Опря А.Т. Статистика (модульний варіант з програмованою формою контролю знань)
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ɉɪɢ ɩɪɚɜɢɥɶɧɨɦɭ ɜɿɞɛɨɪɿ ɨɡɧɚɤ ɧɚ ɤɨɠɧɨɦɭ ɫɬɭɩɟɧɿ ɜɿɞɛɨɪɭ
ɦɨɠɧɚ ɜɢɤɨɪɢɫɬɚɬɢ ɬɿ ɡ ɧɢɯ, ɹɤɿ ɫɩɨɫɬɟɪɿɝɚɸɬɶɫɹ ɭ ɜɢɛɿɪɰɿ ɜɟɥɢɤɨɝɨ
ɨɛ’ɽɦɭ ɹɤ «ɤɨɧɬɪɨɥɶɧɿ», ɬɨɛɬɨ ɤɨɪɢɝɭɸɬɶ ɧɚ ʀɯ ɩɿɞɫɬɚɜɿ ɪɟɡɭɥɶɬɚɬɢ
ɫɩɨɫɬɟɪɟɠɟɧɧɹ ɩɨɜ’ɹɡɚɧɢɯ ɡ ɧɢɦɢ ɨɡɧɚɤ.
Ȼɚɝɚɬɨɫɬɭɩɟɧɟɜɚ ɜɢɛɿɪɤɚ ɡɚɫɬɨɫɨɜɭɽɬɶɫɹ ɩɪɢ ɨɞɟɪɠɚɧɧɿ ɩɪɨɛ ɞɥɹ
ɥɚɛɨɪɚɬɨɪɧɢɯ ɚɧɚɥɿɡɿɜ. ɇɚɩɪɢɤɥɚɞ, ɩɪɢ ɜɿɞɛɨɪɿ ɩɪɨɛɧɢɯ ɡɪɚɡɤɿɜ
ɩɪɨɞɭɤɰɿʀ, ɹɤɚ ɧɚɞɯɨɞɢɬɶ ɜ ɪɿɡɧɿɣ ɬɚɪɿ (ɛɨɱɤɚɯ, ɹɳɢɤɚɯ, ɦɿɲɤɚɯ ɬɨɳɨ).
ɋɩɨɱɚɬɤɭ ɜɿɞɛɢɪɚɸ
ɬɶ ɩɟɜɧɭ ɤɿɥɶɤɿɫɬɶ ɨɞɢɧɢɰɶ ɩɪɨɞɭɤɰɿʀ ɜ ɬɚɪɿ, ɚ
ɩɨɬɿɦ ɛɟɪɭɬɶ ɩɪɨɛɢ ɜ ɫɟɪɟɞɢɧɿ ɡ ɜɿɞɿɛɪɚɧɢɯ ɜɢɞɿɜ ɬɚɪɢ (ɛɨɱɨɤ, ɦɿɲɤɿɜ
ɬɨɳɨ). Ɂɨɤɪɟɦɚ, ɹɤɳɨ ɡɟɪɧɨ ɨɞɟɪɠɭɸɬɶ ɜ ɦɿɲɤɚɯ, ɫɩɨɱɚɬɤɭ
ɜɿɞɛɢɪɚɸɬɶ ɩɟɜɧɭ ɤɿɥɶɤɿɫɬɶ ɦɿɲɤɿɜ, ɚ ɩɨɬɿɦ ɡ ɧɢɯ ɛɟɪɭɬɶ ɩɪɨɛɢ
ɡɟɪɧɚ
ɳɭɩɨɦ ɡɜɟɪɯɭ, ɜɫɟɪɟɞɢɧɿ ɿ ɡɧɢɡɭ ɤɨɠɧɨɝɨ ɦɿɲɤɚ.
ȼ ɨɤɪɟɦɢɯ ɜɢɩɚɞɤɚɯ ɭɬɜɨɪɸɸɬɶ ɬɚɤ ɡɜɚɧɭ ɫɟɪɟɞɧɸ ɩɪɨɛɭ
ɲɥɹɯɨɦ ɡɦɿɲɭɜɚɧɧɹ ɨɤɪɟɦɢɯ ɩɪɨɛ. Ⱦɨɞɚɬɤɨɜɢɦɢ ɫɬɭɩɟɧɹɦɢ ɜɿɞɛɨɪɭ
ɬɭɬ ɛɭɞɭɬɶ ɧɚɫɬɭɩɧɿ ɩɪɨɛɢ, ɹɤɿ ɛɟɪɭɬɶ ɭ ɜɢɝɥɹɞɿ ɞɟɹɤɨʀ ɱɚɫɬɢɧɢ ɞɨ
ɜɢɯɿɞɧɨʀ (ɫɟɪɟɞɧɶɨʀ) ɩɪɨɛɢ; ɡ ɨɫɬɚɧɧɶɨʀ ɛɟɪɭɬɶ ɡɧɨɜ ɩɟɜɧɭ ɱɚɫɬɢɧɭ ɿ
ɬɚɤ ɞɨɬɢ, ɞɨɤɢ ɪɨɡɦɿɪ ɩɪɨɛɢ ɧɟ ɡɦɟɧɲɢɬɶɫɹ ɞɨ ɧɨɪɦɢ, ɩɪɢɞɚɧɨʀ ɞɥɹ
ɥɚɛɨɪɚɬɨɪɧɨɝɨ ɚɧɚɥɿɡɭ.
§ 11.4. ɉɨɦɢɥɤɢ ɜɢɛɿɪɤɢ, ʀɯ ɜɢɡɧɚɱɟɧɧɹ ɩɪɢ ɪɿɡɧɢɯ
ɫɩɨɫɨɛɚɯ ɜɿɞɛɨɪɭ
Ɇɿɠ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɜɢɛɿɪɤɨɜɨʀ ɫɭɤɭɩɧɨɫɬɿ ɿ ɲɭɤɚɧɢɦɢ
ɩɚɪɚɦɟɬɪɚɦɢ ɜɿɞɩɨɜɿɞɧɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɝɟɧɟɪɚɥɶɧɨʀ ɫɭɤɭɩɧɨɫɬɿ
ɿɫɧɭɸɬɶ ɩɟɜɧɿ ɪɨɡɛɿɠɧɨɫɬɿ. Ȳɯ ɧɚɡɢɜɚɸɬɶ ɩɨɦɢɥɤɚɦɢ ɫɩɨɫɬɟɪɟɠɟɧɧɹ.
Ɂɚɝɚɥɶɧɚ ɜɟɥɢɱɢɧɚ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɨɜɨɝɨ ɫɩɨɫɬɟɪɟɠ
ɟɧɧɹ
ɡɭɦɨɜɥɸɽɬɶɫɹ ɦɨɠɥɢɜɿɫɬɸ ɜɢɧɢɤɧɟɧɧɹ ɞɜɨɯ ɜɢɞɿɜ ɩɨɦɢɥɨɤ: ɩɨɦɢɥɤɢ
ɪɟɽɫɬɪɚɰɿʀ ɿ ɩɨɦɢɥɤɢ ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ.
ɉɨɦɢɥɤɢ ɪɟɽɫɬɪɚɰɿʀ ɜɢɧɢɤɚɸɬɶ ɜɧɚɫɥɿɞɨɤ ɧɟɞɨɫɬɚɬɧɶɨɝɨ ɪɿɜɧɹ
ɤɜɚɥɿɮɿɤɚɰɿʀ ɩɪɚɰɿɜɧɢɤɿɜ, ɧɟɬɨɱɧɨɫɬɿ ɩɿɞɪɚɯɭɧɤɿɜ, ɧɟɞɨɫɤɨɧɚɥɨɫɬɿ
ɜɢɦɿɪɸɜɚɥɶɧɢɯ ɩɪɢɥɚɞɿɜ ɿ ɬ.ɿɧ. Ƀɦɨɜɿɪɧɿɫɬɶ ɜɢɧɢɤɧɟɧɧɹ ɩɨɦɢɥɨɤ
ɪɟɽɫɬɪɚɰɿʀ ɩɪɢ ɜɢɛɿɪɤɨɜɨɦɭ ɨɛɫɬɟɠɟɧɧɿ ɡɧɚɱɧɨ ɦɟɧɲɚ, ɧɿɠ ɩɪɢ
ɫɭɰɿɥɶɧɨɦɭ, ɚɞɠɟ ɜɢɛɿɪɤɨɜɟ ɡɞɿɣɫɧɸɽɬɶɫɹ ɤɜɚɥɿɮɿɤɨɜ
ɚɧɿɲɢɦɢ
ɩɪɚɰɿɜɧɢɤɚɦɢ ɿ ɨɪɝɚɧɿɡɨɜɭɽɬɶɫɹ ɛɿɥɶɲ ɪɟɬɟɥɶɧɨ ɿ ɤɨɧɫɬɪɭɤɬɢɜɧɨ, ɧɿɠ
ɫɭɰɿɥɶɧɟ. ɉɪɢ ɜɢɛɿɪɤɨɜɨɦɭ ɫɩɨɫɬɟɪɟɠɟɧɧɿ ɡɚɜɞɹɤɢ ɫɤɨɪɨɱɟɧɧɸ
ɤɿɥɶɤɨɫɬɿ ɞɨɫɥɿɞɠɭɜɚɧɢɯ ɨɞɢɧɢɰɶ ɡɧɚɱɧɨ ɡɦɟɧɲɭɽɬɶɫɹ ɦɨɠɥɢɜɿɫɬɶ
ɨɞɟɪɠɚɬɢ
ɩɨɦɢɥɤɢ ɪɟɽɫɬɪɚɰɿʀ. ɋɩɟɰɿɚɥɶɧɨ ɩɿɞɿɛɪɚɧɿ ɿ ɧɚɜɱɟɧɿ
ɫɩɨɫɬɟɪɟɠɭɜɚɱɿ ɧɟ ɡɚɰɿɤɚɜɥɟɧɿ ɭ ɜɢɤɪɢɜɥɟɧɧɿ ɫɩɨɫɬɟɪɟɠɭɜɚɧɢɯ ɞɚɧɢɯ,
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ɳɨ ɬɚɤɨɠ ɫɩɪɢɹɽ ɨɞɟɪɠɚɧɧɸ ɛɿɥɶɲ ɨɛ’ɽɤɬɢɜɧɨʀ ɿɧɮɨɪɦɚɰɿʀ ɩɪɨ
ɨɛɫɬɟɠɭɜɚɧɭ ɫɭɤɭɩɧɿɫɬɶ ɨɛ’ɽɤɬɿɜ.
ɍ ɬɨɣ ɠɟ ɱɚɫ ɩɪɢ ɜɢɛɿɪɤɨɜɨɦɭ ɫɩɨɫɬɟɪɟɠɟɧɧɿ ɜɢɧɢɤɚɸɬɶ
ɩɨɦɢɥɤɢ, ɹɤɿ ɧɟ ɦɚɸɬɶ ɦɿɫɰɹ ɩɪɢ ɫɭɰɿɥɶɧɨɦɭ ɨɛɫɬɟɠɟɧɧɿ – ɩɨɦɢɥɤɢ
ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ . ȼɨɧɢ ɹɜɥɹɸɬɶ ɫɨɛɨɸ ɪɨɡɛɿɠɧɿɫɬɶ ɦɿɠ
ɜɟɥɢɱɢɧɨɸ ɨɞɟɪɠɚɧɢɯ ɩɨ ɜɢɛɿɪɰɿ ɩɨɤɚɡɧɢɤɿɜ ɿ ɜɟɥɢɱɢɧɨɸ ɬɢɯ
ɩɨɤɚɡɧɢɤɿɜ, ɤɨɬɪɿ ɛɭɥɢ ɛ ɨɞɟɪɠɚɧɿ ɩɪɢ ɩɪɨɜɟɞɟɧɧɿ ɡ ɨɞɧɚɤɨɜɢɦ ɪɿɜɧɟɦ
ɬɨɱɧɨɫɬɿ ɫɭɰɿɥɶɧɨɝɨ ɫɩɨɫɬɟɪɟɠ
ɟɧɧɹ.
Ɉɬɠɟ, ɩɨɦɢɥɤɚ ɜɢɛɿɪɤɢ (ɩɨɦɢɥɤɚ ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ) – ɰɟ
ɚɛɫɨɥɸɬɧɚ ɜɟɥɢɱɢɧɚ ɪɿɡɧɢɰɿ ɦɿɠ ɜɿɞɩɨɜɿɞɧɢɦɢ ɜɢɛɿɪɤɨɜɨɸ ɿ
ɝɟɧɟɪɚɥɶɧɨɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ: (
ɯɯ
~
) – ɩɨɦɢɥɤɚ ɞɥɹ ɫɟɪɟɞɧɶɨʀ; (w-p)
– ɩɨɦɢɥɤɚ ɞɥɹ ɱɚɫɬɤɢ (w, p - ɱɚɫɬɤɚ ɨɡɧɚɤɢ ɜɿɞɩɨɜɿɞɧɨ ɭ ɜɢɛɿɪɤɨɜɿɣ ɿ
ɝɟɧɟɪɚɥɶɧɿɣ ɫɭɤɭɩɧɨɫɬɹɯ). ɉɪɢɪɨɞɚ ɜɢɧɢɤɧɟɧɧɹ ɬɚɤɨʀ ɩɨɦɢɥɤɢ
ɩɨɥɹɝɚɽ ɜ ɬɨɦɭ, ɳɨ ɜɢɛɿɪɤɨɜɚ ɫɭɤɭɩɧɿɫɬɶ ɧɟ ɬɨɱɧɨ ɜɿɞɬɜɨɪɸɽ
ɝɟɧɟɪɚɥɶɧɭ ɫɭɤɭɩɧɿɫɬɶ.
ɉɨɦɢɥɤɢ ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ ɦɨɠɭɬɶ ɛɭɬɢ ɜɢɩɚɞɤɨɜɢɦɢ ɿ
ɫɢɫɬɟɦɚɬɢɱɧɢɦɢ. Ɍɚɤ, ɩɪɢ ɜɢɛɿɪɤɨɜɨɦɭ ɜɢɤɨɩɭɜɚɧɧɿ ɤɨɪɟɧɿɜ
ɰɭɤɪɨɜɢɯ ɛɭɪɹɤɿɜ ɞɥɹ ɜɢɡɧɚɱɟɧɧɹ ʀɯ ɭɪɨɠɚɣɧɨɫɬɿ ɭ ɜɢɛɿɪɤɨɜɭ
ɫɭɤɭɩɧɿɫɬɶ ɜɢɩɚɞɤɨɜɨ ɦɨɠɭɬɶ ɩɨɬɪɚɩɢɬɢ ɞɟɳɨ ɤɪɚɳɿ ɜɿɞ ɫɟɪɟɞɧɿɯ
ɟɤɡɟɦɩɥ
ɹɪɢ. ɍ ɰɶɨɦɭ ɜɢɩɚɞɤɭ ɦɨɠɟ ɣɬɢ ɦɨɜɚ ɩɪɨ ɜɢɩɚɞɤɨɜɭ ɩɨɦɢɥɤɭ
ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ. ɍ ɪɚɡɿ, ɹɤɳɨ ɭ ɜɢɛɿɪɤɭ ɛɭɞɭɬɶ ɫɢɫɬɟɦɚɬɢɱɧɨ
ɜɿɞɛɢɪɚɬɢɫɹ ɤɪɚɳɿ
ɟɤɡɟɦɩɥɹɪɢ, ɬɨ ɦɨɜɚ ɛɭɞɟ ɣɬɢ ɩɪɨ ɫɢɫɬɟɦɚɬɢɱɧɭ
ɩɨɦɢɥɤɭ ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ, ɹɤɚ ɡɭɦɨɜɥɟɧɚ ɧɚɜɦɢɫɧɢɦ ɩɨɪɭɲɟɧɧɹɦ
ɩɪɚɜɢɥ ɜɿɞɛɨɪɭ.
Ɍɚɤɢɦ ɱɢɧɨɦ – ɫɢɫɬɟɦɚɬɢɱɧɿ ɩɨɦɢɥɤɢ ɫɩɪɹɦɨɜɚɧɿ ɜ ɨɞɢɧ ɛɿɤ ɿ
ɦɨɠɭɬɶ ɜɢɧɢɤɚɬɢ ɭ ɡɜ’ɹɡɤɭ ɡ ɨɫɨɛɥɢɜɨɫɬɹɦɢ ɩɪɢɣɧɹɬɨʀ ɫɢɫɬɟɦɢ
ɜɿɞɛɨɪɭ ɿ ɨɛɪɨɛɤɢ ɞɚɧɢɯ ɫɩɨɫɬɟɪɟɠɟɧɧɹ ɚɛɨ ɭ ɡɜ’ɹɡɤɭ ɡ ɩɨɪɭɲɟɧɧɹɦ
ɜɫɬɚɧɨɜɥɟɧɢɯ ɩɪɚɜɢɥ ɿ ɩɪɢɧɰɢɩɿɜ ɜɿɞɛɨɪɭ.
ȼɢɩɚɞɤɨɜɿ ɩɨɦɢɥɤɢ ɧɟ ɦɚɸɬɶ ɩɟɜɧɨɝɨ ɧɚɩɪɹɦɭ. Ȳɯ ɜɢɧɢɤɧɟɧɧɹ
ɩɨɹɫɧɸ
ɽɬɶɫɹ ɧɟɞɨɫɬɚɬɧɶɨ ɪɿɜɧɨɦɿɪɧɢɦ ɩɪɟɞɫɬɚɜɥɟɧɧɹɦ ɭ ɜɢɛɿɪɤɨɜɿɣ
ɫɭɤɭɩɧɨɫɬɿ ɪɿɡɧɢɯ ɤɚɬɟɝɨɪɿɣ ɨɞɢɧɢɰɶ ɝɟɧɟɪɚɥɶɧɨʀ ɫɭɤɭɩɧɨɫɬɿ.
Ɉɫɤɿɥɶɤɢ ɪɨɡɩɨɞɿɥ ɨɞɢɧɢɰɶ ɫɩɨɫɬɟɪɟɠɟɧɧɹ ɜɢɛɿɪɤɨɜɨʀ ɫɭɤɭɩɧɨɫɬɿ ɧɟ
ɡɨɜɫɿɦ ɬɨɱɧɨ ɜɿɞɬɜɨɪɸ
ɽ ɪɨɡɩɨɞɿɥ ɨɞɢɧɢɰɶ ɝɟɧɟɪɚɥɶɧɨʀ ɫɭɤɭɩɧɨɫɬɿ,
ɜɢɛɿɪɤɚ ɧɟ ɦɨɠɟ ɬɨɱɧɨ ɜɿɞɨɛɪɚɠɚɬɢ ɝɟɧɟɪɚɥɶɧɭ ɫɭɤɭɩɧɿɫɬɶ, ɚ ɨɬɠɟ,
ɩɨɜɧɿɫɬɸ ɭɫɭɧɭɬɢ ɜɢɩɚɞɤɨɜɿ ɩɨɦɢɥɤɢ ɧɟɦɨɠɥɢɜɨ, ʀɯ ɦɨɠɧɚ ɡɜɟɫɬɢ ɞɨ
ɧɟɡɧɚɱɧɢɯ ɪɨɡɦɿɪɿɜ.
ɉɢɬɚɧɧɹ ɜɢɡɧɚɱɟɧɧɹ ɦɨɠɥɢɜɨʀ ɿ ɮɚɤɬɢɱɧɨʀ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ ɦɚɽ
ɩɟɪɲɨɱɟɪɝɨɜɟ ɡɧɚɱɟɧɧɹ ɩɪɢ ɨɪɝɚɧɿɡɚɰɿʀ ɿ ɩɪɨɜɟɞɟɧɧɿ ɜɢɛɿɪɤɨɜɨɝɨ
ɨɛɫɬɟɠɟɧɧɹ. Ȳʀ ɜɟɥɢɱɢɧɚ ɯɚɪɚɤɬɟɪɢɡɭɽ ɫɬɭɩɿɧɶ ɧɚɞɿɣɧɨɫɬɿ ɨɞɟɪɠɚɧɢɯ
282
283
ɪɟɡɭɥɶɬɚɬɿɜ ɜɢɛɿɪɤɨɜɨɝɨ ɨɛɫɬɟɠɟɧɧɹ ɿ ɡɭɦɨɜɥɸɽ ɨɛ’ɽɤɬɢɜɧɿɫɬɶ ɨɰɿɧɨɤ
ɩɚɪɚɦɟɬɪɿɜ ɝɟɧɟɪɚɥɶɧɨʀ ɫɭɤɭɩɧɨɫɬɿ. əɤ ɿ ɫɚɦɚ ɜɢɛɿɪɤɨɜɚ
ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ, ɩɨɦɢɥɤɚ ɜɢɛɿɪɤɢ ɽ ɜɢɩɚɞɤɨɜɨɸ ɜɟɥɢɱɢɧɨɸ. Ɋɨɡɦɿɪ
ɜɢɩɚɞɤɨɜɨʀ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ ɜɢɡɧɚɱɚɽɬɶɫɹ ɡɝɿɞɧɨ ɿɡ ɝɪɚɧɢɱɧɢɦɢ
ɬɟɨɪɟɦɚɦɢ ɣɦɨɜɿɪɧɨɫɬɟɣ. Ɋɨɡɪɿɡɧɹɸɬɶ ɫɟɪɟɞɧɸ ɿ ɝɪɚɧɢɱɧɭ ɩɨɦɢɥɤɭ
ɜɢɛɿɪɤɢ. ɉɿɞ ɫɟɪɟɞɧɶɨɸ (ɫɬɚɧɞɚɪɬɧɨɸ) ɩɨɦɢɥɤɨɸ ɜɢɛɿɪɤɢ
ɪɨɡɭɦɿɸɬɶ ɬɚɤɟ ɪɨɡɯɨɞɠɟɧɧɹ ɦɿɠ ɜɢɛɿɪɤɨɜɨɸ ɿ ɝɟɧɟɪɚɥɶɧɨɸ
ɫɟɪɟɞɧɶɨɸ (
ɯɯ
~
), ɹɤɟ ɧɟ ɩɟɪɟɜɢɳɭɽ ɪɨɡɦɿɪ ɫɟɪɟɞɧɶɨɝɨ ɤɜɚɞɪɚɬɢɱɧɨɝɨ
ɜɿɞɯɢɥɟɧɧɹ (
V
r
). Ɇɚɤɫɢɦɚɥɶɧɨ ɦɨɠɥɢɜɟ ɪɨɡɯɨɞɠɟɧɧɹ (
ɯɯ
~
)
ɧɚɡɢɜɚɸɬɶ ɝɪɚɧɢɱɧɨɸ ɩɨɦɢɥɤɨɸ ɜɢɛɿɪɤɢ, ɬɨɛɬɨ – ɰɟ ɦɚɤɫɢɦɭɦ
ɩɨɦɢɥɤɢ ɩɪɢ ɡɚɞɚɧɿɣ ɿɦɨɜɿɪɧɨɫɬɿ ʀʀ ɩɨɹɜɢ.
ȱɫɧɭɸɬɶ ɞɜɿ ɮɨɪɦɭɥɢ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ. Ɉɞɧɚ ɡ ɧɢɯ
ɜɢɤɨɪɢɫɬɨɜɭɽɬɶɫɹ ɩɪɢ ɜɢɦɿɪɸɜɚɧɧɿ ɫɟɪɟɞɧɶɨɝɨ ɡɧɚɱɟɧɧɹ ɨɡɧɚɤɢ
(ɧɚɩɪɢɤɥɚɞ, ɧɚ ɩɿɞɩɪɢɽɦɫɬɜɿ ɜɢɛɿɪɤɨɜɨ ɨɛɫɬɟɠɭɽɬɶɫɹ ɫɟɪɟɞɧɿɣ ɪɨɡɦɿɪ
ɡɚɪɩɥɚɬɢ ɩɪɚɰɸɸɱɢɯ), ɞɪɭɝɚ – ɤɨɥɢ ɜɢɛɿɪɤɨɜɨ ɜɢɦɿɪɸɽɬɶɫɹ ɱɚɫɬɤɚ
ɨɡɧɚɤɢ (ɧɚɩɪɢɤɥɚɞ, ɱɚɫɬɤɚ ɜɢɫɨɤɨɨɩɥɚɱɟɧɢɯ ɩɪɚɰɿɜɧɢɤɿɜ ɧɚ
ɩɿɞɩɪɢɽɦɫɬɜɿ).
Ʉɨɥɢ ɜɢɛɿɪɤɚ ɡɞɿɣɫɧɸ
ɽɬɶɫɹ ɡɚ ɩɪɢɧɰɢɩɨɦ ɩɨɜɬɨɪɧɨɝɨ ɜɿɞɛɨɪɭ, ɬɨ
ɮɨɪɦɭɥɢ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ ɦɚɸɬɶ ɜɢɝɥɹɞ: ɞɥɹ ɫɟɪɟɞɧɶɨʀ -
ɩ
ɬ
2
V
ɞɥɹ ɱɚɫɬɤɢ -
n
ww
ɬ
)1(
.
ɉɨɜɬɨɪɧɭ ɜɢɛɿɪɤɭ ɜɢɤɨɪɢɫɬɨɜɭɸɬɶ ɞɭɠɟ ɪɿɞɤɨ. əɤ ɩɪɚɜɢɥɨ,
ɜɢɛɿɪɤɚ ɨɪɝɚɧɿɡɨɜɭɽɬɶɫɹ ɡɚ ɩɪɢɧɰɢɩɨɦ ɛɟɡɩɨɜɬɨɪɧɨɝɨ ɜɿɞɛɨɪɭ.
ɋɬɨɫɨɜɧɨ ɞɨ ɰɶɨɝɨ ɩɪɢɧɰɢɩɭ ɜɿɞɛɨɪɭ ɜ ɧɚɜɟɞɟɧɢɯ ɜɢɳɟ ɮɨɪɦɭɥɚɯ
ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ ɜ ɩɿɞɤɨɪɟɧɟɜɢɣ ɜɢɪɚɡ ɜɜɨɞɢɬɶɫɹ ɞɨɞɚɬɤɨɜɢɣ
ɦɧɨɠɧɢɤ (
1
ɩ
1
), ɞɟ N - ɱɢɫɟɥɶɧɿɫɬɶ ɝɟɧɟɪɚɥɶɧɨʀ ɫɭɤɭɩɧɨɫɬɿ.
Ɉɬɠɟ, ɞɥɹ ɛɟɡɩɨɜɬɨɪɧɨʀ ɜɢɛɿɪɤɢ ɮɨɪɦɭɥɢ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ
ɧɚɛɭɞɭɬɶ ɜɢɝɥɹɞɭ:
ɚ) ɩɪɢ ɜɢɡɧɚɱɟɧɧɿ ɫɟɪɟɞɧɶɨɝɨ ɡɧɚɱɟɧɧɹ ɨɡɧɚɤɢ -
)1(
2
1
ɩ
ɩ
ɬ
V
;
ɛ) ɩɪɢ ɜɢɡɧɚɱɟɧɧɿ ɱɚɫɬɤɢ ɨɡɧɚɤɢ -
)1(
)1(
N
n
n
ww
ɬ u
.
Ɍɟɨɪɿɽɸ ɣɦɨɜɿɪɧɨɫɬɟɣ ɞɨɜɟɞɟɧɨ : ɫɬɜɟɪɞɠɟɧɧɹ ɩɪɨ ɬɟ, ɳɨ
ɝɟɧɟɪɚɥɶɧɿ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɧɟ ɜɿɞɯɢɥɹɸɬɶɫɹ ɜɿɞ ɜɢɛɿɪɤɨɜɢɯ ɧɚ
ɜɟɥɢɱɢɧɭ ɛɿɥɶɲɭ, ɧɿɠ ɜɟɥɢɱɢɧɚ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ (m), ɡɚɜɠɞɢ ɦɚɽ
ɩɨɫɬɿɣɧɢɣ ɫɬɭɩɿɧɶ ɿɦɨɜɿɪɧɨɫɬɿ 0,683. ȱɦɨɜɿɪɧɿɫɬɶ ɫɬɜɟɪɞɠɟɧɧɹ ɦɨɠɧɚ
283
284
ɩɿɞɜɢɳɢɬɢ, ɩɨɞɜɨʀɜɲɢ ɚɛɨ ɩɨɬɪɨʀɜɲɢ ɫɟɪɟɞɧɸ ɩɨɦɢɥɤɭ (2 m; 3 m). ɍ
ɰɶɨɦɭ ɜɢɩɚɞɤɭ ɣɦɨɜɿɪɧɿɫɬɶ ɫɬɜɟɪɞɠɟɧɶ ɞɨɫɹɝɚɽ ɪɿɜɧɿɜ 0,954 ɚɛɨ 0,997,
ɬɨɛɬɨ ɡ ɬɢɫɹɱɿ ɜɢɩɚɞɤɿɜ ɜɿɞɩɨɜɿɞɧɨ ɜ 954 ɿ 997 ɜɢɩɚɞɤɚɯ ɜɢɛɿɪɤɨɜɿ
ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɛɭɞɭɬɶ ɜɿɞɪɿɡɧɹɬɢɫɹ ɜɿɞ ɝɟɧɟɪɚɥɶɧɢɯ ɧɚ ɜɟɥɢɱɢɧɭ
ɨɛɱɢɫɥɟɧɨʀ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ. ɍ ɪɟɲɬɢ ɜɢɩɚɞɤɿɜ (46 ɿ 3) ɜɿɞɯɢɥɟɧɧɹ
ɝɟɧɟɪɚɥɶɧɢɯ ɿ ɜɢɛɿɪɤɨɜɢɯ ɩɚɪɚɦɟɬɪɿɜ ɦɨɠɟ ɜɢɯɨɞɢɬɢ ɡɚ ɦɟɠɿ
ɨɛɱɢɫɥɟɧɨʀ ɩɨɦɢɥɤɢ.
Ɍɚɤɢɦ ɱɢɧɨɦ, ɳɨɛ ɩɿɞɜɢɳɢɬɢ ɣɦɨɜɿɪɧɿɫɬɶ ɫɬɜɟɪɞɠɟɧɧɹ,
ɧɟɨɛɯɿɞɧɨ ɪɨɡɲɢɪɢɬɢ ɦɟɠɿ ɜɿɞɯɢɥɟɧɶ ɲɥɹɯɨɦ ɡɛɿɥɶɲɟɧɧɹ ɫɟɪɟɞɧɶɨʀ
ɩɨɦɢɥɤɢ ɜ
V
ɯɯ
~
ɪɚɡɿɜ, ɞɟ ɜɿɞɧɨɲɟɧɧɹ ɪɿɡɧɢɰɿ ɫɟɪɟɞɧɿɯ ɞɨ ɫɟɪɟɞɧɶɨɝɨ
ɤɜɚɞɪɚɬɢɱɧɨɝɨ ɜɿɞɯɢɥɟɧɧɹ ɹɜɥɹɽ ɫɨɛɨɸ ɜɟɥɢɱɢɧɭ ɬɚɤ ɡɜɚɧɨɝɨ
ɧɨɪɦɨɜɚɧɨɝɨ ɜɿɞɯɢɥɟɧɧɹ (t).
Ɉɬɠɟ, ɡ ɜɢɡɧɚɱɟɧɨɸ ɣɦɨɜɿɪɧɿɫɬɸ ɦɨɠɧɚ ɫɬɜɟɪɞɠɭɜɚɬɢ, ɳɨ
ɜɿɞɯɢɥɟɧɧɹ ɝɟɧɟɪɚɥɶɧɢɯ ɿ ɜɢɛɿɪɤɨɜɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɧɟ ɩɟɪɟɜɢɳɚɬɶ
ɞɟɹɤɨʀ ɜɟɥɢɱɢɧɢ – ɝɪɚɧɢɱɧɨʀ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ (¨). Ƚɪɚɧɢɱɧɚ ɩɨɦɢɥɤɚ
ɜɢɛɿɪɤɢ ɩɨɜ’ɹɡɚɧɚ ɡ ɫɟɪɟɞɧɶɨɸ ɩɨɦɢɥɤɨɸ ɪɿɜɧɹɧɧɹɦ ¨= t m, ɞɟ t-
ɧɨɪɦɨɜɚɧɟ ɜɿɞɯɢɥɟɧɧɹ (ɤɨɟɮɿɰɿɽɧɬ ɤɪɚɬɧɨɫɬɿ, ɤɨɟɮɿɰɿɽɧɬ ɞɨɜɿɪɢ), ɹɤɟ
ɡɚɥɟɠɢɬɶ ɜɿɞ ɪɿɜɧɹ ɣɦɨɜɿɪɧɨɫɬɿ.
ȼɟɥɢɱɢɧɚ ɣɦɨɜɿɪɧɨɫɬɿ ɡɚɞɚɽɬɶɫɹ ɡɚɥɟɠɧɨ ɜɿɞ ɦɟɬɢ ɿ ɡɚɜɞɚɧɶ
ɞɨɫɥɿɞɠ
ɟɧɧɹ. Ƀɦɨɜɿɪɧɿɫɬɶ ɩɨɬɪɚɩɥɹɧɧɹ ɩɨɦɢɥɤɢ ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ
ɭ ɦɟɠɚɯ
tr
ɜɢɡɧɚɱɚɽɬɶɫɹ ɡɚ ɮɨɪɦɭɥɨɸ ɿɧɬɟɝɪɚɥɚ ɣɦɨɜɿɪɧɨɫɬɟɣ
dtetp
t
t
2
0
2
2
1
)(
³
S
.
Ɍɚɛɥɢɰɹ 87
ȼɢɬɹɝ ɿɡ ɫɬɚɧɞɚɪɬɧɢɯ ɬɚɛɥɢɰɶ «Ɏɭɧɤɰɿɹ Ʌɚɩɥɚɫɚ»
t 1,00 1,96 2,00 2,50 2,58 3,00 3,30
ɪ 0,683 0,950 0,954 0,997 0,990 0,667 0,999
Ɂɧɚɱɟɧɧɹ ɰɶɨɝɨ ɿɧɬɟɝɪɚɥɚ ɦɿɫɬɢɬɶɫɹ ɜ ɫɬɚɧɞɚɪɬɧɢɯ ɦɚɬɟɦɚɬɢɱɧɢɯ
ɬɚɛɥɢɰɹɯ «Ɏɭɧɤɰɿɹ Ʌɚɩɥɚɫɚ» (ɞɢɜ. ɞɨɞɚɬ. 5, ). ȼ ɬɚɛɥɢɰɿ 87 ɧɚɜɟɞɟɧɿ
ɪɿɜɧɿ ɣɦɨɜɿɪɧɨɫɬɟɣ ɪ ɞɥɹ ɞɟɹɤɢɯ ɰɿɥɢɯ ɿ ɞɪɨɛɨɜɢɯ ɡɧɚɱɟɧɶ t.
ɉɪɢɩɭɫɬɢɦɨ, ɳɨ ɩɨɦɢɥɤɭ ɜɢɛɿɪɤɢ ɬɪɟɛɚ ɨɰɿɧɢɬɢ ɡ ɿɦɨɜɿɪɧɿɫɬɸ
0,954. ɐɟ ɨɡɧɚɱɚɽ, ɳɨ ɪɨɡɛɿɠɧɿɫɬɶ ɦɿɠ ɜɢɛɿɪɤɨɜɨɸ ɿ ɝɟɧɟɪɚɥɶɧɨɸ
ɫɟɪɟɞɧɶɨɸ ɧɟ ɩɟɪɟɜɢɳɢɬɶ ɞɜɨɯ ɜɟɥɢɱɢɧ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ, ɬɨɛɬɨ ɜ
95,4 % ɜɢɩɚɞɤɿɜ ɩɨɦɢɥɤɚ ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ ɧɟ ɜɢɣɞɟ ɡɚ ɦɟɠɿ
2r
;
ɩɪɢ ɣɦɨɜɿɪɧɨɫɬɿ 0,997 – ɡɚ ɦɟɠɿ
3r
ɿ ɬ. ɿɧ.
284
285
Ⱦɥɹ ɱɢɫɟɥɶɧɨ ɦɚɥɢɯ ɫɬɚɬɢɫɬɢɱɧɢɯ ɫɭɤɭɩɧɨɫɬɟɣ ɧɟ ɦɨɠɟ ɛɭɬɢ
ɡɚɫɬɨɫɨɜɚɧɚ ɬɟɨɪɟɦɚ Ʌɹɩɭɧɨɜɚ, ɹɤɚ ɡ’ɹɫɨɜɭɽ ɡɚɝɚɥɶɧɿ ɭɦɨɜɢ, ɩɪɢ
ɡɞɿɣɫɧɟɧɧɿ ɤɨɬɪɢɯ ɪɨɡɩɨɞɿɥ ɫɭɦɢ ɧɟɡɚɥɟɠɧɢɯ ɜɢɩɚɞɤɨɜɢɯ ɜɟɥɢɱɢɧ
ɩɪɹɦɭɽ ɞɨ ɧɨɪɦɚɥɶɧɨɝɨ, ɨɫɤɿɥɶɤɢ ɡɧɚɱɟɧɧɹ ɜɢɛɿɪɤɨɜɨʀ ɫɟɪɟɞɧɶɨʀ (
ɯ
)
ɬɭɬ ɡɚɧɚɞɬɨ ɡɚɥɟɠɢɬɶ ɜɿɞ ɜɟɥɢɱɢɧɢ ɤɨɠɧɨʀ ɜɢɩɚɞɤɨɜɨʀ ɡɦɿɧɧɨʀ.
ɏɚɪɚɤɬɟɪ ɪɨɡɩɨɞɿɥɭ
ɯ
ɜ ɰɢɯ ɭɦɨɜɚɯ ɛɭɞɟ ɿɫɬɨɬɧɨ ɜɿɞɪɿɡɧɹɬɢɫɹ ɜɿɞ
ɧɨɪɦɨɜɚɧɨɝɨ ɪɨɡɩɨɞɿɥɭ, ɚ ɞɨɜɿɪɱɿ ɿɧɬɟɪɜɚɥɢ ɿ ɞɨɜɿɪɱɿ ɣɦɨɜɿɪɧɨɫɬɿ (ɩɪɨ
ɧɢɯ ɦɨɜɚ ɩɿɞɟ ɧɢɠɱɟ) ɩɪɢ ɦɚɥɢɯ ɜɢɛɿɪɤɚɯ ɦɨɠɭɬɶ ɛɭɬɢ ɪɨɡɪɚɯɨɜɚɧɿ
ɬɿɥɶɤɢ ɡɚ ɭɦɨɜ ɧɨɪɦɚɥɶɧɨɝɨ ɪɨɡɩɨɞɿɥɭ ɞɨɫɥɿɞɠɭɜɚɧɨʀ ɨɡɧɚɤɢ. Ɂɚ
ɪɨɡɪɚɯɭɧɤɚɦɢ ɋɬ’ɸɞɟɧɬɚ, ɣɦɨɜɿɪɧɿɫɬɶ ɬɨɝɨ, ɳɨ ɚɛɫɨɥɸɬɧɚ ɜɟɥɢɱɢɧɚ
ɪɿɡɧɢɰɿ ɜɢɛɿɪɤɨɜɨʀ ɿ ɝɟɧɟɪɚɥɶɧɨʀ ɫɟɪɟɞɧɶɨʀ ɛɭɞɟ ɦɟɧɲɟ ɝɪɚɧɢɱɧɨʀ
ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ (
P
tɯɯ ¢
~
) ɿ ɹɜɥɹɽ ɫɨɛɨɸ ɮɭɧɤɰɿɸ ɜɿɞ ɧɨɪɦɚɬɢɜɧɨɝɨ
ɜɿɞɯɢɥɟɧɧɹ (t) ɿ ɱɢɫɟɥɶɧɨɫɬɿ ɜɢɛɿɪɤɢ (ɩ). Ɏɨɪɦɭɥɚ ɰɶɨɝɨ ɞɨɜɟɞɟɧɧɹ
ɦɚɽ ɜɢɝɥɹɞ:
dt
n
t
Atɯɯɪ
n
t
t
2
2
2
)
1
1()
~
(
³ ¢
P
,
ɞɟ
)
2
1
(2(
)
2
(
ɩ
Ƚɩɩ
ɩ
Ƚ
A
,
)
2
(
ɩ
Ƚ
- ɝɚɦɦɚ – ɮɭɧɤɰɿɹ.
ɍ ɩɪɚɤɬɢɱɧɢɯ ɪɨɡɪɚɯɭɧɤɚɯ ɜɢɤɨɪɢɫɬɨɜɭɸɬɶɫɹ ɬɚɛɥɢɰɿ ɪɨɡɩɨɞɿɥɭ
ɋ’ɸɞɟɧɬɚ S (t), ɜ ɹɤɢɯ ɞɚɧɨ ɪɿɜɧɿ ɣɦɨɜɿɪɧɨɫɬɟɣ ɞɥɹ ɪɿɡɧɢɯ ɡɧɚɱɟɧɶ ɩ ɿ t
(ɞɨɞɚɬ.2 ).
ɇɚ ɨɫɧɨɜɿ ɬɟɨɪɟɬɢɱɧɢɯ ɪɿɜɧɿɜ ɿɦɨɜɿɪɧɨɫɬɟɣ ɪɨɡɪɚɯɨɜɭɸɬɶ
ɮɚɤɬɢɱɧɿ ʀɯ ɪɿɜɧɿ. ɉɪɢ ɰɶɨɦɭ ɪɨɡɪɚɯɭɧɤɨɜɚ ɣɦɨɜɿɪɧɿɫɬɶ (ɪ)
ɫɬɚɧɨɜɢɬɢɦɟ : ɪ =
>@
25.0)( tS
. ȱɡ ɫɤɚɡɚɧɨɝɨ ɜɢɳɟ ɜɢɩɥɢɜɚɽ, ɳɨ ɩɿɫɥɹ
ɨɛɱɢɫɥɟɧɧɹ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ ɜɢɧɢɤɚɽ ɩɢɬɚɧɧɹ ɨɛɱɢɫɥɟɧɧɹ
ɝɪɚɧɢɱɧɨʀ ɩɨɦɢɥɤɢ ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ (¨) ɪɨɡɦɿɪ ʀʀ ɭ ɜɢɛɿɪɤɨɜɨɦɭ
ɫɩɨɫɬɟɪɟɠɟɧɧɿ ɦɨɠɟ ɛɭɬɢ ɦɟɧɲɢɣ ɚɛɨ ɛɿɥɶɲɢɣ ɜɿɞ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ
ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ (
P
). Ɂɝɿɞɧɨ ɬɟɨɪɟɦɢ ɑɟɛɢɲɟɜɚ ɿ Ʌɹɩɭɧɨɜɚ, ɹɤɚ
ɜɢɡɧɚɱɚɽ ɣɦɨɜɿɪɧɿɫɬɶ ɬɨɝɨ, ɳɨ ɝɪɚɧɢɱɧɚ ɩɨɦɢɥɤɚ ɜɢɛɿɪɤɢ ɧɟ
ɩɟɪɟɜɢɳɢɬɶ t ɪɚɡɿɜ ɜɡɹɬɭ ɫɟɪɟɞɧɸ ɩɨɦɢɥɤɭ ɜɢɛɿɪɤɢ (
P
), ɜɢɪɿɲɭɸɬɶ
ɩɢɬɚɧɧɹ ɩɪɨ ɝɪɚɧɢɱɧɭ ɩɨɦɢɥɤɭ. ɇɚɜɟɞɟɦɨ ɮɨɪɦɭɥɸɜɚɧɧɹ ɬɟɨɪɟɦɢ
ɑɟɛɢɲɟɜɚ: ɡ ɿɦɨɜɿɪɧɿɫɬɸ, ɹɤ ɡɚɜɝɨɞɧɨ ɛɥɢɡɶɤɨɸ ɞɨ ɨɞɢɧɢɰɿ, ɦɨɠɧɚ
ɫɬɜɟɪɞɠɭɜɚɬɢ, ɳɨ ɩɪɢ ɞɨɫɬɚɬɧɶɨ ɜɟɥɢɤɿɣ ɤɿɥɶɤɨɫɬɿ ɧɟɡɚɥɟɠɧɢɯ
ɫɩɨɫɬɟɪɟɠɟɧɶ ɜɢɛɿɪɤɨɜɚ ɫɟɪɟɞɧɹ (
ɯ
~
) ɛɭɞɟ ɹɤ ɡɚɜɝɨɞɧɨ ɦɚɥɨ
ɜɿɞɪɿɡɧɹɬɢɫɹ ɜɿɞ ɝɟɧɟɪɚɥɶɧɨʀ ɫɟɪɟɞɧɶɨʀ (
ɯ
~
). Ɉɬɠɟ, ɝɪɚɧɢɱɧɚ ɩɨɦɢɥɤɚ
ɜɢɛɿɪɤɢ (¨) ɨɛɱɢɫɥɸɽɬɶɫɹ ɡ ɩɟɜɧɨɸ ɣɦɨɜɿɪɧɿɫɬɸ (ɪ), ɹɤɿɣ ɜɿɞɩɨɜɿɞɚɽ t
– ɪɚɡɨɜɟ ɡɧɚɱɟɧɧɹ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ (
P
):¨= t
P
.
285
286
Ɇɟɠɿ ɫɟɪɟɞɧɶɨʀ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɜ ɝɟɧɟɪɚɥɶɧɿɣ ɫɭɤɭɩɧɨɫɬɿ
ɫɬɚɧɨɜɢɬɢɦɭɬɶ: ɞɥɹ ɫɟɪɟɞɧɶɨʀ -
'r ɯɯ
~
;
ɞɥɹ ɱɚɫɬɤɢ -
'r wɪ
.
ɍ ɪɨɡɝɨɪɧɭɬɨɦɭ ɜɢɝɥɹɞɿ ɮɨɪɦɭɥɢ ɝɪɚɧɢɱɧɨʀ ɩɨɦɢɥɤɢ ɞɥɹ
ɩɨɜɬɨɪɧɨʀ ɿ ɛɟɡɩɨɜɬɨɪɧɨʀ ɫɯɟɦɢ ɜɿɞɛɨɪɭ ɧɚɜɟɞɟɧɿ ɭ ɧɚɫɬɭɩɧɨɦɭ
ɩɪɢɤɥɚɞɿ.
ɉɪɢɤɥɚɞ. Ɋɨɡɝɥɹɧɟɦɨ ɤɨɧɤɪɟɬɧɢɣ ɩɪɢɤɥɚɞ ɪɨɡɪɚɯɭɧɤɭ ɝɪɚɧɢɱɧɨʀ ɩɨɦɢɥɤɢ
ɜɢɛɿɪɤɢ ɩɪɢ ɜɢɡɧɚɱɟɧɧɿ ɫɟɪɟɞɧɶɨʀ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɭ ɜɢɛɿɪɤɨɜɿɣ ɫɭɤɭɩɧɨɫɬɿ ɿ
ɱɚɫɬɤɢ ɜɢɛɿɪɤɢ.
ɍɦɨɜɚ. ɍ ɫɿɥɶɫɶɤɨɝɨɫɩɨɞɚɪɫɶɤɢɯ ɩɿɞɩɪɢɽɦɫɬɜɚɯ ɪɚɣɨɧɭ ɩɥɨɳɚ ɡɟɪɧɨɜɢɯ
ɤɭɥɶɬɭɪ ɫɬɚɧɨɜɢɬɶ 20000 ɝɚ. ɉɪɢ 10 % - ɦɭ ɛɟɡɩɨɜɬɨɪɧɨɦɭ ɜɿɞɛɨɪɿ ɜɫɬɚɧɨɜɥɟɧɨ,
ɳɨ ɫɟɪɟɞɧɹ ɭɪɨɠɚɣɧɿɫɬɶ ɡɟɪɧɨɜɢɯ ɜ ɪɚɣɨɧɿ ɞɨɪɿɜɧɸɽ 30 ɰ/ɝɚ,
ɫɟɪɟɞɧɶɨɤɜɚɞɪɚɬɢɱɧɟ ɜɿɞɯɢɥɟɧɧɹ ɭɪɨɠɚɣɧɨɫɬɿ ɫɬɚɧɨɜɢɬɶ 2 ɰ. ɉɢɬɨɦɚ ɜɚɝɚ
ɜɢɫɨɤɨɜɪɨɠɚɣɧɢɯ ɤɭɥɶɬɭɪ 60%. ɉɨɬɪɿɛɧɨ ɜɢɡɧɚɱɢɬɢ ɡ ɣɦɨɜɿɪɧɿɫɬɸ 0,954
ɝɪɚɧɢɱɧɭ ɩɨɦɢɥɤɭ ɫɟɪɟɞɧɶɨʀ ɜɪɨɠɚɣɧɨɫɬɿ ɡɟɪɧɨɜɢɯ ɤɭɥɶɬɭɪ ɩɨ ɜɢɛɿɪɰɿ ɿ
ɝɪɚɧɢɱɧɭ ɩɨɦɢɥɤɭ ɱɚɫɬɤɢ, ɬɨɛɬɨ ɩɢɬɨɦɨʀ ɜɚɝɢ ɜɢɫɨɤɨɜɪɨɠɚɣɧɢɯ ɤɭɥɶɬɭɪ ɜ
ɡɚɝɚɥɶɧɿɣ ɩɥɨɳɿ ɩɨɫɿɜɭ.
ɏɿɞ ɪɿɲɟɧɧɹ. ȼɫɬɚɧɨɜɥɸɽɦɨ ɱɢɫɟɥɶɧɿɫɬɶ ɜɢɛɿɪɤɨɜɨʀ ɫɭɤɭɩɧɨɫɬɿ – 10% ɜɿɞ
20000 ɝɚ, ɜɨɧɚ ɞɨɪɿɜɧɸɽ 2000 ɝɚ. Ɇɚɽɦɨ :
ɯ
~
= 30; N= 20000; ɩ= 2000;
V
= 2; w=
0,60; ɪ= 0,954; t= 2.
08,0)
20000
2000
1(
2000
4
2)1(
2
'
N
n
n
t
ɯ
V
.
Ɉɬɠɟ, ɪɿɡɧɢɰɹ ɦɿɠ ɜɢɛɿɪɤɨɜɨɸ ɫɟɪɟɞɧɶɨɸ ɭɪɨɠɚɣɧɿɫɬɸ ɿ ɝɟɧɟɪɚɥɶɧɨɸ
ɫɟɪɟɞɧɶɨɸ ɛɭɞɟ ɧɟ ɛɿɥɶɲɚ ɡɚ 0,08 ɰ. Ɇɟɠɿ ɫɟɪɟɞɧɶɨʀ ɝɟɧɟɪɚɥɶɧɨʀ ɭɪɨɠɚɣɧɨɫɬɿ ɜ
ɰɟɧɬɧɟɪɚɯ:
08,303092,29 dd
.
Ƚɪɚɧɢɱɧɚ ɩɨɦɢɥɤɚ ɞɥɹ ɱɚɫɬɤɢ ɫɬɚɧɨɜɢɬɶ:
02,0)
20000
2000
1(
2000
)60,01(60,0
2)1(
)1(
'
N
n
n
ww
t
w
.
Ɍɚɤɢɦ ɱɢɧɨɦ, ɩɨɦɢɥɤɚ ɭ ɜɢɡɧɚɱɟɧɧɿ ɱɚɫɬɤɢ ɜɢɫɨɤɨɜɪɨɠɚɣɧɢɯ ɤɭɥɶɬɭɪ ɭ
ɜɢɛɿɪɤɨɜɿɣ ɫɭɤɭɩɧɨɫɬɿ ɧɟ ɩɟɪɟɜɢɳɢɬɶ 2 %, ɬɨɛɬɨ ɩɢɬɨɦɚ ɜɚɝɚ ɜɢɫɨɤɨɜɪɨɠɚɣɧɢɯ
ɤɭɥɶɬɭɪ ɡɧɚɯɨɞɢɬɶɫɹ ɭ ɦɟɠɚɯ
%626058 dd
.
ȼɟɥɢɱɢɧɚ ɜɢɩɚɞɤɨɜɨʀ ɩɨɦɢɥɤɢ ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ ɡɚɥɟɠɢɬɶ: 1) ɜɿɞ
ɫɩɨɫɨɛɭ ɮɨɪɦɭɜɚɧɧɹ (ɜɿɞɛɨɪɭ) ɜɢɛɿɪɤɨɜɨʀ ɫɭɤɭɩɧɨɫɬɿ; 2) ɜɿɞ ɨɛɫɹɝɭ ɜɢɛɿɪɤɢ; 3)
ɜɿɞ ɫɬɭɩɟɧɹ ɜɚɪɿɚɰɿʀ ɞɨɫɥɿɞɠɭɜɚɧɨʀ ɨɡɧɚɤɢ ɭ ɝɟɧɟɪɚɥɶɧɿɣ ɫɭɤɭɩɧɨɫɬɿ.
Ⱥ ɰɟ ɨɡɧɚɱɚɽ, ɳɨ ɞɥɹ ɨɞɟɪɠɚɧɧɹ ɦɿɧɿɦɚɥɶɧɨʀ ɩɨɦɢɥɤɢ ɧɟɨɛɯɿɞɧɨ
ɞɨɬɪɢɦɭɜɚɬɢɫɹ ɬɚɤɢɯ ɦɚɬɟɦɚɬɢɱɧɢɯ ɩɨɥɨɠɟɧɶ: 1) ɱɢɦ ɛɿɥɶɲɢɣ ɨɛɫɹɝ ɜɢɛɿɪɤɢ,
ɬɢɦ ɩɨɜɧɿɲɟ ɜɡɚɽɦɨɩɨɝɚɲɚɸɬɶɫɹ ɜɢɩɚɞɤɨɜɿ ɜɿɞɯɢɥɟɧɧɹ. ȼɟɥɢɱɢɧɚ ɩɨɦɢɥɤɢ
ɜɢɛɿɪɤɢ ɨɛɟɪɧɟɧɨ ɩɪɨɩɨɪɰɿɣɧɚ ɤɨɪɟɧɸ ɤɜɚɞɪɚɬɧɨɦɭ ɡ ɱɢ
ɫɟɥɶɧɨɫɬɿ ɜɢɛɿɪɤɢ. ɉɪɢ
ɡɛɿɥɶɲɟɧɧɿ ɜɢɛɿɪɤɨɜɨʀ ɫɭɤɭɩɧɨɫɬɿ ɭ ɱɨɬɢɪɢ ɪɚɡɢ ɩɨɦɢɥɤɚ ɜɢɛɿɪɤɢ ɡɦɟɧɲɭɽɬɶɫɹ
ɭ ɞɜɚ ɪɚɡɢ; 2) ɡɛɿɥ
ɶɲɟɧɧɹ ɩɨɤɚɡɧɢɤɚ ɜɚɪɿɚɰɿʀ ɞɨɫɥɿɞɠɭɜɚɧɨʀ ɨɡɧɚɤɢ ɡɭɦɨɜɥɸɽ
ɡɛɿɥɶɲɟɧɧɹ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ, ɬɨɛɬɨ ɜɟɥɢɱɢɧɚ ɨɫɬɚɧɧɶɨʀ ɩɪɹɦɨ ɩɪɨɩɨɪɰɿɣɧɚ
ɫɟɪɟɞɧɶɨɦɭ ɤɜɚɞɪɚɬɭ ɜɿɞɯɢɥɟɧɶ.
286
287
ɋɥɿɞ ɩɚɦ’ɹɬɚɬɢ, ɳɨ ɩɪɢ ɜɢɛɿɪɤɨɜɨɦɭ ɨɛɫɬɟɠɟɧɧɿ ɜɿɞɫɭɬɧɹ
ɿɧɮɨɪɦɚɰɿɹ ɩɪɨ ɪɨɡɦɿɪ ɞɢɫɩɟɪɫɿʀ, ɬɨɦɭ ɜɟɥɢɱɧɚ ʀʀ ɩɪɢɣɦɚɽɬɶɫɹ
ɧɚɛɥɢɠɟɧɢɦ ɩɨɤɚɡɧɢɤɨɦ ɭ ɜɢɝɥɹɞɿ ɜɢɛɿɪɤɨɜɨɝɨ ɫɟɪɟɞɧɶɨɝɨ ɤɜɚɞɪɚɬɚ
ɜɿɞɯɢɥɟɧɶ.
Ⱦɥɹ ɤɨɠɧɨɝɨ ɤɨɧɤɪɟɬɧɨɝɨ ɫɩɨɫɨɛɭ ɜɿɞɛɨɪɭ ɭ ɜɢɛɿɪɤɨɜɭ
ɫɭɤɭɩɧɿɫɬɶ ɜɟɥɢɱɢɧɚ ɩɨɦɢɥɤɢ ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ ɦɨɠɟ ɛɭɬɢ
ɜɢɡɧɚɱɟɧɚ ɡɚ ɜɿɞɩɨɜɿɞɧɢɦɢ ɮɨɪɦɭɥɚɦɢ.
ɉɨɜɟɪɧɟɦɨɫɹ ɞɨ ɩɨɜɬɨɪɧɨʀ ɿ ɛɟɡɩɨɜɬɨɪɧɨʀ ɫɯɟɦɢ ɜɿɞɛɨɪɭ ɡ
ɝɟɧɟɪɚɥɶɧɨʀ ɫɭɤɭɩɧɨɫɬɿ. Ɉɫɤɿɥ
ɶɤɢ ɩɪɢ ɛɟɡɩɨɜɬɨɪɧɨɦɭ ɜɿɞɛɨɪɿ
ɱɢɫɟɥɶɧɿɫɬɶ ɝɟɧɟɪɚɥɶɧɨʀ ɫɭɤɭɩɧɨɫɬɿ ɡɦɟɧɲɭɽɬɶɫɹ ( ɩɪɢ ɩɨɜɬɨɪɧɨɦɭ –
ɜɨɧɚ ɡɚɥɢɲɚɽɬɶɫɹ ɧɟɡɦɿɧɧɨɸ), ɩɿɫɥɹ ɤɨɠɧɨɝɨ ɜɿɞɛɨɪɭ ɣɦɨɜɿɪɧɿɫɬɶ
ɩɨɬɪɚɩɢɬɢ ɭ ɜɢɛɿɪɤɭ ɞɥɹ ɨɞɢɧɢɰɶ, ɳɨ ɡɚɥɢɲɚɸɬɶɫɹ, ɩɿɞɜɢɳɭɽɬɶɫɹ.
Ɍɨɦɭ ɫɟɪɟɞɧɹ ɩɨɦɢɥɤɚ ɬɭɬ ɛɭɞɟ ɦɟɧɲɨɸ, ɧɿɠ ɩɪɢ ɩɨɜɬɨɪɧɨɦɭ
ɜɿɞɛɨɪɿ.
ɉɟɪɟɬɜɨɪɟɧɧɹ ɮɨɪɦɭɥɢ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ ɞɥɹ ɫɟɪɟɞɧɶɨʀ ɩɪɢ
ɩɨɜɬɨɪɧɨɦɭ ɜɿɞɛɨɪɿ
ɩ
ɬ
ɯ
2
V
ɭ ɜɢɝɥɹɞ
ɩ
ɬ
ɯ
V
ɞɚɽ ɩɿɞɫɬɚɜɢ
ɫɬɜɟɪɞɠɭɜɚɬɢ, ɳɨ ɫɟɪɟɞɧɹ ɤɜɚɞɪɚɬɢɱɧɚ ɩɨɦɢɥɤɚ (ɫɟɪɟɞɧɽ
ɤɜɚɞɪɚɬɢɱɧɟ ɜɿɞɯɢɥɟɧɧɹ ɜɢɛɿɪɤɨɜɨʀ ɫɟɪɟɞɧɶɨʀ ɜɿɞ ɝɟɧɟɪɚɥɶɧɨʀ) ɩɪɹɦɨ
ɩɪɨɩɨɪɰɿɣɧɚ ɜɚɪɿɚɰɿʀ ɨɡɧɚɤɢ ɭ ɝɟɧɟɪɚɥɶɧɿɣ ɫɭɤɭɩɧɨɫɬɿ ɿ ɨɛɟɪɧɟɧɨ
ɩɪɨɩɨɪɰɿɣɧɚ ɤɨɪɟɧɸ ɤɜɚɞɪɚɬɧɨɦɭ ɡ ɨɛɫɹɝɭ ɜɢɛɿɪɤɢ. Ƚɪɚɧɢɱɧɚ
ɩɨɦɢɥɤɚ ɜɢɛɿɪɤɢ (¨), ɹɤ ɜɢɩɚɞɤɨɜɚ ɜɟɥɢɱɢɧɚ, ɦɨɠɟ ɛɭɬɢ ɜ ɤɨɠɧɨɦɭ
ɤɨɧɤɪɟɬɧɨɦɭ ɜɢɩɚɞɤɭ ɦɟɧɲɚ, ɪɿɜɧɚ ɚɛɨ ɛɿɥɶɲɚ ɡɚ ɫɟɪɟɞɧɸ
ɩɨɦɢɥɤɭ(m) . Ƀɦɨɜɿɪɧɿɫɬɶ ʀʀ ɜɟɥɢɱɢɧɢ ɩɪɢ ɞɨɫɢɬɶ ɜɟɥɢɤɿɣ ɫɭɤɭɩɧɨɫɬɿ
ɜɢɛɿɪɤɢ ɜɢɡɧɚɱɚɸɬɶ ɡɚ ɬɟɨɪɟɦɨɸ Ʌɹɩɭɧɨɜɚ:
)(
2
1
(
2
2
tfdtetmɪ
t
t
t
³ d'
S
.
Ɂɧɚɱɟɧɧɹ ɿɧɬɟɝɪɚɥɚ Ʌɚɩɥɚɫɚ (ɮɭɧɤɰɿɹ ɜɿɞ t) ɦɿɫɬɹɬɶɫɹ ɜ
ɫɬɚɧɞɚɪɬɧɢɯ ɦɚɬɟɦɚɬɢɱɧɢɯ ɬɚɛɥɢɰɹɯ (ɞɨɞɚɬɨɤ ). Ɂɚ ɬɚɤɢɦɢ ɬɚɛɥɢɰɹɦɢ
ɦɨɠɧɚ ɜɫɬɚɧɨɜɢɬɢ, ɳɨ
ɩɪɢ t =
683,0
2
1
)(1
2
2
³ d'
dtemɪ
t
t
t
S
;
ɩɪɢ t =
954,0
2
1
)2(2
2
2
2
2
³ d'
dtemɪ
t
S
.
ɇɚɜɟɞɟɧɿ ɪɨɡɪɚɯɭɧɤɢ ɫɜɿɞɱɚɬɶ ɩɪɨ ɬɟ, ɳɨ ɩɪɚɤɬɢɱɧɨ ɧɟɣɦɨɜɿɪɧɨ
ɨɞɟɪɠɚɬɢ ɩɨɦɢɥɤɭ, ɛɿɥɶɲɭ ɡɚ 3 m, ɬɨɛɬɨ ɛɿɥɶɲɭ, ɧɿɠ
ɩ
V
3
. Ɉɬɠɟ,
ɩɪɚɤɬɢɱɧɨ ɜɿɪɨɝɿɞɧɨ, ɳɨ ɝɟɧɟɪɚɥɶɧɚ ɫɟɪɟɞɧɹ ɧɟ ɜɢɣɞɟ ɡɚ ɝɪɚɧɢɰɿ:
287
288
ɩ
ɯɯ
ɩ
ɯ
V
V
3
~
3
~
dd
.
əɤ ɭɠɟ ɡɚɡɧɚɱɚɥɨɫɹ ɜɢɳɟ, ɩɪɢ ɛɟɡɩɨɜɬɨɪɧɨɦɭ ɫɩɨɫɨɛɿ ɜɿɞɛɨɪɭ
ɞɥɹ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ ɜɜɨɞɹɬɶ ɩɨɩɪɚɜɨɱɧɢɣ ɤɨɟɮɿɰɿɽɧɬ
11
1 ɩ
, ɞɟ ɩ, N- ɜɿɞɩɨɜɿɞɧɨ ɱɢɫɟɥɶɧɿɫɬɶ ɜɢɛɿɪɤɨɜɨʀ ɿ ɝɟɧɟɪɚɥɶɧɨʀ
ɫɭɤɭɩɧɨɫɬɟɣ. Ⱦɥɹ ɞɨɫɢɬɶ ɜɟɥɢɤɢɯ ɨɛɫɹɝɿɜ ɝɟɧɟɪɚɥɶɧɨʀ ɫɭɤɭɩɧɨɫɬɿ
ɡɚɦɿɫɬɶ ɡɧɚɱɟɧɧɹ N- 1 ɜɜɨɞɹɬɶ ɡɧɚɱɟɧɧɹ N, ɬɨɞɿ ɮɨɪɦɭɥɚ ɧɚɛɭɜɚɽ
ɜɢɝɥɹɞɭ:
1
1
1
|
1
1
ɩɩɩ
1
1
.
Ɂ ɜɪɚɯɭɜɚɧɧɹɦ ɧɚɜɟɞɟɧɨʀ ɩɨɩɪɚɜɤɢ ɞɢɫɩɟɪɫɿɹ ɜɢɛɿɪɤɨɜɨʀ
ɫɟɪɟɞɧɶɨʀ ɫɬɚɧɨɜɢɬɶ:
)1(
2
2
1
ɩ
ɩ
ɯ
V
V
.
ɋɟɪɟɞɧɿ ɩɨɦɢɥɤɢ ɫɟɪɟɞɧɶɨʀ ɿ ɱɚɫɬɤɢ ɭ ɜɢɛɿɪɤɨɜɿɣ ɫɭɤɭɩɧɨɫɬɿ ɞɥɹ
ɜɥɚɫɧɟ ɜɢɩɚɞɤɨɜɨɝɨ ɜɿɞɛɨɪɭ ɧɚɜɟɞɟɧɨ ɭ ɡɝɚɞɚɧɿɣ ɜɢɳɟ ɬɚɛɥɢɰɿ.
Ⱦɥɹ ɦɟɯɚɧɿɱɧɨɝɨ ɫɩɨɫɨɛɭ ɜɿɞɛɨɪɭ ɩɨɦɢɥɤɚ ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ
ɪɨɡɪɚɯɨɜɭɽɬɶɫɹ ɚɧɚɥɨɝɿɱɧɨ ɮɨɪɦɭɥɚɦ ɞɥɹ ɜɥɚɫɧɟ ɜɢɩɚɞɤɨɜɨɝɨ
ɜɿɞɛɨɪɭ.
ɉɪɢ ɬɢɩɨɜɨɦɭ ɫɩɨɫɨɛɿ ɜɿɞɛɨɪɭ ɪɨɡɪɚɯɭɧɨɤ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ
ɦɚɽ ɞɟɹɤɿ ɨɫɨɛɥɢɜɨɫɬɿ. Ɋɨɡɝɥɹɧɟɦɨ ʀɯ.
Ɂ ɜɢɤɥɚɞɟɧɨɝɨ ɜɢɳɟ ɡɪɨɡɭɦɿɥɨ, ɳɨ ɫɟɪɟɞɧɹ ɩɨɦɢɥɤɚ ɜɢɛɿɪɤɢ
ɡɚɥɟɠɢɬɶ ɜɿɞ ɫɟɪɟɞɧɶɨɝɨ ɤɜɚɞɪɚɬɚ ɜɿɞɯɢɥɟɧɶ (ɞɢɫɩɟɪɫɿʀ)
ɞɨɫɥɿɞɠ
ɭɜɚɧɨʀ ɨɡɧɚɤɢ. Ɂɝɿɞɧɨ ɩɪɚɜɢɥɭ ɫɤɥɚɞɚɧɧɹ ɿ ɪɨɡɤɥɚɞɚɧɧɹ
ɞɢɫɩɟɪɫɿɣ ɦɚɽɦɨ:
222
=
VVV
ɯɭ
, ɞɟ
2
ɭ
V
- ɡɚɝɚɥɶɧɚ ɞɢɫɩɟɪɫɿɹ;
2
ɯ
V
- ɦɿɠ
ɝɪɭɩɨɜɚ ɞɢɫɩɟɪɫɿɹ;
2
=
V
- ɜɧɭɬɪɿɲɧɶɨɝɪɭɩɨɜɚ ɞɢɫɩɟɪɫɿɹ.
Ⱦɥɹ ɬɢɩɨɜɨʀ ɜɢɛɿɪɤɢ ɦɿɠɝɪɭɩɨɜɚ ɞɢɫɩɟɪɫɿɹ ɜɢɦɿɪɸɽ ɜɚɪɿɚɰɿɸ
ɝɪɭɩɨɜɢɯ ɫɟɪɟɞɧɿɯ (
j
ɯ
~
) ɜɿɞɧɨɫɧɨ ɡɚɝɚɥɶɧɨʀ ɫɟɪɟɞɧɶɨʀ (
x
~
), ɬɨɛɬɨ :
¦
¦
i
ij
x
n
nxx
2
2
)
~~
(
V
.
ɐɟɣ ɜɢɞ ɞɢɫɩɟɪɫɿʀ ɩɨɹɫɧɸɽ ɜɚɪɿɚɰɿɸ, ɜɢɤɥɢɤɚɧɭ ɨɡɧɚɤɨɸ,
ɩɨɤɥɚɞɟɧɨɸ ɜ ɨɫɧɨɜɭ ɝɪɭɩɭɜɚɧɶ ɩɪɢ ɜɢɞɿɥɟɧɧɿ ɬɢɩɨɜɢɯ ɝɪɭɩ ɿ ɧɟ ɦɨɠɟ
ɪɨɡɝɥɹɞɚɬɢɫɹ ɹɤ ɩɨɦɢɥɤɚ ɜɢɛɿɪɤɢ. ɇɚɡɢɜɚɸɬɶ ʀʀ ɫɢɫɬɟɦɚɬɢɱɧɨɸ
ɞɢɫɩɟɪɫɿɽɸ. Ɉɬɠɟ, ɩɪɢ ɪɨɡɪɚɯɭɧɤɚɯ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ ɰɟɣ
ɜɢɞ ɞɢɫɩɟɪɫɿʀ ɜɢɤɥɸɱɚɽɬɶɫɹ.
Ⱥɥɟ ɤɨɠɧɚ ɬɢɩɨɜɚ ɝɪɭɩɚ ɦɚɽ ɜɚɪɿɚɰɿɸ ɨɡɧɚɤɢ, ɜɢɤɥɢɤɚɧɭ
ɜɩɥɢɜɨɦ ɪɿɡɧɢɯ ɧɟɜɪɚɯɨɜɚɧɢɯ ɮɚɤɬɨɪɿɜ - ɜɧɭɬɪɿɲɧɶɨɝɪɭɩɨɜɭ
288
289
(ɡɚɥɢɲɤɨɜɭ) ɜɚɪɿɚɰɿɸ. Ɉɫɬɚɧɧɹ ɪɨɡɪɚɯɨɜɭɽɬɶɫɹ ɹɤ ɫɟɪɟɞɧɹ
ɚɪɢɮɦɟɬɢɱɧɚ ɡ ɝɪɭɩɨɜɢɯ ɞɢɫɩɟɪɫɿɣ:
¦
¦
1
1
=
j
jj
2
2
V
V
,
ɞɟ
2
j
V
- ɝɪɭɩɨɜɿ ɜɢɛɿɪɤɨɜɿ ɞɢɫɩɟɪɫɿʀ.
ɋɚɦɟ ɰɹ ɱɚɫɬɢɧɚ ɜɚɪɿɚɰɿʀ ɡɚɥɢɲɚɽɬɶɫɹ ɧɟɩɨɹɫɧɟɧɨɸ ɿ ɩɨɜɢɧɧɚ
ɪɨɡɝɥɹɞɚɬɢɫɹ ɹɤ ɩɨɦɢɥɤɚ ɜɢɛɿɪɤɢ. Ɍɨɛɬɨ ɮɨɪɦɭɥɚ ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ
ɬɢɩɨɜɨʀ ɜɢɛɿɪɤɢ ɦɚɽ ɜɢɝɥɹɞ:
ɩ
ɬ
ɬɢɩ
2
=
V
,
ɞɟ ɩ – ɡɚɝɚɥɶɧɢɣ ɨɛɫɹɝ ɜɢɛɿɪɤɢ (
¦
j
ɩɩ
).
Ɂɚ ɩɪɚɜɢɥɨɦ ɫɤɥɚɞɚɧɧɹ ɿ ɪɨɡɤɥɚɞɚɧɧɹ ɞɢɫɩɟɪɫɿɣ ɦɚɽɦɨ :
22
ɭ
VV
¢
=
,
ɬɨɦɭ ɫɟɪɟɞɧɹ ɩɨɦɢɥɤɚ ɬɢɩɨɜɨʀ ɜɢɛɿɪɤɢ, ɹɤ ɩɪɚɜɢɥɨ, ɦɟɧɲɚ ɡɚ ɫɟɪɟɞɧɸ
ɩɨɦɢɥɤɭ ɩɪɢ ɜɥɚɫɧɟ ɜɢɩɚɞɤɨɜɿɣ ɜɢɛɿɪɰɿ. Ɉɫɤɿɥɶɤɢ ɫɟɪɟɞɧɹ ɩɨɦɢɥɤɚ
ɬɢɩɨɜɨʀ ɜɢɛɿɪɤɢ ɞɚɽ ɬɨɱɧɿɲɿ ɪɟɡɭɥɶɬɚɬɢ (ɜɢɫɧɨɜɤɢ), ʀʀ ɲɢɪɨɤɨ
ɜɢɤɨɪɢɫɬɨɜɭɸɬɶ ɜ ɞɨɫɥɿɞɠɟɧɿ ɟɤɨɧɨɦɿɱɧɢɯ ɹɜɢɳ.
Ɍɪɟɛɚ ɩɚɦ’ɹɬɚɬɢ, ɳɨ ɨɪɝɚɧɿɡɚɰɿɹ ɬɢɩɨɜɨʀ ɜɢɛɿɪɤɢ ɡɭɦɨɜɥɟɧɚ, ɹɤ
ɩɪɚɜɢɥɨ, ɜɥɚɫɧɟ ɜɢɩɚɞɤɨɜɢɦ ɜɿɞɛɨɪɨɦ. Ⱥɞɠɟ ɜɿɞɛɿɪ ɨɞɢɧɢɰɶ ɡ ɤɨɠɧɨʀ
ɝɪɭɩɢ ɡɞɿɣɫɧɸɸɬɶ ɜɥɚɫɧɟ ɜɢɩɚɞɤɨɜɢɦ ɦɟɬɨɞɨɦ. ɉɪɢ ɰɶɨ
ɦɭ
ɡɚɫɬɨɫɨɜɭɽɬɶɫɹ ɫɯɟɦɚ ɛɟɡɩɨɜɬɨɪɧɨɝɨ ɜɿɞɛɨɪɭ. Ɂ ɰɢɯ ɩɪɢɱɢɧ ɞɨ
ɫɟɪɟɞɧɶɨʀ ɩɨɦɢɥɤɢ ɫɟɪɟɞɧɶɨʀ ɱɢ ɱɚɫɬɤɢ ɩɪɢ ɛɟɡɩɨɜɬɨɪɧɿɣ ɫɯɟɦɿ
ɜɿɞɛɨɪɭ ɜɜɨɞɹɬɶ ɩɨɩɪɚɜɤɭ
1
ɩ
1
.
ɉɪɢ ɫɟɪɿɣɧɨɦɭ ɫɩɨɫɨɛɿ ɜɿɞɛɨɪɭ ɩɨ ɤɨɠɧɿɣ ɜɿɞɿɛɪɚɧɿɣ ɫɟɪɿʀ
ɪɨɡɪɚɯɨɜɭɽɬɶɫɹ ɡɧɚɱɟɧɧɹ ɞɢɫɩɟɪɫɿʀ. ɋɟɪɟɞɧɹ ɚɪɢɮɦɟɬɢɱɧɚ ɡ ɰɢɯ
ɞɢɫɩɟɪɫɿɣ ɫɬɚɧɨɜɢɬɶ ɜɧɭɬɪɿɲɧɶɨɫɟɪɿɣɧɭ (
2
ȼɋ
V
), ɬɨɛɬɨ ɡɚɥɢɲɤɨɜɭ
ɞɢɫɩɟɪɫɿɸ.
ȼɚɪɿɚɰɿɹ ɫɟɪɿɣɧɢɯ ɫɟɪɟɞɧɿɯ (
j
ɯ
~
) ɧɚɜɤɨɥɨ ɡɚɝɚɥɶɧɨʀ ɜɢɛɿɪɤɨɜɨʀ
ɫɟɪɟɞɧɶɨʀ
x
~
ɯɚɪɚɤɬɟɪɢɡɭɽɬɶɫɹ ɦɿɠɫɟɪɿɣɧɨɸ ɞɢɫɩɟɪɫɿɽɸ (
2
Ɇɋ
V
) .
ɋɬɪɭɤɬɭɪɧɚ ɮɨɪɦɭɥɚ ʀʀ ɦɚɽ ɜɢɝɥɹɞ:
ɫ
j
Ɇɋ
n
x
ɯ
¦
2
2
)
~~
(
V
,
ɞɟ
ɫ
ɩ
- ɱɢɫɟɥɶɧɿɫɬɶ ɜɢɛɿɪɤɢ ɜ ɫɟɪɿʀ.
Ɂɝɿɞɧɨ ɡ ɩɪɚɜɢɥɨɦ ɪɨɡɤɥɚɞɚɧɧɹ ɞɢɫɩɟɪɫɿʀ ɦɚɽɦɨ
222
ɦɫɜɫɭ
VVV
.
ȼɧɭɬɪɿɲɧɶɨɫɟɪɿɣɧɚ ɞɢɫɩɟɪɫɿɹ ɪɨɡɪɚɯɨɜɭɽɬɶɫɹ ɧɚ ɨɫɧɨɜɿ ɞɚɧɢɯ
ɫɭɰɿɥɶɧɨɝɨ ɫɩɨɫɬɟɪɟɠɟɧɧɹ ɜɿɞɿɛɪɚɧɢɯ ɫɟɪɿɣ. Ⱥ ɰɟ ɨɡɧɚɱɚɽ, ɳɨ
289
290
ɩɨɦɢɥɤɚ ɪɟɩɪɟɡɟɧɬɚɬɢɜɧɨɫɬɿ ɡɚɥɟɠɢɬɶ ɜɿɞ ɦɿɠɫɟɪɿɣɧɨʀ ɞɢɫɩɟɪɫɿʀ. Ȳʀ
ɪɨɡɪɚɯɨɜɭɸɬɶ ɡɚ ɫɯɟɦɨɸ ɛɟɡɩɨɜɬɨɪɧɨɝɨ ɜɿɞɛɨɪɭ ɫɟɪɿɣ:
)1(
2
1
ɫ
ɫ
ɦɫ
ɩ
ɩ
ɬ
V
.
Ⱦɥɹ ɡɚɡɧɚɱɟɧɢɯ ɜɢɳɟ ɫɩɨɫɨɛɿɜ ɜɿɞɛɨɪɭ ɩɪɢ ɪɨɡɪɚɯɭɧɤɚɯ
ɝɪɚɧɢɱɧɨʀ ɩɨɦɢɥɤɢ ɹɤ ɞɥɹ ɫɟɪɟɞɧɶɨʀ, ɬɚɤ ɿ ɞɥɹ ɱɚɫɬɤɢ, ɫɟɪɟɞɧɸ
ɩɨɦɢɥɤɭ ɦɧɨɠɚɬɶ ɧɚ ɤɨɟɮɿɰɿɽɧɬ ɞɨɜɿɪɢ (t), ɜɟɥɢɱɢɧɚ ɹɤɨɝɨ ɡɚɥɟɠɢɬɶ
ɜɿɞ ɪɿɜɧɹ ɨɛɪɚɧɨʀ ɣɦɨɜɿɪɧɨɫɬɿ. Ⱦɟɬɚɥɶɧɨ ɰɟ ɩɢɬɚɧɧɹ ɜɢɤɥɚɞɟɧɨ ɜ § 4.4.
§ 11.5.Ɉɪɝɚɧɿɡɚɰɿɹ ɜɢɛɿɪɤɨɜɨɝɨ ɫɩɨɫɬɟɪɟɠɟɧɧɹ
Ɂɞɿɣɫɧɟɧɧɹ ɜɢɛɿɪɤɨɜɨɝɨ ɨɛɫɬɟɠɟɧɧɹ ʉɪɭɧɬɭɽɬɶɫɹ, ɧɚɫɚɦɩɟɪɟɞ, ɧɚ
ɡɧɚɧɧɿ ɩɪɢɪɨɞɢ ɞɨɫɥɿɞɠɭɜɚɧɢɯ ɩɪɨɰɟɫɿɜ ɬɚ ɹɜɢɳ ɿ ɝɥɢɛɨɤɨɦɭ
ɬɟɨɪɟɬɢɱɧɨɦɭ ɚɧɚɥɿɡɿ. ȼɢɛɿɪɤɨɜɟ ɨɛɫɬɟɠ
ɟɧɧɹ ɩɨɱɢɧɚɸɬɶ ɡ ɤɨɩɿɬɤɨʀ
ɩɿɞɝɨɬɨɜɤɢ ɪɨɛɨɬɢ, ɹɤɚ ɩɟɪɟɞɛɚɱɚɽ ɜɢɪɿɲɟɧɧɹ ɬɚɤɢɯ ɩɢɬɚɧɶ: ɦɟɬɚ ɿ
ɨɛ’ɽɤɬ ɞɨɫɥɿɞɠɟɧɧɹ; ɩɪɨɝɪɚɦɚ ɬɚ ɿɧɫɬɪɭɦɟɧɬɚɪɿɣ ɨɛɫɬɟɠɟɧɧɹ; ɞɠɟɪɟɥɚ
ɿ ɫɩɨɫɨɛɢ ɡɛɢɪɚɧɧɹ ɧɟɨɛɯɿɞɧɨʀ ɿɧɮɨɪɦɚɰɿʀ; ɩɿɞɛɿɪ ɿ ɩɿɞɝɨɬɨɜɤɚ ɤɚɞɪɿɜ,
ɩɪɨɛɧɿ ɨɛɫɬɟɠɟɧɧɹ ɿ ɪɹɞ ɿɧɲɢɯ ɩɢɬɚɧɶ.
Ɉɪɝɚɧɿɡɚɰɿɹ ɜɢɛɿɪɤɨɜɨɝɨ ɫɩɨɫɬɟɪɟɠ
ɟɧɧɹ ɡ ɦɟɬɨɸ ɜɿɞɬɜɨɪɟɧɧɹ
ɝɟɧɟɪɚɥɶɧɨʀ ɫɭɤɭɩɧɨɫɬɿ ɜɢɫɭɜɚɽ ɪɹɞ ɡɚɜɞɚɧɶ, ɜɢɪɿɲɟɧɧɹ ɹɤɢɯ
ʉɪɭɧɬɭɽɬɶɫɹ ɧɚ ɬɟɨɪɿʀ ɜɢɛɿɪɤɨɜɨɝɨ ɦɟɬɨɞɭ. Ɋɨɡɝɥɹɧɟɦɨ ʀɯ .
1. ɉɨ-ɩɟɪɲɟ, ɰɟ ɜɢɪɿɲɟɧɧɹ ɩɢɬɚɧɧɹ ɳɨɞɨ ɜɫɬɚɧɨɜɥɟɧɧɹ
ɱɢɫɟɥ
ɶɧɨɫɬɿ ɜɢɛɿɪɤɨɜɨʀ ɫɭɤɭɩɧɨɫɬɿ. ɋɭɬɶ ɰɶɨɝɨ ɡɚɜɞɚɧɧɹ ɩɨɥɹɝɚɽ ɜ
ɬɨɦɭ, ɳɨɛ ɡɧɚɣɬɢ ɜɿɞɩɨɜɿɞɶ ɧɚ ɡɚɩɢɬɚɧɧɹ - ɫɤɿɥɶɤɢ ɩɨɬɪɿɛɧɨ ɜɿɞɿɛɪɚɬɢ
ɨɞɢɧɢɰɶ ɫɩɨɫɬɟɪɟɠɟɧɧɹ, ɳɨɛ ɩɨɦɢɥɤɚ ɜɢɛɿɪɤɢ ɡ ɩɟɜɧɢɦ ɪɿɜɧ
ɟɦ
ɣɦɨɜɿɪɧɨɫɬɿ ɧɟ ɩɟɪɟɜɢɳɭɜɚɥɚ ɜɫɬɚɧɨɜɥɟɧɢɣ ɪɨɡɦɿɪ.
2. Ⱦɪɭɝɟ ɡɚ
ɜɞɚɧɧɹ ɜɢɛɿɪɤɨɜɨɝɨ ɫɩɨɫɬɟɪɟɠɟɧɧɹ ɦɚɽ ɧɚ ɦɟɬɿ
ɨɰɿɧɤɭ ɩɨɤɚɡɧɢɤɿɜ, ɨɞɟɪɠɚɧɢɯ ɡɚ ɜɢɛɿɪɤɨɜɢɦɢ ɞɚɧɢɦɢ. ȼɢɪɿɲɟɧɧɹ
ɰɶɨɝɨ ɡɚɜɞɚɧɧɹ ɩɨɥɹɝɚɽ ɭ ɜɢɡɧɚɱɟɧɧɿ ɝɪɚɧɢɱɧɨʀ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɨɜɨʀ
ɫɭɤɭɩɧɨɫɬɿ.
3. Ɍɪɟɬɽ ɡɚɜɞɚɧɧɹ ɜɢɛɿɪɤɢ ɡɜɨɞɢɬɶɫɹ ɞɨ ɜɫɬɚɧɨɜɥɟɧɧɹ
ɣɦɨɜɿɪɧɨɫɬɟɣ ɡ
ɞɿɣɫɧɟɧɧɹ ɩɟɜɧɨɝɨ ɪɨɡɦɿɪɭ ɩɨɦɢɥɤɢ. Ⱦɥɹ ɰɶɨɝɨ
ɧɟɨɛɯɿɞɧɨ ɡɧɚɬɢ ɫɟɪɟɞɧɸ ɿ ɝɪɚɧɢɱɧɭ ɩɨɦɢɥɤɢ ɜɢɛɿɪɤɢ, ɪɨɡɪɚɯɭɜɚɬɢ
ɧɨɪɦɨɜɚɧɟ ɜɿɞɯɢɥɟɧɧɹ, ɧɚ ɩɿɞɫɬɚɜɿ ɹɤɨɝɨ ɡɚ ɫɬɚɧɞɚɪɬɧɢɦɢ ɬɚɛɥɢɰɹɦɢ
ɿɧɬɟɝɪɚɥɚ ɣɦɨɜɿɪɧɨɫɬɿ ɜɢɡɧɚɱɚɽɬɶɫɹ ɪɿɜɟɧɶ ɣɦɨɜɿɪɧɨɫɬɿ.
ȼɢɡɧɚɱɟɧɧɹ ɝɪɚɧɢɰɶ, ɜ ɹɤɢɯ ɡɧɚɯɨɞɹɬɶɫɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɜɫɿɽʀ
ɫɭɤɭɩɧɨɫɬɿ, ɭɫɤɥ
ɚɞɧɸɽɬɶɫɹ ɭ ɜɢɩɚɞɤɚɯ, ɤɨɥɢ ɝɟɧɟɪɚɥɶɧɚ ɫɭɤɭɩɧɿɫɬɶ
ɞɨɫɥɿɞɧɢɤɭ ɧɟɜɿɞɨɦɚ. Ⱥɞɠɟ ɩɪɢ ɜɿɞɨɦɿɣ ɝɟɧɟɪɚɥɶɧɿɣ ɫɭɤɭɩɧɨɫɬɿ
ɡɚɜɠɞɢ ɦɨɠɧɚ ɩɨɛɭɞɭɜɚɬɢ ɫɯɟɦɭ ɜɫɿɯ ɦɨɠɥɢɜɢɯ ɜɢɩɚɞɤɿɜ ɜɢɛɿɪɤɢ ɡ
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