Механика.Методика решения задач
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Ƚɥɚɜɚ 5. Ʉɢɧɟɦɚɬɢɤɚ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ |
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ɉɪɢɧɰɢɩɢɚɥɶɧɨ ɜɚɠɧɨ, ɱɬɨ ɷɬɢ ɞɜɚ ɫɨɛɵɬɢɹ ɩɪɨɢɫɯɨɞɹɬ ɨɞɧɨɜɪɟɦɟɧɧɨ ɜ ɫɢɫɬɟɦɟ S: t1 = t2, 't = t2 –t1 = 0. Ⱦɥɹ ɫɢɫɬɟɦɵ S' ɷɬɢ ɫɨɛɵɬɢɹ ɛɭɞɭɬ ɩɪɨɢɫɯɨɞɢɬɶ ɧɟ ɨɞɧɨɜɪɟɦɟɧɧɨ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ
(5.4) ɞɥɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɢɧɬɟɪɜɚɥɨɜ ǻx x2 x1 ɢ ǻx |
c |
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ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ: |
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ǻxc |
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ǻx |
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ǻx |
Jǻx ɢɥɢ ǻ x |
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1 V / c 2 |
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(5.7) |
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Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɞɥɹ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɜ ɤɨɬɨɪɨɣ ɫɨɛɵɬɢɹ
ɩɪɨɢɫɯɨɞɹɬ ɨɞɧɨɜɪɟɦɟɧɧɨ, ɧɚɛɥɸɞɚɟɬɫɹ ɫɨɤɪɚɳɟɧɢɟ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ɦɟɠɞɭ ɷɬɢɦɢ ɫɨɛɵɬɢɹɦɢ (ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɥɸɛɨɣ ɞɪɭɝɨɣ ɫɢɫɬɟɦɨɣ ɨɬɫɱɟɬɚ) ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦ.
ȿɫɥɢ ɫɨɛɵɬɢɹɦɢ ɹɜɥɹɸɬɫɹ ɢɡɦɟɪɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɥɢɧɟɣɤɢ (ɫɦ.
ɪɢɫ. 5.4), ɬɨ:
l0 |
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J l ɢɥɢ l |
l0 |
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(5.8) |
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V / c 2 |
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Ɂɞɟɫɶ l0 { ǻxc – ɫɨɛɫɬɜɟɧɧɚɹ ɞɥɢɧɚ ɥɢɧɟɣɤɢ (ɞɥɢɧɚ ɥɢɧɟɣɤɢ ɜ ɧɟ-
ɩɨɞɜɢɠɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɢɧɟɣɤɢ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S', ɩɪɢ ɷɬɨɦ ɢɡɦɟɪɟɧɢɟ ɤɨɨɪɞɢɧɚɬ ɤɨɧɰɨɜ ɥɢɧɟɣɤɢ ɦɨɠɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɜ ɪɚɡɧɵɟ
ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ); l { ǻx – ɞɥɢɧɚ ɥɢɧɟɣɤɢ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S,
ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɥɢɧɟɣɤɚ ɞɜɢɠɟɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ V (ɢɡɦɟɪɟɧɢɟ ɤɨɨɪɞɢɧɚɬ ɤɨɧɰɨɜ ɥɢɧɟɣɤɢ ɞɨɥɠɧɨ ɩɪɨɢɫɯɨɞɢɬɶ ɜ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ).
5.1.4. ɉɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ
ɉɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɞɜɭɦɹ ɫɨɛɵɬɢɹɦɢ:
S12 { x2 x1 2 y2 y1 2 z2 z1 2 c2 t2 t1 2
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ǻx2 ǻy2 ǻz2 c2ǻt 2 |
r 2 |
c2t 2 . |
(5.9) |
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Ɂɞɟɫɶ r |
ǻx2 ǻy2 ǻz2 t 0 – |
ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɣ |
ɢɧɬɟɪɜɚɥ |
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(ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɬɨɱɤɚɦɢ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɜ ɤɨɬɨɪɵɯ ɩɪɨɢɡɨɲɥɢ
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɫɨɛɵɬɢɹ) ɢ |
t12 |
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t 0 – ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ |
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ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ.
ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ʌɨɪɟɧɰɚ (5.2), ɥɟɝɤɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɞɜɭ-
ɦɹ ɫɨɛɵɬɢɹɦɢ ɨɞɢɧɚɤɨɜ ɜɨ ɜɫɟɯ ɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟ-
ɬɚ, ɬɨ ɟɫɬɶ ɹɜɥɹɟɬɫɹ ɢɧɜɚɪɢɚɧɬɨɦ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦ Ʌɨɪɟɧɰɚ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɤ ɞɪɭɝɨɣ:
c |
(5.11) |
S12 S12 . |
Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɷɬɨɦ ɩɟɪɟɯɨɞɟ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɣ r12 ɢ ɜɪɟɦɟɧɧɨɣ t12 ɢɧɬɟɪɜɚɥɵ ɥɢɛɨ ɨɛɚ ɭɦɟɧɶɲɚɸɬɫɹ, ɥɢɛɨ ɨɛɚ ɭɜɟɥɢɱɢɜɚɸɬɫɹ.
ɉɪɨɫɬɪɚɧɫɬɜɟɧɧɨɩɨɞɨɛɧɵɣ ɢɧɬɟɪɜɚɥ ɜɟɳɟɫɬɜɟɧɧɵɣ ɩɪɨ-
ɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɞɜɭɦɹ ɫɨɛɵɬɢɹɦɢ, ɞɥɹ ɤɨɬɨɪɨɝɨ S122 ! 0 . ȼ ɷɬɨɦ ɫɥɭɱɚɟ: r12 > ct12.
ɋɜɨɣɫɬɜɚ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɩɨɞɨɛɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ɦɟɠɞɭ ɞɜɭɦɹ ɫɨɛɵɬɢɹɦɢ
1. ɋɭɳɟɫɬɜɭɟɬ ɬɚɤɚɹ ɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ, ɜ ɤɨɬɨɪɨɣ ɫɨɛɵɬɢɹ ɩɪɨɢɫɯɨɞɹɬ ɨɞɧɨɜɪɟɦɟɧɧɨ, ɧɨ ɜ ɪɚɡɧɵɯ ɬɨɱɤɚɯ ɩɪɨɫɬɪɚɧɫɬɜɚ. ȼ ɷɬɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɣ ɢɧɬɟɪɜɚɥ r12c
ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ ɩɪɢɧɢɦɚɟɬ ɦɢɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ:
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c |
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S12 |
r12 |
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t12 |
S12 |
r12 , |
(5.12) |
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r12 |
r12 |
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t12 . |
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(5.13) |
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2.ɇɟ ɫɭɳɟɫɬɜɭɟɬ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɜ ɤɨɬɨɪɨɣ ɫɨɛɵɬɢɹ ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ.
3.ɋɨɛɵɬɢɹ, ɫɜɹɡɚɧɧɵɟ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɩɨɞɨɛɧɵɦ ɢɧɬɟɪɜɚɥɨɦ, ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɟɪɟɯɨɞɚ ɜ ɞɪɭɝɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ ɦɨɝɭɬ ɩɪɨɢɫɯɨɞɢɬɶ ɜɨ ɜɪɟɦɟɧɢ ɜ ɨɛɪɚɬɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ.
4.ɗɬɢ ɫɨɛɵɬɢɹ ɧɟ ɦɨɝɭɬ ɛɵɬɶ ɫɜɹɡɚɧɵ ɩɪɢɱɢɧɧɨɫɥɟɞɫɬɜɟɧɧɨɣ ɫɜɹɡɶɸ, ɩɨɫɤɨɥɶɤɭ ɞɥɹ ɷɬɨɝɨ ɩɨɬɪɟɛɨɜɚɥɚɫɶ ɛɵ ɫɤɨɪɨɫɬɶ ɩɟɪɟɞɚɱɢ ɫɢɝɧɚɥɚ, ɩɪɟɜɵɲɚɸɳɚɹ ɫɤɨɪɨɫɬɶ ɫɜɟɬɚ:
S 2 |
! 0 , |
r |
! ct |
12 |
, X |
r12 |
! c . |
(5.14) |
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t12 |
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5. ɗɬɢ ɫɨɛɵɬɢɹ ɧɟ ɦɨɝɭɬ ɩɪɨɢɫɯɨɞɢɬɶ ɫ ɨɞɧɢɦ ɢ ɬɟɦ ɠɟ ɬɟɥɨɦ (ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɦ, ɱɬɨɛɵ ɫɱɢɬɚɬɶ, ɱɬɨ ɫɨɛɵɬɢɹ ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞ-
Ƚɥɚɜɚ 5. Ʉɢɧɟɦɚɬɢɤɚ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ |
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ɧɨɣ ɢ ɬɨɣ ɠɟ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɷɬɢɦ ɬɟɥɨɦ), ɩɨɫɤɨɥɶɤɭ ɬɟɥɨ ɧɟ ɦɨɠɟɬ ɞɜɢɝɚɬɶɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ, ɩɪɟɜɵɲɚɸɳɟɣ ɫɤɨɪɨɫɬɶ ɫɜɟɬɚ.
ȼɪɟɦɟɧɢɩɨɞɨɛɧɵɣ ɢɧɬɟɪɜɚɥ ɦɧɢɦɵɣ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-
ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɞɜɭɦɹ ɫɨɛɵɬɢɹɦɢ, ɞɥɹ ɤɨɬɨɪɨɝɨ S122 0 . ȼ ɷɬɨɦ ɫɥɭɱɚɟ: r12 < ct12.
ɋɜɨɣɫɬɜɚ ɜɪɟɦɟɧɢɩɨɞɨɛɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ɦɟɠɞɭ ɞɜɭɦɹ ɫɨɛɵɬɢɹɦɢ
1. ɋɭɳɟɫɬɜɭɟɬ ɬɚɤɚɹ ɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ, ɜ ɤɨɬɨɪɨɣ ɨɛɚ ɫɨɛɵɬɢɹ ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɧɨ ɜ ɪɚɡɧɨɟ ɜɪɟɦɹ. ȼ ɷɬɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ t12c
ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ ɩɪɢɧɢɦɚɟɬ ɦɢɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ:
S12 |
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c |
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r12 |
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t12 |
S12 |
ict12 |
(5.15) |
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S12 |
r12 |
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t12 |
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t12 |
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t12 |
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r12 |
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t12 . |
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(5.16) |
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2. ɇɟ ɫɭɳɟɫɬɜɭɟɬ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɜ ɤɨɬɨɪɨɣ ɫɨɛɵɬɢɹ ɩɪɨɢɫ- |
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c |
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ɯɨɞɹɬ ɜ ɨɞɧɨ ɢ ɬɨ ɠɟ ɜɪɟɦɹ: t12 |
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3.ɋɨɛɵɬɢɹ, ɫɜɹɡɚɧɧɵɟ ɜɪɟɦɟɧɢɩɨɞɨɛɧɵɦ ɢɧɬɟɪɜɚɥɨɦ, ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɟɪɟɯɨɞɚ ɜ ɞɪɭɝɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ ɧɟ ɦɨɝɭɬ ɩɪɨɢɫɯɨɞɢɬɶ ɜɨ ɜɪɟɦɟɧɢ ɜ ɨɛɪɚɬɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ.
4.ɗɬɢ ɞɜɚ ɫɨɛɵɬɢɹ ɦɨɝɭɬ ɛɵɬɶ ɫɜɹɡɚɧɵ ɩɪɢɱɢɧɧɨɫɥɟɞɫɬɜɟɧɧɨɣ ɫɜɹɡɶɸ, ɩɨɫɤɨɥɶɤɭ ɞɥɹ ɷɬɨɝɨ ɬɪɟɛɭɟɬɫɹ ɫɤɨɪɨɫɬɶ ɩɟɪɟɞɚɱɢ ɫɢɝɧɚɥɚ ɦɟɧɶɲɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ:
S 2 |
0 |
, r |
ct |
12 |
, X |
r12 |
c . |
(5.17) |
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12 |
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12 |
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t12 |
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5. ɗɬɢ ɫɨɛɵɬɢɹ ɦɨɝɭɬ ɩɪɨɢɫɯɨɞɢɬɶ ɫ ɨɞɧɢɦ ɢ ɬɟɦ ɠɟ ɬɟɥɨɦ, ɩɨɫɤɨɥɶɤɭ ɬɟɥɨ ɦɨɠɟɬ ɞɜɢɝɚɬɶɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ, ɦɟɧɶɲɟɣ ɫɤɨɪɨɫɬɢ
ɫɜɟɬɚ, ɬɨ ct12 > r12 ɢ S122 0 .
ɋɜɟɬɨɩɨɞɨɛɧɵɣ ɢɧɬɟɪɜɚɥ ɧɭɥɟɜɨɣ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɞɜɭɦɹ ɫɨɛɵɬɢɹɦɢ S122 0 .
154 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɋɜɨɣɫɬɜɚ ɫɜɟɬɨɩɨɞɨɛɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ɦɟɠɞɭ ɞɜɭɦɹ ɫɨɛɵɬɢɹɦɢ
1. ȿɫɥɢ ɫɨɛɵɬɢɹ ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɬɨ ɨɧɢ ɩɪɨɢɫɯɨɞɹɬ ɨɞɧɨɜɪɟɦɟɧɧɨ (ɢ ɧɚɨɛɨɪɨɬ) ɜ ɥɸɛɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ:
S |
r 2 |
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(5.18) |
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2.ɇɟ ɫɭɳɟɫɬɜɭɟɬ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɜ ɤɨɬɨɪɨɣ ɞɜɚ ɫɨɛɵɬɢɹ, ɪɚɡɞɟɥɟɧɧɵɟ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɦ ɢɧɬɟɪɜɚɥɨɦ ɩɪɨɢɫɯɨɞɹɬ ɨɞɧɨɜɪɟɦɟɧɧɨ ɢɥɢ ɪɚɡɞɟɥɟɧɧɵɟ ɜɪɟɦɟɧɧɵɦ ɢɧɬɟɪɜɚɥɨɦ, ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ.
3.ɋɨɛɵɬɢɹ, ɫɜɹɡɚɧɧɵɟ ɫɜɟɬɨɩɨɞɨɛɧɵɦ ɢɧɬɟɪɜɚɥɨɦ, ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɟɪɟɯɨɞɚ ɜ ɞɪɭɝɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ ɧɟ ɦɨɝɭɬ ɩɪɨɢɫɯɨɞɢɬɶ ɜɨ ɜɪɟɦɟɧɢ ɜ ɨɛɪɚɬɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ.
Ⱦɨɤɚɠɟɦ ɷɬɨ. Ⱦɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɫɨɛɵɬɢɣ ǻx rcǻt , ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ʌɨɪɟɧɰɚ ɞɥɹ ɢɧɬɟɪɜɚɥɨɜ (5.4), ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ:
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Ʉɚɤ ɜɢɞɢɦ, ɡɧɚɤ ɜɪɟɦɟɧɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ǻtc ɫɨɜɩɚɞɚɟɬ ɫɨ ɡɧɚɤɨɦ ɢɧɬɟɪɜɚɥɚ ǻt ɩɪɢ ɥɸɛɵɯ ɜɨɡɦɨɠɧɵɯ ɫɤɨɪɨɫɬɹɯ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ S' ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S.
4. ɗɬɢ ɞɜɚ ɫɨɛɵɬɢɹ ɦɨɝɭɬ ɛɵɬɶ ɫɜɹɡɚɧɵ ɩɪɢɱɢɧɧɨɫɥɟɞɫɬɜɟɧɧɨɣ ɫɜɹɡɶɸ, ɟɫɥɢ ɢɫɩɨɥɶɡɭɟɬɫɹ ɫɢɝɧɚɥ, ɩɟɪɟɞɚɸɳɢɣɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ ɫɜɟɬɚ:
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5. ɗɬɢ ɫɨɛɵɬɢɹ ɧɟ ɦɨɝɭɬ ɩɪɨɢɫɯɨɞɢɬɶ ɫ ɨɞɧɢɦ ɢ ɬɟɦ ɠɟ ɬɟɥɨɦ, ɢɦɟɸɳɢɦ ɦɚɫɫɭ ɩɨɤɨɹ, ɩɨɫɤɨɥɶɤɭ ɨɧɨ ɧɟ ɦɨɠɟɬ ɞɜɢɝɚɬɶɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ ɫɜɟɬɚ.
ɉɨɧɹɬɢɹ ɜɪɟɦɟɧɢɩɨɞɨɛɧɵɣ, ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɩɨɞɨɛɧɵɣ ɢ ɫɜɟɬɨɩɨɞɨɛɧɵɣ ɢɧɬɟɪɜɚɥɵ – ɩɨɧɹɬɢɹ ɚɛɫɨɥɸɬɧɵɟ, ɧɟ ɡɚɜɢɫɹɳɢɟ ɨɬ ɜɵɛɨɪɚ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
5.1.5.ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ (ɫɥɨɠɟɧɢɟ) ɫɤɨɪɨɫɬɟɣ
ȼɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ʌɨɪɟɧɰɚ (5.2) ɢ ɨɩɪɟɞɟɥɟɧɢɟɦ ɫɤɨɪɨɫɬɢ (ɫɦ. ɩ. 1.1 ɜ Ƚɥɚɜɟ 1)
Ƚɥɚɜɚ 5. Ʉɢɧɟɦɚɬɢɤɚ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ |
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1 V / c 2 Xcz . 1 cV2 Xcx
(5.21)
(5.22)
Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ – ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɨɞɧɨɝɨ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɞɪɭɝɢɦ ɬɟɥɨɦ. ɗɬɚ ɫɤɨɪɨɫɬɶ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɛɨɥɶɲɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ.
ɋɤɨɪɨɫɬɶ ɫɛɥɢɠɟɧɢɹ ɬɟɥ – ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɬɟɥɚɦɢ ɜ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ. ɗɬɚ ɫɤɨɪɨɫɬɶ ɦɨɠɟɬ ɛɵɬɶ ɛɨɥɶɲɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ.
5.2. Ɉɫɧɨɜɧɵɟ ɬɢɩɵ ɡɚɞɚɱ ɢ ɦɟɬɨɞɵ ɢɯ ɪɟɲɟɧɢɹ
5.2.1.Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɡɚɞɚɱ ɤɢɧɟɦɚɬɢɤɢ
ɜɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ
Ȼɨɥɶɲɢɧɫɬɜɨ ɡɚɞɚɱ ɤɢɧɟɦɚɬɢɤɢ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɦɨɠɧɨ ɭɫɥɨɜɧɨ ɨɬɧɟɫɬɢ ɤ ɫɥɟɞɭɸɳɢɦ ɬɢɩɚɦ ɡɚɞɚɱ ɢɥɢ ɢɯ ɤɨɦɛɢɧɚɰɢɹɦ. Ɂɚɞɚɱɢ ɧɚ:
1)ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɢɥɢ ɢɯ ɫɥɟɞɫɬɜɢɹ ("ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɶ ɨɞɧɨɜɪɟɦɟɧɧɨɫɬɢ", "ɡɚɦɟɞɥɟɧɢɟ ɜɪɟɦɟɧɢ" ɢ "ɫɨɤɪɚɳɟɧɢɟ ɞɥɢɧɵ");
2)ɢɧɜɚɪɢɚɧɬɧɨɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɯ ɢɧɬɟɪɜɚɥɨɜ;
156 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
3) ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɫɤɨɪɨɫɬɟɣ.
Ʉɚɤ ɩɪɚɜɢɥɨ, ɨɞɢɧ ɢɡ ɬɢɩɨɜ ɡɚɞɚɱ ɢɦɟɟɬ ɨɫɧɨɜɧɨɟ, ɞɪɭɝɢɟ – ɩɨɞɱɢɧɟɧɧɨɟ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɡɧɚɱɟɧɢɟ.
5.2.2.Ɉɛɳɚɹ ɫɯɟɦɚ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɤɢɧɟɦɚɬɢɤɢ
ɜɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ
I.Ɉɩɪɟɞɟɥɢɬɶɫɹ ɫ ɫɨɛɵɬɢɹɦɢ ɢ ɫɢɫɬɟɦɚɦɢ ɨɬɫɱɟɬɚ.
1.ɇɚɪɢɫɨɜɚɬɶ ɱɟɪɬɟɠ, ɧɚ ɤɨɬɨɪɨɦ ɢɡɨɛɪɚɡɢɬɶ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɬɟɥɚ (ɟɫɥɢ ɷɬɨ ɧɟɨɛɯɨɞɢɦɨ).
2.ȼɵɛɪɚɬɶ ɞɜɢɠɭɳɢɟɫɹ ɞɪɭɝ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝɚ ɢɧɟɪɰɢɚɥɶɧɵɟ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɢ ɢɡɨɛɪɚɡɢɬɶ ɧɚ ɱɟɪɬɟɠɟ ɢɯ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ (ɢɡ ɫɨɨɛɪɚɠɟɧɢɣ ɭɞɨɛɫɬɜɚ).
3.ɂɡɨɛɪɚɡɢɬɶ ɢ ɨɛɨɡɧɚɱɢɬɶ ɫɤɨɪɨɫɬɢ ɬɟɥ.
4.ȼɵɛɪɚɬɶ ɢɧɬɟɪɟɫɭɸɳɢɟ ɧɚɫ ɫɨɛɵɬɢɹ ɢ ɡɚɩɢɫɚɬɶ ɢɯ ɩɪɨ- ɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɵɯ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ.
II.Ɂɚɩɢɫɚɬɶ ɩɨɥɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ.
1.Ɂɚɩɢɫɚɬɶ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɢɥɢ ɢɯ ɫɥɟɞɫɬɜɢɹ (ɞɥɹ ɡɚɞɚɱ ɬɢɩɚ (1)).
2.Ɂɚɩɢɫɚɬɶ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɢɧɬɟɪɜɚɥɵ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ (ɞɥɹ ɡɚɞɚɱ ɬɢɩɚ (2)).
3.Ɂɚɩɢɫɚɬɶ ɮɨɪɦɭɥɵ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɫɤɨɪɨɫɬɟɣ (ɞɥɹ ɡɚɞɚɱ ɬɢɩɚ (3)).
4.ɂɫɩɨɥɶɡɨɜɚɬɶ ɭɫɥɨɜɢɹ ɡɚɞɚɱɢ (ɧɚɩɪɢɦɟɪ, ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɫɢɫɬɟɦɵ ɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-
ɜɪɟɦɟɧɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɫɨɛɵɬɢɣ).
III.ɉɨɥɭɱɢɬɶ ɢɫɤɨɦɵɣ ɪɟɡɭɥɶɬɚɬ ɜ ɚɧɚɥɢɬɢɱɟɫɤɨɦ ɢ ɱɢɫɥɟɧɧɨɦ ɜɢɞɚɯ.
1.Ɋɟɲɢɬɶ ɫɢɫɬɟɦɭ ɩɨɥɭɱɟɧɧɵɯ ɭɪɚɜɧɟɧɢɣ.
2.ɉɪɨɜɟɫɬɢ ɚɧɚɥɢɡ ɪɟɲɟɧɢɹ (ɪɚɫɫɦɨɬɪɟɬɶ ɯɚɪɚɤɬɟɪɧɵɟ ɫɥɭɱɚɢ).
3.ɉɨɥɭɱɢɬɶ ɱɢɫɥɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ.
Ƚɥɚɜɚ 5. Ʉɢɧɟɦɚɬɢɤɚ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ |
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5.3. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 5.1
(ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɢɥɢ ɢɯ ɫɥɟɞɫɬɜɢɹ)
ɋɬɟɪɠɟɧɶ ɩɪɨɥɟɬɚɟɬ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ ɦɢɦɨ ɦɟɬɤɢ, ɧɟɩɨɞɜɢɠɧɨɣ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S. ȼɪɟɦɹ ɩɪɨɥɟɬɚ ɜ ɷɬɨɣ ɫɢɫɬɟɦɟ 't = 20 ɧɫ. ȼ ɫɢɫɬɟɦɟ ɠɟ ɨɬɫɱɟɬɚ S', ɫɜɹɡɚɧɧɨɣ ɫɨ ɫɬɟɪɠɧɟɦ, ɦɟɬɤɚ ɞɜɢɠɟɬɫɹ ɜɞɨɥɶ ɧɟɝɨ ɜ ɬɟɱɟɧɢɟ 't' = 25 ɧɫ. ɇɚɣɬɢ ɫɨɛɫɬɜɟɧɧɭɸ ɞɥɢɧɭ ɫɬɟɪɠɧɹ.
Ɋɟɲɟɧɢɟ
I. ɉɭɫɬɶ ɫɬɟɪɠɟɧɶ, ɚ ɡɧɚɱɢɬ ɢ ɫɢɫɬɟɦɚ S', ɞɜɢɠɭɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ V ɜɞɨɥɶ ɨɫɢ X ɫɢɫɬɟɦɵ S. ɋ ɬɚɤɨɣ ɠɟ ɩɨ ɜɟɥɢɱɢɧɟ ɫɤɨɪɨɫɬɶɸ, ɧɨ ɜ ɨɛɪɚɬɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ, ɞɜɢɠɟɬɫɹ ɦɟɬɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɬɟɪɠɧɹ. Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɪɢɫ. 5.4 (ɫɦ. ɩ. 5.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ). Ɉɩɪɟɞɟɥɢɦ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ, ɤɚɤ ɦɨɦɟɧɬɵ ɩɪɨɥɟɬɚ ɦɟɬɤɢ ɦɢɦɨ ɨɛɨɢɯ ɤɨɧɰɨɜ ɫɬɟɪɠɧɹ. Ɉɛɨɡɧɚɱɢɦ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɷɬɢɯ ɫɨɛɵɬɢɣ, ɤɚɤ ( x1c , t1c )
ɢ ( xc2 , t2c ) ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S' ɢ ɤɚɤ ( x1 , t1 ) ɢ ( x2 , t2 ) ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S. Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɵɛɨɪɨɦ ɫɨɛɵɬɢɣ x1 x2 .
II. ɂɫɤɨɦɚɹ ɫɨɛɫɬɜɟɧɧɚɹ ɞɥɢɧɚ ɫɬɟɪɠɧɹ l0 ɪɚɜɧɚ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɦɭ ɢɧɬɟɪɜɚɥɭ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ Ⱥ ɢ ȼ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ,
ɫɜɹɡɚɧɧɨɣ ɫɨ ɫɬɟɪɠɧɟɦ |
c |
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ɉɨɫɤɨɥɶɤɭ ɦɟɬɤɚ ɞɜɢɠɟɬɫɹ |
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l0 x2 |
x1 . |
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ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɬɟɪɠɧɹ ɫɨ ɫɤɨɪɨɫɬɶɸ V, ɬɨ |
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l0 |
Vǻtc , |
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(5.23) |
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ɝɞɟ ǻt |
c |
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t2 |
t1 ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɦɢ |
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ɫɨɛɵɬɢɹɦɢ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S'. |
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Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ V |
ɡɚɩɢɲɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɞɥɹ |
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ɜɪɟɦɟɧɧɵɯ ɢɧɬɟɪɜɚɥɨɜ ǻtc |
ɢ ǻt |
t2 t1 |
(ɫɦ. (5.5)): |
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ǻt |
c |
ǻt |
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1 V / c 2 . |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɫɨɤɪɚɳɚɟɬɫɹ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ ǻt , ɩɨɫɤɨɥɶɤɭ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ (ɧɚɛɥɸɞɚɟɬɫɹ "ɡɚɦɟɞɥɟɧɢɟ ɜɪɟɦɟɧɢ").
III. ɂɫɩɨɥɶɡɭɹ (5.24), ɨɩɪɟɞɟɥɢɦ ɜɟɥɢɱɢɧɭ ɫɤɨɪɨɫɬɢ V:
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
V c |
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ǻt · |
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ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (5.23) ɢ (5.25) ɢɫɤɨɦɚɹ ɫɨɛɫɬɜɟɧɧɚɹ ɞɥɢɧɚ ɫɬɟɪɠɧɹ (ɞɥɢɧɚ ɫɬɟɪɠɧɹ ɜ ɧɟɩɨɞɜɢɠɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɝɨ ɫɢɫɬɟɦɟ
ɨɬɫɱɟɬɚ S') ɪɚɜɧɚ |
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Vǻtc cǻtc |
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ǻt ·2 |
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ɉɨɞɫɬɚɜɢɜ ɜ (5.26) ɡɧɚɱɟɧɢɹ 't = 20 ɧɫ ɢ 't' = 25 ɧɫ, ɡɚɞɚɧɧɵɟ |
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ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ, ɚ ɬɚɤɠɟ ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ |
c # 3 108 ɦ/ɫ , |
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ɩɨɥɭɱɢɦ: |
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Ɂɚɞɚɱɚ 5.2
(ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɢɥɢ ɢɯ ɫɥɟɞɫɬɜɢɹ) ɋɨɛɫɬɜɟɧɧɨɟ ɜɪɟɦɹ ɠɢɡɧɢ ɧɟɤɨɬɨɪɨɣ ɧɟɫɬɚɛɢɥɶɧɨɣ ɱɚɫɬɢɰɵ
ǻt0 10 ɧɫ. Ʉɚɤɨɣ ɩɭɬɶ ɩɪɨɥɟɬɢɬ ɷɬɚ ɱɚɫɬɢɰɚ, ɞɜɢɝɚɹɫɶ ɫ ɩɨɫɬɨɹɧ-
ɧɨɣ ɫɤɨɪɨɫɬɶɸ, ɞɨ ɪɚɫɩɚɞɚ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɝɞɟ ɟɟ ɜɪɟɦɹ ɠɢɡɧɢ ǻt 20 ɧɫ?
Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ. ɋɜɹɠɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ S' ɫ ɞɜɢɠɭɳɟɣɫɹ ɱɚɫɬɢɰɟɣ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɢɫɬɟɦɚ S' ɞɜɢɠɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ S ɫɨ ɫɤɨɪɨɫɬɶɸ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ V. Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɪɢɫ. 5.3 (ɫɦ. ɩ. 5.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ). Ɉɩɪɟɞɟɥɢɦ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɤɚɤ ɫɨɛɵɬɢɹ, ɫɨɫɬɨɹɳɢɟ ɜ ɪɨɠɞɟɧɢɢ ɢ ɪɚɫɩɚɞɟ ɱɚɫɬɢɰɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɉɭɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɷɬɢɯ ɫɨɛɵɬɢɣ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɪɚɜɧɵ ( x1 , t1 ) ɢ ( x2 , t2 ), ɚ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S' –
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. ɋɨɛɫɬɜɟɧɧɨɟ ɜɪɟɦɹ ɠɢɡɧɢ ɧɟɫɬɚ- |
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x1 |
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ɛɢɥɶɧɨɣ ɱɚɫɬɢɰɵ ǻt0 – ɜɪɟɦɹ ɠɢɡɧɢ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S', ɜ ɤɨɬɨɪɨɣ ɷɬɢ ɞɜɚ ɫɨɛɵɬɢɹ ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ:
ǻt0 ǻtc t2c t1c |
(5.27) |
II. ɂɫɤɨɦɵɣ ɩɭɬɶ l, ɤɨɬɨɪɵɣ ɩɪɨɥɟɬɢɬ ɱɚɫɬɢɰɚ ɞɨ ɫɜɨɟɝɨ ɪɚɫɩɚɞɚ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɨɩɪɟɞɟɥɹɟɬɫɹ ɟɟ ɫɤɨɪɨɫɬɶɸ ɢ ɜɪɟɦɟɧɟɦ ɠɢɡɧɢ ɱɚɫɬɢɰɵ ɜ ɷɬɨɣ ɫɢɫɬɟɦɟ:
l Vǻt . |
(5.28) |
Ƚɥɚɜɚ 5. Ʉɢɧɟɦɚɬɢɤɚ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ |
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Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ (ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢ ɫɤɨɪɨɫɬɢ ɞɜɢɠɭɳɟɣɫɹ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S') ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɫɥɟɞɫɬɜɢɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ – "ɡɚɦɟɞɥɟɧɢɟɦ ɜɪɟɦɟɧɢ". ɉɨɫɤɨɥɶɤɭ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S' ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɬɨ, ɫɨɝɥɚɫɧɨ (5.5), ɞɨɥɠɧɨ ɧɚɛɥɸɞɚɬɶɫɹ ɫɨɤɪɚɳɟɧɢɟ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ ɜ ɫɢɫɬɟɦɟ S' ɦɟɠɞɭ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɦɢ ɫɨɛɵɬɢɹɦɢ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɨɛɫɬɜɟɧɧɨɟ ɜɪɟɦɹ ɠɢɡɧɢ ɢ ɜɪɟɦɹ ɠɢɡɧɢ ɜ ɥɚɛɨɪɚ-
ɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɫɜɹɡɚɧɵ ɫɥɟɞɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ: |
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III. ɂɫɩɨɥɶɡɭɹ (5.29), ɨɩɪɟɞɟɥɢɦ ɫɤɨɪɨɫɬɶ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S': |
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ɉɨɞɫɬɚɜɥɹɹ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ (5.30) ɞɥɹ ɫɤɨɪɨɫɬɢ ɜ (5.28), ɨɩɪɟɞɟɥɢɦ ɢɫɤɨɦɵɣ ɩɭɬɶ l, ɤɨɬɨɪɵɣ ɩɪɨɥɟɬɢɬ ɱɚɫɬɢɰɚ ɞɨ ɫɜɨɟɝɨ ɪɚɫɩɚɞɚ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S:
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ǻt |
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ɉɨɞɫɬɚɜɢɜ ɜ (5.31) ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜɟɥɢɱɢɧ ǻt0 |
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ɡɚɞɚɧɧɵɟ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ, ɩɨɥɭɱɚɟɦ: l = 5,2 ɦ.
Ɂɚɞɚɱɚ 5.3
(ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɢɥɢ ɢɯ ɫɥɟɞɫɬɜɢɹ)
ɋɢɫɬɟɦɚ ɨɬɫɱɟɬɚ S' ɞɜɢɠɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S ɜɞɨɥɶ ɨɫɢ X ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ V = 0,9c. ȼ ɤɚɠɞɨɣ ɫɢɫɬɟɦɟ ɜ ɬɨɱɤɚɯ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ –200 ɦ, –100 ɦ, 0 ɦ, 100 ɦ ɢ 200 ɦ ɧɚɯɨɞɹɬɫɹ ɨɞɢɧɚɤɨɜɵɟ ɫɢɧɯɪɨɧɢɡɨɜɚɧɧɵɟ ɱɚɫɵ. Ɂɚ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜɪɟɦɟɧɢ ɜ ɨɛɟɢɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ ɜɡɹɬ ɬɚɤɨɣ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɱɚɫɵ, ɧɟɩɨɞɜɢɠɧɵɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S ɢ ɢɦɟɸɳɢɟ ɤɨɨɪɞɢɧɚɬɭ x = 0 ɦ, ɨɤɚɠɭɬɫɹ ɧɚɩɪɨɬɢɜ ɱɚɫɨɜ, ɧɟɩɨɞɜɢɠɧɵɯ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S' ɢ ɢɦɟɸɳɢɯ ɤɨɨɪɞɢɧɚɬɭ x' = 0 ɦ. Ɉɩɪɟɞɟɥɢɬɶ ɜɪɟɦɹ, ɤɨɬɨɪɨɟ ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɛɭɞɭɬ ɩɨɤɚɡɵɜɚɬɶ ɱɚɫɵ, ɚ ɬɚɤɠɟ ɢɯ ɤɨɨɪɞɢɧɚɬɵ "ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ" ɧɚɛɥɸɞɚɬɟɥɟɣ, ɧɚɯɨɞɹɳɢɯɫɹ ɤɚɤ ɜ ɫɢɫɬɟɦɟ S, ɬɚɤ ɢ ɜ ɫɢɫɬɟɦɟ S'. ɂɡɨɛɪɚɡɢɬɶ ɪɚɫɩɨɥɨɠɟɧɢɟ ɱɚɫɨɜ ɨɛɟɢɯ ɫɢɫɬɟɦ ɢ ɩɪɢɦɟɪɧɨɟ ɩɨɥɨɠɟɧɢɟ
160 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
ɫɬɪɟɥɨɤ ɷɬɢɯ ɱɚɫɨɜ ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɪɚɡɥɢɱɧɵɯ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ.
Ɋɟɲɟɧɢɟ
I. ɉɭɫɬɶ ɫɨɛɵɬɢɟ Aj (j =1, 2, 3, 4, 5) ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ tA j 0 ɩɨ ɱɚɫɚɦ ɫɢɫɬɟɦɵ S ɮɢɤɫɢɪɭɟɬɫɹ ɩɨɤɚɡɚ-
ɧɢɹ j-ɵɯ ɱɚɫɨɜ, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɜ ɫɢɫɬɟɦɟ S' ɜ ɬɨɱɤɟ ɫ ɤɨɨɪɞɢɧɚɬɨɣ xAc j (ɩɪɢɧɢɦɚɸɳɟɣ ɡɧɚɱɟɧɢɹ –200 ɦ, –100 ɦ, 0 ɦ, 100 ɦ ɢ 200 ɦ ɞɥɹ
ɪɚɡɧɵɯ ɱɚɫɨɜ) ɜ ɫɢɫɬɟɦɟ S'. ɋɨɛɵɬɢɟ Bk (k =1, 2, 3, 4, 5) – ɮɢɤɫɚɰɢɹ ɩɨɤɚɡɚɧɢɹ k-ɵɯ ɱɚɫɨɜ ɫɢɫɬɟɦɵ S, ɢɦɟɸɳɢɯ ɤɨɨɪɞɢɧɚɬɭ xBk
(–200 ɦ, –100 ɦ, 0 ɦ, 100 ɦ ɢ 200 ɦ) ɜ ɫɢɫɬɟɦɟ S ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
c |
0 ɩɨ ɱɚɫɚɦ ɫɢɫɬɟɦɵ S'. Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ, ɱɬɨ ɜɫɟ ɫɨ- |
tBk |
ɛɵɬɢɹ Aj ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɜ ɫɢɫɬɟɦɟ S. ɂ ɧɚɨɛɨɪɨɬ, ɜɫɟ ɫɨɛɵɬɢɹ Bj ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɜ ɫɢɫɬɟɦɟ S'.
II.ɉɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɫɨɛɵɬɢɣ Aj ɢ Bk
ɜɫɢɫɬɟɦɚɯ S ɢ S' ɫɜɹɡɚɧɵ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ʌɨɪɟɧɰɚ (ɫɦ. (5.3)),
ɩɨɥɭɱɟɧɧɵɦɢ ɫ ɭɱɟɬɨɦ tA j |
c |
0 : |
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0 ɢ tBk |
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c |
JxA j |
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V |
JxA j ; |
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(5.32) |
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c2 |
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xA j |
, tA j |
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xBk |
c |
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V |
c |
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(5.33) |
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JxBk |
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c2 JxBk . |
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Ɂɞɟɫɶ |
c |
– ɩɨɤɚɡɚɧɢɹ ɱɚɫɨɜ ɫɢɫɬɟɦɵ S', xA j – ɤɨɨɪɞɢɧɚɬɚ ɱɚɫɨɜ |
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tA j |
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ɫɢɫɬɟɦɵ S' ɜ ɫɢɫɬɟɦɟ S, tBk |
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c |
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– ɩɨɤɚɡɚɧɢɹ ɱɚɫɨɜ ɫɢɫɬɟɦɵ S, xBk – ɤɨ- |
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ɨɪɞɢɧɚɬɚ ɱɚɫɨɜ ɫɢɫɬɟɦɵ S ɜ ɫɢɫɬɟɦɟ S'. |
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ɋɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (5.32), (5.33) ɞɨɩɨɥɧɢɦ ɜɵɪɚɠɟɧɢɟɦ ɞɥɹ |
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Ʌɨɪɟɧɰ-ɮɚɤɬɨɪɚ: |
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J |
1 |
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(5.34) |
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1 V / c 2 |
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III. Ɋɟɲɚɹ ɩɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (5.32) (5.34) ɨɬ- |
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ɧɨɫɢɬɟɥɶɧɨ ɧɟɢɡɜɟɫɬɧɵɯ |
ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɯ |
ɤɨɨɪɞɢɧɚɬ |
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c |
xA j |
ɢ tBk |
c |
) ɢɧɬɟɪɟɫɭɸɳɢɯ ɧɚɫ ɫɨɛɵɬɢɣ A ɢ B ɢ ɦɨɞɭɥɹ |
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( tA j , |
, xBk |
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ɫɤɨɪɨɫɬɢ V ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S' ɨɬɧɨɫɢɬɟɥɶɧɨ S, ɩɨɥɭɱɢɦ: