Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 46 стр.

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46
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ
M
5
ɛɥɨɤɚ Ʉ ɧɚɣɞɟɦ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ
Ɇ, ɩɪɢɧɹɜ ɡɚ ɩɨɥɸɫ ɬɨɱɤɭ N:
vvv
Mx Nx MN x
()
, ɬ. ɟ.
()

vvvss
MN x Mx Nx
12
.
Ɍɚɤ ɤɚɤ
()

vMNr
MN x
M
M
555
2
, ɬɨ
2
55 1 2
rss

M
, ɨɬɤɭɞɚ ɭɝɥɨɜɚɹ
ɫɤɨɪɨɫɬɶ ɛɥɨɤɚ
Ʉ:

M
5
12
5
2
ss
r
. (11)
Ʉɢɧɟɬɢɱɟɫɤɢɟ ɷɧɟɪɝɢɢ ɝɪɭɡɨɜ
Ⱥ ɢ ȼ, ɫɨɜɟɪɲɚɸɳɢɯ ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ
ɞɜɢɠɟɧɢɟ, ɢɦɟɸɬ ɜɢɞ
T
P
g
s
()
1
1
1
2
1
2
,
T
P
g
s
()
2
2
2
2
1
2
. (12)
ȼɵɱɢɫɥɢɦ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɛɥɨɤɨɜ
D ɢ ȿ, ɜɪɚɳɚɸɳɢɯɫɹ ɜɨɤɪɭɝ
ɧɟɩɨɞɜɢɠɧɵɯ ɨɫɟɣ:
2
35
)3(
2
1
M
O
IT
,
2
44
)3(
2
1
M
O
IT
.
ɉɨɞɫɬɚɜɢɜ ɡɧɚɱɟɧɢɹ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ ɛɥɨɤɨɜ
¸
¸
¹
·
¨
¨
©
§
g
rP
I
g
rP
I
OO
2
,
2
2
44
4
2
33
5
ɢ ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ (9), ɧɚɯɨɞɢɦ
T
Ps
g
()
3
31
2
4
,
T
Ps
g
()
4
42
2
4
. (13)
Ʉɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɛɥɨɤɚ
Ʉ, ɫɨɜɟɪɲɚɸɳɟɝɨ ɩɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ,
ɨɩɪɟɞɟɥɢɦ ɩɨ ɮɨɪɦɭɥɟ
2
55
2
5
5
)5(
2
1
2
1
M
OO
Iv
g
P
T
.
ɉɨɞɫɬɚɜɢɜ ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɛɥɨɤɚ
Ʉ
¸
¸
¹
·
¨
¨
©
§
g
rP
I
O
2
2
55
5
ɢ
ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ (10) ɢ (11), ɩɨɥɭɱɢɦ:

T
P
g
ss
P
g
ss
()

5
5
1
2
2
2
5
12
3
16
1
8
. (14)
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɝɪɭɡɚ L, ɞɜɢɠɭɳɟɝɨɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ, ɢɦɟɟɬ ɜɢɞ
T
P
g
v
O
()6
6
2
1
2
5
.
ɉɪɢɦɟɧɢɜ ɮɨɪɦɭɥɭ (10), ɨɩɪɟɞɟɥɹɟɦ
     Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ M5 ɛɥɨɤɚ Ʉ ɧɚɣɞɟɦ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ
Ɇ, ɩɪɢɧɹɜ ɡɚ ɩɨɥɸɫ ɬɨɱɤɭ N:                    v Mx v Nx  (v MN ) x , ɬ. ɟ.
(v MN ) x v Mx  v Nx s1  s2 .
           Ɍɚɤ ɤɚɤ (v MN ) x           MN M5 2r5M5 , ɬɨ 2r5M5                 s1  s2 , ɨɬɤɭɞɚ ɭɝɥɨɜɚɹ
                                       s1  s2
ɫɤɨɪɨɫɬɶ ɛɥɨɤɚ Ʉ: M5
                                          2r5 .                                                      (11)

    Ʉɢɧɟɬɢɱɟɫɤɢɟ ɷɧɟɪɝɢɢ ɝɪɭɡɨɜ Ⱥ ɢ ȼ, ɫɨɜɟɪɲɚɸɳɢɯ ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ
ɞɜɢɠɟɧɢɟ, ɢɦɟɸɬ ɜɢɞ
                    1 P1 2 ( 2 )             1 P2 2
           T (1)        s , T                   s .                                                (12)
                    2 g 1                    2 g 2
    ȼɵɱɢɫɥɢɦ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɛɥɨɤɨɜ D ɢ ȿ, ɜɪɚɳɚɸɳɢɯɫɹ ɜɨɤɪɭɝ
ɧɟɩɨɞɜɢɠɧɵɯ ɨɫɟɣ:
                    1                           1
           T ( 3)     I O 5M 32 , T ( 3)         I O 4M 42 .
                    2                           2
           ɉɨɞɫɬɚɜɢɜ               ɡɧɚɱɟɧɢɹ         ɦɨɦɟɧɬɨɜ      ɢɧɟɪɰɢɢ        ɛɥɨɤɨɜ
§            P3 r32                P4 r42 ·
¨¨ I O 5            , I O4                ¸ ɢ ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ (9), ɧɚɯɨɞɢɦ
 ©            2g                    2 g ¸¹

                    P3 s12                 P4 s22
           T ( 3)           , T
                                (4)
                                                    .                                                (13)
                     4g                      4g
           Ʉɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɛɥɨɤɚ Ʉ, ɫɨɜɟɪɲɚɸɳɟɝɨ ɩɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ,
                                                        1 P5 2 1
ɨɩɪɟɞɟɥɢɦ ɩɨ ɮɨɪɦɭɥɟ                         T (5)          vO 5  I O 5M52 .
                                                        2 g       2
                                                       §                                        P5 r52 ·
           ɉɨɞɫɬɚɜɢɜ ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɛɥɨɤɚ Ʉ ¨¨ I O 5                                         ¸
                                                       ©                                         2 g ¸¹ ɢ
ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ (10) ɢ (11), ɩɨɥɭɱɢɦ:
                     3 P5 2            1 P5
           T ( 5)         s1  s22       s s .                                                  (14)
                    16 g               8 g 1 2
           Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɝɪɭɡɚ L, ɞɜɢɠɭɳɟɝɨɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ, ɢɦɟɟɬ ɜɢɞ
                    1 P6 2
           T ( 6)       v .
                    2 g O5
           ɉɪɢɦɟɧɢɜ ɮɨɪɦɭɥɭ (10), ɨɩɪɟɞɟɥɹɟɦ

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