Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 41 стр.

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Ⱥɤɬɢɜɧɵɦɢ ɫɢɥɚɦɢ ɹɜɥɹɸɬɫɹ: Ɋ – ɜɟɫ ɤɪɢɜɨɲɢɩɚ, Ɋ
2
– ɜɟɫ ɤɨɥɟɫɚ 2,
m
0
– ɜɪɚɳɚɸɳɢɣ ɦɨɦɟɧɬ, ɩɪɢɥɨɠɟɧɧɵɣ ɤ ɤɪɢɜɨɲɢɩɭ ɈȺ. ȼɫɟ ɫɜɹɡɢ,
ɧɚɥɨɠɟɧɧɵɟ ɧɚ ɫɢɫɬɟɦɭ, ɢɞɟɚɥɶɧɵ.
Ⱦɚɞɢɦ ɤɪɢɜɨɲɢɩɭ
ɈȺ ɜɨɡɦɨɠɧɨɟ ɭɝɥɨɜɨɟ ɩɟɪɟɦɟɳɟɧɢɟ
G
M
ɜ
ɧɚɩɪɚɜɥɟɧɢɢ ɜɨɡɪɚɫɬɚɧɢɹ ɭɝɥɚ
M
, ɬ. ɟ. ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɫɢɥɵ
Q
M
ɜɵɱɢɫɥɢɦ ɫɭɦɦɭ ɪɚɛɨɬ
ɚɤɬɢɜɧɵɯ ɫɢɥ ɧɚ ɜɨɡɦɨɠɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ
GM
:
GMMGMMGMG
coscos
20
OAPOCPmA
.
Ɍɚɤ ɤɚɤ
OA=O
P
-A
P
=r
1
-r
2
, ɚ
OC
OA
rr
22
12
, ɬɨ

>@
GMGM
AmPPrr
1
2
22
0212
cos
. (1)
ɍɱɢɬɵɜɚɹ, ɱɬɨ
G
G
M
M
AQ
, ɧɚɯɨɞɢɦ ɨɛɨɛɳɟɧɧɭɸ ɫɢɥɭ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ
ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ
M
:

>@
QmPPrr
M
M
1
2
22
0212
cos
. (2)
ɉɟɪɟɯɨɞɢɦ ɤ ɜɵɱɢɫɥɟɧɢɸ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
Ɍ ɦɟɯɚɧɢɡɦɚ, ɜ ɫɨɫɬɚɜ
ɤɨɬɨɪɨɝɨ ɜɯɨɞɹɬ ɦɚɫɫɵ ɤɪɢɜɨɲɢɩɚ
ɈȺ ɢ ɡɭɛɱɚɬɨɝɨ ɤɨɥɟɫɚ 2 (ɡɭɛɱɚɬɨɟ
ɤɨɥɟɫɨ 1 ɧɟɩɨɞɜɢɠɧɨ), ɬ. ɟ.
TT
T
() ( )12
. (3)
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɤɪɢɜɨɲɢɩɚ
ɈȺ, ɜɪɚɳɚɸɳɟɝɨɫɹ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ
ɨɫɢ
Ɉ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɤ ɩɥɨɫɤɨɫɬɢ ɪɢɫɭɧɤɚ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ
TI
O
()
12
1
2
M
, ɝɞɟ

I
P
g
OA
P
g
rr
O
1
3
1
3
2
12
2
– ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ
ɤɪɢɜɨɲɢɩɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
T
P
g
rr
()
()
1
12
22
1
6
M
. (4)
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɡɭɛɱɚɬɨɝɨ ɤɨɥɟɫɚ 2, ɫɨɜɟɪɲɚɸɳɟɝɨ ɩɥɨɫɤɨɟ
ɞɜɢɠɟɧɢɟ, ɪɚɜɧɚ
T
P
g
vI
AA
()2
2
2
2
2
1
2
1
2
Z
. (5)
ɇɚɣɞɟɦ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ
Ⱥ, ɹɜɥɹɸɳɟɣɫɹ ɤɨɧɰɨɦ ɤɪɢɜɨɲɢɩɚ ɈȺ:
vOA rr
A
()
M
M
12
. (6)
Ɋɚɫɫɦɨɬɪɢɦ ɫɤɨɪɨɫɬɶ ɬɨɣ ɠɟ ɬɨɱɤɢ Ⱥ, ɩɪɢɧɚɞɥɟɠɚɳɟɣ ɡɭɛɱɚɬɨɦɭ ɤɨɥɟɫɭ 2,
ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɦɝɧɨɜɟɧɧɨɦɭ ɰɟɧɬɪɭ ɫɤɨɪɨɫɬɟɣ
P
ɤɨɥɟɫɚ:
22
Z
rv
A
. (7)
ɋɨɩɨɫɬɚɜɥɹɹ ɮɨɪɦɭɥɵ (6) ɢ (7), ɧɚɯɨɞɢɦ:
ZM
2
12
2
rr
r
. (8)
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɡɭɛɱɚɬɨɝɨ ɤɨɥɟɫɚ 2 ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
I
Pr
g
A
22
2
2
. (9) 
    Ⱥɤɬɢɜɧɵɦɢ ɫɢɥɚɦɢ ɹɜɥɹɸɬɫɹ: Ɋ – ɜɟɫ ɤɪɢɜɨɲɢɩɚ, Ɋ2 – ɜɟɫ ɤɨɥɟɫɚ 2,
m0 – ɜɪɚɳɚɸɳɢɣ ɦɨɦɟɧɬ, ɩɪɢɥɨɠɟɧɧɵɣ ɤ ɤɪɢɜɨɲɢɩɭ ɈȺ. ȼɫɟ ɫɜɹɡɢ,
ɧɚɥɨɠɟɧɧɵɟ ɧɚ ɫɢɫɬɟɦɭ, ɢɞɟɚɥɶɧɵ.
    Ⱦɚɞɢɦ ɤɪɢɜɨɲɢɩɭ ɈȺ ɜɨɡɦɨɠɧɨɟ ɭɝɥɨɜɨɟ ɩɟɪɟɦɟɳɟɧɢɟ GM ɜ
ɧɚɩɪɚɜɥɟɧɢɢ ɜɨɡɪɚɫɬɚɧɢɹ ɭɝɥɚ M , ɬ. ɟ. ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ.
    Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɫɢɥɵ QM ɜɵɱɢɫɥɢɦ ɫɭɦɦɭ ɪɚɛɨɬ
ɚɤɬɢɜɧɵɯ ɫɢɥ ɧɚ ɜɨɡɦɨɠɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ GM :
    GA m0GM  P OC cos M ˜ GM  P2 OA cos M ˜ GM .
                                         OA   r1  r2
    Ɍɚɤ ɤɚɤ OA=OP-AP=r1-r2, ɚ OC                      , ɬɨ
                                          2      2
             1
    GA     > 2
                                         @
               2m0  P  2 P2 r1  r2 cosM GM .                      (1)
ɍɱɢɬɵɜɚɹ, ɱɬɨ GA QM GM , ɧɚɯɨɞɢɦ ɨɛɨɛɳɟɧɧɭɸ ɫɢɥɭ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ
                                     1
ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ M : QM       > 2
                                                             @
                                       2m0  P  2 P2 r1  r2 cosM . (2)
ɉɟɪɟɯɨɞɢɦ ɤ ɜɵɱɢɫɥɟɧɢɸ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ Ɍ ɦɟɯɚɧɢɡɦɚ, ɜ ɫɨɫɬɚɜ
ɤɨɬɨɪɨɝɨ ɜɯɨɞɹɬ ɦɚɫɫɵ ɤɪɢɜɨɲɢɩɚ ɈȺ ɢ ɡɭɛɱɚɬɨɝɨ ɤɨɥɟɫɚ 2 (ɡɭɛɱɚɬɨɟ
ɤɨɥɟɫɨ 1 ɧɟɩɨɞɜɢɠɧɨ), ɬ. ɟ. T T (1)  T ( 2 ) .                      (3)
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɤɪɢɜɨɲɢɩɚ ɈȺ, ɜɪɚɳɚɸɳɟɝɨɫɹ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ
ɨɫɢ Ɉ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɤ ɩɥɨɫɤɨɫɬɢ ɪɢɫɭɧɤɚ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ
       1                        1P          1P              2
T (1)     I M 2 , ɝɞɟ    IO         OA 2        r  r – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ
       2 O                      3g          3g 1 2
                                 (1)  1P
ɤɪɢɜɨɲɢɩɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, T               (r  r ) 2 M 2 .          (4)
                                      6g 1 2
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɡɭɛɱɚɬɨɝɨ ɤɨɥɟɫɚ 2, ɫɨɜɟɪɲɚɸɳɟɝɨ ɩɥɨɫɤɨɟ
                     (2) 1 P2 2 1
ɞɜɢɠɟɧɢɟ, ɪɚɜɧɚ T             v  I Z2 .                             (5)
                         2 g A 2 A 2
ɇɚɣɞɟɦ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ Ⱥ, ɹɜɥɹɸɳɟɣɫɹ ɤɨɧɰɨɦ ɤɪɢɜɨɲɢɩɚ ɈȺ:
v A OA M (r1  r2 )M .                                             (6)
Ɋɚɫɫɦɨɬɪɢɦ ɫɤɨɪɨɫɬɶ ɬɨɣ ɠɟ ɬɨɱɤɢ Ⱥ, ɩɪɢɧɚɞɥɟɠɚɳɟɣ ɡɭɛɱɚɬɨɦɭ ɤɨɥɟɫɭ 2,
ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɦɝɧɨɜɟɧɧɨɦɭ ɰɟɧɬɪɭ ɫɤɨɪɨɫɬɟɣ P ɤɨɥɟɫɚ: v A r2Z2 . (7)
                                                    r1  r2
ɋɨɩɨɫɬɚɜɥɹɹ ɮɨɪɦɭɥɵ (6) ɢ (7), ɧɚɯɨɞɢɦ: Z 2                   M .   (8)
                                                       r2
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɡɭɛɱɚɬɨɝɨ ɤɨɥɟɫɚ 2 ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
      P2 r22
IA           .                                                       (9)
       2g


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