ɆɈɋɄɈȼɋɄɂɃ ȽɈɋɍȾȺɊɋɌȼȿɇɇɕɃ ɍɇɂȼȿɊɋɂɌȿɌ ɢɦɟɧɢ Ɇ.ȼ. ɅɈɆɈɇɈɋɈȼȺ Ɇɟɯɚɧɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɮɚɤɭɥɶɬɟɬ
Ɉ.ȼ. Ⱥɥɟɤɫɚɧɞɪɨɜɚ
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ɆɈɋɄɈȼɋɄɂɃ ȽɈɋɍȾȺɊɋɌȼȿɇɇɕɃ ɍɇɂȼȿɊɋɂɌȿɌ ɢɦɟɧɢ Ɇ.ȼ. ɅɈɆɈɇɈɋɈȼȺ Ɇɟɯɚɧɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɮɚɤɭɥɶɬɟɬ
Ɉ.ȼ. Ⱥɥɟɤɫɚɧɞɪɨɜɚ
ɗɅȿɆȿɇɌȺɊɇɕȿ ɎɍɇɄɐɂɂ ɂ ȺɅȽɈɊɂɌɆɕ ɍɉɊȺȼɅȿɇɂə
Ɇɨɫɤɜɚ 2003 ɝɨɞ
Ⱥɥɟɤɫɚɧɞɪɨɜɚ Ɉ.ȼ. ɗɥɟɦɟɧɬɚɪɧɵɟ ɮɭɧɤɰɢɢ ɢ ɚɥɝɨɪɢɬɦɵ ɭɩɪɚɜɥɟɧɢɹ. ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ. – Ɇ.: ɂɡɞ. ɆȽɍ, 2003 – 38ɫ.
ȼ ɩɨɫɨɛɢɢ ɩɪɢɜɟɞɟɧɵ ɩɪɢɦɟɪɵ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɣ ɞɥɹ ɨɩɢɫɚɧɢɹ ɞɜɢɠɟɧɢɹ ɭɩɪɚɜɥɹɟɦɵɯ ɫɢɫɬɟɦ ɢ ɩɨɫɬɪɨɟɧɢɹ ɚɥɝɨɪɢɬɦɨɜ ɫɬɚɛɢɥɢɡɢɪɭɸɳɢɯ ɭɩɪɚɜɥɟɧɢɣ. Ʉɧɢɝɚ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɟɡɧɨɣ ɞɥɹ ɭɱɢɬɟɥɟɣ ɢ ɭɱɚɳɢɯɫɹ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɤɥɚɫɫɨɜ ɢ ɫɬɭɞɟɧɬɨɜ ɬɟɯɧɢɱɟɫɤɢɯ ɜɭɡɨɜ, ɚ ɬɚɤɠɟ ɞɥɹ ɲɢɪɨɤɨɝɨ ɤɪɭɝɚ ɱɢɬɚɬɟɥɟɣ ɢɧɬɟɪɟɫɭɸɳɢɯɫɹ ɜɨɩɪɨɫɚɦɢ ɭɩɪɚɜɥɹɟɦɵɯ ɞɜɢɠɟɧɢɣ
@ Ⱥɥɟɤɫɚɧɞɪɨɜɚ Ɉ.ȼ., 2003 ɝ. @ Ɇɟɯɚɧɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɆȽɍ, 2003 ɝ.
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ɋɨɞɟɪɠɚɧɢɟ ȼɜɟɞɟɧɢɟ……………………………………………………………..3 § 1. Ɏɨɪɦɭɥɚ ɐɢɨɥɤɨɜɫɤɨɝɨ ɞɥɹ ɭɩɪɚɜɥɹɟɦɨɝɨ ɩɨɥɟɬɚ ɢ ɥɨɝɚɪɢɮɦɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ………………………………….. 5 § 2. Ɍɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ ɢ ɩɪɨɫɬɨɣ ɪɟɡɨɧɚɧɫ…………12 § 3. ɉɚɪɚɦɟɬɪɢɱɟɫɤɢɣ ɪɟɡɨɧɚɧɫ ɢ ɩɚɪɚɦɟɬɪɢɱɟɫɤɚɹ ɫɬɚɛɢɥɢɡɚɰɢɹ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɦɚɹɬɧɢɤɚ……………...21 § 4. ɉɨɤɚɡɚɬɟɥɶɧɚɹ ɮɭɧɤɰɢɹ ɢ ɪɟɥɟɣɧɨɟ ɫɬɚɛɢɥɢɡɢɪɭɸɳɟɟ ɭɩɪɚɜɥɟɧɢɟ ɩɟɪɟɜɟɪɧɭɬɵɦ ɦɚɹɬɧɢɤɨɦ……………………………31 Ɂɚɤɥɸɱɟɧɢɟ…………………………………………………………37 Ʌɢɬɟɪɚɬɭɪɚ………………………………………………………….38
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ȼɜɟɞɟɧɢɟ ɍɩɪɚɜɥɟɧɢɟ ɞɢɧɚɦɢɱɟɫɤɢɦɢ ɫɢɫɬɟɦɚɦɢ – ɧɚɢɛɨɥɟɟ ɪɚɡɜɢɜɚɸɳɚɹɫɹ ɨɛɥɚɫɬɶ ɧɚɭɤɢ, ɢɦɟɸɳɚɹ ɦɧɨɝɨɱɢɫɥɟɧɧɵɟ ɩɪɢɦɟɧɟɧɢɹ ɜ ɬɟɯɧɢɤɟ, ɦɟɞɢɰɢɧɟ, ɷɤɨɧɨɦɢɤɟ, ɫɨɰɢɚɥɶɧɨɣ ɠɢɡɧɢ. ɑɚɫɬɶ ɦɨɥɨɞɵɯ ɥɸɞɟɣ ɨɬɩɭɝɢɜɚɟɬ ɨɛɢɥɢɟ ɫɩɟɰɢɚɥɶɧɵɯ ɬɟɪɦɢɧɨɜ, ɫɥɨɠɧɨɫɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɢ ɜɵɱɢɫɥɢɬɟɥɶɧɵɯ ɦɟɬɨɞɨɜ, ɩɪɢɦɟɧɹɟɦɵɯ ɜ ɬɟɨɪɢɢ ɭɩɪɚɜɥɟɧɢɹ. ɇɚ ɫɚɦɨɦ ɞɟɥɟ ɜ ɪɹɞɟ ɫɥɭɱɚɟɜ ɜɨɡɦɨɠɧɨ, ɢɫɩɨɥɶɡɭɹ ɬɨɥɶɤɨ ɷɥɟɦɟɧɬɚɪɧɵɟ ɮɭɧɤɰɢɢ, ɞɚɬɶ ɨɩɢɫɚɧɢɟ ɫɥɨɠɧɵɯ ɞɜɢɠɟɧɢɣ ɭɩɪɚɜɥɹɟɦɵɯ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɢ ɩɨɫɬɪɨɟɧɢɟ ɚɥɝɨɪɢɬɦɨɜ ɫɬɚɛɢɥɢɡɢɪɭɸɳɢɯ ɭɩɪɚɜɥɟɧɢɣ. ɍɞɚɥɨɫɶ ɥɢ ɷɬɨ ɜ ɞɚɧɧɨɣ ɧɟɛɨɥɶɲɨɣ ɤɧɢɝɟ – ɫɭɞɢɬɶ ɱɢɬɚɬɟɥɹɦ. ɉɪɨɫɶɛɚ ɜɫɟ ɡɚɦɟɱɚɧɢɹ ɢ ɩɪɟɞɥɨɠɟɧɢɹ ɩɪɢɫɵɥɚɬɶ ɩɨ ɚɞɪɟɫɭ: 119 992 Ɇɨɫɤɜɚ, ɆȽɍ, ɦɟɯɚɧɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɮɚɤɭɥɶɬɟɬ, ɤɚɛɢɧɟɬ ɷɥɟɦɟɧɬɚɪɧɨɣ ɦɚɬɟɦɚɬɢɤɢ.
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§ 1. Ɏɨɪɦɭɥɚ ɐɢɨɥɤɨɜɫɤɨɝɨ ɞɥɹ ɭɩɪɚɜɥɹɟɦɨɝɨ ɩɨɥɟɬɚ ɢ ɥɨɝɚɪɢɮɦɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ. ɋɪɟɞɢ ɜɟɥɢɤɢɯ ɞɨɫɬɢɠɟɧɢɣ ɦɢɪɨɜɨɣ ɧɚɭɤɢ ɢ ɬɟɯɧɢɤɢ ɤɨɧɰɚ XIX ɢ XX ɫɬɨɥɟɬɢɣ ɨɞɧɨ ɢɡ ɜɚɠɧɟɣɲɢɯ ɦɟɫɬ ɩɪɢɧɚɞɥɟɠɢɬ ɞɨɫɬɢɠɟɧɢɹɦ ɜ ɨɛɥɚɫɬɢ ɪɚɤɟɬɧɨɣ ɬɟɯɧɢɤɢ. Ɍɟɨɪɟɬɢɱɟɫɤɨɣ ɨɫɧɨɜɨɣ ɢɡɭɱɟɧɢɹ ɪɟɚɤɬɢɜɧɨɝɨ ɞɜɢɠɟɧɢɹ ɹɜɥɹɟɬɫɹ ɦɟɯɚɧɢɤɚ ɬɟɥ ɩɟɪɟɦɟɧɧɨɣ ɦɚɫɫɵ. Ɉɫɧɨɜɨɩɨɥɨɠɧɢɤɨɦ ɪɚɤɟɬɨɞɢɧɚɦɢɤɢ ɹɜɥɹɟɬɫɹ Ʉɨɧɫɬɚɧɬɢɧ ɗɞɭɚɪɞɨɜɢɱ ɐɢɨɥɤɨɜɫɤɢɣ. Ɍɟɨɪɢɹ, ɪɚɡɪɚɛɨɬɚɧɧɚɹ ɢɦ, ɫɬɚɥɚ ɬɨɣ ɧɚɭɱɧɨɣ ɛɚɡɨɣ, ɧɚ ɤɨɬɨɪɨɣ ɫɨɡɞɚɜɚɥɢɫɶ ɩɟɪɜɵɟ ɦɟɠɤɨɧɬɢɧɟɧɬɚɥɶɧɵɟ ɛɚɥɥɢɫɬɢɱɟɫɤɢɟ ɪɚɤɟɬɵ, ɩɟɪɜɵɟ ɢɫɤɭɫɫɬɜɟɧɧɵɟ ɫɩɭɬɧɢɤɢ Ɂɟɦɥɢ ɢ ɩɟɪɜɵɟ ɩɢɥɨɬɢɪɭɟɦɵɟ ɤɨɫɦɢɱɟɫɤɢɟ ɤɨɪɚɛɥɢ. ɐɢɨɥɤɨɜɫɤɢɣ ɛɵɥ ɩɟɪɜɵɦ ɜ ɢɫɬɨɪɢɢ ɧɚɭɤɢ, ɤɬɨ ɫɬɪɨɝɨ ɫɮɨɪɦɭɥɢɪɨɜɚɥ ɢ ɢɫɫɥɟɞɨɜɚɥ ɩɪɨɛɥɟɦɭ ɭɩɪɚɜɥɹɟɦɨɝɨ ɞɜɢɠɟɧɢɹ ɪɚɤɟɬ, ɢɫɯɨɞɹ ɢɡ ɡɚɤɨɧɨɜ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ. ɂɦɟɧɧɨ ɫɬɪɨɝɨɟ ɪɚɫɫɦɨɬɪɟɧɢɟ ɪɚɤɟɬɵ ɤɚɤ ɬɟɥɚ ɩɟɪɟɦɟɧɧɨɣ ɦɚɫɫɵ ɹɜɥɹɟɬɫɹ ɞɨɫɬɢɠɟɧɢɟɦ Ʉ. ɗ. ɐɢɨɥɤɨɜɫɤɨɝɨ ɜ ɬɟɨɪɢɢ ɩɨɥɟɬɚ ɪɚɤɟɬ. ɉɪɢ ɢɫɫɥɟɞɨɜɚɧɢɢ ɡɚɤɨɧɨɜ ɞɜɢɠɟɧɢɹ ɪɚɤɟɬ ɐɢɨɥɤɨɜɫɤɢɣ ɢɞɟɬ ɫɬɪɨɝɨ ɧɚɭɱɧɵɦ ɩɭɬɟɦ, ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜɜɨɞɹ ɨɫɧɨɜɧɵɟ ɫɢɥɵ, ɨɬ ɤɨɬɨɪɵɯ ɡɚɜɢɫɢɬ ɞɜɢɠɟɧɢɟ ɪɚɤɟɬɵ. ɋɧɚɱɚɥɚ ɨɧ ɫɬɚɜɢɬ ɩɪɨɫɬɟɣɲɭɸ ɡɚɞɚɱɭ ɨ ɩɪɹɦɨɥɢɧɟɣɧɨɦ ɞɜɢɠɟɧɢɢ ɪɚɤɟɬɵ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɫɢɥɵ ɬɹɠɟɫɬɢ ɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ ɨɬɫɭɬɫɬɜɭɸɬ. ɗɬɭ ɡɚɞɚɱɭ ɫɟɣɱɚɫ ɧɚɡɵɜɚɸɬ ɩɟɪɜɨɣ ɡɚɞɚɱɟɣ ɐɢɨɥɤɨɜɫɤɨɝɨ. Ⱦɜɢɠɟɧɢɟ ɪɚɤɟɬɵ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɨɛɭɫɥɨɜɥɟɧɨ ɬɨɥɶɤɨ ɩɪɨɰɟɫɫɨɦ ɨɬɛɪɚɫɵɜɚɧɢɹ ɱɚɫɬɢɰ ɜɟɳɟɫɬɜɚ ɢɡ ɤɚɦɟɪɵ ɪɟɚɤɬɢɜɧɨɝɨ ɞɜɢɝɚɬɟɥɹ. ɉɪɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɚɯ ɐɢɨɥɤɨɜɫɤɢɣ ɜɜɨɞɢɬ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɨ ɩɨɫɬɨɹɧɫɬɜɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɨɬɛɪɚɫɵɜɚɟɦɵɯ ɱɚɫɬɢɰ. ɗɬɨ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɝɢɩɨɬɟɡɨɣ ɐɢɨɥɤɨɜɫɤɨɝɨ. Ʉɨɧɫɬɚɧɬɢɧ ɗɞɭɚɪɞɨɜɢɱ ɜɟɫɶɦɚ ɩɪɨɫɬɵɦɢ ɪɚɫɫɭɠɞɟɧɢɹɦɢ, ɢɫɩɨɥɶɡɭɸɳɢɦɢ ɬɨɥɶɤɨ ɷɥɟɦɟɧɬɚɪɧɵɟ ɮɭɧɤɰɢɢ ɢ ɢɯ ɫɜɨɣɫɬɜɚ, ɩɨɥɭɱɚɟɬ ɨɫɧɨɜɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɪɚɤɟɬɵ ɜ ɫɪɟɞɟ ɛɟɡ ɞɟɣɫɬɜɢɹ ɜɧɟɲɧɢɯ ɫɢɥ. Ɋɚɫɫɦɨɬɪɢɦ ɛɨɥɟɟ ɩɨɞɪɨɛɧɨ ɷɬɨɬ ɫɥɭɱɚɣ [1]. ɉɭɫɬɶ ɜ ɦɨɦɟɧɬ t 0 ɦɚɫɫɚ ɪɚɤɟɬɵ ɟɫɬɶ M 0 , ɚ ɟɟ ɫɤɨɪɨɫɬɶ v ɪɚɜɧɚ ɧɭɥɸ. ɉɭɫɬɶ ɡɚ ɜɪɟɦɹ dt ɞɜɢɝɚɬɟɥɶ ɪɚɤɟɬɵ ɨɬɛɪɨɫɢɥ ɦɚɫɫɭ dM ɫ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ Vr . ɚ ɪɚɤɟɬɚ ɩɨɥɭɱɢɥɚ ɩɪɢɪɚɳɟɧɢɟ ɫɤɨɪɨɫɬɢ dv . Ɂɚɩɢɲɟɦ ɷɬɢ ɭɫɥɨɜɢɹ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ: t 0 : M M 0 ; Vp 0 t t : M
M 0 dM ; Vɨɬɧ
Vr ; V p
dV .
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ɂɡ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɞɥɹ ɡɚɦɤɧɭɬɵɯ ɦɟɯɚɧɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɢɦɟɟɬ ɦɟɫɬɨ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ. Ⱦɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɫɥɭɱɚɹ ɨɧ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: Vr dM Mdv 0 dM (1) M ɉɪɢɧɹɜ ɝɢɩɨɬɟɡɭ ɐɢɨɥɤɨɜɫɤɨɝɨ Vr const , ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɭɪɚɜɧɟɧɢɟ (1): v Vr ln M C . ɍɱɢɬɵɜɚɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ( t 0; M M 0 ; v 0 ), ɩɨɥɭɱɢɦ: C Vr ln M 0 . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, dv
Vr
M0 . (2) M ɋɤɨɪɨɫɬɶ ɪɚɤɟɬɵ ɞɨɫɬɢɝɧɟɬ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ, ɤɨɝɞɚ ɜɟɫɶ ɡɚɩɚɫ ɬɨɩɥɢɜɚ ɛɭɞɟɬ ɢɡɪɚɫɯɨɞɨɜɚɧ ɢ ɡɧɚɱɟɧɢɟ ɦɚɫɫɵ ɪɚɤɟɬɵ M ɫɬɚɧɟɬ ɦɢɧɢɦɚɥɶɧɵɦ. ȿɫɥɢ ɦɚɫɫɭ ɪɚɤɟɬɵ ɛɟɡ ɬɨɩɥɢɜɚ ɨɛɨɡɧɚɱɢɬɶ ɱɟɪɟɡ M E , ɬɨ ɦɚɤɫɢɦɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɪɚɤɟɬɵ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: v Vr ln
M0 . (3) ME ɗɬɚ ɮɨɪɦɭɥɚ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɮɨɪɦɭɥɵ ɐɢɨɥɤɨɜɫɤɨɝɨ. Ɇɚɫɫɚ ɬɨɩɥɢɜɚ ɜ ɪɚɤɟɬɟ m M 0 M E , ɬɨɝɞɚ ɮɨɪɦɭɥɚ ɐɢɨɥɤɨɜɫɤɨɝɨ ɩɟɪɟɩɢɲɟɬɫɹ ɜ ɜɢɞɟ: M m vmax Vr ln E , vmax Vr ln 1 z , (4) ME vmax
Vr ln
m – ɨɬɧɨɲɟɧɢɟ ɜɟɫɚ ɬɨɩɥɢɜɚ ɤ ɜɟɫɭ ɪɚɤɟɬɵ, ɧɚɡɵɜɚɟɬɫɹ ME ɱɢɫɥɨɦ ɐɢɨɥɤɨɜɫɤɨɝɨ. ɍɱɚɫɬɨɤ ɬɪɚɟɤɬɨɪɢɢ, ɩɪɨɣɞɟɧɧɵɣ ɪɚɤɟɬɨɣ ɡɚ ɜɪɟɦɹ ɪɚɛɨɬɵ ɞɜɢɝɚɬɟɥɹ (ɜɪɟɦɹ, ɤɨɝɞɚ ɩɪɨɢɫɯɨɞɢɬ ɨɬɛɪɚɫɵɜɚɧɢɟ ɱɚɫɬɢɰ) ɧɚɡɵɜɚɟɬɫɹ ɚɤɬɢɜɧɵɦ ɭɱɚɫɬɤɨɦ ɩɨɥɟɬɚ. ɂɡ ɮɨɪɦɭɥɵ (4) ɫɥɟɞɭɟɬ, ɱɬɨ: ɚ) ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɪɚɤɟɬɵ ɜ ɤɨɧɰɟ ɩɪɨɰɟɫɫɚ ɝɨɪɟɧɢɹ (ɜ ɤɨɧɰɟ ɚɤɬɢɜɧɨɝɨ ɭɱɚɫɬɤɚ) ɛɭɞɟɬ ɬɟɦ ɛɨɥɶɲɟ, ɱɟɦ ɛɨɥɶɲɟ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɨɬɛɪɚɫɵɜɚɟɦɵɯ ɱɚɫɬɢɰ; ɛ) ɫɤɨɪɨɫɬɶ ɪɚɤɟɬɵ ɜ ɤɨɧɰɟ ɚɤɬɢɜɧɨɝɨ ɭɱɚɫɬɤɚ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɨɬɧɨɲɟɧɢɹ ɧɚɱɚɥɶɧɨɝɨ ɜɟɫɚ ɪɚɤɟɬɵ ɤ ɜɟɫɭ ɪɚɤɟɬɵ ɜ ɤɨɧɰɟ ɝɨɪɟɧɢɹ (3). Ɂɞɟɫɶ ɭɦɟɫɬɧɨ ɩɪɢɜɟɫɬɢ ɬɟɨɪɟɦɭ ɐɢɨɥɤɨɜɫɤɨɝɨ: ɤɨɝɞɚ ɦɚɫɫɚ ɪɚɤɟɬɵ ɜɦɟɫɬɟ ɫ ɦɚɫɫɨɣ ɬɨɩɥɢɜɚ, ɢɦɟɸɳɢɯɫɹ ɜ ɪɟɚɤɬɢɜɧɨɦ ɩɪɢɛɨɪɟ, ɜɨɡɪɚɫɬɚɟɬ ɜ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ ɫɤɨɪɨɫɬɶ ɪɚɤɟɬɵ ɭɜɟ-
ɝɞɟ z
6
ɥɢɱɢɜɚɟɬɫɹ ɜ ɚɪɢɮɦɟɬɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ. ɗɬɚ ɡɚɜɢɫɢɦɨɫɬɶ ɨɛɭɫɥɨɜɥɟɧɚ ɫɜɨɣɫɬɜɚɦɢ ɮɭɧɤɰɢɢ y ln x . §M · Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɟɫɥɢ ¨ 0 ¸ © M E ¹1 §M · Vr ln ¨ 0 ¸ © M E ¹1 nVr ln b .
ɬɨɝɞɚ vmax 1
vmax 2
§M · b; ¨ 0 ¸ © M E ¹2
§M · 2 b ;... , ¨ 0 ¸ © M E ¹n
Vr ln b , vmax 2
Vr ln b 2
b , n
2Vr ln b ,…,
ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɦɨɠɧɨ ɩɪɢɜɟɫɬɢ ɫɥɟɞɭɸɳɭɸ ɬɚɛɥɢɰɭ ɢ ɝɪɚɮɢɤ ɟɟ ɢɥɥɸɫɬɪɢɪɭɸɳɢɣ (ɪɢɫ.1): Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɦɚɫɫɚ ɪɚɤɟɬɵ
2
4
22
8
2
3
16
2
4
32
2
5
64
2
6
128
2
7
ME m ME M0 ME
Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɪɚɤɟɬɵ
ln 2
a
2a
3a
4a
5a
6a
7a
vmax Vr
vmax Vr
3ln 2 2ln 2 ln 2
1
2
3
4
8
M0 ME
Ɋɢɫ.1.
7
ɂɡ ɬɚɛɥɢɰɵ ɜɢɞɧɨ, ɱɬɨ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɜɟɥɢɱɢɧɵ ɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ ɫɨ ɡɧɚɦɟɧɚɬɟɥɟɦ q
M0 ɜ ɝɟɨɦɟɬɪɢɱɟME
2 , ɜɟɥɢɱɢɧɚ
V p max Vr
ɪɚɫɬɟɬ ɜ
ɚɪɢɮɦɟɬɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ ɫ ɪɚɡɧɨɫɬɶɸ d ln 2 . ɂɡ ɮɨɪɦɭɥɵ ɐɢɨɥɤɨɜɫɤɨɝɨ ɫɥɟɞɭɟɬ ɜɚɠɧɵɣ ɩɪɚɤɬɢɱɟɫɤɢɣ ɪɟɡɭɥɶɬɚɬ: ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɜɨɡɦɨɠɧɨ ɛɨɥɶɲɢɯ ɫɤɨɪɨɫɬɟɣ ɪɚɤɟɬɵ ɜ ɤɨɧɰɟ ɩɪɨɰɟɫɫɚ ɝɨɪɟɧɢɹ ɝɨɪɚɡɞɨ ɜɵɝɨɞɧɟɟ ɢɞɬɢ ɩɨ ɩɭɬɢ ɭɜɟɥɢɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɵɯ ɫɤɨɪɨɫɬɟɣ ɨɬɛɪɚɫɵɜɚɟɦɵɯ ɱɚɫɬɢɰ, ɱɟɦ ɩɨ ɩɭɬɢ ɭɜɟɥɢɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɡɚɩɚɫɚ ɝɨɪɸɱɟɝɨ. Ɍɚɤ, ɧɚɩɪɢɦɟɪ, ɟɫɥɢ ɧɟɨɛɯɨɞɢɦɨ ɭɜɟɥɢɱɢɬɶ ɜ 2 ɪɚɡɚ ɫɤɨɪɨɫɬɶ ɜ ɤɨɧɰɟ ɚɤɬɢɜɧɨɝɨ ɭɱɚɫɬɤɚ ɞɥɹ ɧɟɦɟɰɤɨɣ ɪɚɤɟɬɵ ɎȺɍ-2 (1944 ɝ.), ɢɦɟɸɳɟɣ ɨɬɧɨɲɟɧɢɟ ɧɚɱɚɥɶɧɨɝɨ ɜɟɫɚ ɤ ɜɟɫɭ ɩɭɫɬɨɣ (ɛɟɡ ɝɨɪɸɱɟɝɨ) ɪɚɤɟɬɵ, ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɪɚɜɧɵɦ 3, ɬɨ ɦɨɠɧɨ ɢɞɬɢ ɞɜɭɦɹ ɩɭɬɹɦɢ: 1) ɢɥɢ ɭɜɟɥɢɱɢɬɶ ɨɬɧɨɫɢɬɟɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɢɫɬɟɱɟɧɢɹ ɢɡ ɫɨɩɥɚ ɪɟɚɤɬɢɜɧɨɝɨ ɞɜɢɝɚɬɟɥɹ ɜ 2 ɪɚɡɚ, 2) ɢɥɢ ɭɜɟɥɢɱɢɬɶ ɡɚɩɚɫ ɬɨɩɥɢɜɚ ɧɚɫɬɨɥɶɤɨ, ɱɬɨɛɵ ɨɬɧɨɲɟɧɢɹ ɧɚɱɚɥɶɧɨɝɨ ɜɟɫɚ ɪɚɤɟɬɵ ɛɟɡ ɬɨɩɥɢɜɚ ɛɵɥɨ ɪɚɜɧɵɦ 32 9 . ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɭɜɟɥɢɱɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɵɯ ɫɤɨɪɨɫɬɟɣ ɢɫɬɟɱɟɧɢɹ ɱɚɫɬɢɰ ɬɪɟɛɭɟɬ ɫɨɜɟɪɲɟɧɫɬɜɨɜɚɧɢɹ ɪɟɚɤɬɢɜɧɨɝɨ ɞɜɢɝɚɬɟɥɹ ɢ ɪɚɡɭɦɧɨɝɨ ɜɵɛɨɪɚ ɫɨɫɬɚɜɧɵɯ ɱɚɫɬɟɣ ɩɪɢɦɟɧɹɸɳɢɯɫɹ ɬɨɩɥɢɜ, ɚ ɜɬɨɪɨɣ ɩɭɬɶ, ɫɜɹɡɚɧɧɵɣ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɡɚɩɚɫɚ ɬɨɩɥɢɜɚ, ɬɪɟɛɭɟɬ ɡɧɚɱɢɬɟɥɶɧɨɝɨ ɨɛɥɟɝɱɟɧɢɹ ɤɨɧɫɬɪɭɤɰɢɢ ɤɨɪɩɭɫɚ ɪɚɤɟɬɵ ɢ ɭɦɟɧɶɲɟɧɢɹ ɩɨɥɟɡɧɨɣ ɧɚɝɪɭɡɤɢ. Ɍɟɩɟɪɶ ɪɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ, ɤɨɝɞɚ ɪɚɤɟɬɚ ɞɜɢɠɟɬɫɹ ɜɟɪɬɢɤɚɥɶɧɨ (ɩɨ ɨɫɢ OZ ) ɜ ɨɞɧɨɪɨɞɧɨɦ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ, ɧɟ ɭɱɢɬɵɜɚɹ ɫɢɥ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɫɪɟɞɵ (ɪɢɫ.2). Ɋɟɲɟɧɢɟ ɩɨɥɭɱɚɟɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɫɬɵɦ, ɟɫɥɢ ɩɪɟɞɩɨɥɨɠɢɬɶ, ɱɬɨ ɦɚɫɫɚ ɪɚɤɟɬɵ ɦɟɧɹɟɬɫɹ ɩɨ ɩɨɤɚɡɚɬɟɥɶɧɨɦɭ ɡɚɤɨɧɭ (ɤ ɩɨɤɚɡɚɬɟɥɶɧɨɣ ɮɭɧɤɰɢɢ ɦɵ ɟɳɟ ɜɟɪɧɟɦɫɹ ɜ § 4), ɬ.ɟ. ɧɚ ɚɤɬɢɜɧɨɦ ɭɱɚɫɬɤɟ ɩɨɥɟɬɚ M M 0 eD t , ɝɞɟ M 0 – ɧɚɱɚɥɶɧɚɹ ɦɚɫɫɚ ɪɚɤɟɬɵ ɜɦɟɫɬɟ ɫ ɬɨɩɥɢɜɨɦ, D – ɧɟɤɨɬɨɪɚɹ ɩɨɫɬɨɹɧɧɚɹ ɫ ɪɚɡɦɟɪɧɨɫɬɶɸ c 1 . z ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɆɟɳɟɪɎ ɫɤɨɝɨ, ɨɩɪɟɞɟɥɹɸɳɟɟ ɞɜɢɠɟɧɢɟ ɰɟɧɬɪɚ ɦɚɫɫ ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɫ ɩɟɪɟɦɟɧɧɨɣ ɦɚɫg ɫɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɵɯ ɨɫɟɣ OXYZ [2]. 0 & & & dM dv M FQ Vr , (5) ¦ dt dt Q & Vr ɝɞɟ FQ – ɚɤɬɢɜɧɵɟ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɤɚɠɊɢɫ. 2 ɞɭɸ ɬɨɱɤɭ ɫɢɫɬɟɦɵ. 8
ȼɩɟɪɜɵɟ ɭɪɚɜɧɟɧɢɟ (5) ɛɵɥɨ ɩɨɥɭɱɟɧɨ ɜ 1897 ɝ. Ɇ.ȼ. Ɇɟɳɟɪɫɤɢɦ ɢ ɩɨɷɬɨɦɭ ɧɚɡɵɜɚɟɬɫɹ ɟɝɨ ɢɦɟɧɟɦ. ɂɡɦɟɧɟɧɢɟ ɦɚɫɫɵ ɫɢɫɬɟɦɵ ɩɪɢɜɨɞɢɬ ɤ ɩɨɹɜɥɟɧɢɸ ɞɨɩɨɥɧɢ& dM ɬɟɥɶɧɨɝɨ ɫɥɚɝɚɟɦɨɝɨ Vr ɜ ɭɪɚɜɧɟɧɢɢ (5), ɢɦɟɸɳɟɝɨ ɪɚɡɦɟɪɧɨɫɬɶ dt & & dM ɫɢɥɵ. Ɉɧɨ ɧɚɡɵɜɚɟɬɫɹ ɪɟɚɤɬɢɜɧɨɣ ɫɢɥɨɣ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ: Ɏ Vr . dt Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ Ɇɟɳɟɪɫɤɨɝɨ ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ OZ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: dv dM M Mg Ɏ , ɝɞɟ Ɏ Vr (6) dt dt ɉɪɢɦɟɧɢɦ ɝɢɩɨɬɟɡɭ ɐɢɨɥɤɨɜɫɤɨɝɨ ɨ ɩɨɫɬɨɹɧɫɬɜɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɨɬɛɪɚɫɵɜɚɟɦɵɯ ɱɚɫɬɢɰ, ɬ.ɟ. Vr const . Ɍɚɤ ɤɚɤ M
M 0 eD t , ɬɨ
dM dt
D M 0 e
D t
DM . Ɍɨɝɞɚ Ɏ DVr M ɢ ɭɪɚɜ-
ɧɟɧɢɟ (6) ɩɟɪɟɩɢɲɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: dv M Mg DVr M , dt ɢɥɢ § DV · dv g ¨ r 1¸ . (7) dt © g ¹ ɉɟɪɟɝɪɭɡɤɭ, ɨɛɭɫɥɨɜɥɟɧɧɭɸ ɪɟɚɤɬɢɜɧɨɣ ɫɢɥɨɣ, ɧɚɡɨɜɟɦ ɪɟɚɤɎ DVr ɬɢɜɧɨɣ ɩɟɪɟɝɪɭɡɤɨɣ. ȿɟ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ: n . Ɇg g dv g n 1 . (8) dt ɉɪɢ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ t 0; v 0; z 0 , ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ
Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (7) ɩɪɢɦɟɬ ɜɢɞ: ɭɪɚɜɧɟɧɢɟ (8):
dv
g n 1 dt
1 2 gt n 1 2 ɉɭɫɬɶ ɜ ɤɨɧɰɟ ɚɤɬɢɜɧɨɝɨ ɭɱɚɫɬɤɚ ɩɨɥɟɬɚ t v
M
ME
M 0e
D t1
gt n 1 , z
(9), (10) t1 ; z
z1 ; v
v1 ;
. Ɍɨɝɞɚ ɢɡ ɭɪɚɜɧɟɧɢɣ (9) ɢ (10) ɩɨɥɭɱɢɦ:
9
1 2 gt1 n 1 (11), (12) 2 ɉɚɫɫɢɜɧɵɣ ɭɱɚɫɬɨɤ ɪɚɤɟɬɚ ɩɪɨɯɨɞɢɬ ɤɚɤ ɬɟɥɨ ɩɨɫɬɨɹɧɧɨɣ ɦɚɫɫɵ. ɇɚ ɷɬɨɦ ɭɱɚɫɬɤɟ ɧɚɣɞɟɦ ɞɨɩɨɥɧɢɬɟɥɶɧɭɸ ɜɵɫɨɬɭ z p z 2 z1 , ɧɚɛɢɪɚɟɦɭɸ ɡɚ ɫɱɟɬ ɫɤɨɪɨɫɬɢ
z
v1
z2 v
0
v1 . ȼ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ OZ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɪɚɤɟɬɵ ɢɦɟɟɬ ɜɢɞ: dv ME M E g dt dv dz dv g v g dz dz dt
ME z1
v
gt1 n 1 , z1
v1
Ɋɢɫ. 3 § v2 · d¨ ¸ 2 © 2 ¹ g d v gdz . dz 2 ɂɧɬɟɝɪɢɪɭɹ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɛɭɞɟɦ ɢɦɟɬɶ: v 22 v12 g z2 z1 , ɝɞɟ v2 0; z p z2 z1 . 2 2 Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, v12 1 2 2 zp , ɢɥɢ z p gt1 n 1 (13) 2 2g ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɥɧɚɹ ɜɵɫɨɬɚ ɩɨɞɴɟɦɚ ɪɚɤɟɬɵ ɛɭɞɟɬ ɨɩɪɟ1 2ª 2 gt1 n 1 n 1 º , ɞɟɥɹɬɶɫɹ ɮɨɪɦɭɥɨɣ: H z1 z p ¬ ¼ 2 1 2 2 H gt1 n n (14) 2 ȼɪɟɦɹ t1 ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɩɟɪɟɝɪɭɡɤɭ n . ȼ ɫɚɦɨɦ ɞɟɥɟ,
ME
M 0e
DVr
D t1
, ɢɥɢ
M0 ME
D t1
e
gn g Vr Ʌɨɝɚɪɢɮɦɢɪɭɹ ɜɵɪɚɠɟɧɢɟ (15) ɢ ɢɫɩɨɥɶɡɭɹ (16), ɩɨɥɭɱɢɦ: n
10
, ɢɥɢ D
(15) (16)
M0 M M E Vz t1 , ɝɞɟ Vz Vr ln 0 const . (17) D ME ng ng ɂɡ ɮɨɪɦɭɥɵ (14) ɫ ɭɱɟɬɨɦ (17) ɫɥɟɞɭɟɬ, ɱɬɨ 2 Vz2 § 1 · Vz2 1 Vz 2 H n const . , ɝɞɟ (18) 1 1 ¨ ¸ 2g 2 gn 2 2g © n ¹ ln
M0 ME
Vr ln
Ɇɨɠɧɨ ɡɚɦɟɬɢɬɶ ɫɜɹɡɶ ɦɟɠɞɭ v1 ɢ Vz : ln v1
gt1 n 1
g
M0 ME
D
n 1
Vr ln
M0 ME
§ 1· ¨1 ¸ , ɬ.ɟ. © n¹
§ 1· (19) Vz ¨ 1 ¸ © n¹ M ȼɟɥɢɱɢɧɚ Vz Vr ln 0 ɟɫɬɶ ɫɤɨɪɨɫɬɶ ɩɨ ɐɢɨɥɤɨɜɫɤɨɦɭ, ɜ ME ɫɜɨɛɨɞɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɬ.ɟ. ɦɚɤɫɢɦɚɥɶɧɨ ɜɨɡɦɨɠɧɚɹ ɩɪɢ ɞɚɧɧɨɦ ɡɚɩɚɫɟ ɬɨɩɥɢɜɚ. Ʌɟɝɤɨ ɩɨɧɹɬɶ, ɱɬɨ ɟɫɥɢ ɛɵ ɪɚɤɟɬɚ ɜ ɦɨɦɟɧɬ t 0 ɩɨɥɭɱɢɥɚ ɫɤɨɪɨɫɬɶ v Vz , ɬɨ ɨɧɚ ɩɨɞɧɹɥɚɫɶ ɛɵ ɜ ɨɞɧɨɪɨɞɧɨɦ ɩɨɥɟ ɫɢv1
ɥɵ ɬɹɠɟɫɬɢ ɧɚ ɜɵɫɨɬɭ H max
Vz 2 . ɗɬɚ ɜɵɫɨɬɚ ɩɨɥɭɱɚɟɬɫɹ ɢɡ ɮɨɪɦɭ2g
ɥɵ (18) ɩɪɢ n o f , ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɥɭɱɚɸ t1 0 , ɬ.ɟ. ɦɝɧɨɜɟɧɧɨɦɭ ɫɝɨɪɚɧɢɸ ɜɫɟɝɨ ɢɦɟɸɳɟɝɨɫɹ ɡɚɩɚɫɚ ɬɨɩɥɢɜɚ. ɉɪɢ ɤɨɧɟɱɧɵɯ ɡɧɚɱɟɧɢɹ ɩɟɪɟɝɪɭɡɤɢ n ɜɵɫɨɬɚ ɩɨɞɴɟɦɚ ɪɚɤɟɬɵ ɛɭɞɟɬ ɨɩɪɟɞɟɥɹɬɶɫɹ ɮɨɪɦɭɥɨɣ (18), ɤɨɬɨɪɭɸ ɭɞɨɛɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ: § 1· H H max ¨ 1 ¸ . (20) © n¹ Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɩɨɬɟɪɹ ɜɵɫɨɬɵ ɜ ɩɪɨɰɟɧɬɚɯ ɛɭɞɟɬ ɪɚɜɧɚ H H 100 0 (21) H 0 0 100 max 0 . n H max ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ n f ɩɨɬɟɪɶ ɧɟɬ, ɩɪɢ n 1 , ɬ.ɟ. ɤɨɝɞɚ ɪɟɚɤɬɢɜɧɚɹ ɫɢɥɚ ɪɚɜɧɚ ɫɢɥɟ ɬɹɠɟɫɬɢ, ɜɫɟ ɬɨɩɥɢɜɨ ɪɚɫɯɨɞɭɟɬɫɹ ɛɟɫɩɨɥɟɡɧɨ H 0 ɢ H 100 0 0 .
11
§ 2. Ɍɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ ɢ ɩɪɨɫɬɨɣ ɪɟɡɨɧɚɧɫ Ʉɨɥɟɛɚɧɢɹ, ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɜ ɪɚɡɥɢɱɧɵɯ ɨɛɥɚɫɬɹɯ: ɜ ɦɟɯɚɧɢɤɟ, ɚɤɭɫɬɢɤɟ, ɪɚɞɢɨɬɟɯɧɢɤɟ ɨɬɥɢɱɚɸɬɫɹ ɩɨ ɮɢɡɢɱɟɫɤɨɣ ɩɪɢɪɨɞɟ, ɧɨ ɨɫɧɨɜɧɵɟ ɡɚɤɨɧɵ ɤɨɥɟɛɚɧɢɣ ɨɫɬɚɸɬɫɹ ɨɞɧɢɦɢ ɢ ɬɟɦɟ ɠɟ. Ɋɟɡɭɥɶɬɚɬɵ, ɩɨɥɭɱɟɧɧɵɟ ɩɪɢ ɢɡɭɱɟɧɢɢ ɦɟɯɚɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ, ɢɫɩɨɥɶɡɭɸɬɫɹ ɢ ɞɥɹ ɢɡɭɱɟɧɢɹ ɤɨɥɟɛɚɬɟɥɶɧɵɯ ɹɜɥɟɧɢɣ ɜ ɞɪɭɝɢɯ ɨɛɥɚɫɬɹɯ. ȼ ɦɟɯɚɧɢɤɟ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɨ ɜɨɡɧɢɤɚɸɬ ɡɚɞɚɱɢ ɨ ɪɟɡɨɧɚɧɫɧɵɯ ɹɜɥɟɧɢɹɯ. ȼɨ ɦɧɨɝɢɯ ɢɡ ɧɢɯ ɢɫɫɥɟɞɭɟɦɨɟ ɞɜɢɠɟɧɢɟ ɨɩɢɫɵɜɚɟɬɫɹ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɦɢ ɮɭɧɤɰɢɹɦɢ, ɢ ɜɵɜɨɞɵ ɨ ɯɚɪɚɤɬɟɪɟ ɞɜɢɠɟɧɢɹ ɞɟɥɚɸɬɫɹ ɧɚ ɨɫɧɨɜɚɧɢɢ ɫɜɨɣɫɬɜ ɷɬɢɯ ɮɭɧɤɰɢɣ. Ɋɟɡɨɧɚɧɫɧɵɟ ɹɜɥɟɧɢɹ ɦɨɝɭɬ ɜɨɡɧɢɤɚɬɶ ɜ ɤɨɥɟɛɚɬɟɥɶɧɵɯ ɫɢɫɬɟɦɚɯ (ɞɜɢɠɟɧɢɟ ɬɟɥɚ ɧɚ ɩɪɭɠɢɧɟ, ɪɚɫɤɚɱɢɜɚɧɢɟ ɤɚɱɟɥɟɣ, ɬ.ɞ.). Ɋɚɫɫɦɨɬɪɢɦ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ (ɪɢɫ. 1). Ɇɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ – ɷɬɨ ɝɪɭɡ ɦɚɥɵɯ ɪɚɡɦɟɪɨɜ (ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ɤɚɤ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ), ɩɨɞɜɟɲɟɧɧɵɣ ɧɚ ɧɟɜɟɫɨɦɨɦ ɧɟɪɚɫɬɹɠɢɦɨɦ ɫɬɟɪɠɧɟ ɞɥɢɧɨɣ l . 0
ɉɭɫɬɶ ɦɚɫɫɚ ɝɪɭɡɚ – m . ɇɚ ɝɪɭɡ ɞɟɣɫɬ& & M l ɜɭɟɬ ɫɢɥɚ ɬɹɠɟɫɬɢ P mg , ɧɚɩɪɚɜɥɟɧW ɧɚɹ, ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ. Ɍɨɝɞɚ ɞɜɢɠɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɛɭɞɟɬ ɨɩɢɫɵɜɚɬɶɫɹ ɜɟɤɬɨɪɧɵɦ ɭɪɚɜɧɟɧɢɟɦ & & ɇɶɸɬɨɧɚ: ma mg (1) M ȼ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ W , ɤɚɫɚɬɟɥɶɧɭɸ ɤ & ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɜ ɞɚɧɧɵɣ mg ɦɨɦɟɧɬ, ɭɪɚɜɧɟɧɢɟ (1) ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: Ɋɢɫ.1. mlM mg sin M , (2) ɝɞɟ M – ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɫɬɟɪɠɧɹ ɨɬ ɜɟɪɬɢɤɚɥɢ. Ɍɪɚɟɤɬɨɪɢɟɣ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɨɤɪɭɠɧɨɫɬɶ ɪɚɞɢɭɫɚ l ɫ ɰɟɧɬɪɨɦ ɜ ɬɨɱɤɟ, ɫɨɜɩɚɞɚɸɳɟɣ ɫ ɡɚɤɪɟɩɥɟɧɧɵɦ ɤɨɧɰɨɦ ɫɬɟɪɠɧɹ. ɋɤɨɪɨɫɬɶ ɝɪɭɡɚ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɫɥɭɱɚɟ ɪɚɜɧɚ v lM , ɭɫɤɨɪɟɧɢɟ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ W – aW lM . ɍɪɚɜɧɟɧɢɟ (2) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ: lM g sin M . Ⱦɥɹ ɧɟɛɨɥɶɲɢɯ ɨɬɤɥɨɧɟɧɢɣ, ɬ.ɟ. ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɯ ɡɧɚɱɟɧɢɹɯ ɭɝɥɚ M ɭɪɚɜɧɟɧɢɟ (2) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ: 12
g 0 , ɬɚɤ ɤɚɤ ɡɧɚɱɟɧɢɹ sin M ɛɭɞɭɬ ɞɨɫɬɚɬɨɱɧɨ ɛɥɢɡɤɢ ɤ l ɡɧɚɱɟɧɢɹɦ M .
M M
g , ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɟɟ ɭɪɚɜɧɟɧɢɟ: l M Z 02M 0 .
2
Ɉɛɨɡɧɚɱɢɜ Z 0
ɉɭɫɬɶ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ t
(3)
0 : M 0 M 0 ! 0 – ɭɝɨɥ, ɧɚ ɤɨɬɨɪɵɣ
0 , ɬ.ɟ. ɫɤɨɪɨɫɬɶ ɦɚɹɬɧɢɤɚ ɪɚɜɧɚ ɧɭɥɸ. Ɏɭɧɤ-
ɨɬɤɥɨɧɟɧ ɝɪɭɡ, M 0
ɰɢɹ M t M 0 cos Z 0 t ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɷɬɢɦ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ ɢ ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (3) (ɪɢɫ. 2). Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, M t Z 0M 0 sin Z t M t Z 02M 0 cos Z t
M Z 02M
Z 02M 0 cos Z 0 t Z 02M 0 cos Z 0 t
ɑɟɪɟɡ ɜɪɟɦɹ T
2S
t1
Z0
0 0{0.
ɦɚɹɬɧɢɤ ɜɧɨɜɶ ɨɬɤɥɨɧɢɬɫɹ ɧɚ ɭɝɨɥ M 0 :
M
M0
a ccc M + t / cccc Z 0 sin Z 0t M+ t/
M 0 cos Z 0t
t
Ɋɢɫ. 2. §
M t M 0 cos ¨ Z 0 ©
2S · ¸ M 0 . ȼɟɥɢɱɢɧɚ T Z0 ¹
ɤɨɥɟɛɚɧɢɣ, M t T M t . ȼɟɥɢɱɢɧɚ Z 0
2S
Z0
ɧɚɡɵɜɚɟɬɫɹ ɩɟɪɢɨɞɨɦ 2S T
ɧɚɡɵɜɚɟɬɫɹ ɫɨɛɫɬ-
ɜɟɧɧɨɣ ɱɚɫɬɨɬɨɣ ɤɨɥɟɛɚɧɢɣ. Ɋɚɫɫɦɨɬɪɢɦ ɝɟɨɦɟɬɪɢɱɟɫɤɭɸ ɢɧɬɟɪɩɪɟɬɚɰɢɸ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ (3) ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ M , M . Ɍɚɤ ɤɚɤ M 2
M Z0 M
2
M 0 cos Z 0 t , M 2
2
Z 0M 0 sin Z 0 t , ɬɨ
2
M 0 Z 0 cos Z 0 t M 0 2Z 0 2 sin 2 Z 0 t M 0 2Z 0 2 , ɢɥɢ
13
2 M2 1 M M02 Z02 M0 2
1
(4)
Ʉɪɢɜɚɹ, ɨɩɢɫɵɜɚɟɦɚɹ ɭɪɚɜɧɟɧɢɟɦ (4) ɹɜɥɹɟɬɫɹ ɷɥɥɢɩɫɨɦ ɫ ɩɨɥɭɨɫɹɦɢ M 0 , Z 0M 0 (ɪɢɫ.3). M ɑɟɪɟɡ ɜɪɟɦɹ T ɦɵ ɛɭɞɟɦ ɩɨɩɚɞɚɬɶ
Z 0M 0 M0
ɜ ɨɞɧɭ ɢ ɬɭ ɠɟ ɬɨɱɤɭ ɧɚ ɷɥɥɢɩɫɟ, ɩɨɥɭɱɟɧɧɨɦ ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ. - Ɇɨɠɧɨ ɡɚɦɟɬɢɬɶ, ɬɪɚɟɤɬɨɪɢɹ, ɨɩɢM ɫɵɜɚɟɦɚɹ ɩɟɪɢɨɞɢɱɟɫɤɢɦ ɪɟɲɟɧɢɟɦ M t M 0 cos Z 0 t
ɧɚ ɮɚɡɨɜɨɣ
ɩɥɨɫɤɨɫɬɢ, ɹɜɥɹɟɬɫɹ ɡɚɦɤɧɭɬɨɣ ɬɪɚɊɢɫ.3 ɟɤɬɨɪɢɟɣ. ȼ ɫɥɭɱɚɟ ɟɫɥɢ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɢɦɟɸɬ ɜɢɞ: t 0 , M 0 ɰɢɹ ɜɢɞɚ M t
0; M 0
a
Z0
a , ɬɨ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (3) ɛɭɞɟɬ ɮɭɧɤ-
sin Z 0 t (ɪɢɫ. 2). ɗɬɨ ɪɟɲɟɧɢɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɡɚ-
ɞɚɧɧɵɦ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ. Ɉɛɳɢɣ ɜɢɞ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (3) ɫɥɟɞɭɸɳɢɣ: M t C1 cos Z 0 t C2 sin Z 0 t . ɉɪɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɵɯ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɩɨɥɭɱɚɸɬɫɹ ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɜɵɲɟ ɫɥɭɱɚɢ. Ɍɟɩɟɪɶ ɪɚɫɫɦɨɬɪɢɦ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ, ɤɨɥɟɛɥɸɳɢɣɫɹ ɛɟɡ ɬɪɟɧɢɹ ɢ ɧɚɯɨɞɹɳɢɣɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɧɟɤɨɬɨɪɨɣ ɜɧɟɲɧɟɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɫɢɥɵ. ɇɚɩɪɢɦɟɪ, ɜ ɫɥɭɱɚɟ ɩɟɪɟɦɟɳɟɧɢɹ ɬɨɱɤɢ ɨɩɨɪɵ & ɦɚɹɬɧɢɤɚ ɩɨ ɝɨɪɢɡɨɧɬɚɥɢ ɫ ɭɫɤɨɪɟɧɢɟɦ W ɛɭɞɟɦ ɢɦɟɬɶ ɦɚɫɫɨɜɭɸ & ɫɢɥɭ mW , ɧɚɡɵɜɚɟɦɭɸ ɫɢɥɨɣ ɢɧɟɪɰɢɢ, ɤɨɬɨɪɚɹ ɜɦɟɫɬɟ ɫ ɫɢɥɨɣ
ɬɹɠɟɫɬɢ ɞɟɣɫɬɜɭɟɬ ɧɚ ɦɚɬɟɪɢɚɥɶɧɭɸ ɬɨɱɤɭ (ɪɢɫ. 4). Ɍɨɝɞɚ, ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ W , ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɭɸ ɫɬɟɪɠɧɸ, ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɩɪɢ ɦɚɥɵɯ ɭɝɥɚɯ ɨɬɤɥɨɧɟɧɢɹ sin M | M , cos M | 1 ɩɪɢɨɛɪɟɬɟɬ ɜɢɞ: g l
M M
14
W . l
g W , ɚ ɬɚɤ ɠɟ W1 . l l ɉɨɥɭɱɢɦ ɫɥɟɞɭɸɳɟɟ ɧɟɨɞɧɨɪɨɞɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ: & W M Z 02M W1 (5) 2
Ʉɚɤ ɢ ɪɚɧɟɟ ɨɛɨɡɧɚɱɢɦ Z 0
ȼ ɤɚɱɟɫɬɜɟ ɜɨɡɦɭɳɟɧɢɣ W1 t ɛɭɞɟɦ
M
W
& mW
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɮɭɧɤɰɢɢ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɨɝɪɚɧɢɱɟɧɢɸ W1 t d P . ɉɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (5) ɜ ɜɢɞɟ ɫɢɫɬɟɦɵ: °M t Z . (6) ® 2 °¯Z Z 0 M W1
M & mg
Ɋɢɫ. 4. ɋɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɨɞɧɨɪɨɞɧɚɹ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɜɢɞ: °M t Z . ® 2 °¯Z Z 0 M ȿɟ ɮɭɧɞɚɦɟɧɬɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɪɟɲɟɧɢɣ ɛɵɥɚ ɩɨɥɭɱɟɧɚ ɜɵɲɟ: § · 1 sin Z 0 t , Z cos Z 0 t ¸ . M cos Z0t , Z Z0 sin Z0t , ¨ M Z0 © ¹ Ɍɨɝɞɚ ɥɸɛɨɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɣ ɫɢɫɬɟɦɵ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ 1 °M t C1 cos Z 0 t C2 Z sin Z 0 t . ɜɢɞɟ: 0 ® °Z t C Z sin Z t C cos Z t 1 0 0 2 0 ¯
Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ (6) ɩɪɢɦɟɧɢɦ ɫɬɚɧɞɚɪɬɧɵɣ ɦɟɬɨɞ ɜɚɪɢɚɰɢɢ ɩɨɫɬɨɹɧɧɵɯ. Ⱦɥɹ ɷɬɨɝɨ ɪɚɫɫɦɨɬɪɢɦ C1 ɢ C2 ɤɚɤ ɮɭɧɤɰɢɢ ɜɪɟɦɟɧɢ: 1 °M C1 t cos Z 0 t C2 t Z sin Z 0 t . 0 ® °Z C t Z sin Z t C t cos Z t 1 0 0 2 0 ¯ ɉɨɞɫɬɚɜɥɹɹ M ɢ Z ɜ ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ ɫɢɫɬɟɦɵ (6), ɩɨɥɭɱɢɦ: 1 C1 t cos Z 0 t C 2 t sin Z 0 t C1 t Z 0 sin Z 0 t C2 t cos Z 0 t
Z0
15
C1 t Z 0 sin Z 0 t C2 t cos Z 0 t .
Ⱥɧɚɥɨɝɢɱɧɵɟ ɞɟɣɫɬɜɢɹ ɩɪɨɢɡɜɨɞɢɦ ɫɨ ɜɬɨɪɵɦ ɭɪɚɜɧɟɧɢɟɦ ɫɢɫɬɟɦɵ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɧɫɬɚɧɬ C1 ɢ C2 ɩɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ: 1 °C1 cos Z 0 t C2 Z sin Z 0 t 0 . 0 ® °C Z sin Z t C cos Z t W t 0 2 0 1 ¯ 1 0
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, °C1 ® °C ¯ 2
1
Z0
W1 t sin Z 0 t
.
W1 t cos Z 0 t
ɉɟɪɟɩɢɲɟɦ ɞɚɧɧɭɸ ɫɢɫɬɟɦɭ ɜ ɦɚɬɪɢɱɧɨɦ ɜɢɞɟ &
f t , ɝɞɟ
§ 1 · ¨ Z W1 t sin Z 0 t ¸ 0 ¨ ¸. ¨ W t cos Z t ¸ 1 0 © ¹ Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɪɟɲɚɟɬɫɹ ɤɜɚɞɪɚɬɭɪɨɣ: t § C 0 · & t & t0 ³ f W dW , ɝɞɟ & t0 ¨ 1 ¸. © C2 0 ¹ t0 &
§ C1 · ¨ ¸ ; f t © C2 ¹
ɇɚɣɞɟɦ C1 0 ɢ C2 0 . ȼ ɦɨɦɟɧɬ t
0 : M 0
C1 0 , Z 0
M t C1 t cos Z 0 t C2 t
1
Z0
sin Z 0 t
C2 0 . Ɍɨɝɞɚ t ª º 1 0 M W1 W sin Z 0W » cos Z 0 t « ³ Z0 0 ¬ ¼
ªZ 0 1 t º W1 W cos Z 0W » sin Z 0 t . « ³ Z Z 0 0 ¬ 0 ¼ Ⱦɥɹ ɤɨɨɪɞɢɧɚɬɵ M t ɩɪɢ Z 0 0 , ɩɨɥɭɱɢɦ ɧɭɠɧɭɸ ɞɥɹ ɧɚɫ ɮɨɪ-
ɦɭɥɭ: t
M t M 0 cos Z 0 t ³ 0
1
sin Z 0t cos Z 0W cos Z0t sin Z0W W1 W dW
Z0
ɢɥɢ t
M t M 0 cos Z 0 t ³ 0
16
1
Z0
sin Z 0 t W W1 W dW .
(8)
ɂɡ ɮɨɪɦɭɥɵ (8) ɥɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ ɮɢɤɫɢɪɨɜɚɧɧɨɝɨ ɦɨɦɟɧɬɚ t tk ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ M tk ɛɭɞɟɬ ɞɨɫɬɢɝɚɬɶɫɹ ɩɪɢ W1 W
P sign sin Z 0 tk W ɢ ɪɚɜɧɨ: max M tk W1
P cos Z 0 tk M 0 Z0
tk
³ sin Z t 0
k
W dW .
(9)
0
§S · 1 , ɝɞɟ k 0,1, 2,3,... ɩɨɥɭɱɢɦ ¨ S k ¸ 2 © ¹ Z0 ɩɪɨɫɬɭɸ ɮɨɪɦɭɥɭ ɜɨɡɪɚɫɬɚɧɢɹ ɤɨɥɟɛɚɧɢɣ ɩɨ ɤɨɨɪɞɢɧɚɬɟ M :
ɇɚɩɪɢɦɟɪ, ɩɪɢ tk
max M tk W1
P Z0
tk
³ cos Z W dW .
(10)
0
0
Ɉɛɵɱɧɨ ɚɦɩɥɢɬɭɞɨɣ ɤɨɥɟɛɚɧɢɣ ɧɚɡɵɜɚɸɬ ɡɧɚɱɟɧɢɟ ɦɨɞɭɥɹ
M ti ɜ ɦɨɦɟɧɬɵ ti , ɤɨɝɞɚ ɫɤɨɪɨɫɬɶ Z ti 0 . Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨ ɧɚɯɨɠɞɟɧɢɢ ɦɚɤɫɢɦɚɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɚɦɩɥɢɬɭɞɵ ɩɪɢ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ M 0 M 0 , Z 0 0 ɜ ɩɟɪɜɵɣ ɦɨɦɟɧɬ t1 , ɤɨɝɞɚ Z t1 0 ɢ
Z t 0 ɞɥɹ t 0, t1 . ɂɡ ɮɨɪɦɭɥ (8) ɢ (9) ɜɢɞɧɨ, ɱɬɨ ɜɨɡɦɭɳɟɧɢɟ ɞɨɥɠɧɨ ɛɵɬɶ ɦɚɤɫɢɦɚɥɶɧɵɦ ɩɨ ɦɨɞɭɥɸ ɢ ɦɟɧɹɬɶ ɡɧɚɤ ɱɟɪɟɡ ɩɨɥɭɩɟ-
S ɩɨɥɭɱɢɦ Z0
ɪɢɨɞ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɩɪɢ t1
Z 0 Z t1 0 ɢ Z t 0 t 0, t1 ɢ max M t1 W1
§ S · 1 max cos ¨ Z 0 ¸ M 0 W1 Z Z 0 ¹ 0 © t1
M 0
P sin Z 0W d Z 0W Z 02 ³0 ȼ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t2 max M t2 W1
M0
t1
§
§
³ ¨¨©W t sin ¨© Z 1
0
2P , ɬ.ɟ. Z 02
0
·· S Z 0W ¸ ¸¸dW Z0 ¹¹
max M t1 W1
M0
2P . Z 02
2S ɩɨɥɭɱɢɦ: Z0 P M 0 4 2 , M t2 0 . Z0
t1 T
ɉɪɨɞɨɥɠɚɹ ɬɚɤɢɟ ɩɨɫɬɪɨɟɧɢɹ, ɩɨɥɭɱɢɦ, ɱɬɨ ɚɦɩɥɢɬɭɞɚ ɤɨɥɟ2P ɛɚɧɢɣ ɱɟɪɟɡ ɜɪɟɦɹ tn nT ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: M n tn M 0 n 2 , ɬ.ɟ.
Z0
ɛɭɞɟɬ ɪɚɫɬɢ ɜ ɚɪɢɮɦɟɬɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ.
17
Ɋɚɫɫɦɨɬɪɢɦ ɞɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɩɨɞ ɞɟɣɫɬɜɢɟɦ “ɧɚɢɯɭɞɲɟɝɨ” ɜɧɟɲɧɟɝɨ ɜɨɡɦɭɳɟɧɢɹ ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ M , M : W1 { P
S , Z0
0dt d
M
2P
§ P · P °M t ¨¨ M 0 2 ¸¸ cos Z 0 t 2 Z0 ¹ Z0 ° © , M 0 M 0 , M 0 ® § P · ° °M t Z 0 ¨¨ M 0 Z 2 ¸¸ sin Z 0 t 0 ¹ © ¯ §S · §S · 2P M ¨ ¸ M 0 2 , M ¨ ¸ 0 . Z Z0 © 0¹ © Z0 ¹
M 0 cos Z 0 t ,
Z02
M Z0
2
§ 2 P · § M · M 0 sin Z 0 t ¨¨ M 2 ¸¸ ¨ ¸ Z0 ¹ © Z0 ¹ ©
0,
2
M02
2
§ 2P · ¨¨ M 2 ¸¸ Z0 ¹ M 2 © 2 2
1.
M0 Z02
M0
(11)
ɇɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ M , M ɭɪɚɜɧɟɧɢɟ (11) ɨɩɢɫɵɜɚɟɬ ɷɥɥɢɩɫ ɫ ɩɨɥɭɨɫɹɦɢ M 0 ɢ M 0Z 0 . M
M0 M 0
M0
4P
Z0 2
M
2P
Z02 Ɋɢɫ. 5.
S 2S dt d ɞɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɛɭɞɟɬ ɩɪɨɢɫɇɚ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ Z0 Z0 ɯɨɞɢɬɶ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɧɚɢɯɭɞɲɟɝɨ ɜɧɟɲɧɟɝɨ ɜɨɡɦɭɳɟɧɢɹ W1 { P . S S ɉɟɪɟɣɞɟɦ ɤɨ ɜɪɟɦɟɧɢ W t , 0 dW d . Ɍɨɝɞɚ: Z0 Z0 °M W ® °M W ¯
18
P 2P Z 0 2 , M 0 M 0 2 , M 0 0 , Z0 a0Z 0 sin Z 0W a0 cos Z 0W
ɝɞɟ a0
M 0
3P
Z02
.
M
P Z02
§S · ¸ © Z0 ¹
M¨
a0 cos Z 0W , a0
P Z0 2
M Z0
M0
a0Z sin Z 0W , 4P
Z0
2
§S · , M ¨ ¸ © Z0 ¹
0
2
§ P · 2 ¨M 2 ¸ 2 § · Z0 ¹ M 2 P M 2 © (12) 2 2 1. ¨¨ M 2 ¸¸ 2 a0 2 a0 a0 Z 0 Z0 ¹ Z0 © ɍɪɚɜɧɟɧɢɟ (12) ɨɩɢɫɵɜɚɟɬ ɷɥɥɢɩɫ ɫ ɩɨɥɭɨɫɹɦɢ : § § · 3P · § 3P · 3P ¨¨ M 0 2 ¸¸ ɢ ¨¨ M 0 2 ¸¸ Z 0 , ¨¨ M 0 2 ! M 0 ¸¸ . Z0 ¹ © Z0 ¹ Z0 © © ¹ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ ɪɚɫɤɪɭɱɢɜɚɸɳɟɣɫɹ ɫɩɢɪɚɥɢ (ɪɢɫ.5). ɋ ɩɨɦɨɳɶɸ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɛɵɥɨ ɨɩɢɫɚɧɨ ɹɜɥɟɧɢɟ ɩɪɨɫɬɨɝɨ ɢɥɢ “ɜɧɟɲɧɟɝɨ” ɪɟɡɨɧɚɧɫɚ ɦɚɹɬɧɢɤɚ ɫ ɞɜɢɠɭɳɟɣɫɹ ɩɨ ɝɨɪɢɡɨɧɬɚɥɢ ɬɨɱɤɨɣ ɩɨɞɜɟɫɚ. “ɇɚɢɯɭɞɲɟɟ” ɜɨɡɦɭɳɟɧɢɟ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɪɚɜɧɨ ɩɨ ɦɨɞɭɥɸ ɧɚɢɛɨɥɶɲɟɦɭ ɜɨɡɦɨɠɧɨɦɭ ɡɧɚɱɟɧɢɸ ɢ ɦɟ-
ɧɹɟɬ ɡɧɚɤ ɱɟɪɟɡ ɩɨɥɭɩɟɪɢɨɞ T
S ɢ, ɤɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ, ɚɦɩɥɢɬɭɞɚ Z0
ɤɨɥɟɛɚɧɢɣ ɪɚɫɬɟɬ ɜ ɚɪɢɮɦɟɬɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ. ɇɚɛɥɸɞɚɟɦɨɟ ɹɜɥɟɧɢɟ ɩɪɨɫɬɨɝɨ ɪɟɡɨɧɚɧɫɚ ɨɩɢɫɵɜɚɟɬɫɹ ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ ɪɚɫɤɪɭɱɢɜɚɸɳɟɣɫɹ ɫɩɢɪɚɥɶɸ. ɇɚ ɤɨɨɪɞɢɧɚɬɧɵɯ ɩɥɨɫɤɨɫɬɹɯ W1 ; t ɢ M ; t ɩɨɤɚɡɚɧɨ ɩɨɜɟɞɟɧɢɟ ɜɨɡɦɭɳɟɧɢɹ W1 t ɢ ɨɬɤɥɨɧɟɧɢɹ M t (ɪɢɫ. 6).
19
W1
P 0 t1
S Z0
2S Z0
t2
t
P
M M0
4P
Z0 2
M0
t1 M 0
t2
2P
Z02 Ɋɢɫ. 6.
20
t
§ 3. ɉɚɪɚɦɟɬɪɢɱɟɫɤɢɣ ɪɟɡɨɧɚɧɫ ɢ ɩɚɪɚɦɟɬɪɢɱɟɫɤɚɹ ɫɬɚɛɢɥɢɡɚɰɢɹ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɦɚɹɬɧɢɤɚ Ɋɚɫɫɦɨɬɪɢɦ ɛɨɥɟɟ ɫɥɨɠɧɵɣ ɫɥɭɱɚɣ: ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ ɫ ɩɨɞɜɢɠɧɨɣ ɩɨ ɜɟɪɬɢɤɚɥɢ ɬɨɱɤɨɣ ɩɨɞɜɟɫɚ. Ɍɨɱɤɚ ɩɨɞɜɟɫɚ ɩɟɪɟɦɟɳɚɟɬɫɹ ɩɨ ɜɟɪɬɢɤɚɥɢ ɫ ɩɟɪɟɦɟɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ W t , ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɦ ɭɫɥɨɜɢɹɦ: W t d W0 .
(1)
ȼ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ, ɤɚɫɚɬɟɥɶɧɭɸ ɤ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ, ɚɛɫɨɥɸɬɧɨɟ ɭɫɤɨɪɟɧɢɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: aɚɛɫ lM W sin M . (2) Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɬɭ ɠɟ ɨɫɶ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: mlM mW sin M mg sin M (3) Ⱦɥɹ ɦɚɥɵɯ ɭɝɥɨɜ M ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ (3) ɩɪɢɦɟɬ ɜɢɞ: §
M ¨ Z 02 ©
& W
M
W t · ¸M l ¹
2
0 , ɝɞɟ Z 0
g l
ȼɜɟɞɟɦ ɨɝɪɚɧɢɱɟɧɢɟ Z 2
(4)
Z 02
W t
!0, l ɩɪɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ W t , ɭɞɨɜɥɟɬɜɨɪɹɸ-
W
ɳɢɯ ɭɫɥɨɜɢɸ (1). Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (4) ɩɪɢɦɟɬ ɜɢɞ: M Z 2 t M 0 (5)
& mg
ȼ ɨɛɳɟɦ ɜɢɞɟ ɩɨɥɭɱɢɬɶ ɪɟɲɟɧɢɟ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ɥɸɛɵɯ ɢɡɦɟɧɟɧɢɹɯ ɭɫɤɨɪɟɧɢɹ W t ɧɟɜɨɡɦɨɠɧɨ. ȿɫɥɢ W t ɦɟɧɹɟɬɫɹ ɩɨ ɫɢɧɭɫɨɢɞɚɥɶɧɨɦɭ ɡɚɤɨɧɭ, ɬɨɝɞɚ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɪɟɲɟɧɢɟ, ɨɩɢɫɵɜɚɟɦɨɟ ɮɭɧɤɰɢɹɦɢ Ɇɚɬɶɟ. ȿɫɥɢ W t ɤɭɫɨɱɧɨ-ɩɨɫɬɨɹɧɧɚ, ɦɨɠ-
Ɋɢɫ. 1. ɧɨ ɩɨɥɭɱɢɬɶ ɪɟɲɟɧɢɟ, ɨɩɢɫɵɜɚɟɦɨɟ ɷɥɟɦɟɧɬɚɪɧɵɦɢ ɮɭɧɤɰɢɹɦɢ. ɉɨɞɛɢɪɚɹ ɨɩɪɟɞɟɥɟɧɧɵɦ ɨɛɪɚɡɨɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ, ɬ.ɟ. ɢɡɦɟɧɹɹ ɩɚɪɚɦɟɬɪ Z 2 t ɜ (5) ɦɨɠɧɨ ɞɨɛɢɬɶɫɹ ɭɜɟɥɢɱɟɧɢɹ ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ ɢ ɧɚɛɥɸɞɚɬɶ ɹɜɥɟɧɢɟ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɝɨ ɪɟɡɨɧɚɧɫɚ. ɉɨɤɚ21
ɡɚɧɨ [4], ɱɬɨ ɧɚɢɛɨɥɶɲɢɣ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢɣ ɪɟɡɨɧɚɧɫ ɜɨɡɧɢɤɚɟɬ, ɤɨɝɞɚ ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɜ ɞɜɚ ɪɚɡɚ ɛɨɥɶɲɟ ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ ɦɚɹɬɧɢɤɚ. ɉɟɪɟɣɞɟɦ ɤ ɢɫɫɥɟɞɨɜɚɧɢɸ ɫɥɭɱɚɹ, ɤɨɝɞɚ W t ɹɜɥɹɟɬɫɹ ɤɭɫɨɱɧɨ-ɩɨɫɬɨɹɧɧɨɣ ɮɭɧɤɰɢɟɣ ɫ ɬɚɤɨɣ ɱɚɫɬɨɬɨɣ. ɋɧɚɱɚɥɚ ɪɚɫɫɦɨɬɪɢɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɩɨɞɜɟɫɚ, ɤɨɝɞɚ ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ ɦɚɹɬɧɢɤɚ ɭɜɟɥɢɱɢɜɚɟɬɫɹ. ɉɭɫɬɶ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɦɚɹɬɧɢɤ ɨɬɤɥɨɧɟɧ ɧɚ ɧɟɤɨɬɨɪɵɣ ɭɝɨɥ M 0 : t
0.
0; M 0 M 0 , M 0
Ɂɚɞɚɞɢɦ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t0 ɬɚɤ, ɱɬɨ ɫɬɟɪɠɟɧɶ, ɧɚ ɤɨɬɨɪɨɦ ɩɨɞɜɟɲɟɧ ɦɚɹɬɧɢɤ, ɛɭɞɟɬ ɫɨɜɩɚɞɚɬɶ ɫ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɶɸ: t0 ; M t0 , M t0 0 . ɉɭɫɬɶ ɫ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɬɚɤɨɜɨ, ɱɬɨ W t
2 ɠɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: M Z 0 W0 M
Z 0 2 W0
0 ɞɨ ɦɨɦɟɧɬɚ t0 ɭɫɤɨɪɟɧɢɟ
lW0 , ɢ l 1 . Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ ɞɜɢ0 , 0 d t d t0 ɢɥɢ M Z12M
0 , ɝɞɟ
Z12 . Ɍɚɤɢɦ ɭɪɚɜɧɟɧɢɟɦ ɨɩɢɫɵɜɚɟɬɫɹ ɞɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɫ
ɧɟɩɨɞɜɢɠɧɵɦ ɨɫɧɨɜɚɧɢɟɦ. Ɋɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ɡɚɞɚɧɧɵɯ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɢɦɟɟɬ ɜɢɞ: M t M 0 cos Z1t , M t M 0Z1 sin Z1t . Ɍɨɝɞɚ M t0 M 0 cos Z1t0 . ȿɫɥɢ M t0
0 , ɬɨ t0
S . 2Z1
§ S · M 0Z1 sin ¨ Z1 ¸ M 0Z1 . © 2Z1 ¹ Ⱦɚɥɟɟ ɩɟɪɟɣɞɟɦ ɤɨ ɜɪɟɦɟɧɢ W t t0 . Ɍɨɝɞɚ M W
Ɍɨɝɞɚ M t0
Ɋɚɫɫɦɨɬɪɢɦ 0 d W d W 0 W 0
ɞɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɧɚ t1 t0 , ɝɞɟ M W 0 M1 , M W 0
0
0, M W
ɢɧɬɟɪɜɚɥɟ 0.
0
M 0Z1 .
ɜɪɟɦɟɧɢ
ɍɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɜɧɨɜɶ ɛɭɞɟɬ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɨɣ, ɧɨ ɫ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦ ɡɧɚɤɨɦ: W t lW0 , l 1 . Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: M Z 2 2M 0 , ɝɞɟ Z 2 2 Z 0 2 W0 . Ɋɟɲɟɧɢɟɦ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɮɭɧɤɰɢɹ M W
c sin Z 2W ,
M W cZ 2 cos Z 2W . ɍɱɢɬɵɜɚɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɩɨɥɭɱɢɦ: M W
22
0
M 0Z1
cZ 2 .
M 0Z1
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, c Ɍɨɝɞɚ. M W
Z2
.
M 0Z1 sin Z 2W , M W M 0Z1 cos Z 2W Z2
Ɉɩɪɟɞɟɥɢɦ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ W 0 : M W 0
0 . Ɍɨɝɞɚ M 0Z1 cos Z 2W 0
0.
S . ȼ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ W 0 ɨɬɤɥɨɧɟɧɢɹ ɦɚɹɬɧɢ2Z 2 MZ MZ MZ S 0 1 sin 0 1 . Ɉɛɨɡɧɚɱɢɦ M1 0 1 . Z2 Z2 Z2 2
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, W 0 ɤɚ: M W 0 Ɍɨɝɞɚ
M1 M0
Z 0 2 W0
Z1 Z2
! 1 , ɬ.ɟ. M1 ! M 0 .
Z 0 2 W0
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɭɱɢɦ, ɱɬɨ ɡɚ ɜɪɟɦɹ t1
t0 W 0
S S ɩɪɨ2Z1 2Z 2
ɢɡɨɲɥɨ ɭɜɟɥɢɱɟɧɢɟ ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ. ɉɨɤɚɠɟɦ, ɱɬɨ ɜɨɡɪɚɫɬɚɧɢɟ ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ ɩɪɨɢɫɯɨɞɢɬ ɜ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ. Ⱦɥɹ ɷɬɨɝɨ ɟɳɟ ɪɚɡ ɧɚɣɞɟɦ, ɱɟɦɭ § S S · ɛɭɞɟɬ ɪɚɜɧɨ ɨɬɤɥɨɧɟɧɢɟ ɦɚɹɬɧɢɤɚ M ɱɟɪɟɡ ɜɪɟɦɹ ¨ ¸ ɩɨɫɥɟ © 2Z1 2Z 2 ¹ ɦɨɦɟɧɬɚ t1 . ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ Z12
Z 0 2 W0 , ɬɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (4) ɢɦɟɟɬ
M Z12M
ɜɢɞ: ɉɭɫɬɶ ɜɪɟɦɹ W
t t1 ɢ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɛɭɞɭɬ ɢɦɟɬɶ
MW
ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
0.
0
M1
M0
Z1 ; M W Z2
0
0.
Ɍɨɝɞɚ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɨɩɢɫɵɜɚɬɶɫɹ ɫɥɟɞɭɸɳɟɣ ɮɭɧɤɰɢɟɣ: M W M1 cos Z1W , M W M1Z1 sin Z 2W . Ⱦɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɞɨ ɦɨɦɟɧɬɚ W 1 , ɬɚ-
1
ɤɨɝɨ, ɱɬɨ M W
0 . Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ W 1
M W 1
S ,ɢ 2Z1
§ S · § S · ¸ M1Z1 sin ¨ Z1 ¸ M1Z1 . © 2Z1 ¹ © 2Z1 ¹
M ¨
23
ȼɨɡɜɪɚɳɚɹɫɶ ɤɨ ɜɪɟɦɟɧɢ t , ɩɨɥɭɱɢɦ, ɱɬɨ t10
t1
S . 2Z1
ɇɚ ɫɥɟɞɭɸɳɟɣ “ɱɟɬɜɟɪɬɢ” ɩɟɪɢɨɞɚ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɜɨɡɶɦɟɦ W lW0 , l 1 . Ɍɨɝɞɚ Z 2 2
Z 0 2 W0 ɢ ɭɪɚɜɧɟɧɢɟ (4) ɢɦɟɟɬ ɜɢɞ: M Z 2 2M
ɉɭɫɬɶ ɜɪɟɦɹ W
0.
0 1
t t .
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ: M W
0
0, M W
0
M1Z1 .
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ɡɚɞɚɧɧɵɯ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɢɦɟɟɬ ɜɢɞ:
M W
M1Z1 sin Z 2W
, M W
M1Z1 cos Z 2W
. Z2
1 1 Ⱦɜɢɠɟɧɢɟ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɞɨ ɦɨɦɟɧɬɚ W
, ɤɨɝɞɚ M W
1 Ɉɬɤɭɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ M1Z1 cos Z 2W
1 0 , W
0.
S . 2Z 2
2
§ Z1 · ¨ ¸ . © Z2 ¹ ȼɨɡɜɪɚɳɚɹɫɶ ɤɨ ɜɪɟɦɟɧɢ t , ɩɨɥɭɱɢɦ
Ɍɨɝɞɚ: M 2
1 M W
t2
Ɉɛɨɡɧɚɱɢɦ T
M1Z1 Z2
§ S S · 2¨ ¸ © 2Z1 2Z 2 ¹
t1
S 2Z 2
S S . Z1 Z 2
t2
S S . ȼ ɧɚɲɟɦ ɫɥɭɱɚɟ T ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢZ1 Z 2
ɜɚɬɶ ɤɚɤ “ɩɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ”. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɭɸ ɤɚɪɬɢɧɭ ɢɡɦɟɧɟɧɢɹ ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ:
M 0
§1 · M0 , M ¨ T ¸ ©2 ¹
2
§Z · Z M 0 1 , M T M 0 ¨ 1 ¸ . Z2 © Z2 ¹ n
§Z · Ɋɚɫɫɭɠɞɚɹ ɞɚɥɟɟ ɚɧɚɥɨɝɢɱɧɨ, ɩɨɥɭɱɢɦ M nT M 0 ¨ 1 ¸ . © Z2 ¹ Ɉɬɫɸɞɚ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɜɵɜɨɞ, ɱɬɨ, ɩɪɢɞɚɜɚɹ ɭɫɤɨɪɟɧɢɸ ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɨɞɢɧɚɤɨɜɵɟ ɩɨ ɦɨɞɭɥɸ, ɧɨ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɩɨɫɬɨɹɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢ ɦɟɧɹɹ ɢɯ ɱɟɪɟɡ ɤɚɠɞɭɸ ɱɟɬɜɟɪɬɶ “ɩɟɪɢɨɞɚ”, ɞɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɛɭɞɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨ
24
ɚɦɩɥɢɬɭɞɚ ɟɝɨ ɤɨɥɟɛɚɧɢɣ ɛɭɞɟɬ ɭɜɟɥɢɱɢɜɚɬɶɫɹ ɜ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ (ɫɦ. ɪɢɫ. 2)
M M2
M0
W
W 0
0
t0
t1
t1
t0
t1
t1
t2
t
M1
W W0
0
0
t2
t
W0
Ɋɢɫ. 2. Ɉɬ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɝɨ ɪɟɡɨɧɚɧɫɚ ɩɟɪɟɣɞɟɦ ɬɟɩɟɪɶ ɤ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɦɭ ɭɩɪɚɜɥɟɧɢɸ. Ɂɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɦɨɠɧɨ ɩɨɞɨɛɪɚɬɶ ɢ ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ ɭɦɟɧɶɲɚɥɚɫɶ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɧɚɛɥɸɞɚɬɶ ɹɜɥɟɧɢɟ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɫɬɚɛɢɥɢɡɚɰɢɢ. ɇɟɨɛɯɨɞɢɦɨɫɬɶ ɪɟɲɟɧɢɹ ɬɚɤɨɣ ɡɚɞɚɱɢ ɜɢɞɧɚ ɧɚ ɫɥɟɞɭɸɳɟɦ ɩɪɢɦɟɪɟ. Ʉɨɪɚɛɥɶ ɧɚɯɨɞɢɬɫɹ ɜ ɩɨɪɬɭ. ɇɚ ɧɟɝɨ ɧɭɠɧɨ ɩɨɝɪɭɡɢɬɶ ɤɨɧɬɟɣɧɟɪ. Ʉɨɧɬɟɣɧɟɪ ɭɠɟ ɩɨɞɧɹɬ ɫ ɩɪɢɱɚɥɚ ɦɨɫɬɨɜɵɦ ɤɪɚɧɨɦ, ɭɫɬɚɧɨɜɥɟɧɧɵɦ ɧɚ ɩɚɥɭɛɟ ɤɨɪɚɛɥɹ, ɢ ɩɟɪɟɧɟɫɟɧ ɧɚ ɤɨɪɚɛɥɶ. Ɉɫɬɚɟɬɫɹ ɩɟɪɟɧɟɫɬɢ ɟɝɨ ɜ ɬɪɸɦ. Ɉɞɧɚɤɨ, ɧɚɱɚɥɶɧɨɟ ɨɬɤɥɨɧɟɧɢɟ ɝɪɭɡɚ ɨɬ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɢ ɜɨɥɧɟɧɢɟ ɦɨɪɹ, ɜɵɡɵɜɚɸɳɟɟ ɜɟɪɬɢɤɚɥɶɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ ɤɨɪɚɛɥɹ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɝɪɭɡɚ, ɩɪɢɜɨɞɹɬ ɤ ɤɨɥɟɛɚɧɢɹɦ ɤɨɧɬɟɣɧɟɪɚ ɧɚ ɬɪɨɫɟ ɤɪɚɧɚ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, 25
ɭɩɪɚɜɥɹɹ ɤɪɚɧɨɦ, ɧɟɨɛɯɨɞɢɦɨ ɫɬɚɛɢɥɢɡɢɪɨɜɚɬɶ ɜɟɪɬɢɤɚɥɶɧɨɟ ɩɨɥɨɠɟɧɢɟ ɬɪɨɫɚ ɫ ɝɪɭɡɨɦ. ȿɫɥɢ ɡɧɚɱɟɧɢɹ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɦɟɧɹɬɶ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɩɨɪɹɞɤɟ: ɜ ɦɨɦɟɧɬ t 0 ɩɪɢɞɚɬɶ ɭɫɤɨɪɟɧɢɸ ɨɬɪɢɰɚɬɟɥɶɧɨɟ ɩɨɫɬɨɹɧɧɨɟ ɡɧɚɱɟɧɢɟ W lW0 , l 1 ɢ ɨɫɬɚɜɢɬɶ ɟɝɨ ɧɚ “ɱɟɬɜɟɪɬɶ” ɩɟɪɢɨɞɚ, ɚ ɡɚɬɟɦ ɩɪɢɞɚɬɶ ɭɫɤɨɪɟɧɢɸ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɩɨɫɬɨɹɧɧɨɟ ɡɧɚɱɟɧɢɟ W lW0 , l 1 ɧɚ ɫɥɟɞɭɸɳɭɸ ɱɟɬɜɟɪɬɶ ɩɟɪɢɨɞɚ, ɢ ɬɚɤ ɞɚɥɟɟ, ɬɨ ɚɦɩɥɢɬɭɞɚ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɛɭɞɟɬ ɭɦɟɧɶɲɚɬɶɫɹ ɜ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ. ɗɬɨ ɜɢɞɧɨ ɢɡ ɫɥɟɞɭɸɳɢɯ ɢɫɫɥɟɞɨɜɚɧɢɣ. 1. Ɋɚɫɫɦɨɬɪɢɦ ɞɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɧɚ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ 0 d t d t0 , ɝɞɟ: M 0 M 0 , M 0 0 :, 0 d t d t0 , ɱɬɨ M t0 0 . ɍɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɩɨɞɜɟɫɚ W
2
lW0 , l 1 , ɬɨɝɞɚ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɛɭɞɟɬ
ɢɦɟɬɶ ɜɢɞ M Z 0 W0 M
0 , M Z 2 2M
0.
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ: M t M 0 cos Z 2 t , M t
M 0Z 2 sin Z 2 t . Ɍɚɤ ɤɚɤ
S M t0 M 0Z 2 . 2Z 2
M t0 0 , ɫɥɟɞɨɜɚɬɟɥɶɧɨ, t0
2. Ɋɚɫɫɦɨɬɪɢɦ ɞɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɧɚ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ 0 d W d W 0 , t1 t0 , ɝɞɟ: M W
t t0 , W 0
W
ɩɨɞɜɟɫɚ
W
M Z 0 2 W0 M
lW0 , l 1 ,
Z2 Z1
0, M W
0
ɭɪɚɜɧɟɧɢɟ
0 , M Z12M
ɞɜɢɠɟɧɢɹ
ɦɚɹɬɧɢɤɚ
0.
Z 2 W0 Z 2 W0
1.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɚ “ɩɨɥɭɩɟɪɢɨɞ” ɤɨɥɟɛɚɧɢɣ ɢɡɨɲɥɨ ɭɦɟɧɶɲɟɧɢɟ ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ.
26
M 0Z 2 . ɍɫɤɨɪɟɧɢɟ ɬɨɱɤɢ
M 0Z 2 sin Z1W , M W M 0Z 2 cos Z1W , Z1 Z S 0 . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, W 0 , M W 0 M 0 2 . Z1 2Z1
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ: M W
M W 0
0
T 2
t1
S S ɩɪɨ2Z 2 2Z1
2
ɑɟɪɟɡ “ɩɟɪɢɨɞ” ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ ɛɭɞɟɬ ɪɚɜɧɚ M T
§Z · M0 ¨ 2 ¸ , © Z1 ¹
n
§Z · S S . ɑɟɪɟɡ n ɩɟɪɢɨɞɨɜ M nT M 0 ¨ 2 ¸ . Z 2 Z1 © Z1 ¹
T
M
M0 M2 0
t0
t1
t2
t
t0
t1
t2
t
M1 W W0
0 W0
Ɋɢɫ. 3. Ɍ.ɟ. ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɚɦɩɥɢɬɭɞ ɛɭɞɟɬ ɩɪɟɞɫɬɚɜɥɹɬɶ ɫɨɛɨɣ ɭɛɵɜɚɸɳɭɸ ɝɟɨɦɟɬɪɢɱɟɫɤɭɸ ɩɪɨɝɪɟɫɫɢɸ, ɢ ɦɨɠɟɦ ɧɚɛɥɸɞɚɬɶ ɹɜɥɟɧɢɟ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɫɬɚɛɢɥɢɡɚɰɢɢ (ɪɢɫ. 3). ɂɥɥɸɫɬɪɚɰɢɹ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɝɨ ɪɟɡɨɧɚɧɫɚ ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ M , M . Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɜ ɫɥɭɱɚɟ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɝɨ ɪɟɡɨɧɚɧɫɚ ɜ ɬɟɱɟɧɢɟ ɜɪɟɦɟɧɢ ɪɚɜɧɨɝɨ t
S 2Z1
ɨɩɢɫɵɜɚɟɬɫɹ
ɭɪɚɜɧɟɧɢɹɦɢ:
27
°M t M 0 cos Z1t S . , 0dt d ® Z1 2 M M Z Z t t °¯ 0 1 sin 1 § S · § S · 0, M ¨ ¸ 0 , M ¨ ¸ © 2Z1 ¹ © 2Z1 ¹ ɢ M ɛɭɞɭɬ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɫɥɟɞɭɸɳɟɦɭ ɭɪɚɜɧɟɧɢɸ:
ɉɪɢɱɟɦ M 0 M 0 , M 0
M2 M 2 2 2 2 M 0 M 0 Z1
cos 2 Z1t sin 2 Z1t 1 , ɬ.ɟ.
M 0Z1 . Ɍɨɝɞɚ M
M2 M 2 2 2 2 M 0 M 0 Z1
1.
Ƚɪɚɮɢɤɨɦ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ IV ɤɨɨɪɞɢɧɚɬɧɨɦ ɭɝɥɭ ɩɥɨɫɤɨɫɬɢ M , M ɛɭɞɟɬ ɹɜɥɹɬɶɫɹ ɷɥɥɢɩɫ ɫ ɩɨɥɭɨɫɹɦɢ M0 ɢ M0Z1 . ȼ III ɤɨɨɪɞɢɧɚɬɧɨɦ ɭɝɥɭ ɩɥɨɫɤɨɫɬɢ M , M ɛɭɞɟɦ ɢɦɟɬɶ ɝɪɚɮɢɤ ɭɪɚɜɧɟɧɢɹ: M2 M 2 2 2 2 § Z · M 0 Z1 M 02 ¨ 1 ¸ © Z2 ¹
1 , ɤɨɬɨɪɨɟ ɨɩɢɫɵɜɚɟɬ ɞɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɧɚ
ɫɥɟɞɭɸɳɟɦ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ: § S · ¸ © 2Z1 ¹
M¨
§ S · 0 , M ¨ ¸ © 2Z1 ¹
S S S dt d , 2Z1 2Z 2 2Z1
M 0Z1 . Ɉɛɨɡɧɚɱɢɜ
T 2
S S , ɩɨɥɭɱɢɦ 2Z 2 2Z1
Z Z §T · §T · §T · M ¨ ¸ 0; M ¨ ¸ M 0 1 , M ¨ ¸ ! M 0 , ɬɚɤ ɤɚɤ 1 ! 1 . Z2 Z2 ©2¹ ©2¹ ©2¹ Ɋɚɫɫɭɠɞɚɹ ɚɧɚɥɨɝɢɱɧɨ, ɜɨ II ɢ I ɤɨɨɪɞɢɧɚɬɧɵɯ ɭɝɥɚɯ ɩɥɨɫɤɨɫɬɢ M , M ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɢɟ ɡɚɜɢɫɢɦɨɫɬɢ:
M2
II ɤɨɨɪɞɢɧɚɬɧɵɣ ɭɝɨɥ:
M 2
1. 2 § Z1 · 2 2 § Z1 · M ¨ ¸ M 0 Z1 ¨ ¸ © Z2 ¹ © Z2 ¹ Z T T S §T · §T · dt d , M ¨ ¸ M 0 1 , M ¨ ¸ Z2 2 2 2Z1 ©2¹ ©2¹ 2
2 0
§T S · ¸ © 2 2Z1 ¹
M¨ I ɤɨɨɪɞɢɧɚɬɧɵɣ ɭɝɨɥ:
28
§T Z1 S · 0, M ¨ . ¸ M 0Z1 Z2 © 2 2Z1 ¹
0,
M2 §Z · M ¨ 1¸ © Z2 ¹ 2 0
4
M 2
M 0Z1
2
§ Z1 · ¨ ¸ © Z2 ¹
2
1,
T S d t d T, 2 2Z1
ɝɞɟ
§T §T Z1 S · S · S S ,M ¨ , ¸ 0, M ¨ ¸ M 0Z1 Z Z 2 2 Z 2 Z1 © 2 2Z1 ¹ 1 ¹ © 2
T
2
§ Z1 · §T · ¸ ! M ¨ ¸ , M T Z 2¹ © © 2¹
M T M 0 ¨
0.
Ɇɨɠɧɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɜ ɤɚɠɞɨɣ ɤɨɨɪɞɢɧɚɬɧɨɣ ɱɟɬɜɟɪɬɢ ɞɜɢɠɟɧɢɟ ɨɩɢɫɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɷɥɥɢɩɫɚ, ɢ ɩɨɥɭɨɫɢ ɟɝɨ ɭɜɟɥɢɱɢɜɚɸɬɫɹ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ M , M ɦɵ ɩɨɥɭɱɢɦ ɪɚɫɤɪɭɱɢɜɚɸɳɭɸɫɹ ɫɩɢɪɚɥɶ (ɪɢɫ. 4). ɉɚɪɚɦɟɬɪɢɱɟɫɤɚɹ ɫɬɚɛɢɥɢɡɚɰɢɹ ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ ȼ ɫɥɭɱɚɟ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɫɬɚɛɢɥɢɡɚɰɢɢ ɩɪɢ ɢɡɦɟɧɟɧɢɢ 0 d t d T , ɬ.ɟ. ɞɥɹ ɜɪɟɦɟɧɢ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦɭ “ɩɟɪɢɨɞɭ” ɤɨɥɟɛɚɧɢɣ, ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɢɟ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɤɚɠɞɨɣ ɤɨɨɪɞɢɧɚɬɧɨɣ ɱɟɬɜɟɪɬɢ. M2 M 2 S IV ɤɨɨɪɞɢɧɚɬɧɚɹ ɱɟɬɜɟɪɬɶ: 2 1, 0 d t d . 2Z 2 M 0 M 0Z 2 2 § S · ¸ © 2Z 2 ¹
M 0 M 0 , M 0 0, M ¨
§ S · 0, M ¨ ¸ © 2Z 2 ¹
M 0Z 2 .
III ɤɨɨɪɞɢɧɚɬɧɚɹ ɱɟɬɜɟɪɬɶ: M2 M 2 S T T 1, d t d , ɝɞɟ 2 2 Z 2 2 2 § Z · M 0Z 2 2 M 02 ¨ 2 ¸ © Z1 ¹ § S · § S · M¨ ¸ 0, M ¨ ¸ M 0Z 2 , © 2Z 2 ¹ © 2Z 2 ¹
S 2Z 2
S 2Z1
Z §T · §T · §T · M ¨ ¸ M 0 2 , M ¨ ¸ 0, M ¨ ¸ M 0 . Z1 ©2¹ ©2¹ ©2¹ II ɤɨɨɪɞɢɧɚɬɧɚɹ ɱɟɬɜɟɪɬɶ:
M2 § Z2 · ¸ © Z1 ¹
M 02 ¨
2
M 2
M0Z 2
2
§ Z2 · ¨ ¸ © Z1 ¹
2
1,
T T S dt d . 2 2 2Z 2
29
Z §T · §T · M ¨ ¸ M 0 2 , M ¨ ¸ 0, . Z1 ©2¹ ©2¹ §T § Z2 · S · 0, M ¨ ¸ M 0Z 2 ¨ ¸ © 2 2Z 2 ¹ © Z1 ¹ I ɤɨɨɪɞɢɧɚɬɧɚɹ ɱɟɬɜɟɪɬɶ: M2 M 2 T S 1, dt dT , 4 2 Z2 2 2 2 § Z2 · 2 § Z2 · M 0 ¨ ¸ M 0Z 2 ¨ ¸ © Z1 ¹ © Z1 ¹ §T § Z2 · §T S · S · S S M ¨ ɝɞɟ T , ¸ M 0Z 2 ¨ ¸ , M ¨ ¸ Z 2 Z1 © 2 2Z 2 ¹ © Z1 ¹ © 2 2Z 2 ¹ §T S · ¸ © 2 2Z 2 ¹
M¨
0,
2
§Z · §T · M T M 0 ¨ 2 ¸ M ¨ ¸ , M T 0 . Z ©2¹ © 1¹ ɇɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɟ ɞɜɢɠɟɧɢɟ ɢɦɟɟɬ ɜɢɞ “ɡɚɤɪɭ ɱɢɜɚɸɳɟɣɫɹ” ɫɩɢɪɚɥɢ (ɪɢɫ. 5).
M
M1
0
Ɋɢɫ. 4.
30
M
M0 M 2
M
M1
0
M2
Ɋɢɫ. 5.
M0
M
§ 4. ɉɨɤɚɡɚɬɟɥɶɧɚɹ ɮɭɧɤɰɢɹ ɢ ɪɟɥɟɣɧɨɟ ɫɬɚɛɢɥɢɡɢɪɭɸɳɟɟ ɭɩɪɚɜɥɟɧɢɟ ɩɟɪɟɜɟɪɧɭɬɵɦ ɦɚɹɬɧɢɤɨɦ Ⱦɢɧɚɦɢɤɚ ɩɟɪɟɜɟɪɧɭɬɨɝɨ ɦɚɹɬɧɢɤɚ, ɪɚɡɦɟɳɟɧɧɨɝɨ ɧɚ ɬɟɥɟɠɤɟ – ɩɪɟɤɪɚɫɧɵɣ ɩɪɢɦɟɪ ɩɨɧɹɬɶ ɜɨɡɦɨɠɧɨɫɬɢ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ. ɉɨɩɪɨɛɭɟɦ ɪɟɲɢɬɶ ɡɚɞɚɱɭ ɨɛ ɭɞɟɪɠɚɧɢɢ ɩɟɪɟɜɟɪɧɭɬɨɝɨ ɦɚɹɬɧɢɤɚ ɜ ɜɟɪɬɢɤɚɥɶɧɨɦ ɩɨɥɨɠɟɧɢɢ. əɫɧɨ, ɱɬɨ ɥɸɛɨɟ ɧɚɱɚɥɶɧɨɟ ɨɬɤɥɨɧɟɧɢɟ M 0 z 0 ɩɪɢɜɨɞɢɬ ɤ ɩɨɬɟɪɟ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ, ɬɚɤ ɤɚɤ ɜɨɡɞɟɣɫɬɜɢɟ ɫɢɥɵ ɬɹɠɟɫɬɢ ɭɠɟ ɜ ɩɟɪɜɵɟ ɦɨɦɟɧɬɵ ɩɪɢɜɨɞɢɬ ɤ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɦɭ ɭɜɟɥɢɱɟɧɢɸ ɧɚɱɚɥɶɧɨɝɨ ɨɬɤɥɨɧɟɧɢɹ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɪɢ ɦɚɥɵɯ M t ɢɦɟɟɦ ɭɪɚɜɧɟɧɢɟ ɇɶɸɬɨɧɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ W , ɤɚɫɚɬɟɥɶɧɭɸ ɤ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɜ ɜɢɞɟ (ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɬɟɥɟɠɤɚ ɧɟ ɞɜɢɝɚɟɬɫɹ): mlM mg sin M # mgM ɢɥɢ g 0. l ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɫɩɪɚɜɟɞɥɢɜɨ ɬɨɥɶɤɨ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɧɭɥɹ. ȿɝɨ ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ: g M t C1eZ0t C2 e Z0t , ɝɞɟ Z 0 2 l
M M
Ɍ.ɟ. ɩɪɢ C1 z 0, M t ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ ɜɨɡɪɚɫɬɚɟɬ. . & mg
W
M
& W
Ɋɢɫ. 1. Ɋɚɫɫɦɨɬɪɢɦ ɞɥɹ ɩɪɨɫɬɨɬɵ ɫɥɭɱɚɣ, ɤɨɝɞɚ Ɋɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: M t C1et C2 e t , M t
M2
t 2
t 2
C e C e 1
2
2C1C2 , M t
g l
1.
C1et C2 e t . t 2
t 2
C e C e 1
2
2C1C2 .
31
ɂɡ ɩɨɫɥɟɞɧɢɯ ɞɜɭɯ ɜɵɪɚɠɟɧɢɣ ɩɨɥɭɱɢɦ ɡɚɜɢɫɢɦɨɫɬɶ M ɨɬ M :
M 2 M 2
4C1C2 . ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɞɢɧ ɢɡ ɞɜɭɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɚɜɟɧ 0, ɩɨɥɭɱɢɦ M M M M 0 , M M , ɢɥɢ M M . C1et , ɜɬɨɪɨɣ ɩɪɢ
ɉɟɪɜɵɣ ɢɡ ɷɬɢɯ ɫɥɭɱɚɟɜ ɜɨɡɦɨɠɟɧ, ɤɨɝɞɚ M t
M t C2 e t . ɇɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ ɷɬɢ ɫɥɭɱɚɢ ɛɭɞɭɬ ɢɡɨɛɪɚɠɚɬɶɫɹ ɩɪɹɦɵɦɢ, ɩɪɨɯɨɞɹɳɢɦɢ ɱɟɪɟɡ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ (ɪɢɫ. 2). Ʉɨɝɞɚ C1 C2 z 0 , ɛɭɞɟɦ ɢɦɟɬɶ ɝɢɩɟɪɛɨɥɵ ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ. C1 C2 ! 0, C1 C2 0,
Ɋɢɫ.2. ɱɚɥɶɧɨɦ ɩɨɥɨɠɟɧɢɢ, ɤɨɝɞɚ
M 2 4C1C2
M 2 4C1C2
M2 4C1C2
M2 4C1C2
1,
1.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ ɥɸɛɨɦ ɧɚM 2 0 M 2 0 z 0 , ɦɚɹɬɧɢɤ ɷɤɫɩɨ-
ɧɟɧɰɢɚɥɶɧɨ ɨɬɤɥɨɧɹɟɬɫɹ ɨɬ ɜɟɪɬɢɤɚɥɢ ɢ ɩɚɞɚɟɬ. ɋɭɳɟɫɬɜɭɸɬ ɪɚɡɧɵɟ ɩɭɬɢ, ɱɬɨɛɵ ɜɨɫɩɪɟɩɹɬɫɬɜɨɜɚɬɶ ɷɬɨɦɭ ɨɬɪɢɰɚɬɟɥɶɧɨɦɭ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɜɨɡɞɟɣɫɬɜɢɸ ɜɨɡɦɭɳɚɸɳɟɣ ɝɪɚɜɢɬɚɰɢɨɧɧɨɣ ɫɢɥɵ. ȼɵɛɟɪɟɦ ɨɞɢɧ ɢɡ ɧɢɯ – ɧɚɢɛɨɥɟɟ ɩɪɨɫɬɨɣ. Ɋɚɫɫɦɨɬɪɢɦ ɜɨɡɞɟɣɫɬɜɢɟ ɟɳɟ ɨɞɧɨɣ ɫɢɥɵ – ɫɢɥɵ ɢɧɟɪɰɢɢ, ɜɨɡɧɢɤɚɸɳɟɣ ɜ ɞɜɢɝɚɸɳɟɣɫɹ ɧɟɪɚɜɧɨɦɟɪɧɨ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ. Ⱦɥɹ ɷɬɨɝɨ ɪɚɡɦɟɫɬɢɦ ɦɚɹɬɧɢɤ ɧɚ ɬɟɥɟɠɤɟ ɢ ɩɨɩɪɨɛɭɟɦ ɩɟɪɟɦɟɳɚɬɶ ɟɟ ɫ ɭɫɤɨɪɟɧɢɟɦ W t (ɪɢɫ.1). Ɍɨɝɞɚ ɞɥɹ ɦɚɥɵɯ M ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɩɪɢɦɟɬ ɜɢɞ: W 2 , M Z 0 M u , l g W 2 Z0 ,u . l l ɉɨɩɪɨɛɭɟɦ ɬɟɩɟɪɶ ɩɨɫɬɪɨɢɬɶ ɭɩɪɚɜɥɟɧɢɟ (ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɷɬɨ ɭɫɤɨɪɟɧɢɟ ɬɟɥɟɠɤɢ) ɬɚɤ, ɱɬɨɛɵ ɪɟɲɢɬɶ ɡɚɞɚɱɭ ɫɬɚɛɢɥɢɡɚɰɢɢ ɡɚ ɜɨɡɦɨɠɧɨ ɤɨɪɨɬɤɨɟ ɜɪɟɦɹ. ȿɫɬɟɫɬɜɟɧɧɨ, ɱɬɨ ɭɫɤɨɪɟɧɢɟ ɬɟɥɟɠɤɢ ɨɝmlM mgM
32
mW cos M | mW ɢɥɢ M
g M l
W P d Q . ȿɫɥɢ M 0 z 0 , ɬɨ l l ɫɥɟɞɭɟɬ ɩɪɢɦɟɧɢɬɶ ɦɚɤɫɢɦɚɥɶɧɵɟ ɭɫɢɥɢɹ, ɱɬɨɛɵ ɧɟɣɬɪɚɥɢɡɨɜɚɬɶ ɧɟɝɚɬɢɜɧɨɟ ɜɥɢɹɧɢɟ. ɉɨɫɦɨɬɪɢɦ, ɤɚɤɨɣ ɜɢɞ ɩɪɢɦɭɬ ɮɚɡɨɜɵɟ ɬɪɚɟɤɬɨɪɢɢ ɧɚɲɟɣ ɫɢɫɬɟɦɵ ɩɪɢ u { Q ɢɥɢ u { Q . Ⱦɥɹ ɩɪɨɫɬɨɬɵ ɩɪɢɦɟɦ, ɱɬɨ Z 0 2 { 1; Q 1 ( g 9.8 ɦ ɫ2 , l 9.8 ɦ, P 9.8 ɦ ɫ2 ). Ɍɨɝɞɚ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ u { 1 , ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɪɢɦɟɬ ɜɢɞ: M M 1 . Ɋɟɲɟɧɢɟ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɫɥɟɞɭɸɳɢɦ: M C1et C2 e t 1, M C1et C2 e t . ɇɚɣɞɟɦ ɡɚɜɢɫɢɦɨɫɬɶ ɦɟɠɞɭ M ɢ M .
ɪɚɧɢɱɟɧɨ W t d P . Ɍɨɝɞɚ u
M 2
t 2
t 2
C e C e 1
2
2C1C2 , M 1
ɂɡ ɷɬɢɯ ɫɨɨɬɧɨɲɟɧɢɣ ɩɨɥɭɱɢɦ: M 2 Ɉɛɨɡɧɚɱɢɦ: A
M
2
2
t 2
t 2
C e C e 1
2
2
4C1C2 .
M 1
2C1C2 .
4C1C2 , ɬɨɝɞɚ
M 1
2
A ɢɥɢ M 2 M 1
2
A .
ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ A 0 , ɩɨɥɭɱɢɦ ɞɜɟ ɩɪɹɦɵɟ, ɩɪɨɯɨɞɹɳɢɟ ɱɟɪɟɡ ɬɨɱɤɭ (-1;0) ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ (ɪɢɫ. 3): 2 M 2 M 1 0 , M M 1 M M 1 0 , M 1 .
M M 1 ɢɥɢ M ȼ ɷɬɨɦ ɫɥɭɱɚɟ M
C1e 1 . t
Ɋɢɫ. 3.
Ɋɢɫ. 4.
ɉɪɢ A z 0 , ɩɨɥɭɱɢɦ ɤɪɢɜɵɟ, ɨɩɢɫɵɜɚɟɦɵɟ ɭɪɚɜɧɟɧɢɟɦ:
M
r
M 1
2
A
33
2
§ M · § M 1 · ¸ ¨ ¸ (ɬɨɱɧɟɟ ɷɬɨ ɛɭɞɭɬ ɝɢɩɟɪɛɨɥɵ ¨ ¨ A¸ ¨ ¸ A © ¹ © ¹ 2
2
1 ɢɥɢ
2
§ M · § M 1 · ¸ ¨ ¸ 1 ). ¨ ¨ A¸ ¨ ¸ A © ¹ © ¹ ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ u1 { 1 , ɬ.ɟ. ɞɜɢɠɟɧɢɟ ɬɟɥɟɠɤɢ ɩɪɨɢɫɯɨɞɢɬ ɜ ɞɪɭɝɭɸ ɫɬɨɪɨɧɭ. Ⱥɧɚɥɨɝɢɱɧɨ ɩɨɥɭɱɢɦ M M 1 . Ƚɪɚɮɢɤɨɦ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɧɚ ɮɚɡɨɜɨɣ ɩɥɨɫɤɨɫɬɢ ɛɭɞɭɬ ɢɥɢ ɩɪɹɦɵɟ ɩɪɨɯɨɞɹɳɢɟ ɱɟɪɟɡ ɬɨɱɤɭ (1;0) ɢɥɢ ɝɢɩɟɪɛɨɥɵ (ɪɢɫ. 4): 2 M C1et C2 e t 1, M C1et C2 e t , M 2 M 1 4C1C2 .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɜɢɠɟɧɢɟ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ ɧɟ ɩɨɦɨɝɚɟɬ ɫɬɚɛɢɥɢɡɚɰɢɢ ɦɚɹɬɧɢɤɚ ɜ ɜɟɪɬɢɤɚɥɶɧɨɦ ɩɨɥɨɠɟɧɢɢ. ɉɨɫɦɨɬɪɢɦ, ɱɬɨ ɩɨɥɭɱɢɬɫɹ, ɟɫɥɢ ɨɪɝɚɧɢɡɨɜɚɬɶ ɞɜɢɠɟɧɢɟ ɬɟɥɟɠɤɢ ɫ ɩɟɪɟɦɟɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ. Ɇɨɠɧɨ ɭɜɢɞɟɬɶ, ɱɬɨ ɱɟɪɟɡ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɩɪɨɯɨɞɹɬ ɞɜɟ ɬɪɚɟɤɬɨɪɢɢ; ɨɞɧɚ ɫ u { Q u { 1 ɢ ɞɪɭɝɚɹ ɫ u { Q u { 1 (ɪɢɫ. 5). ɗɬɨ ɧɚɦ ɞɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɝɨɜɨɪɢɬɶ ɨ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɞɥɹ ɬɨɱɟɤ, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɧɚ ɷɬɢɯ ɤɪɢɜɵɯ, ɢɫɯɨɞɹ, ɢɡ ɤɨɬɨɪɵɯ ɫɢɫɬɟɦɚ ɩɪɢɯɨɞɢɬ ɜ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ. Ⱦɥɹ ɞɪɭɝɢɯ ɬɨɱɟɤ, ɤɨɬɨɪɵɟ ɧɚɯɨɞɹɬɫɹ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɞɜɭɯ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɬɪɚɟɤɬɨɪɢɣ ɦɨɠɧɨ ɜɵɛɪɚɬɶ ɫɥɟɞɭɸɳɭɸ ɫɬɪɚɬɟɝɢɸ ɭɩɪɚɜɥɟɧɢɹ.
Ɋɢɫ. 5.
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Ɋɢɫ. 6.
ȼɵɛɟɪɟɦ ɭɩɪɚɜɥɟɧɢɟ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦ ɩɨ ɡɧɚɤɭ. ȼ ɨɛɪɚɬɧɨɦ ɜɪɟɦɟɧɢ W t0 t ɧɚɣɞɟɦ ɜɫɟ ɬɨɱɤɢ, ɢɡ ɤɨɬɨɪɵɯ ɦɨɠɧɨ ɩɨɩɚɫɬɶ ɜ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɢ ɜɵɫɬɪɨɢɦ ɬɪɚɟɤɬɨɪɢɸ (ɪɢɫ.6). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɭɱɢɦ ɛɨɥɟɟ ɛɨɝɚɬɨɟ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ, ɢɡ ɤɨɬɨɪɵɯ ɦɨɠɧɨ ɞɨɫɬɢɱɶ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ. ɗɬɢ ɬɨɱɤɢ ɡɚɩɨɥɧɹɸɬ ɰɟɥɭɸ ɩɨɥɨɫɭ (ɪɢɫ.6). Ʉ ɫɨɠɚɥɟɧɢɸ, ɞɥɹ ɬɨɱɟɤ, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɜɧɟ ɩɨɥɨɫɵ ɡɚɞɚɱɚ ɧɟ ɪɟɲɚɟɬɫɹ. ȼ ɫɜɹɡɢ ɫ ɬɟɦ, ɱɬɨ ɪɟɫɭɪɫɵ ɭɩɪɚɜɥɟɧɢɹ ɨɝɪɚɧɢɱɟɧɵ, ɨɛɥɚɫɬɶ ɭɩɪɚɜɥɹɟɦɨɫɬɢ, ɬ.ɟ. ɨɛɥɚɫɬɶ, ɢɡ ɤɚɠɞɨɣ ɬɨɱɤɢ ɤɨɬɨɪɨɣ ɦɨɠɧɨ ɩɨɩɚɫɬɶ ɜ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ, ɨɝɪɚɧɢɱɟɧɚ ɞɜɭɦɹ ɩɪɹɦɵɦɢ ɤɚɤ ɷɬɨ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 6. ɇɨ ɷɬɨ ɧɟ ɞɨɥɠɧɨ ɧɚɫ ɨɝɨɪɱɚɬɶ, ɬɚɤ ɤɚɤ ɢɫɯɨɞɧɵɟ ɭɪɚɜɧɟɧɢɹ ɧɚɩɢɫɚɧɵ ɞɥɹ ɦɚɥɵɯ “ M ”. Ⱦɥɹ ɫɥɭɱɚɹ ɛɨɥɶɲɢɯ ɧɚɱɚɥɶɧɵɯ ɨɬɤɥɨɧɟɧɢɣ ɧɟɨɛɯɨɞɢɦɨ ɪɚɫɫɦɨɬɪɟɬɶ ɧɟɥɢɧɟɣɧɭɸ ɦɚɬɟɦɚɬɢɱɟɫɤɭɸ ɦɨɞɟɥɶ ɢ ɪɟɲɚɬɶ ɧɟɥɢɧɟɣɧɭɸ ɡɚɞɚɱɭ ɩɨ ɭɩɪɚɜɥɟɧɢɸ. Ɉɫɬɚɥɨɫɶ ɞɨɛɚɜɢɬɶ, ɱɬɨ ɩɨɥɭɱɟɧɧɚɹ ɤɚɪɬɢɧɚ ɬɪɚɟɤɬɨɪɢɣ, ɩɪɢɜɨɞɹɳɢɯ ɜ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ, ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɨ ɜɪɟɦɟɧɢ ɭɩɪɚɜɥɟɧɢɹ, ɩɪɨɬɢɜɨɞɟɣɫɬɜɭɸɳɟɝɨ ɧɟɝɚɬɢɜɧɨɦɭ ɜɨɡɞɟɣɫɬɜɢɸ ɫɢɥɵ ɬɹɠɟɫɬɢ [3]. ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɡɧɚɱɟɧɢɣ M 0 ɢ M 0 ɬɟɥɟɠɤɚ ɞɨɥɠɧɚ ɞɜɢɝɚɬɶɫɹ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦ ɢɥɢ ɨɬɪɢɰɚɬɟɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ, ɞɟɥɚɹ ɬɨɥɶɤɨ ɨɞɢɧ ɪɚɡ ɩɟɪɟɤɥɸɱɟɧɢɟ.
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Ɂɚɤɥɸɱɟɧɢɟ Ⱦɢɧɚɦɢɤɚ ɭɩɪɚɜɥɹɟɦɵɯ ɫɢɫɬɟɦ ɬɚɢɬ ɜ ɫɟɛɟ ɦɧɨɝɨ ɢɧɬɟɪɟɫɧɵɯ ɫɜɨɣɫɬɜ (ɤɪɨɦɟ ɩɟɪɟɱɢɫɥɟɧɧɵɯ ɜɵɲɟ), ɬɚɤɢɯ ɤɚɤ: ɪɟɝɭɥɹɪɧɨɟ ɨɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ, ɨɫɨɛɵɟ ɬɪɚɟɤɬɨɪɢɢ, chattering-ɪɟɠɢɦ, ɫɬɚɛɢɥɢɡɚɰɢɹ ɫɨɫɬɨɹɧɢɣ ɩɨɤɨɹ ɢ ɩɪɢɞɟɥɶɧɵɯ ɰɢɤɥɨɜ, ɯɚɨɫ ɜ ɭɩɪɚɜɥɹɟɦɵɯ ɫɢɫɬɟɦɚɯ ɢ ɭɩɪɚɜɥɟɧɢɟ ɯɚɨɫɨɦ. ȿɫɥɢ ɛɪɨɲɸɪɚ ȼɚɦɢ ɩɪɨɱɢɬɚɧɚ ɫ ɢɧɬɟɪɟɫɨɦ, ɬɨ ɦɨɠɧɨ ɩɪɢɫɬɭɩɚɬɶ ɤ ɢɡɭɱɟɧɢɸ ɢ ɛɨɥɟɟ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɯ ɭɱɟɛɧɵɯ ɩɨɫɨɛɢɣ [5].
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Ʌɢɬɟɪɚɬɭɪɚ 1. 2. 3. 4. 5.
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Ⱥ.Ⱥ. Ʉɨɫɦɨɞɟɦɶɹɧɫɤɢɣ, Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ ɢ ɫɨɜɪɟɦɟɧɧɚɹ ɬɟɯɧɢɤɚ. ɂɡɞ. ɉɪɨɫɜɟɳɟɧɢɟ, 1969 ȿ.ɇ. Ȼɟɪɟɡɤɢɧ, Ʌɟɤɰɢɢ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ, ɂɡɞ. ɆȽɍ, 1968, II ɱ. ȼ.ȼ. Ⱥɥɟɤɫɚɧɞɪɨɜ, ɋ.ɂ. Ɂɥɨɱɟɜɫɤɢɣ, ɋ.ɋ. Ʌɟɦɚɤ, ɇ.Ⱥ. ɉɚɪɭɫɧɢɤɨɜ, ȼɜɟɞɟɧɢɟ ɜ ɞɢɧɚɦɢɤɭ ɭɩɪɚɜɥɹɟɦɵɯ ɫɢɫɬɟɦ, ɂɡɞ. ɆȽɍ, 1993 Ɉ.ȼ. Ⱥɥɟɤɫɚɧɞɪɨɜɚ, Ɉɛɨɛɳɟɧɧɵɣ ɪɟɡɨɧɚɧɫ ɜ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɟ. ȼɟɫɬɧ. Ɇɨɫɤ. ɍɧɢɜɟɪɫɢɬɟɬɚ. ɋɟɪ. 1. Ɇɚɬɟɦ., ɦɟɯ. 1991ɝ. ʋ3 ɫ. 89-92. ȼ.ȼ. Ⱥɥɟɤɫɚɧɞɪɨɜ, ȼ.Ƚ. Ȼɨɥɬɹɧɫɤɢɣ, ɋ.ɋ. Ʌɟɦɚɤ, ɇ.Ⱥ. ɉɚɪɭɫɧɢɤɨɜ, ȼ.Ɇ. Ɍɢɯɨɦɢɪɨɜ, Ɉɩɬɢɦɢɡɚɰɢɹ ɞɢɧɚɦɢɤɢ ɭɩɪɚɜɥɹɟɦɵɯ ɫɢɫɬɟɦ. ɂɡɞ. ɆȽɍ, 2000ɝ. 304 ɫ.
Ⱥɥɟɤɫɚɧɞɪɨɜɚ Ɉɥɶɝɚ ȼɥɚɞɢɦɢɪɨɜɧɚ ɗɥɟɦɟɧɬɚɪɧɵɟ ɮɭɧɤɰɢɢ ɢ ɚɥɝɨɪɢɬɦɵ ɭɩɪɚɜɥɟɧɢɹ. Ɇ:ɂɡɞ-ɜɨ ɦɟɯɚɧɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɮɚɤɭɥɶɬɟɬɚ ɆȽɍ, 2003.
ɉɨɞɩɢɫɚɧɨ ɜ ɩɟɱɚɬɶ 28.04.2003 Ɏɨɪɦɚɬ 60 ɯ 90 1/6 Ɉɛɴɟɦ 2,5 ɩ.ɥ. Ɂɚɤɚɡ 10 Ɍɢɪɚɠ 50 ɷɤɡ. ________________________________________________________ ɂɡɞɚɬɟɥɶɫɬɜɨ ɐɉɂ ɩɪɢ ɦɟɯɚɧɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɨɦ ɮɚɤɭɥɶɬɟɬɟ ɆȽɍ, ɝ. Ɇɨɫɤɜɚ , ȼɨɪɨɛɶɟɜɵ ɝɨɪɵ. Ʌɢɰɟɧɡɢɹ ɧɚ ɢɡɞɚɬɟɥɶɫɤɭɸ ɞɟɹɬɟɥɶɧɨɫɬɶ ɂȾ ʋ 04059 ɨɬ 30.02.200 Ɉɬɩɟɱɚɬɚɧɨ ɧɚ ɬɢɩɨɝɪɚɮɫɤɨɦ ɨɛɨɪɭɞɨɜɚɧɢɢ ɦɟɯɚɧɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɮɚɤɭɥɶɬɟɬɚ ɢ Ɏɪɚɧɤɨ-ɪɭɫɫɤɨɝɨ ɰɟɧɬɪɚ ɢɦ. Ⱥ.Ɇ. Ʌɹɩɭɧɨɜɚ
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