Механика.Методика решения задач
.pdfȽɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
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m R2 . |
(6.138) |
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ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ Ⱥɦɨɧɬɨɧɚ–Ʉɭɥɨɧɚ (ɫɦ. ɩ. 2.1.2 ɜ Ƚɥɚɜɟ 2) ɞɥɹ ɫɢɥ ɬɪɟɧɢɹ ɩɨɤɨɹ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɰɢɥɢɧɞɪɵ, ɫɩɪɚɜɟɞɥɢɜɵ ɧɟɪɚɜɟɧɫɬɜɚ:
Fɬɪ1 |
d PN1 . |
(6.139) |
Fɬɪ2 |
d PN2 . |
(6.140) |
III. ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɫɨɨɬɧɨɲɟɧɢɹɦɢ (6.130) (6.138), ɜɵɪɚɡɢɦ ɫɢɥɵ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɧɚɤɥɨɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ ɢ ɫɢɥ ɬɪɟɧɢɹ ɩɨɤɨɹ ɫɨ ɫɬɨɪɨɧɵ ɷɬɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɧɚ ɨɛɚ ɰɢɥɢɧɞɪɚ ɱɟɪɟɡ ɜɟɥɢɱɢɧɵ, ɡɚɞɚɧɧɵɟ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ, ɢ ɢɫɤɨɦɵɣ ɭɝɨɥ ɩɪɢ
ɨɫɧɨɜɚɧɢɢ ɤɥɢɧɚ D : |
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N1 |
m1g cosD , |
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(6.141) |
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N2 |
m2 g cosD , |
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(6.142) |
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Fɬɪ1 |
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m1 |
m2 |
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m1g sin D , |
(6.143) |
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3m1 |
4m2 |
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F |
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2 m1 m2 |
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m g sinD . |
(6.144) |
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ɬɪ2 |
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3m1 4m2 |
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ɉɨɞɫɬɚɜɥɹɹ ɜɵɪɚɠɟɧɢɹ (6.141) (6.144) ɜ ɧɟɪɚɜɟɧɫɬɜɚ (6.139) ɢ (6.140), ɩɨɥɭɱɢɦ ɭɫɥɨɜɢɹ, ɩɪɢ ɤɨɬɨɪɵɯ ɤɚɱɟɧɢɟ ɰɢɥɢɧɞɪɨɜ ɩɪɨ-
ɢɫɯɨɞɢɬ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ: |
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tgD d P |
3m1 |
4m2 |
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(6.145) |
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tgD d P |
3m1 |
4m2 |
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(6.146) |
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2 m m |
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ɉɨɫɤɨɥɶɤɭ ɩɪɚɜɚɹ ɱɚɫɬɶ ɧɟɪɚɜɟɧɫɬɜɚ (6.146) ɦɟɧɶɲɟ ɩɪɚɜɨɣ ɱɚɫɬɢ ɧɟɪɚɜɟɧɫɬɜɚ (6.145) ɩɪɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɪɟɧɢɹ P ɢ ɦɚɫɫ ɰɢɥɢɧɞɪɨɜ m1 ɢ m2, ɬɨ ɢɫɤɨɦɚɹ ɨɛɥɚɫɬɶ ɡɧɚɱɟɧɢɣ ɭɝɥɚ ɩɪɢ ɨɫɧɨɜɚɧɢɢ ɤɥɢɧɚ D, ɩɪɢ ɤɨɬɨɪɵɯ ɰɢɥɢɧɞɪɵ ɛɭɞɭɬ ɫɤɚɬɵɜɚɬɶɫɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨɦ (6.146).
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (6.146) ɨɛɥɚɫɬɶ ɜɨɡɦɨɠɧɵɯ ɡɧɚɱɟɧɢɣ ɭɝɥɚ D ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɨɬɧɨɲɟɧɢɟɦ ɦɚɫɫ ɰɢɥɢɧɞɪɨɜ ɩɪɢ ɡɚɞɚɧ-
ɧɨɦ ɡɧɚɱɟɧɢɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɪɟɧɢɹ P: |
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3m1 / m2 4 |
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D d arctg¨P |
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¸ . |
(6.147) |
2 m / m |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɇɚ ɪɢɫ. 6.18 ɢɡɨɛɪɚɠɟɧ ɝɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɩɪɟɞɟɥɶɧɨɣ ɜɟɥɢɱɢɧɵ tgDɩɪ / P ɨɬ ɨɬɧɨɲɟɧɢɹ ɦɚɫɫɵ ɫɩɥɨɲɧɨɝɨ ɰɢɥɢɧɞɪɚ ɤ ɦɚɫ-
ɫɟ ɬɨɧɤɨɫɬɟɧɧɨɝɨ m1 / m2 .
tgDɩɪ / P |
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m1 / m2
Ɋɢɫ. 6.18
Ʉɚɤ ɜɢɞɢɦ, ɨɛɥɚɫɬɶ ɡɧɚɱɟɧɢɣ ɭɝɥɚ D, ɩɪɢ ɤɨɬɨɪɵɯ ɰɢɥɢɧɞɪɵ
ɛɭɞɭɬ ɫɤɚɬɵɜɚɬɶɫɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ, ɨɝɪɚɧɢɱɟɧɚ ɫɜɟɪɯɭ ɩɪɟ- |
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ɞɟɥɶɧɵɦ ɡɧɚɱɟɧɢɟɦ Dɩɪ , ɤɨɬɨɪɨɟ ɪɚɜɧɨ arctg 2P |
ɩɪɢ m1 m2 ɢ |
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ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɫɬɪɟɦɢɬɫɹ ɤ ɡɧɚɱɟɧɢɸ arctg¨ |
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¸ ɩɪɢ ɧɟɨɝɪɚɧɢ- |
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ɱɟɧɧɨɦ ɭɜɟɥɢɱɟɧɢɢ ɨɬɧɨɲɟɧɢɹ ɦɚɫɫ ɰɢɥɢɧɞɪɨɜ m1 / m2 (ɪɢɫ. 6.18).
Ɂɚɞɚɱɚ 6.10
ɐɢɥɢɧɞɪɢɱɟɫɤɚɹ ɲɚɣɛɚ ɪɚɞɢɭɫɨɦ r = 3 ɫɦ ɤɚɫɚɟɬɫɹ ɛɨɪɬɚ ɝɥɚɞɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɳɚɞɤɢ, ɢɦɟɸɳɟɣ ɮɨɪɦɭ ɤɪɭɝɚ ɪɚɞɢɭɫɨɦ R = 10 ɦ. ɒɚɣɛɟ ɩɪɢɞɚɥɢ ɫɤɨɪɨɫɬɶ X0 30 ɦ/ɫ, ɧɚɩɪɚɜɥɟɧɧɭɸ
ɜɞɨɥɶ ɛɨɪɬɚ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɦɟɠɞɭ ɛɨɪɬɨɦ ɢ ɲɚɣɛɨɣ ɪɚɜɟɧ P 0,1 . Ɉɩɪɟɞɟɥɢɬɶ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɲɚɣɛɵ Xɤ ɩɨɫɥɟ ɬɨɝɨ, ɤɚɤ
ɩɪɟɤɪɚɬɢɬɫɹ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɟ ɦɟɠɞɭ ɛɨɪɬɨɦ ɢ ɲɚɣɛɨɣ, ɚ ɬɚɤɠɟ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ tɤ, ɱɟɪɟɡ ɤɨɬɨɪɵɣ ɷɬɨ ɩɪɨɢɡɨɣɞɟɬ.
Ɋɟɲɟɧɢɟ
I. Ⱦɜɢɠɟɧɢɟ ɲɚɣɛɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɨɣ ɫ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɳɚɞɤɨɣ. ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɣɛɵ ɛɭɞɟɦ ɡɚɩɢɫɵɜɚɬɶ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɧɨɪɦɚɥɶɧɭɸ n ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɭɸ
Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
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IJ ɨɫɢ (ɫɦ. ɩ. 1.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ ɜ Ƚɥɚɜɟ 1), ɫɜɹɡɚɧɧɵɟ ɫ
ɰɟɧɬɪɨɦ ɞɜɢɠɭɳɟɣɫɹ ɲɚɣɛɵ (ɫɦ. ɪɢɫ. 6.19). |
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ɉɪɢ ɩɥɨɫɤɨɦ ɞɜɢɠɟɧɢɢ ɲɚɣɛɵ |
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ɜɞɨɥɶ ɛɨɪɬɚ ɩɥɨɳɚɞɤɢ ɧɚ ɧɟɟ ɞɟɣɫɬɜɭɸɬ |
Xɤ |
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ɫɢɥɚ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ N ɢ ɫɢɥɚ ɬɪɟ- |
IJ |
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ɧɢɹ Fɬɪ ɫɨ ɫɬɨɪɨɧɵ ɛɨɪɬɚ. ɉɪɢ ɷɬɨɦ ɦɨ- |
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ɦɟɧɬ ɫɢɥɵ ɬɪɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨ- |
n |
X(t) |
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ɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ ɲɚɣɛɵ, ɜɵɡɵ- |
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ɜɚɟɬ ɟɟ ɜɪɚɳɟɧɢɟ ɜɨɤɪɭɝ ɭɤɚɡɚɧɧɨɣ ɨɫɢ. |
N |
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ɋɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɣɛɵ ɛɭɞɟɬ |
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ɭɦɟɧɶɲɚɬɶɫɹ, ɚ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɟɟ ɜɪɚ- |
X0 |
Fɬɪ |
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ɳɟɧɢɹ ɭɜɟɥɢɱɢɜɚɬɶɫɹ, ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ |
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ɧɟ ɩɪɟɤɪɚɬɢɬɫɹ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɟ ɲɚɣɛɵ |
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ɨ ɛɨɪɬ ɩɥɨɳɚɞɤɢ. ɇɚɱɢɧɚɹ ɫ ɷɬɨɝɨ ɦɨɦɟɧɬɚ ɫɢɥɚ ɬɪɟɧɢɹ ɲɚɣɛɵ ɨ ɛɨɪɬ ɪɚɜɧɚ ɧɭɥɸ, ɚ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɣɛɵ ɧɟ ɢɡɦɟɧɹɟɬɫɹ.
II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɣɛɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɜɵɛɪɚɧɧɵɟ
ɧɨɪɦɚɥɶɧɭɸ n ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɭɸ IJ |
ɨɫɢ: |
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man |
N , |
(6.148) |
maW |
Fɬɪ . |
(6.149) |
ɇɨɪɦɚɥɶɧɚɹ ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɚɹ ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɣɛɵ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɨɨɬɧɨɲɟɧɢɹɦɢ (ɫɦ. ɩ. 1.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ ɜ Ƚɥɚɜɟ 1):
an |
X |
2 |
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R r |
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aW |
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dX |
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dt |
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ɝɞɟ X ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɣɛɵ ɢ (R r)
(6.150)
(6.151)
ɪɚɞɢɭɫ
ɤɪɢɜɢɡɧɵ ɟɝɨ ɬɪɚɟɤɬɨɪɢɢ.
ɍɪɚɜɧɟɧɢɟ ɦɨɦɟɧɬɨɜ (6.47) ɞɥɹ ɜɪɚɳɚɸɳɟɣɫɹ ɲɚɣɛɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɟ ɰɟɧɬɪ ɦɚɫɫ, ɢɦɟɟɬ ɜɢɞ:
J |
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dZ |
F r . |
(6.152) |
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dt |
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ɬɪ |
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Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɲɚɣɛɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɤɚɡɚɧɧɨɣ ɨɫɢ (6.44) ɪɚɜɟɧ:
224 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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J0 |
mr 2 |
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. |
(6.153) |
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ɇɚ ɧɚɱɚɥɶɧɨɦ ɷɬɚɩɟ ( t tɤ ) ɲɚɣɛɚ ɞɜɢɠɟɬɫɹ ɫ ɩɪɨɫɤɚɥɶɡɵɜɚ-
ɧɢɟɦ ɢ ɧɚ ɧɟɟ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ Ⱥɦɨɧɬɨɧɚ Ʉɭɥɨɧɚ (ɫɦ. ɩ. 2.1.2.ȼ ɜ Ƚɥɚɜɟ 2) ɞɟɣɫɬɜɭɟɬ ɫɢɥɚ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ, ɪɚɜɧɚɹ:
Fɬɪ PN . |
(6.154) |
Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ ɦɟɠɞɭ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ ɜɪɚɳɟɧɢɹ ɢ ɫɤɨɪɨɫɬɶɸ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɣɛɵ ɩɨɫɥɟ ɩɪɟ-
ɤɪɚɳɟɧɢɹ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ (ɩɪɢ t t tɤ ): |
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X rZ . |
(6.155) |
III. ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɫɨɨɬɧɨɲɟɧɢɹɦɢ |
(6.148) (6.151) ɢ |
(6.154), ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɦɨɞɭɥɹ ɫɤɨɪɨ-
ɫɬɢ ɰɟɧɬɪɚ |
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ɦɚɫɫ ɲɚɣɛɵ ɧɚ |
ɧɚɱɚɥɶɧɨɦ ɷɬɚɩɟ |
ɞɜɢɠɟɧɢɹ ɲɚɣɛɵ |
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( t tɤ ): |
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Ɋɟɲɚɟɦ ɭɪɚɜɧɟɧɢɟ (6.156) ɫ ɩɨɦɨɳɶɸ ɦɟɬɨɞɚ ɪɚɡɞɟɥɟɧɢɹ ɩɟ- |
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ɪɟɦɟɧɧɵɯ: |
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³dt , |
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(6.158) |
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R r X0 Pt |
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ɋɜɹɡɶ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ ɲɚɣɛɵ ɫ ɦɨɞɭɥɟɦ ɫɤɨɪɨ- |
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ɫɬɢ ɟɟ ɰɟɧɬɪɚ ɦɚɫɫ ɩɨɥɭɱɚɟɦ ɢɡ (6.148) (6.154): |
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ɉɨɞɫɬɚɜɥɹɹ (6.158) ɜ (6.159), ɩɨɥɭɱɚɟɦ: |
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2PX02 R r t |
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Z(t) |
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³0 |
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r |
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ȼ ɪɟɡɭɥɶɬɚɬɟ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɲɚɣɛɵ ɧɚ ɧɚɱɚɥɶɧɨɦ ɷɬɚɩɟ ɟɟ |
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ɞɜɢɠɟɧɢɹ ɪɚɜɧɚ: |
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Z(t) |
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(6.161) |
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r R r X0 Pt |
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Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
225 |
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (6.158) ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɣɛɵ ɭɦɟɧɶɲɚɟɬɫɹ ɨɬ ɡɧɚɱɟɧɢɹ X0 (ɩɪɢ t = 0), ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɭɝɥɨɜɚɹ
ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɲɚɣɛɵ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɩɨ ɡɚɤɨɧɭ (6.161). ȼ ɦɨɦɟɧɬ ɩɪɟɤɪɚɳɟɧɢɹ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ (ɱɟɪɟɡ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ tɤ ɩɨɫɥɟ ɫɨɨɛɳɟɧɢɹ ɲɚɣɛɟ ɫɤɨɪɨɫɬɢ X0 ) ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ
ɲɚɣɛɵ ɢ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɟɟ ɰɟɧɬɪɚ ɦɚɫɫ ɫɜɹɡɚɧɵ ɭɪɚɜɧɟɧɢɟɦ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ (6.157).
ɇɚ ɪɢɫ. 6.20 ɩɪɟɞɫɬɚɜɥɟɧɵ ɡɚɜɢɫɢɦɨɫɬɢ ɦɨɞɭɥɹ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɣɛɵ X ɢ ɩɪɨɢɡɜɟɞɟɧɢɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ ɲɚɣɛɵ ɧɚ ɟɟ ɪɚɞɢɭɫ Z r .
X,Zr, ɦ/c
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tɤ 2 |
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Ɋɢɫ. 6.20 |
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Ƚɪɚɮɢɤɢ ɩɪɟɞɫɬɚɜɥɟɧɧɵɯ ɡɚɜɢɫɢɦɨɫɬɟɣ ɩɟɪɟɫɟɤɚɸɬɫɹ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ tɤ (ɫɦ. ɪɢɫ. 6.20). ɉɨɞɫɬɚɜɥɹɹ (6.158) (6.161) ɜ ɭɪɚɜɧɟɧɢɟ (6.157) ɩɨɥɭɱɚɟɦ ɡɧɚɱɟɧɢɟ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ tɤ:
tɤ |
R r |
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ɉɨɫɥɟ ɩɪɟɤɪɚɳɟɧɢɹ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɩɪɢ ɞɚɥɶɧɟɣɲɟɦ ɞɜɢɠɟɧɢɢ ɲɚɣɛɵ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɟɟ ɰɟɧɬɪɚ ɦɚɫɫ ɧɟ ɦɟɧɹɟɬɫɹ ɢ ɪɚɜɟɧ:
Xɤ {X(t tɤ ) |
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X0 . |
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Ʉɚɤ ɜɢɞɢɦ, ɢɫɤɨɦɵɣ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɲɚɣɛɵ ɨɩɪɟɞɟɥɹɟɬɫɹ |
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ɬɨɥɶɤɨ ɟɟ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ X0 |
ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɞɪɭɝɢɯ ɜɟɥɢɱɢɧ, |
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ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ. |
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ɉɨɞɫɬɚɜɥɹɹ ɡɚɞɚɧɧɵɟ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ (6.162) ɢ (6.163), |
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ɧɚɯɨɞɢɦ ɢɫɤɨɦɵɟ ɜɟɥɢɱɢɧɵ: |
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tɤ #1,66 ɫ, Xɤ |
20 ɦ/ɫ. |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
6.4. Ɂɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ
Ɂɚɞɚɱɚ 1
Ⱦɜɟ ɩɚɪɚɥɥɟɥɶɧɵɟ ɪɟɣɤɢ ɞɜɢɠɭɬɫɹ ɜ ɨɞɧɭ ɫɬɨɪɨɧɭ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɫɤɨɪɨɫɬɹɦɢ X1 ɢ X2 ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ XY. Ɇɟɠɞɭ ɪɟɣɤɚɦɢ ɡɚɠɚɬɚ ɤɚɬɭɲɤɚ ɫ ɪɚɞɢɭɫɚɦɢ R ɢ r (ɫɦ. ɪɢɫ.), ɤɨɬɨɪɚɹ ɞɜɢɠɟɬɫɹ ɜɞɨɥɶ ɪɟɟɤ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ.
Y
X1
Rr
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ɇɚɣɬɢ ɤɨɨɪɞɢɧɚɬɭ yɦ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ Z ɤɚɬɭɲɤɢ ɢ ɫɤɨɪɨɫɬɶ X ɟɟ ɨɫɢ.
Ɉɬɜɟɬ: yɦ |
X2 (R r) , Z |
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Ɂɚɞɚɱɚ 2
Ʉɪɭɝɥɵɣ ɤɨɧɭɫ ɜɵɫɨɬɨɣ h ɢ ɪɚɞɢɭɫɨɦ ɨɫɧɨɜɚɧɢɹ r ɤɚɬɢɬɫɹ ɛɟɡ ɫɤɨɥɶɠɟɧɢɹ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ (ɫɦ. ɪɢɫ.). ȼɟɪɲɢɧɚ ɤɨɧɭɫɚ ɲɚɪɧɢɪɧɨ ɡɚɤɪɟɩɥɟɧɚ ɜ ɬɨɱɤɟ O ɧɚ ɭɪɨɜɧɟ ɰɟɧɬɪɚ ɨɫɧɨɜɚɧɢɹ ɤɨɧɭɫɚ, ɤɨɬɨɪɵɣ ɞɜɢɠɟɬɫɹ ɫ ɩɨɫɬɨɹɧɧɨɣ ɩɨ ɦɨɞɭɥɸ ɫɤɨɪɨɫɬɶɸ X . ɇɚɣɬɢ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɤɨɧɭɫɚ Z ɢ ɟɝɨ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ E.
O
h
X
r
Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
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Ɉɬɜɟɬ: Z |
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Ɂɚɞɚɱɚ 3
Ɉɞɧɨɪɨɞɧɵɣ ɞɢɫɤ ɪɚɞɢɭɫɨɦ R ɪɚɫɤɪɭɬɢɥɢ ɜɨɤɪɭɝ ɟɝɨ ɨɫɢ ɞɨ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ Z ɢ ɩɨɥɨɠɢɥɢ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɦɟɠɞɭ ɩɨɜɟɪɯɧɨɫɬɶɸ ɢ ɞɢɫɤɨɦ ɪɚɜɟɧ P . ɑɟ-
ɪɟɡ ɤɚɤɨɟ ɜɪɟɦɹ W ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɭɦɟɧɶɲɢɬɫɹ ɜ n = 2 ɪɚɡɚ.
Ɉɬɜɟɬ: W |
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Ɂɚɞɚɱɚ 4 |
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Ʉɪɢɜɨɲɢɩ, ɫɨɟɞɢɧɹɸɳɢɣ ɨɫɢ ɞɜɭɯ |
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ɡɭɛɱɚɬɵɯ ɤɨɥɟɫ ɪɚɞɢɭɫɚɦɢ R ɢ r, ɜɪɚɳɚɟɬɫɹ ɫ |
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ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ : (ɫɦ. ɪɢɫ.). ȼɧɭɬɪɟɧɧɟɟ |
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ɤɨɥɟɫɨ ɧɟɩɨɞɜɢɠɧɨ. ɇɚɣɬɢ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ |
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ɜɪɚɳɟɧɢɹ ɜɧɟɲɧɟɝɨ ɤɨɥɟɫɚ Z ɢ ɟɝɨ ɨɬɧɨɫɢ- |
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ɬɟɥɶɧɭɸ (ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɤɪɢɜɨɲɢɩɭ) ɭɝɥɨ- |
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ɜɭɸ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ Zɨɬɧ . |
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Ɉɬɜɟɬ: Z |
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Ɂɚɞɚɱɚ 5
ɇɚ ɞɜɚ ɨɞɢɧɚɤɨɜɵɯ ɨɞɧɨɪɨɞɧɵɯ ɛɥɨɤɚ ɪɚɞɢɭɫɚɦɢ R ɧɚɦɨɬɚɧɚ ɥɟɝɤɚɹ ɧɟɪɚɫɬɹɠɢɦɚɹ ɧɢɬɶ (ɫɦ. ɪɢɫ.). ȼ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ ɨɫɢ ɛɥɨɤɨɜ ɨɫɬɚɸɬɫɹ ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɢ ɧɚɯɨɞɹɬɫɹ ɜ ɜɟɪɬɢɤɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ. Ɍɪɟɧɢɟɦ ɜ ɡɚɤɪɟɩɥɟɧɧɨɣ ɨɫɢ ɜɟɪɯɧɟɝɨ ɛɥɨɤɚ, ɚ ɬɚɤɠɟ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɟɦ ɧɢɬɢ ɩɨ ɛɥɨɤɚɦ ɩɪɟɧɟɛɪɟɱɶ. ɇɚɣɬɢ ɦɨɞɭɥɶ ɭɫɤɨɪɟɧɢɹ ɨɫɢ ɧɢɠɧɟɝɨ ɛɥɨɤɚ a ɢ ɦɨɞɭɥɶ ɟɝɨ ɭɝɥɨɜɨɝɨ ɭɫɤɨɪɟɧɢɹ E.
Ɉɬɜɟɬ: a |
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g , E |
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5 |
5 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Ɂɚɞɚɱɚ 6
Ɍɨɧɤɨɫɬɟɧɧɵɣ ɰɢɥɢɧɞɪ ɦɚɫɫɨɣ m ɫɤɚɬɵɜɚɟɬɫɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ ɫ ɭɝɥɨɦ ɩɪɢ ɨɫɧɨɜɚɧɢɢ
D (ɫɦ. ɪɢɫ.).
D
ɇɚɣɬɢ ɭɫɤɨɪɟɧɢɟ a ɨɫɢ ɰɢɥɢɧɞɪɚ ɢ ɫɢɥɭ ɬɪɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɧɟɝɨ ɫɨ ɫɬɨɪɨɧɵ ɧɚɤɥɨɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ.
Ɉɬɜɟɬ: a |
1 |
g sinD , F |
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mg sin D . |
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Ɂɚɞɚɱɚ 7
Ɋɚɞɢɭɫɵ ɰɢɥɢɧɞɪɨɜ ɨɞɢɧɚɤɨɜɵ, ɦɚɫɫɚ ɤɚɠɞɨɝɨ ɰɢɥɢɧɞɪɚ m. Ɉɩɪɟɞɟɥɢɬɶ ɫɢɥɭ F ɪɟɚɤɰɢɢ ɫɬɟɪɠɧɹ.
mg sinD . 7
Ɂɚɞɚɱɚ 8
ɋɩɥɨɲɧɨɦɭ ɨɞɧɨɪɨɞɧɨɦɭ ɰɢɥɢɧɞɪɭ ɦɚɫɫɨɣ m ɢ ɪɚɞɢɭɫɨɦ R ɫɨɨɛɳɢɥɢ ɜɪɚɳɟɧɢɟ ɜɨɤɪɭɝ ɟɝɨ ɨɫɢ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Z . Ɂɚɬɟɦ ɩɨɥɨɠɢɥɢ ɟɝɨ ɛɨɤɨɜɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɩɥɨɫɤɨɫɬɶ ɢ ɩɪɟɞɨɫɬɚɜɢɥɢ ɫɚɦɨɦɭ ɫɟɛɟ. ɇɚ ɤɚɤɨɟ ɪɚɫɫɬɨɹɧɢɟ ɩɟɪɟɦɟɫɬɢɬɫɹ ɰɢɥɢɧɞɪ ɡɚ ɜɪɟɦɹ, ɜ ɬɟɱɟɧɢɟ ɤɨɬɨɪɨɝɨ ɞɜɢɠɟɧɢɟ ɰɢɥɢɧɞɪɚ ɩɪɨɢɫ-
Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
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ɯɨɞɢɥɨ ɫɨ ɫɤɨɥɶɠɟɧɢɟɦ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɦɟɠɞɭ ɩɨɜɟɪɯɧɨɫɬɶɸ ɢ ɰɢɥɢɧɞɪɨɦ ɪɚɜɟɧ P .
Ɉɬɜɟɬ: ǻx Z2 R2 . 18Pg
Ɂɚɞɚɱɚ 9
Ⱦɜɚ ɬɟɥɚ ɦɚɫɫɚɦɢ m1 ɢ m2 ɫɨɟɞɢɧɟɧɵ ɧɟɜɟɫɨɦɨɣ ɧɟɪɚɫɬɹɠɢɦɨɣ ɧɢɬɶɸ, ɩɟɪɟɤɢɧɭɬɨɣ ɱɟɪɟɡ ɨɞɧɨɪɨɞɧɵɣ ɛɥɨɤ ɦɚɫɫɨɣ m (ɫɦ. ɪɢɫ.).
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Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɦɟɠɞɭ ɩɟɪɜɵɦ ɬɟɥɨɦ ɢ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ ɪɚɜɟɧ P . ȼ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ ɬɟɥ ɧɟ ɩɪɨɢɫɯɨɞɢɬ
ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɧɢɬɢ ɩɨ ɩɨɜɟɪɯɧɨɫɬɢ ɛɥɨɤɚ. ɇɚɣɬɢ ɭɫɤɨɪɟɧɢɟ ɜɬɨɪɨɝɨ ɬɟɥɚ, ɩɪɟɧɟɛɪɟɝɚɹ ɬɪɟɧɢɟɦ ɜ ɨɫɢ ɛɥɨɤɚ.
Ɉɬɜɟɬ: a |
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0 , ɩɪɢ m2 d Pm1 . |
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Ɂɚɞɚɱɚ 10 |
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Ɉɞɧɨɪɨɞɧɵɣ ɫɩɥɨɲɧɨɣ ɰɢɥɢɧɞɪ ɦɚɫɫɨɣ M |
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ɦɨɠɟɬ ɫɜɨɛɨɞɧɨ ɜɪɚɳɚɬɶɫɹ ɜɨɤɪɭɝ ɫɜɨɟɣ ɧɟɩɨɞ- |
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ɜɢɠɧɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ (ɫɦ. ɪɢɫ.). ɇɚ ɰɢ- |
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ɥɢɧɞɪ ɧɚɦɨɬɚɧɚ ɬɨɧɤɚɹ ɧɢɬɶ ɞɥɢɧɨɣ L ɢ ɦɚɫɫɨɣ |
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m. ɇɚɣɬɢ ɭɫɤɨɪɟɧɢɟ a ɫɜɟɲɢɜɚɸɳɟɣɫɹ ɱɚɫɬɢ |
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ɧɢɬɢ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɟɟ ɞɥɢɧɵ x. |
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Ɉɬɜɟɬ: a |
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ML 2m(L x) |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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Ɂɚɞɚɱɚ 11 |
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ɋɢɫɬɟɦɚ ɬɟɥ, ɫɨɫɬɨɹɳɚɹ ɢɡ ɝɪɭɡɚ ɢ ɞɜɭɯ |
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ɨɞɢɧɚɤɨɜɵɯ ɛɥɨɤɨɜ, ɢɡɨɛɪɚɠɟɧɚ ɧɚ ɪɢɫɭɧɤɟ. |
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Ɉɫɶ ɥɟɜɨɝɨ ɛɥɨɤɚ ɡɚɤɪɟɩɥɟɧɚ, ɚ ɩɪɚɜɵɣ ɛɥɨɤ |
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ɫɜɨɛɨɞɧɨ ɥɟɠɢɬ ɧɚ ɧɢɬɢ. ɉɪɢ ɞɜɢɠɟɧɢɢ ɬɟɥ |
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ɫɢɫɬɟɦɵ ɧɟ ɩɪɨɢɫɯɨɞɢɬ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɧɢ- |
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ɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɜɟɪɯɧɨɫɬɟɣ ɛɥɨɤɨɜ. ɋɱɢ- |
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ɬɚɹ ɡɚɞɚɧɧɵɦɢ ɦɚɫɫɭ ɝɪɭɡɚ m, ɦɚɫɫɵ ɛɥɨɤɨɜ |
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M ɢ ɢɯ ɪɚɞɢɭɫɵ R, ɨɩɪɟɞɟɥɢɬɶ ɭɫɤɨɪɟɧɢɟ ɝɪɭ- |
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ɡɚ a. Ɍɪɟɧɢɟɦ ɜ ɨɫɢ ɛɥɨɤɚ ɩɪɟɧɟɛɪɟɱɶ. |
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Ɉɬɜɟɬ: a |
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