Механика.Методика решения задач
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Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
141 |
II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɰɢɥɢɧɞɪɚ ɜ |
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ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɏ': |
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mx' ma Fɬɪ . |
(4.90) |
ɍɪɚɜɧɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ (ɫɦ. (6.30) |
ɜ Ƚɥɚɜɟ 6) |
ɰɢɥɢɧɞɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɫɨɜɩɚɞɚɸɳɟɣ ɫ ɟɝɨ ɨɫɶɸ ɢ ɧɚɩɪɚɜɥɟɧɧɨɣ ɡɚ ɩɥɨɫɤɨɫɬɶ ɱɟɪɬɟɠɚ (ɫɦ. ɪɢɫ. 4.14), ɢɦɟɟɬ ɜɢɞ:
JM |
Fɬɪ R , |
(4.91) |
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ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɰɢɥɢɧɞɪɚ, R – |
ɟɝɨ ɪɚɞɢɭɫ. Ɇɨɦɟɧɬ |
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ɝɞɟ M – |
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ɢɧɟɪɰɢɢ ɰɢɥɢɧɞɪɚ ȳ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɤɚɡɚɧɧɨɣ ɨɫɢ ɪɚɜɟɧ |
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mR2 |
(4.92) |
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ɂɡ ɭɫɥɨɜɢɹ ɤɚɱɟɧɢɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɫɜɹɡɢ ɭɫɤɨɪɟɧɢɹ ɨɫɢ ɰɢɥɢɧɞɪɚ (ɫɨɜɩɚɞɚɸɳɟɝɨ ɫ ɭɫɤɨɪɟɧɢɟɦ
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c |
) ɫ ɟɝɨ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ: |
ɰɟɧɬɪɚ ɦɚɫɫ x |
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c |
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(4.93) |
x |
MR . |
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Ɂɚɩɢɲɟɦ ɬɚɤɠɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɫɨɨɬɧɨɲɟɧɢɹ ɩɪɢ ɪɚɜɧɨɭɫɤɨɪɟɧɧɨɦ ɞɜɢɠɟɧɢɢ ɰɟɧɬɪɚ ɦɚɫɫ ɰɢɥɢɧɞɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɢɧɟɪ-
ɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ: |
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Vɰɦ |
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c |
(4.94) |
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x t . |
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c |
2 |
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L |
x t |
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(4.95) |
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ɉɪɢ ɡɚɩɢɫɢ ɩɨɫɥɟɞɧɢɯ ɫɨɨɬɧɨɲɟɧɢɣ ɭɱɬɟɧɨ, ɱɬɨ ɞɜɢɠɟɧɢɟ ɧɚɱɢɧɚɟɬɫɹ ɫ ɧɭɥɟɜɨɣ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ.
III. Ɋɟɲɚɹ ɫɨɜɦɟɫɬɧɨ ɭɪɚɜɧɟɧɢɹ (4.90) – (4.93), ɩɨɥɭɱɚɟɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɭɫɤɨɪɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɰɢɥɢɧɞɪɚ:
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3 a . |
(4.96) |
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x' |
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ɂɫɤɨɦɭɸ ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɦɚɫɫ ɰɢɥɢɧɞɪɚ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɨɧ ɛɭɞɟɬ ɧɚɯɨɞɢɬɶɫɹ ɧɚɞ ɤɪɚɟɦ ɞɨɫɤɢ, ɩɨɥɭɱɢɦ ɢɡ (4.94) –
(4.96): |
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aL . |
(4.97) |
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ɰɦ |
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142 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
4.4. Ɂɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ
Ɂɚɞɚɱɚ 1
ɋ ɤɚɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɫɢɥɨɣ F ɫɥɟɞɭɟɬ ɞɜɢɝɚɬɶ ɤɥɢɧ ɫ ɭɝɥɨɦ D ɩɪɢ ɨɫɧɨɜɚɧɢɢ ɢ ɦɚɫɫɨɣ Ɇ, ɱɬɨɛɵ ɥɟɠɚɳɢɣ ɧɚ ɧɟɦ ɛɪɭɫɨɤ ɦɚɫɫɨɣ m ɧɟ ɩɟɪɟɦɟɳɚɥɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɥɢɧɚ? ɋɢɥɚ F ɧɚɩɪɚɜɥɟɧɚ ɬɚɤ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫɭɧɤɟ.
F
D
Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɦɟɠɞɭ ɛɪɭɫɤɨɦ ɢ ɤɥɢɧɨɦ ɪɚɜɟɧ P .
Ɉɬɜɟɬ: F d (M m)g |
P tgD |
ɩɪɢ P ! tgD , ɚ ɩɪɢ P d tgD ɩɪɨ- |
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1 PtgD |
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ɫɤɚɥɶɡɵɜɚɧɢɟ ɛɭɞɟɬ ɩɪɢ ɥɸɛɨɦ ɡɧɚɱɟɧɢɢ ɦɨɞɭɥɹ ɫɢɥɵ F.
Ɂɚɞɚɱɚ 2
ɋ ɤɚɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɫɢɥɨɣ F ɫɥɟɞɭɟɬ ɞɜɢɝɚɬɶ ɤɥɢɧ ɫ ɭɝɥɨɦ D ɩɪɢ ɨɫɧɨɜɚɧɢɢ ɢ ɦɚɫɫɨɣ Ɇ, ɱɬɨɛɵ ɥɟɠɚɳɢɣ ɧɚ ɧɟɦ ɛɪɭɫɨɤ ɦɚɫɫɨɣ m ɧɟ ɩɟɪɟɦɟɳɚɥɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɥɢɧɚ? ɋɢɥɚ F ɧɚɩɪɚɜɥɟɧɚ ɬɚɤ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫɭɧɤɟ.
F |
D |
Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɦɟɠɞɭ ɛɪɭɫɤɨɦ ɢ ɤɥɢɧɨɦ ɪɚɜɟɧ P .
Ɉɬɜɟɬ:
(M m)g |
tgD P |
d F d (M m)g |
tgD P |
ɩɪɢ P ctgD ; |
1 PtgD |
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1 PtgD |
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Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
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(M m)g |
tgD P |
d F d f ɩɪɢ P t ctgD . |
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1 PtgD |
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Ɂɚɞɚɱɚ 3
ɋɩɥɨɲɧɨɣ ɰɢɥɢɧɞɪ ɫɤɚɬɵɜɚɟɬɫɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɫ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɫ ɭɝɥɨɦ D ɤ ɝɨɪɢɡɨɧɬɭ. ɇɚɤɥɨɧɧɚɹ ɩɥɨɫɤɨɫɬɶ ɨɩɭɫɤɚɟɬɫɹ ɜ ɥɢɮɬɟ ɫ ɭɫɤɨɪɟɧɢɟɦ ɚ0. Ɉɩɪɟɞɟɥɢɬɶ ɭɫɤɨɪɟɧɢɟ ɨɫɢ ɰɢɥɢɧɞɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ.
Ɉɬɜɟɬ: a |
2 |
(g a0 ) sin D . |
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Ɂɚɞɚɱɚ 4
Ƚɨɪɢɡɨɧɬɚɥɶɧɵɣ ɞɢɫɤ ɪɚɞɢɭɫɚ R ɜɪɚɳɚɸɬ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Ȧ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɤɪɚɣ. ɉɨ ɤɪɚɸ ɞɢɫɤɚ ɪɚɜɧɨɦɟɪɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɝɨ ɞɜɢɠɟɬɫɹ ɱɚɫɬɢɰɚ ɦɚɫɫɨɣ m. ȼ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɨɧɚ ɨɤɚɡɵɜɚɟɬɫɹ ɧɚ ɦɚɤɫɢɦɚɥɶɧɨɦ ɪɚɫɫɬɨɹɧɢɢ ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɫɭɦɦɚ ɜɫɟɯ ɫɢɥ ɢɧɟɪɰɢɢ Fɢɧ , ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɱɚɫɬɢɰɭ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ
ɞɢɫɤɨɦ, ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ. ɇɚɣɬɢ ɡɚɜɢɫɢɦɨɫɬɶ ɦɨɞɭɥɹ ɫɢɥɵ Fɢɧ ɨɬ ɪɚɫɫɬɨɹɧɢɹ r ɨɬ ɱɚɫɬɢɰɵ ɞɨ ɨɫɢ ɜɪɚɳɟɧɢɹ.
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2R ·2 |
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Ɉɬɜɟɬ: Fɢɧ mZ |
r |
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r ¹ |
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Ɂɚɞɚɱɚ 5
ɀɟɫɬɤɢɟ ɫɬɟɪɠɧɢ ɨɛɪɚɡɭɸɬ ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ ɩɪɹɦɨɭɝɨɥɶɧɵɣ ɬɪɟɭɝɨɥɶɧɢɤ, ɤɨɬɨɪɵɣ ɜɪɚɳɚɟɬɫɹ ɫ ɩɨɫɬɨɹɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ
Ȧ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɤɚɬɟɬɚ Ⱥȼ (ɫɦ. |
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ɪɢɫ.). ɉɨ ɫɬɟɪɠɧɸ Ⱥɋ ɫɤɨɥɶɡɢɬ ɛɟɡ |
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ɋ |
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ɬɪɟɧɢɹ ɦɭɮɬɚ |
ɦɚɫɫɨɣ |
ɬ, ɫɜɹɡɚɧɧɚɹ ȼ |
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ɩɪɭɠɢɧɨɣ ɠɟɫɬɤɨɫɬɶɸ k ɫ ɜɟɪɲɢɧɨɣ Ⱥ |
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ɬɪɟɭɝɨɥɶɧɢɤɚ. |
Ⱦɥɢɧɚ |
ɧɟɪɚɫɬɹɧɭɬɨɣ |
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ɩɪɭɠɢɧɵ l. Ɉɩɪɟɞɟɥɢɬɶ ɩɪɢ ɤɚɤɨɦ |
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ɡɧɚɱɟɧɢɢ ɦɨɞɭɥɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ Ȧ |
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ɦɭɮɬɚ ɛɭɞɟɬ ɜ ɪɚɜɧɨɜɟɫɢɢ ɩɪɢ |
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ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ ɩɪɭɠɢɧɟ? Ȼɭɞɟɬ ɥɢ |
Ⱥ |
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ɷɬɨ ɪɚɜɧɨɜɟɫɢɟ ɭɫɬɨɣɱɢɜɵɦ? |
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144 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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Ɉɬɜɟɬ: Z |
g |
2 |
, ɪɚɜɧɨɜɟɫɢɟ ɭɫɬɨɣɱɢɜɨ, ɟɫɥɢ kl ! |
mg |
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l |
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Ɂɚɞɚɱɚ 6
ɂɡ ɪɭɠɶɹ ɩɪɨɢɡɜɟɞɟɧ ɜɵɫɬɪɟɥ ɜɜɟɪɯ (ɩɚɪɚɥɥɟɥɶɧɨ ɥɢɧɢɢ ɨɬɜɟɫɚ). Ƚɟɨɝɪɚɮɢɱɟɫɤɚɹ ɲɢɪɨɬɚ ɦɟɫɬɚ M = 60q, ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɩɭɥɢ V0 100 ɦ/ɫ. Ɉɩɪɟɞɟɥɢɬɶ ɧɚɫɤɨɥɶɤɨ ɜɨɫɬɨɱɧɟɟ ɢɥɢ ɡɚɩɚɞɧɟɟ
ɨɬ ɦɟɫɬɚ ɜɵɫɬɪɟɥɚ ɭɩɚɞɟɬ ɩɭɥɹ.
Ɉɬɜɟɬ: ɩɭɥɹ ɨɬɤɥɨɧɢɬɫɹ ɤ ɡɚɩɚɞɭ ɧɚ ɪɚɫɫɬɨɹɧɢɟ
x4 V03Z cosM | 0,5 ɦ. 3 g 2
Ɂɚɞɚɱɚ 7
ɉɨɟɡɞ ɦɚɫɫɨɣ m ɞɜɢɠɟɬɫɹ ɜɞɨɥɶ ɦɟɪɢɞɢɚɧɚ ɧɚ ɫɟɜɟɪɧɨɣ ɲɢɪɨɬɟ M ɫɨ ɫɤɨɪɨɫɬɶɸ V . Ɉɩɪɟɞɟɥɢɬɶ ɜɟɥɢɱɢɧɭ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɫɢɥɵ
ɛɨɤɨɜɨɝɨ ɞɚɜɥɟɧɢɹ ɩɨɟɡɞɚ ɧɚ ɪɟɥɶɫɵ.
Ɉɬɜɟɬ: F 2mVZ sin M (ɧɚ ɩɪɚɜɵɣ ɩɨ ɯɨɞɭ ɩɨɟɡɞɚ ɪɟɥɶɫ).
Ɂɚɞɚɱɚ 8
ɇɚ ɷɤɜɚɬɨɪɟ ɧɚ ɪɟɥɶɫɚɯ ɫɬɨɢɬ ɩɭɲɤɚ. Ɋɟɥɶɫɵ ɧɚɩɪɚɜɥɟɧɵ ɫ ɡɚɩɚɞɚ ɧɚ ɜɨɫɬɨɤ, ɢ ɩɭɲɤɚ ɦɨɠɟɬ ɞɜɢɝɚɬɶɫɹ ɩɨ ɧɢɦ ɛɟɡ ɬɪɟɧɢɹ. ɉɭɲɤɚ ɫɬɪɟɥɹɟɬ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ. Ʉɚɤɭɸ ɫɤɨɪɨɫɬɶ ɛɭɞɟɬ ɢɦɟɬɶ ɩɭɲɤɚ ɩɨɫɥɟ ɜɵɫɬɪɟɥɚ? Ɇɚɫɫɚ ɩɭɲɤɢ Ɇ, ɦɚɫɫɚ ɫɧɚɪɹɞɚ m, ɞɥɢɧɚ ɫɬɜɨɥɚ l. ɋɱɢɬɚɬɶ, ɱɬɨ ɜ ɫɬɜɨɥɟ ɫɧɚɪɹɞ ɞɜɢɠɟɬɫɹ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ ɚ.
Ɉɬɜɟɬ: V |
2mlZ0 |
, Z0 – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ Ɂɟɦɥɢ. |
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Ɂɚɞɚɱɚ 9
ɉɨɞ ɤɚɤɢɦ ɭɝɥɨɦ ɤ ɜɟɪɬɢɤɚɥɢ ɧɚɞɨ ɩɪɨɢɡɜɟɫɬɢ ɜɵɫɬɪɟɥ ɜɜɟɪɯ, ɱɬɨɛɵ ɩɭɥɹ ɭɩɚɥɚ ɨɛɪɚɬɧɨ ɜ ɬɨɱɤɭ, ɢɡ ɤɨɬɨɪɨɣ ɛɵɥ ɩɪɨɢɡɜɟɞɟɧ ɜɵɫɬɪɟɥ? ɇɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɩɭɥɢ V0 = 100 ɦ/c, ɝɟɨɝɪɚɮɢɱɟɫɤɚɹ ɲɢɪɨɬɚ ɦɟɫɬɚ ij = 60q.
Ɉɬɜɟɬ: ɫɬɜɨɥ ɪɭɠɶɹ ɧɚɞɨ ɧɚɤɥɨɧɢɬɶ ɤ ɜɨɫɬɨɤɭ ɩɨɞ ɭɝɥɨɦ
D |
2V0Z cosM |
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Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
145 |
Ɂɚɞɚɱɚ 10
ɇɚ ɷɤɜɚɬɨɪɟ ɫ ɜɵɫɨɬɵ H ɧɚ ɩɨɜɟɪɯɧɨɫɬɶ Ɂɟɦɥɢ ɩɚɞɚɟɬ ɬɟɥɨ ɫ ɧɭɥɟɜɨɣ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ. ɉɪɟɧɟɛɪɟɝɚɹ ɫɢɥɨɣ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ, ɨɩɪɟɞɟɥɢɬɶ ɜ ɤɚɤɭɸ ɫɬɨɪɨɧɭ ɢ ɧɚ ɤɚɤɨɟ ɪɚɫɫɬɨɹɧɢɟ ɨɬɤɥɨɧɢɬɫɹ ɬɟɥɨ ɩɪɢ ɩɚɞɟɧɢɢ ɨɬ ɜɟɪɬɢɤɚɥɢ. ɍɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ Ɂɟɦɥɢ Ȧ ɫɱɢɬɚɬɶ ɡɚɞɚɧɧɨɣ.
Ɉɬɜɟɬ: Ɉɬɤɥɨɧɢɬɫɹ ɧɚ ɜɨɫɬɨɤ ɧɚ ɪɚɫɫɬɨɹɧɢɟ x | |
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Ɂɚɞɚɱɚ 11
ɉɨɟɡɞ ɦɚɫɫɨɣ m ɞɜɢɠɟɬɫɹ ɜɞɨɥɶ ɷɤɜɚɬɨɪɚ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ ȣ. Ɉɩɪɟɞɟɥɢɬɶ ɫɢɥɭ N ɧɨɪɦɚɥɶɧɨɝɨ ɞɚɜɥɟɧɢɹ ɩɨɟɡɞɚ ɧɚ ɪɟɥɶɫɵ. Ɋɟɲɢɬɶ ɡɚɞɚɱɭ ɜ ɞɜɭɯ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ: ɜ ɫɢɫɬɟɦɟ, ɫɜɹɡɚɧɧɨɣ ɫ ɩɨɜɟɪɯɧɨɫɬɶɸ Ɂɟɦɥɢ, ɢ ɜ ɫɢɫɬɟɦɟ, ɫɜɹɡɚɧɧɨɣ ɫ ɩɨɟɡɞɨɦ. Ɋɚɞɢɭɫ Ɂɟɦɥɢ R ɢ ɟɟ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ Ȧ ɫɱɢɬɚɬɶ ɡɚɞɚɧɧɵɦɢ.
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Ɉɬɜɟɬ: N m¨ g |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ȽɅȺȼȺ 5 ɄɂɇȿɆȺɌɂɄȺ ȼ ɌȿɈɊɂɂ ɈɌɇɈɋɂɌȿɅɖɇɈɋɌɂ
5.1.Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ
5.1.1.ɉɨɫɬɭɥɚɬɵ ɢ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ (ɫɩɟɰɢɚɥɶɧɨɣ) ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ
I.ɉɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ: ɥɸɛɨɟ ɮɢɡɢɱɟɫɤɨɟ ɹɜɥɟɧɢɟ ɜ ɩɪɢɪɨɞɟ ɩɪɨɬɟɤɚɟɬ ɨɞɢɧɚɤɨɜɵɦ ɨɛɪɚɡɨɦ ɜɨ ɜɫɟɯ ɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɥɸɛɨɣ ɡɚɤɨɧ ɩɪɢɪɨɞɵ ɨɞɢɧɚɤɨɜɨ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɜɨ ɜɫɟɯ ɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ (ɭɪɚɜɧɟɧɢɹ, ɨɩɢɫɵɜɚɸɳɢɟ ɡɚɤɨɧɵ ɩɪɢɪɨɞɵ ɜ ɪɚɡɥɢɱɧɵɯ ɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ, ɢɦɟɸɬ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɜɢɞ).
II. ɉɪɢɧɰɢɩ ɩɨɫɬɨɹɧɫɬɜɚ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ: ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨ-
ɫɬɪɚɧɟɧɢɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɨɥɧ (ɜ ɬɨɦ ɱɢɫɥɟ ɫɜɟɬɚ) ɜ ɜɚɤɭɭɦɟ ɨɞɢɧɚɤɨɜɚ ɜɨ ɜɫɟɯ ɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɫɤɨɪɨɫɬɟɣ ɞɜɢɠɟɧɢɹ ɢɫɬɨɱɧɢɤɚ ɢ ɩɪɢɟɦɧɢɤɚ ɢɡɥɭɱɟɧɢɹ.
ɋɨɛɵɬɢɟ
Ʌɸɛɨɟ ɫɨɛɵɬɢɟ, ɩɪɨɢɡɨɲɟɞɲɟɟ ɜ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ (x,y,z) ɷɬɨɣ ɬɨɱɤɢ ɢ ɦɨɦɟɧɬɨɦ ɜɪɟɦɟɧɢ t, ɤɨɝɞɚ ɨɧɨ ɩɪɨɢɡɨɲɥɨ.
ɉɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɫɨɛɵɬɢɹ –
(x,y,z,t) ɢɥɢ ( r ,t).
ɋɢɧɯɪɨɧɢɡɚɰɢɹ ɱɚɫɨɜ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɱɚɫɵ, ɧɟɩɨɞɜɢɠɧɨ ɪɚɫɩɨɥɨɠɟɧɧɵɟ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S, ɩɨɤɚɡɵɜɚɥɢ ɨɞɧɨ ɢ ɬɨ ɠɟ ɜɪɟɦɹ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ ɧɚɛɥɸɞɚɬɟɥɹ, ɧɟɩɨɞɜɢɠɧɨɝɨ ɜ ɬɨɣ ɠɟ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɧɟɨɛɯɨɞɢɦɨ ɢɯ ɫɢɧɯɪɨɧɢɡɨɜɚɬɶ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɝɨɜɨɪɢɬɶ ɨ ɟɞɢɧɨɦ ɜɪɟɦɟɧɢ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ.
ɍɫɥɨɜɢɟ ɫɢɧɯɪɨɧɢɡɚɰɢɢ ɱɚɫɨɜ A ɢ B, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɜ ɩɪɨɢɡɜɨɥɶɧɵɯ ɬɨɱɤɚɯ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S (ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɨɛ ɢɡɨɬɪɨɩɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ):
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Ƚɥɚɜɚ 5. Ʉɢɧɟɦɚɬɢɤɚ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ |
147 |
Ɂɞɟɫɶ (ɫɦ. ɪɢɫ. 5.1) t1A – ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɢɡɥɭɱɟɧɢɹ ɢɡ ɬɨɱɤɢ A ɫɜɟɬɨɜɨɝɨ ɫɢɝɧɚɥɚ (ɤɜɚɧɬɚ ɫɜɟɬɚ) ɩɨ ɱɚɫɚɦ ɜ ɬɨɱɤɟ A, t B – ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɟɝɢɫɬɪɚɰɢɢ ɷɬɨɝɨ ɫɢɝɧɚɥɚ ɜ ɬɨɱɤɟ B ɩɨ ɱɚɫɚɦ ɜ ɬɨɱɤɟ B, t2A –
ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɟɝɢɫɬɪɚɰɢɢ ɜ ɬɨɱɤɟ A ɨɬɪɚɠɟɧɧɨɝɨ ɜ ɬɨɱɤɟ B ɫɢɝɧɚɥɚ ɩɨ ɱɚɫɚɦ ɜ ɬɨɱɤɟ A.
S
t A |
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Ɋɢɫ. 5.1. ɋɢɧɯɪɨɧɢɡɚɰɢɹ |
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ɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S
A
5.1.2. ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ
ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ – ɷɬɨ ɜɡɚɢɦɨɫɜɹɡɶ ɩɪɨɫɬɪɚɧɫɬɜɟɧ- ɧɨ-ɜɪɟɦɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɫɨɛɵɬɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɪɚɡɥɢɱɧɵɯ ɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ (ɫɦ. ɪɢɫ. 5.2).
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Ɋɢɫ. 5.2. ȼɡɚɢɦɧɚɹ ɨɪɢɟɧɬɚɰɢɹ ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ ɞɜɢɠɭɳɢɯɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɚ ɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ ɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɫɨɛɵɬɢɹ
ɉɭɫɬɶ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ S' ɞɜɢɠɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ V ɜɞɨɥɶ ɨɫɢ X (ɪɢɫ. 5.2). ɉɪɢ ɷɬɨɦ ɨɫɢ ɫɢɫ-
ɬɟɦ ɨɪɢɟɧɬɢɪɨɜɚɧɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɨɞɢɧɚɤɨɜɨ ɢ ɱɚɫɵ ɫɢɧɯɪɨɧɢ-
ɡɨɜɚɧɵ ɬɚɤ, ɱɬɨ ɫɨɛɵɬɢɟ ɫ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ( r 0 , t = 0) ɜ ɫɢɫɬɟɦɟ S ɢɦɟɟɬ ɤɨɨɪɞɢɧɚɬɵ ( r' 0 , t' = 0) ɜ ɫɢɫɬɟɦɟ S'. Ɍɨɝɞɚ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɨɫɬɭɥɚɬɚɦɢ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶ-
148 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɧɨɫɬɢ ɢ ɢɡ ɨɞɧɨɪɨɞɧɨɫɬɢ ɜɪɟɦɟɧɢ, ɚ ɬɚɤɠɟ ɨɞɧɨɪɨɞɧɨɫɬɢ ɢ ɢɡɨɬɪɨɩɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɥɸɛɨɝɨ ɫɨɛɵɬɢɹ (x,y,z,t) ɢ (x',y',z',t') ɜ ɷɬɢɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ ɫɜɹɡɚ-
ɧɵ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ʌɨɪɟɧɰɚ:
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ȼ ɫɢɥɭ ɥɢɧɟɣɧɨɫɬɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ (5.3) ɚɧɚɥɨɝɢɱɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɢ ɞɥɹ ɢɧɬɟɪɜɚɥɨɜ ɩɪɨɫɬɪɚɧɫɬ-
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɟɥɢɱɢɧɵ ɢɧɬɟɪɜɚɥɨɜ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɢ ɜɪɟɦɟɧɢ ɡɚɜɢɫɹɬ ɨɬ ɜɵɛɨɪɚ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
Ƚɥɚɜɚ 5. Ʉɢɧɟɦɚɬɢɤɚ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ |
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5.1.3.ɋɥɟɞɫɬɜɢɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ
1.ɉɪɟɞɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ
ɋɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɥɸɛɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ (ɚ ɡɧɚɱɢɬ ɢ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɨɛɴɟɤɬɨɜ) ɜ ɩɪɢɪɨɞɟ ɧɟ ɩɪɟɜɵɲɚɟɬ ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɨɥɧ (ɜ ɬɨɦ ɱɢɫɥɟ ɫɜɟɬɚ) ɜ ɜɚɤɭɭɦɟ.
2. "Ɉɬɧɨɫɢɬɟɥɶɧɨɫɬɶ ɨɞɧɨɜɪɟɦɟɧɧɨɫɬɢ"
ɋɨɛɵɬɢɹ, ɩɪɨɢɫɯɨɞɹɳɢɟ ɨɞɧɨɜɪɟɦɟɧɧɨ ɜ ɨɞɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɢ ɢɦɟɸɳɢɟ ɪɚɡɥɢɱɧɵɟ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɞɜɢɠɟɧɢɹ ɞɪɭɝɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ, ɧɟ ɹɜɥɹɸɬɫɹ ɜ ɧɟɣ ɨɞɧɨɜɪɟɦɟɧɧɵɦɢ.
ɗɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɥɟɞɭɟɬ ɢɡ (5.4).
3. "Ɂɚɦɟɞɥɟɧɢɟ ɜɪɟɦɟɧɢ"
Ɋɚɫɫɦɨɬɪɢɦ ɞɜɚ ɫɨɛɵɬɢɹ, ɩɪɨɢɫɯɨɞɹɳɢɟ ɜ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜ ɞɜɢɠɭɳɟɣɫɹ ɫɢɫɬɟɦɟ S' (ɧɚɩɪɢɦɟɪ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɟ "ɬɢɤɚɧɶɟ" ɱɚɫɨɜ ɫɢɫɬɟɦɵ S') (ɪɢɫ. 5.3).
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Ɋɢɫ. 5.3. ȼɡɚɢɦɧɚɹ ɨɪɢɟɧɬɚɰɢɹ ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ ɞɜɢɠɭɳɢɯɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɚ ɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ ɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɫɨɛɵɬɢɣ
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ɉɪɢɧɰɢɩɢɚɥɶɧɨ ɜɚɠɧɨ, ɱɬɨ ɷɬɢ ɞɜɚ ɫɨɛɵɬɢɹ ɩɪɨɢɫɯɨɞɹɬ ɜ |
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ɪɚɡɧɵɯ ɬɨɱɤɚɯ ɩɪɨɫɬɪɚɧɫɬɜɚ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ (ɫɦ. (5.4)):
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Ɂɞɟɫɶ 't' – ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ, ɩɪɨɢɫɯɨɞɹɳɢɦɢ ɜ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜ ɫɢɫɬɟɦɟ S', ɩɨ ɱɚɫɚɦ ɫɢɫɬɟɦɵ S', 't – ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɬɟɦɢ ɠɟ ɫɨɛɵɬɢɹɦɢ ɩɨ ɱɚɫɚɦ ɫɢɫɬɟɦɵ S.
Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɞɥɹ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɜ ɤɨɬɨɪɨɣ ɫɨɛɵɬɢɹ
ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɧɚɛɥɸɞɚɟɬɫɹ ɫɨɤɪɚɳɟɧɢɟ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɷɬɢɦɢ ɫɨɛɵɬɢɹɦɢ (ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɥɸɛɨɣ ɞɪɭɝɨɣ ɫɢɫɬɟɦɨɣ ɨɬɫɱɟɬɚ).
ȿɫɥɢ T0 { 't' – ɩɟɪɢɨɞ ɯɨɞɚ ɞɜɢɠɭɳɢɯɫɹ ɱɚɫɨɜ (ɱɚɫɨɜ ɫɢɫɬɟɦɵ S') ɩɨ ɱɚɫɚɦ ɫɢɫɬɟɦɵ S', ɚ T { 't – ɩɟɪɢɨɞ ɯɨɞɚ ɞɜɢɠɭɳɢɯɫɹ ɱɚɫɨɜ ɩɨ ɱɚɫɚɦ ɫɢɫɬɟɦɵ S, ɬɨ ɦɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɞɜɢɠɭɳɢɟɫɹ ɱɚɫɵ ɢɞɭɬ ɦɟɞɥɟɧɧɟɟ ɧɟɩɨɞɜɢɠɧɵɯ ɱɚɫɨɜ:
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4. "ɋɨɤɪɚɳɟɧɢɟ ɞɥɢɧɵ"
Ɋɚɫɫɦɨɬɪɢɦ ɞɜɚ ɫɨɛɵɬɢɹ, ɩɪɨɢɫɯɨɞɹɳɢɟ ɨɞɧɨɜɪɟɦɟɧɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S. ɗɬɢɦɢ ɫɨɛɵɬɢɹɦɢ ɦɨɝɭɬ ɛɵɬɶ, ɧɚɩɪɢɦɟɪ, ɢɡɦɟɪɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɩɪɚɜɨɝɨ x2 ɢ ɥɟɜɨɝɨ x1 ɤɨɧɰɨɜ ɞɜɢɠɭɳɟɣɫɹ ɜɦɟɫɬɟ ɫ ɫɢɫɬɟɦɨɣ S' ɥɢɧɟɣɤɢ, ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɜɞɨɥɶ ɨɫɟɣ X ɢ X' (ɫɦ. ɪɢɫ. 5.4).
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Ɋɢɫ. 5.4. ȼɡɚɢɦɧɚɹ ɨɪɢɟɧɬɚɰɢɹ ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ ɞɜɢɠɭɳɢɯɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɚ ɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ ɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɫɨɛɵɬɢɣ