Механика.Методика решения задач
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Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
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Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɥɚɛɨɪɚɬɨɪɧɭɸ ɢɧɟɪɰɢɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ. ɇɚɩɪɚɜɢɦ ɨɫɢ X ɢ Y ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɬɚɤ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 6.5.
ɉɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ ɤɨɥɟɫɚ ɜ ɬɟɱɟɧɢɟ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɝɨ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ, ɤɚɤ "ɱɢɫɬɵɣ" ɩɨɜɨɪɨɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ ɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɩɟɪɟɫɟɱɟɧɢɹ ɩɪɹɦɵɯ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɫɤɨɪɨɫɬɹɦ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɤɨɥɟɫɚ (ɫɦ. ɩ. 6.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ). ɉɨɫɤɨɥɶɤɭ ɩɪɢ ɞɜɢɠɟɧɢɢ ɤɨɥɟɫɚ ɧɟ ɩɪɨɢɫɯɨɞɢɬ ɟɝɨ ɨɬɪɵɜɚ ɨɬ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɬɨ ɫɤɨɪɨɫɬɶ ɧɢɠɧɟɣ ɬɨɱɤɢ ɨɛɨɞɚ ɤɨɥɟɫɚ, ɤɨɬɨɪɚɹ ɫɨɩɪɢɤɚɫɚɟɬɫɹ ɫ ɩɨɜɟɪɯɧɨɫɬɶɸ, ɦɨɠɟɬ ɛɵɬɶ ɧɚɩɪɚɜɥɟɧɚ ɬɨɥɶɤɨ ɜɞɨɥɶ ɩɨɜɟɪɯɧɨɫɬɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɝɧɨɜɟɧɧɚɹ ɨɫɶ ɜɪɚɳɟɧɢɹ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɨɞɧɭ ɢɡ ɬɨɱɟɤ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɞɢɚɦɟɬɪɚ ɤɨɥɟɫɚ. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɦɝɧɨɜɟɧɧɚɹ ɨɫɶ ɜɪɚɳɟɧɢɹ ɦɨɠɟɬ ɧɚɯɨɞɢɬɶɫɹ ɤɚɤ ɜɵɲɟ, ɬɚɤ ɢ ɧɢɠɟ ɩɨɜɟɪɯɧɨɫɬɢ, ɩɨ ɤɨɬɨɪɨɣ ɤɚɬɢɬɫɹ ɤɨɥɟɫɨ.
ɉɭɫɬɶ yɆ – ɤɨɨɪɞɢɧɚɬɚ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ (ɫɦ. ɪɢɫ. 6.5) ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ. Ⱦɥɹ ɭɞɨɛɫɬɜɚ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɜɜɟɞɟɦ ɜɬɨɪɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɞɜɢɠɭɳɭɸɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɜɦɟɫɬɟ ɫ ɰɟɧɬɪɨɦ ɤɨɥɟɫɚ ɫɨ ɫɤɨɪɨɫɬɶɸ X0 ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚ-
ɬɨɪɧɨɣ ɫɢɫɬɟɦɵ, ɫ ɨɫɹɦɢ ɤɨɨɪɞɢɧɚɬ, ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɨɫɹɦ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ X ɢ Y.
II. ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ (6.2), ɫɜɹɡɵɜɚɸɳɟɣ ɫɤɨɪɨɫɬɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢ ɞɜɢɠɭɳɟɣɫɹ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ (ɫɦ. ɩ. 6.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ). ȼ ɞɜɢɠɭɳɟɣɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ X0 ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɦɨɞɭɥɢ ɫɤɨɪɨɫɬɟɣ
ɬɨɱɟɤ ɨɛɨɞɚ ɤɨɥɟɫɚ A ɢ B ɨɞɢɧɚɤɨɜɵ ɢ ɪɚɜɧɵ ZR . Ⱦɥɹ ɦɨɞɭɥɟɣ
ɷɬɢɯ ɫɤɨɪɨɫɬɟɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ (ɫɦ. ɪɢɫ. 6.5):
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Ɂɞɟɫɶ ɢ ɞɚɥɟɟ ɩɨɥɨɠɢɬɟɥɶɧɵɦ ɡɧɚɱɟɧɢɹɦ Z ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɜɪɚɳɟɧɢɟ ɤɨɥɟɫɚ ɩɨ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɟ.
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɨɱɟɜɢɞɧɵɦɢ ɝɟɨɦɟɬɪɢɱɟɫɤɢɦɢ ɫɨɨɬɧɨɲɟɧɢɹ-
ɦɢ (ɫɦ. ɪɢɫ. 6.5):
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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ɝɞɟ D – ɭɝɨɥ ɦɟɠɞɭ ɫɤɨɪɨɫɬɶɸ ɬɨɱɤɢ B ɢ ɧɚɩɪɚɜɥɟɧɢɟɦ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɤɨɥɟɫɚ.
III. ɉɪɟɨɛɪɚɡɭɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (6.49) ɢ (6.50) ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ ɤɨɥɟɫɚ Z:
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Ɋɟɲɚɹ ɩɨɥɭɱɟɧɧɨɟ ɤɜɚɞɪɚɬɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɚɟɦ ɞɜɚ ɡɧɚ- |
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ɱɟɧɢɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ: |
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5XA , ɫɥɟɞɨɜɚɬɟɥɶɧɨ: |
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ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ XB |
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ɋɨɝɥɚɫɧɨ (6.49) ɷɬɢɦ ɡɧɚɱɟɧɢɹɦ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ |
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ɤɨɥɟɫɚ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɞɜɚ ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɤɨɥɟɫɚ: |
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X01 |
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ɂɫɩɨɥɶɡɭɹ (6.51) ɢ (6.52) ɞɥɹ ɤɨɨɪɞɢɧɚɬɵ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ ɩɨɥɭɱɚɟɦ ɫɥɟɞɭɸɳɟɟ ɜɵɪɚɠɟɧɢɟ:
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ɉɨɞɫɬɚɜɥɹɹ (6.55) ɢ (6.56) ɜ (6.57), ɩɨɥɭɱɚɟɦ ɞɜɚ ɡɧɚɱɟɧɢɹ ɤɨɨɪɞɢɧɚɬɵ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ:
yM1 |
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ɂɬɚɤ, ɡɚɞɚɱɚ ɢɦɟɟɬ ɞɜɚ ɪɟɲɟɧɢɹ.
1. ɋɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɤɨɥɟɫɚ ɧɚɩɪɚɜɥɟɧɚ ɜ ɨɬɪɢɰɚɬɟɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ X, ɜɪɚɳɟɧɢɟ ɤɨɥɟɫɚ ɩɪɨɢɫɯɨɞɢɬ ɩɨ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɟ, ɦɝɧɨɜɟɧɧɚɹ ɨɫɶ ɜɪɚɳɟɧɢɹ ɪɚɫɩɨɥɨɠɟɧɚ ɧɚ ɜɟɪɬɢɤɚɥɶɧɨɦ ɞɢɚɦɟɬɪɟ ɧɢɠɟ ɬɨɱɤɢ A, ɧɨ ɜɵɲɟ ɰɟɧɬɪɚ ɤɨɥɟɫɚ (ɫɦ. ɪɢɫ. 6.6):
X |
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Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
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2. ɋɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɤɨɥɟɫɚ ɧɚɩɪɚɜɥɟɧɚ ɜ ɩɨɥɨɠɢɬɟɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ X, ɜɪɚɳɟɧɢɟ ɤɨɥɟɫɚ ɩɪɨɢɫɯɨɞɢɬ ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ, ɦɝɧɨɜɟɧɧɚɹ ɨɫɶ ɜɪɚɳɟɧɢɹ ɪɚɫɩɨɥɨɠɟɧɚ ɜɵɲɟ ɬɨɱɤɢ A (ɫɦ.
ɪɢɫ. 6.7):
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Ɋɢɫ. 6.6 |
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Ɋɢɫ. 6.7
Ɂɚɞɚɱɚ 6.2
Ʉɨɧɭɫ, ɜɵɫɨɬɚ ɤɨɬɨɪɨɝɨ h = 4 ɫɦ ɢ ɪɚɞɢɭɫ ɨɫɧɨɜɚɧɢɹ r = 3 ɫɦ, ɤɚɬɢɬɫɹ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ, ɢɦɟɹ ɧɟɩɨɞɜɢɠɧɭɸ ɜɟɪɲɢɧɭ ɜ ɬɨɱɤɟ O (ɪɢɫ. 6.8).
Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɤɨɧɭɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɩɨɜɟɪɯɧɨɫɬɶɸ, ɟɫɥɢ ɤɨɧɭɫ ɞɟɥɚɟɬ ɨɞɢɧ ɨɛɨɪɨɬ ɜɨɤɪɭɝ ɨɫɢ OZ ɡɚ ɜɪɟɦɹ T = 3 ɫ.
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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A Ȧ
Ɋɢɫ. 6.8
Ɋɟɲɟɧɢɟ
I. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɜɵɛɟɪɟɦ ɥɚɛɨɪɚɬɨɪɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɭɸ ɫ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ. ɉɪɢ ɷɬɨɦ ɨɫɶ Z ɫɢɫɬɟɦɵ ɧɚɩɪɚɜɢɦ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɨɜɟɪɯɧɨɫɬɢ, ɚ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɫɨɜɦɟɫɬɢɦ ɫ ɧɟɩɨɞɜɢɠɧɨɣ ɜɟɪɲɢɧɨɣ ɤɨɧɭ-
ɫɚ O (ɫɦ. ɪɢɫ. 6.8).
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɢɧɰɢɩɨɦ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɢɠɟɧɢɣ ɞɜɢɠɟɧɢɟ ɤɚɠɞɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɤɨɧɭɫɚ (ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ ɬɨɱɟɤ, ɥɟɠɚɳɢɯ ɧɚ ɨɫɢ ɤɨɧɭɫɚ OC) ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɨɣ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɫɭɩɟɪɩɨɡɢɰɢɸ ɞɜɭɯ ɞɜɢɠɟɧɢɣ – ɜɪɚɳɟɧɢɟ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Ȧ1 ɜɨɤɪɭɝ ɨɫɢ ɤɨɧɭɫɚ
OC ɢ ɜɪɚɳɟɧɢɟ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Ȧ2 ɜɨɤɪɭɝ ɨɫɢ Z. Ɍɨɱɤɢ, ɥɟ-
ɠɚɳɢɟ ɧɚ ɩɪɹɦɨɣ OA ɫɨɩɪɢɤɨɫɧɨɜɟɧɢɹ ɤɨɧɭɫɚ ɫ ɩɨɜɟɪɯɧɨɫɬɶɸ, ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɩɨɤɨɹɬɫɹ, ɬɚɤ ɤɚɤ ɧɟɬ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ. ɗɬɚ ɩɪɹɦɚɹ ɹɜɥɹɟɬɫɹ ɦɝɧɨɜɟɧɧɨɣ ɨɫɶɸ ɜɪɚɳɟɧɢɹ, ɜɨɤɪɭɝ ɤɨɬɨɪɨɣ ɤɨɧɭɫ ɜɪɚɳɚɟɬɫɹ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Ȧ Ȧ1 Ȧ2 .
II. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ C ɜ ɰɟɧɬɪɟ ɨɫɧɨɜɚɧɢɹ ɤɨɧɭɫɚ ɭɱɚɫɬɜɭɟɬ ɬɨɥɶɤɨ ɜ ɨɞɧɨɦ ɞɜɢɠɟɧɢɢ ɜɪɚɳɟɧɢɢ ɜɨɤɪɭɝ ɨɫɢ Z ɫ ɪɚɞɢɭɫɨɦ R (ɫɦ. ɪɢɫ. 6.9). ɉɪɢ ɷɬɨɦ ɟɟ ɫɤɨɪɨɫɬɶ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɪɚɜɧɚ:
XC Z2 R |
2S |
R . |
(6.59) |
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Ȧ |
O |
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Ɋɢɫ. 6.9 |
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Ɇɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɜ ɬɟɱɟɧɢɟ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɝɨ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ ɬɨɱɤɚ C ɜɪɚɳɚɟɬɫɹ ɜɨɤɪɭɝ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ OA ɫ
Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
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ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Ȧ . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɤɨɪɨɫɬɶ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ C ɜ ɰɟɧɬɪɟ ɨɫɧɨɜɚɧɢɹ ɤɨɧɭɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɪɚɜɧɚ:
XC Zf , |
(6.60) |
Ʉɚɤ ɜɢɞɧɨ ɧɚ ɪɢɫ. 6.9, ɞɥɹ R ɢ f ɜɵɩɨɥɧɹɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɫɨɨɬɧɨɲɟɧɢɹ:
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(6.61) |
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h2 f 2 |
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h2 f 2 . |
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(6.62) |
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III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (6.59) ɢ (6.60), ɩɨɥɭɱɚɟɦ: |
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2S |
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ɂɡ (6.61) ɢ (6.62) ɨɩɪɟɞɟɥɹɟɦ R ɢ f:
f |
rh |
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h2 |
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(6.64) |
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r 2 h2 |
r 2 h2 |
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ɂɫɤɨɦɚɹ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɤɨɧɭɫɚ ɜɨɤɪɭɝ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɪɚɜɧɚ:
Z |
h |
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2S |
2,79 ɪɚɞ/ɫ. |
(6.65) |
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ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ ɩɨɥɭɱɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ. ȼ ɱɚɫɬɧɨɫɬɢ, ɭɛɟɞɢɦɫɹ, ɱɬɨ ɜɵɩɨɥɧɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɭɝɥɨɜɵɦɢ ɫɤɨɪɨɫɬɹɦɢ ɜɪɚɳɟɧɢɹ ɤɨɧɭɫɚ:
Ȧ1 Ȧ2 Ȧ . |
(6.66) |
Ɉɩɪɟɞɟɥɢɦ ɦɨɞɭɥɶ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ Ȧ1 . Ⱦɥɹ ɷɬɨɝɨ ɜɨɫɩɨɥɶ-
ɡɭɟɦɫɹ ɬɟɦ, ɱɬɨ ɬɨɱɤɚ A, ɥɟɠɚɳɚɹ ɧɚ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɭɱɚɫɬɜɭɟɬ ɜ ɞɜɭɯ ɞɜɢɠɟɧɢɹɯ, ɩɪɢ ɷɬɨɦ ɟɟ ɫɤɨɪɨɫɬɶ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɪɚɜɧɚ ɧɭɥɸ:
X OA Z2 Z1r 0 . |
(6.67) |
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɤɨɧɭɫɚ ɜɨɤɪɭɝ ɨɫɢ OZ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (6.67) ɢ ɪɢɫ. 6.9 ɪɚɜɧɚ
Z1 Z2 |
h2 r 2 |
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(6.68) |
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ɇɚɩɪɚɜɥɟɧɢɹ ɜɟɤɬɨɪɨɜ Ȧ1 , Ȧ2 , ɢ Ȧ ɩɨɤɚɡɚɧɵ ɧɚ ɪɢɫ. 6.10.
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Ȧ1
Z
Ȧ
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ɋ |
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Ȧ1 |
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Ɋɢɫ. 6.10 |
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ȼɟɤɬɨɪ Ȧ1 ɧɚɩɪɚɜɥɟɧ ɜɞɨɥɶ ɨɫɢ ɤɨɧɭɫɚ OC, Ȧ2 – ɜɞɨɥɶ ɨɫɢ Z |
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ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, |
Ȧ – ɜɞɨɥɶ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟ- |
ɧɢɹ. ȼɫɟ ɬɪɢ ɜɟɤɬɨɪɚ ɥɟɠɚɬ ɜ ɨɞɧɨɣ ɩɥɨɫɤɨɫɬɢ ɢ ɫɨɫɬɚɜɥɹɸɬ ɩɪɹɦɨɭɝɨɥɶɧɵɣ ɬɪɟɭɝɨɥɶɧɢɤ (ɫɦ. ɪɢɫ. 6.10). ɂɫɩɨɥɶɡɭɹ ɫɨɨɬɧɨɲɟɧɢɹ
(6.67) ɢ (6.68), ɭɛɟɠɞɚɟɦɫɹ, ɱɬɨ
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Ɂɚɞɚɱɚ 6.3
Ⱦɜɚ ɫɨɨɫɧɵɯ ɤɨɥɟɫɚ ɫ ɪɚɞɢɭɫɚɦɢ r1 ɢ r2 (r1 < r2) ɜɪɚɳɚɸɬɫɹ ɜ
ɨɞɧɭ ɫɬɨɪɨɧɭ ɫ ɩɨɫɬɨɹɧɧɵɦɢ |
ɭɝɥɨɜɵɦɢ ɫɤɨɪɨɫɬɹɦɢ |
Ȧ1 ɢ Ȧ2 |
( Ȧ1 ! Z2 ). Ɇɟɠɞɭ ɤɨɥɟɫɚɦɢ |
ɡɚɠɚɬɨ ɬɪɟɬɶɟ ɤɨɥɟɫɨ |
ɪɚɞɢɭɫɨɦ |
r3 = (r2 – r1)/2, ɞɜɢɠɭɳɟɟɫɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ (ɪɢɫ. 6.11). ɇɚɣɬɢ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ Ȧ ɜɪɚɳɟɧɢɹ ɬɪɟɬɶɟɝɨ ɤɨɥɟɫɚ ɢ ɫɤɨɪɨɫɬɶ X0 ɟɝɨ ɰɟɧɬɪɚ.
Ɋɟɲɟɧɢɟ
I. Ɉɛɨɡɧɚɱɢɦ ɬɨɱɤɢ ɫɨɩɪɢɤɨɫɧɨɜɟɧɢɹ ɬɪɟɬɶɟɝɨ ɤɨɥɟɫɚ ɫ ɩɟɪɜɵɦ ɢ ɜɬɨɪɵɦ ɬɨɱɤɚɦɢ B ɢ A ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ⱦɥɹ ɬɪɟɬɶɟɝɨ ɤɨɥɟɫɚ, ɡɚɠɚɬɨɝɨ ɦɟɠɞɭ ɞɜɭɦɹ ɜɪɚɳɚɸɳɢɦɢɫɹ ɤɨɥɟɫɚɦɢ, ɫɤɨɪɨɫɬɢ ɬɨɱɟɤ A ɢ B ɫɨɜɩɚɞɚɸɬ ɫɨ ɫɤɨɪɨɫɬɹɦɢ ɬɨɱɟɤ, ɧɚɯɨɞɹɳɢɯɫɹ ɧɚ ɨɛɨɞɚɯ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɨɥɟɫ (ɫɦ. ɪɢɫ. 6.11).
Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
207 |
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Ɋɢɫ. 6.11 |
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II. ɉɨɫɤɨɥɶɤɭ ɩɟɪɜɨɟ ɢ ɜɬɨɪɨɟ ɤɨɥɟɫɚ ɜɪɚɳɚɸɬɫɹ ɫ ɭɝɥɨɜɵɦɢ |
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ɫɤɨɪɨɫɬɹɦɢ Ȧ1 |
ɢ Ȧ2 , ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɬɨɱɟɤ A ɢ B ɩɪɢ ɜɪɚɳɟ- |
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ɧɢɢ ɷɬɢɯ ɤɨɥɟɫ ɪɚɜɧɵ: |
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XA |
Z2r2 , |
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(6.70) |
XB Z1r1 . |
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(6.71) |
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ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɭɞɨɛɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɩɨɧɹɬɢɟɦ ɦɝɧɨ- |
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ɜɟɧɧɨɣ |
ɨɫɢ |
ɜɪɚɳɟɧɢɹ |
ɞɥɹ |
ɬɪɟɬɶɟɝɨ |
ɤɨɥɟɫɚ |
(ɫɦ. |
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ɩ. 6.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ), |
ɨɬɧɨɫɢɬɟɥɶɧɨ |
ɤɨɬɨɪɨɣ |
ɤɨɥɟɫɨ |
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ɜɪɚɳɚɟɬɫɹ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ |
Ȧ . ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɷɬɚ ɨɫɶ ɩɟɪ- |
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ɩɟɧɞɢɤɭɥɹɪɧɚ ɩɥɨɫɤɨɫɬɢ ɱɟɪɬɟɠɚ ɢ ɩɟɪɟɫɟɤɚɟɬ ɩɪɹɦɭɸ, ɩɪɨɯɨɞɹɳɭɸ ɱɟɪɟɡ ɬɨɱɤɢ O, B ɢ A. ɉɭɫɬɶ ɦɝɧɨɜɟɧɧɚɹ ɨɫɶ ɜɪɚɳɟɧɢɹ ɥɟɠɢɬ ɦɟɠɞɭ ɬɨɱɤɚɦɢ O ɢ B ɧɚ ɪɚɫɫɬɨɹɧɢɢ rx ɨɬ ɬɨɱɤɢ O, ɬɨɝɞɚ:
XA |
Z r1 rx 2r3 , |
(6.72) |
XB |
Z r1 rx , |
(6.73) |
ɩɪɢ ɷɬɨɦ ɞɥɹ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɬɪɟɬɶɟɝɨ ɤɨɥɟɫɚ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ: |
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X0 |
Z r1 rx r3 . |
(6.74) |
III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (6.70) (6.73) ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ ɬɪɟɬɶɟɝɨ ɤɨɥɟɫɚ Ȧ , ɩɨɥɭɱɚɟɦ:
Z |
Z2r2 Z1r1 . |
(6.75) |
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2r3 |
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ɉɨɞɫɬɚɜɥɹɹ (6.75) ɜ (6.74), ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɭɸ ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɬɪɟɬɶɟɝɨ ɤɨɥɟɫɚ:
208 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
X0 |
Z1r1 Z2r2 . |
(6.76) |
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2 |
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6.3.2. Ⱦɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
Ɂɚɞɚɱɚ 6.4
ɋ ɧɚɤɥɨɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ ɫ ɭɝɥɨɦ D ɩɪɢ ɜɟɪɲɢɧɟ ɫɤɚɬɵɜɚɟɬɫɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɨɞɧɨɪɨɞɧɵɣ ɲɚɪ ɦɚɫɫɨɣ m ɢ ɪɚɞɢɭɫɨɦ R. ɇɚɣɬɢ ɭɫɤɨɪɟɧɢɟ a ɰɟɧɬɪɚ ɲɚɪɚ.
Ɋɟɲɟɧɢɟ
I. ɉɪɢ ɫɚɦɨɩɪɨɢɡɜɨɥɶɧɨɦ ɫɤɚɬɵɜɚɧɢɢ ɲɚɪɚ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ ɜ ɫɬɨɪɨɧɭ ɟɝɨ ɪɟɛɪɚ ɞɜɢɠɟɧɢɟ ɲɚɪɚ ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɢɦ, ɩɨɫɤɨɥɶɤɭ ɜɫɟ ɟɝɨ ɦɚɬɟɪɢɚɥɶɧɵɟ ɬɨɱɤɢ ɞɜɢɠɭɬɫɹ ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɥɨɫɤɨɫɬɹɯ. Ɇɝɧɨɜɟɧɧɚɹ ɨɫɶ ɜɪɚɳɟɧɢɹ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɭ M (ɫɦ. ɪɢɫ. 6.12)
N
Fɬɪ M |
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X |
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Y |
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mg |
D |
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Ɋɢɫ. 6.12
ɫɨɩɪɢɤɨɫɧɨɜɟɧɢɹ ɲɚɪɚ ɫ ɩɨɜɟɪɯɧɨɫɬɶɸ ɩɚɪɚɥɥɟɥɶɧɨ ɪɟɛɪɭ ɤɥɢɧɚ (ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɪɚ).
ȼɵɛɟɪɟɦ ɥɚɛɨɪɚɬɨɪɧɭɸ ɢɧɟɪɰɢɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɭɸ ɫ ɧɚɤɥɨɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ, ɨɫɶ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɥɟɧɚ ɜɞɨɥɶ ɧɚɤɥɨɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ, ɚ ɨɫɶ Y ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ, ɩɪɢɱɟɦ ɩɥɨɫɤɨɫɬɶ XY ɩɚɪɚɥɥɟɥɶɧɚ ɩɥɨɫɤɨɫɬɹɦ, ɜ ɤɨɬɨɪɵɯ ɞɜɢɠɭɬɫɹ ɦɚɬɟɪɢɚɥɶɧɵɟ ɬɨɱɤɢ ɲɚɪɚ
(ɪɢɫ. 6.12).
ɇɚ ɲɚɪ ɜ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ ɞɟɣɫɬɜɭɸɬ ɬɪɢ ɫɢɥɵ – ɫɢɥɚ ɬɹɠɟɫɬɢ mg , ɫɢɥɚ ɬɪɟɧɢɹ ɩɨɤɨɹ Fɬɪ (ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɧɟɬ) ɢ ɫɢɥɚ
ɪɟɚɤɰɢɢ ɨɩɨɪɵ N (ɪɢɫ. 6.12).
Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
209 |
II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ (ɭɪɚɜɧɟɧɢɟ ɦɨɦɟɧɬɨɜ (6.48)) ɞɥɹ ɲɚɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ, ɫɨɜɩɚɞɚɸɳɟɣ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫ ɦɝɧɨɜɟɧɧɨɣ ɨɫɶɸ ɜɪɚɳɟɧɢɹ (ɫɦ. ɪɢɫ. 6.12), ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ:
J |
dZ |
mgR sin D , |
(6.77) |
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dt |
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ɝɞɟ Z ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɲɚɪɚ, J ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɲɚɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ. Ɇɨɦɟɧɬɵ ɫɢɥɵ ɬɪɟɧɢɹ ɢ ɫɢɥɵ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɨɩɨɪɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɨɫɢ ɪɚɜɧɵ ɧɭɥɸ.
Ɂɚɩɢɲɟɦ ɬɚɤɠɟ ɭɪɚɜɧɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ ɦɟɠɞɭ ɭɫ-
ɤɨɪɟɧɢɟɦ ɰɟɧɬɪɚ ɦɚɫɫ a ɢ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ ɲɚɪɚ ddZt (ɜɫɥɟɞɫɬ-
ɜɢɟ ɞɜɢɠɟɧɢɹ ɲɚɪɚ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ):
a |
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R |
dZ |
. |
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(6.78) |
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dt |
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Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɲɚɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟ- |
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ɧɢɹ ɧɚɣɞɟɦ, ɢɫɩɨɥɶɡɭɹ ɬɟɨɪɟɦɭ Ƚɸɣɝɟɧɫɚ-ɒɬɟɣɧɟɪɚ (6.42): |
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J |
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J0 mR2 , |
(6.79) |
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ɝɞɟ J0 ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɲɚɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟ- |
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ɪɟɡ ɟɝɨ ɰɟɧɬɪ ɦɚɫɫ (6.45), ɪɚɜɧɵɣ |
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J0 |
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2 |
mR2 . |
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(6.80) |
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5 |
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III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (6.77) – (6.79) ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɫ- |
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ɤɨɪɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɪɚ, ɩɨɥɭɱɢɦ: |
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a |
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1 |
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g sinD . |
(6.81) |
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1 |
J0 |
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mR2 |
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ɍɱɢɬɵɜɚɹ ɜɵɪɚɠɟɧɢɟ (6.80) ɞɥɹ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɲɚɪɚ, ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɨɟ ɭɫɤɨɪɟɧɢɟ ɰɟɧɬɪɚ ɲɚɪɚ:
a |
5 |
g sin D . |
(6.82) |
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7 |
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Ɉɩɪɟɞɟɥɢɦ ɭɫɥɨɜɢɹ, ɩɪɢ ɤɨɬɨɪɵɯ ɩɨɥɭɱɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ ɹɜɥɹɟɬɫɹ ɜɟɪɧɵɦ. Ⱦɥɹ ɞɜɢɠɟɧɢɹ ɲɚɪɚ ɩɨ ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɧɟɨɛɯɨɞɢɦɨ, ɱɬɨɛɵ ɫɢɥɚ ɬɪɟɧɢɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚ-
210 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
ɤɨɧɨɦ Ⱥɦɨɧɬɨɧɚ–Ʉɭɥɨɧɚ (ɫɦ. ɩ. 2.1.2 ɜ Ƚɥɚɜɟ 2) ɭɞɨɜɥɟɬɜɨɪɹɥɚ ɧɟɪɚɜɟɧɫɬɜɭ:
Fɬɪ d PN . |
(6.83) |
Ɂɞɟɫɶ P – ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ.
Ɇɨɞɭɥɢ ɫɢɥ ɬɪɟɧɢɹ ɢ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɨɩɨɪɵ ɦɨɠɧɨ ɧɚɣɬɢ ɢɡ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɲɚɪɚ, ɡɚɩɢɫɚɧɧɨɝɨ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ X ɢ Y ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ (ɫɦ. ɪɢɫ. 6.12):
ma |
mg sin D Fɬɪ , |
(6.84) |
0 |
mg cosD N . |
(6.85) |
Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɞɜɭɯ ɭɪɚɜɧɟɧɢɣ (6.84) – (6.85) ɢ ɨɞɧɨɝɨ ɧɟɪɚɜɟɧɫɬɜɚ (6.83) ɫ ɭɱɟɬɨɦ ɧɚɣɞɟɧɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɰɟɧɬɪɚ ɲɚɪɚ (6.82), ɩɨɥɭɱɚɟɦ ɭɫɥɨɜɢɟ ɤɚɱɟɧɢɹ ɲɚɪɚ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɩɨ ɧɚɤɥɨɧɧɨɣ
ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ: |
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tgD d |
7 |
P . |
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(6.86) |
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2 |
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5 |
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7 |
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Ɉɬɜɟɬ: a |
g sin D ɩɪɢ tgD d P |
. |
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7 |
2 |
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Ɂɚɞɚɱɚ 6.5
ɇɚ ɝɥɚɞɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɥɟɠɢɬ ɞɨɫɤɚ ɦɚɫɫɨɣ m1 ɢ ɧɚ ɧɟɣ ɨɞɧɨɪɨɞɧɵɣ ɲɚɪ ɦɚɫɫɨɣ m2. Ʉ ɞɨɫɤɟ ɩɪɢɥɨɠɢɥɢ ɩɨɫɬɨɹɧɧɭɸ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɫɢɥɭ F. ɋ ɤɚɤɢɦɢ ɭɫɤɨɪɟɧɢɹɦɢ ɛɭɞɭɬ ɞɜɢɝɚɬɶɫɹ ɞɨɫɤɚ ɚ1 ɢ ɰɟɧɬɪ ɲɚɪɚ ɚ2 ɜ ɨɬɫɭɬɫɬɜɢɟ ɫɤɨɥɶɠɟɧɢɹ ɦɟɠɞɭ ɧɢɦɢ?
Ɋɟɲɟɧɢɟ
I ȼɜɟɞɟɦ ɥɚɛɨɪɚɬɨɪɧɭɸ ɢɧɟɪɰɢɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɨɫɢ X ɢ Y ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɥɟɧɵ ɬɚɤ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 6.13. ɉɪɢ ɞɜɢɠɟɧɢɢ ɲɚɪɚ ɩɨ ɞɨɫɤɟ ɜɞɨɥɶ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɫɢɥɵ F ɞɜɢɠɟɧɢɟ ɲɚɪɚ ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɢɦ.
ȼ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɞɜɢɠɟɧɢɟ ɲɚɪɚ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɫɭɩɟɪɩɨɡɢɰɢɸ ɟɝɨ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɢ ɜɪɚɳɟɧɢɹ ɲɚɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɰɟɧɬɪ ɦɚɫɫ.