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2015
ɇȺɍɑɇɕɃ ȼȿɋɌɇɂɄ ɆȽɌɍ ȽȺ
ʋ 219
ɍȾɄ 531.8
ɆɈɓɇɈɋɌɖ ɋɂɅɕ ɋ ɉȿɊȿɆȿɇɇɈɃ ɌɈɑɄɈɃ ɉɊɂɅɈɀȿɇɂə
ȼ.ȼ. ɉȿɊɆəɄɈȼȺ
Ɋɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɨɫɨɛɟɧɧɨɫɬɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɜ ɨɛɥɚɫɬɢ ɬɟɯɧɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ, ɫɜɹɡɚɧɧɵɟ ɫ ɜɵɱɢɫɥɟɧɢɟɦ ɪɚɛɨɬɵ ɫɢɥɵ, ɬɨɱɤɚ ɩɪɢɥɨɠɟɧɢɹ ɤɨɬɨɪɨɣ ɢɡɦɟɧɹɟɬ ɫɜɨɟ ɩɨɥɨɠɟɧɢɟ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ. ɉɪɟɞɫɬɚɜɥɟɧɨ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ
ɬɟɨɪɟɦɵ ɫ ɜɵɱɢɫɥɟɧɢɟɦ ɦɨɳɧɨɫɬɢ ɬɚɤɨɣ ɫɢɥɵ. ɉɪɢɦɟɧɟɧɢɟ ɬɟɨɪɟɦɵ ɢ ɜɵɬɟɤɚɸɳɢɯ ɩɪɢ ɷɬɨɦ ɫɥɟɞɫɬɜɢɣ ɞɟɦɨɧɫɬɪɢɪɭɟɬɫɹ ɧɚ ɤɨɧɤɪɟɬɧɵɯ ɩɪɢɦɟɪɚɯ.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɪɚɛɨɬɚ ɫɢɥɵ, ɦɨɳɧɨɫɬɶ ɫɢɥɵ, ɜɟɤɬɨɪ ɫɢɥɵ, ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ, ɬɟɨɪɟɦɚ, ɩɪɢɦɟɪɵ
ɪɚɫɱɟɬɨɜ.
Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɷɥɟɦɟɧɬɚɪɧɚɹ ɪɚɛɨɬɚ ɫɢɥɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɤɚɥɹɪɧɵɦ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɜɟɤɬɨɪɚ
ɫɢɥɵ ɧɚ ɞɢɮɮɟɪɟɧɰɢɚɥ ɪɚɞɢɭɫɚ-ɜɟɤɬɨɪɚ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ
݀‫ ܣ‬ൌ ‫ܨ‬Ԧ ή ݀‫ݎ‬ԦǤ
(1)
Ɉɞɧɚɤɨ ɞɚɧɧɚɹ ɮɨɪɦɭɥɚ ɜɟɪɧɚ ɥɢɲɶ ɞɥɹ ɫɢɥɵ, ɜɟɤɬɨɪ ɤɨɬɨɪɨɣ ɡɚɤɪɟɩɥɟɧ ɜ ɬɨɱɤɟ ɬɟɥɚ. ȿɫɥɢ
ɠɟ ɬɨɱɤɚ ɩɪɢɥɨɠɟɧɢɹ ɜɟɤɬɨɪɚ ɫɢɥɵ ɦɟɧɹɟɬ ɫɜɨɟ ɩɨɥɨɠɟɧɢɟ ɧɚ ɬɟɥɟ, ɬɨ ɧɟ ɜɫɟ ɨɤɚɡɵɜɚɟɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɨɩɪɟɞɟɥɟɧɧɵɦ.
ȼ ɞɚɥɶɧɟɣɲɟɦ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɨɩɪɟɞɟɥɹɬɶ ɧɟ ɪɚɛɨɬɭ ɫɢɥɵ, ɚ ɟɟ ɦɨɳɧɨɫɬɶ, ɬɚɤ ɤɚɤ ɜɟɤɬɨɪ
ɫɤɨɪɨɫɬɢ ɧɚɝɥɹɞɧɟɟ ɢ ɩɪɢɜɵɱɧɟɟ ɜɟɤɬɨɪɚ ɩɟɪɟɦɟɳɟɧɢɹ.
Ⱦɥɹ ɡɚɤɪɟɩɥɟɧɧɨɝɨ ɜɟɤɬɨɪɚ ɫɢɥɵ ɦɨɳɧɨɫɬɶ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
ܹ ൌ ‫ܨ‬Ԧ ή ‫ݒ‬Ԧ,
(2)
ɝɞɟ ‫ݒ‬Ԧ - ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ ɬɟɥɚ, ɤ ɤɨɬɨɪɨɣ ɩɪɢɥɨɠɟɧ ɜɟɤɬɨɪ ɫɢɥɵ ‫ܨ‬Ԧ .
Ɍɟɨɪɟɦɚ. ɉɭɫɬɶ ɧɚ ɬɜɟɪɞɨɟ ɬɟɥɨ ɞɟɣɫɬɜɭɟɬ ɧɟɫɤɨɥɶɤɨ ɡɚɤɪɟɩɥɟɧɧɵɯ ɫɢɥ ሬሬԦ
‫ܨ‬ప , ɢɦɟɸɳɢɯ ɪɚɜɧɨԦ
ɞɟɣɫɬɜɭɸɳɭɸ ‫( ܨ‬ɪɢɫ. 1). Ɇɨɳɧɨɫɬɶ ɤɚɠɞɨɣ ɫɢɥɵ
ሬሬԦప ή ‫ݒ‬
ܹ௜ ൌ ‫ܨ‬
ሬሬሬԦ.
ప
Ʉɚɤ ɢɡɜɟɫɬɧɨ ɢɡ ɤɢɧɟɦɚɬɢɤɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
ሬሬሬԦ
ሬሬሬሬԦ଴ ൅ ߱
ሬԦ ൈ ߩ
ሬሬሬԦప ,
‫ݒ‬ప ൌ ‫ݒ‬
ɝɞɟ ‫ݒ‬
ሬሬሬሬԦ଴ – ɫɤɨɪɨɫɬɶ ɩɨɥɸɫɚ Ɉ;
ߩ
ሬሬሬԦప – ɪɚɞɢɭɫ-ɜɟɤɬɨɪ, ɨɩɪɟɞɟɥɹɸɳɢɣ ɬɨɱɤɭ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ ɧɚ ɬɟɥɟ.
Ɋɢɫ. 1. ɉɪɢɦɟɪ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɦɨɳɧɨɫɬɢ W
ɉɨɷɬɨɦɭ
ሬሬԦప ή ሺ‫ݒ‬
ሬሬԦ ሬሬሬሬԦ௢ ൅ ߱
ሬሬԦప ൯.
ܹ௜ ൌ ‫ܨ‬
ሬሬሬሬԦ௢ ൅ ߱
ሬԦ ൈ ߩ
ሬሬሬԦሻ
ሬԦ൫ߩ
ሬሬሬԦప ൈ ‫ܨ‬
ప ൌ ‫ܨ‬ప ή ‫ݒ‬
128
ȼ.ȼ. ɉɟɪɦɹɤɨɜɚ
Ɇɨɳɧɨɫɬɶ ɜɫɟɯ ɫɢɥ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɜɵɪɚɠɟɧɢɸ
ሬሬԦప ൅ ߱
ሬሬԦప ൯,
ܹ ൌ σ ܹ௜ ൌ ‫ݒ‬
ሬሬሬሬԦ଴ ή σ ‫ܨ‬
ሬԦ ή σ൫ߩ
ሬሬሬԦప ൈ ‫ܨ‬
ɝɞɟ σ ሬሬԦ
‫ܨ‬ప ൌ ‫ܨ‬Ԧ - ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɫɢɥ;
ሬሬԦప ൯ ൌ ߩ
σ൫ߩ
ሬሬሬԦప ൈ ‫ܨ‬
ሬሬሬሬԦ஼ ൈ ‫ܨ‬Ԧ – ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɬɟɨɪɟɦɟ ȼɚɪɢɧɶɨɧɚ,
ɝɞɟ ߩ
ሬሬሬሬԦ஼ - ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɨɩɪɟɞɟɥɟɧɧɨɣ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ (ɬɨɱɤɢ, ɩɪɢɧɚɞɥɟɠɚɳɟɣ ɬɟɥɭ).
Ɍɨɝɞɚ ɦɨɳɧɨɫɬɶ ɜɫɟɯ ɫɢɥ, ɚ ɡɧɚɱɢɬ ɢ ɫɢɥɵ ‫ܨ‬Ԧ
Ԧ ሬሬሬሬԦ଴ ൅ ߱
Ԧ ሬሬሬሬԦ.
ܹ ൌ ‫ܨ‬Ԧ ή ‫ݒ‬
ሬሬሬሬԦ଴ ൅ ߱
ሬԦ ή ൫ߩ
ሬሬሬሬԦ஼ ൈ ‫ܨ‬Ԧ ൯ ൌ ‫ܨ‬Ԧ ή ‫ݒ‬
ሬሬሬሬԦ଴ ൅ ‫ܨ‬Ԧ ή ሺ߱
ሬԦ ൈ ߩ
ሬሬሬሬԦሻ
ሬԦ ൈ ߩ
ሬሬሬሬԦሻ
(3)
஼ ൌ ‫ ܨ‬ή ሺ‫ݒ‬
஼ ൌ‫ܨ‬ή‫ݒ‬
஼
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɡɚɤɪɟɩɥɟɧɧɨɣ ɫɢɥɵ ‫ܨ‬Ԧ ɦɨɳɧɨɫɬɶ ɪɚɜɧɚ ɫɤɚɥɹɪɧɨɦɭ ɩɪɨɢɡɜɟɞɟɧɢɸ ɫɢɥɵ
ɧɚ ɫɤɨɪɨɫɬɶ ɬɨɣ ɬɨɱɤɢ ɬɟɥɚ, ɤ ɤɨɬɨɪɨɣ ɩɪɢɥɨɠɟɧ ɜɟɤɬɨɪ ɫɢɥɵ.
ɇɨ, ɩɭɫɬɶ ɜɟɤɬɨɪɵ ɫɢɥ, ɨɫɬɚɜɚɹɫɶ ɡɚɤɪɟɩɥɟɧɧɵɦɢ, ɧɟɩɪɟɪɵɜɧɨ ɢɡɦɟɧɹɸɬɫɹ ɢ ɩɨ ɜɟɥɢɱɢɧɟ, ɢ
ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ. Ɍɨɝɞɚ ɬɨɱɤɚ ɩɪɢɥɨɠɟɧɢɹ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ ɛɭɞɟɬ ɦɟɧɹɬɶ ɫɜɨɟ ɩɨɥɨɠɟɧɢɟ ɧɚ
ɬɟɥɟ ɢ ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ ɩɨ ɥɢɧɢɢ ɞɟɣɫɬɜɢɹ, ɚ ɮɨɪɦɭɥɚ (3) ɩɪɨɞɨɥɠɚɟɬ ɞɟɣɫɬɜɨɜɚɬɶ, ɬɚɤ ɤɚɤ ɫɢɥɵ
ɨɫɬɚɸɬɫɹ ɡɚɤɪɟɩɥɟɧɧɵɦɢ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɳɧɨɫɬɶ ɧɟɤɨɬɨɪɨɣ ɫɢɥɵ ‫ܨ‬Ԧ , ɬɨɱɤɚ ɩɪɢɥɨɠɟɧɢɹ ɤɨɬɨɪɨɣ ɡɚɧɢɦɚɟɬ ɪɚɡɥɢɱɧɵɟ ɩɨɥɨɠɟɧɢɹ ɜ ɬɟɥɟ, ɟɫɬɶ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɤɬɨɪɚ ɷɬɨɣ ɫɢɥɵ ɧɚ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɬɨɣ ɬɨɱɤɢ ɬɟɥɚ, ɫ ɤɨɬɨɪɨɣ ɫɨɜɩɚɞɚɟɬ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɧɚɱɚɥɨ ɜɟɤɬɨɪɚ ɫɢɥɵ.
ɍɞɨɛɧɨ ɜɜɟɫɬɢ ɩɨɧɹɬɢɟ ɫɤɨɪɨɫɬɢ, ɫ ɤɨɬɨɪɨɣ ɬɨɱɤɚ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ ɦɟɧɹɟɬ ɫɜɨɟ ɩɨɥɨɠɟɧɢɟ
ɜ ɬɟɥɟ – ɨɬɧɨɫɢɬɟɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ሬሬሬԦ
‫ݒ‬௥ (ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɫɤɪɟɩɥɟɧɧɨɣ ɫ ɬɟɥɨɦ) ɢ
ɚɛɫɨɥɸɬɧɭɸ ɫɤɨɪɨɫɬɶ ‫ݒ‬Ԧ – ɫɤɨɪɨɫɬɶ ɩɟɪɟɦɟɳɟɧɢɹ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ (ɧɚɱɚɥɚ ɜɟɤɬɨɪɚ ɫɢɥɵ)
ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɣ ɧɟɩɨɞɜɢɠɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ. ɉɟɪɟɧɨɫɧɨɣ ɫɤɨɪɨɫɬɶɸ ɛɭɞɟɬ ɬɨɝɞɚ
ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ ɋ ɬɟɥɚ, ɫ ɤɨɬɨɪɨɣ ɬɨɱɤɚ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ ɫɨɜɩɚɞɚɟɬ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ (‫ݒ‬
ሬሬሬԦ௘ ൌ ሬሬሬሬԦ).
‫ݒ‬஼
ɗɬɢ ɩɨɧɹɬɢɹ ɩɨɡɜɨɥɹɸɬ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɬɪɢ ɫɥɟɞɫɬɜɢɹ, ɜɵɬɟɤɚɸɳɢɟ ɢɡ ɬɟɨɪɟɦɵ.
1. Ɍɚɤ ɤɚɤ ሬሬሬሬԦ
‫ݒ‬஼ ൌ ‫ݒ‬Ԧ െ ‫ݒ‬
ሬሬሬԦ,
௥ ɬɨ
ܹ ൌ ‫ܨ‬Ԧ ή ‫ݒ‬Ԧ െ ‫ܨ‬Ԧ ή ‫ݒ‬
ሬሬሬԦ௥ .
Ԧ
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɳɧɨɫɬɶ ɧɟɤɨɬɨɪɨɣ ɫɢɥɵ ‫ ܨ‬ɫ ɩɟɪɟɦɟɧɧɨɣ ɬɨɱɤɨɣ ɩɪɢɥɨɠɟɧɢɹ ɪɚɜɧɚ ɦɨɳɧɨɫɬɢ ɩɪɢ ɚɛɫɨɥɸɬɧɨɦ ɞɜɢɠɟɧɢɢ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɜɟɤɬɨɪɚ ɫɢɥɵ ɦɢɧɭɫ ɦɨɳɧɨɫɬɶ ɟɟ ɩɪɢ
ɞɜɢɠɟɧɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɟɥɚ.
Ɂɧɚɱɟɧɢɟ W ɩɪɢ ɡɚɤɪɟɩɥɟɧɧɨɦ ɜɟɤɬɨɪɟ ɫɢɥɵ (‫ݒ‬
ሬሬሬԦ௥ ൌ Ͳ) ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
ܹ ൌ ‫ܨ‬Ԧ ή ‫ݒ‬Ԧ.
(4)
2. ȿɫɥɢ ɬɟɥɨ ɞɜɢɠɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ (ɫɤɨɪɨɫɬɢ ɜɫɟɯ ɬɨɱɟɤ ɨɞɢɧɚɤɨɜɵ), ɬɨ ɥɸɛɨɟ ɩɟɪɟɦɟɳɟɧɢɟ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɜɟɤɬɨɪɚ ɫɢɥɵ ɩɨ ɬɟɥɭ ɦɨɠɧɨ ɧɟ ɭɱɢɬɵɜɚɬɶ ɢ ɩɪɟɞɩɨɥɚɝɚɬɶ ɷɬɨɬ
ɜɟɤɬɨɪ ɡɚɤɪɟɩɥɟɧɧɵɦ ɜ ɩɪɨɢɡɜɨɥɶɧɨ ɜɵɛɪɚɧɧɨɣ ɬɨɱɤɟ ɬɟɥɚ.
3. ȿɫɥɢ ɩɨɥɨɠɟɧɢɟ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɜɟɤɬɨɪɚ ɫɢɥɵ ɦɟɧɹɟɬɫɹ ɧɚ ɬɟɥɟ ɜɞɨɥɶ ɥɢɧɢɢ ɞɟɣɫɬɜɢɹ, ɦɨɳɧɨɫɬɶ, ɚ ɡɧɚɱɢɬ ɢ ɪɚɛɨɬɚ, ɧɟ ɢɡɦɟɧɹɟɬɫɹ, ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɬɚɤɨɝɨ ɢɡɦɟɧɟɧɢɹ ɬɨɱɤɢ
ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ.
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɭɫɬɶ ɬɨɱɤɚ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ ɩɟɪɟɦɟɫɬɢɥɚɫɶ ɩɨ ɥɢɧɢɢ ɞɟɣɫɬɜɢɹ ɢɡ M1 ɜ M2
(ɪɢɫ. 2), ɩɪɢ ɷɬɨɦ:
- ɜ ɩɟɪɜɨɦ ɩɨɥɨɠɟɧɢɢ ɟɟ ɦɨɳɧɨɫɬɶ
Ԧ ሬሬሬሬԦ଴ ൅ ߱
ሬሬሬሬԦଵ ൌ ‫ܨ‬Ԧ ή ሺ‫ݒ‬
ሬሬሬሬԦ଴ ൅ ߱
ሬԦ ൈ ߩ
ሬሬሬሬԦሻ
ሬԦ ή ൫ߩ
ሬሬሬሬԦଵ ൈ ‫ܨ‬Ԧ ൯;
ܹଵ ൌ ‫ܨ‬Ԧ ή ‫ݒ‬
ଵ ൌ‫ܨ‬ή‫ݒ‬
- ɜɨ ɜɬɨɪɨɦ ɩɨɥɨɠɟɧɢɢ
Ԧ ሬሬሬሬԦ଴ ൅ ߱
ሬሬሬሬԦଶ ൌ ‫ܨ‬Ԧ ή ሺ‫ݒ‬
ሬሬሬሬԦ଴ ൅ ߱
ሬԦ ൈ ߩ
ሬሬሬሬԦሻ
ሬԦ ή ൫ߩ
ሬሬሬሬԦଶ ൈ ‫ܨ‬Ԧ ൯.
ܹଶ ൌ ‫ܨ‬Ԧ ή ‫ݒ‬
ଶ ൌ‫ܨ‬ή‫ݒ‬
ɇɨ ሬሬሬሬԦ
ߩଵ ൈ ‫ܨ‬Ԧ ɢ ሬሬሬሬԦ
ߩଶ ൈ ‫ܨ‬Ԧ – ɟɫɬɶ ɦɨɦɟɧɬɵ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ Ɉ ɜ ɩɟɪɜɨɦ ɢ ɜɬɨɪɨɦ ɩɨɥɨɠɟɧɢɹɯ ɫɢɥɵ. Ɇɨɦɟɧɬɵ ɷɬɢ ɪɚɜɧɵ, ɬ.ɟ. ܹଵ ൌ ܹଶ .
ɉɨɷɬɨɦɭ ɢɡɦɟɧɟɧɢɟ ɩɨɥɨɠɟɧɢɹ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ ɜɞɨɥɶ ɥɢɧɢɢ ɞɟɣɫɬɜɢɹ ɦɨɠɧɨ ɧɟ
ɭɱɢɬɵɜɚɬɶ ɩɪɢ ɜɵɱɢɫɥɟɧɢɢ ɪɚɛɨɬɵ ɢ ɦɨɳɧɨɫɬɢ (ɫɢɥɭ ɦɨɠɧɨ ɩɟɪɟɧɨɫɢɬɶ ɩɨ ɥɢɧɢɢ ɞɟɣɫɬɜɢɹ ɜ
ɩɪɨɢɡɜɨɥɶɧɭɸ ɬɨɱɤɭ ɬɟɥɚ).
Ɇɨɳɧɨɫɬɶ ɫɢɥɵ ɫ ɩɟɪɟɦɟɧɧɨɣ ɬɨɱɤɨɣ ɩɪɢɥɨɠɟɧɢɹ
129
Ɋɢɫ. 2. ɉɪɢɦɟɪ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɦɨɳɧɨɫɬɢ W1
ɋɢɥɵ, ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɤɨɬɨɪɵɯ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɟɥɚ ɩɟɪɟɦɟɳɚɸɬɫɹ, ɜɫɬɪɟɱɚɸɬɫɹ ɧɟɪɟɞɤɨ.
Ɉɛɵɱɧɨ ɷɬɨ ɫɢɥɵ ɬɪɟɧɢɹ, ɜɨɡɧɢɤɚɸɳɢɟ ɩɪɢ ɫɤɨɥɶɠɟɧɢɢ ɨɞɧɨɝɨ ɬɟɥɚ ɩɨ ɩɨɜɟɪɯɧɨɫɬɢ ɞɪɭɝɨɝɨ;
ɫɢɥɵ ɬɹɠɟɫɬɢ ɬɟɥɚ ɩɟɪɟɦɟɧɧɨɣ ɦɚɫɫɵ; ɧɚɬɹɠɟɧɢɟ ɧɢɬɢ, ɧɚɦɨɬɚɧɧɨɣ ɧɚ ɬɟɥɨ ɢ ɞɪ.
ɉɪɢɦɟɪ. Ɍɟɥɨ 1 ɫɤɨɥɶɡɢɬ ɜɥɟɜɨ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɝɥɚɞɤɨɣ ɩɥɨɫɤɨɫɬɢ. Ʉ ɧɟɦɭ ɩɪɢɠɢɦɚɟɬɫɹ
ɧɟɩɨɞɜɢɠɧɨɟ ɬɟɥɨ 2 (ɪɢɫ. 3). ȼɨɡɧɢɤɚɸɬ ɫɢɥɵ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ. Ɉɞɧɚ ሬሬሬԦ
‫ܨ‬ଵ ɩɪɢɥɨɠɟɧɚ ɤ ɬɟɥɭ 1,
ɜɬɨɪɚɹ ሬሬሬԦ
‫ܨ‬ଶ – ɤ ɬɟɥɭ 2. Ʉɨɧɟɱɧɨ, ɷɬɢ ɫɢɥɵ ɩɨ ɜɟɥɢɱɢɧɟ ɪɚɜɧɵ. Ɍɪɟɛɭɟɬɫɹ ɜɵɱɢɫɥɢɬɶ ɪɚɛɨɬɭ ɫɢɥ
ɬɪɟɧɢɹ ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ ɬɟɥɚ 1 ɧɚ ɪɚɫɫɬɨɹɧɢɟ S.
Ɋɢɫ. 3. Ʉ ɨɩɪɟɞɟɥɟɧɢɸ ɪɚɛɨɬɵ ɫɢɥ ɬɪɟɧɢɹ
Ɋɚɛɨɬɚ ɫɢɥɵ ሬሬሬԦ
‫ܨ‬ଶ ɪɚɜɧɚ ɧɭɥɸ, ɬɚɤ ɤɚɤ ɬɟɥɨ 2, ɤ ɤɨɬɨɪɨɦɭ ɨɧɚ ɩɪɢɥɨɠɟɧɚ, ɧɟɩɨɞɜɢɠɧɨ. Ɋɚɛɨɬɚ
ሬሬሬԦ
ɠɟ ɫɢɥɵ ‫ܨ‬ଵ , ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɞɜɢɠɭɳɟɟɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɬɟɥɨ 1, ɜɵɱɢɫɥɹɟɬɫɹ ɫ ɭɱɟɬɨɦ ɬɨɝɨ, ɱɬɨ
ɫɢɥɚ ሬሬሬԦ
‫ܨ‬ଵ ɩɪɢɤɪɟɩɥɟɧɚ ɤ ɬɟɥɭ ɢ ɞɜɢɠɟɬɫɹ ɜɦɟɫɬɟ ɫ ɧɢɦ (ɫɥɟɞɫɬɜɢɟ 3) ɢ ɫɢɥɵ ɬɪɟɧɢɹ ɩɪɢɧɢɦɚɸɬɫɹ
ɩɨɫɬɨɹɧɧɵɦɢ ɩɨ ɮɨɪɦɭɥɟ
‫ ܣ‬ൌ െ‫ܨ‬ଵ ή ܵ.
ȿɫɥɢ ɬɟɥɨ 1 ɢɦɟɟɬ ɮɨɪɦɭ ɤɥɢɧɚ (ɪɢɫ. 4), ɬɨ ɪɟɡɭɥɶɬɚɬ ɩɨɥɭɱɚɟɬɫɹ ɢɧɨɝɨ ɫɨɞɟɪɠɚɧɢɹ.
ሬሬሬԦଵ ɞɟɣɫɬɜɭɟɬ ɧɚ ɬɟɥɨ 1 ɢ ɬɨɱɤɚ ɩɪɢɥɨɠɟɧɢɹ ɟɟ ɩɟɪɟɦɟɳɚɟɬɫɹ ɩɨ ɟɝɨ ɩɨɜɟɪɯɧɨɫɬɢ. ɋɚɦɨ
ɋɢɥɚ ‫ܨ‬
ɬɟɥɨ ɞɜɢɠɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ ‫ݒ‬
ሬሬሬሬԦ,
ଵ ɜ ɫɜɹɡɢ ɫ ɱɟɦ, ɭɱɢɬɵɜɚɹ ɫɥɟɞɫɬɜɢɟ 3, ɡɚɤɪɟɩɥɹɟɦ
ɫɢɥɭ ɢ ɨɩɪɟɞɟɥɹɟɦ ɟɟ ɦɨɳɧɨɫɬɶ
ܹଵ ൌ െ‫ܨ‬ଵ ή ‫ݒ‬ଵ ή …‘• ߙ.
ሬሬሬԦ
ɋɢɥɚ ‫ܨ‬ଶ ɩɪɢɥɨɠɟɧɚ ɤ ɬɟɥɭ 2 (ɹɜɥɹɟɬɫɹ ɡɚɤɪɟɩɥɟɧɧɨɣ ɫɢɥɨɣ). Ɇɨɳɧɨɫɬɶ ɟɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ
ɮɨɪɦɭɥɟ
ܹଶ ൌ െ‫ܨ‬ଶ ή ‫ݒ‬ଶ ή •‹ ߙ ൌ െ‫ܨ‬ଶ ή ‫ݒ‬ଵ ή –ƒ ߙ ή •‹ ߙ.
130
ȼ.ȼ. ɉɟɪɦɹɤɨɜɚ
Ɋɢɫ. 4. Ʉ ɨɩɪɟɞɟɥɟɧɢɸ ɪɚɛɨɬɵ ɫɢɥ ɬɪɟɧɢɹ (ɬɟɥɨ ɢɦɟɟɬ ɮɨɪɦɭ ɤɥɢɧɚ)
ɋɭɦɦɚɪɧɚɹ ɦɨɳɧɨɫɬɶ ɫɢɥ ɬɪɟɧɢɹ (‫ܨ‬ଵ ൌ ‫ܨ‬ଶ ൌ ‫)ܨ‬
௩
ܹ ൌ ܹଵ ൅ ܹଶ ൌ െ‫ ܨ‬ή ‫ݒ‬ଵ ሺ…‘• ߙ ൅ –ƒ ߙ ή •‹ ߙሻ ൌ െ‫ ܨ‬ή భ ൌ െ‫ ܨ‬ή ‫ݒ‬ˑ˕ː .
ୡ୭ୱ ఈ
ɉɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɹ ɦɨɳɧɨɫɬɢ ɧɨɪɦɚɥɶɧɵɯ ɪɟɚɤɰɢɣ ܰଵ ɢ ܰଶ . Ɍɨɱɤɚ
ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ ܰଵ ɩɟɪɟɦɟɳɚɟɬɫɹ ɩɨ ɬɟɥɭ 1. ɉɨɷɬɨɦɭ ɦɨɳɧɨɫɬɶ ɷɬɨɣ ɫɢɥɵ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ
ɜɵɪɚɠɟɧɢɸ
ܹଵ ൌ െܰଵ ή ‫ݒ‬ଵ ή •‹ ߙ.
ȼɬɨɪɚɹ ɫɢɥɚ ܰଶ ɩɪɢɥɨɠɟɧɚ ɤ ɬɟɥɭ 2. Ɇɨɳɧɨɫɬɶ ɟɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
ܹଶ ൌ ܰଶ ή ‫ݒ‬ଶ ή …‘• ߙ ൌ ܰଶ ή ‫ݒ‬ଵ ή –ƒ ߙ ή …‘• ߙ ൌ ܰଶ ή ‫ݒ‬ଵ ή •‹ ߙ.
Ɍɚɤ ɤɚɤ ܰଵ ൌ ܰଶ , ɬɨ ɫɭɦɦɚɪɧɚɹ ɦɨɳɧɨɫɬɶ, ɤɚɤ ɢ ɫɥɟɞɨɜɚɥɨ ɨɠɢɞɚɬɶ, ɪɚɜɧɚ ɧɭɥɸ.
POWER FORCES WITH VARIABLE APPLICATION POINT
Permyakova V.V.
The article shows algorithm of solving problems in calculating work of force applied to the floating point in a solid. A
theorem proof is provided calculating power of such a force. The use of the theorem and its consequence is shown on some
examples.
Keywords: work of force, power of force, point of force, floating action point of force, forces with floating action
point.
ɋɜɟɞɟɧɢɹ ɨɛ ɚɜɬɨɪɟ
ɉɟɪɦɹɤɨɜɚ ȼɟɪɚ ȼɥɚɞɢɦɢɪɨɜɧɚ, ɨɤɨɧɱɢɥɚ ɇɨɪɢɥɶɫɤɢɣ ɢɧɞɭɫɬɪɢɚɥɶɧɵɣ ɢɧɫɬɢɬɭɬ (1979), ɤɚɧɞɢɞɚɬ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ, ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ ɬɟɯɧɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ ɢ ɢɧɠɟɧɟɪɧɨɣ ɝɪɚɮɢɤɢ ɆȽɌɍ ȽȺ, ɚɜɬɨɪ
31 ɧɚɭɱɧɨɣ ɪɚɛɨɬɵ, ɨɛɥɚɫɬɶ ɧɚɭɱɧɵɯ ɢɧɬɟɪɟɫɨɜ – ɦɟɯɚɧɢɤɚ, ɤɨɧɫɬɪɭɢɪɨɜɚɧɢɟ ɦɟɯɚɧɢɱɟɫɤɨɝɨ ɨɛɨɪɭɞɨɜɚɧɢɹ.
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