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ɅȺȻɈɊȺɌɈɊɇȺə ɊȺȻɈɌȺ ʋ ɆɨɞɆ-06

ɉɈ ɂɁɍɑȿɇɂɘ ɆɈȾȿɅɃ ɎɂɁɂɑȿɋɄɂɏ

ɉɊɈɐȿɋɋɈȼ ɂ əȼɅȿɇɂɃ ɇȺ ɄɈɆɉɖɘɌȿɊȿ

Ɋɟɚɤɬɢɜɧɨɟ ɞɜɢɠɟɧɢɟ

ɐɟɥɶ ɪɚɛɨɬɵ: ɢɡɭɱɟɧɢɟ ɪɟɚɤɬɢɜɧɨɝɨ ɞɜɢɠɟɧɢɹ ɤɚɤ ɞɜɢɠɟɧɢɹ ɬɟɥɚ

ɫ ɩɟɪɟɦɟɧɧɨɣ ɦɚɫɫɨɣ. ɂɫɫɥɟɞɨɜɚɧɢɟ ɡɚɜɢɫɢɦɨɫɬɢ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚ-

ɪɚɤɬɟɪɢɫɬɢɤ (ɤɨɨɪɞɢɧɚɬɵ, ɫɤɨɪɨɫɬɢ, ɭɫɤɨɪɟɧɢɹ) ɩɪɹɦɨɥɢɧɟɣɧɨɝɨ ɪɟɚɤ-

ɬɢɜɧɨɝɨ ɞɜɢɠɟɧɢɹ ɜ ɜɚɤɭɭɦɟ ɨɬ ɦɚɫɫɵ ɬɟɥɚ ɢ ɡɚɤɨɧɨɜ ɢɯ ɢɡɦɟɧɟɧɢɹ ɜ

ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ.

1. Ɍɟɨɪɟɬɢɱɟɫɤɨɟ ɫɨɞɟɪɠɚɧɢɟ

1.1. Ɋɟɚɤɬɢɜɧɨɟ ɞɜɢɠɟɧɢɟ

ɉɨɞ ɪɟɚɤɬɢɜɧɵɦ ɞɜɢɠɟɧɢɟɦ ɩɨɧɢɦɚɸɬ ɞɜɢɠɟɧɢɟ ɬɟɥɚ ɡɚ ɫɱɟɬ

ɢɡɦɟɧɟɧɢɹ ɟɝɨ ɦɚɫɫɵ. ɂɞɟɹ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɪɟɚɤɬɢɜɧɨɝɨ ɞɜɢɠɟɧɢɹ ɞɥɹ

ɨɫɭɳɟɫɬɜɥɟɧɢɹ ɤɨɫɦɢɱɟɫɤɢɯ ɩɨɥɟɬɨɜ ɩɪɢɧɚɞɥɟɠɢɬ ɪɭɫɫɤɨɦɭ ɭɱɟɧɨɦɭ

Ʉ.ɗ. ɐɢɨɥɤɨɜɫɤɨɦɭ (1857 – 1935). Ɉɫɧɨɜɭ ɪɟɚɤɬɢɜɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫɨ-

ɫɬɚɜɥɹɟɬ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɡɚɦɤɧɭɬɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ.

ɋɨɝɥɚɫɧɨ ɜɬɨɪɨɦɭ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ ɢɦɩɭɥɶɫ

ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɵ

ɦɨɠɟɬ ɛɵɬɶ ɢɡɦɟɧɟɧ ɬɨɥɶɤɨ ɡɚ ɫɱɟɬ ɜɧɟɲɧɟɝɨ ɜɨɡɞɟɣɫɬɜɢɹ

ɜɧɟɲ

ȿɫɥɢ ɜɧɟɲɧɢɟ ɫɢɥɵ

ɜɧɟɲ

ɨɬɫɭɬɫɬɜɭɸɬ, ɢɦɩɭɥɶɫ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟ-

ɦɵ ɫɨɯɪɚɧɹɟɬɫɹ

ȼ ɪɚɦɤɚɯ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɪɚɫɫɦɨɬɪɢɦ ɞɜɟ ɩɨɞɫɢɫɬɟɦɵ: 1) ɧɟɤɨɬɨ-

ɪɨɟ ɬɟɥɨ, ɨɛɥɚɞɚɸɳɟɟ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ

ɦɚɫɫɨɣ m ɢ ɫɤɨɪɨɫɬɶɸ v

ɧɟɤɨɬɨɪɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, 2) ɨɬɞɟɥɹɸɳɟɟɫɹ ɨɬ ɬɟɥɚ ɜɟ-

ɳɟɫɬɜɨ (ɫɤɚɠɟɦ, ɝɚɡ), ɞɜɢɠɭɳɟɟɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɟɥɚ ɫɨ ɫɤɨɪɨɫɬɶɸ

ɫɨ ɫɤɨɪɨɫɬɶɸ

v ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɣ ɠɟ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟ-

ɬɚ.

Ɇɟɯɚɧɢɤɚ

ɂɡ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɢɡɦɟɧɟɧɢɟ ɢɦɩɭɥɶɫɚ

ɨɞɧɨɣ ɩɨɞɫɢɫɬɟɦɵ

ɪɚɜɧɨ ɢɡɦɟɧɟɧɢɸ ɢɦɩɭɥɶɫɚ ɞɪɭɝɨɣ ɩɨɞɫɢɫɬɟɦɵ

ɢ ɧɚɩɪɚɜɥɟɧɨ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɫɬɨɪɨɧɭ:

0)(

 PPdPd



PdPd

 .

ɂɦɩɭɥɶɫ ɩɟɪɜɨɣ ɩɨɞɫɢɫɬɟɦɵ

. Ɍɨɝɞɚ ɢɡɦɟɧɟɧɢɟ ɢɦɩɭɥɶɫɚ

ɩɟɪɜɨɣ ɩɨɞɫɢɫɬɟɦɵ (ɬɟɥɚ) ɪɚɜɧɨ

dmmdPd vv



ɂɡɦɟɧɟɧɢɟ ɢɦɩɭɥɶɫɚ ɜɬɨɪɨɣ ɩɨɞɫɢɫɬɟɦɵ (ɝɚɡɚ) ɩɪɨɢɫɯɨɞɢɬ ɡɚ ɫɱɟɬ

ɢɡɦɟɧɟɧɢɹ (ɭɜɟɥɢɱɟɧɢɹ) ɧɚ ɜɟɥɢɱɢɧɭ

ɦɚɫɫɵ ɨɬɞɟɥɢɜɲɟɝɨɫɹ ɜɟɳɟ-

ɫɬɜɚ (ɢɥɢ ɭɦɟɧɶɲɟɧɢɹ ɦɚɫɫɵ ɬɟɥɚ

dmdm  )

dmudmuPd

)()(

  .

Ɍɨɝɞɚ

PdPd



 dmudmmd )( vvv

  ,

dmumd

v ɢɥɢ

ɉɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ Ɇɟɳɟɪɫɤɨɝɨ ɢ

ɨɩɢɫɵɜɚɟɬ ɞɜɢɠɟɧɢɟ ɬɟɥɚ ɩɟɪɟɦɟɧɧɨɣ ɦɚɫɫɵ ɜ ɨɬɫɭɬɫɬɜɢɢ ɜɧɟɲɧɢɯ ɫɢɥ.

ȼɟɤɬɨɪ

ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ Ɇɟɳɟɪɫɤɨɝɨ ɧɚɡɵ-

ɜɚɸɬ ɪɟɚɤɬɢɜɧɨɣ ɫɢɥɨɣ, ɤɨɬɨɪɚɹ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɦɟɯɚɧɢɱɟɫɤɨɟ ɜɨɡɞɟɣ-

ɫɬɜɢɟ ɧɚ ɬɟɥɨ ɨɬɞɟɥɹɸɳɟɝɨɫɹ ɨɬ ɧɟɝɨ ɜɟɳɟɫɬɜɚ.

ɍɪɚɜɧɟɧɢɟ Ɇɟɳɟɪɫɤɨɝɨ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɜ ɨɬɫɭɬɫɬɜɢɢ ɜɧɟɲɧɢɯ

ɫɢɥ ɢɡɦɟɧɟɧɢɟ ɫɤɨɪɨɫɬɢ ɬɟɥɚ ɜɨɡɦɨɠɧɨ ɡɚ ɫɱɟɬ ɢɡɦɟɧɟɧɢɹ ɟɝɨ ɦɚɫɫɵ

ɢɥɢ, ɞɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɡɚ ɫɱɟɬ ɞɟɣɫɬɜɢɹ ɪɟɚɤɬɢɜɧɨɣ ɫɢɥɵ. Ɋɟɚɤɬɢɜɧɚɹ

ɫɢɥɚ ɫɭɳɟɫɬɜɭɟɬ, ɬɨɥɶɤɨ ɟɫɥɢ

ɦɚɫɫɚ ɬɟɥɚ ɦɟɧɹɟɬɫɹ. Ʉɨɝɞɚ ɦɚɫɫɚ ɬɟɥɚ ɩɟ-

ɪɟɫɬɚɟɬ ɢɡɦɟɧɹɬɶɫɹ – ɞɟɣɫɬɜɢɟ ɪɟɚɤɬɢɜɧɨɣ ɫɢɥɵ ɩɪɟɤɪɚɳɚɟɬɫɹ.

ɍɪɚɜɧɟɧɢɟ Ɇɟɳɟɪɫɤɨɝɨ – ɜɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɞɥɹ ɪɟɚɤɬɢɜɧɨ

ɞɜɢɠɭɳɟɝɨɫɹ ɬɟɥɚ:

{

ɢɥɢ

Fam

Ⱦɥɹ ɩɨɥɧɨɣ ɫɢɫɬɟɦɵ (ɬɟɥɨ + ɝɚɡ) ɪɟɚɤɬɢɜɧɚɹ ɫɢɥɚ ɹɜɥɹɟɬɫɹ ɜɧɭɬ-

ɪɟɧɧɟɣ.

ɆɨɞɆ-06 Ɋɟɚɤɬɢɜɧɨɟ ɞɜɢɠɟɧɢɟ

Ɋɚɫɫɦɨɬɪɢɦ ɪɟɲɟɧɢɟ ɩɨɥɭɱɟɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ Ɇɟɳɟɪɫɤɨɝɨ ɞɥɹ

ɫɥɭɱɚɹ, ɤɨɝɞɚ ɫɤɨɪɨɫɬɶ ɝɚɡɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɟɥɚ

(ɫɤɨɪɨɫɬɶ ɢɫɬɟɱɟɧɢɹ

ɝɚɡɚ) ɩɨɫɬɨɹɧɧɚ (ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ):

ɩɪɢ const u

ɂɡ ɭɪɚɜɧɟɧɢɹ ɜɢɞɧɨ, ɱɬɨ ɭɫ-

ɤɨɪɟɧɢɟ ɬɟɥɚ

ɢ ɫɤɨɪɨɫɬɶ

ɢɫɬɟɱɟɧɢɹ ɝɚɡɚ

ɧɚɩɪɚɜɥɟɧɵ

ɜɞɨɥɶ ɨɞɧɨɣ ɩɪɹɦɨɣ. ȿɫɥɢ ɜ ɧɚ-

ɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫɤɨɪɨɫɬɶ

ɬɟɥɚ

ɛɵɥɚ ɧɚɩɪɚɜɥɟɧɚ ɜɞɨɥɶ ɬɨɣ

ɠɟ ɩɪɹɦɨɣ (ɢɥɢ ɪɚɜɧɚ ɧɭɥɸ), ɬɨ

ɞɜɢɠɟɧɢɟ ɬɟɥɚ ɛɭɞɟɬ ɩɪɹɦɨɥɢɧɟɣ-

ɧɵɦ. ɉɨɷɬɨɦɭ ɜɵɛɟɪɟɦ ɨɫɶ ɤɨɨɪɞɢɧɚɬ OX, ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɞɜɢɠɟɧɢɹ

ɬɟɥɚ (ɪɢɫ. 1).

Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɩɟɪɟɦɟɧɧɨɣ ɦɚɫɫɵ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ

ɨɫɶ OX ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ

dmumd

v  udmmd  v 

ud  v

ɉɪɢ

const u ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨɥɭɱɟɧɧɨɝɨ ɜɵɪɚɠɟɧɢɹ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭ-

ɱɢɬɶ

)(

ln)(

ut 

vv ,

ɝɞɟ

v – ɫɤɨɪɨɫɬɶ, ɚ

m – ɦɚɫɫɚ ɬɟɥɚ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ

(

0 t ).

ɋ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɦɚɫɫɚ ɬɟɥɚ

)(tm

ɭɦɟɧɶɲɚɟɬɫɹ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,

ɫɤɨɪɨɫɬɶ

)(tv ɭɜɟɥɢɱɢɜɚɟɬɫɹ. Ʉɚɤ ɩɪɚɜɢɥɨ, ɦɚɫɫɚ ɝɚɡɚ, ɤɨɬɨɪɵɣ ɦɨɠɟɬ

ɨɬɞɟɥɢɬɶɫɹ ɨɬ ɬɟɥɚ, ɨɝɪɚɧɢɱɟɧɚ. Ɇɚɫɫɚ ɬɟɥɚ

)(tm ɫɬɚɧɟɬ ɦɢɧɢɦɚɥɶɧɨɣ,

ɤɨɝɞɚ ɢɡ ɬɟɥɚ ɜɵɬɟɱɟɬ ɜɟɫɶ ɝɚɡ. ɉɨɫɥɟ ɷɬɨɝɨ ɦɚɫɫɚ ɬɟɥɚ ɩɟɪɟɫɬɚɧɟɬ ɦɟ-

ɧɹɬɶɫɹ, ɜɨɡɞɟɣɫɬɜɢɟ ɪɟɚɤɬɢɜɧɨɣ ɫɢɥɵ ɩɪɟɤɪɚɬɢɬɫɹ. Ɍɟɥɨ ɩɪɨɞɨɥɠɢɬ

ɞɜɢɠɟɧɢɟ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ.

Ɍɚɤ ɤɚɤ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɫ ɭɦɟɧɶɲɟɧɢɟɦ ɦɚɫɫɵ ɭɜɟɥɢɱɢɜɚɟɬɫɹ, ɝɨɜɨ-

ɪɹɬ, ɱɬɨ ɬɟɥɨ ɪɚɡɝɨɧɹɟɬɫɹ.

ɉɨɥɭɱɟɧɧɚɹ ɮɨɪɦɭɥɚ ɩɨɡɜɨɥɹɟɬ ɪɚɫɫɱɢɬɚɬɶ, ɤɚɤɭɸ ɩɪɟɞɟɥɶɧɭɸ

(ɦɚɤɫɢɦɚɥɶɧɭɸ) ɫɤɨɪɨɫɬɶ ɦɨɠɟɬ ɪɚɡɜɢɬɶ ɬɟɥɨ ɢɡ

ɫɨɫɬɨɹɧɢɹ ɩɨɤɨɹ

m a

Ɋɢɫ. 1

Ɇɟɯɚɧɢɤɚ

(

v ), ɤɨɝɞɚ ɢɡ ɧɟɝɨ ɜɵɬɟɱɟɬ ɜɟɫɶ ɢɦɟɸɳɢɣɫɹ ɝɚɡ. ȿɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ

ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɦɚɫɫɚ ɝɚɡɚ ɜɧɭɬɪɢ ɬɟɥɚ ɛɵɥɚ ɪɚɜɧɚ

, ɬɨ ɦɚɤɫɢɦɚɥɶɧɚɹ

ɫɤɨɪɨɫɬɶ

max

, ɤɨɬɨɪɭɸ ɫɦɨɠɟɬ ɪɚɡɜɢɬɶ ɬɟɥɨ ɩɪɢ ɪɟɚɤɬɢɜɧɨɦ ɞɜɢɠɟɧɢɢ,

ɪɚɜɧɚ



max

lnv

ɗɬɚ ɮɨɪɦɭɥɚ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɮɨɪɦɭɥɵ ɐɢɨɥɤɨɜɫɤɨɝɨ. ɂɡ ɮɨɪɦɭ-

ɥɵ ɐɢɨɥɤɨɜɫɤɨɝɨ ɥɟɝɤɨ ɫɞɟɥɚɬɶ ɜɵɜɨɞ, ɱɬɨ ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɬɨɝɨ, ɩɨ ɤɚɤɨ-

ɦɭ ɡɚɤɨɧɭ ɭɦɟɧɶɲɚɟɬɫɹ ɦɚɫɫɚ ɝɚɡɚ, ɦɚɤɫɢɦɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɬɟɥɚ

max

ɛɭɞɟɬ ɨɞɢɧɚɤɨɜɨɣ.

1.2. ɋɤɨɪɨɫɬɶ ɢɫɬɟɱɟɧɢɹ ɝɚɡɚ

Ɋɚɫɫɦɨɬɪɢɦ, ɩɪɢ ɤɚɤɢɯ ɭɫɥɨɜɢɹɯ ɫɤɨɪɨɫɬɶ ɢɫɬɟɱɟɧɢɹ ɝɚɡɚ ɦɨɠɧɨ

ɫɱɢɬɚɬɶ ɩɨɫɬɨɹɧɧɨɣ.

ɉɭɫɬɶ ɨɬɞɟɥɹɸɳɟɟɫɹ ɨɬ ɬɟɥɚ

ɜɟɳɟɫɬɜɨ (ɝɚɡ) ɧɚɯɨɞɢɬɫɹ ɜ ɰɢɥɢɧ-

ɞɪɢɱɟɫɤɨɦ ɪɟɡɟɪɜɭɚɪɟ ɨɛɴɟɦɨɦ

ɫ ɩɥɨɳɚɞɶɸ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ

S , ɚ ɩɥɨɳɚɞɶ ɨɬɜɟɪɫɬɢɹ (ɫɨɩɥɚ),

ɱɟɪɟɡ ɤɨɬɨɪɨɟ ɝɚɡ ɜɵɬɟɤɚɟɬ ɢɡ ɪɟ-

ɡɟɪɜɭɚɪɚ, ɪɚɜɧɚ

(ɪɢɫ. 2). Ɍɟɦɩɟ-

ɪɚɬɭɪɚ ɝɚɡɚ

ɞɨɫɬɚɬɨɱɧɨ ɜɟɥɢɤɚ,

ɱɬɨɛɵ ɞɚɜɥɟɧɢɟ

ɢ ɩɥɨɬɧɨɫɬɶ

ɜɵɪɚɜɧɢɜɚɥɢɫɶ ɦɝɧɨɜɟɧɧɨ ɜɨ ɜɫɟɦ ɫɨɫɭɞɟ ɩɨɫɥɟ ɤɚɠɞɨɝɨ ɜɵɛɪɨɫɚ ɝɚɡɚ ɢ

ɫɬɚɧɨɜɢɥɢɫɶ ɨɞɢɧɚɤɨɜɵɦɢ ɜɨ ɜɫɟɦ ɪɟɡɟɪɜɭɚɪɟ. Ɍɨɝɞɚ ɩɪɢ ɜɵɬɟɤɚɧɢɢ ɝɚ-

ɡɚ ɬɟɱɟɧɢɟ ɫɭɳɟɫɬɜɭɟɬ ɬɨɥɶɤɨ ɜ ɧɟɛɨɥɶɲɨɦ ɫɥɨɟ ɜɛɥɢɡɢ ɫɨɩɥɚ (ɬɨɥɳɢ-

ɧɨɣ G). Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɥɢɧɢɢ ɬɨɤɚ ɧɟɩɪɟɪɵɜɧɵ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,

ɩɥɨɬɧɨɫɬɶ ɝɚɡɚ, ɩɪɨɯɨɞɹɳɟɝɨ ɱɟɪɟɡ ɫɨɩɥɨ, ɢ ɩɥɨɬɧɨɫɬɶ ɝɚɡɚ ɧɚ ɪɚɫɫɬɨɹ-

ɧɢɢ G

ɨɬ ɫɨɩɥɚ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɨɞɢɧɚɤɨɜɨɣ. Ɍɨɝɞɚ ɞɥɹ ɬɟɱɟɧɢɹ ɜ ɫɥɨɟ

ɬɨɥɳɢɧɨɣ G ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɟ ɧɟɪɚɡɪɵɜɧɨɫɬɢ:

uSSu U U

ɢɥɢ

uSSu

ɝɞɟ

– ɫɤɨɪɨɫɬɶ ɬɟɱɟɧɢɹ ɱɟɪɟɡ ɫɨɩɥɨ, ɚ

– ɫɤɨɪɨɫɬɶ ɬɟɱɟɧɢɹ ɜ G-ɫɥɨɟ

ɜɛɥɢɡɢ ɫɨɩɥɚ.

ȿɫɥɢ ɪɟɚɤɬɢɜɧɨɟ ɞɜɢɠɟɧɢɟ ɫɨɜɟɪɲɚɟɬɫɹ ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦ ɧɚ-

ɩɪɚɜɥɟɧɢɢ, ɬɨ ɨɛɚ ɫɟɱɟɧɢɹ

S ɧɚɯɨɞɹɬɫɹ ɧɚ ɨɞɢɧɚɤɨɜɨɣ ɜɵɫɨɬɟ

ɨɬ

= 0

Ɋɢɫ. 2

ɆɨɞɆ-06 Ɋɟɚɤɬɢɜɧɨɟ ɞɜɢɠɟɧɢɟ

ɩɨɜɟɪɯɧɨɫɬɢ Ɂɟɦɥɢ. Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ Ȼɟɪɧɭɥɥɢ ɞɥɹ ɷɬɨɝɨ ɬɟɱɟɧɢɹ ɝɚɡɚ

ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ

ghp

UU UU

ɝɞɟ

p – ɞɚɜɥɟɧɢɟ ɝɚɡɚ ɜɧɭɬɪɢ ɪɟɡɟɪɜɭɚɪɚ,

p – ɜɧɟɲɧɟɟ ɞɚɜɥɟɧɢɟ. ɉɪɢ

ɞɜɢɠɟɧɢɢ ɜ ɜɚɤɭɭɦɟ ɜɧɟɲɧɟɟ ɞɚɜɥɟɧɢɟ ɪɚɜɧɨ ɧɭɥɸ

p .

ɋɥɟɞɨɜɚɬɟɥɶɧɨ,

U U

. ɍɱɢɬɵɜɚɹ, ɱɬɨ

uSSu , ɩɨɥɭ-

ɱɢɦ

U U

ɢɥɢ

U

ɉɥɨɬɧɨɫɬɶ ɝɚɡɚ

U , ɬɨɝɞɚ



ɋ ɩɨɦɨɳɶɸ ɭɪɚɜɧɟɧɢɹ Ɇɟɧɞɟɥɟɟɜɚ-Ʉɥɚɣɩɟɪɨɧɚ

ɦɨɠɧɨ ɫɜɹɡɚɬɶ ɫɤɨɪɨɫɬɶ ɢɫɬɟɱɟɧɢɹ ɝɚɡɚ

ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ ɝɚɡɚ

ɢ ɟɝɨ

ɦɨɥɹɪɧɨɣ ɦɚɫɫɨɣ

P (

= 8,31441 Ⱦɠ/(ɦɨɥɶɄ) – ɭɧɢɜɟɪɫɚɥɶɧɚɹ ɝɚɡɨɜɚɹ

ɩɨɫɬɨɹɧɧɚɹ)



P

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,

P

ɂɡ ɩɨɥɭɱɟɧɧɨɝɨ ɜɵɪɚɠɟɧɢɹ ɜɢɞɧɨ, ɱɬɨ ɟɫɥɢ ɜ ɩɪɨɰɟɫɫɟ ɜɵɬɟɤɚɧɢɹ

ɝɚɡɚ ɟɝɨ ɬɟɦɩɟɪɚɬɭɪɚ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ (

const

), ɬɨ ɢ ɫɤɨɪɨɫɬɶ ɢɫ-

ɬɟɱɟɧɢɹ ɝɚɡɚ ɹɜɥɹɟɬɫɹ ɤɨɧɫɬɚɧɬɨɣ

const u .

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɥɭɱɟɧɧɚɹ ɜ ɩɚɪɚɝɪɚɮɟ 1.1 ɡɚɜɢɫɢɦɨɫɬɶ ɫɤɨɪɨɫɬɢ

ɬɟɥɚ ɨɬ ɜɪɟɦɟɧɢ ɦɨɠɟɬ ɧɚɛɥɸɞɚɬɶɫɹ, ɧɚɩɪɢɦɟɪ, ɜ ɫɥɭɱɚɟ, ɟɫɥɢ ɜ ɩɪɨɰɟɫ-

ɫɟ ɢɫɬɟɱɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɚ ɝɚɡɚ ɜɧɭɬɪɢ ɬɟɥɚ (ɜ ɪɟɡɟɪɜɭɚɪɟ) ɨɫɬɚɟɬɫɹ ɩɨ-

ɫɬɨɹɧɧɨɣ.

Ɇɟɯɚɧɢɤɚ

ȼ ɩɪɨɰɟɫɫɟ ɢɫɬɟɱɟɧɢɹ ɦɚɫɫɚ ɝɚɡɚ ɜ ɪɟɡɟɪɜɭɚɪɟ ɩɨɫɬɨɹɧɧɨ ɭɦɟɧɶ-

ɲɚɟɬɫɹ. ȿɫɥɢ ɬɟɦɩɟɪɚɬɭɪɚ ɝɚɡɚ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ, ɬɨ ɫɨɝɥɚɫɧɨ ɭɪɚɜ-

ɧɟɧɢɸ Ɇɟɧɞɟɥɟɟɜɚ-Ʉɥɚɣɩɟɪɨɧɚ

ɢɡɦɟɧɟɧɢɟ ɦɚɫɫɵ ɝɚɡɚ ɦɨɠɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɥɢɛɨ ɡɚ ɫɱɟɬ ɢɡɦɟɧɟɧɢɹ ɞɚɜ-

ɥɟɧɢɹ, ɥɢɛɨ ɡɚ ɫɱɟɬ ɢɡɦɟɧɟɧɢɹ ɨɛɴɟɦɚ (ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɞɚɥɶɧɟɣɲɢɯ ɪɚɫ-

ɱɟɬɨɜ ɫɥɭɱɚɣ ɨɞɧɨɜɪɟɦɟɧɧɨɝɨ ɢɡɦɟɧɟɧɢɹ ɢ ɞɚɜɥɟɧɢɹ, ɢ ɨɛɴɟɦɚ ɧɟ ɪɚɫ-

ɫɦɚɬɪɢɜɚɟɬɫɹ):

)()( tpV

ɢɥɢ dVp

(

const p

Vtp

)()(

ɢɥɢ

dpV

( const V ).

Ɂɞɟɫɶ

m – ɦɚɫɫɚ ɝɚɡɚ ɜɧɭɬɪɢ ɬɟɥɚ (ɦɚɫɫɚ ɨɫɬɚɜɲɟɝɨɫɹ ɝɚɡɚ). ɉɨɷɬɨɦɭ

ɢɡɦɟɧɟɧɢɟ ɦɚɫɫɵ ɝɚɡɚ

dm ɪɚɜɧɨ ɢɡɦɟɧɟɧɢɸ ɦɚɫɫɵ ɜɫɟɝɨ ɬɟɥɚ:

dmdm

1.3. Ɂɚɜɢɫɢɦɨɫɬɶ ɦɚɫɫɵ ɝɚɡɚ ɨɬ ɜɪɟɦɟɧɢ ɩɪɢ ɢɫɬɟɱɟɧɢɢ

ɍɦɟɧɶɲɟɧɢɟ ɦɚɫɫɵ ɝɚɡɚ ɜɧɭɬɪɢ ɬɟɥɚ (ɜ ɪɟɡɟɪɜɭɚɪɟ) ɩɪɨɢɫɯɨɞɢɬ

ɱɟɪɟɡ ɫɨɩɥɨ ɩɥɨɳɚɞɶɸ

S . Ƚɚɡ ɜɵɬɟɤɚɟɬ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ u , ɩɨ-

ɷɬɨɦɭ ɡɚ ɜɪɟɦɹ

ɨɛɴɟɦ ɝɚɡɚ ɭɦɟɧɶɲɢɬɫɹ ɧɚ

dtSu

, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɚɫ-

ɫɚ ɝɚɡɚ ɭɦɟɧɶɲɢɬɫɹ ɧɚ

dtSudm U .

ɂɡ ɭɪɚɜɧɟɧɢɹ Ɇɟɧɞɟɥɟɟɜɚ-Ʉɥɚɣɩɟɪɨɧɚ

RTV

Ɍɨɝɞɚ

dtp

Sudm 

 .

1.3.1. ȼɵɬɟɤɚɧɢɟ ɝɚɡɚ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ

Ɋɚɫɫɦɨɬɪɢɦ, ɤɚɤ ɦɟɧɹɟɬɫɹ ɦɚɫɫɚ ɝɚɡɚ, ɟɫɥɢ ɜɧɭɬɪɢ ɪɟɡɟɪɜɭɚɪɚ ɩɨɞ-

ɞɟɪɠɢɜɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɟ ɞɚɜɥɟɧɢɟ

const p .

Ɏɚɤɬɢɱɟɫɤɢ ɷɬɨ ɝɨɜɨɪɢɬ ɨ ɬɨɦ, ɱɬɨ ɜ ɩɪɨɰɟɫɫɟ ɢɫɬɟɱɟɧɢɹ ɞɥɢɧɚ ɪɟ-

ɡɟɪɜɭɚɪɚ ɫ ɝɚɡɨɦ ɞɨɥɠɧɚ ɩɨɫɬɨɹɧɧɨ ɭɦɟɧɶɲɚɬɶɫɹ, ɱɬɨɛɵ ɩɨɞɞɟɪɠɢɜɚɬɶ ɜ

ɧɟɦ ɩɨɫɬɨɹɧɧɨɟ ɞɚɜɥɟɧɢɟ.

Ɍɨɝɞɚ ɢɡɦɟɧɟɧɢɟ ɦɚɫɫɵ ɝɚɡɚ ɫ ɨɞɧɨɣ ɫɬɨɪɨɧɵ ɡɚɜɢɫɢɬ ɨɬ ɨɛɴɟɦɚ, ɚ

ɫ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ – ɨɬ ɜɪɟɦɟɧɢ

ɆɨɞɆ-06 Ɋɟɚɤɬɢɜɧɨɟ ɞɜɢɠɟɧɢɟ

dVp

ɢ dtp

Sudm 

 .

ɉɪɢɪɚɜɧɹɜ ɞɚɧɧɵɟ ɜɵɪɚɠɟɧɢɹ, ɩɨɥɭɱɢɦ ɡɚɜɢɫɢɦɨɫɬɶ ɨɛɴɟɦɚ ɝɚɡɚ

ɨɬ ɜɪɟɦɟɧɢ

dtp

SudVp





 dtSudV  .

ɉɪɨɢɧɬɟɝɪɢɪɨɜɚɜ, ɩɨɥɭɱɢɦ

uStVV 

max

ɢɥɢ

)1(

max

tVV E

, ɝɞɟ

max

E .

Ɉɛɴɟɦ ɝɚɡɚ ɭɦɟɧɶɲɚɟɬɫɹ ɥɢɧɟɣɧɨ ɩɨ ɜɪɟɦɟɧɢ ɨɬ

max

V (ɜ ɧɚɱɚɥɶ-

ɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ) ɞɨ ɧɭɥɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɚɫɫɚ ɝɚɡɚ ɬɨɠɟ ɭɦɟɧɶ-

ɲɚɟɬɫɹ ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɜɪɟɦɟɧɢ

)1(

max

tmm

E

, ɝɞɟ

maxmax

ɉɪɢ ɬɚɤɨɦ ɪɟɠɢɦɟ ɜɵɬɟɤɚɧɢɹ ɜɟɫɶ ɝɚɡ (

m ) ɜɵɬɟɱɟɬ ɢɡ ɪɟɡɟɪ-

ɜɭɚɪɚ ɡɚ ɜɪɟɦɹ

E /1

max

t . ɋɩɭɫɬɹ ɜɪɟɦɹ

max

t ɬɟɥɨ ɪɚɡɨɜɶɟɬ ɦɚɤɫɢɦɚɥɶ-

ɧɭɸ ɫɤɨɪɨɫɬɶ

max

v , ɩɪɟɞɫɤɚɡɚɧɧɭɸ ɐɢɨɥɤɨɜɫɤɢɦ.

ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɞɥɹ ɫɨɛɥɸɞɟɧɢɹ ɭɫɥɨɜɢɹ

const u

ɧɟɨɛɯɨ-

ɞɢɦɨ, ɱɬɨɛɵ ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɪɟɡɟɪɜɭɚɪɚ ɧɟ ɦɟɧɹɥɚɫɶ ɫɨ

ɜɪɟɦɟɧɟɦ. ɉɨɷɬɨɦɭ ɢɡɦɟɧɟɧɢɟ ɨɛɴɟɦɚ ɪɟɡɟɪɜɭɚɪɚ ɜɨɡɦɨɠɧɨ ɬɨɥɶɤɨ ɡɚ

ɫɱɟɬ ɭɦɟɧɶɲɟɧɢɹ ɟɝɨ ɞɥɢɧɵ

)1(

max

tLL E

. Ⱦɥɹ ɩɨɞɞɟɪɠɚɧɢɹ ɜ ɪɟɡɟɪ-

ɜɭɚɪɟ ɩɨɫɬɨɹɧɧɨɝɨ ɞɚɜɥɟɧɢɹ ɜ ɷɬɢɯ ɭɫɥɨɜɢɹɯ ɞɨɫɬɚɬɨɱɧɨ ɭɦɟɧɶɲɚɬɶ ɟɝɨ

ɞɥɢɧɭ ɪɚɜɧɨɦɟɪɧɨ, ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ

u .

1.3.2. ȼɵɬɟɤɚɧɢɟ ɝɚɡɚ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɨɛɴɟɦɟ

Ɋɚɫɫɦɨɬɪɢɦ, ɤɚɤ ɦɟɧɹɟɬɫɹ ɦɚɫɫɚ ɝɚɡɚ, ɟɫɥɢ ɨɛɴɟɦ ɪɟɡɟɪɜɭɚɪɚ ɨɫɬɚ-

ɟɬɫɹ ɩɨɫɬɨɹɧɧɵɦ ɩɪɢ ɜɵɬɟɤɚɧɢɢ ɝɚɡɚ

const V

Ɍɨɝɞɚ ɢɡɦɟɧɟɧɢɟ ɦɚɫɫɵ ɝɚɡɚ ɫ ɨɞɧɨɣ ɫɬɨɪɨɧɵ ɡɚɜɢɫɢɬ ɨɬ ɞɚɜɥɟɧɢɹ,

ɚ ɫ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ – ɨɬ ɜɪɟɦɟɧɢ

dpV

ɢ dtp

Sudm 

 .

ɉɪɢɪɚɜɧɹɜ ɞɚɧɧɵɟ ɜɵɪɚɠɟɧɢɹ, ɩɨɥɭɱɢɦ ɡɚɜɢɫɢɦɨɫɬɶ ɞɚɜɥɟɧɢɹ ɝɚ-

ɡɚ ɨɬ ɜɪɟɦɟɧɢ

dtp

SudpV





 dt

 .

Ɇɟɯɚɧɢɤɚ

Ɉɛɴɟɦ ɝɚɡɚ ɧɟ ɦɟɧɹɟɬɫɹ, ɩɨɷɬɨɦɭ

const

max

VV . Ɍɨɝɞɚ E

max

ɚ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ

E

ɉɪɨɢɧɬɟɝɪɢɪɨɜɚɜ, ɩɨɥɭɱɢɦ

E

max

ln ɢɥɢ

epp

E

max

Ⱦɚɜɥɟɧɢɟ ɝɚɡɚ ɭɦɟɧɶɲɚɟɬɫɹ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ ɩɨ ɜɪɟɦɟɧɢ ɨɬ

max

ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɞɨ ɧɭɥɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɚɫɫɚ ɝɚɡɚ ɬɨɠɟ

ɭɦɟɧɶɲɚɟɬɫɹ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ

emm

E

max

, ɝɞɟ Vp

maxmax

ɉɪɢ ɬɚɤɨɦ ɪɟɠɢɦɟ ɜɵɬɟɤɚɧɢɹ ɡɚ ɜɪɟɦɹ

ɦɚɫɫɚ ɝɚɡɚ ɭɦɟɧɶɲɢɬ-

ɫɹ ɜ

ɪɚɡ. Ɍɟɥɨ ɪɚɡɨɜɶɟɬ ɦɚɤɫɢɦɚɥɶɧɭɸ ɫɤɨɪɨɫɬɶ

max

, ɩɪɟɞɫɤɚɡɚɧɧɭɸ

ɐɢɨɥɤɨɜɫɤɢɦ, ɧɨ ɡɧɚɱɢɬɟɥɶɧɨ ɩɨɡɠɟ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɜɵɬɟɤɚɧɢɟɦ ɝɚɡɚ

ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ. ȼɪɟɦɹ ɪɚɡɝɨɧɚ ɫɬɪɟɦɢɬɫɹ ɤ ɛɟɫɤɨɧɟɱɧɨɫɬɢ.

1.4. Ʉɢɧɟɦɚɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɪɟɚɤɬɢɜɧɨɝɨ

ɞɜɢɠɟɧɢɹ

Ɋɚɫɫɦɨɬɪɢɦ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɜɪɟɦɟɧɢ ɨɫɧɨɜɧɵɯ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ

ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɪɟɚɤɬɢɜɧɨɝɨ ɞɜɢɠɟɧɢɹ (ɫɤɨɪɨɫɬɢ, ɤɨɨɪɞɢɧɚɬɵ ɢ ɭɫɤɨɪɟ-

ɧɢɹ) ɞɥɹ ɩɨɥɭɱɟɧɧɵɯ ɜɪɟɦɟɧɧ

ɵɯ ɡɚɜɢɫɢɦɨɫɬɟɣ ɦɚɫɫɵ ɝɚɡɚ.

Ʉɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɜ ɩɚɪɚɝɪɚɮɟ 1.1, ɩɪɢ ɞɜɢɠɟɧɢɢ ɢɡ ɫɨɫɬɨɹɧɢɹ

ɩɨɤɨɹ (

v ) ɫɤɨɪɨɫɬɶ ɪɟɚɤɬɢɜɧɨ ɞɜɢɠɭɳɟɝɨɫɹ ɬɟɥɚ ɡɚɜɢɫɢɬ ɨɬ ɦɚɫɫɵ

ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ

)(

ln)(

v .

ȼ ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɦɨɞɟɥɢ ɪɟɚɤɬɢɜɧɨ ɞɜɢɠɭɳɟɟɫɹ ɬɟɥɨ ɫɨɫɬɨɢɬ ɧɟ

ɬɨɥɶɤɨ ɢɡ ɩɨɫɬɨɹɧɧɨ ɜɵɬɟɤɚɸɳɟɝɨ ɝɚɡɚ, ɧɨ ɢɡ ɧɟɤɨɬɨɪɨɣ ɩɨɫɬɨɹɧɧɨɣ ɧɚ-

ɝɪɭɡɤɢ. Ɍɨ ɟɫɬɶ

ɝɚɡ ɜɵɫɬɭɩɚɟɬ ɜ ɪɨɥɢ ɬɨɩɥɢɜɚ, ɚ ɫɚɦɚ ɪɚɤɟɬɚ – ɜ ɪɨɥɢ

ɩɨɫɬɨɹɧɧɨɣ ɧɚɝɪɭɡɤɢ

. Ɍɨɝɞɚ ɦɚɫɫɚ ɬɟɥɚ m ɜɤɥɸɱɚɟɬ ɧɟ ɬɨɥɶɤɨ ɩɨɫɬɨ-

ɹɧɧɨ ɭɦɟɧɶɲɚɸɳɭɸɫɹ ɦɚɫɫɭ ɝɚɡɚ

m , ɧɨ ɢ ɧɟɤɨɬɨɪɭɸ ɧɟɢɡɦɟɧɧɭɸ ɦɚɫ-

ɫɭ ɪɚɤɟɬɵ

const

)()( tmMtm

ɝɪ



Ⱥ ɫɤɨɪɨɫɬɶ ɪɚɤɟɬɵ ɩɪɢ ɪɚɡɝɨɧɟ

ɆɨɞɆ-06 Ɋɟɚɤɬɢɜɧɨɟ ɞɜɢɠɟɧɢɟ

)(

)0(

ln)(

tmM

ɝɪ



ɍɫɤɨɪɟɧɢɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɭɬɟɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɫɤɨɪɨɫɬɢ, ɚ

ɤɨɨɪɞɢɧɚɬɚ – ɩɭɬɟɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɨ ɜɪɟɦɟɧɢ

)(



dttxx

)(v .

Ʉɨɝɞɚ ɬɨɩɥɢɜɨ ɜɵɬɟɤɚɟɬ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ

const p , ɦɚɫ-

ɫɚ ɥɢɧɟɣɧɨ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ

)1(

max

tmm

E . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ

ɪɚɡɝɨɧɟ ɢɡ ɫɨɫɬɨɹɧɢɹ ɩɨɤɨɹ ɢ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ (

)

ɫɤɨɪɨɫɬɶ ɪɚɤɟɬɵ

)1(

ln)(

max

tmM

E



ɭɫɤɨɪɟɧɢɟ ɪɚɤɟɬɵ

)1(

)(

max

tmM

E

ɤɨɨɪɞɢɧɚɬɚ ɪɚɤɟɬɵ

)1(

)(

max

tmM

uttx

ɪɪ

E



E



ȿɫɥɢ

ɦɚɫɫɚ ɝɚɡɚ ɦɧɨɝɨ ɦɟɧɶɲɟ ɦɚɫɫɵ ɪɚɤɟɬɵ

Mm 

max

, ɬɨ

ɜɬɨɪɵɦ ɫɥɚɝɚɟɦɵɦ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɜ ɜɵɪɚɠɟɧɢɢ ɞɥɹ ɭɫɤɨɪɟɧɢɹ ɦɨɠɧɨ

ɩɪɟɧɟɛɪɟɱɶ:

max

. Ɍɨɝɞɚ ɪɚɤɟɬɚ ɞɜɢɠɟɬɫɹ ɩɪɚɤɬɢɱɟɫɤɢ ɫ ɩɨɫɬɨ-

ɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ. ɂɡɜɟɫɬɧɨ, ɱɬɨ ɩɪɢ ɞɜɢɠɟɧɢɢ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟ-

ɧɢɟɦ ɫɤɨɪɨɫɬɶ ɜɨɡɪɚɫɬɚɟɬ ɥɢɧɟɣɧɨ, ɚ ɤɨɨɪɞɢɧɚɬɚ – ɤɜɚɞɪɚɬɢɱɧɨ.

ȼ ɷɬɢɯ ɭɫɥɨɜɢɹɯ ɝɪɚɮɢɤɢ ɫɤɨɪɨɫɬɢ ɢ ɭɫɤɨɪɟɧɢɹ ɧɚɩɨɦɢɧɚɸɬ ɥɢ-

ɧɟɣɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ (ɪɢɫ. 3

ɚ). Ƚɪɚɮɢɤ ɤɨɨɪɞɢɧɚɬɵ ɨɬɪɚɠɚɟɬ ɭɫɤɨɪɟɧ-

ɧɵɣ (ɤɜɚɞɪɚɬɢɱɧɵɣ) ɯɚɪɚɤɬɟɪ ɞɜɢɠɟɧɢɹ. ɍɫɤɨɪɟɧɢɟ ɫɥɚɛɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ

ɤɨɧɫɬɚɧɬɵ, ɚ ɞɜɢɠɟɧɢɟ ɫɥɚɛɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɪɚɜɧɨɭɫɤɨɪɟɧɧɨɝɨ.

ɉɨɫɥɟ ɜɵɬɟɤɚɧɢɹ ɜɫɟɝɨ ɝɚɡɚ (ɬɨɩɥɢɜɚ) ɯɚɪɚɤɬɟɪ ɞɜɢɠɟɧɢɹ ɪɟɡɤɨ

ɦɟɧɹɟɬɫɹ: ɭɫɤɨɪɟɧɢɟ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ (ɪɟɚɤɬɢɜɧɚɹ ɫɢɥɚ ɩɪɟɤɪɚɳɚɟɬ

ɞɟɣɫɬɜɨɜɚɬɶ), ɫɤɨɪɨɫɬɶ ɫɬɚɧɨɜɢɬɫɹ ɩɨɫɬɨɹɧɧɨɣ, ɚ ɤɨɨɪɞɢɧɚɬɚ ɜɨɡɪɚɫɬɚɟɬ

ɥɢɧɟɣɧɨ. ɇɚ ɪɢɫɭɧɤɟ 3

ɚ ɢɡɨɛɪɚɠɟɧɵ ɝɪɚɮɢɤɢ )(ta , )(tv , )(t

ɜ ɬɟɱɟɧɢɟ

ɪɚɡɝɨɧɚ, ɪɚɜɧɨɦɟɪɧɨɟ ɞɜɢɠɟɧɢɟ ɩɨɫɥɟ ɪɚɡɝɨɧɚ ɇȿ ɩɪɟɞɫɬɚɜɥɟɧɨ.

Ɇɟɯɚɧɢɤɚ

x(t)

v(t)

a(t)

x(t)

v(t)

a(t)

Ɋɢɫ. 3

Ʉɨɝɞɚ ɝɚɡ (ɬɨɩɥɢɜɨ) ɜɵɬɟɤɚɟɬ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɨɛɴɟɦɟ

const V ,

ɦɚɫɫɚ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ

emm

E

max

. ɋɥɟɞɨɜɚ-

ɬɟɥɶɧɨ, ɩɪɢ ɪɚɡɝɨɧɟ ɢɡ ɫɨɫɬɨɹɧɢɹ ɩɨɤɨɹ ɢ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ (

v ,

x )

ɫɤɨɪɨɫɬɶ ɪɚɤɟɬɵ

emM

E



max

ln)(v ,

ɭɫɤɨɪɟɧɢɟ ɪɚɤɟɬɵ

E



max

)(

ɤɨɨɪɞɢɧɚɬɚ ɪɚɤɟɬɵ

E



utx

max

ln)( .

ȿɫɥɢ

ɦɚɫɫɚ ɝɚɡɚ ɦɧɨɝɨ ɦɟɧɶɲɟ ɦɚɫɫɵ ɪɚɤɟɬɵ

Mm 

max

, ɬɨ

ɜɬɨɪɵɦ ɫɥɚɝɚɟɦɵɦ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɜ ɜɵɪɚɠɟɧɢɢ ɞɥɹ ɭɫɤɨɪɟɧɢɹ ɬɚɤɠɟ

ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ:

E

max

)( . ɍɫɤɨɪɟɧɢɟ ɪɚɤɟɬɵ ɩɨɫɬɟɩɟɧɧɨ

ɭɦɟɧɶɲɚɟɬɫɹ ɞɨ ɧɭɥɹ.

Ƚɪɚɮɢɤɢ ɫɤɨɪɨɫɬɢ ɢ ɭɫɤɨɪɟɧɢɹ ɧɚɩɨɦɢɧɚɸɬ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɵɟ

ɡɚɜɢɫɢɦɨɫɬɢ (ɪɢɫ. 3

ɛ). ɍɫɤɨɪɟɧɢɟ ɩɨɫɬɨɹɧɧɨ ɭɦɟɧɶɲɚɟɬɫɹ, ɫɤɨɪɨɫɬɶ

ɫɬɪɟɦɢɬɫɹ ɤ ɤɨɧɫɬɚɧɬɟ, ɤɨɨɪɞɢɧɚɬɚ – ɤ ɥɢɧɟɣɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ. ɍɫɤɨɪɟɧ-

ɧɨɟ ɞɜɢɠɟɧɢɟ ɩɥɚɜɧɨ ɩɟɪɟɯɨɞɢɬ ɜ ɪɚɜɧɨɦɟɪɧɨɟ.

ɋɥɟɞɭɟɬ ɬɚɤɠɟ ɩɨɦɧɢɬɶ, ɱɬɨ ɩɨɫɥɟ ɨɤɨɧɱɚɧɢɹ ɪɚɡɝɨɧɚ (ɩɨɫɥɟ ɢɫɬɟ-

ɱɟɧɢɹ ɜɫɟɝɨ ɬɨɩɥɢɜɚ) ɪɚɤɟɬɚ ɛɭɞɟɬ ɞɜɢɝɚɬɶɫɹ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ