Механика.Методика решения задач
.pdfȽɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
131 |
ɞɪɭɝɭɸ – ɜ ɦɟɪɢɞɢɨɧɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ, ɜ ɤɨɬɨɪɨɣ ɥɟɠɢɬ ɜɟɤɬɨɪ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɩɭɥɢ V (ɪɢɫ. 4.7). Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɩɨɜɟɪɯɧɨɫɬɶ Ɂɟɦɥɢ ɹɜɥɹɟɬɫɹ ɫɮɟɪɢɱɟɫɤɨɣ, ɢ Ɂɟɦɥɹ ɜɦɟɫɬɟ ɫɨ ɫɜɹɡɚɧɧɨɣ ɫ ɧɟɣ ɫɢɫɬɟɦɨɣ ɨɬɫɱɟɬɚ X'Y'Z' ɜɪɚɳɚɟɬɫɹ ɫ ɩɨɫɬɨɹɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨ-
ɪɨɫɬɶɸ Z |
2S |
, ɝɞɟ T = 24 ɱ. |
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Ɍ |
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ɇɚ ɩɭɥɸ ɜ ɩɪɨɰɟɫɫɟ ɩɨɥɟɬɚ ɞɟɣɫɬɜɭɸɬ ɫɢɥɚ ɝɪɚɜɢɬɚɰɢɨɧɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫ Ɂɟɦɥɟɣ Fɝɪ ɢ ɫɢɥɵ ɢɧɟɪɰɢɢ – ɰɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɚ
ɢɧɟɪɰɢɢ Fɰɛ ɢ ɫɢɥɚ ɢɧɟɪɰɢɢ Ʉɨɪɢɨɥɢɫɚ FɄɨɪ , ɢɡɨɛɪɚɠɟɧɧɵɟ ɧɚ
ɪɢɫ. 4.7 ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (4.16) ɢ (4.17). Ɉɬɤɥɨɧɟɧɢɟ ɩɭɥɢ ɨɬ ɜɟɪɬɢɤɚɥɶɧɨɣ ɱɟɪɬɵ ɦɢɲɟɧɢ ɜɵɡɵɜɚɟɬ ɫɢɥɚ ɢɧɟɪɰɢɢ Ʉɨɪɢɨɥɢɫɚ.
ɐɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ ɢɦɟɟɬ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɫɨɫɬɚɜɥɹɸɳɭɸ, ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɡɦɟɧɹɟɬ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɩɪɨɟɤɰɢɸ ɫɤɨɪɨɫɬɢ ɩɭɥɢ. Ɉɞɧɚɤɨ ɭɱɟɬ ɰɟɧɬɪɨɛɟɠɧɨɣ ɫɢɥɵ ɢɧɟɪɰɢɢ ɞɚɫɬ ɦɚɥɵɟ ɩɨɩɪɚɜɤɢ ɤ ɜɟɥɢɱɢɧɟ ɢ ɧɚɩɪɚɜɥɟɧɢɸ ɫɤɨɪɨɫɬɢ ɩɨɥɟɬɚ ɩɭɥɢ.
ɋɢɥɚ ɢɧɟɪɰɢɢ Ʉɨɪɢɨɥɢɫɚ ɧɟ ɦɟɧɹɹ ɜɟɥɢɱɢɧɵ ɫɤɨɪɨɫɬɢ ɩɭɥɢ, ɢɡɦɟɧɹɟɬ ɧɚɩɪɚɜɥɟɧɢɟ ɟɟ ɩɨɥɟɬɚ. ɉɪɢ ɷɬɨɦ ɩɪɨɟɤɰɢɹ ɫɤɨɪɨɫɬɢ ɩɨɥɟɬɚ ɩɭɥɢ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɜɵɫɬɪɟɥɚ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɦɟɧɹɟɬɫɹ. ɉɨɷɬɨɦɭ ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɞɜɢɠɟɧɢɟ ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɩɪɨɢɫɯɨɞɢɬ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ, ɪɚɜɧɨɣ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɩɭɥɢ.
II. ȼɪɟɦɹ ɩɨɥɟɬɚ ɩɭɥɢ t ɧɚɯɨɞɢɦ ɢɡ ɭɫɥɨɜɢɹ ɪɚɜɧɨɦɟɪɧɨɝɨ ɞɜɢɠɟɧɢɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɜɵɫɬɪɟɥɚ:
t |
s |
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(4.51) |
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V |
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ɍɫɤɨɪɟɧɢɟ ɜ ɜɨɫɬɨɱɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɦ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɩɭɥɢ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɢɥɨɣ ɢɧɟɪɰɢɢ Ʉɨɪɢɨɥɢɫɚ
(ɫɦ. ɪɢɫ. 4.7) ɢ ɪɚɜɧɨ |
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a 2ZV sin M . |
(4.52) |
Ɉɬɤɥɨɧɟɧɢɟ ɩɭɥɢ ɨɬ ɜɟɪɬɢɤɚɥɶɧɨɣ ɱɟɪɬɵ ɦɢɲɟɧɢ ɩɪɢ ɭɫɤɨɪɟɧɧɨɦ ɞɜɢɠɟɧɢɢ ɜ ɬɟɱɟɧɢɟ ɜɪɟɦɟɧɢ W ɪɚɜɧɨ
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at 2 |
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(4.53) |
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III. Ɋɟɲɚɹ ɩɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (4.51) – (4.53), ɧɚɯɨɞɢɦ ɢɫɤɨɦɨɟ ɡɧɚɱɟɧɢɟ ɨɬɤɥɨɧɟɧɢɹ ɩɭɥɢ ɨɬ ɜɟɪɬɢɤɚɥɶɧɨɣ ɱɟɪɬɵ ɦɢɲɟɧɢ:
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
l |
Z s2 sin M |
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Ɉɰɟɧɢɦ ɢɡɦɟɧɟɧɢɟ ɝɨɪɢɡɨɧ- |
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ɬɚɥɶɧɨɣ |
ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɩɭɥɢ |
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ǻVW ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɰɟɧɬɪɨɛɟɠɧɨɣ ɫɢɥɵ ɢɧɟɪɰɢɢ FɰɛW . Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ
ɞɜɢɠɟɧɢɹ ɩɭɥɢ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɬɚɧɝɟɧɰɢɚɥɶɧɭɸ ɨɫɶ (ɫɦ. ɪɢɫ. 4.8):
ma |
F W . |
(4.55) |
W |
ɰɛ |
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ɂɫɩɨɥɶɡɭɹ (4.16) ɢ (4.55), ɩɨɥɭɱɚɟɦ ɞɥɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ ɩɭɥɢ:
aW Z2r sinM
Z2 R cosM sin M .
(4.54)
Z' IJ
r
Fɰɛ
Z |
R |
W |
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Fɰɛ |
ij
Y'
Ɋɢɫ. 4.8
(4.56)
ɂɡɦɟɧɟɧɢɟ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɩɭɥɢ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɰɟɧɬɪɨɛɟɠɧɨɣ ɫɢɥɵ ɢɧɟɪɰɢɢ ɫ ɭɱɟɬɨɦ (4.51) ɪɚɜɧɨ:
ǻVW |
# aW t Z2 R cosM sinM |
s |
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(4.57) |
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V |
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ɉɨɞɫɬɚɧɨɜɤɚ ɱɢɫɥɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɜɟɥɢɱɢɧ ɜ (4.57) ɞɚɟɬ ǻVW # 0,01ɦ/ɫ V 900 ɦ/ɫ , ɱɬɨ ɩɨɞɬɜɟɪɠɞɚɟɬ ɫɩɪɚɜɟɞɥɢɜɨɫɬɶ ɩɪɢɧɹɬɨɝɨ ɞɨɩɭɳɟɧɢɹ.
Ɂɚɞɚɱɚ 4.6
(ȼɪɚɳɚɸɳɚɹɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ) ȼɪɚɳɟɧɢɟ Ɂɟɦɥɢ ɜɵɡɵɜɚɟɬ ɨɬɤɥɨɧɟɧɢɟ ɩɨɜɟɪɯɧɨɫɬɢ ɜɨɞɵ ɜ
ɪɟɤɚɯ ɨɬ ɝɨɪɢɡɨɧɬɚɥɶɧɨɝɨ ɩɨɥɨɠɟɧɢɹ. Ɉɩɪɟɞɟɥɢɬɶ, ɭ ɤɚɤɨɝɨ ɛɟɪɟɝɚ ɢ ɧɚ ɤɚɤɭɸ ɜɟɥɢɱɢɧɭ h ɭɪɨɜɟɧɶ ɜɨɞɵ ɜ ɪɟɤɟ ɛɭɞɟɬ ɜɵɲɟ. Ɋɟɤɚ ɬɟɱɟɬ
ɜ ɫɟɜɟɪɧɨɦ ɩɨɥɭɲɚɪɢɢ ɧɚ ɲɢɪɨɬɟ M 60o ɫ ɫɟɜɟɪɚ ɧɚ ɸɝ. ɒɢɪɢɧɚ ɪɟɤɢ L 1ɤɦ , ɫɤɨɪɨɫɬɶ ɬɟɱɟɧɢɹ V 1ɦ/ɫ , ɩɟɪɢɨɞ ɨɛɪɚɳɟɧɢɹ Ɂɟɦɥɢ ɜɨɤɪɭɝ ɫɜɨɟɣ ɨɫɢ T 24 ɱ . ɋɱɢɬɚɬɶ ɭɫɤɨɪɟɧɢɟ ɫɜɨɛɨɞɧɨɝɨ ɩɚɞɟɧɢɹ ɧɚ ɞɚɧɧɨɣ ɲɢɪɨɬɟ ɪɚɜɧɵɦ g 9,8ɦ/ɫ2 .
Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
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Ɋɟɲɟɧɢɟ
I. ȼɵɞɟɥɢɦ ɦɵɫɥɟɧɧɨ ɧɟɛɨɥɶɲɨɣ ɨɛɴɟɦ ɠɢɞɤɨɫɬɢ ɜɛɥɢɡɢ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɪɚɫɫɦɨɬɪɢɦ ɟɝɨ ɞɜɢɠɟɧɢɟ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ Ɂɟɦɥɟɣ. ɇɚɩɪɚɜɢɦ ɨɫɶ Y ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ (ɜɞɨɥɶ ɥɢɧɢɢ ɨɬɜɟɫɚ), ɚ ɨɫɶ X – ɝɨɪɢɡɨɧɬɚɥɶɧɨ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɫɤɨɪɨɫɬɢ ɬɟɱɟɧɢɹ ɪɟɤɢ ɜ ɫɬɨɪɨɧɭ ɩɪɚɜɨɝɨ ɛɟɪɟɝɚ (ɧɚ ɡɚɩɚɞ).
ɇɚ ɪɢɫ. 4.9 ɢɡɨɛɪɚɠɟɧɵ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɷɥɟɦɟɧɬ ɨɛɴɟɦɚ ɠɢɞɤɨɫɬɢ – ɫɢɥɚ ɬɹɠɟɫɬɢ mg , ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɫɢɥ ɞɚɜɥɟɧɢɹ
ɫɨ ɫɬɨɪɨɧɵ ɜɫɟɣ ɨɫɬɚɥɶɧɨɣ ɜɨɞɵ N ɢ ɫɢɥɚ ɢɧɟɪɰɢɢ Ʉɨɪɢɨɥɢɫɚ
FɄɨɪ .
Y
N D |
FɄɨɪ |
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Ɋɢɫ. 4.9 |
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ɀɢɞɤɨɫɬɶ ɫɱɢɬɚɟɦ ɧɟɫɠɢɦɚɟɦɨɣ, ɫɢɥ ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ ɧɟɬ. ɇɚɩɪɚɜɥɟɧɢɟ ɬɟɱɟɧɢɹ ɪɟɤɢ ɧɚɩɪɚɜɥɟɧɨ ɡɚ ɩɥɨɫɤɨɫɬɶ ɱɟɪɬɟɠɚ. Ɋɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɫɢɥ ɞɚɜɥɟɧɢɹ ɧɚɩɪɚɜɥɟɧɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɨɜɟɪɯɧɨɫɬɢ ɠɢɞɤɨɫɬɢ ɢ ɨɛɪɚɡɭɟɬ ɭɝɨɥ D ɫ ɜɟɪɬɢɤɚɥɶɸ.
II. ɀɢɞɤɨɫɬɶ ɞɜɢɠɟɬɫɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɦ ɩɥɨɫɤɨɫɬɢ ɱɟɪɬɟɠɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɟɤɬɨɪɧɚɹ ɫɭɦɦɚ ɫɢɥ, ɥɟɠɚɳɢɯ ɜ ɩɥɨɫɤɨɫɬɢ ɱɟɪɬɟɠɚ ɢ ɢɡɨɛɪɚɠɟɧɧɵɯ ɧɚ ɪɢɫɭɧɤɟ, ɪɚɜɧɚ ɧɭɥɸ:
mg N FɄɨɪ |
0 . |
(4.58) |
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ɍɪɚɜɧɟɧɢɟ (4.58) ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨ- |
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ɨɪɞɢɧɚɬ ɩɪɢɧɢɦɚɟɬ ɜɢɞ: |
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N sinD FɄɨɪ |
0, |
(4.59) |
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N cosD mg |
0. |
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Ɋɚɡɧɨɫɬɶ ɜɵɫɨɬ ɩɪɚɜɨɝɨ ɢ ɥɟɜɨɝɨ ɛɟɪɟɝɨɜ ɪɟɤɢ, ɤɚɤ ɜɢɞɧɨ ɧɚ ɪɢɫ. 4.9, ɪɚɜɧɚ
h L tgD . |
(4.60) |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Ⱦɥɹ ɫɢɥɵ ɢɧɟɪɰɢɢ Ʉɨɪɢɨɥɢɫɚ (ɫɦ. (4.17)) ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ |
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ɫɩɪɚɜɟɞɥɢɜɨ ɜɵɪɚɠɟɧɢɟ: |
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Fɤɨɪ 2mZV sinM . |
(4.61) |
III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (4.59) – (4.61), |
ɩɨɥɭɱɚɟɦ, ɱɬɨ |
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ɭɪɨɜɟɧɶ ɜɨɞɵ ɭ ɩɪɚɜɨɝɨ ɛɟɪɟɝɚ ɛɭɞɟɬ ɜɵɲɟ ɧɚ ɜɟɥɢɱɢɧɭ |
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h |
2ZVL sin M |
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(4.62) |
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g |
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ɉɨɞɫɬɚɧɨɜɤɚ ɱɢɫɥɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɜɟɥɢɱɢɧ ɜ (4.62) ɞɚɟɬ ɢɫɤɨɦɨɟ ɡɧɚɱɟɧɢɟ ɪɚɡɧɨɫɬɢ ɜɵɫɨɬ ɩɪɚɜɨɝɨ ɢ ɥɟɜɨɝɨ ɛɟɪɟɝɨɜ h #1.3ɫɦ .
Ɂɚɞɚɱɚ 4.7
(ȼɪɚɳɚɸɳɚɹɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ)
Ɍɨɧɤɢɣ ɨɞɧɨɪɨɞɧɵɣ ɫɬɟɪɠɟɧɶ ɞɥɢɧɨɣ L ɢ ɦɚɫɫɨɣ m, ɲɚɪɧɢɪɧɨ ɡɚɤɪɟɩɥɟɧɧɵɣ ɜ ɜɟɪɯɧɟɣ ɬɨɱɤɟ Ɉ, ɜɪɚɳɚɟɬɫɹ ɫ ɩɨɫɬɨɹɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Z ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɞɚɧɧɭɸ ɬɨɱɤɭ (ɫɦ. ɪɢɫ. 4.10). Ɉɩɪɟɞɟɥɢɬɶ ɭɝɨɥ ɭɫɬɨɣɱɢɜɨɝɨ ɜɪɚɳɟɧɢɹ ɫɬɟɪɠɧɹ.
Ɋɟɲɟɧɢɟ
I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫɨ ɫɬɟɪɠɧɟɦ ɢ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɶɸ ɜɪɚɳɟɧɢɹ. Ȼɭɞɟɦ ɫɱɢɬɚɬɶ ɫɬɟɪɠɟɧɶ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɵɦ ɬɟɥɨɦ. ɇɚ ɫɬɟɪɠɟɧɶ ɞɟɣɫɬɜɭɸɬ ɬɪɢ ɫɢɥɵ: ɫɢɥɚ ɬɹɠɟɫɬɢ mg, ɫɢɥɚ ɪɟɚɤɰɢɢ ɫɨ ɫɬɨɪɨɧɵ ɲɚɪɧɢɪɚ ɢ ɰɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ. ɋɢɥɨɣ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ ɩɪɟɧɟɛɪɟ-
ɝɚɟɦ. ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɬɹɠɟɫɬɢ ɢ ɫɢɥɵ |
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ɪɟɚɤɰɢɢ ɫɨ ɫɬɨɪɨɧɵ ɲɚɪɧɢɪɚ ɫɬɟɪɠɟɧɶ ɫɨ- |
Z |
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ɜɟɪɲɚɟɬ ɜɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɨɬɧɨɫɢ- |
O |
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ɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟ- |
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ɦɵ ɨɬɫɱɟɬɚ, ɩɪɢɱɟɦ ɪɚɡɥɢɱɧɵɟ ɭɱɚɫɬɤɢ |
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ɫɬɟɪɠɧɹ ɞɜɢɠɭɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɹɦ ɪɚɡɧɵɯ |
D |
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ɪɚɞɢɭɫɨɜ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɚɡɥɢɱɧɚ ɢ ɫɢɥɚ |
dFɢɧ |
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ɢɧɟɪɰɢɢ dFɢɧ , ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɨɬɞɟɥɶɧɵɟ |
dl |
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ɷɥɟɦɟɧɬɵ ɫɬɟɪɠɧɹ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫ- |
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ɬɟɦɟ ɨɬɫɱɟɬɚ. |
mg |
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II. ɉɪɢ ɭɫɬɨɣɱɢɜɨɦ ɜɪɚɳɟɧɢɢ ɫɬɟɪ- |
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ɠɟɧɶ ɨɬɤɥɨɧɟɧ ɨɬ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ ɧɚ ɩɨ- |
Ɋɢɫ. 4.10 |
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ɫɬɨɹɧɧɵɣ ɭɝɨɥ D . ɉɨɫɤɨɥɶɤɭ ɫɬɟɪɠɟɧɶ ɩɨ- |
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Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
135 |
ɤɨɢɬɫɹ ɜ ɜɵɛɪɚɧɧɨɣ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɭɦɦɚ ɦɨɦɟɧɬɨɜ ɜɫɟɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɫɬɟɪɠɟɧɶ ɜ ɷɬɨɣ ɫɢɫɬɟɦɟ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ Ɉ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɥɨɫɤɨɫɬɢ ɱɟɪɬɟɠɚ (ɪɢɫ. 4.10) ɢ ɧɚɩɪɚɜɥɟɧɧɨɣ ɢɡ ɩɥɨɫɤɨɫɬɢ ɱɟɪɬɟɠɚ, ɪɚɜɧɚ ɧɭɥɸ:
M mg M ɢɧ 0 . |
(4.63) |
ɉɪɢ ɡɚɩɢɫɢ (4.63) ɭɱɬɟɧɨ, ɱɬɨ ɦɨɦɟɧɬ ɫɢɥɵ ɪɟɚɤɰɢɢ ɲɚɪɧɢɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɤɚɡɚɧɧɨɣ ɨɫɢ ɪɚɜɟɧ ɧɭɥɸ.
Ɂɚɩɢɲɟɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɦɨɦɟɧɬɚ ɫɢɥɵ ɬɹɠɟɫɬɢ: |
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M mg |
mg |
L |
sinD . |
(4.64) |
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2 |
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Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɫɭɦɦɚɪɧɨɝɨ ɦɨɦɟɧɬɚ ɫɢɥ ɢɧɟɪɰɢɢ ɪɚɫɫɦɨɬɪɢɦ ɷɥɟɦɟɧɬ ɫɬɟɪɠɧɹ ɞɥɢɧɨɣ dl, ɧɚɯɨɞɹɳɢɣɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ l ɨɬ ɬɨɱɤɢ Ɉ. ɐɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ (ɫɦ. (4.16)), ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɷɬɨɬ ɷɥɟɦɟɧɬ, ɪɚɜɧɚ
dF |
USdlZ2r , |
(4.65) |
ɢɧ |
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ɝɞɟ U – ɩɥɨɬɧɨɫɬɶ ɫɬɟɪɠɧɹ, S – ɩɥɨɳɚɞɶ ɟɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ,
r – ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɷɥɟɦɟɧɬɚ ɫɬɟɪɠɧɹ ɞɨ ɨɫɢ ɜɪɚɳɟɧɢɹ.
Ɇɨɦɟɧɬ ɫɢɥɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ:
dM |
ɢɧ |
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dF |
l cosD USdlZ2l sin D l cosD . |
(4.66) |
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ɢɧ |
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ɋɭɦɦɚɪɧɵɣ ɦɨɦɟɧɬ ɫɢɥ ɢɧɟɪɰɢɢ ɪɚɜɟɧ |
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L |
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M ɢɧ |
³dM ɢɧ |
USZ2 sin D cosD ³l 2dl |
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L Z |
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m sinD cosD . |
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(4.67) |
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III. Ɋɟɲɚɹ ɩɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (4.63), (4.64) ɢ |
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(4.67) ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɝɥɚ D, ɩɨɥɭɱɚɟɦ: |
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cosD |
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3g |
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(4.68) |
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2LZ2 |
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sinD |
0 . |
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(4.69) |
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ɇɟɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɩɪɢ Z ! |
3g |
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ɡɧɚɱɟɧɢɟ ɭɝɥɚ |
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2L |
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136 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
D |
§ |
3g · |
(4.70) |
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arccos©¨ |
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2LZ2 |
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ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɭɫɬɨɣɱɢɜɨɦɭ ɜɪɚɳɟɧɢɸ ɫɬɟɪɠɧɹ, ɚ ɡɧɚɱɟɧɢɟ D = 0, ɤɨɬɨɪɨɟ ɫɥɟɞɭɟɬ ɢɡ ɭɪɚɜɧɟɧɢɹ (4.69), ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɟɭɫɬɨɣɱɢɜɨɦɭ
ɜɪɚɳɟɧɢɸ ɫɬɟɪɠɧɹ. ȼ ɫɥɭɱɚɟ Z d |
3g |
ɭɫɬɨɣɱɢɜɵɦ ɹɜɥɹɟɬɫɹ ɜɟɪ- |
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ɬɢɤɚɥɶɧɨɟ ɩɨɥɨɠɟɧɢɟ ɫɬɟɪɠɧɹ (D = 0).
Ɂɚɞɚɱɚ 4.8
(ȼɪɚɳɚɸɳɚɹɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ)
Ƚɥɚɞɤɚɹ ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɬɪɭɛɤɚ ɞɥɢɧɨɣ L ɪɚɜɧɨɦɟɪɧɨ ɜɪɚɳɚɟɬɫɹ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Ȧ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɨɞɢɧ ɢɡ ɟɟ ɤɨɧɰɨɜ (ɫɦ. ɪɢɫ. 4.11).
Y'
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Ɋɢɫ. 4.11 |
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Ɉɩɪɟɞɟɥɢɬɶ, ɡɚ ɤɚɤɨɟ ɜɪɟɦɹ t0 ɦɚɥɟɧɶɤɢɣ ɲɚɪɢɤ, ɧɚɯɨɞɹɳɢɣɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ l ɨɬ ɨɫɢ ɬɪɭɛɤɢ, ɞɨɫɬɢɝɧɟɬ ɟɟ ɤɨɧɰɚ, ɚ ɬɚɤɠɟ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɲɚɪɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɡɟɦɥɢ X0 ɜ ɦɨɦɟɧɬ ɟɝɨ ɜɵɥɟɬɚ
ɢɡ ɬɪɭɛɤɢ. ɇɚɱɚɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɲɚɪɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɪɭɛɤɢ ɩɪɢɧɹɬɶ ɪɚɜɧɨɣ ɧɭɥɸ.
Ɋɟɲɟɧɢɟ
I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɬɪɭɛɤɨɣ ɢ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɶɸ ɜɪɚɳɟɧɢɹ. Ɉɫɢ X' ɢ Y' ɩɪɚɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ X'Y'Z' ɧɚɩɪɚɜɢɦ ɜɞɨɥɶ ɬɪɭɛɤɢ ɢ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɟɟ ɜɪɚɳɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ (ɫɦ. ɪɢɫ. 4.11). ɒɚɪɢɤ ɫɱɢɬɚɟɦ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ, ɚ ɬɪɭɛɤɭ – ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɵɦ ɬɟɥɨɦ. ɇɚ ɲɚɪɢɤ ɜ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ ɞɟɣɫɬɜɭɸɬ ɱɟɬɵɪɟ ɫɢɥɵ: ɫɢɥɚ ɬɹɠɟɫɬɢ mg, ɫɢɥɚ ɪɟɚɤɰɢɢ ɬɪɭɛɤɢ, ɰɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ, ɚ ɬɚɤɠɟ ɫɢɥɚ ɢɧɟɪɰɢɢ Ʉɨɪɢɨɥɢɫɚ. ɋɢɥɚɦɢ ɬɪɟɧɢɹ ɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ ɩɪɟ-
Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
137 |
ɧɟɛɪɟɝɚɟɦ. ɒɚɪɢɤ ɞɜɢɠɟɬɫɹ ɜɞɨɥɶ ɬɪɭɛɤɢ, ɩɪɢɱɟɦ ɟɝɨ ɭɫɤɨɪɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɶɤɨ ɰɟɧɬɪɨɛɟɠɧɨɣ ɫɢɥɨɣ ɢɧɟɪɰɢɢ, ɩɨɫɤɨɥɶɤɭ ɧɚɩɪɚɜɥɟɧɢɟ ɞɟɣɫɬɜɢɹ ɨɫɬɚɥɶɧɵɯ ɫɢɥ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɟɝɨ ɞɜɢɠɟɧɢɸ.
II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɲɚɪɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɨɣ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ, ɫɨɜɩɚɞɚɸɳɭɸ ɫ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɨɫɶɸ ɬɪɭɛɤɢ (ɫɦ. ɪɢɫ. 4.11):
ma' mZ2r' . |
(4.71) |
ɋ ɰɟɥɶɸ ɭɩɪɨɳɟɧɢɹ ɞɚɥɶɧɟɣɲɟɝɨ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɭɪɚɜɧɟɧɢɹ (4.71) ɜɵɪɚɡɢɦ ɩɪɨɟɤɰɢɸ ɭɫɤɨɪɟɧɢɹ ɲɚɪɢɤɚ a' ɱɟɪɟɡ ɩɪɨɢɡɜɨɞɧɭɸ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɩɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ r' ɲɚɪɢɤɚ:
a' |
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III. ɂɫɩɨɥɶɡɭɹ ɫɨɨɬɧɨɲɟɧɢɟ (4.72), ɢɫɤɥɸɱɚɟɦ ɩɪɨɟɤɰɢɸ ɭɫ- |
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ɤɨɪɟɧɢɹ ɢɡ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɲɚɪɢɤɚ (4.71): |
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Z2r' . |
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ɉɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ (4.73) ɪɟɲɚɟɦ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ |
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ɩɟɪɟɦɟɧɧɵɯ: |
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X' |
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³X'dX' |
³Z2r'dr' . |
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0l
ɂɧɬɟɝɪɢɪɭɹ (4.74), ɩɨɥɭɱɚɟɦ ɫɜɹɡɶ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɲɚɪɢɤɚ X' ɢ ɟɝɨ ɤɨɨɪɞɢɧɚɬɵ r':
X' |
Z r'2 l 2 . |
(4.75) |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ |
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ɍɪɚɜɧɟɧɢɟ (4.76) ɪɟɲɚɟɦ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ: |
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³Z dt . |
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ȼ ɪɟɡɭɥɶɬɚɬɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɭɪɚɜɧɟɧɢɹ (4.77) ɧɚɯɨɞɢɦ ɢɫɤɨɦɨɟ ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ ɲɚɪɢɤɚ ɜ ɬɪɭɛɤɟ:
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138 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɫɤɨɪɨɫɬɢ ɲɚɪɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɪɭɛɤɢ ɜɨɫ- |
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ɩɨɥɶɡɭɟɦɫɹ (4.75): |
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ȣ' Z r'2 l 2 eX' , |
(4.79) |
ɝɞɟ eX' ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɜɞɨɥɶ ɨɫɢ X'.
ɋɤɨɪɨɫɬɶ ɲɚɪɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (4.8) ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɚ ɜ ɜɢɞɟ:
ȣ |
c |
c |
Zr'eZ' Z |
r' |
2 |
l |
2 |
eX' , |
(4.80) |
>Ȧr |
@ ȣ |
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ɝɞɟ eZ' |
ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɜɞɨɥɶ ɨɫɢ Z'. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɤɨɦɵɣ |
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ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɲɚɪɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɪɚɜɟɧ:
X Z 2r'2 l 2 . |
(4.81) |
Ɂɚɞɚɱɚ 4.91
(ɉɨɫɬɭɩɚɬɟɥɶɧɨ ɞɜɢɠɭɳɚɹɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ) Ɉɞɧɨɪɨɞɧɵɣ ɰɢɥɢɧɞɪ ɦɚɫɫɨɣ m ɫɤɚɬɵɜɚɟɬɫɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵ-
ɜɚɧɢɹ ɫ ɤɥɢɧɚ ɦɚɫɫɨɣ M ɢ ɭɝɥɨɦ ɩɪɢ ɨɫɧɨɜɚɧɢɢ D , ɫɬɨɹɳɟɝɨ ɧɚ ɝɥɚɞɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ. ɇɚɣɬɢ ɭɫɤɨɪɟɧɢɟ ɤɥɢɧɚ.
Ɋɟɲɟɧɢɟ
I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɤɥɢɧɨɦ, ɨɫɢ ɤɨɬɨɪɨɣ X' ɢ Y' ɧɚɩɪɚɜɢɦ ɜɞɨɥɶ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɢ ɜɟɪɬɢɤɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɟɣ ɤɥɢɧɚ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ (ɫɦ. ɪɢɫ. 4.12 ɢ 4.13).
Y' |
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Fɬɪ |
ma0 |
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mg |
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ɬɪ |
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Ɋɢɫ. 4.12 |
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Ɋɢɫ. 4.13 |
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ɇɚ ɰɢɥɢɧɞɪ ɞɟɣɫɬɜɭɸɬ ɱɟɬɵɪɟ ɫɢɥɵ: ɫɢɥɚ ɬɹɠɟɫɬɢ mg, ɫɢɥɚ ɬɪɟɧɢɹ ɩɨɤɨɹ Fɬɪ (ɤɚɱɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ) ɢ ɫɢɥɚ
1 Ɂɚɞɚɱɢ 4.9 ɢ 4.10, ɚ ɬɚɤɠɟ ɡɚɞɚɱɭ 2 ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ɪɟɤɨɦɟɧɞɭɟɦ ɪɟɲɚɬɶ ɩɨɫɥɟ ɨɡɧɚɤɨɦɥɟɧɢɹ ɫ ɫɨɞɟɪɠɚɧɢɟɦ Ƚɥɚɜɵ 6.
Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
139 |
ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ N ɫɨ ɫɬɨɪɨɧɵ ɤɥɢɧɚ, ɚ ɬɚɤɠɟ ɩɟɪɟɧɨɫɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ ma0 (ɪɢɫ. 4.12), ɨɛɭɫɥɨɜɥɟɧɧɚɹ ɞɜɢɠɟɧɢɟɦ ɤɥɢɧɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɫ ɭɫɤɨɪɟɧɢɟɦ a0 , ɧɚɩɪɚɜɥɟɧɧɵɦ ɜ ɨɬɪɢɰɚɬɟɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ X'. ɋɢɥɨɣ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ ɩɪɟɧɟɛɪɟɝɚɟɦ.
ɇɚ ɤɥɢɧ ɞɟɣɫɬɜɭɸɬ: ɫɢɥɚ ɬɹɠɟɫɬɢ Mg, ɫɢɥɵ ɧɨɪɦɚɥɶɧɨɝɨ ɞɚɜɥɟɧɢɹ N ɢ ɬɪɟɧɢɹ ɩɨɤɨɹ Fɬɪ ɫɨ ɫɬɨɪɨɧɵ ɰɢɥɢɧɞɪɚ, ɫɢɥɚ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɨɩɨɪɵ R ɢ ɩɟɪɟɧɨɫɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ Ma0 (ɪɢɫ. 4.13).
II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɰɢɥɢɧɞɪɚ ɜ
ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ: |
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max' |
N sin D Fɬɪ cosD ma0 , |
(4.82) |
may' |
N cosD Fɬɪ sin D mg , |
(4.83) |
ɝɞɟ ax' , ay' |
– ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ a ɰɟɧɬɪɚ ɦɚɫɫ ɰɢɥɢɧɞɪɚ, ɞɥɹ ɤɨ- |
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ɬɨɪɵɯ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ: |
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a cosD , ay' a sin D . |
(4.84) |
ɍɪɚɜɧɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ (ɫɦ. (6.30) ɜ Ƚɥɚɜɟ 6) ɰɢɥɢɧɞɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɫɨɜɩɚɞɚɸɳɟɣ ɫ ɟɝɨ ɨɫɶɸ ɢ ɧɚɩɪɚɜɥɟɧ-
ɧɨɣ ɡɚ ɩɥɨɫɤɨɫɬɶ ɱɟɪɬɟɠɚ (ɫɦ. ɪɢɫ. 4.12), ɢɦɟɟɬ ɜɢɞ: |
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Fɬɪ r , |
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ɝɞɟ M – ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɰɢɥɢɧɞɪɚ, r – ɟɝɨ ɪɚɞɢɭɫ. ɉɪɢ ɡɚɩɢɫɢ |
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(4.85) ɭɱɬɟɧɨ ɢɡɜɟɫɬɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ J ɰɢɥɢɧɞ- |
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ɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɫɨɜɩɚɞɚɸɳɟɣ ɫ ɨɫɶɸ ɰɢɥɢɧɞɪɚ – J |
mr 2 |
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ɬɚɤɠɟ ɪɚɜɟɧɫɬɜɨ ɧɭɥɸ ɦɨɦɟɧɬɨɜ ɫɢɥ ɢɧɟɪɰɢɢ, ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɢ ɬɹɠɟɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɤɚɡɚɧɧɨɣ ɨɫɢ.
ɂɡ ɭɫɥɨɜɢɹ ɤɚɱɟɧɢɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɫɜɹɡɢ ɭɫɤɨɪɟɧɢɹ ɨɫɢ ɰɢɥɢɧɞɪɚ (ɫɨɜɩɚɞɚɸɳɟɝɨ ɫ ɭɫɤɨɪɟɧɢɟɦ ɰɟɧɬɪɚ ɦɚɫɫ a) ɫ ɟɝɨ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ:
a |
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(4.86) |
Mr . |
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Ɂɚɩɢɲɟɦ ɬɚɤɠɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɤɥɢɧɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ |
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N sinD Fɬɪ cosD Ma0 . |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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III. Ɋɟɲɚɹ ɫɨɜɦɟɫɬɧɨ ɭɪɚɜɧɟɧɢɹ (4.82) – (4.87), ɩɨɥɭɱɚɟɦ ɜɵ- |
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ɪɚɠɟɧɢɟ ɞɥɹ ɦɨɞɭɥɹ ɭɫɤɨɪɟɧɢɹ ɤɥɢɧɚ: |
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a0 |
mg sin 2D |
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(4.88) |
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3M m(1 |
2sin2 D) |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɤɨɦɨɟ ɭɫɤɨɪɟɧɢɟ ɤɥɢɧɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɪɚɜɧɨ:
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mg sin 2D |
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a0 |
3M m(1 2sin2 D) eX , |
(4.89) |
ɝɞɟ eX ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ
ɨɬɫɱɟɬɚ, ɫɨɜɩɚɞɚɸɳɢɣ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɨɫɢ X' ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
Ɂɚɞɚɱɚ 4.10
(ɉɨɫɬɭɩɚɬɟɥɶɧɨ ɞɜɢɠɭɳɚɹɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ) ɐɢɥɢɧɞɪ ɦɚɫɫɨɣ m ɢ ɪɚɞɢɭɫɨɦ R ɧɚɯɨɞɢɬɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ L
ɨɬ ɩɪɚɜɨɝɨ ɤɪɚɹ ɞɨɫɤɢ. Ⱦɨɫɤɚ ɧɚɱɢɧɚɟɬ ɞɜɢɝɚɬɶɫɹ ɫ ɭɫɤɨɪɟɧɢɟɦ a ɜɥɟɜɨ (ɪɢɫ. 4.14), ɩɪɢ ɷɬɨɦ ɰɢɥɢɧɞɪ ɤɚɬɢɬɫɹ ɩɨ ɞɨɫɤɟ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ. ɋ ɤɚɤɨɣ ɫɤɨɪɨɫɬɶɸ Vɰɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɨɫɤɢ ɛɭɞɟɬ ɞɜɢ-
ɝɚɬɶɫɹ ɰɟɧɬɪ ɦɚɫɫ ɰɢɥɢɧɞɪɚ ɜ ɬɨɬ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɨɧ ɛɭɞɟɬ ɧɚɯɨɞɢɬɶɫɹ ɧɚɞ ɤɪɚɟɦ ɞɨɫɤɢ?
N
a Fɬɪ 
ma
X'
mg
Ɋɢɫ. 4.14
Ɋɟɲɟɧɢɟ
I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɞɨɫɤɨɣ. ɇɚɩɪɚɜɢɦ ɨɫɶ X' ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɭɫɤɨɪɟɧɢɸ ɞɨɫɤɢ. ɇɚ ɰɢɥɢɧɞɪ ɜ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ ɞɟɣɫɬɜɭɸɬ ɱɟɬɵɪɟ ɫɢɥɵ: ɫɢɥɚ ɬɹɠɟɫɬɢ mg , ɫɢɥɚ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɨɩɨɪɵ N ,
ɫɢɥɚ ɬɪɟɧɢɹ ɩɨɤɨɹ Fɬɪ (ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɩɪɢ ɤɚɱɟɧɢɢ ɧɟɬ) ɢ ɩɟɪɟ-
ɧɨɫɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ ma . ɋɢɥɨɣ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ ɩɪɟɧɟɛɪɟɝɚɟɦ.