (t ) = x (t ) i + y (t ) j = y ( t )
(2.17)
2.6
m : x ( t ) = 1.736 t s m m y ( t ) = 9.848 t 4.905 2 t 2 s s 1.162s 1.796s.
2.1
121
: ( ) . : : x(t) : y(t) .
m (.. x ( t ) = 1.736 t s m m (.. y ( t ) = 9.848 t 4.905 2 t 2 s s . (2.17). 2.1.5
tA R A ( = x ( t A ) i + y ( t A ) j) , tB, R B ( = x ( t B ) i + y ( t B ) j) o 2.8. = R B R A () , 2.8(). tA tB :
= ( x ( t B ) i + y ( t B ) j) ( x ( t A ) i + y ( t A ) j) = ( x ( tB ) x ( t A ) ) i + ( y ( tB ) y ( t A ) ) j
(2.18)
(2.18) () t= tB tA. ( U, Ux Uy):
U=
y ( tB ) y ( t A ) x ( tB ) x ( t A ) = i + j t t tUx Uy
(2.19)
Ux Uy 2.8() 2.8() .
2.7
122
2:
, , ( 1.1.5 1), Ux Uy , , [tA, tB]. U ( 2.8[]) [tA, tB]. , (Ux Uy) . t = t B t A . 2.8() 2.8() . AB 2.8() 2.8(). , . [5], 2.8() ( ). (2.19) U : t = t B t A 0
V = lim
t 0
x ( tB ) x ( t A ) y ( tB ) y ( t A ) = lim i + lim j t 0 t 0 t t tVx Vy
(2.20)
V, (2.20), ( lim ) t 0 , 2.8(). , Vx Vy, .
[5]
. 2.8() 2.8() , .
2.1
123
V , .
2.8 () X ;
2.8
Vx = 1.736 m s, , , m m Vy ( t ) = 9.848 9.81 2 t. s s 1.162s 1.796s. .
2.9
. , (2.20) x(t) y(t).
t = t B t A Vx = limt 0
x ( tB ) x ( t A ) t y ( tB ) y ( t A ) t
= =
dx ( t ) dt dy ( t ) dt
(2.21)
Vy = lim
t 0
, R(t) :
R A = R ( t A ) = x ( t A ) i + y ( t A ) j R B = R ( t B ) = x ( t B ) i + y ( t B ) j V = limt 0
= RB R A
(2.22)
R ( t B ) R ( t A ) dR ( t ) = lim = t 0 dt t t
124
2:
(2.21) (2.22), : V (t ) =
dR ( t ) dt
=
dx ( t ) dt
i +
dy ( t ) dt
j
(2.23)
2.10 :
m m m R ( t ) = 1.736 t i + 9.848 t 4.905 2 t 2 j s s s 2.8. t = 0.3s. t = 0.5s. t = 0.9s , t = 1.7s . : .
2.1.6
2.9 , 2.1.1, . , , . , .
V ( t ) = Vx ( t ) i + Vy ( t ) j
(2.24)
Vx Vy ax ay ( 1.1.7 1).
t = t B t A
a x = lim
Vx ( t B ) Vx ( t A ) t Vy ( t B ) Vy ( t A ) t
t 0
= =
dVx ( t ) dt dVy ( t ) dt
(2.25)
a y = lim
t 0
( ) :
a = ax i + ay j
(2.26)
2.1
125
2.9
(2.26) (2.25), :
t = t B t A a = lim Vx ( t B ) Vx ( t A )t 0
t 0 t dVy ( t ) dV ( t ) dV ( t ) a= x i + j= dt dt dt
i + lim
Vy ( t B ) V y ( t A ) t
j
(2.27)
126
2:
: . . t :
R (t ) = x (t ) i + y (t ) j
(2.28)
x(t) y(t) . ( 1.2.4 1), :
1 x ( t ) = Vx 0 t + a x t 2 + x0 2 1 y ( t ) = Vy 0 t + a y t 2 + y0 2
(2.29)
Vx 0 , Vy 0 x0, y0 , t = 0 . ax ay .
V ( t ) = Vx ( t ) i + Vy ( t ) j :
(2.30)
Vx ( t ) = Vx 0 + a x t Vy ( t ) = V y 0 + a y t
(2.31)
:
a = ax i + ay j
(2.32)
, :
a (t ) = a x (t ) i + a y (t ) j
(2.33)
, .
2.1
127
. . 2.1.7
. . 2.10, . R :
R = rx i + ry j + rz k
(2.34)
i, j k , .
2.10
, .
128
2:
rx = x ( t ) ry = y ( t ) rz = z ( t ) 2.1.6 : , . . t : (2.35)
R (t ) = x (t ) i + y (t ) j + z (t ) k x(t), y(t) z(t) , Y Z . ( ), :
1 x ( t ) = Vx 0 t + a x t 2 + x0 2 1 y ( t ) = Vy 0 t + a y t 2 + y0 2 1 z ( t ) = Vz 0 t + a z t 2 + z0 2 Vx 0 , Vy 0 , Vz 0 x0, y0, z0 t= 0. ax, ay az .
V ( t ) = Vx ( t ) i + Vy ( t ) j + Vz ( t ) k :
Vx ( t ) = Vx 0 + a x t Vy ( t ) = V y 0 + a y t Vz ( t ) = Vz 0 + a z t :
2.1
129
a = ax i + ay j + az k , :
a (t ) = a x (t ) i + a y (t ) j + a z (t ) k
. . t :
R (t ) = x (t ) i + y (t ) j x(t) y(t) . , :
1 x ( t ) = Vx 0 t + a x t 2 + x0 2 1 y ( t ) = Vy 0 t + a y t 2 + y0 2 Vx 0 , Vy 0 x0, y0 , t= 0, . ax ay .
V ( t ) = Vx ( t ) i + Vy ( t ) j :
Vx ( t ) = Vx 0 + a x t Vy ( t ) = Vy 0 + a y t
130
2:
:
a = ax i + ay j , :
a (t ) = a x (t ) i + a y (t ) j , . . t :
R (t ) = x (t ) i + y (t ) j + z (t ) k x(t), y(t) z(t) , Y Z . ( ), :
1 x ( t ) = Vx 0 t + a x t 2 + x0 2 1 y ( t ) = Vy 0 t + a y t 2 + y0 2 1 z ( t ) = Vz 0 t + a z t 2 + z0 2 Vx 0 , Vy 0 , Vz 0 x0, y0, z0 t = 0. ax, ay az .
V ( t ) = Vx ( t ) i + Vy ( t ) j + Vz ( t ) k :
2.1
131
Vx ( t ) = Vx 0 + a x t Vy ( t ) = Vy 0 + a y t Vz ( t ) = Vz 0 + a z t :
a = ax i + ay j + az k , :
a (t ) = a x (t ) i + a y (t ) j + a z (t ) k
2.1 r0 = ( 0.5m ) i + ( 0.6 m ) j t = 1.4s, v = (1m s ) i + ( 0.3m s ) j. , r ( t ) .
2.2 :
r (t ) = x (t ) i + y (t ) j x ( t ) = A cos (t ) y ( t ) = 2 A sin (t ) A = 4 m = s. 0, 0.5, 1, 1.5s. .
132
2:
2.3 t = 0
r0 = ( 2 m ) i + (10 m ) j + ( 5m ) k
v 0 = ( 0.5m s ) i + ( 2 m s ) j ( 0.8m s ) k
a = ( 0.5m s 2 ) j + (1.1m s 2 ) k . 10s.
2.4 , , 2.4m/s 1.1m/s2 3s. 5.7m/s. .
2.5 ;
2.6 , 260 i m s a = 0.38 i m s 2 + 0.72 j m s 2 24s. : ) ) .
2.7 :
r ( t ) = ( 3.2 t + 1.8 t 2 ) i + (1.7 t 2.4 t 2 ) j m, ( t ) . .
2.2
133
2.2
1.3 1 . .
.
1, 2 3 ( , , ) 2.2.1
, 2.11.
2.11
134
2:
, , . , . ; . ( 2.11[]) .
2.11 ; . , ( ), .[6] () . , , ( G) B F . G N.
2.12
[6]
, . , , . 2.11() . .
2.2
135
, : . . 2.11 , 2.12, () , , . . [7] , , : B = Bx i + By j ( 2.12), :
2.12
Bx = B cos By = B sin .
:
S x = Bx F S y = N By S = S x i + S y j . 2.2.2 1
, , ( ) .
[7]
.
136
2:
2.13 . O , = 90.
2.11, F . , , . , 2.13(), [8], . , : ) , , . ) , m, . 2.13(). . , F T.[9] , . : S x = S y = 0[8]
( ) , . , T, , ( ).
[9]
2.2
137
, , , < m ( s ).
S x = Bx Ts = B cos ( 90 ) Ts = 0 Ts = B sin
Ts = tan N S y = N By = N B sin ( 90 ) = 0 ( ) N = B cos
( )
( )
(2.36)
(2.36) , m 0, , : ( 1.3.5 1) ( (2.36[]) . , . , tan m N . , s, m. Ts s N ( ), , . , ( ), , . , k, . . (1 ). , :
S x = Bx Tk = B cos ( 90 k ) Tk = 0 Tk = B sin k
Tk = tan k N S y = N By = N B sin ( 90 k ) = 0 ( ) N = B cos k
( )
( )
(2.37)
(2.37) k .
138
2:
k k. :
Tk = k N
(2.38)
[10] , .
2.13 k < s . , , k, a. , :
S x = B cos ( 90 k ) T = m a T = B sin m a ( ), . , k. > k (2.38).[11]
2.14 , . k > k, k = tan k . : 2 (2.38).
[10]
. . , .
[11]
2.2
139
: , . : Ts s N . s . , . Tk = k N , k ( ) . . , . , .
; : .
2.15
2.2.3
1
2.14. . .
140
2:
2.14
2.15() , . F1 , ( ) F2. , F1 F2 , F2, . F1 = F2 . ( ) . , . , , [12] . -
[12]
, .
2.2
141
. 2.15() , . . .
F1 cos 1 = F2 F1 sin 1 = B
() ( )
(2.39)
30kgr (F1) 1000, ( F1) . (2.39[]) ( 10m/s2), :
m 30 ( kgr ) 10 2 s = 0.3 = 17.46 sin 1 = 1 1000 ( N ) ( F2); 2.15() . tan 2 =B . 2 F1
2.16
2.17
(30kgr) 1000. : , . 2.15() . , :
B = F1 sin 1 + F2 sin 2 F1 ( sin 1 + cos 1 tan 2 ) = B F1 cos 1 = F2 cos 2
142
2:
2.18 1 = 72 2 = 18, F1 F2 (30kgr) .
2.2.4
2 3
F, a, ( ) :
F = ma = m
dv d2R = m 2 dt dt
(2.40)
v R . ( i, j k), :
F = Fx i + Fy j + Fz k v = v x i + v y j + vz k R = xi + y j+ z k (2.40) , : (2.41)
dvx d2 x = m 2 dt dt dv y d2 y Fy = m a y = m = m 2 dt dt dv d2 z Fz = m a z = m z = m 2 dt dt Fx = m a x = m
(2.42)
, , . (2.42), 1.3.4 1, . . F, G = F. , .
2.2
143
2.2.5
2 3
2.16, m1 = 50kgr, , m2 = 15kgr. , , , 45o. ( ) = 0.5, , a, . ( , g = 9.81m/s2.)
2.16
. , : 1. , , , .
. : t h. x. , x = h. .
2.19
2. . , , .
144
2:
3. . 2 . , . , , 2.16(), . 4. . , , 2.16. 1: , a, . 2: 2.16() 2.16() . , , F ( 2.16[]) F ( 2.16[]). ( ), , F = F. . 3: , F (W = m2 g). a. F , F = F, :
F W = m2 a F m2 g = m2 a F = m2 ( a + g ) ( 2.16[]):
(2.43)
) F, ) B ( B = m1 g ) , ) N ) T (T = N ) . . , ,
2.2
145
( ), , . , . . , :
m1 g sin F N = m1 a m1 g cos N = 0
(2.44) (2.45)
4: (2.43), (2.44) (2.45) . a, :a =g
m1 sin m2 m1 cos m1 + m2
(2.46)
.
2.20
2.2.6
2.1.1 . 2.1 V0, . , . , , . ( ) , 2.17.
146
2:
2.17
, (1 ) . , (2 ) . . , . 2.1.6 2.1.7 , . , . : ) : . , V0 x :
V0 x = V0 cos
(2.47)
t ( ) :
x = V0 cos t .) :
(2.48)
2.2
147
. , ay. 2 , ( 1.3.6 1) ay .
ay = g
(2.49)
t :
Vy = V0 sin g t
(2.50)
gt (2.50). : .
2.21
t :
1 y = V0 sin t g t 2 2
(2.51)
(2.48), (2.49), (2.50) (2.51) , . 2.17, , , . , , . . . , Vy = 0: ( [2.48], [2.49], [2.50] [2.51]) (2.50) . , th, , :
148
2:
0 = V0 sin g th th =
V0 sin g
(2.52)
(2.51) . (2.52) (2.51), . (), .
V sin 1 V0 sin (V0 sin ) h = V0 sin 0 g = g g 2g 2 2
2
(2.53)
2.22 10m/s 80 , .
, R, , . 2.17, , , . , (2.51) y = 0, . (t = 0), . , .2 V0 sin + (V0 sin ) t1 = =0 g 1 2 0 = V0 sin t g t 2 2 V0 sin (V0 sin ) 2 V0 sin = t2 = g g
(2.54)
t2. (2.54) (2.48), :
R=
2 V02 sin cos g
(2.55)
2.2
149
, (V0), = 45 . : 2 sin cos = sin ( 2 ) (2.55) :
2.23
R=
V02 sin ( 2 ) g
(, , 2.4). , , . (2.48), , , . , , (2.48), , t, x :
t=
x V0 cos
(2.56)
(2.51) . (2.56), ():
y = x tan x 2
g 2 (V0 cos )2
(2.57)
2.1 2.1.1 = 80 V0 = 10m/s. 2.1 ( ). (2.57) (2.57) .
2.24
150
2:
2.2.7 .
2.18
2.18, . h ( ) ( ) V0. , , R, , VT, . . . . , ( [2.18]) :
Bx = B sin By = B cos
(2.58)
2.2
151
( ). :
V0 x = V0 cos V0 y = V0 sin
(2.59)
, ax ay .
Bx = B sin = m a x a x = g sin By = B cos = m a y a y = g cos :
(2.60)
Vx = V0 cos g sin t 1 x = V0 cos t g sin t 2 2 : (2.61)
Vy = V0 sin g cos t 1 y = V0 sin t g cos t 2 2(2.62)
( 2.18) h. (2.62) y= h. :
V sin t = 0 1 1 h = V0 sin t g cos t 2 2 V0 sin + t2 =
(V0 sin )g cos
2
+ 2g cos h + 2g cos h(2.63)
(V0 sin )g cos
2
152
2:
2.25 t2 ( h = 0). t2; : :V0 sin (V0 sin ) + 2 g cos h .2
. t2 , t1 . , R, (2.61) t1.1 R = V0 cos t1 g sin t12 2
(2.64)
t1 (2.63). (2.61) (2.62):
Vx = V0 cos g sin t1 Vy = V0 sin g cos t1 t1 (2.63).
(2.65)
2.26 :) , h. ) , h. ) , h = 0.
2.2
153
1 ( ) , ( ) 2 F, a, :
F = ma = m
dv d2R = m 2 dt dt
3 F, G = F. , . Ts s N , . s . , . Tk = k N . . k ( ) . . , . , .
154
2:
2.8 4kg . 30 45 . .
2.9 30kg ( 0.25) , 45 . : ) , 0.08m/s2.
2.10 675m/s 511m . ;
2.11 :
F = ma 1 x = x0 + v0t + a 0t 2 2 x = x0 + v0 cos 0 t 1 y = y0 + v0 sin 0t gt 2 2 2 v R = 0 sin 2 0 2g : ) ; ) ;
2.2
155
2.12 10m/s, 30 . 18m . .
2.13 h. x . h, x . .
2.19
2.14 2.20 45 60 . ) , , . ) 100m/s 1.3km, .
2.20
156
2:
2.15 ( ) 90000 60 . , . .
2.16 10kg ( ), 2.21.
2.21
2.17 , , . ( .)
2.22
2.2
157
2.18 2.23 F . .
2.23
2.19 () 0.8. ( ) .
2.24
158
2:
2.20 F 2.25, m=5kg, .
2.25
) . ) . ( 30, F , 0.8 0.4.)
2.21 , m1 m2, 2.26. : ) m2 > m1 ) 1 2 .
2.26
2.3
159
2.3
. , , . , .
2.3.1
( 2.17). , . , , ( ). , cartoon 2.27. ( 2.28), ( ) .
160
2:
2.27
, . , , . , , . . .
2.28
2.3
161
. 2.29() r = 20m. , . , i, j k , . , R, . ( , , 2.29[]) :
R = r cos i + r sin jx y
(2.66)
( x = r cos y = r sin ) 2.29[] 2.29[]. . , , r.
162
2:
, :
= lim
d = t 0 t dt
(2.67)
, , . ( 2.29[]) ( : ) ( 2.30). 2.29[] 2.29[] . , , . , , . . , .
2.30
2.27 2.29[] / 2.29[] .
2.3
163
2.3.2
. 2.31 , R, t, , . , t , R . , :
V = lim
R t 0 t
(2.68)
.[13] , . . t0 , R, , S, . , :
V = lim
t 0
R S = lim t 0 t t
(2.69)
2.31
[13]
, 2.2.
164
2:
( ) :
S = r
(2.70)
(2.67), (2.69) (2.70), :V = limt 0
r = r lim = r t 0 t t
(2.71)
2.28 (, ) . : (2.71) .
, ( ) (2 ).
=
2 T
(2.72)
(2.71):
V=
2 r T
(2.73)
: : () , V , , : =V 2 = r
2.3
165
2.3.3
t. , (VA VB ) [14], , , , 2.32(). , V = (VB VA), . V 2.32(). V .
V = 2 V sin
2
(2.74)
V VA VB.
2.32
[14]
. , .
166
2:
, , . ( ) .
a = lim
V t 0 t
(2.75)
( [2.75]) . : a = a = lim
V V V = lim = lim t 0 t t 0 t t 0 t
(2.76)
(t0) , () . , VA VB ( 2.32[]) V VA. , . 2.32 . , r, ar. t . , : 0 t 0
lim sin
= 2 2
(2.77)
(2.74) (2.77) (2.76), :
V = lim a r = lim t 0 t t 0
2 V sin
2 = lim V = V t 0 t t
(2.78)
(2.71) (2.72), (2.78) :
ar =
V2 4 2 r = 2 r = r T2
(2.79)
2.3
167
: ) ; ) ; ) ;
2.29
: , . :
ar =
V2 4 2 r = 2 r = r T2
, m ar Fr, :
Fr = m a r
(2.80)
(2.80), Fr ( ) . , , , . , (2.79) (2.80) :
Fr = m
V2 4 2 r = m 2 r = m r T2
(2.81)
, , ( ). :
Fr = m
V2 4 2 r = m 2 r = m r T2
168
2:
2.3.4
2.3.1 . ( ) ( m) h ( [1.50] 1.3.6 1):
Fr = G
M m
( R + h)
2
(2.82)
R .[15] , Fr, . , :
Fr = G
M m
( R + h)
2
=
m V 2 = m 2 ( R + h) R + h) (
(2.83)
2.33
[15]
.
2.3
169
; ; ; : (2.81) (2.82). . () 2.84 108 m 2.66 106 rad s. ( G = 6.673 1011 Nm 2 kg 2 ) .
2.30
2.31
( ) . . 2.34() . ( 2.34[]) r. . , , . ; , .
2.34
170
2:
2.34[] t. . () . , , , . , :
N cos = B
(2.84)
:
Fr = N sin =
m V 2 r
(2.85)
(2.84) (2.85) . , = m g, :
r=
V2 g tan
(2.86)
. , , , . (2.86) . , , (2.86), ! , .
2.3
171
. . r, V. , . , . , , ( ) 180 ( 2.35[]) . 2.35 t. t, , . , .
2.35
172
2:
. F , . . , :
B = m g = F cos
(2.87)
. :
F sin =
m V 2 r V2 gr
(2.88)
(2.87) (2.88), :
tan =
(2.89)
( [2.86]), . 2.3.5
, . , , 2.36, m . . , , L. ( B = m g ) (T), , .
2.3
173
2.36 , , . ( r) ( t). :
Fr = m g cos + T :
(2.90)
Ft = m g sin
(2.91)
, V, .
2.36
, , , . , , . : Fr (2.90).
174
2:
ar :
ar = :
V2 L
(2.92)
Fr = m g cos + T =
m V 2 L
(2.93)
Ft (2.91). . at () :
at = :
dV dt
(2.94)
Ft = m g sin = m
dV dt
, :
a=
dV dV V2 = et + er dt dt Lat ar
(2.95)
et er . , . , .
2.3
175
2.3.6
2.37 V(t) t. , , u(t) . , . . 1.2.6 1 , . 2.1.3 , . ( ) , , , 2.37. , .
2.37 u(t) V(t). , .
176
2:
R ( t ) , R ( t ) R ( t ) , t . : () . :
V ( t ) = lim
R ( t + t ) R ( t ) t
t 0
=
dR ( t ) dt
(2.96)
() . :
u ( t ) = lim
R ( t + t ) R ( t ) t
t 0
=
dR ( t ) dt
(2.97)
2.37 :
R ( t ) = R ( t ) R ( t )
(2.98)
(2.98) , :
dR ( t )
dt dt dt V ( t ) = V ( t ) u ( t ) V ( t ) = V ( t ) + u ( t )
=
dR ( t )
dR ( t )(2.99)
, u(t) V ( t ) , V ( t ) :
V ( t ) = V ( t ) u ( t )
(2.100)
(2.100) .
2.3
177
.
u V :
2.32
u = 2 km h i + 0.2 km h j V = 0.9 km h i + 12 km h j i j , V .
, (2.100), . , . , L, V . u. . . , . L ( ) V, L/V . , 2 L/V. . , , , . , , , . u () . V ( ) ().
!!!
178
2:
( [2.99]) V+u. : L/(V+u). B A V ( ) (). ( [2.99]) Vu. : L/(Vu). , , :
T=
L L 2 V 2 L + = L 2 = 2 V +u V u V u V
1 u2 1 2 V
2.33 ; . , , . . . , , ( ). , Lorenz, Einstein. 2.3.7
, (2.100) . ; ( ), ;
2.3
179
, . . . . . . , 2 . . , , , ( ) : , . ( , ) . , ( ) . , . ( ) . . ( ) ; . . , . .
180
2:
2.38 m , a. , () , , . : , , a. ( = m g) (). , a. , . , , .
2.38 a. : ) ) .
2.3
181
(2.101).
2.34
T cos m g = 0 a tan = T sin = m a g
(2.101)
. . , () FI, 2.38(). : FI =ma ( ) ( ), :
T cos m g = 0 a tan = T sin m a = 0 g
(2.102)
.
( a) ; : F = m a , m .
2.35
1.3.7 1 . .
182
2:
2.39 a
2.36 2.39 a. F ( ). :
F = m (g a )
(2.103)
, , (2.103), ( a = g), . , , .
2.37 ( ) ; : a > g (2.103) . . , , . Atwod, 1.3.9 1.
2.3
183
2.40 twood
, , , mA mB (mA < mB) , . a, (a < g, a = g). 2.40, . . , , , . 2.40 , , . , ( ), ( ), . , 2 , :
184
2:
: :
mB g mB a FB = mB mA a FA + mA g = mA
(2.104) (2.105)
: . 3 , ( ).
FA = FB = T
(2.106)
(2.104), (2.105) (2.106) , :
mB mA (g a ) mB + mA 2 mB mA (g a ) T= mB + m A
=
(2.107) (2.108)
. (2.107) (2.108) . , . , , . , , : . , , , .
2.38 , a > g, (2.107) (2.108) , , , . , ( ) . 2.41.
2.3
185
2.41
2.3.8
. ( ) , . , , 2.42, . m, . () ( ) . . : ;
186
2:
2.42
() , (), , R R (R< R) , 2.43. : . . , , , , 2.43.
2.43
2.3
187
() () 2.43. , R. , :
2.39
tan =
2 rg
(2.109)
, , 2.44. , FI, .
2.44
188
2:
() , . , . , , , . , , . , FI , , , R , , ( 2.44) :
FI = m 2 R
(2.110)
, , , . , :
T sin = m 2 R T cos = m g
(2.111)
, , ( [2.109]) . , . , . , .
2.3
189
, . ; ;
2.40
2.45
. R= 6.37 106m = 7.292 105rad/s. m ( ) , 2.45. [16] . r, :
r = R cos
(2.112)
[16]
. , . , .
190
2:
, , . ( B0 = G
mM , M ) R2 , . , , , ( 2.46[]).
2.41 .
, , , (.. ) , T, . , , , . , , , . . . , FI, 2.46.
2.42 , , .
191
2.46 ()
. , :
= lim
t 0
d = t dt
, , . .
192
2:
: , V , , , =
V 2 = . r T
, . :
ar =
V2 4 2 r = 2 r = r T2
, , , ( ). :Fr = m V2 4 2 r = m 2 r = m r T2
: Fr Ft. , :
a=
dV dV V2 = et + er dt dt Lat ar
193
et er .
u(t) V ( t ) , V ( t ) , :
V ( t ) = V ( t ) u ( t )
. ( ) . . . . : FI = m a
194
2:
2.22 , ( ) , , 8.2 108. 5.3 1011m, .
2.23 132 . . .
2.24 15m . 1m/s, 1.5m/s. ) . . ) ( ), ;
2.25 m . .
2.47
195
2.26 2.48 F. 0.5, 0.75. .
2.48
2.27 , , r m ; ( : m = 1kg, = 10kg r = 0.5m.)
2.49
196
2:
2.28 2.50 . m r v. . : ) ) .
2.50
2.29 ( ). . ( 6.38 106m 5.975 1024kg , G = 6.672 1011m2/kg2.)
2.30 . . m v0, 2.51, t = 0. , . s. : )
197
, ) ) 10s .
2.51
2.31 1200km/h. . ( 2.52) . .
2.52
2.32 65km/h 120m. , , , 0.65m/s2. 90.
198
2:
2.33 940g 1.3m. ) 120, . ) .
2.34 TSS (Tethered Satellite System) SA 500kg, , 20km . 250km , 92.6% . TSS . 230km, 93.2% . TSS . ( TSS .)
2.53
2.35 40km/h 130m, 60km/h. ;
199
2.1 m F. v ( t ) = ( b t 2 ) i + ( c t + d ) j , t , , b, c d . b, c d, . : b, c d m/s3, m/s2 m/s. : a (t ) = ( 2 b t ) i + (c) j
2.2 21km/s. , . 0.035km/s2, . 250s, . : 22.6
2.3 , 1.9m . 4.5m, . : 7.23m/s
2.4 v0, h , 2.54. a .2 : a = tan 1 gh 2v0
2.5 5m 5m , 2.55. -
200
2:
2.54
; ; 1m . : ) 12.7m/s, ) 51.3
2.55
201
2.6 , 1968, . , . . (g = 9.786m/s2) . , Robert Beamon 8.90m , . , 25 . Beamon , g = 9.81m/s2, . ( ). : 8.88m
2.7 2.5kg 1.6m/s. , F1 F2. , F1 F1 = 15 j. 3s, {4.8m, 10.8m}. F2. : F2 = 9 j N
2.8 3.1kg . , a = 0.91 i 0.27 j. : F1 = 1.2 i 2.5 j , ; : 4.2 i + 1.66 j N
2.9 3700kg , 2.56.
202
2:
1100 25 . : ) ) 0.16m/s2. : ) 1990N, ) 1400
2.56
2.10 15kg , 2.57. . : 528
2.57
203
2.11 10kg , , 2.58. . : 1 = 139, 2 = 98
2.58
2.12 2.59, a1 a2, . . : a 2 = 2 a1 , a1 = m1g ( m1 + 4m2 )
2.59
204
2:
2.13 22kg 35 . 0.68, ; . : 342N
2.14 2.5kg 3.1kg , 2.60. 0.51, 0.23. . : 1.63m/s2 3.29
2.60
2.15 75m . : 97.6 km/h
2.16 7.2 kg . : ) 0.95m/s2, )
205
14m/s, ) 9s 14m/s. ( ) 450. : ) 77.47 ) 70.63 ) 59.4 52.69m/s2.
2.17 m1 m2, 2.61. F . . : F = ( m1 + m2 ) g tan
2.61
1.1 . : {0.5, 0, 0}. , Q, {1.0, 0.5, 0.8}, Y : {1.0(0.5), 0.5, 0.8} = {1.5, 0.5, 0.8}.
1.3 1.1 12s 22m.
1.4 x, t. (1.2). , , ,
( x2 x ) ( x1 x ) ( x2 ) ( x1 ) = , ( t2 t ) ( t1 t ) ( t2 ) ( t1 )1.5
.
x2 x1. (1.2) :
( x2 ) ( x1 ) ( x2 ) ( x1 ) = ( t2 ) ( t1 ) ( t2 ) ( t1 )1.6 . , , . V V , , .
1.7 : , , .
208
:
. , . , .
1.8) t= 0 ) .
1.9 2.
1.10, , (1.28) (1.29), (1.27). x x1. (1.27) x x (1.28), : x x1 = v 1 0 V v
1.11x (t ) = t + v (t ) = dx ( t ) dt = d d ( t ) + ( ) = dt dt
1.12 . t, , . , t , t .
v ( t ) = ( 0.094 m s ) + 2 ( 0.1m s 2 ) t + 3 ( 0.0009 m s3 ) t 2
1.13a ( t ) = 2 ( 0.1m s 2 ) + 2 3 ( 0.0009 m s3 ) t
209
1.14x (t ) =1 ( 3m s 2 ) t 2 2 v ( t ) = ( 3m s 2 ) t
1.15 .
1.16 .
1.17 .
1.18 .
1.19 . .
1.21 , , 10m/s. 1.25() . , , . 10m : . . , 10m/s ( ) . .
210
:
1.22 :
1 2 ( 2 m s 2 ) t 1 2 2 2 ( 2 m s ) t = ( 0.01m s ) t + 90 m 2 x = ( 0.01m s ) t + 90 m x =
( m s ) t ( 0.01m s ) t2 2
t 9.48s 90 m = 0 1 t2 +9.49s
. t1 (.. ), .
x = ( 0.01m s ) t1 + 90 m = ( 0.01m s ) 9.48s + 90 m = 90.0948m
1.23 1.26.
1.24V2 = V2 V1 = 8km h 20 km h = 12 km h , .
1.25V2 = V2 V1 = 0 km h 20 km h = 20 km h
1.26V2 = V2 V1 V2 = V2 + V1 = 10 km h 20 km h = 10 km h
1.27 : +90km/h
1.28 , . , , , , . , .
211
1.31 .
1.32 .
1.33 .
1.34 (1.54) m:
M m R2 M m B = G 2 R M m G 2 6 2 22 R2 M ( 6.37 10 ) ( 7.36 10 ) B R = = 2 = 0.165 2 M m R M B 1.74 106 ) ( 5.98 1024 ) ( G R2 B = G
1.35 .
1.36 . , V = a t . .
1.37 .
1.38 .
212
:
1.39v ( t ) = a1 t a1 = a 2 v (t ) = a 2 t
1.40 .
2.12 2 R = ( 2.935951) + ( 2.629045 ) 12
= 3.941026
2.2
2.3cos = R = 2.935951 3.941026 0.75, sin = R 0.661, 2.1 R = R cos i + R sin j
2.4A+B =
( x + x )
2
+ ( y + y ) , A B =2
( x x )
2
+ ( y y )
2
2.5 2.7, .
2.6x (1.162s ) = 1.736 1.162 = 2.017232 m y (1.162s ) = 9.848 1.162 4.905 (1.1622 ) = 4.82043m.: R (1.162s ) = ( 2.017232 m ) i + ( 4.82043m ) j t = 1.796s.
(
)
2.7 . , x() .
213
2.8 X , . , Y () , , . , , Y , .
2.9Vy(1.162s) = 1.55122m/s, Vy(1.796s) = 7.77076m/s. x . t = 1.162s :
|V(1.162s) | = [(1.736)2 + (1.55122)2]1/2m/s 2.326m/s. X (2.5) : cos = (1.736 ) 2.32 = 0.748(2.113)
, . : 270 < < 360 (2.113) 318.42. t = 1.796s.
2.10 (2.23) :V ( t ) = (1.736 m s ) i + 9.848m s ( 9.81m s 2 ) t j
(
)
(2.114)
(2.114) . t=0.3s V(0.3s)=(1.736 m/s) i+(6.905 m/s) j t=0.5s V(0.5s)=(1.736 m/s) i+(4.943 m/s) j. :
Vy ( t ) = 9.848m s ( 9.81m s 2 ) t j
(
)
(2.115)
j . , , Vy j. (2.115) :
Vy ( 0.9s ) = (1.019 m s ) j , Vy (1.7s ) = ( 6.829 m s ) j
214
:
2.11 B . . F .
2.12 .
2.13 . Ts = s N = B sin (2.116)
, . : Tk = k N < B sin (2.116) (2.117): s > k . (2.117)
2.14m a = B sin k N N = B cos m a = B sin k B cos
a = g ( sin k cos )
(2.118)
( [0,90]), . , (2.118), .
2.15 (.. ) . . , , . , . , , . , , .
215
. ( , ). : , , , . , .
2.16 (2.39) .
2.17 .
2.18 (2.39) .
2.19 .
2.20 !
2.21 .
2.22 (2.53) .
2.23 (2.55) 2 = 90.
2.24 (2.57) .
216
:
2.25 : V0 sin ,
(V0 sin )
2
+ 2 g cos h g cos ( 0 90
)
. . , 1.21 1 .
2.26 (2.63), (2.64) (2.65). ) = 0, sin = 0, cos = 1 ) = 0, = = 0 ) = 0, h = 0. (2.52) (2.55) . : Vx = V0cos Vy = V0sin gt
2.27 20 .
2.28 (2.71) V r = V/r. , . , , , . , .
2.29 . .
2.30 (2.81) (2.82) :
217
Fr = G T=
M m
( R + h)
2
= m3
4 2 ( R + h)T2
4 2 ( R + h) GM
.
2.31 m , M R , (2.83) :
G
M m
( R)
2
= m 2 R M = ( 2 R 3 ) G
(2.119)
, :
Fr = G
M m
( R)
2
= m ar ar = G
M
( R)
2
(2.120)
, . : 1N = 1kg 1m/s2
2.32 (2.99) V(t) .
2.33 . L : t = L (V u ) (2.121) . (2.121) ( ) . , ( ) . , , , .
2.34 .
218
:
2.35 , F, . , , , .
2.36 (), () . , :
B = T + m a T = m g m a = m(g a ), 3 : F = T (2.122) (2.123): F = m ( g a )
(2.