Механика.Методика решения задач
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Ƚɥɚɜɚ 7. Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
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ɞɜɢɠɟɧɢɹ ɢɯ ɰɟɧɬɪɨɜ ɦɚɫɫ ɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫ ɨɞɢɧɚɤɨɜɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɟɣ, ɩɪɨɯɨɞɹɳɢɯ ɱɟɪɟɡ ɢɯ ɰɟɧɬɪɵ ɦɚɫɫ (ɫɦ. ɪɢɫ. 7.12).
D
Z
X2c
C
A Z
X1c
B
Ɋɢɫ. 7.12
ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɦɚɫɫɚ ɝɚɧɬɟɥɢ ɫɨɫɪɟɞɨɬɨɱɟɧɚ ɧɚ ɟɟ ɤɨɧɰɚɯ, ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ, ɪɚɜɟɧ:
J |
ml |
2 |
. |
(7.71) |
4 |
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ɉɪɢ ɷɬɨɦ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɨɜ ɦɚɫɫ ɝɚɧɬɟɥɟɣ ɩɨɫɥɟ ɩɟɪɜɨɝɨ ɫɨɭɞɚɪɟɧɢɹ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɢ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɢɯ ɜɪɚɳɟɧɢɹ ɪɚɜɧɵ:
X1c |
X2c |
X1 X2 |
, |
(7.72) |
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2 |
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Z |
X1 X2 . |
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(7.73) |
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l |
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ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (7.65) ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɨɜ ɦɚɫɫ ɝɚɧɬɟɥɟɣ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ ɜ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɦɚɫɫ ɫɢɫɬɟɦɵ ɬɟɥ ɪɚɜɧɵ ɧɭɥɸ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɝɚɧɬɟɥɢ, ɜɪɚɳɚɹɫɶ ɫ ɨɞɢɧɚɤɨɜɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ, ɢɫɩɵɬɚɸɬ ɩɨɜɬɨɪɧɨɟ ɫɨɭɞɚɪɟɧɢɟ (ɫɦ. ɪɟɲɟɧɢɟ Ɂɚɞɚɱɢ 8 ɜ Ƚɥɚɜɟ 3).
Ɂɚɞɚɱɚ. 7.5
Ɍɨɧɤɢɣ ɨɞɧɨɪɨɞɧɵɣ ɫɬɟɪɠɟɧɶ ɞɥɢɧɨɣ l0 ɢ ɦɚɫɫɨɣ m0 10 ɝ
ɜɪɚɳɚɟɬɫɹ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɟɝɨ ɩɨɞɜɟɫɚ, ɨɩɢɫɵɜɚɹ ɩɪɢ ɷɬɨɦ ɤɨɧɢɱɟɫɤɭɸ ɩɨɜɟɪɯɧɨɫɬɶ (ɫɦ. ɪɢɫ. 7.13). ɀɭɤ, ɫɢɞɹɳɢɣ ɧɚ ɫɬɟɪɠɧɟ, ɧɚɱɢɧɚɟɬ ɦɟɞɥɟɧɧɨ ɩɨɥɡɬɢ ɫ ɜɟɪɯɧɟɝɨ ɡɚɤɪɟɩɥɟɧɧɨɝɨ ɟɝɨ ɤɨɧɰɚ ɤ ɧɢɠɧɟɦɭ ɤɨɧɰɭ. ɇɚɱɚɥɶɧɵɣ
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɫɬɟɪɠɧɹ ɨɬ ɜɟɪɬɢɤɚɥɢ |
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ɫɨɫɬɚɜɥɹɟɬ D0 = 60q. ɉɪɢ ɤɚɤɨɣ ɦɚɫɫɟ ɠɭɤɚ |
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m ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɫɬɟɪɠɧɹ ɨɬ ɜɟɪɬɢɤɚɥɢ |
D m |
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ɫɨɫɬɚɜɢɬ D1 = 45q ɩɨɫɥɟ ɬɨɝɨ, ɤɚɤ ɠɭɤ ɞɨɫ- |
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ɬɢɝɧɟɬ ɧɢɠɧɟɝɨ ɤɨɧɰɚ ɫɬɟɪɠɧɹ?
Ɋɟɲɟɧɢɟ
I. ɉɪɢ ɞɜɢɠɟɧɢɢ ɠɭɤɚ ɜɞɨɥɶ ɫɬɟɪɠɧɹ ɧɚ ɫɢɫɬɟɦɭ ɬɟɥ «ɫɬɟɪɠɟɧɶ + ɠɭɤ» ɞɟɣɫɬɜɭɸɬ ɫɢɥɵ ɬɹɠɟɫɬɢ ɢ ɫɢɥɵ ɪɟɚɤɰɢɢ ɫɨ ɫɬɨɪɨɧɵ ɩɨɞɜɟɫɚ. ɉɨɫɤɨɥɶɤɭ ɦɨɦɟɧɬɵ ɷɬɢɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ,
ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɩɨɞɜɟɫɚ, ɪɚɜɧɵ ɧɭɥɸ, ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ ɬɟɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɨɫɢ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɧɟ ɢɡɦɟɧɹɟɬɫɹ. ɉɨɷɬɨɦɭ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɛɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ ɞɥɹ ɞɜɭɯ ɦɨɦɟɧɬɨɜ ɜɪɟɦɟɧɢ: ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ (l = 0, D D0 ) ɢ ɦɨɦɟɧɬɚ ɞɨɫɬɢɠɟɧɢɹ ɠɭ-
ɤɨɦ ɤɨɧɰɚ ɫɬɟɪɠɧɹ (l = l0, D D1 ). Ɂɞɟɫɶ ɜɜɟɞɟɧɵ ɨɛɨɡɧɚɱɟɧɢɹ: l
ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɠɭɤɚ ɞɨ ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɫɬɟɪɠɧɹ ɢ D ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɫɬɟɪɠɧɹ ɨɬ ɜɟɪɬɢɤɚɥɢ.
ȼ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ ɠɭɤɚ ɪɚɫɫɬɨɹɧɢɟ l, ɭɝɨɥ D , ɚ ɬɚɤɠɟ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɫɬɟɪɠɧɹ Z ɢɡɦɟɧɹɸɬɫɹ. Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɜɡɚɢɦɨɫɜɹɡɢ ɦɟɠɞɭ ɭɝɥɨɦ ɨɬɤɥɨɧɟɧɢɹ D ɢ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Z ɭɞɨɛɧɨ ɩɟɪɟɣɬɢ ɤ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɨɣ ɫ ɜɪɚɳɚɸɳɢɦɫɹ ɫɬɟɪɠɧɟɦ. ȼ ɷɬɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɧɚ ɫɢɫɬɟɦɭ ɬɟɥ «ɫɬɟɪɠɟɧɶ + ɦɭɮɬɚ» ɞɟɣɫɬɜɭɸɬ ɫɢɥɵ ɬɹɠɟɫɬɢ, ɫɢɥɚ ɪɟɚɤɰɢɢ ɩɨɞɜɟɫɚ ɜ ɬɨɱɤɟ ɤɪɟɩɥɟɧɢɹ ɫɬɟɪɠɧɹ ɢ ɫɢɥɵ ɢɧɟɪɰɢɢ.
ɉɨɫɤɨɥɶɤɭ ɠɭɤ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɞɜɢɠɟɬɫɹ ɩɨ ɫɬɟɪɠɧɸ ɦɟɞɥɟɧɧɨ, ɬɨ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɫɬɟɪɠɧɹ Z ɬɚɤɠɟ ɦɟɞɥɟɧɧɨ ɢɡɦɟɧɹɟɬɫɹ ( Ȧ # 0 ), ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɟɪɟɧɨɫɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ (ɫɦ. (4.16) ɜ Ƚɥɚɜɟ 4) ɩɪɢɦɟɪɧɨ ɪɚɜɧɚ ɰɟɧɬɪɨɛɟɠɧɨɣ ɫɢɥɟ ɢɧɟɪɰɢɢ: Fɩɟɪ # Fɰɛ . ɋɢɥɨɣ ɢɧɟɪɰɢɢ Ʉɨɪɢɨɥɢɫɚ (ɫɦ. (4.17) ɜ
Ƚɥɚɜɟ 4), ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɦɟɞɥɟɧɧɨ ɞɜɢɠɭɳɟɝɨɫɹ ɠɭɤɚ, ɬɚɤɠɟ ɩɪɟɧɟɛɪɟɝɚɟɦ: FɄɨɪ # 0 .
Ƚɥɚɜɚ 7. Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
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ɋɭɦɦɚ ɦɨɦɟɧɬɨɜ ɜɧɟɲɧɢɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɫɢɫɬɟɦɭ ɬɟɥ «ɫɬɟɪɠɟɧɶ + ɠɭɤ, ɪɚɜɧɚ ɧɭɥɸ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɸɛɨɣ ɨɫɢ, ɧɟɩɨɞɜɢɠɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɨɣ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɰɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɪɚɡɥɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɫɬɟɪɠɧɹ, ɡɚɜɢɫɢɬ ɨɬ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɞɚɧɧɵɦ ɷɥɟɦɟɧɬɨɦ ɢ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɶɸ ɜɪɚɳɟɧɢɹ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɪɚɫɱɟɬɚ ɫɭɦɦɚɪɧɨɝɨ ɦɨɦɟɧɬɚ ɫɢɥ ɢɧɟɪɰɢɢ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɫɬɟɪɠɟɧɶ, ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɫɭɦɦɢɪɨɜɚɬɶ ɦɨɦɟɧɬɵ ɫɢɥ ɢɧɟɪɰɢɢ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɤɚɠɞɵɣ ɢɡ ɷɥɟɦɟɧɬɨɜ ɫɬɟɪɠɧɹ. Ɂɚɦɟɬɢɦ ɬɚɤɠɟ, ɱɬɨ ɦɨɦɟɧɬ ɫɢɥɵ ɪɟɚɤɰɢɢ ɩɨɞɜɟɫɚ ɜ ɬɨɱɤɟ ɤɪɟɩɥɟɧɢɹ ɫɬɟɪɠɧɹ ɪɚɜɟɧ ɧɭɥɸ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɩɨɞɜɟɫɚ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɡɚɤɨɧɨɦ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɬɟɥ «ɫɬɟɪɠɟɧɶ + ɠɭɤ» ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɢ ɭɫɥɨɜɢɟɦ ɪɚɜɟɧɫɬɜɚ ɧɭɥɸ ɫɭɦɦɚɪɧɨɝɨ ɦɨɦɟɧɬɚ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɷɬɭ ɫɢɫɬɟɦɭ ɬɟɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɨɣ ɫ ɜɪɚɳɚɸɳɢɦɫɹ ɫɬɟɪɠɧɟɦ ɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɩɨɞɜɟɫɚ,
ɜɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ.
II. Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɬɟɥ «ɫɬɟɪɠɟɧɶ + ɠɭɤ» L ɫɤɥɚ-
ɞɵɜɚɟɬɫɹ ɢɡ ɦɨɦɟɧɬɨɜ ɢɦɩɭɥɶɫɨɜ ɫɬɟɪɠɧɹ Lɫɬ ɢ ɠɭɤɚ Lɠ ɜ ɨɬɞɟɥɶ-
ɧɨɫɬɢ. Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɷɥɟɦɟɧɬɚ ɫɬɟɪɠɧɹ Lɫɬ ɡɚɜɢɫɢɬ ɨɬ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɞɚɧɧɵɦ ɷɥɟɦɟɧɬɨɦ ɢ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɶɸ ɜɪɚɳɟɧɢɹ ɫɬɟɪɠɧɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɪɚɫɱɟɬɚ ɫɭɦɦɚɪɧɨɝɨ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɬɟɪɠɧɹ ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɫɭɦɦɢɪɨɜɚɬɶ ɦɨɦɟɧɬɵ ɢɦɩɭɥɶɫɨɜ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ ɫɬɟɪɠɧɹ. Ɂɚɩɢɲɟɦ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɬɟɥ «ɫɬɟɪɠɟɧɶ + ɠɭɤ» ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ:
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L Lɫɬ Lɠ |
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x sin D |
dxZx sin D l sin D mZl sinD |
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Z sin |
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m0l0 |
ml |
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¸ , |
(7.74) |
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ɝɞɟ x ɤɨɨɪɞɢɧɚɬɚ ɷɥɟɦɟɧɬɚ ɫɬɟɪɠɧɹ ɞɥɢɧɨɣ dx ɜɞɨɥɶ ɫɬɟɪɠɧɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɩɨɞɜɟɫɚ.
Ɂɚɩɢɲɟɦ ɬɚɤɠɟ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɬɟɥ «ɫɬɟɪɠɟɧɶ + ɠɭɤ» ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɧɚ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ ɨɬ ɧɚɱɚɥɚ
254 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
ɞɜɢɠɟɧɢɹ ɠɭɤɚ (l = 0, D |
D0 ) ɞɨ ɦɨɦɟɧɬɚ ɞɨɫɬɢɠɟɧɢɹ ɢɦ ɧɢɠɧɟɝɨ |
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ɤɨɧɰɚ ɫɬɟɪɠɧɹ (l = l0, D |
D1 ): |
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L L |
1 |
Z sin |
2 D |
m 3m l 2 |
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Z |
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sin 2 D m l 2 |
0 , (7.75) |
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0 0 |
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ɝɞɟ L0 ɢ Z0 ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɬɟɥ ɢ |
O |
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ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɢɯ ɜɪɚɳɟɧɢɹ ɜ ɧɚɱɚɥɶɧɵɣ |
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ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɚ ɡɧɚɱɟɧɢɹ ɷɬɢɯ ɜɟɥɢɱɢɧ ɜ |
D |
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ɤɨɧɟɱɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ L1 ɢ Z1. |
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Ɂɚɩɢɲɟɦ ɪɚɜɟɧɫɬɜɨ ɧɭɥɸ ɫɭɦɦɵ ɦɨ- |
mg |
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ɦɟɧɬɨɜ ɜɧɟɲɧɢɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɫɢɫɬɟ- |
dF ɫɬ |
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ɦɭ ɬɟɥ «ɫɬɟɪɠɟɧɶ + ɠɭɤ», ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɨɪɢ- |
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ɡɨɧɬɚɥɶɧɨɣ ɨɫɢ, ɧɟɩɨɞɜɢɠɧɨɣ ɜ ɜɵɛɪɚɧɧɨɣ |
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ɰɛ |
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ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɢ ɩɪɨɯɨɞɹ- |
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ɳɟɣ ɱɟɪɟɡ ɜɟɪɯɧɢɣ ɤɨɧɟɰ ɫɬɟɪɠɧɹ O (ɫɦ. |
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Ɋɢɫ. 7.14 |
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ɪɢɫ. 7.14): |
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M ɬɠɫɬ M ɬɠɠ M ɰɛɫɬ M ɰɛɠ |
0 . |
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(7.76) |
Ɇɨɦɟɧɬɵ ɫɢɥ ɬɹɠɟɫɬɢ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɫɬɟɪɠɟɧɶ M ɬɠɫɬ |
ɢ ɠɭɤɚ |
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M ɬɠɠ , ɪɚɜɧɵ: |
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M ɫɬ |
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m gl |
0 |
sin D , |
(7.77) |
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ɬɠ |
2 |
0 |
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M ɬɠɠ |
mgl sinD . |
(7.78) |
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Ɇɨɦɟɧɬ ɰɟɧɬɪɨɛɟɠɧɨɣ ɫɢɥɵ ɢɧɟɪɰɢɢ M ɰɛɠ (ɫɦ. (4.16) ɜ Ƚɥɚɜɟ 4), ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɠɭɤɚ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ:
M ɰɛɠ mZ2l 2 sinD cosD . |
(7.79) |
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɫɭɦɦɚɪɧɨɝɨ ɦɨɦɟɧɬɚ ɫɢɥ ɢɧɟɪɰɢɢ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɫɬɟɪɠɟɧɶ M ɰɛɫɬ , ɪɚɫɫɦɨɬɪɢɦ ɷɥɟɦɟɧɬ ɫɬɟɪɠɧɹ ɞɥɢɧɨɣ dx,
ɧɚɯɨɞɹɳɢɣɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ x ɨɬ ɜɟɪɯɧɟɝɨ ɤɨɧɰɚ ɫɬɟɪɠɧɹ. ɐɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ (ɫɦ. (4.16)), ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɷɬɨɬ ɷɥɟɦɟɧɬ, ɪɚɜɧɚ
dF ɫɬ |
m0 |
dxZ2 x sin D , |
(7.80) |
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ɰɛ |
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ɝɞɟ x sin D – ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɷɥɟɦɟɧɬɚ ɫɬɟɪɠɧɹ ɞɨ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ ɜɵɛɪɚɧɧɨɣ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
256 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɱɢɫɥɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ, ɧɚɯɨɞɢɦ
m | 0,32 m0 3,2 ɝ .
Ɂɚɞɚɱɚ 7.6
ɋɩɭɬɧɢɤ ɦɚɫɫɨɣ m ɞɜɢɠɟɬɫɹ ɩɨ ɷɥɥɢɩɬɢɱɟɫɤɨɣ ɬɪɚɟɤɬɨɪɢɢ ɜɨɤɪɭɝ ɩɥɚɧɟɬɵ, ɧɚɯɨɞɹɳɟɣɫɹ ɜ ɨɞɧɨɦ ɢɡ ɟɟ ɮɨɤɭɫɨɜ (ɫɦ. ɪɢɫ. 7.15).
ȣ1
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Ɋɢɫ. 7.15
ɂɡɜɟɫɬɧɵ ɧɚɢɦɟɧɶɲɟɟ r1 ɢ ɧɚɢɛɨɥɶɲɟɟ r2 ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɫɩɭɬɧɢɤɚ ɞɨ ɰɟɧɬɪɚ ɩɥɚɧɟɬɵ, ɚ ɬɚɤɠɟ ɦɨɞɭɥɶ ɟɝɨ ɫɤɨɪɨɫɬɢ X1 ɜ ɧɚɢɛɨɥɟɟ ɛɥɢɡɤɨɣ ɤ ɩɥɚɧɟɬɟ ɬɨɱɤɟ ɬɪɚɟɤɬɨɪɢɢ. ɇɚɣɬɢ ɦɚɫɫɭ ɩɥɚɧɟɬɵ M, ɚ ɬɚɤɠɟ ɪɚɞɢɭɫɵ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ ɫɩɭɬɧɢɤɚ ɢ ɜ ɧɚɢɛɨɥɟɟ ɛɥɢɡɤɨɣ R1 ɢ ɧɚɢɛɨɥɟɟ ɭɞɚɥɟɧɧɨɣ R2 ɨɬ ɩɥɚɧɟɬɵ ɬɨɱɤɚɯ ɟɝɨ ɬɪɚɟɤɬɨɪɢɢ.
Ɋɟɲɟɧɢɟ
I. Ʌɚɛɨɪɚɬɨɪɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ ɩɥɚɧɟɬɨɣ, ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɢɧɟɪɰɢɚɥɶɧɨɣ. ɉɪɢ ɞɚɥɶɧɟɣɲɟɦ ɪɚɫɫɦɨɬɪɟɧɢɢ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɫɩɭɬɧɢɤ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ, ɚ ɩɥɚɧɟɬɭ ɫɮɟɪɢɱɟɫɤɢ ɫɢɦɦɟɬɪɢɱɧɵɦ ɬɟɥɨɦ. Ⱦɜɢɠɟɧɢɟ ɫɩɭɬɧɢɤɚ ɩɨ ɷɥɥɢɩɬɢɱɟɫɤɨɣ ɬɪɚɟɤɬɨɪɢɢ ɩɪɨɢɫɯɨɞɢɬ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɨɞɧɨɣ ɫɢɥɵ – ɫɢɥɵ ɝɪɚɜɢɬɚɰɢɨɧɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ (ɫɦ. ɩ. 2.1.2.Ⱥ ɜ Ƚɥɚɜɟ 2) ɫɩɭɬɧɢɤɚ ɢ ɩɥɚɧɟɬɵ. ɉɨɫɤɨɥɶɤɭ ɷɬɚ ɫɢɥɚ ɹɜɥɹɟɬɫɹ ɰɟɧɬɪɚɥɶɧɨɣ (ɩ. 3.1.2.Ⱥ ɜ Ƚɥɚɜɟ 3), ɬɨ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɩɭɬɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɩɥɚɧɟɬɵ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɥɨɫɤɨɫɬɢ ɬɪɚɟɤɬɨɪɢɢ ɫɩɭɬɧɢɤɚ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ (ɫɦ. ɩ. 7.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ) ɧɟ ɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ.
Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɫɢɫɬɟɦɚ ɬɟɥ «ɫɩɭɬɧɢɤ + ɩɥɚɧɟɬɚ» ɹɜɥɹɟɬɫɹ ɢɡɨɥɢɪɨɜɚɧɧɨɣ, ɚ ɰɟɧɬɪɚɥɶɧɵɟ ɫɢɥɵ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɬɟɥ ɫɢɫɬɟɦɵ
Ƚɥɚɜɚ 7. Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
257 |
ɩɨɬɟɧɰɢɚɥɶɧɵ (ɩ. 3.1.2.Ⱥ ɜ Ƚɥɚɜɟ 3). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɡɚɤɨɧɨɦ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ (ɩ. 7.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ).
II. Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɩɭɬɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɩɥɚɧɟɬɵ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɥɨɫɤɨɫɬɢ ɬɪɚɟɤɬɨɪɢɢ ɫɩɭɬɧɢɤɚ (ɫɦ. ɪɢɫ. 7.15), ɞɥɹ ɦɨɦɟɧɬɨɜ ɜɪɟɦɟɧɢ ɧɚɯɨɠɞɟɧɢɹ ɫɩɭɬɧɢɤɚ ɧɚ ɦɢɧɢɦɚɥɶɧɨɦ r1 ɢ ɦɚɤɫɢɦɚɥɶɧɨɦ r2 ɪɚɫɫɬɨɹɧɢɢ ɨɬ ɩɥɚɧɟɬɵ:
r2mX2 r1mX1 0 , |
(7.89) |
ɝɞɟ X2 ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɫɩɭɬɧɢɤɚ ɩɪɢ ɦɚɤɫɢɦɚɥɶɧɨɦ ɭɞɚɥɟɧɢɢ ɨɬ
ɩɥɚɧɟɬɵ. ɉɪɢ ɡɚɩɢɫɢ (7.89) ɛɵɥɨ ɭɱɬɟɧɨ, ɱɬɨ ɫɤɨɪɨɫɬɶ ȣ ɢ ɪɚɞɢɭɫɜɟɤɬɨɪ r ɫɩɭɬɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɩɥɚɧɟɬɵ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɜɡɚɢɦɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɨɫɬɚɥɶɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɭɝɨɥ ɦɟɠɞɭ ɫɤɨɪɨɫɬɶɸ ȣ ɢ ɪɚɞɢɭɫɜɟɤɬɨɪɨɦ r ɧɟ ɪɚɜɟɧ S / 2 (ɪɢɫ. 7.15).
Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɬɟɥ «ɫɩɭɬɧɢɤ + ɩɥɚɧɟɬɚ» ɞɥɹ ɦɨɦɟɧɬɨɜ ɜɪɟɦɟɧɢ ɧɚɯɨɠɞɟɧɢɹ ɫɩɭɬɧɢɤɚ ɧɚ ɦɢɧɢɦɚɥɶɧɨɦ r1 ɢ ɦɚɤɫɢɦɚɥɶɧɨɦ r2 ɪɚɫɫɬɨɹɧɢɢ ɨɬ ɩɥɚɧɟɬɵ:
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mX |
2 |
E p |
mX2 |
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2 |
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0 . |
(7.90) |
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Ɂɞɟɫɶ E p |
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ɢ E p |
ɩɨɬɟɧɰɢɚɥɶɧɵɟ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ |
ɬɟɥ «ɫɩɭɬ- |
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ɧɢɤ + ɩɥɚɧɟɬɚ» ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ.
Ɉɩɪɟɞɟɥɢɦ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɫɢɫɬɟɦɵ ɩɪɢ ɩɪɨɢɡɜɨɥɶɧɨɦ ɪɚɫɫɬɨɹɧɢɢ r ɦɟɠɞɭ ɫɩɭɬɧɢɤɨɦ ɢ ɰɟɧɬɪɨɦ ɩɥɚɧɟɬɵ.
ȼɵɛɟɪɟɦ ɧɨɥɶ ɨɬɫɱɟɬɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɩɨɥɨɠɟɧɢɸ ɫɩɭɬɧɢɤɚ ɧɚ ɮɢɡɢɱɟɫɤɢ ɛɟɫɤɨɧɟɱɧɨ ɛɨɥɶɲɨɦ ɪɚɫɫɬɨɹɧɢɢ ɨɬ ɩɥɚɧɟɬɵ. Ɍɨɝɞɚ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ (ɫɦ. (3.32) ɜ Ƚɥɚɜɟ 3) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ:
f |
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mM |
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mM |
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E p ³ |
G |
d r |
G |
. |
(7.91) |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɬɟɥ «ɫɩɭɬɧɢɤ + ɩɥɚɧɟɬɚ» ɜ ɦɨɦɟɧɬɵ ɧɚɯɨɠɞɟɧɢɹ ɫɩɭɬɧɢɤɚ ɧɚ ɦɢɧɢɦɚɥɶɧɨɦ r1 ɢ ɦɚɤɫɢɦɚɥɶɧɨɦ r2 ɪɚɫɫɬɨɹɧɢɢ ɨɬ ɩɥɚɧɟɬɵ ɪɚɜɧɚ:
E P |
G |
mM |
, |
(7.92) |
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1 |
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r1 |
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258 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
E2P G |
mM |
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(7.93) |
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r2 |
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Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɪɚɞɢɭɫɚ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ ɫɩɭɬɧɢɤɚ ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɟɝɨ ɞɜɢɠɟɧɢɹ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɧɨɪɦɚɥɶɧɭɸ ɨɫɶ, ɧɚɩɪɚɜɥɟɧɧɭɸ ɤ ɰɟɧɬɪɭ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɫɤɨɪɨɫɬɢ ɫɩɭɬɧɢɤɚ, ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ:
m |
X2 |
G |
mM |
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1 |
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(7.94) |
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r 2 |
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1 |
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m |
X2 |
G |
mM |
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2 |
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(7.95) |
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r 2 |
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III. Ɋɟɲɚɹ |
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ɡɚɩɢɫɚɧɧɭɸ ɜ ɩ. II ɫɢɫɬɟɦɭ |
ɭɪɚɜɧɟɧɢɣ |
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(7.89) (7.95) ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɨɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɦɚɫɫɵ ɩɥɚɧɟɬɵ |
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2 |
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r1 |
· |
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X1 |
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¨ |
¸ |
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M |
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2G r2 |
r1 |
¨ r |
¸ |
(7.96) |
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© 2 |
¹ |
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ɢ ɪɚɞɢɭɫɨɜ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ
R1 R2 |
2 |
r1r2 |
. |
(7.97) |
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r1 |
r2 |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɨɛɟ ɢɫɤɨɦɵɟ ɮɢɡɢɱɟɫɤɢɟ ɜɟɥɢɱɢɧɵ ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɦɚɫɫɵ ɫɩɭɬɧɢɤɚ, ɱɬɨ ɞɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɧɚɯɨɠɞɟɧɢɹ ɦɚɫɫɵ ɩɥɚɧɟɬɵ, ɢɫɯɨɞɹ ɢɡ ɢɡɦɟɪɟɧɢɣ ɬɨɥɶɤɨ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɫɩɭɬɧɢɤɚ.
7.3.2. Ƚɢɪɨɫɤɨɩɵ. Ƚɢɪɨɫɤɨɩɢɱɟɫɤɢɟ ɫɢɥɵ
Ɂɚɞɚɱɚ 7.7
ɗɥɟɤɬɪɨɞɜɢɝɚɬɟɥɶ ɡɚɤɪɟɩɥɟɧ ɧɚ ɩɨɞɫɬɚɜɤɟ ɬɚɤ, ɱɬɨ ɟɝɨ ɨɫɶ ɢ ɨɛɳɢɣ ɰɟɧɬɪ ɦɚɫɫ ɧɚɯɨɞɹɬɫɹ ɩɨɫɟɪɟɞɢɧɟ ɦɟɠɞɭ ɨɩɨɪɚɦɢ ɩɨɞɫɬɚɜɤɢ, ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɤɨɬɨɪɵɦɢ ɪɚɜɧɨ l. Ⱦɜɢɝɚɬɟɥɶ ɫ ɩɨɞɫɬɚɜɤɨɣ ɩɨɫɬɚɜɢɥɢ ɧɚ ɝɥɚɞɤɭɸ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ. ɇɚɣɬɢ ɫɢɥɵ ɞɚɜɥɟɧɢɹ ɨɩɨɪ ɩɨɞɫɬɚɜɤɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɶ, ɟɫɥɢ ɩɨɫɥɟ ɜɤɥɸɱɟɧɢɹ ɪɨɬɨɪ ɞɜɢɝɚɬɟɥɹ ɪɚɫɤɪɭɱɢɜɚɟɬɫɹ ɫ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ E ɜɨɤɪɭɝ
ɟɝɨ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɨɫɢ, ɚ ɟɝɨ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɨɫɢ ɪɚɜɟɧ J. Ɇɚɫɫɚ ɞɜɢɝɚɬɟɥɹ ɫ ɩɨɞɫɬɚɜɤɨɣ ɪɚɜɧɚ m.
Ƚɥɚɜɚ 7. Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
259 |
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Ɋɟɲɟɧɢɟ |
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I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ |
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ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ. Ɉɫɶ Z ɞɟɤɚɪɬɨɜɨɣ ɫɢɫ- |
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ɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɧɚɩɪɚɜɢɦ ɜɟɪɬɢɤɚɥɶɧɨ |
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ɜɜɟɪɯ. Ɉɫɶ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɡɚ- |
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ɩɢɫɵɜɚɟɦ ɦɨɦɟɧɬɵ ɫɢɥ ɢ ɢɦɩɭɥɶɫɚ ɬɟɥ, |
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ɜɵɛɟɪɟɦ ɫɨɜɩɚɞɚɸɳɟɣ ɫ ɝɟɨɦɟɬɪɢɱɟ- |
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ɫɤɨɣ ɨɫɶɸ ɪɨɬɨɪɚ ɢ ɧɚɩɪɚɜɥɟɧɧɨɣ ɡɚ |
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ɩɥɨɫɤɨɫɬɶ ɱɟɪɬɟɠɚ (ɫɦ. ɪɢɫ. 7.16). |
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ɉɪɢ ɜɤɥɸɱɟɧɢɢ ɷɥɟɤɬɪɨɞɜɢɝɚɬɟɥɹ |
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ɦɨɦɟɧɬ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɪɨɬɨɪ ɫɨ |
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ɫɬɨɪɨɧɵ ɫɬɚɬɨɪɚ, ɡɚɤɪɟɩɥɟɧɧɨɝɨ ɧɚ ɩɨɞɫɬɚɜɤɟ, ɢɡɦɟɧɹɟɬ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ
ɪɨɬɨɪɚ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɪɟɬɶɢɦ ɡɚɤɨɧɨɦ ɇɶɸɬɨɧɚ ɬɚɤɨɣ ɠɟ ɩɨ ɜɟɥɢɱɢɧɟ ɦɨɦɟɧɬ ɫɢɥ ɞɟɣɫɬɜɭɟɬ ɫɨ ɫɬɨɪɨɧɵ ɪɨɬɨɪɚ ɧɚ ɫɬɚɬɨɪ ɫ ɩɨɞɫɬɚɜɤɨɣ. ɉɪɢ ɷɬɨɦ ɧɚ ɩɨɞɫɬɚɜɤɭ ɞɟɣɫɬɜɭɸɬ ɬɚɤɠɟ ɦɨɦɟɧɬɵ ɫɢɥ ɪɟɚɤɰɢɢ ɫɨ ɫɬɨɪɨɧɵ ɩɨɜɟɪɯɧɨɫɬɢ, ɧɚ ɤɨɬɨɪɨɣ ɨɧɚ ɧɚɯɨɞɢɬɫɹ. ɉɨɞɫɬɚɜɤɚ ɫ ɡɚɤɪɟɩɥɟɧɧɵɦ ɧɚ ɧɟɣ ɫɬɚɬɨɪɨɦ ɨɫɬɚɟɬɫɹ ɜ ɩɨɤɨɟ, ɩɨɷɬɨɦɭ ɫɭɦɦɚɪɧɵɣ ɦɨɦɟɧɬ ɜɫɟɯ ɜɧɟɲɧɢɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɧɢɯ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɢɡɜɨɥɶɧɨ ɜɵɛɪɚɧɧɨɣ ɨɫɢ, ɪɚɜɟɧ ɧɭɥɸ.
II. Ⱦɥɹ ɪɨɬɨɪɚ ɞɜɢɝɚɬɟɥɹ ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɦɨɦɟɧɬɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɨɣ ɨɫɢ:
dL |
M , |
(7.98) |
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ɝɞɟ L ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɪɨɬɨɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɟɝɨ ɜɪɚɳɟɧɢɹ. ɉɨɫɤɨɥɶɤɭ ɦɨɦɟɧɬ ɫɢɥɵ ɬɹɠɟɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɪɨɬɨɪɚ
ɪɚɜɟɧ ɧɭɥɸ, ɫɭɦɦɚɪɧɵɣ ɦɨɦɟɧɬ ɜɧɟɲɧɢɯ ɫɢɥ M ɪɚɜɟɧ ɦɨɦɟɧɬɭ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɪɨɬɨɪ ɫɨ ɫɬɨɪɨɧɵ ɫɬɚɬɨɪɚ ɫ ɩɨɞɫɬɚɜɤɨɣ.
ɍɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɞɥɹ ɫɬɚɬɨɪɚ ɢ ɩɨɞɫɬɚɜɤɢ ɡɚɩɢɲɟɦ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
M F |
l |
F |
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0 , |
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F1 F2 mg |
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ɝɞɟ F1 ɢ F2 – ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɩɨɞɫɬɚɜɤɭ ɫɨ ɫɬɨɪɨɧɵ ɩɨɜɟɪɯɧɨɫɬɢ (ɫɦ. ɪɢɫ. 7.16).
Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɪɨɬɨɪɚ, ɜɪɚɳɚɸɳɟɝɨɫɹ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Z ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɜɨɟɣ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɨɫɢ, ɪɚɜɟɧ
L JZ . |
(7.101) |
260 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɭɝɥɨɜɨɝɨ ɭɫɤɨɪɟɧɢɹ ɡɚɩɢɲɟɦ:
E |
dZ |
. |
(7.102) |
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dt |
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ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɟɧɚ ɩɨɥɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɩɨɞɫɬɚɜɤɭ ɫɨ ɫɬɨɪɨɧɵ ɩɨɜɟɪɯɧɨɫɬɢ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɪɟɬɶɢɦ ɡɚɤɨɧɨɦ ɇɶɸɬɨɧɚ ɷɬɢ ɫɢɥɵ ɪɚɜɧɵ ɩɨ ɜɟɥɢɱɢɧɟ ɢɫɤɨɦɵɦ ɫɢɥɚɦ, ɞɟɣɫɬɜɭɸɳɢɦ ɫɨ ɫɬɨɪɨɧɵ ɩɨɞɫɬɚɜɤɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɶ.
III. Ɋɟɲɚɹ ɩɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (7.98) – (7.102) ɨɬ-
ɧɨɫɢɬɟɥɶɧɨ F1 ɢ F2, ɩɨɥɭɱɚɟɦ: |
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F |
mg |
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JE |
, |
(7.103) |
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1 |
2 |
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l |
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F |
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mg |
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JE |
. |
(7.104) |
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2 |
2 |
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l |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɭɝɥɨɜɨɦ ɭɫɤɨɪɟɧɢɢ ɪɨɬɨɪɚ E t mgl2J ɩɪɚɜɚɹ ɨɩɨɪɚ ɩɨɞɫɬɚɜɤɢ ɞɜɢɝɚɬɟɥɹ ɨɬɪɵɜɚɟɬɫɹ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ.
Ɂɚɞɚɱɚ 7.8
Ɇɚɫɫɢɜɧɵɣ ɰɢɥɢɧɞɪɢɱɟɫɤɢɣ ɤɚɬɨɤ (ɛɟɝɭɧ) ɦɚɫɫɨɣ m, ɤɨɬɨɪɵɣ ɦɨɠɟɬ ɜɪɚɳɚɬɶɫɹ ɜɨɤɪɭɝ ɫɜɨɟɣ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɨɫɢ, ɩɪɢɜɟɞɟɧ ɜɨ ɜɪɚɳɟɧɢɟ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ ɫ ɩɨɫɬɨɹɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Z ɢ ɤɚɬɢɬɫɹ ɛɟɡ ɫɤɨɥɶɠɟɧɢɹ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɩɨɪɧɨɣ ɩɥɢɬɟ
(ɫɦ. ɪɢɫ. 7.17).
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m, J0 |
Z |
r |
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Ɋɢɫ. 7.17 |
Ɋɚɞɢɭɫ ɤɚɬɤɚ r. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɤɚɬɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɨɫɢ ɪɚɜɟɧ J0. ȼɵɱɢɫɥɢɬɶ ɩɨɥɧɭɸ ɫɢɥɭ ɞɚɜɥɟɧɢɹ ɤɚɬɤɚ ɧɚ ɨɩɨɪɧɭɸ ɩɥɢɬɭ.