Теоретическая механика_в_Вопросах_и_Ответах

ȼ. Ⱦ. Ȼɟɪɬɹɟɜ, Ʌ. Ⱥ. Ȼɭɥɚɬɨɜ

ȼ. ɋ. Ʉɭɬɟɩɨɜ, Ʌ. Ɇ. ɇɟɱɚɟɜ

ɌȿɈɊȿɌɂɑȿɋɄȺə

ɆȿɏȺɇɂɄȺ

ȼ ȼɈɉɊɈɋȺɏ ɂ ɈɌȼȿɌȺɏ

ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɡɚɨɱɧɨɣ ɮɨɪɦɵ ɨɛɭɱɟɧɢɹ

Ɍɭɥɚ 2005

ȼȼȿȾȿɇɂȿ

ȼ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɡɚɞɚɱɚ ɩɨɜɵɲɟɧɢɹ ɤɚɱɟɫɬɜɚ ɨɛɪɚɡɨɜɚɧɢɹ ɩɨɫɬɚɜ-

ɥɟɧɚ ɤɨɧɰɟɩɰɢɟɣ ɦɨɞɟɪɧɢɡɚɰɢɢ ɪɨɫɫɢɣɫɤɨɝɨ ɨɛɪɚɡɨɜɚɧɢɹ, ɨɞɧɢɦ ɢɡ ɲɚɝɨɜ ɤɨɬɨɪɨɣ ɹɜɥɹɟɬɫɹ ɜɜɟɞɟɧɢɟ ɪɟɣɬɢɧɝɨɜɨɣ ɫɢɫɬɟɦɵ ɨɞɧɨɜɪɟɦɟɧɧɨ ɫ ɲɢɪɨɤɢɦ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɬɟɫɬɢɪɨɜɚɧɢɹ, ɭɫɢɥɢɜɚɸɳɟɝɨ ɪɨɥɶ ɢ ɡɧɚɱɟɧɢɟ ɫɚɦɨɫɬɨɹ-

ɬɟɥɶɧɨɣ ɪɚɛɨɬɵ ɫɬɭɞɟɧɬɚ.

Ɍɟɫɬɢɪɨɜɚɧɢɟ ɩɨɡɜɨɥɹɟɬ ɧɚɢɛɨɥɟɟ ɩɨɥɧɨ ɢ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɨ ɨɰɟ-

ɧɢɬɶ ɭɪɨɜɟɧɶ ɡɧɚɧɢɣ ɢ ɭɦɟɧɢɣ ɤɚɠɞɨɝɨ ɫɬɭɞɟɧɬɚ ɩɨ ɭɱɟɛɧɨɣ ɞɢɫɰɢɩɥɢɧɟ.

Ʉɨɧɬɪɨɥɶ ɤɚɱɟɫɬɜɚ ɭɫɜɨɟɧɢɹ ɢ ɩɨɧɢɦɚɧɢɹ ɢɡɭɱɚɟɦɨɝɨ ɩɪɟɞɦɟɬɚ, ɱɟɪɟɡ ɦɵɫɥɢɬɟɥɶɧɭɸ ɞɟɹɬɟɥɶɧɨɫɬɶ ɬɟɫɬɢɪɭɟɦɨɝɨ, ɫɥɭɠɢɬ ɫɚɦɨɨɛɭɱɚɸɳɢɦ ɢɫ-

ɬɨɱɧɢɤɨɦ ɡɧɚɧɢɣ. Ɉɪɝɚɧɢɡɚɰɢɹ ɬɟɫɬɨɜɨɝɨ ɤɨɧɬɪɨɥɹ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟ-

ɯɚɧɢɤɟ ɢɦɟɟɬ ɫɜɨɸ ɨɫɨɛɟɧɧɨɫɬɶ ɜ ɫɢɥɭ ɨɛɢɥɢɹ ɬɟɨɪɟɦ ɢ ɡɚɤɨɧɨɜ, ɡɚɩɢɫɚɧ-

ɧɵɯ ɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɨɪɦɟ. Ɍɟɫɬɵ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɫɭɳɟɫɬ-

ɜɟɧɧɨ ɨɬɥɢɱɚɸɬɫɹ ɨɬ ɜɟɪɛɚɥɶɧɵɯ ɬɟɫɬɨɜ, ɬɚɤ ɤɚɤ ɜ ɧɢɯ ɢɫɩɨɥɶɡɭɟɬɫɹ ɛɨɥɶ-

ɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɜɟɤɬɨɪɨɜ, ɢɡɨɛɪɚɠɚɸɳɢɯ ɮɢɡɢɱɟɫɤɢɟ ɜɟɥɢɱɢɧɵ.

Ɍɟɫɬɨɜɨɟ ɡɚɞɚɧɢɟ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɦɭ ɦɚɬɟɪɢɚɥɭ ɤɭɪɫɚ, ɨɬɪɚɠɚɟɬ ɨɫ-

ɧɨɜɧɵɟ ɩɨɧɹɬɢɹ, ɡɚɤɨɧɵ ɢ ɭɪɚɜɧɟɧɢɹ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ.

ɇɚɫɬɨɹɳɟɟ ɭɱɟɛɧɨɟ ɩɨɫɨɛɢɟ ɩɪɟɞɧɚɡɧɚɱɟɧɨ ɞɥɹ ɩɪɨɜɟɞɟɧɢɹ ɫɚɦɨɤɨɧ-

ɬɪɨɥɹ ɩɟɪɟɞ ɬɟɤɭɳɢɦ ɢ ɩɪɨɦɟɠɭɬɨɱɧɵɦ ɬɟɫɬɢɪɨɜɚɧɢɟɦ. ȼ ɨɫɧɨɜɭ ɩɨɫɨɛɢɹ ɩɨɥɨɠɟɧɵ ɜɨɩɪɨɫɵ ɫɚɦɨɤɨɧɬɪɨɥɹ ɢɡ "Ʉɭɪɫɚ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ" əɛ-

ɥɨɧɫɤɨɝɨ Ⱥ. Ⱥ.

Ɋɟɤɨɦɟɧɞɭɟɬɫɹ ɩɟɪɟɞ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɩɨɫɨɛɢɹ ɢɡɭɱɢɬɶ ɤɭɪɫ ɬɟɨɪɟɬɢ-

ɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ ɩɨ ɨɞɧɨɦɭ ɢɡ ɭɱɟɛɧɢɤɨɜ, ɩɪɢɜɟɞɟɧɧɵɯ ɜ ɛɢɛɥɢɨɝɪɚɮɢɱɟ-

ɫɤɨɦ ɫɩɢɫɤɟ.

ɉɨɫɨɛɢɟ ɪɚɫɫɱɢɬɚɧɨ ɧɚ ɫɬɭɞɟɧɬɨɜ ɨɱɧɨɣ ɢ ɡɚɨɱɧɨɣ ɮɨɪɦ ɫɢɫɬɟɦɵ ɨɛɭɱɟɧɢɹ ɢ ɦɨɠɟɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɨ ɜ ɤɚɱɟɫɬɜɟ ɫɩɪɚɜɨɱɧɨɝɨ ɩɨɫɨɛɢɹ ɞɥɹ ɢɧɠɟɧɟɪɨɜ ɢ ɞɪɭɝɢɯ ɫɩɟɰɢɚɥɢɫɬɨɜ.

2

ȼɈɉɊɈɋɕ ɂ ɈɌȼȿɌɕ ɉɈ ɋɌȺɌɂɄȿ

ɑɬɨ ɧɚɡɵɜɚɟɬɫɹ ɫɜɹɡɶɸ?

Ɍɟɥɨ, ɨɝɪɚɧɢɱɢɜɚɸɳɟɟ ɫɜɨɛɨɞɭ ɞɜɢɠɟɧɢɹ ɞɚɧɧɨɝɨ ɬɜɺɪɞɨɝɨ ɬɟɥɚ, ɹɜɥɹ-

ɟɬɫɹ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɧɟɦɭ ɫɜɹɡɶɸ.

ȼ ɱɺɦ ɡɚɤɥɸɱɚɟɬɫɹ ɩɪɢɧɰɢɩ ɨɫɜɨɛɨɠɞɚɟɦɨɫɬɢ ɨɬ ɫɜɹɡɟɣ?

ɇɟɫɜɨɛɨɞɧɨɟ ɬɜɺɪɞɨɟ ɬɟɥɨ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɫɜɨɛɨɞɧɨɟ, ɧɚ ɤɨɬɨ-

ɪɨɟ, ɤɪɨɦɟ ɡɚɞɚɜɚɟɦɵɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɬ ɪɟɚɤɰɢɢ ɫɜɹɡɟɣ.

ɉɟɪɟɱɢɫɥɢɬɟ ɨɫɧɨɜɧɵɟ ɬɢɩɵ ɨɩɨɪ, ɞɥɹ ɤɨɬɨɪɵɯ ɥɢɧɢɢ ɞɟɣɫɬɜɢɹ ɪɟɚɤɰɢɢ ɢɡɜɟɫɬɧɵ.

ɚ) ɝɥɚɞɤɚɹ ɩɥɨɫɤɨɫɬɶ (ɪɟɚɤɰɢɹɧɚɩɪɚɜɥɟɧɚɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɤɩɥɨɫɤɨɫɬɢ);

ɛ) ɧɢɬɶ, ɤɚɧɚɬ (ɪɟɚɤɰɢɹ ɧɚɩɪɚɜɥɟɧɚ ɜɞɨɥɶ ɧɢɬɢ);

ɜ) ɲɚɪɧɢɪɧɨ-ɩɨɞɜɢɠɧɚɹ ɨɩɨɪɚ (ɪɟɚɤɰɢɹ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɰɟɧɬɪ ɲɚɪɧɢɪɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɤ ɨɩɨɪɧɨɣ ɩɥɨɫɤɨɫɬɢ);

ɝ) ɨɩɨɪɧɵɣ ɫɬɟɪɠɟɧɶ ɫ ɲɚɪɧɢɪɚɦɢ ɧɚ ɤɨɧɰɚɯ (ɪɟɚɤɰɢɹ ɧɚɩɪɚɜɥɟɧɚ ɩɨ ɫɬɟɪɠɧɸ).

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɚɩɪɚɜɥɟɧɢɟ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ ɫɢɫɬɟɦɵ ɫɯɨɞɹɳɢɯɫɹ ɫɢɥ?

Ɋɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ R ɫɯɨɞɹɳɢɯɫɹ ɫɢɥ ɩɪɢɥɨɠɟɧɚ ɜ ɬɨɱɤɟ ɩɟɪɟɫɟɱɟɧɢɹ ɥɢɧɢɣ ɞɟɣɫɬɜɢɹ ɫɢɥ Fk ɢ ɪɚɜɧɚ ɢɯ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ.

n

R ¦Fk F1 F2 ... Fn

k 1

Ɋɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɧɚɩɪɚɜɥɟɧɚ ɩɨ ɩɪɹɦɨɣ, ɫɨɟɞɢɧɹɸɳɟɣ ɧɚɱɚɥɨ ɩɟɪɜɨɣ ɢ ɤɨɧɟɰ ɩɨɫɥɟɞɧɟɣ ɫɢɥɵ.

Ʉɚɤɨɜɵ ɭɫɥɨɜɢɹ ɢ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɩɥɨɫɤɨɣ ɫɢɫɬɟɦɵ ɫɯɨɞɹɳɢɯɫɹ ɫɢɥ?

ɍɫɥɨɜɢɹɦɢ ɪɚɜɧɨɜɟɫɢɹ ɜɵɪɚɠɚɸɬɫɹ ɡɚɦɤɧɭɬɨɫɬɶɸ ɫɢɥɨɜɨɝɨ ɦɧɨɝɨ-

ɭɝɨɥɶɧɢɤɚ, ɬ. ɟ. ɧɚɱɚɥɨ ɩɟɪɜɨɣ ɢ ɤɨɧɟɰ ɩɨɫɥɟɞɧɟɣ ɫɢɥɵ ɫɨɜɩɚɞɚɸɬ.

3

Ⱦɥɹ ɩɥɨɫɤɨɣ ɫɯɨɞɹɳɟɣɫɹ ɫɢɫɬɟɦɵ ɫɢɥ ɢɦɟɟɦ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ.

¦Xk 0, ¦Yk 0.

Ʉɚɤ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɩɥɚɧ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɫɬɚɬɢɤɢ ɧɚ ɪɚɜɧɨɜɟɫɢɟ ɫɢɥ?

ȼɫɟ ɡɚɞɚɱɢ ɧɚ ɪɚɜɧɨɜɟɫɢɟ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɧɟɤɨɬɨɪɨɦɭ ɬɟɥɭ, ɪɟɲɚ-

ɸɬɫɹ ɩɨ ɫɥɟɞɭɸɳɟɦɭ ɩɥɚɧɭ:

ɚ) ɉɨɤɚɡɵɜɚɟɦ ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɬɟɥɨ ɡɚɞɚɜɚɟɦɵɟ ɫɢɥɵ.

ɛ) Ɇɵɫɥɟɧɧɨ ɨɫɜɨɛɨɠɞɚɟɦ ɬɟɥɨ ɨɬ ɫɜɹɡɟɣ, ɡɚɦɟɧɹɹ ɢɯ ɞɟɣɫɬɜɢɟ ɪɟɚɤɰɢɹ-

ɦɢ ɫɜɹɡɟɣ.

ɜ) Ʉ ɩɨɥɭɱɟɧɧɨɣ ɫɢɫɬɟɦɟ ɫɢɥ ɩɪɢɦɟɧɹɟɦ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ, ɫɨɨɬɜɟɬɫɬ-

ɜɭɸɳɢɟ ɷɬɨɣ ɫɢɫɬɟɦɟ.

ɝ) Ɉɩɪɟɞɟɥɹɟɦ ɢɫɤɨɦɵɟ ɜɟɥɢɱɢɧɵ.

Ʉɚɤ ɮɨɪɦɢɪɭɟɬɫɹ ɬɟɨɪɟɦɚ ɨ ɪɚɜɧɨɜɟɫɢɢ ɬɪɺɯ ɧɟɩɚɪɚɥɥɟɥɶɧɵɯ ɫɢɥ?

Ʌɢɧɢɢ ɞɟɣɫɬɜɢɹ ɬɪɺɯ ɧɟɩɚɪɚɥɥɟɥɶɧɵɯ, ɜɡɚɢɦɧɨ ɭɪɚɜɧɨɜɟɲɢɜɚɸɳɢɯɫɹ ɫɢɥ ɩɟɪɟɫɟɤɚɸɬɫɹ ɜ ɨɞɧɨɣ ɬɨɱɤɟ.

ȼ ɱɺɦ ɡɚɤɥɸɱɚɟɬɫɹ ɫɭɳɧɨɫɬɶ ɫɩɨɫɨɛɚ ɜɵɪɟɡɚɧɢɹ ɭɡɥɨɜ?

ɋɩɨɫɨɛ ɜɵɪɟɡɚɧɢɹ ɭɡɥɨɜ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɦɵɫɥɟɧɧɨ ɜɵɪɟɡɚɸɬ ɭɡɥɵ ɮɟɪɦɵ, ɩɪɢɤɥɚɞɵɜɚɸɬ ɤ ɧɢɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɜɧɟɲɧɢɟ ɫɢɥɵ ɢ ɪɟɚɤɰɢɢ ɫɬɟɪɠɧɟɣ ɢ ɫɨɫɬɚɜɥɹɸɬ ɩɨ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɞɚɧɧɨɦɭ ɭɡɥɭ. Ɋɟɚɤɰɢɢ ɫɬɟɪɠɧɟɣ ɧɚɩɪɚɜɥɹɸɬ ɨɬ ɭɡɥɨɜ. ȿɫɥɢ ɜ ɪɟɡɭɥɶ-

ɬɚɬɟ ɜɵɱɢɫɥɟɧɢɣ ɩɨɥɭɱɚɬ ɨɬɜɟɬ ɫɨ ɡɧɚɤɨɦ ɦɢɧɭɫ, ɬɨ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɫɬɟɪɠɟɧɶ ɫɠɚɬ.

Ʉɚɤɨɜɵ ɥɟɦɦɵ ɨ ɧɭɥɟɜɵɯ ɫɬɟɪɠɧɹɯ?

ȿɫɥɢ ɭɫɢɥɢɹ ɜ ɨɬɞɟɥɶɧɵɯ ɫɬɟɪɠɧɹɯ ɡɚɝɪɭɠɟɧɧɨɣ ɮɟɪɦɵ ɪɚɜɧɵ ɧɭɥɸ,

ɫɬɟɪɠɧɢ ɩɪɢɧɹɬɨ ɧɚɡɵɜɚɬɶ ɧɭɥɟɜɵɦɢ ɫɬɟɪɠɧɹɦɢ.

Ʌɟɦɦɚ 1. ȿɫɥɢ ɜ ɧɟɡɚɝɪɭɠɟɧɧɨɦ ɭɡɥɟ ɮɟɪɦɵ ɫɯɨɞɹɬɫɹ ɞɜɚ ɫɬɟɪɠɧɹ, ɬɨ ɭɫɢɥɢɹ ɜ ɷɬɢɯ ɫɬɟɪɠɧɹɯ ɪɚɜɧɵ ɧɭɥɸ.

4

Ʌɟɦɦɚ 2. ȿɫɥɢ ɜ ɧɟɡɚɝɪɭɠɟɧɧɨɦ ɭɡɥɟ ɮɟɪɦɵ ɫɯɨɞɹɬɫɹ ɬɪɢ ɫɬɟɪɠɧɹ, ɢɡ ɤɨɬɨɪɵɯ ɞɜɚ ɪɚɫɩɨɥɨɠɟɧɵ ɩɨ ɨɞɧɨɣ ɩɪɹɦɨɣ, ɬɨ ɭɫɢɥɢɟ ɜ ɬɪɟɬɶɟɦ ɫɬɟɪɠ-

ɧɟ ɪɚɜɧɨ ɧɭɥɸ. ɍɫɢɥɢɹ ɜ ɞɜɭɯ ɩɟɪɜɵɯ ɫɬɟɪɠɧɹɯ ɪɚɜɧɵ ɦɟɠɞɭ ɫɨɛɨɣ.

Ʌɟɦɦɚ 3. ȿɫɥɢ ɜ ɭɡɥɟ ɮɟɪɦɵ ɫɯɨɞɹɬɫɹ ɞɜɚ ɫɬɟɪɠɧɹ ɢ ɤ ɭɡɥɭ ɩɪɢɥɨɠɟɧɚ ɜɧɟɲɧɹɹ ɫɢɥɚ, ɥɢɧɢɹ ɞɟɣɫɬɜɢɹ ɤɨɬɨɪɨɣ ɫɨɜɩɚɞɚɟɬ ɫ ɨɫɶɸ ɨɞɧɨɝɨ ɢɡ ɫɬɟɪɠɧɟɣ, ɬɨ ɭɫɢɥɢɟ ɜ ɷɬɨɦ ɫɬɟɪɠɧɟ ɪɚɜɧɨ ɩɨ ɦɨɞɭɥɸ ɞɚɧɧɨɣ ɫɢɥɟ, ɚ

ɭɫɢɥɢɟ ɜ ɞɪɭɝɨɦ ɫɬɟɪɠɧɟ ɪɚɜɧɨ ɧɭɥɸ.

ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɬɟɨɪɟɦɵ ɨ ɩɚɪɚɯ ɫɢɥ ɧɚ ɩɥɨɫɤɨɫɬɢ.

ɚ) ɉɚɪɭ ɫɢɥ ɦɨɠɧɨ ɩɟɪɟɦɟɳɚɬɶ ɜ ɥɸɛɨɟ ɩɨɥɨɠɟɧɢɟ ɜ ɩɥɨɫɤɨɫɬɢ ɟɺ ɞɟɣ-

ɫɬɜɢɹ.

ɛ) ɉɚɪɵ ɫɢɥ, ɦɨɦɟɧɬɵ ɤɨɬɨɪɵɯ ɱɢɫɥɟɧɧɨ ɪɚɜɧɵ ɢ ɨɞɢɧɚɤɨɜɵ ɩɨ ɡɧɚɤɭ,

ɷɤɜɢɜɚɥɟɧɬɧɵ.

ɜ) Ɇɨɦɟɧɬ ɩɚɪɵ ɫɢɥ, ɷɤɜɢɜɚɥɟɧɬɧɨɣ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɟ ɩɚɪ ɧɚ ɩɥɨɫɤɨɫɬɢ, ɪɚɜɟɧ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɫɨɫɬɚɜɥɹɸɳɢɯ ɩɚɪ.

ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɩɚɪ ɧɚ ɩɥɨɫɤɨɫɬɢ.

ɉɚɪɵ ɫɢɥ ɧɚ ɩɥɨɫɤɨɫɬɢ ɭɪɚɜɧɨɜɟɲɢɜɚɸɬɫɹ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ ɚɥɝɟɛɪɚɢ-

ɱɟɫɤɚɹ ɫɭɦɦɚ ɢɯ ɦɨɦɟɧɬɨɜ ɪɚɜɧɚ ɧɭɥɸ.

Ʉɚɤ ɧɚɩɪɚɜɥɟɧɵ ɪɟɚɤɰɢɢ ɨɩɨɪ ɛɚɥɤɢ, ɧɚɝɪɭɠɟɧɧɨɣ ɩɚɪɨɣ ɫɢɥ ɢ ɥɟɠɚɳɟɣ ɧɚ ɞɜɭɯ ɨɩɨɪɚɯ, ɨɞɧɚ ɢɡ ɤɨɬɨɪɵɯ – ɲɚɪɧɢɪɧɨ-ɧɟɩɨɞɜɢɠɧɚɹ, ɚ ɞɪɭɝɚɹ – ɧɚ ɤɚɬɤɚɯ?

Ɍɚɤ ɤɚɤ ɡɚɞɚɧɧɚɹ ɧɚɝɪɭɡɤɚ ɫɨɫɬɨɢɬ ɬɨɥɶɤɨ ɢɡ ɩɚɪɵ ɫɢɥ, ɬɨ ɪɟɚɤɰɢɢ ɨɩɨɪ ɞɨɥɠɧɵ ɫɨɫɬɚɜɢɬɶ ɩɚɪɭ ɫɢɥ, ɩɚɪɚɥɥɟɥɶɧɵɯ ɪɟɚɤɰɢɢ ɨɩɨɪɵ ɧɚ ɤɚɬɤɚɯ ɢ ɧɚɩɪɚɜɥɟɧɧɭɸ ɜ ɫɬɨɪɨɧɭ ɨɛɪɚɬɧɭɸ ɧɚɩɪɚɜɥɟɧɢɸ ɩɪɢɥɨɠɟɧɧɨɣ ɩɚɪɵ ɫɢɥ.

Ɂɚɜɢɫɢɬ ɥɢ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɢ ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ ɡɚɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɫɢɥ ɨɬ ɜɵɛɨɪɚ ɰɟɧɬɪɚ ɩɪɢɜɟɞɟɧɢɹ?

Ƚɥɚɜɧɵɣ ɜɟɤɬɨɪ, ɪɚɜɧɵɣ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɜɫɟɯ ɫɢɥ ɫɢɫɬɟɦɵ, ɧɟ ɡɚ-

ɜɢɫɢɬ ɨɬ ɜɵɛɨɪɚ ɰɟɧɬɪɚ ɩɪɢɜɢɞɟɧɢɹ.

5

Ƚɥɚɜɧɵɣ ɦɨɦɟɧɬ, ɪɚɜɧɵɣ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɜɫɟɯ ɫɢɥ ɫɢɫ-

ɬɟɦɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɩɪɢɜɢɞɟɧɢɹ, ɡɚɜɢɫɢɬ ɨɬ ɜɵɛɨɪɚ ɷɬɨɝɨ ɰɟɧɬɪɚ,

ɬɚɤ ɤɚɤ ɜɵɛɨɪ ɰɟɧɬɪɚ ɩɪɢɜɢɞɟɧɢɹ ɜɥɢɹɟɬ ɧɚ ɜɟɥɢɱɢɧɭ ɢ ɡɧɚɤ ɝɥɚɜɧɨɝɨ ɦɨɦɟɧɬɚ.

Ʉɚɤɨɜɵ ɜɨɡɦɨɠɧɵɟ ɫɥɭɱɚɢ ɩɪɢɜɟɞɟɧɢɹ ɫɢɥ, ɩɪɨɢɡɜɨɥɶɧɨ ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɧɚ ɩɥɨɫɤɨɫɬɢ?

ɞɢɬɫɹ ɤ ɩɚɪɟ ɫɢɥ, ɦɨɦɟɧɬ ɤɨɬɨɪɨɣ ɪɚɜɟɧ ɝɥɚɜɧɨɦɭ ɦɨɦɟɧɬɭ ɫɢɥ ɨɬɧɨɫɢ-

ɬɟɥɶɧɨ ɰɟɧɬɪɚ ɩɪɢɜɟɞɟɧɢɹ.

ɜ) Ƚɥɚɜɧɵɣ ɜɟɤɬɨɪ P z 0, ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ MO 0 . ɋɢɫɬɟɦɚ ɫɢɥ ɩɪɢɜɨ-

ɞɢɬɫɹ ɤ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ ɫɢɥɟ, ɪɚɜɧɨɣ ɝɥɚɜɧɨɦɭ ɜɟɤɬɨɪɭ ɫɢɥ, ɥɢɧɢɹ ɞɟɣɫɬɜɢɹ ɤɨɬɨɪɨɣ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɰɟɧɬɪ ɩɪɢɜɟɞɟɧɢɹ.

ɝ) Ƚɥɚɜɧɵɣ ɜɟɤɬɨɪ P z 0, ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ MO z 0. ɋɢɫɬɟɦɚ ɫɢɥ ɩɪɢɜɨ-

ɞɢɬɫɹ ɤ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ, ɥɢɧɢɹ ɞɟɣɫɬɜɢɹ ɤɨɬɨɪɨɣ ɩɪɨɯɨɞɢɬ ɨɬ ɰɟɧɬɪɚ ɩɪɢɜɟɞɟɧɢɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ d MO P .

ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɬɟɨɪɟɦɭ ȼɚɪɢɧɶɨɧɚ.

Ɇɨɦɟɧɬ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɸɛɨɣ ɬɨɱɤɢ ɧɚ ɩɥɨɫɤɨ-

ɫɬɢ ɪɚɜɟɧ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɫɨɫɬɚɜɥɹɸɳɢɯ ɫɢɥ ɨɬɧɨɫɢ-

ɬɟɥɶɧɨ ɬɨɣ ɠɟ ɬɨɱɤɢ.

MO R ¦MO Fk .

k

Ɂɚɩɢɲɢɬɟɫɢɫɬɟɦɵɭɪɚɜɧɟɧɢɣɪɚɜɧɨɜɟɫɢɹɩɪɨɢɡɜɨɥɶɧɨɣɩɥɨɫɤɨɣɫɢɫɬɟɦɵɫɢɥ.

6

ɜ) ¦Mk A 0, ¦Mk B 0, ¦Mk C 0 .

Ʉɚɤɢɟ ɡɚɞɚɱɢ ɧɚɡɵɜɚɸɬɫɹ ɫɬɚɬɢɱɟɫɤɢ ɨɩɪɟɞɟɥɢɦɵɦɢ?

Ɂɚɞɚɱɢ, ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɪɟɲɚɬɶ ɦɟɬɨɞɚɦɢ ɫɬɚɬɢɤɢ ɬɜɺɪɞɨɝɨ ɬɟɥɚ, ɬɨ ɟɫɬɶ ɡɚɞɚɱɢ, ɜ ɤɨɬɨɪɵɯ ɱɢɫɥɨ ɧɟɢɡɜɟɫɬɧɵɯ ɧɟ ɩɪɟɜɵɲɚɟɬ ɱɢɫɥɚ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ ɫɢɥ, ɧɚɡɵɜɚɸɬɫɹ ɫɬɚɬɢɱɟɫɤɢ ɨɩɪɟɞɟɥɢɦɵɦɢ.

ȼ ɱɺɦ ɫɨɫɬɨɢɬ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɪɵɱɚɝɭ?

ɉɪɢ ɪɚɜɧɨɜɟɫɢɢ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɪɵɱɚɝɭ, ɚɥɝɟɛɪɚɢɱɟɫɤɚɹ ɫɭɦɦɚ ɦɨ-

ɦɟɧɬɨɜ ɜɫɟɯ ɡɚɞɚɜɚɟɦɵɯ ɫɢɥ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɩɨɪɧɨɣ ɬɨɱɤɢ ɪɚɜɧɚ ɧɭɥɸ.

¦M k o 0.

ɑɬɨ ɧɚɡɵɜɚɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɭɫɬɨɣɱɢɜɨɫɬɢ?

Ʉɨɷɮɮɢɰɢɟɧɬɨɦ ɭɫɬɨɣɱɢɜɨɫɬɢ ɩɪɢ ɨɩɪɨɤɢɞɵɜɚɧɢɢ ɩɪɢɧɹɬɨ ɨɩɪɟɞɟɥɹɬɶ ɨɬɧɨɲɟɧɢɟɦ ɜɟɥɢɱɢɧɵ ɭɞɟɪɠɢɜɚɸɳɟɝɨ ɦɨɦɟɧɬɚ ɤ ɜɟɥɢɱɢɧɟ ɨɩɪɨɤɢɞɵ-

ɜɚɸɳɟɝɨ ɦɨɦɟɧɬɚ.

kɆɭɞ

Ɇɨɩɪ

ȼɫɥɭɱɚɟ ɭɫɬɨɣɱɢɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ k !1.

ȼɱɺɦ ɡɚɤɥɸɱɚɟɬɫɹ ɫɭɳɧɨɫɬɶ ɫɩɨɫɨɛɚ Ɋɢɬɬɟɪɚ?

ȼɮɟɪɦɟ ɩɪɨɜɨɞɢɬɫɹ ɫɟɱɟɧɢɟ, ɪɚɫɫɟɤɚɸɳɟɟ ɧɟ ɛɨɥɟɟ ɬɪɺɯ ɫɬɟɪɠɧɟɣ.

Ɇɵɫɥɟɧɧɨ ɨɬɛɪɚɫɵɜɚɟɦ ɨɞɧɭ ɢɡ ɱɚɫɬɟɣ ɮɟɪɦɵ, ɡɚɦɟɧɹɹ ɟɺ ɞɟɣɫɬɜɢɟ ɪɟ-

ɚɤɰɢɹɦɢ, ɧɚɩɪɚɜɥɟɧɧɵɦɢ ɩɨ ɫɬɟɪɠɧɹɦ ɜ ɫɬɨɪɨɧɭ ɨɬɛɪɨɲɟɧɧɨɣ ɱɚɫɬɢ.

Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɢɡɜɟɫɬɧɵɯ ɭɫɢɥɢɣ ɜ ɫɬɟɪɠɧɹɯ ɫɨɫɬɚɜɥɹɟɦ ɭɪɚɜɧɟɧɢɹ ɦɨɦɟɧɬɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɫɬɟɪɠɧɟɣ (ɬɨɱɤɢ Ɋɢɬɬɟɪɚ).

Ɂɧɚɤ ɦɢɧɭɫ ɩɪɢ ɪɟɲɟɧɢɢ ɭɪɚɜɧɟɧɢɣ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɫɬɟɪɠɟɧɶ ɫɠɚɬ.

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɥɢɱɢɧɚ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɫɢɥɵ ɬɪɟɧɢɹ?

7

ɉɪɢ ɫɤɨɥɶɠɟɧɢɢ ɬɟɥɚ ɩɨ ɲɟɪɨɯɨɜɚɬɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤ ɧɟɦɭ ɩɪɢɥɨɠɟɧɚ ɫɢɥɚ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ, ɦɨɞɭɥɶ ɤɨɬɨɪɨɣ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ ɧɨɪɦɚɥɶɧɨɦɭ ɞɜɢɠɟɧɢɸ N , ɚ ɧɚɩɪɚɜɥɟɧɢɟ ɷɬɨɣ ɫɢɥɵ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɧɚɩɪɚɜɥɟɧɢɸ ɫɤɨɪɨɫɬɢ ɬɟɥɚ, ɪɚɜɧɚɹ Fɬɪ f N , ɝɞɟ f — ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶ-

ɠɟɧɢɹ.

Ʉɚɤ ɚɧɚɥɢɬɢɱɟɫɤɢ ɨɩɪɟɞɟɥɢɬɶ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɭɸ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɫɢɫɬɟɦɵ ɫɯɨɞɹɳɢɯɫɹ ɫɢɥ?

Ɇɨɞɭɥɶ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ ɪɚɜɟɧ:

ɇɚɩɪɚɜɥɟɧɢɟɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣɨɩɪɟɞɟɥɹɟɬɫɹɧɚɩɪɚɜɥɹɸɳɢɦɢɤɨɫɢɧɭɫɚɦɢ

Ɂɚɩɢɲɢɬɟ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɫɢɫɬɟɦɵ ɫɯɨɞɹɳɢɯɫɹ ɫɢɥ.

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɦɨɦɟɧɬ ɩɚɪɵ ɫɢɥ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ?

Ɇɨɦɟɧɬ ɩɚɪɵ ɫɢɥ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬ ɤɚɤ ɜɟɤɬɨɪ, ɧɚɩɪɚɜɥɟɧ-

ɧɵɣ ɩɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɭ ɤ ɩɥɨɫɤɨɫɬɢ ɩɚɪɵ ɫɢɥ ɜ ɬɚɤɭɸ ɫɬɨɪɨɧɭ, ɱɬɨɛɵ,

ɫɦɨɬɪɹ ɧɚɜɫɬɪɟɱɭ ɷɬɨɦɭ ɜɟɤɬɨɪɭ, ɜɢɞɟɬɶ ɩɚɪɭ ɫɬɪɟɦɹɳɟɣɫɹ ɜɪɚɳɚɬɶ ɩɥɨɫɤɨɫɬɶ ɩɪɨɬɢɜ ɞɜɢɠɟɧɢɹ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ:

M F; Fc BAu F AB u Fc MB F M A Fc .

Ɇɨɞɭɥɶ ɜɟɤɬɨɪɚ ɦɨɦɟɧɬɚ ɩɚɪɵ ɪɚɜɟɧ ɩɪɨɢɡɜɟɞɟɧɢɸ ɨɞɧɨɣ ɢɡ ɫɢɥ ɩɚɪɵ

ɧɚ ɤɪɚɬɱɚɣɲɟɟ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɥɢɧɢɹɦɢ ɞɟɣɫɬɜɢɹ ɫɢɥ ɩɚɪɵ

M F; Fc rF d rFc d .

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ɉɟɪɟɱɢɫɥɢɬɟ ɬɟɨɪɟɦɵ ɨ ɩɚɪɚɯ ɫɢɥ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ.

ɚ) Ɂɚɞɚɧɧɭɸ ɩɚɪɭ ɫɢɥ, ɧɟ ɢɡɦɟɧɹɹ ɟɺ ɞɟɣɫɬɜɢɹ ɧɚ ɬɜɺɪɞɨɟ ɬɟɥɨ, ɦɨɠɧɨ ɩɟɪɟɧɟɫɬɢ ɜ ɥɸɛɭɸ ɩɥɨɫɤɨɫɬɶ, ɩɚɪɚɥɥɟɥɶɧɭɸ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ ɩɚɪɵ.

ɛ) ɉɚɪɵ ɫɢɥ ɷɤɜɢɜɚɥɟɧɬɧɵ, ɟɫɥɢ ɢɯ ɦɨɦɟɧɬɵ ɝɟɨɦɟɬɪɢɱɟɫɤɢ ɪɚɜɧɵ.

ɜ) Ɇɨɦɟɧɬ ɩɚɪɵ ɫɢɥ, ɷɤɜɢɜɚɥɟɧɬɧɨɣ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɩɚɪ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ,

ɪɚɜɟɧ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɫɨɫɬɚɜɥɹɸɳɢɯ ɫɢɫɬɟɦɭ ɩɚɪ.

Ɂɚɩɢɲɢɬɟ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɩɚɪ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ.

ɉɚɪɵ ɫɢɥ, ɪɚɫɩɨɥɨɠɟɧɧɵɟ ɩɪɨɢɡɜɨɥɶɧɨ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɭɪɚɜɧɨɜɟɲɢɜɚ-

ɸɬɫɹ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ ɝɟɨɦɟɬɪɢɱɟɫɤɚɹ ɫɭɦɦɚ ɢɯ ɦɨɦɟɧɬɨɜ ɪɚɜɧɚ ɧɭɥɸ

¦Mk 0.

k

Ɂɚɩɢɲɢɬɟ ɦɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɤɚɤ ɜɟɤɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ.

ȼɟɤɬɨɪ ɦɨɦɟɧɬɚ ɫɢɥɵ Mo ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɜɟɤɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɪɚɞɢɭɫɚ ɜɟɤɬɨɪɚ r , ɩɪɨɜɟɞɺɧɧɨɝɨ ɢɡ ɰɟɧɬɪɚ ɦɨ-

ɦɟɧɬɚ ɜ ɬɨɱɤɭ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ, ɧɚ ɜɟɤɬɨɪ ɫɢɥɵ F .

Mo r u F .

Ʉɚɤ ɜɵɱɢɫɥɹɟɬɫɹ ɦɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ?

Ɇɨɦɟɧɬɨɦ ɫɢɥɵ F ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z ɧɚɡɵɜɚɟɬɫɹ ɜɡɹɬɨɟ ɫɨ ɡɧɚɤɨɦ ɩɥɸɫ ɢɥɢ ɦɢɧɭɫ ɩɪɨɢɡɜɟɞɟɧɢɟ ɦɨɞɭɥɹ ɩɪɨɟɤɰɢɢ F1 ɫɢɥɵ F ɧɚ ɩɥɨɫ-

ɤɨɫɬɶ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɭɸ ɨɫɢ, ɧɚ ɟɺ ɩɥɟɱɨ d1 ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ O ɩɟ-

ɪɟɫɟɱɟɧɢɹ ɨɫɢ ɫ ɩɥɨɫɤɨɫɬɶɸ:

Mz rF1 d1 .

Ʉɚɤɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɫɭɳɟɫɬɜɭɟɬ ɦɟɠɞɭ ɦɨɦɟɧɬɚɦɢ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɢ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɷɬɭ ɬɨɱɤɭ?

ɉɪɨɟɤɰɢɹ ɦɨɦɟɧɬɚ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɧɚ ɨɫɶ, ɩɪɨɯɨɞɹɳɭɸ ɱɟɪɟɡ

ɷɬɭ ɬɨɱɤɭ, ɪɚɜɧɚ ɦɨɦɟɧɬɭ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ:

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Mz Mo cos Mo ,k .

ȼɵɪɚɡɢɬɟ ɦɨɦɟɧɬɵ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ ɱɟɪɟɡ ɩɪɨɟɤ-

ɰɢɢ ɫɢɥɵ ɧɚ ɷɬɢ ɨɫɢ.

M x yZ zY , M y zX xZ , M z xY yX .

Ɂɚɩɢɲɢɬɟ ɝɥɚɜɧɵɟ ɦɨɦɟɧɬɵ ɫɢɫɬɟɦɵ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɢ ɨɬɧɨ-

ɫɢɬɟɥɶɧɨ ɨɫɢ.

Ɇɨɦɟɧɬ, ɪɚɜɧɵɣ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɜɫɟɯ ɡɚɞɚɧɧɵɯ ɫɢɥ ɨɬ-

ɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ Ɉ, ɧɚɡɵɜɚɟɬɫɹ ɝɥɚɜɧɵɦ ɦɨɦɟɧɬɨɦ ɫɢɫɬɟɦɵ ɫɢɥ ɨɬɧɨ-

ɫɢɬɟɥɶɧɨ ɷɬɨɣ ɬɨɱɤɢ.

Mo ¦MOK ¦rk u Fk .

kk

Ɇɨɦɟɧɬ, ɪɚɜɧɵɣ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɜɫɟɯ ɡɚɞɚɧɧɵɯ ɫɢɥ ɨɬ-

ɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z, ɧɚɡɵɜɚɟɬɫɹ ɝɥɚɜɧɵɦ ɦɨɦɟɧɬɨɦ ɫɢɫɬɟɦɵ ɫɢɥ ɨɬɧɨɫɢ-

ɬɟɥɶɧɨ ɨɫɢ z.

MZ ¦MZK ¦rkXY u FkXY .

kk

Ʉɚɤɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɫɭɳɟɫɬɜɭɟɬ ɦɟɠɞɭ ɦɨɦɟɧɬɚɦɢ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ

ɬɨɱɤɢ ɢ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɷɬɭ ɬɨɱɤɭ?

ɉɪɨɟɤɰɢɹ ɝɥɚɜɧɨɝɨ ɦɨɦɟɧɬɚ ɫɢɫɬɟɦɵ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɢ ɧɚ ɨɫɶ, ɩɪɨɯɨɞɹɳɭɸ ɱɟɪɟɡ ɷɬɭ ɬɨɱɤɭ, ɪɚɜɧɚ ɝɥɚɜɧɨɦɭ ɦɨɦɟɧɬɭ ɡɚɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɨɫɢ:

Mo cos Mo ,k Mz .

Ʉ ɱɟɦɭ ɦɨɝɭɬ ɛɵɬɶ ɩɪɢɜɟɞɟɧɵ ɫɢɥɵ, ɩɪɨɢɡɜɨɥɶɧɨ ɪɚɫɩɨɥɨɠɟɧɧɵɟ ɜ ɩɪɨ-

ɫɬɪɚɧɫɬɜɟ?

ɋɢɥɵ, ɩɪɨɢɡɜɨɥɶɧɨ ɪɚɫɩɨɥɨɠɟɧɧɵɟ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɜɫɟɝɞɚ ɦɨɝɭɬ ɛɵɬɶ

ɩɪɢɜɟɞɟɧɵ ɤ ɨɞɧɨɣ ɫɢɥɟ, ɪɚɜɧɨɣ ɢɯ ɝɥɚɜɧɨɦɭ ɜɟɤɬɨɪɭ, ɩɪɢɥɨɠɟɧɧɨɣ ɜ

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