Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 35 стр.

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ɉɨɞɫɬɚɜɢɜ ɩɨɥɭɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ G
2
ɜ ɮɨɪɦɭɥɭ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɫɢɥɵ F,
ɩɨɥɭɱɢɦ
S
h
GR
S
mV
F
2
2
0
.
Ɂɚɬɟɦ ɜɵɱɢɫɥɹɟɦ ɜɟɥɢɱɢɧɭ ɫɢɥɵ F, ɭɱɢɬɵɜɚɹ, ɱɬɨ G=mg,
ɇ
S
mgh
R
S
mV
F 85100
500
281,910
19600
5002
1210
2
626
2
0
.
§11. ɍɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ
Ⱦɥɹ ɫɢɫɬɟɦɵ ɫ ɝɨɥɨɧɨɦɧɵɦɢ ɢɞɟɚɥɶɧɵɦɢ ɢ ɭɞɟɪɠɢɜɚɸɳɢɦɢ ɫɜɹɡɹɦɢ
ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɢɦɟɸɬ ɜɢɞ [1], [5]
j
jj
Q
q
T
q
T
dt
d
w
w
¸
¸
¹
·
¨
¨
©
§
w
w
(j = 1, 2, ... , s),
ɝɞɟ q
1
, q
2
,..., q
S
– ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɫɢɫɬɟɦɵ;
s – ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɫɢɫɬɟɦɵ;
,
1
q
,
2
q
…,
S
q
– ɨɛɨɛɳɟɧɧɵɟ ɫɤɨɪɨɫɬɢ;
Ɍ – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ;
Q
1
, Q
2
, ..., Q
S
– ɨɛɨɛɳɟɧɧɵɟ ɫɢɥɵ.
Ⱦɥɹ ɫɢɫɬɟɦ, ɞɜɢɠɭɳɢɯɫɹ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ, ɭɪɚɜɧɟɧɢɹ
Ʌɚɝɪɚɧɠɚ ɛɭɞɭɬ
0
w
w
¸
¸
¹
·
¨
¨
©
§
w
w
jj
q
L
q
L
dt
d
,
ɝɞɟ L = T – ɉ – ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ.
Ⱦɥɹ ɫɨɫɬɚɜɥɟɧɢɹ ɭɪɚɜɧɟɧɢɣ Ʌɚɝɪɚɧɠɚ ɫɥɟɞɭɟɬ:
x ɭɫɬɚɧɨɜɢɬɶ ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɢ ɜɵɛɪɚɬɶ ɨɛɨɛɳɟɧɧɵɟ
ɤɨɨɪɞɢɧɚɬɵ;
x ɩɪɟɞɩɨɥɨɠɢɜ, ɱɬɨ ɫɢɫɬɟɦɚ ɞɜɢɠɟɬɫɹ ɬɚɤ, ɱɬɨ ɜɫɟ ɨɛɨɛɳɟɧɧɵɟ
ɤɨɨɪɞɢɧɚɬɵ ɭɜɟɥɢɱɢɜɚɸɬɫɹ, ɫɨɫɬɚɜɢɬɶ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɤɢɧɟɬɢɱɟɫɤɨɣ
ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ, ɩɪɢ ɷɬɨɦ ɜɫɟ ɩɟɪɟɦɟɧɧɵɟ ɜɟɥɢɱɢɧɵ, ɜɯɨɞɹɳɢɟ ɜ Ɍ,
ɞɨɥɠɧɵ ɛɵɬɶ ɜɵɪɚɠɟɧɵ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ
ɢ ɨɛɨɛɳɟɧɧɵɟ
ɫɤɨɪɨɫɬɢ, ɬ.ɟ.
);,...,,;,...,,(
2121
tqqqqqqTT
SS
(ɜ ɫɥɭɱɚɟ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɜɹɡɟɣ ɜɪɟɦɹ t
ɧɟ ɜɯɨɞɢɬ ɜ ɜɵɪɚɠɟɧɢɟ T);
ɉɨɞɫɬɚɜɢɜ ɩɨɥɭɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ G2 ɜ ɮɨɪɦɭɥɭ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɫɢɥɵ F,
             mV02         h
ɩɨɥɭɱɢɦ F          RG .
              2S         S
Ɂɚɬɟɦ ɜɵɱɢɫɥɹɟɦ ɜɟɥɢɱɢɧɭ ɫɢɥɵ F, ɭɱɢɬɵɜɚɹ, ɱɬɨ G=mg,
      mV02     mgh            10 6 ˜ 12 2           10 6 ˜ 9,81 ˜ 2
F          R                             19600                       85100 ɇ .
       2S       S              2 ˜ 500                   500

                         §11. ɍɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ

     Ⱦɥɹ ɫɢɫɬɟɦɵ ɫ ɝɨɥɨɧɨɦɧɵɦɢ ɢɞɟɚɥɶɧɵɦɢ ɢ ɭɞɟɪɠɢɜɚɸɳɢɦɢ ɫɜɹɡɹɦɢ
ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɢɦɟɸɬ ɜɢɞ [1], [5]
      d §¨ wT ·¸ wT
                             Q j (j = 1, 2, ... , s),
      dt ¨© wq j ¸¹ wq j
ɝɞɟ q1, q2,..., qS – ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɫɢɫɬɟɦɵ;
     s – ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɫɢɫɬɟɦɵ;
     q1 , q 2 , …, q S – ɨɛɨɛɳɟɧɧɵɟ ɫɤɨɪɨɫɬɢ;
     Ɍ – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ;
     Q1, Q2, ..., QS – ɨɛɨɛɳɟɧɧɵɟ ɫɢɥɵ.
     Ⱦɥɹ ɫɢɫɬɟɦ, ɞɜɢɠɭɳɢɯɫɹ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ, ɭɪɚɜɧɟɧɢɹ
Ʌɚɝɪɚɧɠɚ ɛɭɞɭɬ
      d §¨ wL ·¸ wL
                             0,
      dt ¨© wq j ¸¹ wq j
     ɝɞɟ L = T – ɉ – ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ.
     Ⱦɥɹ ɫɨɫɬɚɜɥɟɧɢɹ ɭɪɚɜɧɟɧɢɣ Ʌɚɝɪɚɧɠɚ ɫɥɟɞɭɟɬ:
   x ɭɫɬɚɧɨɜɢɬɶ ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɢ ɜɵɛɪɚɬɶ ɨɛɨɛɳɟɧɧɵɟ
       ɤɨɨɪɞɢɧɚɬɵ;
   x ɩɪɟɞɩɨɥɨɠɢɜ, ɱɬɨ ɫɢɫɬɟɦɚ ɞɜɢɠɟɬɫɹ ɬɚɤ, ɱɬɨ ɜɫɟ ɨɛɨɛɳɟɧɧɵɟ
       ɤɨɨɪɞɢɧɚɬɵ ɭɜɟɥɢɱɢɜɚɸɬɫɹ, ɫɨɫɬɚɜɢɬɶ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɤɢɧɟɬɢɱɟɫɤɨɣ
       ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ, ɩɪɢ ɷɬɨɦ ɜɫɟ ɩɟɪɟɦɟɧɧɵɟ ɜɟɥɢɱɢɧɵ, ɜɯɨɞɹɳɢɟ ɜ Ɍ,
       ɞɨɥɠɧɵ ɛɵɬɶ ɜɵɪɚɠɟɧɵ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢ ɨɛɨɛɳɟɧɧɵɟ
       ɫɤɨɪɨɫɬɢ,                                                                              ɬ.ɟ.
       T T (q1 , q2 ,..., q S ; q1 , q 2 ,..., q S ; t ) (ɜ ɫɥɭɱɚɟ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɜɹɡɟɣ ɜɪɟɦɹ t
       ɧɟ ɜɯɨɞɢɬ ɜ ɜɵɪɚɠɟɧɢɟ T);




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