Лебединская. Динамика материальной точки
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M .
4- . 9.2.
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'. 9.2 |
. 9.2 4 x. !, |
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ω, L = I ω x. , |
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, z, |
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, , - |
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y, . + |
M = [rF ], |
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M L , 2 ,
, L . + , ( L , , 4 -
x, z. , 4
( z.
, ( * - Ω. |
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Δϕ, |
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t |
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Ω = ϕ/ t . |
(9.2) |
& (
- 4 . '
dt. &, : 2
, dϕ , 2 dL
, - 2 ( L .
. 9.2 ,
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dϕ = |
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= dL , |
(9.3) |
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r |
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L |
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L |
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= dL – dL , |
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= L – L . |
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dL |
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C (9.3) (9.1), (9.2) |
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Ω = M . |
(9.4) |
L |
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, 4 ω |
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( Ω ( - .
F , -4 ,
, m.
mg . , 2 l
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- r |
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. + , |
M = mgl . |
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0 (9.4) , |
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Ω = mgl , |
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I ω |
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4 |
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ω = mgl . |
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(9.5) |
I Ω |
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! 4 , :
, (9.5) ,
( .
+!# ! , !& # ! ,
1. - , : - 4- . 8, ,
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-
2., * . /
4(().
3.! 4- , , -
t, ϕ
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Ω . / * .
4..
5.' - 4 ω (9.5).
6., *1±5 , .
7.' (.
8., (
. , -
, $
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9./( Ω ω ( - -
).
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8, |
* *, |
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± / . % *
: 360° = 2π .
( I = 0,01 · 2.
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2.%( 4 ),
4
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3.9 ( ? / ?
4.9 ? 1 ? 1
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7.(9.5) 4 . /
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( .
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1.! . . 1 . +. 1. ± .: , 1989. ± !. 94±116.
2.+ +. . 1 . ± .: . ., 2001. ± !. 34±46.
55
' 1.10
+: + :+ 7 ; = 0 :
9 &) # ! ,: I ±
4- ; II ±
4- .
+# !#, # & . !: ,
*( , .
&. ( !#! ! !"!
! mg - (-
( ) Fg ( ( F( , - -42 48 ( . 10.1):
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mg = Fg + F( . |
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(10.1) |
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! Fg |
( 8, |
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Fg = G |
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8m |
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(10.2) |
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R2 |
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G = 6,67 ×10−11 |
2 -2 ± ( , M |
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R ± |
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8 . - - 8 ( -
22 ) .
'. 10.1
) ( F( ( ,
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= mω2 R , |
(10.3) |
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ω8 ± 4 8, |
R ± , |
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56
) ( , (,
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. 10.1 , , ,
* 4 * 8 j, |
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R = R8 os j. |
(10.4) |
, (10.4) (10.3), *, ( -
-* 2. 2 ,
48 (ω8 = 2p / 86400 = 7,27×10-5 -1)
1/291 .
. 10.1 , ( ,
, -4 ,
, ( 8,
b, 4 . ' ,
sinβ = 0,0018sin 2ϕ ,
, 2(j = 0°) -*(j = 90°) b = 0°
( 8.
45°, b » 6¢
, ( , 2
- - - , (10.2),
g = GM 8 / R82 . / *
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(.
4 -
4- .
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( ! * !'* # %
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( . 10.2). % |
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'. 10.2 |
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57
1 a ( . 10.3),
mg , -4- , -4: Fn ,
- , Fτ , - . !-
-4
α
F
Fn F ,
-4 Fτ . /
4 .
, Fτ = mg . 0 a , sina
a,
*. + a = /l , , 4-4
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x |
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Fτ = −mg x , |
(10.4) |
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± 4 - |
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Fn |
, l ± , ²±² , |
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- . |
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mg |
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'. 10.3 |
+ II |
- : |
- mg x = mx&, |
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(10.5) |
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l |
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&x ± 4 ,
( τ , , Fτ . /
g / l = ω2 , |
(10.6) |
0 |
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(10.5) |
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&x + ω2 x = 0 . |
(10.7) |
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(10.7) (2 ),
4 x -4- (- :
x = Acos(ω0t + ϕ) , |
(10.8) |
# j ± , .
, * *
, ( ). , (10.8), *, # ±
( ), j ±
, ω0 ± (( ) ,
* :
w = |
2π |
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(10.9) |
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0 |
T |
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! (10.6)
*
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T = 2p |
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(10.10) |
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g |
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, g, 4-
8
g = |
4π2l |
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(10.11) |
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T 2 |
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, , t,
n :
T = t / n . |
(10.12) |
+!# ! , !& # ! , |
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1. / l |
( |
( ). |
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2./ 10±15° . / 15±30
* . , 3 .
3.n (10.12).
4., (10.11)
8.
5.% (.
5 |
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6. /( g * -
. g ,
(,.7) ²' ¼ ².
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1.1 ? ,
? 8 .
2.8 ( (. 1
.
3.1 ? 8
II - ( .
4.8 . 1
-4 ?
5.9 ? 1
( ?
6.%, ((10.8) (10.7).
7.% : (10.11).
59
II
# &( !# ! ! !"!
!*!-)5 2 ' !"! *
( ! * !'* # %
- ( . 10.4),
&, * 4 ( %. 0
α ,
mg , 4-4 . ,
4 (+.13). (+.13)
−mgl sinα = I d 2α |
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(10.13) |
dt2 |
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m – , g – , l –
4 ( ,
h = l sin α – , I – (
4, d2α /dt2 = α&= ε – .
+ *
*, sinα α, (10.13)
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α&+ mgl α = 0 , |
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α&+ ω2α = 0, |
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(10.14) |
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0 |
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mgl |
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= ω2 . |
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(10.15) |
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(10.14) |
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-4 ( ( 2 |
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α |
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α = αmax cos(ω0t + ϕ0 ) . |
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(10.16) |
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. . (10.16) , αmax |
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ϕ0 – , ω0 – |
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mg |
( , |
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'. 10.4 |
ω0 = 2π /T |
(10.17) |
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* = 2π |
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(10.18) |
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mgl |
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( I &
4- 3
I = I0 + ml 2 , |
(10.19) |
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I0 – ( , * 4
( % &.
! (10.19) (10.18) :
T = 2π |
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I0 |
+ ml 2 |
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(10.20) |
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mgl |
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-, ( I0 . %, - I0 g, -
. 4 ( . 10.5)
1,
2, 3 * /1 /2, * ,
4 . / ,
4 - ,
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% (10.20)
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T1 = 2π |
I0 + ml 12 |
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T = 2π |
I0 + ml 22 , |
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mgl 2 |
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l 1 – &1 ( , l 2 – |
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4&2 ( . |
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O2 |
- I0 (10.21), * |
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g = |
4π2 |
(l 2 |
− l 2 ) |
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(10.22) |
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T 2l |
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− T 2l |
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'. 10.5 |
(10.22) - . , |
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* (10.12), 20–30 * . %
l 1 l 2 - -
( , -4 - . , 4 ,
( . ' *
l 1 l 2 . 0 l 1 l 2 -
, T1 T2 , , ,
g * l 1 l 2 -.
+!# ! , !& # ! ,
1.' -- .
2.& - 4. /
10±15°, t 20±30 * . /
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* . . , -- T1 (10.12).
3., , - 4 ,
, 2, T2 .
4.! 4 4-
( l 1 l 2 .
5.' (10. 9).
6.% (.
5 n |
t1 |
<t1> <T1> |
t2 <t2> <T2> |
l 1 |
l 2 |
g |
1
2
3
7. ' * . % -
, (,.7) ²' ¼ ².
! #!&) ,! #! ,
1.1 ? 9 -
? 8 . 1 Fg ?
2.8 ( (. 1
**8 ?
3.9 ( (( ) . 1 ?
4.% ( . / ( ? 1
? 0 ( . ! 3 .
5.% . 9 , -4
? 1 ?
6.8 4 .
.
7.%, ((10.16) (10.14). 1
-4 ?
8.9 ? 1
( ?
9.% : (10.22).
& 9,780 / 2 2 9,832 / 2 -*. 8 g = 9,806 / 2
.
# (#
1.! . . 1 4 . +. 1. ± .: , 1989. ± !. 118±122.
2.+ +. . 1 . ± .: . ., 2001. ± !. 34±46.
62