Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 47 стр.

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47

T
P
g
ss
()

6
6
12
2
1
8
. (15)
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
Ɍ ɦɚɬɟɪɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ
ɩɨɞɫɬɚɜɥɹɟɦ ɜ ɮɨɪɦɭɥɭ (8) ɜɵɪɚɠɟɧɢɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɢɡ ɮɨɪɦɭɥ
(12)–(15). ȼ ɢɬɨɝɟ ɩɨɥɭɱɚɟɦ:
2222
3
12 4
1212
22 2
556
12 12 12
1111
2244
311
() (),
16 8 8
P
PP P
Ts s s s
gggg
PPP
s
sssss
ggg




ɬ. ɟ.
T
PPPP
g
s
PPPP
g
s
PP
g
ss


8432
16
8432
16
2
8
1356
1
2
2456
2
2
56
12

. (16)
Ⱦɥɹ ɫɨɫɬɚɜɥɟɧɢɹ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ ɫɥɟɞɭɟɬ
ɜɵɱɢɫɥɢɬɶ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
Ɍ ɩɨ ɨɛɨɛɳɟɧɧɵɦ
ɫɤɨɪɨɫɬɹɦ
s
1
ɢ
s
2
1
65
2
6542
2
2
65
1
6531
1
8
2
8
2348
,
8
2
8
2348
s
g
PP
s
g
PPPP
s
T
s
g
PP
s
g
PPPP
s
T
w
w
w
w
ɢ ɜɡɹɬɶ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨɥɭɱɟɧɧɵɯ ɪɟɡɭɥɶɬɚɬɨɜ ɩɨ ɜɪɟɦɟɧɢ
°
°
¿
°
°
¾
½
1
65
2
6542
2
2
65
1
6531
1
8
2
8
2348
8
2
8
2348
s
g
PP
s
g
PPPP
s
T
dt
d
s
g
PP
s
g
PPPP
s
T
dt
d
w
w
w
w
. (17)
ɍɱɢɬɵɜɚɹ, ɱɬɨ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ
Ɍ, ɨɩɪɟɞɟɥɟɧɧɚɹ ɮɨɪɦɭɥɨɣ (16)
ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ
s
1
ɢ
s
2
, ɢɦɟɟɦ
0
1
s
T
w
w
,
0
2
s
T
w
w
. (18)
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɮɨɪɦɭɥ (4), (7), (17) ɢ (18) ɜ ɭɪɚɜɧɟɧɢɟ (1) ɩɨɥɭɱɢɦ
ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ ɞɥɹ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ
s
1
ɢ
s
2
:
             1 P6           2
    T ( 6)        s1  s2 .                                                 (15)
             8 g
     Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ Ɍ ɦɚɬɟɪɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ
ɩɨɞɫɬɚɜɥɹɟɦ ɜ ɮɨɪɦɭɥɭ (8) ɜɵɪɚɠɟɧɢɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɢɡ ɮɨɪɦɭɥ
(12)–(15). ȼ ɢɬɨɝɟ ɩɨɥɭɱɚɟɦ:
        1 P1 2 1 P2 2 1 P3 2 1 P4 2
    T         s1         s2         s1      s2 
        2 g         2 g          4 g          4 g
       3 P5 2 2 1 P5                       1P
          ( s1  s2 )        s1s2  6 ( s1  s2 ) 2 ,
      16 g                  8 g            8 g
    ɬ. ɟ.
    8 P1  4 P3  3 P5  2 P6 2 8 P2  4 P4  3 P5  2 P6 2 P5  2 P6
T                            s1                        s2         s1 s2 . (16)
              16 g                        16 g                 8g
    Ⱦɥɹ ɫɨɫɬɚɜɥɟɧɢɹ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ ɫɥɟɞɭɟɬ
ɜɵɱɢɫɥɢɬɶ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ Ɍ ɩɨ ɨɛɨɛɳɟɧɧɵɦ
ɫɤɨɪɨɫɬɹɦ s1 ɢ s2
    wT         8P1  4 P3  3P5  2 P6      P  2 P6
                                       s1  5       s2 ,
    w s1                8g                    8g
    wT         8 P2  4 P4  3P5  2 P6      P  2 P6
                                        s2  5       s1
    w s2                 8g                    8g
    ɢ ɜɡɹɬɶ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨɥɭɱɟɧɧɵɯ ɪɟɡɭɥɶɬɚɬɨɜ ɩɨ ɜɪɟɦɟɧɢ
    d wT            8 P1  4 P3  3P5  2 P6          P  2 P6 ½
                                             s1  5          s2 °
    dt w s1                   8g                       8g            °
    d wT            8P2  4 P4  3P5  2 P6           P5  2 P6 ¾ .           (17)
                                               s2             s1 °
    dt w s2                   8g                        8g           °¿
     ɍɱɢɬɵɜɚɹ, ɱɬɨ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ Ɍ, ɨɩɪɟɞɟɥɟɧɧɚɹ ɮɨɪɦɭɥɨɣ (16)
ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ s1 ɢ s2 , ɢɦɟɟɦ
    wT               wT
               0,             0.                                              (18)
    w s1             w s2
    ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɮɨɪɦɭɥ (4), (7), (17) ɢ (18) ɜ ɭɪɚɜɧɟɧɢɟ (1) ɩɨɥɭɱɢɦ
ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ ɞɥɹ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ s1 ɢ s2 :




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