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ɆɨɞɆ-04 Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ

Ɍɚɛɥɢɰɚ 3

Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ

ɢɧɟɪɰɢɢ, J

= < J > – m

((x

– X

)

+ (y

– Y

)

), ɤɝ



ɫɦ

– x

ɫɦ

–

– y

, ɫɦ

6. Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ

1. ɉɟɪɟɱɢɫɥɢɬɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢ-

ɠɟɧɢɹ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ.

ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɨɫɧɨɜɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɢɧɚɦɢɤɢ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ

ɬɟɥɚ.

Ɉɩɢɲɢɬɟ ɩɪɢɧɰɢɩɢɚɥɶɧɭɸ ɫɯɟɦɭ ɷɤɫɩɟɪɢɦɟɧɬɚ ɢ ɡɚɩɢɲɢɬɟ ɭɪɚɜ-

ɧɟɧɢɟ ɞɢɧɚɦɢɤɢ ɞɥɹ ɧɟɟ.

ɉɨɥɭɱɢɬɟ ɫɜɹɡɶ ɦɟɠɞɭ ɦɨɦɟɧɬɨɦ ɢɧɟɪɰɢɢ ɬɟɥɚ ɢ ɭɝɥɨɜɨɣ ɫɤɨɪɨ-

ɫɬɶɸ ɜɪɚɳɟɧɢɹ ɜ ɩɪɟɞɫɬɨɹɳɟɦ ɷɤɫɩɟɪɢɦɟɧɬɟ.

Ʉɚɤ ɬɟɨɪɟɬɢɱɟɫɤɢ ɦɨɠɧɨ ɪɚɫɫɱɢɬɚɬɶ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɢ ɩɨɥɨɠɟɧɢɟ

ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɬɟɥɚ?

Ɉɩɢɲɢɬɟ ɩɨɪɹɞɨɤ ɜɵɩɨɥɧɟɧɢɹ ɪɚɛɨɬɵ.

ɅȺȻɈɊȺɌɈɊɇȺə ɊȺȻɈɌȺ ʋ ɆɨɞɆ-05

ɉɈ ɂɁɍɑȿɇɂɘ ɆɈȾȿɅȿɃ ɎɂɁɂɑȿɋɄɂɏ

ɉɊɈɐȿɋɋɈȼ ɂ əȼɅȿɇɂɃ ɇȺ ɄɈɆɉɖɘɌȿɊȿ

Ɋɚɛɨɬɚ ɢ ɷɧɟɪɝɢɹ

ɐɟɥɶ ɪɚɛɨɬɵ: ɢɡɭɱɟɧɢɟ ɪɚɛɨɬɵ ɧɟɤɨɧɫɟɪɜɚɬɢɜɧɵɯ ɫɢɥ ɧɚ ɩɪɢɦɟ-

ɪɟ ɫɢɥɵ ɬɪɟɧɢɹ ɩɪɢ ɩɪɹɦɨɥɢɧɟɣɧɨɦ ɞɜɢɠɟɧɢɢ ɢ ɞɜɢɠɟɧɢɢ ɩɨ ɨɤɪɭɠɧɨ-

ɫɬɢ. ɂɡɭɱɟɧɢɟ ɡɚɜɢɫɢɦɨɫɬɢ ɪɚɛɨɬɵ ɫɢɥɵ ɬɪɟɧɢɹ ɨɬ ɦɚɫɫɵ ɞɜɢɠɭɳɟɝɨɫɹ

ɬɟɥɚ, ɮɨɪɦɵ ɢ ɞɥɢɧɵ ɩɪɨɣɞɟɧɧɨɝɨ ɢɦ ɩɭɬɢ. Ɉɩɪɟɞɟɥɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ

ɬɪɟɧɢɹ.

1. Ɍɟɨɪɟɬɢɱɟɫɤɨɟ ɫɨɞɟɪɠɚɧɢɟ

ɗɧɟɪɝɢɹ – ɷɬɨ ɫɤɚɥɹɪɧɚɹ ɮɢɡɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ

ɫɨɫɬɨɹɧɢɟ ɬɟɥɚ ɢɥɢ ɫɢɫɬɟɦɵ ɬɟɥ, ɢɡɦɟɧɟɧɢɟ ɤɨɬɨɪɨɣ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɱɟ-

ɪɟɡ ɫɨɜɟɪɲɟɧɢɟ ɪɚɛɨɬɵ. Ɍɨ ɟɫɬɶ, ɱɬɨɛɵ ɬɟɥɨ ɢɥɢ ɫɢɫɬɟɦɭ ɬɟɥ ɩɟɪɟɜɟɫɬɢ

ɢɡ ɨɞɧɨɝɨ ɫɨɫɬɨɹɧɢɟ ɜ ɞɪɭɝɨɟ (ɢɡ ɫɨɫɬɨɹɧɢɹ ɫ ɨɞɧɨɣ ɷɧɟɪɝɢɟɣ ɜ ɫɨɫɬɨɹ-

ɧɢɟ ɫ ɞɪɭɝɨɣ ɷɧɟɪɝɢɟɣ), ɤ ɬɟɥɭ ɢɥɢ ɫɢɫɬɟɦɟ ɬɟɥ ɧɟɨɛɯɨɞɢɦɨ ɩɪɢɥɨɠɢɬɶ

ɧɟɤɨɬɨɪɭɸ ɫɢɥɭ

, ɫɨɜɟɪɲɚɸɳɭɸ ɧɟɧɭɥɟɜɭɸ ɪɚɛɨɬɭ.

ȼ ɤɥɚɫɫɢɱɟɫɤɨɣ ɧɟɪɟɥɹɬɢɜɢɫɬɫɤɨɣ ɦɟɯɚɧɢɤɟ ɫɨɫɬɨɹɧɢɟ ɬɟɥɚ ɯɚɪɚɤ-

ɬɟɪɢɡɭɟɬɫɹ ɟɝɨ ɩɨɥɨɠɟɧɢɟɦ

ɢ ɫɤɨɪɨɫɬɶɸ v

. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɷɬɨɦ

ɫɥɭɱɚɟ ɷɧɟɪɝɢɹ (ɩɨɥɧɚɹ ɷɧɟɪɝɢɹ), ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɫɨɫɬɨɹɧɢɟ ɬɟɥɚ, ɹɜ-

ɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɤɨɨɪɞɢɧɚɬ ɢ ɫɤɨɪɨɫɬɢ. Ʉɪɨɦɟ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ, ɯɚɪɚɤ-

ɬɟɪɢɡɭɸɳɟɣ ɫɨɫɬɨɹɧɢɟ ɬɟɥɚ ɜ ɰɟɥɨɦ, ɜɵɞɟɥɹɸɬ ɨɬɞɟɥɶɧɵɟ ɜɢɞɵ ɷɧɟɪ-

ɝɢɢ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟ ɫɨɫɬɨɹɧɢɟ ɬɟɥɚ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɨɬɞɟɥɶɧɵɦ ɜɢ-

ɞɚɦ ɞɜɢɠɟɧɢɹ ɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫ ɞɪɭɝɢɦɢ ɬɟɥɚɦɢ. ȼ ɱɚɫɬɧɨɫɬɢ ɩɪɢɧɹɬɨ

ɜɵɞɟɥɹɬɶ ɤɢɧɟɬɢɱɟɫɤɭɸ

ɷɧɟɪɝɢɸ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ, ɩɨɬɟɧɰɢ-

ɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɩɨɥɹ ɫɢɥɵ ɬɹɠɟɫɬɢ, ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɭɩɪɭɝɨɣ

ɞɟɮɨɪɦɚɰɢɢ ɢ ɬ.ɞ.

ȼ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɷɧɟɪɝɢɢ ɡɚɜɢɫɢɬ ɨɬ ɜɵ-

ɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ. Ɉɞɧɚɤɨ ɜ ɮɢɡɢɤɟ ɨɫɧɨɜɧɨɟ ɡɧɚɱɟɧɢɟ ɢɦɟɟɬ ɧɟ

ɚɛɫɨɥɸɬɧɨɟ ɡɧɚɱɟɧɢɟ ɷɧɟɪɝɢɢ, ɚ ɟɟ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɢɡɦɟɧɟɧɢɟ, ɤɨɬɨɪɨɟ

ɪɚɜɧɨ ɪɚɛɨɬɟ ɜɫɟɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɨ.

1.1. Ɋɚɛɨɬɚ ɫɢɥɵ

Ɋɚɛɨɬɚ – ɷɬɨ ɦɟɪɚ ɢɡɦɟɧɟɧɢɹ ɷɧɟɪɝɢɢ ɬɟɥɚ ɢɥɢ ɫɢɫɬɟɦɵ ɬɟɥ.

ɆɨɞɆ-05 Ɋɚɛɨɬɚ ɢ ɷɧɟɪɝɢɹ

ȼ ɦɟɯɚɧɢɤɟ ɪɚɛɨɬɚ

ɫɢɥɵ

–

ɷɬɨ ɫɤɚɥɹɪɧɚɹ ɮɢɡɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ,

ɪɚɜɧɚɹ ɫɤɚɥɹɪɧɨɦɭ ɩɪɨɢɡɜɟɞɟɧɢɸ ɷɬɨɣ

ɫɢɥɵ ɧɚ ɷɥɟɦɟɧɬɚɪɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ

ɬɟɥɚ ɜɞɨɥɶ ɭɱɚɫɬɤɚ ɩɭɬɢ

)( rdFdA

 .

Ɋɚɛɨɬɚ

A ɫɢɥɵ

ɧɚ ɭɱɚɫɬɤɟ

ɬɪɚɟɤɬɨɪɢɢ ɦɟɠɞɭ ɬɨɱɤɚɦɢ 1 ɢ 2 (ɪɢɫ. 1)

ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɩɭɬɟɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɷɥɟɦɟɧɬɚɪɧɨɣ ɪɚɛɨɬɵ

dA ɩɨ ɜɫɟ-

ɦɭ ɭɱɚɫɬɤɭ ɬɪɚɟɤɬɨɪɢɢ



)( rdFA

Ɉɱɟɜɢɞɧɨ, ɫɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɧɚɩɪɚɜɥɟɧɢɸ

ɞɜɢɠɟɧɢɹ ɬɟɥɚ, ɧɟ ɫɨɜɟɪɲɚɟɬ ɪɚɛɨɬɭ (

0)(  rdFdA

, ɬ.ɤ.

A rd

) ɢ ɧɟ

ɢɡɦɟɧɹɟɬ ɦɟɯɚɧɢɱɟɫɤɨɟ ɫɨɫɬɨɹɧɢɟ (ɷɧɟɪɝɢɸ) ɬɟɥɚ.

Ɋɚɛɨɬɚ, ɤɚɤ ɢ ɷɧɟɪɝɢɹ, ɢɡɦɟɪɹɟɬɫɹ ɜ ɞɠɨɭɥɹɯ:

1 Ⱦɠ = 1 ɇ·ɦ =

ɦɤɝ 

ɋɢɥɵ ɪɚɡɧɨɣ ɩɪɢɪɨɞɵ ɩɪɢ ɞɜɢɠɟɧɢɢ ɩɨ ɨɞɧɨɦɭ ɢ ɬɨɦɭ ɠɟ ɭɱɚɫɬ-

ɤɭ ɬɪɚɟɤɬɨɪɢɢ ɦɨɝɭɬ ɫɨɜɟɪɲɚɬɶ ɪɚɡɧɭɸ ɪɚɛɨɬɭ.

1.1.1. Ɋɚɛɨɬɚ ɫɢɥɵ ɬɹɠɟɫɬɢ

Ɋɚɫɫɱɢɬɚɟɦ ɪɚɛɨɬɭ ɫɢɥɵ

ɬɹɠɟɫɬɢ

ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ

ɬɟɥɚ ɜɞɨɥɶ ɭɱɚɫɬɤɚ ɬɪɚɟɤɬɨɪɢɢ

ɦɟɠɞɭ ɬɨɱɤɚɦɢ 1 ɢ 2.

³³

 

)()( rdgmrdFA

GGG

ȼ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨ-

ɨɪɞɢɧɚɬ ɜɟɤɬɨɪ ɷɥɟɦɟɧɬɚɪɧɨɝɨ

ɩɟɪɟɦɟɳɟɧɢɹ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ

ɜɢɞɟ



dr dx i dy j dz k 



ɋɢɥɚ ɬɹɠɟɫɬɢ ɧɚɩɪɚɜɥɟɧɚ ɜɟɪɬɢ-

ɤɚɥɶɧɨ ɜɧɢɡ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ

ɨɫɢ

z (ɪɢɫ. 2):

mg iA



mg jA

Ɋɢɫ. 1

Ɋɢɫ. 2

Ɇɟɯɚɧɢɤɚ

mg knp

. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,

)( mgzmgzdzmgdzmgrdgmA

   

³³³

ɢɥɢ

2112

mghmghA  ,

ɟɫɥɢ

ɢ h

– ɜɵɫɨɬɚ, ɧɚ ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɥɨɫɶ ɬɟɥɨ ɜ ɬɨɱɤɚɯ 1 ɢ 2 ɫɨɨɬ-

ɜɟɬɫɬɜɟɧɧɨ.

Ɋɚɛɨɬɚ ɫɢɥɵ ɬɹɠɟɫɬɢ ɡɚɜɢɫɢɬ ɨɬ ɪɚɡɧɨɫɬɢ ɜɵɫɨɬ (

– h

) ɧɚ ɤɨɬɨ-

ɪɵɯ ɪɚɫɩɨɥɨɠɟɧɵ ɧɚɱɚɥɶɧɚɹ ɢ ɤɨɧɟɱɧɚɹ ɬɨɱɤɢ ɭɱɚɫɬɤɚ ɬɪɚɟɤɬɨɪɢɢ, ɢ ɧɟ

ɡɚɜɢɫɢɬ ɧɢ ɨɬ ɮɨɪɦɵ, ɧɢ ɨɬ ɞɥɢɧɵ ɩɭɬɢ, ɩɨ ɤɨɬɨɪɨɦɭ ɞɜɢɝɚɥɨɫɶ ɬɟɥɨ.

1.1.2. Ɋɚɛɨɬɚ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ

ɉɭɫɬɶ

xxx  ' – ɜɟɥɢɱɢɧɚ ɞɟɮɨɪɦɚɰɢɢ ɩɪɭɠɢɧɵ ɠɟɫɬɤɨɫɬɶɸ

k. Ɋɚɫɫɱɢɬɚɟɦ ɪɚɛɨɬɭ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ, ɩɨɞɱɢɧɹɸɳɟɣɫɹ ɡɚɤɨɧɭ Ƚɭɤɚ

'

, ɧɚɩɪɚɜɥɟɧɧɨɣ ɜɞɨɥɶ ɨɫɢ x, ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ ɬɟɥɚ ɜɞɨɥɶ ɭɱɚ-

ɫɬɤɚ ɬɪɚɟɤɬɨɪɢɢ ɦɟɠɞɭ ɬɨɱɤɚɦɢ 1 ɢ 2.

³³

' 

)()( rdxkrdFA

GGG

ȼ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨ-

ɨɪɞɢɧɚɬ ɜɟɤɬɨɪ ɷɥɟɦɟɧɬɚɪɧɨɝɨ

ɩɟɪɟɦɟɳɟɧɢɹ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ

ɜɢɞɟ

dr dx i dy j dz k 



. ɋɢ-

ɥɚ ɭɩɪɭɝɨɫɬɢ

ɧɚɩɪɚɜɥɟɧɚ ɝɨ-

ɪɢɡɨɧɬɚɥɶɧɨ, ɜɞɨɥɶ ɨɫɢ x

(ɪɢɫ. 3):

inn



ȼɟɤɬɨɪ ɞɟɮɨɪɦɚɰɢɢ

ɧɚɩɪɚɜɥɟɧ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɨɫɢ

x (

i'np



) ɢɡ ɩɨɥɨɠɟɧɢɹ x

, ɤɨ-

ɝɞɚ ɩɪɭɠɢɧɚ ɧɟ ɞɟɮɨɪɦɢɪɨɜɚɧɚ,

ɜ ɫɬɨɪɨɧɭ ɞɟɮɨɪɦɚɰɢɢ. ɇɚɩɪɢ-

ɦɟɪ, ɟɫɥɢ ɩɪɭɠɢɧɚ ɫɠɚɬɚ (x < x

ɬɨ

()

xx i xxi xi'      ' 



. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,

³³

' '

)( dxxkrdxkA

Ɋɢɫ. 3

ɆɨɞɆ-05 Ɋɚɛɨɬɚ ɢ ɷɧɟɪɝɢɹ

ɍɱɢɬɵɜɚɹ, ɱɬɨ dx = d(x – x

) = – d(x

– x) = – d('x), ɪɚɛɨɬɚ ɫɢɥɵ ɭɩ-

ɪɭɝɨɫɬɢ ɪɚɜɧɚ

³³

'' '

)(

xxdkxdxk



Ɋɚɛɨɬɚ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ ɡɚɜɢɫɢɬ ɨɬ ɜɟɥɢɱɢɧɵ ɞɟɮɨɪɦɚɰɢɢ ɜ ɧɚ-

ɱɚɥɶɧɨɦ 'x

ɢ ɤɨɧɟɱɧɨɦ 'x

ɫɨɫɬɨɹɧɢɢ, ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɬɨɝɨ ɞɜɢɝɚɥɨɫɶ

ɥɢ ɬɟɥɨ ɜ ɞɪɭɝɢɯ ɧɚɩɪɚɜɥɟɧɢɹɯ ɢɥɢ ɧɟɬ.

1.1.3. Ɋɚɛɨɬɚ ɫɢɥɵ ɬɪɟɧɢɹ

Ɋɚɫɫɱɢɬɚɟɦ ɪɚɛɨɬɭ ɫɢɥɵ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ NF P ɩɪɢ ɩɟɪɟ-

ɦɟɳɟɧɢɢ ɬɟɥɚ ɜɞɨɥɶ ɧɟɤɨɬɨɪɨɣ ɬɪɚɟɤɬɨɪɢɢ ɦɟɠɞɭ ɬɨɱɤɚɦɢ 1 ɢ 2 (|N| –

ɦɨɞɭɥɶ ɫɢɥɵ ɧɨɪɦɚɥɶɧɨɝɨ ɞɚɜɥɟɧɢɹ ɬɟɥɚ ɧɚ ɨɩɨɪɭ, P – ɤɨɷɮɮɢɰɢɟɧɬ

ɬɪɟɧɢɹ).



)( rdFA

ɋɢɥɚ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɜɫɟɝɞɚ ɧɚɩɪɚɜɥɟɧɚ ɜ ɫɬɨɪɨɧɭ, ɩɪɨɬɢɜɨɩɨ-

ɥɨɠɧɭɸ ɞɜɢɠɟɧɢɸ (ɩɟɪɟɦɟɳɟɧɢɸ):

np . ɋɥɟɞɨɜɚɬɟɥɶɧɨ,

³³³

P  

)( drNdrFrdFA

Ɇɨɞɭɥɶ ɫɢɥɵ ɧɨɪɦɚɥɶɧɨɝɨ ɞɚɜɥɟɧɢɹ

ɩɨ ɬɪɟɬɶɟɦɭ ɡɚɤɨɧɭ ɇɶɸ-

ɬɨɧɚ ɪɚɜɟɧ ɦɨɞɭɥɸ ɫɢɥɵ ɪɟɚɤɰɢɢ ɨɩɨɪɵ

ɨɩ

 . ɋɢɥɚ ɪɟɚɤɰɢɢ

ɨɩɨɪɵ ɡɚɜɢɫɢɬ ɨɬ ɬɨɝɨ, ɩɨ ɤɚɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɞɜɢɠɟɬɫɹ ɬɟɥɨ.

ȿɫɥɢ ɬɟɥɨ ɞɜɢɠɟɬɫɹ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɫɢɥɚ ɪɟɚɤɰɢɢ

ɨɩɨɪɵ ɧɚɩɪɚɜɥɟɧɚ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ (ɪɢɫ. 4ɚ), ɪɚɜɧɚ mg ɢ ɩɨɫɬɨɹɧɧɚ

ɜɞɨɥɶ ɜɫɟɣ ɬɪɚɟɤɬɨɪɢɢ (ɢɡ ɜɬɨɪɨɝɨ ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ,

ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɭɸ ɩɥɨɫɤɨɫɬɢ: R

ɨɩ

– mg = 0). Ɍɨɝɞɚ ɦɨɞɭɥɶ ɫɢɥɵ ɧɨɪ-

ɦɚɥɶɧɨɝɨ ɞɚɜɥɟɧɢɹ |N| = mg, ɚ ɪɚɛɨɬɚ ɫɢɥɵ ɬɪɟɧɢɹ ɪɚɜɧɚ

P P

1212

mgSdrmgA

ɝɞɟ S

– ɞɥɢɧɚ ɭɱɚɫɬɤɚ ɬɪɚɟɤɬɨɪɢɢ ɦɟɠɞɭ ɬɨɱɤɚɦɢ 1 ɢ 2.

ȿɫɥɢ ɬɟɥɨ ɞɜɢɠɟɬɫɹ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɨɞ ɭɝɥɨɦ D ɤ ɝɨɪɢ-

ɡɨɧɬɭ, ɫɢɥɚ ɪɟɚɤɰɢɢ ɨɩɨɪɵ ɧɚɩɪɚɜɥɟɧɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɧɚɤɥɨɧɧɨɣ

ɩɥɨɫɤɨɫɬɢ (ɪɢɫ. 4ɛ), ɪɚɜɧɚ mg cos D ɢ ɬɨɠɟ ɩɨɫɬɨɹɧɧɚ ɜɞɨɥɶ ɜɫɟɣ ɬɪɚɟɤ-

ɬɨɪɢɢ (ɢɡ ɜɬɨɪɨɝɨ ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ, ɩɟɪɩɟɧɞɢɤɭɥɹɪ-

Ɇɟɯɚɧɢɤɚ

ɧɭɸ ɩɥɨɫɤɨɫɬɢ: R

ɨɩ

– mg cos D = 0). Ɍɨɝɞɚ ɦɨɞɭɥɶ ɫɢɥɵ ɧɨɪɦɚɥɶɧɨɝɨ

ɞɚɜɥɟɧɢɹ |N| = mg cos D, ɚ ɪɚɛɨɬɚ ɫɢɥɵ ɬɪɟɧɢɹ ɪɚɜɧɚ

DP DP

1212

coscos SmgdrmgA ,

ɝɞɟ S

– ɞɥɢɧɚ ɭɱɚɫɬɤɚ ɬɪɚɟɤɬɨɪɢɢ ɦɟɠɞɭ ɬɨɱɤɚɦɢ 1 ɢ 2.

ɂɡ ɩɪɢɜɟɞɟɧɧɵɯ ɩɪɢɦɟɪɨɜ ɜɢɞɧɨ, ɱɬɨ ɩɪɢ ɞɜɢɠɟɧɢɢ ɬɟɥɚ ɩɨ ɩɥɨɫ-

ɤɨɫɬɢ (ɧɚɤɥɨɧɧɨɣ ɢɥɢ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ) ɪɚɛɨɬɚ ɫɢɥɵ ɬɪɟɧɢɹ A

ɩɪɹɦɨ

ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɞɥɢɧɟ ɩɭɬɢ S

, ɩɪɨɣɞɟɧɧɨɝɨ ɬɟɥɨɦ:

const

P mg

(ɩɪɢ ɞɜɢɠɟɧɢɢ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ);

constcos

DP mg

(ɩɪɢ ɞɜɢɠɟɧɢɢ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ).

ɭɦɟɧɶɲɟɧɧɵɣ

ɜɢɞ ɫɜɟɪɯɭ

ɨɩ

= –N

ɬɪ

ɭɦɟɧɶɲɟɧɧɵɣ

ɜɢɞ ɫɜɟɪɯɭ

ɨɩ

= –N

ɬɪ

Ɋɢɫ. 4

ȿɫɥɢ ɜɨ ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ ɦɟɠɞɭ ɬɨɱɤɚɦɢ 1 ɢ 2 ɩɨɜɟɪɯɧɨɫɬɶ, ɩɨ ɤɨ-

ɬɨɪɨɣ ɞɜɢɠɟɬɫɹ ɬɟɥɨ, ɦɟɧɹɟɬ ɫɜɨɣ ɧɚɤɥɨɧ, ɬɨ ɫɢɥɚ ɧɨɪɦɚɥɶɧɨɝɨ ɞɚɜɥɟ-

ɧɢɹ (ɫɢɥɚ ɪɟɚɤɰɢɢ ɨɩɨɪɵ) ɛɭɞɟɬ ɪɚɡɥɢɱɧɚ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɬɪɚɟɤɬɨɪɢɢ.

Ɍɨɝɞɚ ɪɚɛɨɬɭ

P

)( drrNA

ɧɭɠɧɨ ɜɵɱɢɫɥɹɬɶ ɫ ɭɱɟɬɨɦ ɢɡɦɟɧɟɧɢɹ

ɧɚɤɥɨɧɚ ɬɪɚɟɤɬɨɪɢɢ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɧɚ ɢɧɬɟɪɟɫɭɸɳɟɦ ɧɚɫ ɭɱɚɫɬɤɟ ɩɭɬɢ.

ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɪɚɛɨɬɚ ɫɢɥɵ ɬɪɟɧɢɹ ɡɚɜɢɫɢɬ ɧɟ ɬɨɥɶɤɨ ɨɬ ɧɚɱɚɥɶ-

ɧɨɝɨ ɢ ɤɨɧɟɱɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɬɟɥɚ, ɧɨ ɢ ɨɬ ɞɥɢɧɵ ɩɭɬɢ, ɩɨ ɤɨɬɨɪɨɦɭ ɨɧɨ

ɞɜɢɝɚɥɨɫɶ. Ɂɚɜɢɫɢɦɨɫɬɶ ɪɚɛɨɬɵ ɫɢɥɵ ɬɪɟɧɢɹ ɨɬ ɞɥɢɧɵ ɩɭɬɢ ɹɜɥɹɟɬɫɹ

ɆɨɞɆ-05 Ɋɚɛɨɬɚ ɢ ɷɧɟɪɝɢɹ

ɥɢɧɟɣɧɨɣ (ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ) ɬɨɥɶɤɨ ɩɪɢ ɞɜɢɠɟɧɢɢ ɬɟɥɚ ɩɨ

ɩɥɨɫɤɨɫɬɢ. ɉɪɢ ɞɜɢɠɟɧɢɢ ɩɨ ɨɩɨɪɟ ɞɪɭɝɨɣ ɮɨɪɦɵ ɡɚɜɢɫɢɦɨɫɬɶ ɪɚɛɨɬɵ

ɫɢɥɵ ɬɪɟɧɢɹ ɨɬ ɞɥɢɧɵ ɩɭɬɢ ɦɨɠɟɬ ɧɨɫɢɬɶ ɧɟɥɢɧɟɣɧɵɣ ɯɚɪɚɤɬɟɪ.

1.2. Ʉɨɧɫɟɪɜɚɬɢɜɧɵɟ ɢ ɧɟɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɥɵ

ɂɡ ɩɪɢɜɟɞɟɧɧɵɯ ɜɵɲɟ ɩɪɢɦɟɪɨɜ ɜɢɞɧɨ, ɱɬɨ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɪɚ-

ɛɨɬɟ ɫɢɥɵ ɦɨɠɧɨ ɪɚɡɞɟɥɢɬɶ ɧɚ ɞɜɟ ɝɪɭɩɩɵ. Ⱦɥɹ ɨɞɧɢɯ ɪɚɛɨɬɚ ɧɚ ɧɟɤɨɬɨ-

ɪɨɦ ɭɱɚɫɬɤɟ ɩɭɬɢ ɡɚɜɢɫɢɬ ɨɬ ɮɨɪɦɵ ɢ ɞɥɢɧɵ ɩɭɬɢ, ɞɥɹ ɞɪɭɝɢɯ – ɧɟ ɡɚɜɢ-

ɫɢɬ. ɋɢɥɵ, ɪɚɛɨɬɚ ɤɨɬɨɪɵɯ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɮɨɪɦɵ ɢ ɞɥɢɧɵ ɩɭɬɢ, ɧɚɡɵɜɚ-

ɸɬɫɹ

ɤɨɧɫɟɪɜɚɬɢɜɧɵɦɢ. ɋɢɥɵ, ɪɚɛɨɬɚ ɤɨɬɨɪɵɯ ɡɚɜɢɫɢɬ ɨɬ ɮɨɪɦɵ ɢ

ɞɥɢɧɵ ɩɭɬɢ, ɧɚɡɵɜɚɸɬɫɹ

ɧɟɤɨɧɫɟɪɜɚɬɢɜɧɵɦɢ.

ȼ ɱɚɫɬɧɨɫɬɢ, ɫɢɥɚ ɬɹɠɟɫɬɢ ɢ ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ (ɨɩɢɫɵɜɚɟɦɚɹ ɡɚɤɨ-

ɧɨɦ Ƚɭɤɚ) ɹɜɥɹɸɬɫɹ ɤɨɧɫɟɪɜɚɬɢɜɧɵɦɢ ɫɢɥɚɦɢ, ɚ ɫɢɥɚ ɬɪɟɧɢɹ – ɧɟɤɨɧ-

ɫɟɪɜɚɬɢɜɧɨɣ.

ɉɨɥɟ ɤɨɧɫɟɪɜɚɬɢɜɧɨɣ ɫɢɥɵ ɬɚɤɠɟ ɧɚɡɵɜɚɟɬɫɹ ɤɨɧɫɟɪɜɚɬɢɜɧɵɦ.

1.3. ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ

Ɋɚɛɨɬɚ ɤɨɧɫɟɪɜɚɬɢɜɧɵɯ ɫɢɥ ɧɚ ɭɱɚɫɬɤɟ ɬɪɚɟɤɬɨɪɢɢ ɦɟɠɞɭ ɬɨɱɤɚɦɢ

1 ɢ 2 ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɚɡɧɨɫɬɶ ɞɜɭɯ ɨɞɧɨɬɢɩɧɵɯ ɜɟɥɢɱɢɧ, ɤɚɠɞɚɹ ɢɡ

ɤɨɬɨɪɵɯ ɡɚɜɢɫɢɬ ɨɬ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɜ ɬɨɱɤɚɯ 1 ɢ 2.

ɇɚɩɪɢɦɟɪ, ɞɥɹ ɫɢɥɵ ɬɹɠɟɫɬɢ – ɷɬɨ ɜɟɥɢɱɢɧɚ mgh, ɡɚɜɢɫɹɳɚɹ ɨɬ ɜɵɫɨɬɵ

ɧɚɞ ɩɨɜɟɪɯɧɨɫɬɶɸ Ɂɟɦɥɢ, ɞɥɹ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ – ɷɬɨ ɜɟɥɢɱɢɧɚ

, ɡɚ-

ɜɢɫɹɳɚɹ ɨɬ ɫɠɚɬɢɹ ɩɪɭɠɢɧɵ.

Ɋɚɛɨɬɚ – ɷɬɨ ɦɟɪɚ ɢɡɦɟɧɟɧɢɹ ɷɧɟɪɝɢɢ ɬɟɥɚ ɢɥɢ ɫɢɫɬɟɦɵ ɬɟɥ. ɋɥɟ-

ɞɨɜɚɬɟɥɶɧɨ, ɜɟɥɢɱɢɧɚ, ɤɨɬɨɪɚɹ ɢɡɦɟɧɹɟɬɫɹ ɩɪɢ ɫɨɜɟɪɲɟɧɢɢ ɪɚɛɨɬɵ, ɢ

ɟɫɬɶ ɷɧɟɪɝɢɹ. Ɋɚɫɫɦɚɬɪɢɜɚɹ ɨɬɞɟɥɶɧɵɟ ɫɢɥɵ, ɩɨɥɭɱɚɟɦ ɢɡɦɟɧɟɧɢɟ ɨɬ-

ɞɟɥɶɧɵɯ ɜɢɞɨɜ ɷɧɟɪɝɢɢ.

Ʉɨɝɞɚ ɪɚɛɨɬɭ ɫɨɜɟɪɲɚɟɬ ɫɢɥɚ ɬɹɠɟɫɬɢ, ɢɡɦɟɧɹɟɬɫɹ ɜɟɥɢɱɢɧɚ

U = mgh – ɷɧɟɪɝɢɹ ɩɨɥɹ ɫɢɥɵ ɬɹɠɟɫɬɢ.

ɬɹɠ

UUUmghmghA '  

212112

Ʉɨɝɞɚ ɪɚɛɨɬɭ ɫɨɜɟɪɲɚɟɬ ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ, ɢɡɦɟɧɹɟɬɫɹ ɜɟɥɢɱɢɧɚ

– ɷɧɟɪɝɢɹ ɫɠɚɬɨɣ ɩɪɭɠɢɧɵ.

ɭɩɪ

UUU

kA ' 



Ɇɟɯɚɧɢɤɚ

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɟɫɥɢ ɬɟɥɨ ɧɚɯɨɞɢɬɫɹ ɜ ɩɨɥɟ ɤɨɧɫɟɪɜɚɬɢɜɧɨɣ ɫɢɥɵ,

ɬɨ ɝɨɜɨɪɹɬ, ɱɬɨ ɨɧɨ ɨɛɥɚɞɚɟɬ ɨɩɪɟɞɟɥɟɧɧɨɣ ɷɧɟɪɝɢɟɣ: ɜ ɬɨɱɤɟ 1 – ɷɧɟɪ-

ɝɢɟɣ U

, ɜ ɬɨɱɤɟ 2 – ɷɧɟɪɝɢɟɣ U

ȼ ɦɟɯɚɧɢɤɟ ɧɟɧɭɥɟɜɨɟ ɡɧɚɱɟɧɢɟ ɷɧɟɪɝɢɢ U ɬɟɥɚ ɜ ɩɨɥɟ ɤɨɧɫɟɪɜɚ-

ɬɢɜɧɵɯ ɫɢɥ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɜɨɡɦɨɠɧɨɫɬɶ ɞɜɢɠɟɧɢɹ. ɇɚ-

ɩɪɢɦɟɪ, ɟɫɥɢ ɬɟɥɨ ɩɨɞɧɹɬɨ ɧɚ ɧɟɤɨɬɨɪɭɸ ɜɵɫɨɬɭ h > 0, ɬɨ ɩɪɟɞɨɫɬɚɜɥɟɧ-

ɧɨɟ ɫɚɦɨ ɫɟɛɟ ɬɟɥɨ ɧɚɱɧɟɬ ɞɜɢɝɚɬɶɫɹ ɜɧɢɡ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɬɹɠɟɫɬɢ.

ȿɫɥɢ ɫɠɚɬɢɟ (ɪɚɫɬɹɠɟɧɢɟ) ɩɪɭɠɢɧɵ ɨɬɥɢɱɧɨ ɨɬ ɧɭɥɹ 'x z 0, ɬɨ ɩɪɟɞɨɫ-

ɬɚɜɥɟɧɧɵɣ

ɫɚɦ ɫɟɛɟ ɧɟɡɚɤɪɟɩɥɟɧɧɵɣ ɤɨɧɟɰ ɩɪɭɠɢɧɵ ɬɨɠɟ ɧɚɱɧɟɬ ɞɜɢ-

ɝɚɬɶɫɹ. ɉɨɷɬɨɦɭ ɷɧɟɪɝɢɸ ɬɟɥɚ ɜ ɩɨɥɟ ɤɨɧɫɟɪɜɚɬɢɜɧɵɯ ɫɢɥ ɧɚɡɵɜɚɸɬ

ɩɨɬɟɧɰɢɚɥɶɧɨɣ.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ ɯɚ-

ɪɚɤɬɟɪɢɡɭɟɬ ɫɨɫɬɨɹɧɢɟ ɬɟɥɚ ɢɥɢ ɫɢɫɬɟɦɵ ɬɟɥ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɜɨɡɦɨɠ-

ɧɨɫɬɢ ɞɜɢɠɟɧɢɹ. ȼ ɤɚɱɟɫɬɜɟ ɧɭɥɟɜɨɝɨ ɨɬɫɱɟɬɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ,

ɤɚɤ ɩɪɚɜɢɥɨ, ɜɵɛɢɪɚɟɬɫɹ ɬɚɤɨɟ ɫɨɫɬɨɹɧɢɟ ɬɟɥɚ ɢɥɢ ɫɢɫɬɟɦɵ, ɢɡ ɤɨɬɨɪɨɝɨ

ɫɚɦɨɩɪɨɢɡɜɨɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɧɟɜɨɡɦɨɠɧɨ. ɇɚɩɪɢɦɟɪ, ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟ-

ɫɬɢ – ɷɬɨ ɩɨɥɨɠɟɧɢɟ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ

Ɂɟɦɥɢ. ɇɨ ɟɫɥɢ ɪɟɱɶ ɢɞɟɬ ɬɨɥɶɤɨ ɨɛ

ɢɡɦɟɧɟɧɢɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ, ɧɭɥɟɜɨɣ ɨɬɫɱɟɬ ɦɨɠɟɬ ɛɵɬɶ ɜɵɛɪɚɧ

ɩɪɨɢɡɜɨɥɶɧɨ, ɢɫɯɨɞɹ ɢɡ ɭɞɨɛɫɬɜɚ ɪɟɲɟɧɢɹ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ.

1.4. Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ

ȿɫɥɢ ɧɚ ɬɟɥɨ ɦɚɫɫɵ m ɞɟɣɫɬɜɭɸɬ ɫɢɥɵ

, ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ

ɤɨɬɨɪɵɯ ɨɬɥɢɱɧɚ ɨɬ ɧɭɥɹ, ɬɨ ɫɨɝɥɚɫɧɨ ɜɬɨɪɨɦɭ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ ɫɤɨ-

ɪɨɫɬɶ ɬɟɥɚ ɢɡɦɟɧɹɟɬɫɹ

Ɋɚɫɫɱɢɬɚɟɦ ɪɚɛɨɬɭ

ɜɫɟɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɨ, ɩɪɢ ɞɜɢɠɟ-

ɧɢɢ ɩɨ ɭɱɚɫɬɤɭ ɬɪɚɟɤɬɨɪɢɢ ɦɟɠɞɭ ɬɨɱɤɚɦɢ 1 ɢ 2. ɋ ɨɞɧɨɣ ɫɬɨɪɨɧɵ, ɪɚ-

ɛɨɬɚ – ɫɤɚɥɹɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɩɨɷɬɨɦɭ ɪɚɛɨɬɚ ɫɭɦɦɚɪɧɨɣ ɫɢɥɵ ɪɚɜɧɚ ɫɭɦ-

ɦɟ ɪɚɛɨɬ ɜɫɟɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɨ

¦¦

³³

 

6 12

)()( ArdFrdFA

ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɢɡ ɜɬɨɪɨɝɨ ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ



³³³³



 

)( vvv

GGG

dmdt

mrd

mrdFA .

ɆɨɞɆ-05 Ɋɚɛɨɬɚ ɢ ɷɧɟɪɝɢɹ

Ɋɚɫɫɦɨɬɪɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥ

   

vvvvvvvv

dddd    2 .

Ɍɨɝɞɚ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ





vvvvv ddd  

. ɋɥɟɞɨɜɚ-

ɬɟɥɶɧɨ, ɫɭɦɦɚɪɧɭɸ ɪɚɛɨɬɭ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ



222

vvv

dmA  

³³

vv mm

A 

Ɋɚɛɨɬɚ – ɦɟɪɚ ɢɡɦɟɧɟɧɢɹ ɷɧɟɪɝɢɢ. ɋɭɦɦɚɪɧɚɹ ɪɚɛɨɬɚ ɜɫɟɯ ɫɢɥ,

ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɨ, ɬɚɤɠɟ ɛɵɥɚ ɜɵɪɚɠɟɧɚ ɱɟɪɟɡ ɢɡɦɟɧɟɧɢɟ ɧɟɤɨɬɨɪɨɣ

ɜɟɥɢɱɢɧɵ

, ɤɨɬɨɪɚɹ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɤɚɠɞɨɟ ɢɡ ɫɨɫɬɨɹɧɢɣ 1 ɢ 2 ɱɟɪɟɡ

ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɬɟɥɚ ɜ ɷɬɢɯ ɬɨɱɤɚɯ ɬɪɚɟɤɬɨɪɢɢ. ɉɨɫɤɨɥɶɤɭ ɪɚɛɨɬɚ ɩɪɢ-

ɜɨɞɢɬ ɤ ɢɡɦɟɧɟɧɢɸ ɜɟɥɢɱɢɧɵ

, ɬɨ ɜɟɥɢɱɢɧɚ

– ɷɧɟɪɝɢɹ.

ɗɬɚ ɷɧɟɪɝɢɹ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɤɨɪɨɫɬɶ ɩɟɪɟɦɟɳɟɧɢɹ ɬɟɥɚ (ɤɢɧɟɬɢɤɭ) ɜ

ɞɚɧɧɨɣ ɬɨɱɤɟ, ɩɨɷɬɨɦɭ ɟɟ ɧɚɡɵɜɚɸɬ

ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɟɣ.

A ' 

1.5. Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ

Ʉɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɪɚɛɨɬɚ ɜɫɟɯ ɫɢɥ

, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ

ɬɟɥɨ, ɩɪɢ ɞɜɢɠɟɧɢɢ ɩɨ ɭɱɚɫɬɤɭ ɬɪɚɟɤɬɨɪɢɢ ɦɟɠɞɭ ɬɨɱɤɚɦɢ 1 ɢ 2 ɫ ɨɞɧɨɣ

ɫɬɨɪɨɧɵ ɪɚɜɧɚ ɫɭɦɦɟ ɪɚɛɨɬ ɷɬɢɯ ɫɢɥ

6 12

ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɷɬɚ ɪɚɛɨɬɚ ɜɟɞɟɬ ɤ ɢɡɦɟɧɟɧɢɸ ɤɢɧɟɬɢɱɟɫɤɨɣ

ɷɧɟɪɝɢɢ ɬɟɥɚ

EA '



ɗɬɨ ɨɞɢɧ ɢɡ ɜɚɪɢɚɧɬɨɜ ɡɚɩɢɫɢ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ: ɱɬɨɛɵ

ɢɡɦɟɧɢɬɶ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɬɟɥɚ (ɢɥɢ ɫɢɫɬɟɦɵ ɬɟɥ) ɧɟɨɛɯɨɞɢɦɨ

ɫɨɜɟɪɲɢɬɶ ɧɟɧɭɥɟɜɭɸ ɪɚɛɨɬɭ.

Ɇɟɯɚɧɢɤɚ

ȿɫɥɢ ɫɪɟɞɢ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɨ, ɟɫɬɶ ɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɢ ɧɟ-

ɤɨɧɫɟɪɜɚɬɢɜɧɵɟ,

¦¦¦



ɇȿɄɈɇɄɈɇ

AAA

121212

. Ɋɚɛɨɬɚ ɤɨɧɫɟɪɜɚɬɢɜɧɵɯ

ɫɢɥ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɢɡɦɟɧɟɧɢɟ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɬɟɥɚ

' UA

ɄɈɇ

Ɍɨɝɞɚ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ:

E' =

'

ɇȿɄɈɇ

ɂɡɦɟɧɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɬɟɥɚ (ɢɥɢ ɫɢɫɬɟɦɵ ɬɟɥ) ɩɪɨɢɫ-

ɯɨɞɢɬ ɡɚ ɫɱɟɬ ɢɡɦɟɧɟɧɢɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɬɟɥɚ ɢ ɡɚ ɫɱɟɬ ɪɚɛɨɬɵ

ɧɟɤɨɧɫɟɪɜɚɬɢɜɧɵɯ ɫɢɥ.

Ɂɚɩɢɲɟɦ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɱɟɪɟɡ ɡɧɚɱɟɧɢɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ

ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɜ ɬɨɱɤɚɯ 1 ɢ 2:

KKK

EEE  ' ,

UUU  ' . Ɍɨɝɞɚ

 

ɇȿɄɈɇ

AUUEE

1221

ɢɥɢ

 

ɇȿɄɈɇ

AUEUE

1212

ɉɨɥɧɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɟɣ E ɬɟɥɚ (ɫɢɫɬɟɦɵ ɬɟɥ) ɜ ɞɚɧɧɨɣ

ɬɨɱɤɟ ɧɚɡɵɜɚɟɬɫɹ ɫɭɦɦɚ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɷɬɨɝɨ

ɬɟɥɚ ɫ ɭɱɟɬɨɦ ɜɫɟɯ ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɨ.

UEE



ȿɫɥɢ ɧɚ ɬɟɥɨ ɧɟ ɞɟɣɫɬɜɭɸɬ ɧɟɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɥɵ

(

ɇȿɄɈɇ

A ), ɬɨ ɩɨɥɧɚɹ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɨɯɪɚɧɹɟɬɫɹ:

UEUE

  

ȿɫɥɢ ɧɚ ɬɟɥɨ ɞɟɣɫɬɜɭɸɬ ɧɟɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɥɵ, ɬɨ ɩɨɥɧɚɹ ɦɟɯɚ-

ɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ (ɫɢɫɬɟɦɵ ɬɟɥ) ɭɦɟɧɶɲɚɟɬɫɹ:

 

ɇȿɄɈɇ

AUEUE

1212





ɇȿɄɈɇ

AEE

1212

ɢɥɢ



ɇȿɄɈɇ

AEE

1212

Ɋɚɛɨɬɚ ɧɟɤɨɧɫɟɪɜɚɬɢɜɧɵɯ ɫɢɥ ɜɫɟɝɞɚ ɨɬɪɢɰɚɬɟɥɶɧɚ. Ɉɧɚ ɜɟɞɟɬ ɤ

ɭɦɟɧɶɲɟɧɢɸ ɩɨɥɧɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ, ɬɨ ɟɫɬɶ ɤ ɩɨɬɟɪɹɦ

(ɞɢɫɫɢɩɚɰɢɢ) ɷɧɟɪɝɢɢ. ɉɨɷɬɨɦɭ ɧɟɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɥɵ ɧɚɡɵɜɚɸɬ ɞɢɫ-

ɫɢɩɚɬɢɜɧɵɦɢ. ɉɪɢɦɟɪɨɦ ɞɢɫɫɢɩɚɬɢɜɧɵɯ ɫɢɥ ɹɜɥɹɟɬɫɹ ɫɢɥɚ ɬɪɟɧɢɹ.