Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 40 стр.

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ɂɧɬɟɝɪɢɪɭɹ ɟɳɟ ɪɚɡ ɢ ɨɬɛɪɚɫɵɜɚɹ ɧɟɫɭɳɟɫɬɜɟɧɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɩɨɥɭɱɢɦ
ɢɫɤɨɦɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ:
.
48
3
;
8
1
;
16
3
222
gtzgtygtx
Ɂɚɞɚɱɚ ʋ 2. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɤɪɢɜɨɲɢɩɚ ɈȺ ɩɥɚɧɟɬɚɪɧɨɣ
ɩɟɪɟɞɚɱɢ, ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɜ ɜɟɪɬɢɤɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ. Ɂɭɛɱɚɬɨɟ ɤɨɥɟɫɨ 2
ɧɚɯɨɞɢɬɫɹ ɜɨ ɜɧɭɬɪɟɧɧɟɦ ɡɚɰɟɩɥɟɧɢɢ ɫ ɧɟɩɨɞɜɢɠɧɵɦ ɡɭɛɱɚɬɵɦ ɤɨɥɟɫɨɦ 1.
Ʉɨɥɟɫɨ 2 ɩɪɢɜɨɞɢɬɫɹ ɜ ɞɜɢɠɟɧɢɟ ɩɨɫɪɟɞɫɬɜɨɦ ɤɪɢɜɨɲɢɩɚ ɈȺ, ɤ ɤɨɬɨɪɨɦɭ
ɩɪɢɥɨɠɟɧ ɜɪɚɳɚɸɳɢɣ ɦɨɦɟɧɬ m
0
. ȼɟɫ ɤɪɢɜɨɲɢɩɚ ɈȺ ɪɚɜɟɧ Ɋ, Ɋ
2
– ɜɟɫ
ɤɨɥɟɫɚ 2, r
2
– ɪɚɞɢɭɫ ɤɨɥɟɫɚ 2, r
1
– ɪɚɞɢɭɫ ɧɟɩɨɞɜɢɠɧɨɝɨ ɤɨɥɟɫɚ 1.
Ʉɨɥɟɫɨ 2 ɫɱɢɬɚɬɶ ɨɞɧɨɪɨɞɧɵɦ ɤɪɭɝɥɵɦ ɞɢɫɤɨɦ, ɚ ɤɪɢɜɨɲɢɩ ɈȺ – ɬɨɧɤɢɦ
ɨɞɧɨɪɨɞɧɵɦ ɫɬɟɪɠɧɟɦ. ɋɢɥɚɦɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ ɩɪɟɧɟɛɪɟɱɶ.
Ɋɟɲɟɧɢɟ.
ɉɥɚɧɟɬɚɪɧɚɹ ɩɟɪɟɞɚɱɚ ɢɦɟɟɬ ɨɞɧɭ ɫɬɟɩɟɧɶ
ɫɜɨɛɨɞɵ, ɬɚɤ ɤɚɤ ɭɝɨɥ ɩɨɜɨɪɨɬɚ
M
ɤɪɢɜɨɲɢɩɚ ɈȺ
ɨɩɪɟɞɟɥɹɟɬ ɩɨɥɨɠɟɧɢɟ ɜɫɟɯ ɬɨɱɟɤ ɦɟɯɚɧɢɡɦɚ. ȼ
ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ ɜɵɛɢɪɚɟɦ ɭɝɨɥ
M
ɨɬɫɱɢɬɵɜɚɟɦɵɣ ɨɬ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ ɩɪɨɬɢɜ
ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ.
ɍɪɚɜɧɟɧɢɟ
Ʌɚɝɪɚɧɠɚ ɞɥɹ
ɨɛɨɛɳɟɧɧɨɣ
ɤɨɨɪɞɢɧɚɬɵ
M
ɢɦɟɟɬ ɜɢɞ
M
wM
w
Mw
w
Q
TT
dt
d
.
    ɂɧɬɟɝɪɢɪɭɹ ɟɳɟ ɪɚɡ ɢ ɨɬɛɪɚɫɵɜɚɹ ɧɟɫɭɳɟɫɬɜɟɧɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɩɨɥɭɱɢɦ
ɢɫɤɨɦɵɟ              ɭɪɚɜɧɟɧɢɹ           ɞɜɢɠɟɧɢɹ             ɫɢɫɬɟɦɵ:
     3 2        1 2          3 2
x      gt ; y     gt ; z       gt .
    16          8           48
    Ɂɚɞɚɱɚ ʋ 2. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɤɪɢɜɨɲɢɩɚ ɈȺ ɩɥɚɧɟɬɚɪɧɨɣ
ɩɟɪɟɞɚɱɢ, ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɜ ɜɟɪɬɢɤɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ. Ɂɭɛɱɚɬɨɟ ɤɨɥɟɫɨ 2
ɧɚɯɨɞɢɬɫɹ ɜɨ ɜɧɭɬɪɟɧɧɟɦ ɡɚɰɟɩɥɟɧɢɢ ɫ ɧɟɩɨɞɜɢɠɧɵɦ ɡɭɛɱɚɬɵɦ ɤɨɥɟɫɨɦ 1.
Ʉɨɥɟɫɨ 2 ɩɪɢɜɨɞɢɬɫɹ ɜ ɞɜɢɠɟɧɢɟ ɩɨɫɪɟɞɫɬɜɨɦ ɤɪɢɜɨɲɢɩɚ ɈȺ, ɤ ɤɨɬɨɪɨɦɭ
ɩɪɢɥɨɠɟɧ ɜɪɚɳɚɸɳɢɣ ɦɨɦɟɧɬ m0. ȼɟɫ ɤɪɢɜɨɲɢɩɚ ɈȺ ɪɚɜɟɧ Ɋ, Ɋ2 – ɜɟɫ
ɤɨɥɟɫɚ 2, r2 – ɪɚɞɢɭɫ ɤɨɥɟɫɚ 2, r1 – ɪɚɞɢɭɫ ɧɟɩɨɞɜɢɠɧɨɝɨ ɤɨɥɟɫɚ 1.
Ʉɨɥɟɫɨ 2 ɫɱɢɬɚɬɶ ɨɞɧɨɪɨɞɧɵɦ ɤɪɭɝɥɵɦ ɞɢɫɤɨɦ, ɚ ɤɪɢɜɨɲɢɩ ɈȺ – ɬɨɧɤɢɦ
ɨɞɧɨɪɨɞɧɵɦ ɫɬɟɪɠɧɟɦ. ɋɢɥɚɦɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ ɩɪɟɧɟɛɪɟɱɶ.
         Ɋɟɲɟɧɢɟ.
                                                        ɍɪɚɜɧɟɧɢɟ
                                                        Ʌɚɝɪɚɧɠɚ ɞɥɹ
                                                        ɨɛɨɛɳɟɧɧɨɣ
                                                        ɤɨɨɪɞɢɧɚɬɵ M
                                                        ɢɦɟɟɬ ɜɢɞ
                                                      d wT wT
                                                                  QM .
                                                     dt wM wM




ɉɥɚɧɟɬɚɪɧɚɹ ɩɟɪɟɞɚɱɚ ɢɦɟɟɬ ɨɞɧɭ ɫɬɟɩɟɧɶ
ɫɜɨɛɨɞɵ, ɬɚɤ ɤɚɤ ɭɝɨɥ ɩɨɜɨɪɨɬɚ M ɤɪɢɜɨɲɢɩɚ ɈȺ
ɨɩɪɟɞɟɥɹɟɬ ɩɨɥɨɠɟɧɢɟ ɜɫɟɯ ɬɨɱɟɤ ɦɟɯɚɧɢɡɦɚ. ȼ
ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ ɜɵɛɢɪɚɟɦ ɭɝɨɥ
M ɨɬɫɱɢɬɵɜɚɟɦɵɣ ɨɬ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ ɩɪɨɬɢɜ
ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ.



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