Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 42 стр.

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ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɡɧɚɱɟɧɢɣ
v
A
,
Z
2
ɢ
I
A
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɢɡ ɮɨɪɦɭɥ (6),
(8) ɢ (9) ɜɵɪɚɠɟɧɢɟ (5) ɩɪɢɦɟɬ ɜɢɞ
T
P
g
rr
()
()
2
2
12
22
3
4
M
. (10)
ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ (3), (4) ɢ (10), ɡɚɩɢɲɟɦ ɜɵɪɚɠɟɧɢɟ
ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɩɥɚɧɟɬɚɪɧɨɝɨ ɦɟɯɚɧɢɡɦɚ:
T
PP
g
rr
29
12
2
12
22
()
M
. (11)
ȼɵɱɢɫɥɢɦ ɱɚɫɬɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
Ɍ ɩɨ ɨɛɨɛɳɟɧɧɨɣ
ɫɤɨɪɨɫɬɢ
M
:
M
Mw
w
2
21
2
)(
6
92
rr
g
PPT
ɢ ɜɨɡɶɦɟɦ ɩɪɨɢɡɜɨɞɧɭɸ
ɩɨɥɭɱɟɧɧɨɝɨ ɪɟɡɭɥɶɬɚɬɚ ɩɨ ɜɪɟɦɟɧɢ:
M
Mw
w
2
21
2
)(
12
92
rr
g
PPT
dt
d
. (12)
Ɂɚɦɟɬɢɜ, ɱɬɨ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ
Ɍ ɫɢɫɬɟɦɵ, ɨɩɪɟɞɟɥɟɧɧɚɹ
ɮɨɪɦɭɥɨɣ (11), ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ
M
, ɧɚɯɨɞɢɦ:
0
wM
w
T
. (13)
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɜɵɪɚɠɟɧɢɣ (2), (12), (13) ɜ ɭɪɚɜɧɟɧɢɟ Ʌɚɝɪɚɧɠɚ
ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɟɯɚɧɢɡɦɚ ɞɥɹ
ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ
M
:
 
>@
29
6
1
2
22
2
12
2
0212
PP
g
rr m P Prr

cos
MM
,
ɨɬɤɭɞɚ ɨɩɪɟɞɟɥɹɟɦ ɢɫɤɨɦɨɟ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ

M
ɤɪɢɜɨɲɢɩɚ ɈȺ:


cos
M
M

3
22
29
0212
2
g
mPPrr
PP
. (14)
Ɋɚɜɧɨɦɟɪɧɨɟ ɜɪɚɳɟɧɢɟ ɤɪɢɜɨɲɢɩɚ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ
ɭɫɥɨɜɢɹ:

mPPrr
0212
1
2
2 cos
M
.
Ɂɚɞɚɱɚ ʋ 3. Ʉ ɤɨɧɰɚɦ ɬɨɧɤɨɣ ɧɟɪɚɫɬɹɠɢɦɨɣ ɧɢɬɢ ɩɪɢɜɹɡɚɧɵ ɝɪɭɡ Ⱥ
ɜɟɫɨɦ Ɋ
1
ɢ ɝɪɭɡ ȼ ɜɟɫɨɦ Ɋ
2
. ɇɢɬɶ ɩɟɪɟɛɪɨɲɟɧɚ ɱɟɪɟɡ ɛɥɨɤɢ D ɢȿɢ
ɨɯɜɚɬɵɜɚɟɬ ɫɧɢɡɭ ɩɨɞɜɢɠɧɨɣ ɛɥɨɤ Ʉ. ɄɨɫɢɈ
5
ɩɨɞɜɢɠɧɨɝɨ ɛɥɨɤɚ Ʉ
ɩɪɢɤɪɟɩɥɟɧ ɝɪɭɡ L ɜɟɫɨɦ Ɋ
6
; Ɋ
3
– ɜɟɫ ɛɥɨɤɚ D, Ɋ
4
– ɜɟɫ ɛɥɨɤɚ ȿ, Ɋ
5
– ɜɟɫ
ɛɥɨɤɚ Ʉ. Ƚɪɭɡɵ Ⱥ ɢ ȼ ɞɜɢɠɭɬɫɹ ɩɨ ɧɚɤɥɨɧɧɵɦ ɩɥɨɫɤɨɫɬɹɦ,
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɚɫɩɨɥɨɠɟɧɧɵɦ ɩɨɞ ɭɝɥɚɦɢ
D
ɢ
E
ɤ ɝɨɪɢɡɨɧɬɭ.
Ɉɩɪɟɞɟɥɢɬɶ ɭɫɤɨɪɟɧɢɹ ɝɪɭɡɨɜ Ⱥ, ȼɢ L. Ȼɥɨɤɢ ɫɱɢɬɚɬɶ ɨɞɧɨɪɨɞɧɵɦɢ
ɤɪɭɝɥɵɦɢ ɞɢɫɤɚɦɢ. ɋɢɥɚɦɢ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɝɪɭɡɨɜ ɨ ɧɚɤɥɨɧɧɵɟ
ɩɥɨɫɤɨɫɬɢ ɢ ɦɚɫɫɨɣ ɧɢɬɢ ɩɪɟɧɟɛɪɟɱɶ.
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɡɧɚɱɟɧɢɣ v A , Z 2 ɢ I A ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɢɡ ɮɨɪɦɭɥ (6),
                                      (2)    3 P2
(8) ɢ (9) ɜɵɪɚɠɟɧɢɟ (5) ɩɪɢɦɟɬ ɜɢɞ T              (r  r ) 2 M 2 .      (10)
                                            4 g 1 2
ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ (3), (4) ɢ (10), ɡɚɩɢɲɟɦ ɜɵɪɚɠɟɧɢɟ
ɤɢɧɟɬɢɱɟɫɤɨɣ                  ɷɧɟɪɝɢɢ     ɩɥɚɧɟɬɚɪɧɨɝɨ              ɦɟɯɚɧɢɡɦɚ:
     2 P  9 P2
T               (r1  r2 ) 2 M 2 .                                      (11)
        12 g
ȼɵɱɢɫɥɢɦ ɱɚɫɬɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ Ɍ ɩɨ ɨɛɨɛɳɟɧɧɨɣ
                  w T 2 P  9 P2
ɫɤɨɪɨɫɬɢ M :                    (r1  r2 ) 2 M ɢ ɜɨɡɶɦɟɦ ɩɪɨɢɡɜɨɞɧɭɸ
                  wM      6g
                                      d w T 2 P  9 P2
ɩɨɥɭɱɟɧɧɨɝɨ ɪɟɡɭɥɶɬɚɬɚ ɩɨ ɜɪɟɦɟɧɢ:                     (r1  r2 ) 2 M . (12)
                                     dt wM       12 g
    Ɂɚɦɟɬɢɜ, ɱɬɨ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ Ɍ ɫɢɫɬɟɦɵ, ɨɩɪɟɞɟɥɟɧɧɚɹ
ɮɨɪɦɭɥɨɣ (11), ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ M , ɧɚɯɨɞɢɦ:
wT
      0.                                                                  (13)
wM
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɜɵɪɚɠɟɧɢɣ (2), (12), (13) ɜ ɭɪɚɜɧɟɧɢɟ Ʌɚɝɪɚɧɠɚ
ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɟɯɚɧɢɡɦɚ ɞɥɹ
ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ M :
2 P  9 P2             1
   6g
                   2
           r1  r2 M
                       2
                         >
                         2m0  P  2 P2 r1  r2 cosM ,@
ɨɬɤɭɞɚ ɨɩɪɟɞɟɥɹɟɦ ɢɫɤɨɦɨɟ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ M ɤɪɢɜɨɲɢɩɚ ɈȺ:
         2m0  P  2 P2 r1  r2 cos M
M 3g                             .                    (14)
              2 P  9 P2
Ɋɚɜɧɨɦɟɪɧɨɟ ɜɪɚɳɟɧɢɟ ɤɪɢɜɨɲɢɩɚ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ
                1
ɭɫɥɨɜɢɹ:   m0     P  2 P2 r1  r2 cosM .
               2


    Ɂɚɞɚɱɚ ʋ 3. Ʉ ɤɨɧɰɚɦ ɬɨɧɤɨɣ ɧɟɪɚɫɬɹɠɢɦɨɣ ɧɢɬɢ ɩɪɢɜɹɡɚɧɵ ɝɪɭɡ Ⱥ
ɜɟɫɨɦ Ɋ1 ɢ ɝɪɭɡ ȼ ɜɟɫɨɦ Ɋ2. ɇɢɬɶ ɩɟɪɟɛɪɨɲɟɧɚ ɱɟɪɟɡ ɛɥɨɤɢ D ɢ ȿ ɢ
ɨɯɜɚɬɵɜɚɟɬ ɫɧɢɡɭ ɩɨɞɜɢɠɧɨɣ ɛɥɨɤ Ʉ. Ʉ ɨɫɢ Ɉ5 ɩɨɞɜɢɠɧɨɝɨ ɛɥɨɤɚ Ʉ
ɩɪɢɤɪɟɩɥɟɧ ɝɪɭɡ L ɜɟɫɨɦ Ɋ6; Ɋ3 – ɜɟɫ ɛɥɨɤɚ D, Ɋ4 – ɜɟɫ ɛɥɨɤɚ ȿ, Ɋ5 – ɜɟɫ
ɛɥɨɤɚ Ʉ. Ƚɪɭɡɵ Ⱥ ɢ ȼ ɞɜɢɠɭɬɫɹ ɩɨ ɧɚɤɥɨɧɧɵɦ ɩɥɨɫɤɨɫɬɹɦ,
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɚɫɩɨɥɨɠɟɧɧɵɦ ɩɨɞ ɭɝɥɚɦɢ D ɢ E ɤ ɝɨɪɢɡɨɧɬɭ.
Ɉɩɪɟɞɟɥɢɬɶ ɭɫɤɨɪɟɧɢɹ ɝɪɭɡɨɜ Ⱥ, ȼ ɢ L. Ȼɥɨɤɢ ɫɱɢɬɚɬɶ ɨɞɧɨɪɨɞɧɵɦɢ
ɤɪɭɝɥɵɦɢ ɞɢɫɤɚɦɢ. ɋɢɥɚɦɢ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɝɪɭɡɨɜ ɨ ɧɚɤɥɨɧɧɵɟ
ɩɥɨɫɤɨɫɬɢ ɢ ɦɚɫɫɨɣ ɧɢɬɢ ɩɪɟɧɟɛɪɟɱɶ.

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