Механика.Методика решения задач
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ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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Ɋɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: |
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Acos(Z0t M0 ) , |
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(8.229) |
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ɝɞɟ ɭɝɥɨɜɚɹ ɱɚɫɬɨɬɚ Z0 |
ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ ɪɚɜɧɚ |
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Z0 |
2k , |
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(8.230) |
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ɚ ɚɦɩɥɢɬɭɞɚ A ɢ ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ M0 |
ɨɩɪɟɞɟɥɹɸɬɫɹ ɧɚɱɚɥɶɧɵɦɢ ɭɫ- |
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ɥɨɜɢɹɦɢ, ɡɚɞɚɧɧɵɦɢ ɜ ɡɚɞɚɱɟ: |
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l0 , |
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0) l0 , |
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(8.231) |
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(8.232) |
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ȼ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (8.224), (8.227), |
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(8.231) ɢ (8.232) ɩɨɥɭɱɚɟɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɞɥɢɧɵ ɩɪɭɠɢɧɤɢ: |
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l(t) |
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(1 cosZ0t) l0 . |
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(8.233) |
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2k |
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ɇɚ ɪɢɫ. 8.35 ɢɡɨɛɪɚɠɟɧɚ |
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l(t) |
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ɡɚɜɢɫɢɦɨɫɬɶ ɞɥɢɧɵ ɩɪɭɠɢɧɤɢ ɨɬ |
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ɜɪɟɦɟɧɢ l(t). Ʉɚɤ ɜɢɞɢɦ, ɩɪɭ- |
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ɠɢɧɤɚ ɜ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ ɲɚ- l |
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ɪɢɤɨɜ ɧɚɯɨɞɢɬɫɹ |
ɜ |
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ɪɚɫɬɹɧɭɬɨɦ |
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ɫɨɫɬɨɹɧɢɢ, ɩɟɪɢɨɞɢɱɟɫɤɢ |
ɦɟɧɹɹ |
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ɫɜɨɸ ɞɥɢɧɭ ɩɨ ɝɚɪɦɨɧɢɱɟɫɤɨɦɭ |
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l0 |
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ɡɚɤɨɧɭ ɨɬ |
l0 |
(ɞɥɢɧɵ ɧɟɪɚɫɬɹɧɭ- |
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ɬɨɣ |
ɩɪɭɠɢɧɤɢ) |
ɞɨ |
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ɡɧɚɱɟɧɢɹ |
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T0 |
t |
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F . |
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l0 |
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Ɋɢɫ. 8.35 |
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ɋɥɨɠɟɧɢɟ ɭɪɚɜɧɟɧɢɣ (8.222) ɢ (8.223) ɫ ɭɱɟɬɨɦ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɦɚɫɫ (8.225) ɞɚɟɬ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɭɫɤɨɪɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ:
x |
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(8.234) |
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ɰɦ |
2m |
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Ɋɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɭɱɟɬɨɦ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ (8.231), (8.232) ɢɦɟɟɬ ɜɢɞ:
x (t) |
Ft 2 |
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(8.235) |
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ɰɦ |
4m |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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ɉɟɪɟɯɨɞɹ ɨɬ ɩɟɪɟɦɟɧɧɵɯ l(t) ɢ xɰɦ (t) |
ɤ ɤɨɨɪɞɢɧɚɬɚɦ ɲɚɪɢ- |
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ɤɨɜ ɫ ɩɨɦɨɳɶɸ (8.224) ɢ (8.225) ɢɡ (8.233) ɢ (8.235) ɩɨɥɭɱɚɟɦ ɡɚɤɨ- |
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ɧɵ ɞɜɢɠɟɧɢɹ ɲɚɪɢɤɨɜ: |
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t 2 |
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l0 , |
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x1 (t) |
xɰɦ (t) l(t) |
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1 cos(Z0t) |
(8.236) |
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xɰɦ (t) l(t) |
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1 cos(Z0t) l0 . |
(8.237) |
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ɇɚ ɪɢɫ. 8.36 ɢɡɨɛɪɚɠɟ- |
x(t) |
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ɧɵ |
ɡɚɜɢɫɢɦɨɫɬɢ |
ɤɨɨɪɞɢɧɚɬ |
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x2 (t) |
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ɲɚɪɢɤɨɜ ɨɬ ɜɪɟɦɟɧɢ. Ʉɚɤ ɜɢ- |
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ɞɢɦ, ɞɜɢɠɟɧɢɟ ɲɚɪɢɤɨɜ ɹɜɥɹ- |
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ɟɬɫɹ |
ɫɭɩɟɪɩɨɡɢɰɢɟɣ ɪɚɜɧɨɭɫ- |
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x1 (t) |
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ɤɨɪɟɧɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫ ɭɫɤɨ- |
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ɪɟɧɢɟɦ ɰɟɧɬɪɚ ɦɚɫɫ ɫɢɫɬɟɦɵ |
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xɰɦ |
F |
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ɤɨɥɟɛɚɬɟɥɶɧɨɝɨ |
l0 / 2 |
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2m |
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ɞɜɢɠɟɧɢɹ |
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ɫ |
ɱɚɫɬɨɬɨɣ |
0 |
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2k |
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Z0 |
, ɩɪɢ ɷɬɨɦ ɤɨɥɟɛɚ- |
l0 / 2 |
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ɧɢɹ |
ɲɚɪɢɤɨɜ ɩɪɨɢɫɯɨɞɹɬ ɜ |
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ɩɪɨɬɢɜɨɮɚɡɟ. |
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T0 |
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Ɂɚɦɟɬɢɦ, ɱɬɨ, ɟɫɥɢ ɩɪɢ- |
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Ɋɢɫ. 8.36 |
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ɥɨɠɢɬɶ ɫɢɥɭ F ɤ ɡɚɞɧɟɦɭ ɩɨ |
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ɨɬɧɨɲɟɧɢɸ ɤ ɟɟ ɧɚɩɪɚɜɥɟɧɢɸ ɲɚɪɢɤɭ, ɬɨ ɩɪɭɠɢɧɤɚ ɜ ɩɪɨɰɟɫɫɟ |
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ɞɜɢɠɟɧɢɹ ɲɚɪɢɤɨɜ ɧɚɯɨɞɢɬɫɹ ɜ ɫɠɚɬɨɦ ɫɨɫɬɨɹɧɢɢ, ɩɟɪɢɨɞɢɱɟɫɤɢ |
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ɦɟɧɹɹ ɫɜɨɸ ɞɥɢɧɭ ɩɨ ɝɚɪɦɨɧɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ ɨɬ l0 |
ɞɨ ɡɧɚɱɟɧɢɹ |
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l0 |
F . |
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8.4. Ɂɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ |
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Ɂɚɞɚɱɚ 1
ȼ ɛɨɱɤɟ ɫ ɠɢɞɤɨɫɬɶɸ ɩɥɨɬɧɨɫɬɶɸ U ɜ ɜɟɪɬɢɤɚɥɶɧɨɦ ɩɨɥɨɠɟɧɢɢ ɩɥɚɜɚɟɬ ɩɪɨɛɢɪɤɚ ɦɚɫɫɨɣ Ɇ. ȼ ɩɪɨɛɢɪɤɭ ɩɚɞɚɟɬ ɤɭɫɨɱɟɤ ɩɥɚɫɬɢɥɢɧɚ ɦɚɫɫɨɣ ò . ɉɪɨɥɟɬɟɜ ɩɨ ɜɟɪɬɢɤɚɥɢ ɪɚɫɫɬɨɹɧɢɟ h , ɨɧ ɩɪɢɥɢɩɚɟɬ ɤ ɞɧɭ ɩɪɨɛɢɪɤɢ. ɉɪɟɧɟɛɪɟɝɚɹ ɬɪɟɧɢɟɦ, ɧɚɣɬɢ ɚɦɩɥɢɬɭɞɭ
ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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ɤɨɥɟɛɚɧɢɣ ɩɪɨɛɢɪɤɢ, ɟɫɥɢ ɩɥɨɳɚɞɶ ɟɟ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɪɚɜɧɚ
S . |
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Ɉɬɜɟɬ: A |
m § |
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2mh |
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¨ |
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¸ . |
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¨ |
US |
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US © |
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Ɂɚɞɚɱɚ 2
ɇɚ ɬɟɥɟɠɤɟ ɦɚɫɫɨɣ Ɇ ɡɚɤɪɟɩɥɟɧ ɝɨɪɢɡɨɧɬɚɥɶɧɵɣ ɫɬɟɪɠɟɧɶ, ɩɨ ɤɨɬɨɪɨɦɭ ɦɨɠɟɬ ɛɟɡ ɬɪɟɧɢɹ ɫɤɨɥɶɡɢɬɶ ɦɭɮɬɚ ɦɚɫɫɨɣ ɬ. Ⱦɜɟ ɩɪɭɠɢɧɵ, ɧɚɞɟɬɵɟ ɧɚ ɫɬɟɪɠɟɧɶ, ɨɞɧɢɦ ɤɨɧɰɨɦ ɩɪɢɤɪɟɩɥɟɧɵ ɤ ɦɭɮɬɟ, ɚ ɞɪɭɝɢɦ – ɤ ɬɟɥɟɠɤɟ. Ɉɛɳɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɠɟɫɬɤɨɫɬɢ ɩɪɭɠɢɧ ɪɚɜɟɧ k. ȼ ɫɨɫɬɨɹɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɰɟɧɬɪɵ ɦɚɫɫ ɦɭɮɬɵ ɢ ɬɟɥɟɠɤɢ ɧɚɯɨɞɹɬɫɹ ɧɚ ɨɞɧɨɣ ɜɟɪɬɢɤɚɥɢ. Ɇɭɮɬɭ ɫɦɟɳɚɸɬ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɧɚ ɧɟɛɨɥɶɲɨɟ ɪɚɫɫɬɨɹɧɢɟ l ɢ ɨɬɩɭɫɤɚɸɬ ɫ ɧɭɥɟɜɨɣ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ. Ɉɩɪɟɞɟɥɢɬɶ ɱɚɫɬɨɬɭ Z ɢ ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ ɦɭɮɬɵ Aɦ ɢ ɬɟɥɟɠɤɢ Aɬ. Ɍɪɟɧɢɟɦ ɩɪɟɧɟɛɪɟɱɶ.
Ɉɬɜɟɬ: Z |
k(m M ) |
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l |
M |
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l |
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Ɂɚɞɚɱɚ 3
ȼ ɫɩɥɨɲɧɨɦ ɰɢɥɢɧɞɪɟ ɪɚɞɢɭɫɨɦ R ɫɞɟɥɚɧɚ ɰɢɥɢɧɞɪɢɱɟɫɤɚɹ ɩɨɥɨɫɬɶ ɪɚɞɢɭɫɨɦ R/2 ɫ ɨɫɶɸ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɫɟɪɟɞɢɧɭ ɪɚɞɢɭɫɚ ɰɢɥɢɧɞɪɚ ɩɚɪɚɥɥɟɥɶɧɨ ɟɝɨ ɨɫɢ. Ɉɩɪɟɞɟɥɢɬɶ ɩɟɪɢɨɞ ɦɚɥɵɯ ɤɨɥɟɛɚɧɢɣ, ɤɨɬɨɪɵɟ ɜɨɡɧɢɤɧɭɬ, ɟɫɥɢ ɩɨɥɨɠɢɬɶ ɰɢɥɢɧɞɪ ɧɚ ɲɟɪɨɯɨɜɚɬɭɸ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɢ ɜɵɜɟɫɬɢ ɟɝɨ ɢɡ ɫɨɫɬɨɹɧɢɹ ɪɚɜɧɨɜɟɫɢɹ.
Ɉɬɜɟɬ: T |
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29R |
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Ɂɚɞɚɱɚ 4 |
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Ɉɞɧɨɪɨɞɧɵɣ ɫɬɟɪɠɟɧɶ ɦɚɫɫɨɣ m ɫɨɜɟɪɲɚɟɬ ɦɚɥɵɟ ɤɨɥɟɛɚɧɢɹ |
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ɜɨɤɪɭɝ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ |
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ɱɟɪɟɡ ɬɨɱɤɭ Ɉ. ɉɪɚɜɵɣ ɤɨɧɟɰ ɫɬɟɪɠɧɹ |
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ɩɨɞɜɟɲɟɧ ɧɚ ɧɟɜɟɫɨɦɨɣ ɩɪɭɠɢɧɟ ɠɟɫɬ- |
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ɤɨɫɬɶɸ k |
(ɫɦ. ɪɢɫ.). ɇɚɣɬɢ ɩɟɪɢɨɞ ɤɨɥɟ- |
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ɛɚɧɢɣ ɫɬɟɪɠɧɹ, ɟɫɥɢ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨ- |
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ɜɟɫɢɹ ɨɧ ɝɨɪɢɡɨɧɬɚɥɟɧ. |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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Ɉɬɜɟɬ: T 2S |
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Ɂɚɞɚɱɚ 5 |
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ɇɚɣɬɢ ɱɚɫɬɨɬɭ ɦɚɥɵɯ ɤɨɥɟɛɚɧɢɣ ɬɨɧɤɨɝɨ |
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ɨɞɧɨɪɨɞɧɨɝɨ ɫɬɟɪɠɧɹ ɦɚɫɫɨɣ m ɢ ɞɥɢɧɨɣ l ɜɨ- |
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ɤɪɭɝ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱ- |
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ɤɭ Ɉ (ɫɦ. ɪɢɫ.). ɀɟɫɬɤɨɫɬɶ ɩɪɭɠɢɧɵ ɪɚɜɧɚ k , ɟɟ |
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ɦɚɫɫɚ ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɚ. ȼ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨ- |
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ɜɟɫɢɹ ɫɬɟɪɠɟɧɶ ɜɟɪɬɢɤɚɥɟɧ. |
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Ɉɬɜɟɬ: Z |
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Ɂɚɞɚɱɚ 6
ȼ ɫɩɥɨɲɧɨɦ ɲɚɪɟ ɪɚɞɢɭɫɨɦ R ɫɞɟɥɚɧɚ ɲɚɪɨɨɛɪɚɡɧɚɹ ɩɨɥɨɫɬɶ ɪɚɞɢɭɫɨɦ R/2 ɫ ɰɟɧɬɪɨɦ, ɪɚɫɩɨɥɨɠɟɧɧɵɦ ɜ ɫɟɪɟɞɢɧɟ ɪɚɞɢɭɫɚ ɲɚɪɚ. Ɉɩɪɟɞɟɥɢɬɶ ɩɟɪɢɨɞ ɦɚɥɵɯ ɤɨɥɟɛɚɧɢɣ, ɤɨɬɨɪɵɟ ɜɨɡɧɢɤɧɭɬ, ɟɫɥɢ ɩɨɥɨɠɢɬɶ ɲɚɪ ɧɚ ɲɟɪɨɯɨɜɚɬɭɸ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɢ ɜɵɜɟɫɬɢ ɟɝɨ ɢɡ ɫɨɫɬɨɹɧɢɹ ɪɚɜɧɨɜɟɫɢɹ.
Ɉɬɜɟɬ: T 2S 177R . 10g
Ɂɚɞɚɱɚ 7
ɇɚɣɬɢ ɞɨɛɪɨɬɧɨɫɬɶ Q ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɞɥɢɧɨɣ l = 50 ɫɦ, ɟɫɥɢ ɡɚ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ IJ = 5,2 ɦɢɧ ɟɝɨ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɭɦɟɧɶɲɢɥɚɫɶ ɜ Ș = 4·104 ɪɚɡ.
Ɉɬɜɟɬ: Q # |
W |
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#1,3 102 . |
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Ɂɚɞɚɱɚ 8
ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɦɨɦɟɧɬɚ ɜɧɟɲɧɢɯ ɫɢɥ M z M 0 cosZt ɬɟɥɨ, ɩɨɞɜɟɲɟɧɧɨɟ ɧɚ ɭɩɪɭɝɨɣ ɧɢɬɢ, ɫɨɜɟɪɲɚɟɬ ɭɫɬɚɧɨɜɢɜɲɢɟɫɹ ɜɵɧɭɠɞɟɧɧɵɟ ɤɪɭɬɢɥɶɧɵɟ ɤɨɥɟɛɚɧɢɹ ɩɨ ɡɚɤɨɧɭ M M0 cos(Zt D) . ɇɚɣɬɢ
ɪɚɛɨɬɭ ɫɢɥ ɬɪɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɨ, ɡɚ ɩɟɪɢɨɞ ɤɨɥɟɛɚɧɢɹ.
Ɉɬɜɟɬ: Ⱥɬɪ SM0 Ɇ 0 sin D .
ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
325 |
Ɂɚɞɚɱɚ 9
ɉɪɢ ɱɚɫɬɨɬɚɯ ɜɵɧɭɠɞɚɸɳɟɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɫɢɥɵ Z1 ɢ Z2
ɚɦɩɥɢɬɭɞɚ ɫɤɨɪɨɫɬɢ ɨɫɰɢɥɥɹɬɨɪɚ ɪɚɜɧɚ ɩɨɥɨɜɢɧɟ ɫɜɨɟɝɨ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɩɪɢ ɪɟɡɨɧɚɧɫɟ. ɇɚɣɬɢ ɱɚɫɬɨɬɭ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɪɟɡɨɧɚɧɫɭ ɫɤɨɪɨɫɬɢ.
Ɉɬɜɟɬ: Zɪ Z1Z2 .
Ɂɚɞɚɱɚ 10
ȼ ɭɫɥɨɜɢɹɯ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɢ ɨɩɪɟɞɟɥɢɬɶ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ ɢ ɱɚɫɬɨɬɭ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ.
Ɉɬɜɟɬ: G |
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(Z Z |
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326 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ȽɅȺȼȺ 9 ȻȿȽɍɓɂȿ ɂ ɋɌɈəɑɂȿ ȼɈɅɇɕ.
ɆɈȾɕ ɂ ɇɈɊɆȺɅɖɇɕȿ ɑȺɋɌɈɌɕ
9.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ
ȼɨɡɦɭɳɟɧɢɟ – ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ ɥɨɤɚɥɶɧɨɟ, ɧɟɪɚɜɧɨɜɟɫɧɨɟ ɞɥɹ ɜɫɟɣ ɫɪɟɞɵ ɢɡɦɟɧɟɧɢɟ ɟɟ ɫɨɫɬɨɹɧɢɹ – ɢɡɦɟɧɟɧɢɟ ɮɢɡɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ (ɫɤɚɥɹɪɧɨɣ – [ (t, r) ɢɥɢ ɜɟɤɬɨɪɧɨɣ – ȟ (t, r) ), ɨɩɢɫɵɜɚɸ-
ɳɟɣ ɷɬɨ ɫɨɫɬɨɹɧɢɟ.
ȼɨɥɧɚ – ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɡɦɭɳɟɧɢɹ ɜ ɩɪɨɫɬɪɚɧɫɬ-
ɜɟ.
ȼɟɤɬɨɪɧɨɟ ɜɨɥɧɨɜɨɟ ɩɨɥɟ ȟ (t, r) – ɜɟɤɬɨɪɧɚɹ ɮɭɧɤɰɢɹ ɜɪɟ-
ɦɟɧɢ t ɢ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɬɨɱɤɢ ɧɚɛɥɸɞɟɧɢɹ r , ɨɩɢɫɵɜɚɸɳɚɹ ɜɨɡɦɭɳɟɧɢɟ ɫɪɟɞɵ, ɜ ɤɨɬɨɪɨɣ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɜɨɥɧɚ.
ɋɤɚɥɹɪɧɨɟ ɜɨɥɧɨɜɨɟ ɩɨɥɟ [ (t, r) – ɫɤɚɥɹɪɧɚɹ ɮɭɧɤɰɢɹ ɜɪɟ-
ɦɟɧɢ t ɢ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɬɨɱɤɢ ɧɚɛɥɸɞɟɧɢɹ r , ɨɩɢɫɵɜɚɸɳɚɹ ɜɨɡɦɭɳɟɧɢɟ ɫɪɟɞɵ, ɜ ɤɨɬɨɪɨɣ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɜɨɥɧɚ.
ɋɤɨɪɨɫɬɶ ɜɨɥɧɵ – ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɡɦɭɳɟɧɢɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ.
ɉɪɨɞɨɥɶɧɵɟ ɢ ɩɨɩɟɪɟɱɧɵɟ ɜɨɥɧɵ – ɜɨɥɧɵ, ɜ ɤɨɬɨɪɵɯ ɜɟɤ-
ɬɨɪɧɨɟ ɜɨɥɧɨɜɨɟ ɩɨɥɟ ȟ (t, r) ɧɚɩɪɚɜɥɟɧɨ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜɞɨɥɶ ɢɥɢ
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɧɚɩɪɚɜɥɟɧɢɸ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ.
ɍɩɪɭɝɚɹ (ɚɤɭɫɬɢɱɟɫɤɚɹ) ɜɨɥɧɚ – ɜɨɥɧɚ ɭɩɪɭɝɢɯ ɞɟɮɨɪɦɚ-
ɰɢɣ (ɧɚɩɪɹɠɟɧɢɣ, ɞɚɜɥɟɧɢɣ, ɫɦɟɳɟɧɢɣ ɱɚɫɬɢɰ, ɚ ɬɚɤɠɟ ɢɯ ɫɤɨɪɨɫɬɟɣ ɢ ɭɫɤɨɪɟɧɢɣ) ɜ ɫɪɟɞɟ. ɋɤɨɪɨɫɬɶ ɭɩɪɭɝɨɣ ɜɨɥɧɵ, ɤɚɤ ɩɪɚɜɢɥɨ, ɡɧɚɱɢɬɟɥɶɧɨ ɛɨɥɶɲɟ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɜ ɫɪɟɞɟ.
9.1.1. ȼɨɥɧɨɜɨɟ ɭɪɚɜɧɟɧɢɟ
ȼɟɤɬɨɪɧɨɟ ɜɨɥɧɨɜɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɥɢɧɟɣɧɨɣ, ɢɡɨɬɪɨɩɧɨɣ ɢ ɨɞɧɨɪɨɞɧɨɣ ɫɪɟɞɵ:
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ǻȟ (t, r) , |
(9.1) |
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ɫ – ɫɤɨɪɨɫɬɶ ɜɨɥɧɵ, |
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ǻ |
w2 |
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– ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ Ʌɚɩɥɚɫɚ. |
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Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
327 |
ɋɤɚɥɹɪɧɨɟ ɜɨɥɧɨɜɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɥɢɧɟɣɧɨɣ, ɢɡɨɬɪɨɩɧɨɣ ɢ ɨɞɧɨɪɨɞɧɨɣ ɫɪɟɞɵ:
w2[(t, r) |
c2ǻ[(t, r) , |
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ɝɞɟ [(t, r) – ɫɤɚɥɹɪɧɨɟ ɜɨɥɧɨɜɨɟ ɩɨɥɟ (ɜ ɬɨɦ ɱɢɫɥɟ ɢ ɩɪɨɟɤɰɢɹ ɜɟɤ-
ɬɨɪɧɨɝɨ ɜɨɥɧɨɜɨɝɨ ɩɨɥɹ ɧɚ ɨɫɶ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ).
Ɂɚɤɨɧ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ (ɭɪɚɜɧɟɧɢɟ) ɜɨɥɧɵ – ɪɟɲɟɧɢɟ ɜɨɥ-
ɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ (9.1) ɢɥɢ (9.2).
ɉɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ ɜɨɥɧɨɜɵɯ ɩɨɥɟɣ – ɜɨɥɧɨɜɨɟ (ɜɟɤ-
ɬɨɪɧɨɟ ɢɥɢ ɫɤɚɥɹɪɧɨɟ) ɩɨɥɟ ɞɥɹ ɫɨɜɨɤɭɩɧɨɫɬɢ ɜɨɥɧ ɪɚɜɧɨ ɫɭɦɦɟ
ɜɨɥɧɨɜɵɯ ɩɨɥɟɣ ɞɥɹ ɤɚɠɞɨɣ ɜɨɥɧɵ ɜ ɨɬɞɟɥɶɧɨɫɬɢ: |
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ȟ |
t, r |
¦ȟi t, r , [ t, r |
¦[i t, r . |
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ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ, ɟɫɥɢ ɩɨɥɹ |
ȟi (t, r) (ɢɥɢ |
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[i (t, r) ) ɹɜɥɹɸɬɫɹ ɪɟɲɟɧɢɹɦɢ ɥɢɧɟɣɧɨɝɨ ɢ ɨɞɧɨɪɨɞɧɨɝɨ ɜɨɥɧɨɜɨɝɨ
ɭɪɚɜɧɟɧɢɹ (9.1) (ɢɥɢ (9.2)), ɨɩɢɫɵɜɚɸɳɟɝɨ ɜɨɥɧɨɜɨɣ ɩɪɨɰɟɫɫ, ɬɨ ɟɝɨ ɪɟɲɟɧɢɟɦ ɹɜɥɹɟɬɫɹ ɢ ɢɯ ɥɸɛɚɹ ɥɢɧɟɣɧɚɹ ɤɨɦɛɢɧɚɰɢɹ.
ɂɡ ɩɪɢɧɰɢɩɚ ɫɭɩɟɪɩɨɡɢɰɢɢ ɫɥɟɞɭɟɬ, ɱɬɨ ɪɚɡɥɢɱɧɵɟ ɜɨɥɧɵ ɪɚɫɩɪɨɫɬɪɚɧɹɸɬɫɹ ɧɟɡɚɜɢɫɢɦɨ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɜ ɥɢɧɟɣɧɨɣ, ɢɡɨɬɪɨɩɧɨɣ ɢ ɨɞɧɨɪɨɞɧɨɣ ɫɪɟɞɟ. Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ – ɜɫɹɤɨɟ ɜɨɡɦɭɳɟɧɢɟ, ɫɭɳɟɫɬɜɭɸɳɟɟ ɜ ɥɢɧɟɣɧɨɣ, ɢɡɨɬɪɨɩɧɨɣ ɢ ɨɞɧɨɪɨɞɧɨɣ ɫɪɟɞɟ, ɧɟ ɜɥɢɹɟɬ ɧɚ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɞɪɭɝɨɝɨ ɜɨɡɦɭɳɟɧɢɹ.
ȼɨɥɧɨɜɨɣ ɮɪɨɧɬ ɩɨɜɟɪɯɧɨɫɬɶ, ɨɛɪɚɡɨɜɚɧɧɚɹ ɬɨɱɤɚɦɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɫ ɨɞɢɧɚɤɨɜɵɦ ɜɨɡɦɭɳɟɧɢɟɦ (ɡɧɚɱɟɧɢɟɦ ɜɟɤɬɨɪɧɨɝɨ ȟ (t, r) ɢɥɢ ɫɤɚɥɹɪɧɨɝɨ [(t, r) ɜɨɥɧɨɜɨɝɨ ɩɨɥɹ).
9.1.2. ɉɥɨɫɤɢɟ ɜɨɥɧɵ
ɉɭɫɬɶ ɫɤɚɥɹɪɧɨɟ ɜɨɥɧɨɜɨɟ ɩɨɥɟ [(t, r) ɡɚɜɢɫɢɬ ɥɢɲɶ ɨɬ ɨɞɧɨɣ
ɢɡ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬ, ɧɚɩɪɢɦɟɪ, ɤɨɨɪɞɢɧɚɬɵ x ɜɞɨɥɶ ɨɫɢ X [(t, x) . ɉɪɢ ɷɬɨɦ ɜɨɥɧɨɜɵɦ ɮɪɨɧɬɨɦ ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɨɫɬɶ, ɜ ɫɪɟɞɟ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɩɥɨɫɤɚɹ ɜɨɥɧɚ.
ȼɨɥɧɨɜɨɟ ɭɪɚɜɧɟɧɢɟ ɩɥɨɫɤɨɣ ɜɨɥɧɵ: |
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w2[(t, x) |
c2 w2[(t, x) |
ɢɥɢ [ (t, x) c2[x'' (t, x) , |
(9.4) |
wt 2 |
wx2 |
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ɝɞɟ ɫ – ɫɤɨɪɨɫɬɶ ɜɨɥɧɵ.
Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
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ɝɞɟ [0 – ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ ɜɨɥɧɨɜɨɝɨ ɩɨɥɹ ɜ ɬɨɱɤɚɯ ɩɥɨɫɤɨɫɬɢ YZ ɫ ɤɨɨɪɞɢɧɚɬɨɣ x = 0, Z – ɭɝɥɨɜɚɹ ɱɚɫɬɨɬɚ, ɫɜɹɡɚɧɧɚɹ ɫ ɱɚɫɬɨɬɨɣ
ɤɨɥɟɛɚɧɢɣ Q ɢ ɩɟɪɢɨɞɨɦ T |
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ɫɨɨɬɧɨɲɟɧɢɹɦɢ Z 2SQ |
2S |
; M0 – |
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T |
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ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ ɤɨɥɟɛɚɧɢɣ (ɜ ɬɨɱɤɟ x = 0 ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t = 0).
ȼɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɛɳɢɦ ɪɟɲɟɧɢɟɦ ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ (9.5)
ɜɨɛɥɚɫɬɢ x t 0 ɛɭɞɟɬ ɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ ɩɥɨɫɤɚɹ ɝɚɪɦɨɧɢɱɟɫɤɚɹ ɜɨɥɧɚ.
Ɂɚɤɨɧ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɩɥɨɫɤɨɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɥɧɵ:
[ (t, x) |
[0 cos Z(t x / c) M0 [0 cos Zt kx M0 |
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[0 cos Zt M(x) [0 cos ) (t, x) . |
(9.8) |
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Ɂɞɟɫɶ k |
Z |
2SQ |
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2S |
– ɜɨɥɧɨɜɨɟ ɱɢɫɥɨ, O – ɞɥɢɧɚ ɜɨɥɧɵ, kx – |
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ɧɚɛɟɝ ɮɚɡɵ (ɢɥɢ ɮɚɡɨɜɵɣ ɫɞɜɢɝ) ɩɪɢ ɩɪɨɯɨɠɞɟɧɢɢ ɜɨɥɧɨɣ ɪɚɫ-
ɫɬɨɹɧɢɹ x, ) (t, x) Zt kx M0 – (ɩɨɥɧɚɹ) ɮɚɡɚ.
Ⱦɥɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɨɩɪɟɞɟɥɹɟɦɨɣ ɪɚɞɢɭɫ-
ɜɟɤɬɨɪɨɦ r , ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ: |
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[ t, r |
[0 cos Zt kn r M0 |
[0 cos Zt k r M0 , |
(9.9) |
ɝɞɟ k { kn – ɜɨɥɧɨɜɨɣ ɜɟɤɬɨɪ, ɪɚɜɧɵɣ ɩɨ ɦɨɞɭɥɸ ɜɨɥɧɨɜɨɦɭ ɱɢɫɥɭ k ɢ ɧɚɩɪɚɜɥɟɧɧɵɣ ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ (ɜɨɥɧɨɜɨɝɨ ɮɪɨɧɬɚ).
Ɏɚɡɨɜɵɣ ɮɪɨɧɬ (ɜɨɥɧɨɜɨɣ ɮɪɨɧɬ ɞɥɹ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɥɧɵ)
– ɩɨɜɟɪɯɧɨɫɬɶ, ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɤɨɬɨɪɨɣ ɜɨɥɧɨɜɨɟ ɜɨɡɦɭɳɟɧɢɟ [ t, r
ɢɦɟɟɬ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɨɞɢɧɚɤɨɜɭɸ ɮɚɡɭ ).
ȼ ɫɥɭɱɚɟ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɝɨ ɡɚɬɭɯɚɧɢɹ ɩɥɨɫɤɨɣ ɝɚɪɦɨɧɢ-
ɱɟɫɤɨɣ ɜɨɥɧɵ ɡɚɤɨɧ ɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɡɚɩɢɲɟɬɫɹ ɜ ɜɢɞɟ:
[ t, r [0e Gr cos Zt k r M0 , |
(9.10) |
ɝɞɟ G – ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ.
9.1.3. ɋɮɟɪɢɱɟɫɤɢɟ ɜɨɥɧɵ
ɉɭɫɬɶ ɫɤɚɥɹɪɧɨɟ ɜɨɥɧɨɜɨɟ ɩɨɥɟ [(t, r) ɡɚɜɢɫɢɬ ɥɢɲɶ ɨɬ ɨɞɧɨɣ
ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɩɟɪɟɦɟɧɧɨɣ – ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɬɨɱɤɢ ɧɚɛɥɸɞɟɧɢɹ ɞɨ ɨɩɪɟɞɟɥɟɧɧɨɣ ɬɨɱɤɢ ɩɪɨɫɬɪɚɧɫɬɜɚ S. ɉɪɢ ɷɬɨɦ ɜɨɥɧɨɜɵɦ ɮɪɨɧ-
330 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɬɨɦ ɹɜɥɹɟɬɫɹ ɫɮɟɪɚ ɫ ɰɟɧɬɪɨɦ ɜ ɬɨɱɤɟ S, ɜ ɫɪɟɞɟ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ
ɫɮɟɪɢɱɟɫɤɚɹ ɜɨɥɧɚ.
Ɂɚɤɨɧ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɫɮɟɪɢɱɟɫɤɨɣ ɜɨɥɧɵ (ɨɛɳɟɟ ɪɟɲɟ-
ɧɢɟ ɫɤɚɥɹɪɧɨɝɨ ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ (9.2) ɩɪɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɹɯ):
[ (t, r) |
[1 (t r / c) [2 (t r / c) , |
(9.11) |
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ɝɞɟ [1(t r / c) |
ɢ [2 (t r / c) – ɩɪɨɢɡɜɨɥɶɧɵɟ ɞɜɚɠɞɵ ɞɢɮɮɟɪɟɧɰɢ- |
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ɪɭɟɦɵɟ ɫɤɚɥɹɪɧɵɟ ɮɭɧɤɰɢɢ, |
r { x2 y2 z 2 |
– ɦɨɞɭɥɶ ɪɚɞɢɭɫ- |
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ɜɟɤɬɨɪɚ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ ɩɪɨɫɬɪɚɧɫɬɜɚ r ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ S, ɧɚɯɨɞɹɳɟɣɫɹ ɜ ɧɚɱɚɥɟ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ (ɫɦ. ɪɢɫ. 9.2).
Z r0
S 
XY
Ɋɢɫ. 9.2. ɉɨɥɨɠɟɧɢɟ ɜɨɥɧɨɜɵɯ ɮɪɨɧɬɨɜ ɫ ɪɚɞɢɭɫɚɦɢ r0 ɢ r ɜ ɫɥɭɱɚɟ ɫɮɟɪɢɱɟɫɤɨɣ ɜɨɥɧɵ
Ⱦɚɧɧɨɟ ɪɟɲɟɧɢɟ ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɨɩɢɫɵɜɚɟɬ ɫɭɩɟɪɩɨɡɢ-
ɰɢɸ ɞɜɭɯ ɜɨɥɧ. ɉɟɪɜɨɟ ɢɡ ɫɥɚɝɚɟɦɵɯ |
[1 (t |
r / c) |
– ɪɚɫɯɨɞɹɳɭɸɫɹ |
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ɨɬ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɫɮɟɪɢɱɟɫɤɭɸ ɜɨɥɧɭ, |
ɚ ɜɬɨɪɨɟ |
– |
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ɫɮɟɪɢɱɟɫɤɭɸ ɜɨɥɧɭ, ɫɯɨɞɹɳɭɸɫɹ ɤ ɧɚɱɚɥɭ ɤɨɨɪɞɢɧɚɬ. |
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ɉɭɫɬɶ ɧɚ ɫɮɟɪɟ ɪɚɞɢɭɫɚ r0 ɡɚɞɚɧɨ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ ɜ ɜɢɞɟ |
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ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɜɨɡɦɭɳɟɧɢɹ, ɫɢɧɮɚɡɧɨɝɨ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɫɮɟɪɵ: |
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[(t, r0 ) [0 (r0 ) cos Z(t r0 / c) M0 |
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(9.12) |
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Ɍɨɝɞɚ ɜ ɨɛɥɚɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜɧɟ ɫɮɟɪɵ ɪɚɞɢɭɫɚ r0 ɛɭɞɟɬ ɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ ɪɚɫɯɨɞɹɳɚɹɫɹ ɫɮɟɪɢɱɟɫɤɚɹ ɝɚɪɦɨɧɢɱɟɫɤɚɹ ɜɨɥɧɚ.
Ɂɚɤɨɧ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɪɚɫɯɨɞɹɳɟɣɫɹ ɫɮɟɪɢɱɟɫɤɨɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɥɧɵ: