Теоретическая механика_в_Вопросах_и_Ответах

ɰɟɧɬɪɟ ɩɪɢɜɟɞɟɧɢɹ, ɢ ɤ ɩɚɪɟ ɫɢɥ ɫ ɦɨɦɟɧɬɨɦ, ɪɚɜɧɵɦ ɝɥɚɜɧɨɦɭ ɦɨɦɟɧɬɭ

ɜɫɟɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɩɪɢɜɟɞɟɧɢɹ.

Ʉɚɤ ɜɵɱɢɫɥɹɸɬɫɹ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɢ ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ

ɫɢɫɬɟɦɵ ɫɢɥ?

Ɇɨɞɭɥɶ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɝɥɚɜɧɨɝɨ ɜɟɤɬɨɪɚ P ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ:

PX 2 Y 2 Z2 ;

X ¦Xk , Y ¦Yk , Z ¦Zk .

Ɇɨɞɭɥɶ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɝɥɚɜɧɨɝɨ ɦɨɦɟɧɬɚ Mo ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ:

Ʉɚɤɨɜɵ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɫɢɫɬɟɦɵ ɫɢɥ?

Ɋɚɜɧɨɜɟɫɢɸ ɩɪɨɢɡɜɨɥɶɧɨɣ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɫɢɫɬɟɦɵ ɫɢɥ ɫɨɨɬɜɟɬɫɬɜɭ-

ɸɬ ɞɜɚ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ

MMo 0 , P 0,

ɤɨɬɨɪɵɦ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɲɟɫɬɶ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ

Ʉɚɤɨɜɵ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɫɢɫɬɟɦɵ ɩɚɪɚɥɥɟɥɶɧɵɯ ɫɢɥ?

11

Ɋɚɜɧɨɜɟɫɢɸ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɫɢɫɬɟɦɵ ɩɚɪɚɥɥɟɥɶɧɵɯ ɫɢɥ ɫɨɨɬɜɟɬɫɬɜɭ-

ɸɬ ɬɪɢ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ:

¦Zk 0, ¦Mkx 0, ¦Mk y 0.

Ʉɚɤ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɬɟɨɪɟɦɚ ȼɚɪɢɧɶɨɧɚ ɞɥɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɫɢɫɬɟ-

ɦɵ ɦɢɥ?

Ɇɨɦɟɧɬ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɸɛɨɣ ɬɨɱɤɢ ɪɚɜɟɧ ɝɟɨɦɟɬɪɢ-

ɱɟɫɤɨɣ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɫɨɫɬɚɜɥɹɸɳɢɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɬɨɱɤɢ, ɚ

ɦɨɦɟɧɬ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɸɛɨɣ ɨɫɢ ɪɚɜɟɧ ɚɥɝɟɛ-

ɪɚɢɱɟɫɤɨɣ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɫɨɫɬɚɜɥɹɸɳɢɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɨɫɢ.

MO R ¦MO Fk .

k

Ʉ ɤɚɤɨɦɭ ɩɪɨɫɬɟɣɲɟɦɭ ɜɢɞɭ ɦɨɠɧɨ ɩɪɢɜɟɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɭɸ ɫɢɫɬɟɦɭ ɫɢɥ, ɟɫɥɢ ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɪɚɡɥɢɱɧɵɯ ɬɨɱɟɤ:

ɚ) ɢɦɟɟɬ ɨɞɧɨ ɢ ɬɨ ɠɟ ɡɧɚɱɟɧɢɟ, ɧɟ ɪɚɜɧɨɟ ɧɭɥɸ;

ɛ) ɪɚɜɟɧ ɧɭɥɸ;

ɜ) ɢɦɟɟɬ ɪɚɡɥɢɱɧɵɟ ɡɧɚɱɟɧɢɹ ɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɟɧ ɤ ɝɥɚɜɧɨɦɭ ɜɟɤɬɨɪɭ;

ɝ) ɢɦɟɟɬ ɪɚɡɥɢɱɧɵɟ ɡɧɚɱɟɧɢɹ ɢ ɧɟ ɩɟɪɩɟɧɞɢɤɭɥɹɪɟɧ ɝɥɚɜɧɨɦɭ ɜɟɤɬɨɪɭ?

ɚ) ȼ ɫɥɭɱɚɟ, ɟɫɥɢ Mo const , ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ P 0 ɢ ɫɢɥɵ ɩɪɢɜɨɞɹɬɫɹ ɤ ɩɚɪɟ ɫɢɥ ɫ ɦɨɦɟɧɬɨɦ ɪɚɜɧɵɦ ɝɥɚɜɧɨɦɭ ɦɨɦɟɧɬɭ ɡɚɞɚɧɧɵɯ ɫɢɥ ɨɬɧɨɫɢ-

ɬɟɥɶɧɨ ɰɟɧɬɪɚ ɩɪɢɜɟɞɟɧɢɹ.

ɜ) ȿɫɥɢ Mo A P , ɫɢɥɵ ɩɪɢɜɨɞɹɬɫɹ ɤ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ ɪɚɜɧɨɣ ɝɥɚɜɧɨɦɭ

12

ɝ) ȿɫɥɢ Mo ɧɟ ɩɟɪɩɟɧɞɢɤɭɥɹɪɟɧ P , ɫɢɫɬɟɦɭ ɫɢɥ ɦɨɠɧɨ ɩɪɢɜɟɫɬɢ ɤ ɫɢɥɨ-

ɜɨɦɭ ɜɢɧɬɭ – ɞɢɧɚɦɟ, ɩɪɟɞɫɬɚɜɥɹɸɳɟɣ ɫɨɛɨɣ ɫɨɜɨɤɭɩɧɨɫɬɶ ɫɢɥɵ ɢ ɩɚɪɵ ɫɢɥ ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɜ ɩɥɨɫɤɨɫɬɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɤ ɥɢɧɢɢ ɞɟɣɫɬɜɢɹ ɷɬɨɣ ɫɢɥɵ, ɫ ɦɨɦɟɧɬɨɦ ɪɚɜɧɵɦ:

ɚ) Ƚɥɚɜɧɵɣ ɜɟɤɬɨɪ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɫɢɥ ɢɧɜɚɪɢɚɧɬɟɧ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɰɟɧɬɪɭ ɩɪɢɜɟɞɟɧɢɹ.

ɛ) ɋɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɝɥɚɜɧɨɝɨ ɜɟɤɬɨɪɚ ɧɚ ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɫɢɥ ɢɧɜɚɪɢɚɧɬɧɨ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɰɟɧɬɪɭ ɩɪɢɜɟɞɟɧɢɹ:

P Mo const .

ɜ) ɉɪɨɟɤɰɢɹ ɝɥɚɜɧɨɝɨ ɦɨɦɟɧɬɚ ɫɢɫɬɟɦɵ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɸɛɨɝɨ ɰɟɧɬɪɚ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɝɥɚɜɧɨɝɨ ɜɟɤɬɨɪɚ ɟɫɬɶ ɜɟɥɢɱɢɧɚ ɩɨɫɬɨɹɧɧɚɹ:

ɪɚɦɟɬɪɨɦ ɜɢɧɬɚ ɢɥɢ ɞɢɧɚɦɵ.

Ʉɚɤɢɦ ɫɜɨɣɫɬɜɨɦ ɨɛɥɚɞɚɟɬ ɰɟɧɬɪ ɬɹɠɟɫɬɢ?

ɋɢɥɚ, ɫ ɤɨɬɨɪɨɣ ɬɟɥɨ ɩɪɢɬɹɝɢɜɚɟɬɫɹ ɤ Ɂɟɦɥɟ, ɧɚɡɵɜɚɟɬɫɹ ɰɟɧɬɪɨɦ ɬɹɠɟ-

ɫɬɢ. ɇɚɩɪɚɜɥɟɧɢɟ ɫɢɥ ɩɪɢɬɹɠɟɧɢɹ ɨɬɞɟɥɶɧɵɯ ɱɚɫɬɢɰ ɬɟɥɚ ɤ Ɂɟɦɥɟ ɩɪɚɤ-

ɬɢɱɟɫɤɢ ɩɚɪɚɥɥɟɥɶɧɵ ɦɟɠɞɭ ɫɨɛɨɣ. Ɋɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɷɬɢɯ ɩɚɪɚɥɥɟɥɶ-

13

ɧɵɯ ɫɢɥ, ɪɚɜɧɚɹ ɢɯ ɫɭɦɦɟ, ɟɫɬɶ ɜɟɫ ɬɟɥɚ, ɚ ɰɟɧɬɪ ɷɬɨɣ ɫɢɫɬɟɦɵ ɫɢɥ, ɜ ɤɨ-

ɬɨɪɨɦ ɩɪɢɥɨɠɟɧ ɜɟɫ ɬɟɥɚ, ɧɚɡɵɜɚɟɬɫɹ ɰɟɧɬɪɨɦ ɬɹɠɟɫɬɢ. ȼ ɬɜɟɪɞɨɦ ɬɟɥɟ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɪɚɫɩɨɥɨɠɟɧɢɹ ɬɟɥɚ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ.

ɉɨ ɤɚɤɢɦ ɮɨɪɦɭɥɚɦ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɨɞɧɨɪɨɞ-

ɧɨɝɨ ɬɟɥɚ?

Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ ɢ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɨɞɧɨɪɨɞɧɨɝɨ ɬɟɥɚ ɨɩɪɟɞɟ-

ɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ:

ɝɞɟ rk , xk , yk , zk — ɪɚɞɢɭɫ-ɜɟɤɬɨɪɵ ɢ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɨɜ ɬɹɠɟɫɬɢ ɨɬ-

ɞɟɥɶɧɵɯ ɱɚɫɬɟɣ ɬɟɥɚ.

ɉɨ ɤɚɤɢɦ ɮɨɪɦɭɥɚɦ ɨɩɪɟɞɟɥɹɸɬɫɹ ɤɨɨɪɞɢɧɚɬɵ ɨɛɴɟɦɚ ɬɟɥɚ, ɩɥɨɫɤɢɯ ɮɢ-

ɝɭɪ ɤ ɥɢɧɢɢ?

Ʉɨɨɪɞɢɧɚɬɵ ɨɛɴɺɦɚ ɬɟɥɚ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ:

Ʉɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɩɥɚɫɬɢɧɨɤ (ɩɥɨɫɤɢɯ ɮɢɝɭɪ) ɨɩɪɟɞɟɥɹɸɬɫɹ

ɩɨ ɮɨɪɦɭɥɚɦ:

1 1 1

rc S ³S r dS ɢɥɢ xc S ³S x dS , yc S ³S y dS .

Ʉɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɥɢɧɢɣ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ:

ɑɬɨ ɧɚɡɵɜɚɟɬɫɹ ɫɬɚɬɢɱɟɫɤɢɦ ɦɨɦɟɧɬɨɦ ɩɥɨɳɚɞɢ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɨɬɧɨ-

ɫɢɬɟɥɶɧɨ ɨɫɢ? Ʉɚɤ ɨɧ ɜɵɱɢɫɥɹɟɬɫɹ ɢ ɤɚɤɭɸ ɪɚɡɦɟɪɧɨɫɬɶ ɢɦɟɸɬ?

14

ɋɭɦɦɚ ɩɪɨɢɡɜɟɞɟɧɢɣ ɷɥɟɦɟɧɬɚɪɧɵɯ ɩɥɨɳɚɞɟɣ, ɜɯɨɞɹɳɢɯ ɜ ɫɨɫɬɚɜ ɩɥɨ-

ɳɚɞɢ ɮɢɝɭɪɵ, ɧɚ ɚɥɝɟɛɪɚɢɱɟɫɤɢɟ ɡɧɚɱɟɧɢɹ ɢɯ ɪɚɫɫɬɨɹɧɢɣ ɞɨ ɧɟɤɨɬɨɪɨɣ ɨɫɢ, ɧɚɡɵɜɚɟɬɫɹ ɫɬɚɬɢɱɟɫɤɢɦ ɦɨɦɟɧɬɨɦ ɩɥɨɳɚɞɢ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɨɬɧɨ-

ɫɢɬɟɥɶɧɨ ɷɬɨɣ ɨɫɢ.

Sx ¦yk Sk S yc ; Sy ¦xk Sk S xC

ɋɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɩɥɨɳɚɞɢ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɢɡɦɟ-

ɪɹɟɬɫɹ ɜ ɫɦ3 .

ɋɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɩɥɨɳɚɞɢ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨ-

ɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɮɢɝɭɪɵ, ɪɚɜɟɧ ɧɭɥɸ.

Ʉɚɤɢɦɢ ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɦɢ ɬɟɨɪɟɦɚɦɢ ɩɨɥɶɡɭɸɬɫɹ ɩɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɩɨ-

ɥɨɠɟɧɢɹ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ?

Ɍɟɨɪɟɦɚ 1. ȿɫɥɢ ɨɞɧɨɪɨɞɧɨɟ ɬɟɥɨ ɢɦɟɟɬ ɨɫɶ ɫɢɦɦɟɬɪɢɢ, ɬɨ ɰɟɧɬɪ ɬɹɠɟ-

ɫɬɢ ɬɟɥɚ ɧɚɯɨɞɢɬɫɹ ɧɚ ɷɬɨɣ ɨɫɢ.

Ɍɟɨɪɟɦɚ 2. ȿɫɥɢ ɨɞɧɨɪɨɞɧɨɟ ɬɟɥɨ ɢɦɟɟɬ ɩɥɨɫɤɨɫɬɶ ɫɢɦɦɟɬɪɢɢ, ɬɨ ɟɝɨ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɧɚɯɨɞɢɬɫɹ ɜ ɷɬɨɣ ɩɥɨɫɤɨɫɬɢ.

Ɍɟɨɪɟɦɚ 3. Ɉɛɴɺɦ ɬɟɥɚ ɜɪɚɳɟɧɢɹ, ɩɨɥɭɱɟɧɧɨɝɨ ɜɪɚɳɟɧɢɟɦ ɩɥɨɫɤɨɣ ɮɢ-

ɝɭɪɵ ɜɨɤɪɭɝ ɨɫɢ, ɥɟɠɚɳɟɣ ɜ ɩɥɨɫɤɨɫɬɢ ɮɢɝɭɪɵ, ɧɨ ɧɟ ɩɟɪɟɫɟɤɚɸɳɟɣ ɟɺ,

ɪɚɜɟɧ ɩɪɨɢɡɜɟɞɟɧɢɸ ɩɥɨɳɚɞɢ ɮɢɝɭɪɵ ɧɚ ɞɥɢɧɭ ɨɤɪɭɠɧɨɫɬɢ, ɨɩɢɫɚɧɧɨɣ ɟɺ ɰɟɧɬɪɨɦ ɬɹɠɟɫɬɢ.

Ɍɟɨɪɟɦɚ 4. ɉɥɨɳɚɞɶ ɩɨɜɟɪɯɧɨɫɬɢ ɜɪɚɳɟɧɢɹ, ɩɨɥɭɱɟɧɧɨɣ ɜɪɚɳɟɧɢɟɦ ɩɥɨɫɤɨɣ ɤɪɢɜɨɣ ɜɨɤɪɭɝ ɨɫɢ, ɥɟɠɚɳɟɣ ɜ ɩɥɨɫɤɨɫɬɢ ɷɬɨɣ ɤɪɢɜɨɣ, ɧɨ ɟɺ ɧɟ ɩɟɪɟɫɟɤɚɸɳɟɣ, ɪɚɜɧɚ ɩɪɨɢɡɜɟɞɟɧɢɸ ɞɥɢɧɵ ɷɬɨɣ ɤɪɢɜɨɣ ɧɚ ɞɥɢɧɭ ɨɤ-

ɪɭɠɧɨɫɬɢ, ɨɩɢɫɚɧɧɨɣ ɟɺ ɰɟɧɬɪɨɦ ɬɹɠɟɫɬɢ.

Ʉɚɤɢɦɢ ɫɩɨɫɨɛɚɦɢ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɩɥɨ-

ɳɚɞɢ ɜ ɫɥɭɱɚɟ, ɟɫɥɢ ɢɡɜɟɫɬɧɵ ɩɨɥɨɠɟɧɢɹ ɰɟɧɬɪɨɜ ɬɹɠɟɫɬɢ ɨɬɞɟɥɶɧɵɯ ɟɺ ɱɚɫɬɟɣ?

ɚ) Ɇɟɬɨɞ ɝɪɭɩɩɢɪɨɜɤɢ ɢɥɢ ɪɚɡɛɢɟɧɢɹ.

15

ȿɫɥɢ ɨɩɪɟɞɟɥɢɬɶ ɰɟɧɬɪɵ ɬɹɠɟɫɬɢ ɨɬɞɟɥɶɧɵɯ ɱɚɫɬɟɣ ɮɢɝɭɪɵ, ɬɨ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɨ ɮɨɪɦɭɥɚɦ:

ɛ) Ɇɟɬɨɞ ɨɬɪɢɰɚɬɟɥɶɧɵɯ ɩɥɨɳɚɞɟɣ.

ȿɫɥɢ ɜ ɩɥɚɫɬɢɧɟ ɢɦɟɟɬɫɹ ɨɬɜɟɪɫɬɢɟ, ɬɨ ɨɬɜɟɪɫɬɢɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɩɥɨɳɚɞɶ ɫ ɨɬɪɢɰɚɬɟɥɶɧɨɣ ɦɚɫɫɨɣ.

ɇɚɡɨɜɢɬɟ ɨɫɧɨɜɧɵɟ ɚɤɫɢɨɦɵ ɫɬɚɬɢɤɢ.

ɚ) ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɜɡɚɢɦɧɨ ɭɪɚɜɧɨɜɟɲɢɜɚɸɳɢɯɫɹ ɫɢɥ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱ-

ɤɚ (ɬɟɥɨ) ɧɚɯɨɞɢɬɫɹ ɜ ɫɨɫɬɨɹɧɢɢ ɩɨɤɨɹ ɢɥɢ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ.

ɛ) Ⱦɜɟ ɫɢɥɵ, ɩɪɢɥɨɠɟɧɧɵɟ ɤ ɬɜɺɪɞɨɦɭ ɬɟɥɭ, ɜɡɚɢɦɧɨ ɭɪɚɜɧɨɜɟɲɢɜɚɸɬɫɹ ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ ɢɯ ɦɨɞɭɥɢ ɪɚɜɧɵ, ɢ ɨɧɢ ɧɚɩɪɚɜɥɟɧɵ ɩɨ ɨɞɧɨɣ ɩɪɹɦɨɣ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɫɬɨɪɨɧɵ.

ɜ) ȿɫɥɢ ɤ ɬɜɺɪɞɨɦɭ ɬɟɥɭ, ɧɚɯɨɞɹɳɟɦɭɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɧɟɤɨɬɨɪɨɣ ɫɢɫɬɟ-

ɦɵ ɫɢɥ, ɩɪɢɥɨɠɢɬɶ ɭɪɚɜɧɨɜɟɲɟɧɧɭɸ ɫɢɫɬɟɦɭ ɢɥɢ ɢɫɤɥɸɱɢɬɶ ɬɚɤɭɸ ɫɢɫ-

ɬɟɦɭ ɫɢɥ, ɬɨ ɩɨɥɭɱɢɬɫɹ ɫɢɫɬɟɦɚ ɫɢɥ ɷɤɜɢɜɚɥɟɧɬɧɚɹ ɡɚɞɚɧɧɨɣ ɫɢɫɬɟɦɟ.

ɝ) Ɋɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɞɜɭɯ ɩɟɪɟɫɟɤɚɸɳɢɯɫɹ ɫɢɥ ɩɪɢɥɨɠɟɧɚ ɜ ɬɨɱɤɟ ɢɯ ɩɟɪɟɫɟɱɟɧɢɹ ɢ ɢɡɨɛɪɚɠɚɟɬɫɹ ɞɢɚɝɨɧɚɥɶɸ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ, ɩɨɫɬɪɨɟɧɧɨ-

ɝɨ ɧɚ ɷɬɢɯ ɫɢɥɚɯ.

ɞ) ȼɫɹɤɨɦɭ ɞɟɣɫɬɜɢɸ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɚɜɧɨɟ ɢ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɧɚɩɪɚɜ-

ɥɟɧɧɨɟ ɩɪɨɬɢɜɨɞɟɣɫɬɜɢɟ.

ɟ) Ɋɚɜɧɨɜɟɫɢɟ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɞɟɮɨɪɦɢɪɭɸɳɟɦɭɫɹ ɬɟɥɭ, ɫɨɯɪɚɧɹɟɬ-

ɫɹ ɩɪɢ ɟɝɨ ɡɚɬɜɟɪɞɟɜɚɧɢɢ.

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɪɨɟɤɰɢɹ ɫɢɥɵ ɧɚ ɨɫɶ?

ɉɪɨɟɤɰɢɹ ɫɢɥɵ F ɧɚ ɨɫɶ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɦɨɞɭɥɹ ɫɢɥɵ ɧɚ ɤɨɫɢɧɭɫ ɭɝɥɚ D ɦɟɠɞɭ ɩɨɥɨɠɢɬɟɥɶɧɵɦ ɧɚɩɪɚɜɥɟɧɢɹɦ ɨɫɢ ɢ ɫɢɥɵ

16

Fx F cosD .

— ɨɫɬɪɵɣ ɭɝɨɥ ɦɟɠɞɭ ɥɢɧɢɟɣ ɞɟɣɫɬɜɢɹ ɫɢɥɵ ɫ ɨɫɶɸ.

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɦɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɧɚ ɩɥɨɫɤɨɫɬɢ?

Ɇɨɦɟɧɬɨɦ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɢ Ɉ ɧɚ ɩɥɨɫɤɨɫɬɢ ɧɚɡɵɜɚ-

ɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟ ɦɨɞɭɥɹ ɫɢɥɵ ɧɟ ɟɺ ɩɥɟɱɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɬɨɱɤɢ,

ɜɡɹɬɨɟ ɫɨ ɡɧɚɤɨɦ ɩɥɸɫ ɢɥɢ ɦɢɧɭɫ:

Mo rFd .

ɉɥɟɱɨɦ d ɫɢɥɵ F ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ O ɧɚɡɵɜɚɸɬ ɞɥɢɧɭ ɩɟɪɩɟɧɞɢɤɭ-

ɥɹɪɚ, ɨɩɭɳɟɧɧɨɝɨ ɢɡ ɬɨɱɤɢ O ɧɚ ɥɢɧɢɸ ɞɟɣɫɬɜɢɹ ɫɢɥɵ.

Ɇɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ O ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɩɨɥɨɠɢɬɟɥɶɧɵɦ, ɟɫ-

ɥɢ ɫɢɥɚ F ɫɬɪɟɦɢɬɫɹ ɩɨɜɟɪɧɭɬɶ ɩɥɨɫɤɨɫɬɶ ɱɟɪɬɟɠɚ ɜɨɤɪɭɝ ɬɨɱɤɢ O ɜ

ɫɬɨɪɨɧɭ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɞɜɢɠɟɧɢɸ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ, ɢ ɨɬɪɢɰɚɬɟɥɶ-

ɧɵɦ – ɜ ɨɛɪɚɬɧɨɦ ɫɥɭɱɚɟ.

17

ȼɈɉɊɈɋɕ ɂ ɈɌȼȿɌɕ ɉɈ ɄɂɇȿɆȺɌɂɄȿ

Ʉɚɤɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɫɩɨɫɨɛɵ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɫɭɳɟɫɬɜɭɸɬ ɢ ɜ ɱɺɦ ɫɨɫɬɨɢɬ ɤɚɠɞɵɣ ɢɡ ɷɬɢɯ ɫɩɨɫɨɛɨɜ?

ɋɭɳɟɫɬɜɭɸɬ: ɟɫɬɟɫɬɜɟɧɧɵɣ, ɜɟɤɬɨɪɧɵɣ ɢ ɤɨɨɪɞɢɧɚɬɧɵɣ ɫɩɨɫɨɛɵ ɡɚɞɚ-

ɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ.

ɚ). ȿɫɬɟɫɬɜɟɧɧɵɣ ɫɩɨɫɨɛ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɩɪɢɦɟɧɹɟɬɫɹ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɬɪɚɟɤɬɨɪɢɹ ɬɨɱɤɢ ɡɚɪɚɧɟɟ ɢɡɜɟɫɬɧɚ (ɩɪɹɦɚɹ ɢɥɢ ɤɪɢɜɚɹ ɥɢɧɢɹ). ɉɨɥɨɠɟ-

ɧɢɟ ɞɜɢɠɭɳɟɣɫɹ ɬɨɱɤɢ ɧɚ ɬɪɚɟɤɬɨɪɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɞɭɝɨɜɨɣ ɤɨɨɪɞɢɧɚ-

ɬɨɣ, ɨɬɫɱɢɬɵɜɚɟɦɨɣ ɨɬ ɧɚɱɚɥɚ ɨɬɫɱɺɬɚ:

Sf (t).

ɛ). ɉɪɢ ɜɟɤɬɨɪɧɨɦ ɫɩɨɫɨɛɟ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɩɨɥɨɠɟɧɢɟ ɬɨɱɤɢ ɜ ɩɪɨ-

ɫɬɪɚɧɫɬɜɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɚɞɚɧɢɟɦ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ r , ɩɪɨɜɟɞɺɧɧɨɝɨ ɢɡ ɧɟɩɨɞɜɢɠɧɨɝɨ ɰɟɧɬɪɚ ɜ ɞɚɧɧɭɸ ɬɨɱɤɭ.

rr(t).

ɜ). ɉɪɢ ɤɨɨɪɞɢɧɚɬɧɨɦ ɫɩɨɫɨɛɟ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɩɨɥɨɠɟɧɢɟ ɬɨɱɤɢ ɜ ɞɟ-

ɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ Ɉɯ, Oɭ, Oz ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɪɟɦɹ ɤɨɨɪɞɢɧɚ-

ɬɚɦɢ.

x f1(t), y f2 (t), z f3 (t) .

ɑɟɦ ɹɜɥɹɟɬɫɹ ɬɪɚɟɤɬɨɪɢɹ ɬɨɱɤɢ ɩɪɢ ɜɟɤɬɨɪɧɨɦ ɫɩɨɫɨɛɟ ɡɚɞɚɧɢɹ ɞɜɢɠɟ-

ɧɢɹ ɬɨɱɤɢ?

Ɍɪɚɟɤɬɨɪɢɹ ɬɨɱɤɢ ɹɜɥɹɟɬɫɹ ɝɨɞɨɝɪɚɮɨɦ ɟɺ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ r .

Ʉɚɤ ɩɨ ɭɪɚɜɧɟɧɢɹɦ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɤɨɨɪɞɢɧɚɬɧɨɣ ɮɨɪɦɟ ɨɩɪɟɞɟɥɢɬɶ ɟɺ ɬɪɚɟɤɬɨɪɢɸ?

Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɭɪɚɜɧɟɧɢɹ ɬɪɚɟɤɬɨɪɢɢ ɧɟɨɛɯɨɞɢɦɨ ɢɫɤɥɸɱɢɬɶ ɢɡ ɭɪɚɜɧɟ-

ɧɢɣ ɞɜɢɠɟɧɢɹ ɩɚɪɚɦɟɬɪ t (ɜɪɟɦɹ).

ȿɫɥɢ ɞɜɢɠɟɧɢɟ ɬɨɱɤɢ ɜ ɩɥɨɫɤɨɫɬɢ ɡɚɞɚɧɨ ɭɪɚɜɧɟɧɢɹɦɢ:

18

xf1(t), y f2 (t).

ɬɨ, ɪɟɲɢɜ ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ t, ɩɨɥɭɱɢɦ t M (x) . ɉɨɞɫɬɚ-

ɜɢɜ t ɜɨ ɜɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ:

yf2 >M(x)@.

ɑɟɦɭ ɪɚɜɟɧ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɢ ɤɚɤɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɨɧ ɢɦɟɟɬ?

ɋɤɨɪɨɫɬɶ v – ɷɬɨ ɜɟɤɬɨɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɛɵɫɬɪɨɬɭ ɢ ɧɚ-

ɩɪɚɜɥɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɺɬɚ. ȼɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɪɚɜɟɧ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɬɨɱɤɢ ɩɨ ɜɪɟɦɟɧɢ:

vr .

ȼɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ v ɧɚɩɪɚɜɥɟɧ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɬɪɚɟɤɬɨɪɢɢ ɜ ɫɬɨ-

ɪɨɧɭ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ.

Ʉɚɤ ɫɜɹɡɚɧ ɨɪɬ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɤɪɢɜɨɣ ɫ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɨɦ ɞɜɢɠɭɳɟɣɫɹ ɬɨɱɤɢ?

Wd r . d s

ɑɟɦɭ ɪɚɜɧɚ ɩɪɨɟɤɰɢɹ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɧɚ ɤɚɫɚɬɟɥɶɧɭɸ ɤ ɟɺ ɬɪɚɟɤɬɨɪɢɢ ɢ ɦɨɞɭɥɶ ɟɺ ɫɤɨɪɨɫɬɢ?

ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɞɭɝɨɜɨɣ ɤɨɨɪɞɢɧɚɬɵ ɩɨ ɜɪɟɦɟɧɢ d s d t ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɪɨɟɤɰɢɸ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ v ɧɚ ɤɚɫɚɬɟɥɶɧɭɸ ɤ ɬɪɚɟɤɬɨɪɢɢ:

v W d s dt

Ɇɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɪɚɜɟɧ ɚɛɫɨɥɸɬɧɨɦɭ ɡɧɚɱɟɧɢɸ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɞɭɝɨɜɨɣ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɩɨ ɜɪɟɦɟɧɢ v s .

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Ʉɚɤ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɧɚ ɧɟɩɨɞɜɢɠɧɵɟ ɨɫɢ ɞɟɤɚɪ-

ɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ?

vx x ; vy y ; vz z .

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɥɢɱɢɧɚ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɩɪɢ ɤɨɨɪɞɢ-

ɧɚɬɧɨɦ ɫɩɨɫɨɛɟ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ?

ȼɟɥɢɱɢɧɚ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɱɟɪɟɡ ɟɝɨ ɩɪɨɟɤɰɢɢ:

ɇɚɩɪɚɜɥɟɧɢɟ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɚɩɪɚɜɥɹɸɳɢɦɢ ɤɨɫɢɧɭɫɚɦɢ:

ɑɟɦɭ ɪɚɜɟɧ ɜɟɤɬɨɪ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɢ ɤɚɤ ɨɧ ɧɚɩɪɚɜɥɟɧ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɝɨɞɨɝɪɚɮɭ ɫɤɨɪɨɫɬɢ?

ɍɫɤɨɪɟɧɢɟ a — ɷɬɨ ɜɟɤɬɨɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɛɵɫɬɪɨɬɭ ɢɡɦɟɧɟɧɢɹ ɦɨɞɭɥɹ ɢ ɧɚɩɪɚɜɥɟɧɢɹ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ.

ȼɟɤɬɨɪ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɪɚɜɟɧ ɩɟɪɜɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɫɤɨɪɨɫɬɢ ɢɥɢ ɜɬɨ-

ɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɬɨɱɤɢ ɩɨ ɜɪɟɦɟɧɢ.

av r .

Ʉɚɤ ɧɚɩɪɚɜɥɟɧɵ ɟɫɬɟɫɬɜɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɧɵɟ ɨɫɢ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɤɪɢɜɨɣ?

ȿɫɬɟɫɬɜɟɧɧɵɦɢ ɤɨɨɪɞɢɧɚɬɧɵɦɢ ɨɫɹɦɢ ɧɚɡɵɜɚɸɬɫɹ ɬɪɢ ɜɡɚɢɦɧɨ ɩɟɪɩɟɧ-

ɞɢɤɭɥɹɪɧɵɟ ɨɫɢ: ɤɚɫɚɬɟɥɶɧɚɹ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜ ɫɬɨɪɨɧɭ ɜɨɡɪɚɫɬɚɧɢɹ ɞɭɝɨ-

ɜɨɣ ɤɨɨɪɞɢɧɚɬɵ, ɝɥɚɜɧɚɹ ɧɨɪɦɚɥɶ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜ ɫɬɨɪɨɧɭ ɜɨɝɧɭɬɨɫɬɢ ɤɪɢɜɨɣ ɢ ɛɢɧɨɪɦɚɥɶ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɥɨɫɤɨɫɬɢ ɩɪɨɜɟ-

ɞɺɧɧɨɣ ɱɟɪɟɡ ɤɚɫɚɬɟɥɶɧɭɸ ɢ ɝɥɚɜɧɭɸ ɧɨɪɦɚɥɶ.

Ʉɚɤɨɜɵ ɜɟɥɢɱɢɧɚ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɜɟɤɬɨɪɚ ɤɪɢɜɢɡɧɵ F ɤɪɢɜɨɣ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ?

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