Механика.Методика решения задач
.pdfȽɥɚɜɚ 3. Ɂɚɤɨɧɵ ɢɡɦɟɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
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3.3. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 3.1
(Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ)
ɋɬɜɨɥ ɢɝɪɭɲɟɱɧɨɣ ɩɭɲɤɢ ɧɚɩɪɚɜɥɟɧ ɩɨɞ ɭɝɥɨɦ D = 45q ɤ ɝɨɪɢɡɨɧɬɭ. ɇɚɣɬɢ ɫɤɨɪɨɫɬɶ ɩɭɲɤɢ ɫɪɚɡɭ ɩɨɫɥɟ ɜɵɫɬɪɟɥɚ, ɟɫɥɢ ɨɧɚ ɧɟ ɡɚɤɪɟɩɥɟɧɚ ɢ ɦɨɠɟɬ ɫɤɨɥɶɡɢɬɶ ɩɨ ɚɛɫɨɥɸɬɧɨ ɝɥɚɞɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ. Ɇɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɫɧɚɪɹɞɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɭɲɤɢ ɫɪɚɡɭ ɩɨɫɥɟ ɜɵɫɬɪɟɥɚ ɪɚɜɟɧ X0 = 2,2 ɦ/ɫ, ɚ ɟɝɨ ɦɚɫɫɚ ɜ k = 10 ɪɚɡ ɦɟɧɶɲɟ ɦɚɫɫɵ ɩɭɲɤɢ.
Ɋɟɲɟɧɢɟ
Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɨɛɳɟɣ ɫɯɟɦɨɣ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɦɟɯɚɧɢɤɢ ɫ ɩɨɦɨɳɶɸ ɡɚɤɨɧɨɜ ɫɨɯɪɚɧɟɧɢɹ.
I. ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ. Ɉɫɶ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɧɚɩɪɚɜɢɦ ɝɨɪɢɡɨɧɬɚɥɶɧɨ, ɚ ɨɫɶ Y – ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ. Ɉɩɪɟɞɟɥɢɦɫɹ ɫ ɦɨɞɟɥɹɦɢ ɦɚɬɟɪɢɚɥɶɧɵɯ ɨɛɴɟɤɬɨɜ ɢ ɹɜɥɟɧɢɣ. ɋɢɫɬɟɦɚ ɬɟɥ «ɩɭɲɤɚ + ɫɧɚɪɹɞ» ɹɜɥɹɟɬɫɹ ɡɚɦɤɧɭɬɨɣ ɜɞɨɥɶ ɨɫɢ X ɜ ɬɟɱɟɧɢɟ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ ɨɬ ɦɨɦɟɧɬɚ, ɩɪɟɞɲɟɫɬɜɭɸɳɟɝɨ ɜɵɫɬɪɟɥɭ ɩɭɲɤɢ, ɞɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ ɫɪɚɡɭ ɩɨɫɥɟ ɜɵɫɬɪɟɥɚ, ɩɨɫɤɨɥɶɤɭ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɹɦɢ ɡɚɞɚɱɢ ɫɢɥ ɬɪɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɚ ɫɢɫɬɟɦɵ, ɧɟɬ.
II. Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɢɦɩɭɥɶɫɚ (3.13) ɧɚ ɨɫɶ X ɞɥɹ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɬɟɥ ɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ:
mɩXɩ mɫXɫx 0 . |
(3.41) |
Ɂɞɟɫɶ Xɩ ɢ Xɫx – ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɟɣ ɩɭɲɤɢ ɢ ɫɧɚɪɹɞɚ ɩɨɫɥɟ ɜɵ-
ɫɬɪɟɥɚ ɧɚ ɨɫɶ X.
ɉɪɨɟɤɰɢɹ ɧɚ ɨɫɶ X ɧɟɢɡɜɟɫɬɧɨɣ ɫɤɨɪɨɫɬɢ ɫɧɚɪɹɞɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ Xɫx ɫɜɹɡɚɧɚ ɫ ɩɪɨɟɤɰɢɹɦɢ
ɫɤɨɪɨɫɬɢ ɩɭɲɤɢ Xɩ ɢ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɫɧɚɪɹɞɚ ȣ0 |
ɫɥɟɞɭɸ- |
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ɳɢɦ ɨɛɪɚɡɨɦ: |
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Xɫx Xɩ X0 cos Į . |
(3.42) |
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ɂɫɩɨɥɶɡɭɟɦ ɬɚɤɠɟ ɡɚɞɚɧɧɨɟ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɫɨɨɬɧɨɲɟɧɢɟ |
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ɦɟɠɞɭ ɦɚɫɫɚɦɢ ɫɧɚɪɹɞɚ ɢ ɩɭɲɤɢ: |
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mɩ |
k . |
(3.43) |
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mɫ |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (3.41) – (3.43), ɧɚɯɨɞɢɦ ɢɫɤɨɦɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɩɭɲɤɢ ɧɚ ɨɫɶ X ɩɨɫɥɟ ɜɵɫɬɪɟɥɚ:
Xɩ |
X0 cos Į |
mɫ |
X0 cos Į |
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(3.44) |
mɫ mɩ |
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ɉɨɞɫɬɚɜɥɹɹ ɜ (3.44) ɡɧɚɱɟɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ, ɩɨɥɭɱɚɟɦ
Xɩ # 0,14 ɦ/ɫɟɤ . |
(3.45) |
Ɂɚɞɚɱɚ 3.2
(Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ)
Ⱦɜɟ ɨɞɢɧɚɤɨɜɵɟ ɬɟɥɟɠɤɢ, ɧɚ ɤɚɠɞɨɣ ɢɡ ɤɨɬɨɪɵɯ ɧɚɯɨɞɢɬɫɹ ɩɨ ɱɟɥɨɜɟɤɭ, ɞɜɢɠɭɬɫɹ ɛɟɡ ɬɪɟɧɢɹ ɩɨ ɢɧɟɪɰɢɢ ɧɚɜɫɬɪɟɱɭ ɞɪɭɝ ɞɪɭɝɭ ɩɨ ɩɚɪɚɥɥɟɥɶɧɵɦ ɪɟɥɶɫɚɦ. Ʉɨɝɞɚ ɬɟɥɟɠɤɢ ɩɨɪɚɜɧɹɥɢɫɶ, ɫ ɤɚɠɞɨɣ ɢɡ ɧɢɯ ɧɚ ɞɪɭɝɭɸ ɩɟɪɟɩɪɵɝɧɭɥ ɱɟɥɨɜɟɤ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɦ ɤ ɞɜɢɠɟɧɢɸ ɬɟɥɟɠɟɤ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɟɪɜɚɹ ɬɟɥɟɠɤɚ ɨɫɬɚɧɨɜɢɥɚɫɶ, ɚ ɫɤɨɪɨɫɬɶ ɜɬɨɪɨɣ ɫɬɚɥɚ ɪɚɜɧɚ V. ɇɚɣɬɢ ɦɨɞɭɥɢ ɩɟɪɜɨɧɚɱɚɥɶɧɵɯ ɫɤɨɪɨɫɬɟɣ ɬɟɥɟɠɟɤ V1 ɢ V2 , ɟɫɥɢ ɦɚɫɫɚ ɤɚɠɞɨɣ ɬɟɥɟɠɤɢ
ɪɚɜɧɚ Ɇ, ɚ ɦɚɫɫɚ ɤɚɠɞɨɝɨ ɱɟɥɨɜɟɤɚ – m .
Ɋɟɲɟɧɢɟ
I. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɛɳɟɣ ɫɯɟɦɨɣ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɧɚ ɡɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɨɩɪɟɞɟɥɢɦɫɹ ɫ ɦɨɞɟɥɹɦɢ ɦɚɬɟɪɢɚɥɶɧɵɯ ɨɛɴɟɤɬɨɜ ɢ ɹɜɥɟɧɢɣ. ɉɪɟɧɟɛɪɟɝɚɹ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ ɜɨɡɞɭɯɚ, ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɫɤɨɪɨɫɬɶ ɤɚɠɞɨɝɨ ɱɟɥɨɜɟɤɚ ɫɪɚɡɭ ɩɨɫɥɟ ɩɪɵɠɤɚ ɪɚɜɧɚ ɟɝɨ ɫɤɨɪɨɫɬɢ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɟɪɟɞ ɩɪɢɡɟɦɥɟɧɢɟɦ ɧɚ ɞɪɭɝɭɸ ɬɟɥɟɠɤɭ.
ɇɚ ɪɢɫ. 3.4 ɩɨɤɚɡɚɧɨ ɫɨɫɬɨɹɧɢɟ ɫɢɫɬɟɦɵ ɬɟɥ ɞɥɹ ɬɪɟɯ ɦɨɦɟɧɬɨɜ ɜɪɟɦɟɧɢ: t1 – ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɟɪɟɞ ɩɪɵɠɤɨɦ, t2 – ɦɨɦɟɧɬ,
ɤɨɝɞɚ ɨɛɚ ɱɟɥɨɜɟɤɚ ɧɚɯɨɞɹɬɫɹ ɜ ɩɨɥɟɬɟ, t3 – ɫɪɚɡɭ ɩɨɫɥɟ ɩɪɢɡɟɦɥɟ-
ɧɢɹ.
ɇɚ ɪɢɫ. 3.4 ɬɚɤɠɟ ɢɡɨɛɪɚɠɟɧɚ ɜɵɛɪɚɧɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ XY, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɚɹ ɫ ɪɟɥɶɫɚɦɢ, ɢ ɨɛɨɡɧɚɱɟɧɵ ɫɤɨɪɨɫɬɢ ɜɫɟɯ ɬɟɥ ɜ ɭɤɚɡɚɧɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ.
ɇɚ ɜɪɟɦɟɧɧɨɦ ɢɧɬɟɪɜɚɥɟ (t1, t2) ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɞɜɟ ɫɢɫɬɟɦɵ ɬɟɥ: «ɩɟɪɜɵɣ ɱɟɥɨɜɟɤ + ɩɟɪɜɚɹ ɬɟɥɟɠɤɚ» ɢ «ɜɬɨɪɨɣ ɱɟɥɨɜɟɤ + ɜɬɨɪɚɹ ɬɟɥɟɠɤɚ». ɉɨɫɤɨɥɶɤɭ ɱɟɥɨɜɟɤ ɩɪɵɝɧɭɥ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɦ ɞɜɢɠɟɧɢɸ ɬɟɥɟɠɤɢ, ɬɨ ɩɨɫɥɟ ɨɬɪɵɜɚ ɨɬ ɬɟɥɟɠɤɢ ɩɪɨɟɤɰɢɹ ɟɝɨ ɫɤɨɪɨɫɬɢ ɧɚ ɨɫɶ X (ɫɨɜɩɚɞɚɸɳɭɸ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ
Ƚɥɚɜɚ 3. Ɂɚɤɨɧɵ ɢɡɦɟɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
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ɞɜɢɠɟɧɢɹ ɩɟɪɜɨɣ ɬɟɥɟɠɤɢ; ɫɦ. ɪɢɫ. 3.4) ɪɚɜɧɚ ɫɤɨɪɨɫɬɢ ɬɟɥɟɠɤɢ ɩɨɫɥɟ ɟɝɨ ɩɪɵɠɤɚ.
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V12y |
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V22x |
V22y |
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Ɋɢɫ. 3.4 |
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ɇɚ ɜɪɟɦɟɧɧɨɦ ɢɧɬɟɪɜɚɥɟ (t2, t3) ɬɚɤɠɟ ɪɚɫɫɦɨɬɪɢɦ ɞɜɟ ɫɢɫɬɟɦɵ ɬɟɥ: «ɩɟɪɜɵɣ ɱɟɥɨɜɟɤ + ɜɬɨɪɚɹ ɬɟɥɟɠɤɚ», «ɜɬɨɪɨɣ ɱɟɥɨɜɟɤ + ɩɟɪɜɚɹ ɬɟɥɟɠɤɚ».
ɉɨɫɤɨɥɶɤɭ ɜɫɟ ɜɧɟɲɧɢɟ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɦ ɫɢɫɬɟɦɚɦ ɬɟɥ ɫɢɥɵ (ɫɢɥɵ ɬɹɠɟɫɬɢ ɢ ɫɢɥɵ ɪɟɚɤɰɢɢ ɪɟɥɶɫɨɜ) ɧɚɩɪɚɜɥɟɧɵ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɧɚɩɪɚɜɥɟɧɢɸ ɨɫɢ X, ɬɨ ɷɬɢ ɫɢɫɬɟɦɵ ɬɟɥ ɡɚɦɤɧɭɬɵ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɞɚɧɧɨɣ ɨɫɢ, ɢ ɞɥɹ ɧɢɯ ɜɵɩɨɥɧɹɟɬɫɹ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɢɦɩɭɥɶɫɚ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɢɦ ɜɪɟɦɟɧɧɵɯ ɢɧɬɟɪɜɚɥɚɯ.
II. Ɂɚɩɢɲɟɦ ɡɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɢɦɩɭɥɶɫɚ ɞɥɹ ɜɵɛɪɚɧɧɵɯ ɫɢɫɬɟɦ ɬɟɥ ɢ ɜɵɛɪɚɧɧɵɯ ɜɪɟɦɟɧɧɵɯ ɢɧɬɟɪɜɚɥɨɜ.
ɋɢɫɬɟɦɚ ɬɟɥ «ɩɟɪɜɵɣ ɱɟɥɨɜɟɤ + ɩɟɪɜɚɹ ɬɟɥɟɠɤɚ, ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ (t1,t2 ) :
(M m)V1 MV12 x mV12 x . |
(3.46) |
ɂɡ ɭɪɚɜɧɟɧɢɹ (3.46) ɫɥɟɞɭɟɬ, ɱɬɨ ɫɤɨɪɨɫɬɶ ɬɟɥɟɠɤɢ ɩɨɫɥɟ ɩɪɵɠɤɚ ɱɟɥɨɜɟɤɚ ɧɟ ɢɡɦɟɧɢɬɫɹ:
V12 x V1 . (3.47)
Ⱥɧɚɥɨɝɢɱɧɵɣ ɜɵɜɨɞ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɢ ɞɥɹ ɜɬɨɪɨɣ ɬɟɥɟɠɤɢ, ɪɚɫɫɦɚɬɪɢɜɚɹ ɬɨɬ ɠɟ ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ ɢ ɫɢɫɬɟɦɭ ɬɟɥ «ɜɬɨɪɨɣ ɱɟɥɨɜɟɤ + ɜɬɨɪɚɹ ɬɟɥɟɠɤɚ»:
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
V22 x V2 . |
(3.48) |
ɋɢɫɬɟɦɚ ɬɟɥ «ɜɬɨɪɨɣ ɱɟɥɨɜɟɤ + ɩɟɪɜɚɹ ɬɟɥɟɠɤɚ, ɜɪɟɦɟɧɧɨɣ |
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ɢɧɬɟɪɜɚɥ (t2 , t3 ) : |
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MV1 mV2 0 . |
(3.49) |
ȼ (3.49) ɭɱɬɟɧɨ, ɱɬɨ ɩɟɪɜɚɹ ɬɟɥɟɠɤɚ ɨɫɬɚɧɨɜɢɥɚɫɶ ɩɨɫɥɟ ɩɪɢɡɟɦɥɟɧɢɹ ɜɬɨɪɨɝɨ ɱɟɥɨɜɟɤɚ.
Ⱦɥɹ ɫɢɫɬɟɦɵ ɬɟɥ «ɩɟɪɜɵɣ ɱɟɥɨɜɟɤ + ɜɬɨɪɚɹ ɬɟɥɟɠɤɚ» ɢ ɬɨɝɨ
ɠɟ ɜɪɟɦɟɧɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ɢɦɟɟɦ: |
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mV1 MV2 |
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III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (3.49) – (3.50), |
ɧɚɯɨɞɢɦ ɢɫɤɨ- |
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ɦɵɟ ɦɨɞɭɥɢ ɫɤɨɪɨɫɬɟɣ ɬɟɥɟɠɟɤ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ: |
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V1 |
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ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ. ȿɫɥɢ ɦɚɫɫɵ ɬɟɥɟɠɟɤ ɪɚɡɧɵɟ, ɬɨ (3.51) ɞɚɟɬ ɨɞɧɨɡɧɚɱɧɵɣ ɨɬɜɟɬ ɧɚ ɜɨɩɪɨɫ ɡɚɞɚɱɢ. ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ M = m ɨɛɟ ɬɟɥɟɠɤɢ ɨɫɬɚɧɚɜɥɢɜɚɸɬɫɹ (V = 0). ɉɪɢ ɷɬɨɦ ɧɚɱɚɥɶɧɵɟ ɫɤɨɪɨɫɬɢ ɬɟɥɟɠɟɤ ɦɨɝɭɬ ɛɵɬɶ ɥɸɛɵɦɢ ɩɨ ɜɟɥɢɱɢɧɟ, ɧɨ ɪɚɜɧɵɦɢ ɞɪɭɝ ɞɪɭɝɭ: V1 V2 .
Ɂɚɞɚɱɚ 3.3
(Ⱦɜɢɠɟɧɢɟ ɬɟɥ ɫ ɩɟɪɟɦɟɧɧɨɣ ɦɚɫɫɨɣ)
ɉɨ ɞɜɭɦ ɝɨɪɢɡɨɧɬɚɥɶɧɵɦ ɪɟɥɶɫɚɦ ɞɜɢɠɭɬɫɹ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ X0 = 1 ɦ/ɫ ɛɟɡ ɬɪɟɧɢɹ (ɩɨ ɢɧɟɪɰɢɢ) ɞɜɟ ɨɞɢɧɚɤɨɜɵɟ ɬɟɥɟɠɤɢ ɦɚɫɫɨɣ M0 = 100 ɤɝ ɤɚɠɞɚɹ. ȼ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t0 = 0 ɧɚ ɨɛɟ ɬɟɥɟɠɤɢ ɫɜɟɪɯɭ ɧɟɩɪɟɪɵɜɧɨɣ ɫɬɪɭɣɤɨɣ ɧɚɱɢɧɚɟɬ ɫɵɩɚɬɶɫɹ ɩɟɫɨɤ ɬɚɤ, ɱɬɨ ɦɚɫɫɚ ɫɵɩɥɸɳɟɝɨɫɹ ɩɟɫɤɚ ɪɚɫɬɟɬ ɥɢɧɟɣɧɨ ɩɨ ɡɚɤɨɧɭ m = kt, ɝɞɟ k = 10 ɤɝ/ɫ. ȼ ɩɟɪɜɨɣ ɬɟɥɟɠɤɟ ɟɫɬɶ ɭɫɬɪɨɣɫɬɜɨ ɞɥɹ ɧɟɩɪɟɪɵɜɧɨɝɨ ɜɵɛɪɨɫɚ ɜɫɟɝɨ ɫɫɵɩɚɧɧɨɝɨ ɧɚ ɧɟɟ ɩɟɫɤɚ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɦ ɫɤɨɪɨɫɬɢ ɬɟɥɟɠɤɢ. ɂɡ ɜɬɨɪɨɣ ɬɟɥɟɠɤɢ ɩɟɫɨɤ ɧɟ ɜɵɛɪɚɫɵɜɚɟɬɫɹ. Ʉɚɤ ɛɭɞɭɬ ɡɚɜɢɫɟɬɶ ɨɬ ɜɪɟɦɟɧɢ ɫɤɨɪɨɫɬɶ ɢ ɩɟɪɟɦɟɳɟɧɢɟ ɤɚɠɞɨɣ ɬɟɥɟɠɤɢ? Ɂɚ ɤɚɤɨɟ ɜɪɟɦɹ ɤɚɠɞɚɹ ɬɟɥɟɠɤɚ ɩɪɨɣɞɟɬ ɪɚɫɫɬɨɹɧɢɟ L = 9 ɦ?
Ɋɟɲɟɧɢɟ
I. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɛɳɟɣ ɫɯɟɦɨɣ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɧɚ ɡɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɪɚɫɫɦɨɬɪɢɦ ɨɫɨɛɟɧɧɨɫɬɢ ɩɪɨɰɟɫɫɨɜ ɞɥɹ ɨɛɟɢɯ ɬɟɥɟɠɟɤ. ȼ ɨɛɨɢɯ ɫɥɭɱɚɹɯ ɫɢɫɬɟɦɚ ɬɟɥ «ɬɟɥɟɠɤɚ + ɫɫɵɩɚɸɳɢɣɫɹ ɧɚ ɧɟɟ ɡɚ ɜɪɟɦɹ dt ɩɟɫɨɤ» ɹɜɥɹɟɬɫɹ ɡɚɦɤɧɭɬɨɣ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɞɜɢɠɟɧɢɹ ɬɟ-
Ƚɥɚɜɚ 3. Ɂɚɤɨɧɵ ɢɡɦɟɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
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ɥɟɠɤɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɢɦɩɭɥɶɫɚ. ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɬɚɤ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 3.5.
II. Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɢɦɩɭɥɶɫɚ ɧɚ ɨɫɶ X ɞɥɹ ɫɢɫɬɟɦɵ ɬɟɥ «ɬɟɥɟɠɤɚ + ɫɫɵɩɚɸɳɢɣɫɹ ɧɚ ɧɟɟ ɡɚ ɜɪɟɦɹ dt ɩɟɫɨɤ» ɢ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ [t, t + dt] ɜ ɨɛɨɢɯ ɫɥɭɱɚɹɯ ɢɦɟɟɬ ɜɢɞ:
M (t)X(t) d m 0 M (t) d m X(t) dX . (3.52)
Ɂɞɟɫɶ M(t) ɢ X(t) – ɦɝɧɨɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɦɚɫɫɵ ɢ ɫɤɨɪɨɫɬɢ ɬɟɥɟɠɤɢ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t; dm – ɩɪɢɪɚɳɟɧɢɟ ɦɚɫɫɵ ɬɟɥɟɠɤɢ ɡɚ ɦɚɥɵɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ dt, dX – ɩɪɢɪɚɳɟɧɢɟ ɫɤɨɪɨɫɬɢ ɬɟɥɟɠɤɢ.
Ɋɢɫ. 3.5 |
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ɂɡ ɭɫɥɨɜɢɹ ɡɚɞɚɱɢ ɫɥɟɞɭɟɬ, ɱɬɨ |
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dm kdt . |
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Ⱥɧɚɥɨɝɢɱɧɨ ɬɨɦɭ, ɤɚɤ ɷɬɨ ɛɵɥɨ ɫɞɟɥɚɧɨ ɜ ɬɟɨɪɟɬɢɱɟɫɤɨɦ ɜɜɟɞɟɧɢɢ ɩɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɞɜɢɠɟɧɢɹ ɬɟɥ ɫ ɩɟɪɟɦɟɧɧɨɣ ɦɚɫɫɨɣ, ɩɪɟ-
ɧɟɛɪɟɠɟɦ ɜ (3.52) ɱɥɟɧɚɦɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɦɚɥɨɫɬɢ: |
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0 M (t)dX(t) X(t)dm . |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ: |
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Ⱦɚɥɟɟ ɪɚɫɫɦɨɬɪɢɦ ɞɜɢɠɟɧɢɟ ɤɚɠɞɨɣ ɬɟɥɟɠɤɢ ɜ ɨɬɞɟɥɶɧɨɫɬɢ. Ɍɟɥɟɠɤɚ ʋ1. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɦɚɫɫɚ ɩɟɪɜɨɣ
ɬɟɥɟɠɤɢ ɧɟ ɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ (M(t) = M0), ɩɨɷɬɨɦɭ, ɢɧɬɟɝɪɢɪɭɹ
(3.55), ɩɨɥɭɱɢɦ:
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɚɜɢɫɢɦɨɫɬɶ ɫɤɨɪɨɫɬɢ ɩɟɪɜɨɣ ɬɟɥɟɠɤɢ ɨɬ ɜɪɟɦɟɧɢ ɢɦɟɟɬ ɜɢɞ:
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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Ɂɚɤɨɧ ɞɜɢɠɟɧɢɹ ɩɟɪɜɨɣ ɬɟɥɟɠɤɢ ɩɨɥɭɱɚɟɦ, ɢɧɬɟɝɪɢɪɭɹ (3.57):
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ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɡɚɤɨɧɨɦ ɞɜɢɠɟɧɢɹ (3.58), ɧɚɯɨɞɢɦ ɜɪɟɦɹ, |
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ɡɚ ɤɨɬɨɪɨɟ ɩɟɪɜɚɹ ɬɟɥɟɠɤɚ ɩɪɨɣɞɟɬ ɪɚɫɫɬɨɹɧɢɟ L = 9 ɦ: |
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# 23,0 ɫ. |
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Ɍɟɥɟɠɤɚ 2. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɞɥɹ ɦɚɫɫɵ ɜɬɨ- |
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ɪɨɣ ɬɟɥɟɠɤɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ: |
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ɢ ɭɪɚɜɧɟɧɢɟ (3.55) ɩɪɢɧɢɦɚɟɬ ɜɢɞ: |
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ɉɨɫɥɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ (3.61) ɩɨɥɭɱɚɟɦ: |
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Ɂɚɜɢɫɢɦɨɫɬɶ ɫɤɨɪɨɫɬɢ ɜɬɨɪɨɣ ɬɟɥɟɠɤɢ ɨɬ ɜɪɟɦɟɧɢ ɢɦɟɟɬ ɜɢɞ: |
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ɂɫɩɨɥɶɡɭɹ (3.63) ɧɚɯɨɞɢɦ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɜɬɨɪɨɣ ɬɟɥɟɠɤɢ: |
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ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɡɚɤɨɧɨɦ ɞɜɢɠɟɧɢɹ (3.64), ɧɚɯɨɞɢɦ ɜɪɟɦɹ, |
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ɡɚ ɤɨɬɨɪɨɟ ɜɬɨɪɚɹ ɬɟɥɟɠɤɚ ɩɪɨɣɞɟɬ ɪɚɫɫɬɨɹɧɢɟ L = 9 ɦ: |
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k¨ ¸ © ¹
III. ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ. Ɇɚɫɫɚ ɜɬɨɪɨɣ ɬɟ-
ɥɟɠɤɢ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ, ɩɨɷɬɨɦɭ ɩɪɢ ɩɚɞɟɧɢɢ ɧɚ ɧɟɟ ɨɱɟɪɟɞɧɨɣ ɩɨɪɰɢɢ ɩɟɫɤɚ ɟɟ ɫɤɨɪɨɫɬɶ ɭɦɟɧɶɲɚɟɬɫɹ ɦɟɞɥɟɧɧɟɟ, ɱɟɦe 1 14,6 ɫ.t (3.65)
Ƚɥɚɜɚ 3. Ɂɚɤɨɧɵ ɢɡɦɟɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
97 |
ɫɤɨɪɨɫɬɶ ɩɟɪɜɨɣ ɬɟɥɟɠɤɢ. Ƚɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɟɣ ɤɨɨɪɞɢɧɚɬ ɬɟɥɟɠɟɤ ɨɬ ɜɪɟɦɟɧɢ ɩɨɤɚɡɚɧɵ ɧɚ ɪɢɫ. 3.6.
x, ɦ |
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Ɋɢɫ. 3.6 |
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ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (3.58) ɤɨɨɪɞɢɧɚɬɚ ɩɟɪɜɨɣ ɬɟɥɟɠɤɢ ɚɫɢɦɩɬɨ-
ɬɢɱɟɫɤɢ ɫɬɪɟɦɢɬɫɹ ɤ ɡɧɚɱɟɧɢɸ x |
M 0X0 |
= 10 ɦ. ɋɤɨɪɨɫɬɶ ɜɬɨ- |
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ɪɨɣ ɬɟɥɟɠɤɢ ɬɚɤɠɟ ɭɦɟɧɶɲɚɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ, ɨɞɧɚɤɨ ɟɟ ɤɨɨɪɞɢɧɚɬɚ ɛɭɞɟɬ ɜɫɟ ɜɪɟɦɹ ɭɜɟɥɢɱɢɜɚɬɶɫɹ.
Ɂɚɞɚɱɚ 3.4
(Ⱦɜɢɠɟɧɢɟ ɬɟɥ ɫ ɩɟɪɟɦɟɧɧɨɣ ɦɚɫɫɨɣ)
Ɋɚɤɟɬɚ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɜ ɜɨɡɞɭɯɟ ɧɚ ɩɨɫɬɨɹɧɧɨɣ ɜɵɫɨɬɟ ɫ ɩɨɦɨɳɶɸ ɠɢɞɤɨɫɬɧɨɝɨ ɪɚɤɟɬɧɨɝɨ ɞɜɢɝɚɬɟɥɹ. ɇɚɱɚɥɶɧɚɹ ɦɚɫɫɚ ɪɚɤɟɬɵ (ɫ ɬɨɩɥɢɜɨɦ) ɪɚɜɧɚ M0 = 105 ɤɝ, ɚ ɫɤɨɪɨɫɬɶ ɜɵɛɪɚɫɵɜɚɟɦɵɯ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɝɚɡɨɜ ɪɚɜɧɚ u = 2600 ɦ/ɫ. ɇɚɣɬɢ ɪɚɫɯɨɞ ɬɨɩɥɢɜɚ ȝ(t) ɢ ɦɚɫɫɭ ɜɵɛɪɨɲɟɧɧɵɯ ɪɚɤɟɬɨɣ ɝɚɡɨɜ ɜ ɩɟɪɜɭɸ ɫɟɤɭɧɞɭ ɩɨɥɟɬɚ.
Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɫɜɹɡɚɧɧɭɸ ɫ ɩɨɜɟɪɯɧɨɫɬɶɸ Ɂɟɦɥɢ, ɨɫɶ X ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɢɦ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ. Ⱦɥɹ ɚɧɚɥɢɡɚ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɛɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɞɥɹ ɫɢɫɬɟɦɵ ɬɟɥ «ɪɚɤɟɬɚ + ɜɵɥɟɬɟɜɲɢɟ ɢɡ ɧɟɟ ɝɚɡɵ». ɇɚ ɷɬɭ ɫɢɫɬɟɦɭ ɬɟɥ ɞɟɣɫɬɜɭɟɬ ɜɧɟɲɧɹɹ ɫɢɥɚ – ɫɢɥɚ ɬɹɠɟɫɬɢ.
II. Ɂɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɢɦɩɭɥɶɫɚ (ɫɦ. (3.7)) ɪɚɤɟɬɵ ɡɚ-
ɩɢɲɟɦ ɜ ɜɢɞɟ |
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M (t) 0 d m u M (t)g d t . |
(3.66) |
Ɂɞɟɫɶ M(t) – ɦɚɫɫɚ ɪɚɤɟɬɵ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t;
98 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
dm P(t)dt – |
(3.67) |
ɢɡɦɟɧɟɧɢɟ ɦɚɫɫɵ ɪɚɤɟɬɵ ɡɚ ɦɚɥɵɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ dt. Ⱦɨɩɨɥɧɢɦ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɭɫɥɨɜɢɟɦ ɫɨɯɪɚɧɟɧɢɹ ɫɭɦɦɚɪɧɨɣ
ɦɚɫɫɵ ɪɚɤɟɬɵ ɢ ɜɵɥɟɬɚɸɳɟɝɨ ɝɚɡɚ:
d M d m 0 . (3.68) III. Ɋɟɲɢɦ ɩɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (3.66) – (3.68) ɨɬ-
ɧɨɫɢɬɟɥɶɧɨ ɪɚɫɯɨɞɚ ɬɨɩɥɢɜɚ P(t) .
ɂɫɤɥɸɱɚɹ ɢɡ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɦɚɫɫɭ ɪɚɤɟɬɵ M ɢ ɦɚɫɫɭ ɢɫɬɟɤɚɸɳɢɯ ɢɯ ɧɟɟ ɝɚɡɨɜ m, ɩɨɥɭɱɚɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ P(t) :
d P(t) |
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Ɋɟɲɚɟɦ (3.69) ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ: |
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ln P(t) |
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t const , |
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Ʉɨɧɫɬɚɧɬɭ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ A ɜ (3.73) ɨɩɪɟɞɟɥɹɟɦ ɢɡ ɭɪɚɜɧɟɧɢɹ (3.66), ɡɚɩɢɫɚɧɧɨɝɨ ɞɥɹ ɧɚɱɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ:
A P(0) |
M 0 g |
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ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɪɚɫɯɨɞɚ ɬɨɩ-
ɥɢɜɚ:
g
P(t) M 0 g e u t . (3.73) u
Ɇɚɫɫɭ ɝɚɡɨɜ, ɜɵɛɪɨɲɟɧɧɵɯ ɪɚɤɟɬɨɣ ɡɚ ɜɪɟɦɹ t, ɧɚɯɨɞɢɦ ɜ ɪɟɡɭɥɶɬɚɬɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ P(t) ɩɨ ɜɪɟɦɟɧɢ:
t
m(t) ³P(t
0
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ɉɪɢ t ug ɦɚɫɫɚ ɜɵɛɪɚɫɵɜɚɟɦɵɯ ɝɚɡɨɜ ɨɤɚɡɵɜɚɟɬɫɹ ɩɪɨ-
ɩɨɪɰɢɨɧɚɥɶɧɚ ɜɪɟɦɟɧɢ ɢɯ ɢɫɬɟɱɟɧɢɹ: |
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Ƚɥɚɜɚ 3. Ɂɚɤɨɧɵ ɢɡɦɟɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
99 |
ɉɨɫɤɨɥɶɤɭ ɩɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ug # 26009,8 c # 260 c , ɜɨɫɩɨɥɶɡɭ-
ɟɦɫɹ ɜɵɪɚɠɟɧɢɟɦ (3.75) ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɢɫɤɨɦɨɣ ɦɚɫɫɵ ɜɵɛɪɨɲɟɧɧɵɯ ɪɚɤɟɬɨɣ ɝɚɡɨɜ ɜ ɩɟɪɜɭɸ ɫɟɤɭɧɞɭ ɩɨɥɟɬɚ:
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Ɂɚɞɚɱɚ 3.5
(Ɂɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ)
Ⱦɜɚ ɲɚɪɢɤɚ ɫ ɨɞɢɧɚɤɨɜɨɣ ɦɚɫɫɨɣ m, ɫɨɟɞɢɧɟɧɧɵɟ ɧɟɪɚɫɬɹɧɭɬɨɣ ɩɪɭɠɢɧɤɨɣ ɞɥɢɧɨɣ l0, ɥɟɠɚɬ ɧɚ ɝɥɚɞɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ. ɇɚ ɨɞɢɧ ɢɡ ɲɚɪɢɤɨɜ ɧɚɱɢɧɚɟɬ ɞɟɣɫɬɜɨɜɚɬɶ ɩɨɫɬɨɹɧɧɚɹ ɫɢɥɚ F, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɞɨɥɶ ɨɫɢ ɩɪɭɠɢɧɤɢ. ɑɟɪɟɡ ɧɟɤɨɬɨɪɨɟ ɜɪɟɦɹ ɞɥɢɧɚ ɩɪɭɠɢɧɤɢ ɫɬɚɧɨɜɢɬɫɹ ɦɚɤɫɢɦɚɥɶɧɨɣ ɢ ɪɚɜɧɨɣ lmax. Ɉɩɪɟɞɟɥɢɬɶ ɤɨɷɮɮɢɰɢɟɧɬ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɤɢ k.
Ɋɟɲɟɧɢɟ
ɉɪɢɥɨɠɢɦ ɫɢɥɭ F ɤ ɩɟɪɟɞɧɟɦɭ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɞɟɣɫɬɜɢɹ ɫɢɥɵ ɲɚɪɢɤɭ (ɫɦ. ɪɢɫ. 3.7), ɩɨɫɤɨɥɶɤɭ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɜ ɪɟɡɭɥɶɬɚɬɟ ɞɟɣɫɬɜɢɹ ɫɢɥɵ ɩɪɨɢɫɯɨɞɢɬ ɪɚɫɬɹɠɟɧɢɟ ɩɪɭɠɢɧɤɢ.
F
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Ɋɢɫ. 3.7 |
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ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɫɜɹɡɚɧɧɭɸ ɫ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ, ɧɚɩɪɚɜɢɜ ɨɫɶ X ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɞɟɣɫɬɜɢɹ ɫɢɥɵ, ɢ ɨɛɨɡɧɚɱɢɦ ɤɨɨɪɞɢɧɚɬɵ ɲɚɪɢɤɨɜ x10, x20 ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɢ x11, x21 ɜ ɦɨɦɟɧɬ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɫɠɚɬɢɹ ɩɪɭɠɢɧɵ (ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 3.7). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɞɥɢɧɚ ɧɟɪɚɫɬɹɧɭɬɨɣ ɩɪɭɠɢɧɤɢ ɜ ɢɫɯɨɞɧɨɦ ɫɨɫɬɨɹɧɢɢ l0 x20 x10 , ɚ ɟɟ ɞɥɢɧɚ ɜ ɦɨɦɟɧɬ ɦɚɤɫɢɦɚɥɶɧɨɝɨ
ɪɚɫɬɹɠɟɧɢɹ lmax x21 x11 .
Ⱦɜɢɠɟɧɢɟ ɬɟɥ ɫɢɫɬɟɦɵ, ɫɨɫɬɨɹɳɟɣ ɢɡ ɞɜɭɯ ɫɜɹɡɚɧɧɵɯ ɩɪɭɠɢɧɤɨɣ ɲɚɪɢɤɨɜ, ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɟɣ ɫɢɥɵ F ɢɡ-ɡɚ ɢɡɦɟɧɹɸ-
100 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɳɢɯɫɹ ɜɨ ɜɪɟɦɟɧɢ ɜɧɭɬɪɟɧɧɢɯ ɭɩɪɭɝɢɯ ɫɢɥ ɛɭɞɟɬ ɞɨɫɬɚɬɨɱɧɨ ɫɥɨɠɧɵɦ. Ɉɞɧɚɤɨ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɲɚɪɢɤɚɦɢ ɦɚɤɫɢɦɚɥɶɧɨ ɢ ɪɚɜɧɨ lmax, ɫɤɨɪɨɫɬɢ ɢɯ ɛɭɞɭɬ ɪɚɜɧɵ, ɱɬɨ ɫɭɳɟɫɬɜɟɧɧɨ ɭɩɪɨɳɚɟɬ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɩɪɟɧɟɛɪɟɠɟɦ ɫɢɥɚɦɢ ɬɪɟɧɢɹ ɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ, ɦɚɫɫɨɣ ɩɪɭɠɢɧɤɢ ɢ ɪɚɡɦɟɪɚɦɢ ɲɚɪɢɤɨɜ. ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɡɚɤɨɧɨɦ ɢɡɦɟɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɢ ɬɟɨɪɟɦɨɣ ɨ ɞɜɢɠɟɧɢɢ ɰɟɧɬɪɚ ɦɚɫɫ ɞɥɹ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɬɟɥ (ɫɦ. ɩ. 3.1).
II. Ɂɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ (3.39) ɞɥɹ ɫɢɫɬɟɦɵ «ɞɜɚ ɲɚɪɢɤɚ + ɩɪɭɠɢɧɤɚ» ɧɚ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ ɨɬ ɧɚɱɚɥɚ ɞɟɣɫɬɜɢɹ ɫɢɥɵ ɞɨ ɦɨɦɟɧɬɚ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɪɚɫɬɹɠɟɧɢɹ ɩɪɭɠɢɧɤɢ ɢɦɟɟɬ ɜɢɞ:
ǻ(E k E p ) 2 |
mX2 |
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k(l |
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l |
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A , |
(3.77) |
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ɝɞɟ X – ɫɤɨɪɨɫɬɶ ɲɚɪɢɤɨɜ ɜ ɦɨɦɟɧɬ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɪɚɫɬɹɠɟɧɢɹ |
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ɩɪɭɠɢɧɵ, ɚ ɪɚɛɨɬɚ ɜɧɟɲɧɟɣ ɫɢɥɵ ɪɚɜɧɚ |
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A F (x21 x20 ) . |
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(3.78) |
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Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ (3.6) ɞɥɹ ɪɚɫɫɦɚɬ- |
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ɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ «ɞɜɚ ɲɚɪɢɤɚ + ɩɪɭɠɢɧɤɚ»: |
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2maɰɦ F . |
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ɉɨɫɤɨɥɶɤɭ ɰɟɧɬɪ ɦɚɫɫ ɫɢɫɬɟɦɵ ɞɜɢɠɟɬɫɹ ɪɚɜɧɨɭɫɤɨɪɟɧɧɨ ɫ ɭɫɤɨɪɟɧɢɟɦ aɰɦ , ɢɡɦɟɧɟɧɢɟ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɦɚɫɫ ɛɭɞɟɬ ɪɚɜɧɨ
x11 x21 |
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x10 x20 |
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Xɰɦ2 |
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ɝɞɟ Xɰɦ X – ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɦɚɫɫ ɜ ɦɨɦɟɧɬ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɪɚɫɬɹ-
ɠɟɧɢɹ ɩɪɭɠɢɧɵ.
III. ɉɨɞɫɬɚɜɥɹɹ (3.79) ɜ (3.80), ɜɵɪɚɠɚɟɦ ɤɜɚɞɪɚɬ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɦɚɫɫ ɱɟɪɟɡ ɤɨɨɪɞɢɧɚɬɵ ɲɚɪɢɤɨɜ:
Xɰɦ2 |
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(3.81) |
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Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (3.77), (3.78) ɢ (3.81), ɩɨɥɭɱɚɟɦ ɢɫ- |
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ɤɨɦɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɤɢ: |
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