Теоретическая механика_в_Вопросах_и_Ответах

ȼɟɤɬɨɪ ɤɪɢɜɢɡɧɵ ɤɪɢɜɨɣ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ ɪɚɜɟɧ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɨɪɬɚ ɤɚ-

ɫɚɬɟɥɶɧɨɣ ɤ ɤɪɢɜɨɣ ɩɨ ɞɭɝɨɜɨɣ ɤɨɨɪɞɢɧɚɬɟ:

FdW . d t

ȼɟɤɬɨɪ ɤɪɢɜɢɡɧɵ F ɪɚɫɩɨɥɨɠɟɧ ɜ ɫɨɩɪɢɤɚɫɚɸɳɟɣɫɹ ɩɥɨɫɤɨɫɬɢ ɢ ɧɚɩɪɚɜ-

ɥɟɧ ɩɨ ɝɥɚɜɧɨɣ ɧɨɪɦɚɥɢ ɤ ɰɟɧɬɪɭ ɤɪɢɜɢɡɧɵ ɤɪɢɜɨɣ:

Fn 1 ,

U

ɝɞɟ U — ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɤɪɢɜɨɣ.

ȼ ɤɚɤɨɣ ɩɥɨɫɤɨɫɬɢ ɪɚɫɩɨɥɨɠɟɧɨ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɢ ɱɟɦɭ ɪɚɜɧɵ ɟɝɨ ɩɪɨɟɤ-

ɰɢɢ ɧɚ ɟɫɬɟɫɬɜɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɧɵɟ ɨɫɢ?

ȼɟɤɬɨɪ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɚ ɥɟɠɢɬ ɜ ɫɨɩɪɢɤɚɫɚɸɳɟɣɫɹ ɩɥɨɫɤɨɫɬɢ ɢ ɪɚɜɟɧ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɞɜɭɯ ɜɟɤɬɨɪɨɜ, ɨɞɢɧ ɢɡ ɤɨɬɨɪɵɯ ɧɚɩɪɚɜɥɟɧɧɵɣ ɩɨ ɝɥɚɜɧɨɣ ɧɨɪɦɚɥɢ, ɧɚɡɵɜɚɟɬɫɹ ɧɨɪɦɚɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ, ɚ ɞɪɭɝɨɣ, ɧɚ-

ɩɪɚɜɥɟɧ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ, ɧɚɡɵɜɚɟɬɫɹ ɤɚɫɚɬɟɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ.

aan aW .

ɉɪɨɟɤɰɢɹ ɭɫɤɨɪɟɧɢɹ ɧɚ ɝɥɚɜɧɭɸ ɧɨɪɦɚɥɶ ɪɚɜɧɚ ɤɜɚɞɪɚɬɭ ɦɨɞɭɥɹ ɫɤɨɪɨ-

ɫɬɢ ɬɨɱɤɢ ɧɚ ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ.

v2 an U .

ɉɪɨɟɤɰɢɹ ɭɫɤɨɪɟɧɢɹ ɧɚ ɤɚɫɚɬɟɥɶɧɭɸ ɪɚɜɧɚ ɩɟɪɜɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɚɥ-

ɝɟɛɪɚɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ ɫɤɨɪɨɫɬɢ ɢɥɢ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɞɭɝɨɜɨɣ ɤɨɨɪɞɢɧɚɬɵ ɩɨ ɜɪɟɦɟɧɢ

aW v s .

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɥɢɱɢɧɚ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɩɪɢ ɟɫɬɟɫɬ-

ɜɟɧɧɨɦ ɫɩɨɫɨɛɟ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ?

21

ȼɟɥɢɱɢɧɚ ɭɫɤɨɪɟɧɢɹ ɪɚɜɧɚ:

a an2 aW 2 .

ɇɚɩɪɚɜɥɟɧɢɟ ɭɫɤɨɪɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɝɥɨɦ E ɦɟɠɞɭ ɜɟɤɬɨɪɨɦ aGɢ

ɝɥɚɜɧɨɣ ɧɨɪɦɚɥɶɸ:

E arctg aW . an

ɑɬɨ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɨɛɨɣ ɤɚɫɚɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ?

Ʉɚɫɚɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɫɭɳɟɫɬɜɭɟɬ ɥɢɲɶ ɩɪɢ ɧɟɪɚɜɧɨɦɟɪɧɨɦ ɞɜɢɠɟɧɢɢ ɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɢɡɦɟɧɟɧɢɟ ɫɤɨɪɨɫɬɢ ɩɨ ɜɟɥɢɱɢɧɟ.

ɉɪɢ ɤɚɤɨɦ ɞɜɢɠɟɧɢɢ ɬɨɱɤɢ ɪɚɜɧɨ ɧɭɥɸ ɤɚɫɚɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ, ɢ ɩɪɢ ɤɚ-

ɤɨɦ – ɧɨɪɦɚɥɶɧɨɟ?

ȿɫɥɢ ɤɚɫɚɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɜɨ ɜɫɺ ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ ɩɨɫɬɨɹɧɧɨ aW const , ɬɨ ɞɜɢɠɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨɩɟɪɟɦɟɧɧɵɦ.

Ʉɚɤ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ ɧɚ ɧɟɩɨɞɜɢɠɧɵɟ ɨɫɢ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬ?

ɉɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ ɧɚ ɧɟɩɨɞɜɢɠɧɵɟ ɨɫɢ ɪɚɜɧɵ ɩɟɪɜɵɦ ɩɪɨɢɡɜɨɞɧɵɦ ɩɨ ɜɪɟɦɟɧɢ ɨɬ ɩɪɨɟɤɰɢɣ ɫɤɨɪɨɫɬɟɣ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɫɢ ɢɥɢ ɜɬɨɪɵɦ ɩɪɨɢɡɜɨɞɧɵɦ ɨɬ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɨɨɪɞɢɧɚɬ ɬɨɱɤɢ:

22

ax vx x , ay vy y , az vz z .

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɦɨɞɭɥɶ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɭɫɤɨɪɟɧɢɹ ɩɪɢ ɤɨɨɪɞɢɧɚɬɧɨɦ ɫɩɨ-

ɫɨɛɟ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ?

Ɇɨɞɭɥɶ ɭɫɤɨɪɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɱɟɪɟɡ ɩɪɨɟɤɰɢɢ:

a ax 2 ay 2 az 2 .

ɇɚɩɪɚɜɥɟɧɢɟ ɭɫɤɨɪɟɧɢɹ a ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɚɩɪɚɜɥɹɸɳɢɦɢ ɤɨɫɢɧɭɫɚɦɢ:

ɑɬɨ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɨɛɨɣ ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ?

ɇɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɫɭɳɟɫɬɜɭɟɬ ɥɢɲɶ ɩɪɢ ɤɪɢɜɨɥɢɧɟɣɧɨɦ ɞɜɢɠɟɧɢɢ

ɢɯɚɪɚɤɬɟɪɢɡɭɟɬ ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɚɜɥɟɧɢɹ ɫɤɨɪɨɫɬɢ.

ȼɤɚɤɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɜ ɤɪɢɜɨɥɢɧɟɣɧɨɦ ɞɜɢɠɟ-

ɧɢɢ ɦɨɠɟɬ ɨɛɪɚɬɢɬɶɫɹ ɜ ɧɭɥɶ?

ɇɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɦɨɠɟɬ ɛɵɬɶ ɪɚɜɧɨ ɧɭ-

ɥɸ an 0 ɜ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ ɨɛɪɚɳɚ-

ɟɬɫɹ ɜ ɧɭɥɶ (ɬɨɱɤɚ ɦɟɧɹɟɬ ɧɚɩɪɚɜɥɟɧɢɟ ɞɜɢɠɟɧɢɹ), ɢɥɢ, ɤɨɝɞɚ ɞɜɢɠɭɳɚɹ-

ɫɹ ɬɨɱɤɚ ɧɚɯɨɞɢɬɫɹ ɜ ɬɨɱɤɟ ɩɟɪɟɝɢɛɚ ɫɜɨɟɣ ɬɪɚɟɤɬɨɪɢɢ U f.

ȼ ɤɚɤɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɤɚɫɚɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɜ ɧɟɪɚɜɧɨɦɟɪɧɨɦ ɞɜɢ-

ɠɟɧɢɢ ɦɨɠɟɬ ɨɛɪɚɬɢɬɶɫɹ ɜ ɧɭɥɶ?

ȿɫɥɢ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ aW 0 , ɬɨ ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɜɟɥɢɱɢɧɚ ɫɤɨ-

ɪɨɫɬɢ ɞɨɫɬɢɝɚɟɬ ɦɚɤɫɢɦɭɦɚ ɢɥɢ ɦɢɧɢɦɭɦɚ.

ȿɫɥɢ aW 0 ɜ ɬɟɱɟɧɢɟ ɧɟɤɨɬɨɪɨɝɨ ɩɪɨɦɟɠɭɬɤɚ ɜɪɟɦɟɧɢ, ɬɨ ɧɚ ɷɬɨɦ ɢɧ-

ɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ ɱɢɫɥɟɧɧɚɹ ɜɟɥɢɱɢɧɚ ɫɤɨɪɨɫɬɢ ɩɨɫɬɨɹɧɧɚ ɢ ɞɜɢɠɟɧɢɟ ɹɜɥɹɟɬɫɹ ɪɚɜɧɨɦɟɪɧɵɦ ɤɪɢɜɨɥɢɧɟɣɧɵɦ, ɚ ɭɫɤɨɪɟɧɢɟ ɧɚɩɪɚɜɥɟɧɨ ɩɨ ɝɥɚɜɧɨɣ ɧɨɪɦɚɥɢ.

23

ɑɟɦ ɨɬɥɢɱɚɟɬɫɹ ɝɪɚɮɢɤ ɩɭɬɢ ɨɬ ɝɪɚɮɢɤɚ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ?

Ƚɪɚɮɢɤɨɦ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɧɚɡɵɜɚɟɬɫɹ ɝɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɟɺ ɞɭɝɨɜɨɣ ɤɨɨɪɞɢɧɚɬɵ s ɨɬ ɜɪɟɦɟɧɢ t .

ɉɭɬɶ, ɩɪɨɣɞɟɧɧɵɣ ɬɨɱɤɨɣ ɡɚ ɧɟɤɨɬɨɪɵɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ, ɩɪɟɞɫɬɚɜ-

ɥɹɟɬ ɫɨɛɨɣ ɫɭɦɦɭ ɚɛɫɨɥɸɬɧɵɯ ɡɧɚɱɟɧɢɣ ɷɥɟɦɟɧɬɚɪɧɵɯ ɩɟɪɟɦɟɳɟɧɢɣ ɡɚ ɷɬɨɬ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ, ɬ. ɟ. ɥɢɧɢɹ ɷɬɨɝɨ ɝɪɚɮɢɤɚ ɧɟɩɪɟɪɵɜɧɨ ɩɨɞɧɢ-

ɦɚɟɬɫɹ ɜɜɟɪɯ ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɧɚɩɪɚɜɥɟɧɢɹ ɞɜɢɠɟɧɢɹ.

Ʉɚɤ ɩɨ ɝɪɚɮɢɤɭ ɞɜɢɠɟɧɢɹ ɨɩɪɟɞɟɥɢɬɶ ɚɥɝɟɛɪɚɢɱɟɫɤɭɸ ɜɟɥɢɱɢɧɭ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ?

Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫɥɟɞɭɟɬ ɩɪɨ-

ɜɟɫɬɢ ɤɚɫɚɬɟɥɶɧɭɸ ɤ ɝɪɚɮɢɤɭ ɞɜɢɠɟɧɢɹ ɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɬɨɱɤɟ, ɨɩɪɟ-

ɞɟɥɢɬɶ ɭɝɨɥ D ɧɚɤɥɨɧɚ ɷɬɨɣ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɨɫɢ t ɢ ɨɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ v tg D .

Ʉɚɤ ɩɨ ɝɪɚɮɢɤɭ ɫɤɨɪɨɫɬɢ ɨɩɪɟɞɟɥɢɬɶ ɚɥɝɟɛɪɚɢɱɟɫɤɭɸ ɜɟɥɢɱɢɧɭ ɤɚɫɚɬɟɥɶ-

ɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ?

Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɚɫɚɬɟɥɶɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɫɥɟɞɭɟɬ ɩɪɨɜɟɫɬɢ ɤɚɫɚ-

ɬɟɥɶɧɭɸ ɤ ɝɪɚɮɢɤɭ ɫɤɨɪɨɫɬɢ ɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɬɨɱɤɟ ɢ ɧɚɣɬɢ ɭɝɨɥ ɧɚ-

ɤɥɨɧɚ E ɷɬɨɣ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɨɫɢ t . Ɍɚɧɝɟɧɫ ɭɝɥɚ E ɨɩɪɟɞɟɥɹɟɬ ɚɥɝɟɛ-

ɪɚɢɱɟɫɤɭɸ ɜɟɥɢɱɢɧɭ ɤɚɫɚɬɟɥɶɧɨɝɨ ɭɫɤɨɪɟɧɢɹ:

aW tg E .

ȼ ɤɚɤɨɦ ɫɥɭɱɚɟ ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɜ ɬɟɱɟɧɢɟ ɧɟɤɨɬɨɪɨɝɨ ɩɪɨɦɟɠɭɬ-

ɤɚ ɜɪɟɦɟɧɢ ɦɨɠɟɬ ɛɵɬɶ ɪɚɜɧɨ ɧɭɥɸ.

ɉɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ ɦɨɠɟɬ ɛɵɬɶ ɪɚɜɧɨ ɧɭɥɸ a 0, ɤɨɝɞɚ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɺɬɚ ɪɚɜɧɨɦɟɪɧɨ ɢ ɩɪɹɦɨɥɢɧɟɣɧɨ.

ɇɚɡɨɜɢɬɟ ɨɫɧɨɜɧɵɟ ɜɢɞɵ ɞɜɢɠɟɧɢɹ ɬɜɺɪɞɨɝɨ ɬɟɥɚ.

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Ɋɚɡɥɢɱɚɸɬ ɩɹɬɶ ɜɢɞɨɜ ɞɜɢɠɟɧɢɹ ɬɜɺɪɞɨɝɨ ɬɟɥɚ: ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ, ɜɪɚɳɚ-

ɬɟɥɶɧɨɟ, ɩɥɨɫɤɨɩɚɪɚɥɥɟɥɶɧɨɟ (ɩɥɨɫɤɨɟ), ɫɮɟɪɢɱɟɫɤɨɟ ɢ ɨɛɳɢɣ ɫɥɭɱɚɣ ɞɜɢɠɟɧɢɹ ɬɜɺɪɞɨɝɨ ɬɟɥɚ.

Ʉɚɤɨɟ ɞɜɢɠɟɧɢɟ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɧɚɡɵɜɚɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɵɦ?

ɉɨɫɬɭɩɚɬɟɥɶɧɵɦ ɞɜɢɠɟɧɢɟɦ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɧɚɡɵɜɚɟɬɫɹ ɬɚɤɨɟ ɞɜɢɠɟɧɢɟ,

ɩɪɢ ɤɨɬɨɪɨɦ ɥɸɛɚɹ ɩɪɹɦɚɹ, ɫɨɟɞɢɧɹɸɳɚɹ ɞɜɟ ɬɨɱɤɢ ɬɟɥɚ, ɞɜɢɠɟɬɫɹ ɩɚ-

ɪɚɥɥɟɥɶɧɨ ɫɚɦɨɣ ɫɟɛɟ.

ȼɫɟ ɬɨɱɤɢ ɬɜɺɪɞɨɝɨ ɬɟɥɚ, ɞɜɢɠɭɳɟɝɨɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ, ɨɩɢɫɵɜɚɸɬ ɬɨɠ-

ɞɟɫɬɜɟɧɧɵɟ ɢ ɩɚɪɚɥɥɟɥɶɧɵɟ ɦɟɠɞɭ ɫɨɛɨɣ ɬɪɚɟɤɬɨɪɢɢ ɢ ɜ ɤɚɠɞɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɢɦɟɸɬ ɝɟɨɦɟɬɪɢɱɟɫɤɢ ɪɚɜɧɵɟ ɫɤɨɪɨɫɬɢ ɢ ɭɫɤɨɪɟɧɢɹ.

Ʉɚɤɨɟ ɞɜɢɠɟɧɢɟ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɧɚɡɵɜɚɟɬɫɹ ɜɪɚɳɚɬɟɥɶɧɵɦ?

ȼɪɚɳɚɬɟɥɶɧɵɦ ɧɚɡɵɜɚɟɬɫɹ ɬɚɤɨɟ ɞɜɢɠɟɧɢɟ ɬɜɺɪɞɨɝɨ ɬɟɥɚ, ɩɪɢ ɤɨɬɨɪɨɦ ɨɫɬɚɸɬɫɹ ɧɟɩɨɞɜɢɠɧɵɦɢ ɜɫɟ ɟɝɨ ɬɨɱɤɢ, ɥɟɠɚɳɢɟ ɧɚ ɩɪɹɦɨɣ, ɧɚɡɵɜɚɟɦɨɣ ɨɫɶɸ ɜɪɚɳɟɧɢɹ.

ɉɪɢ ɷɬɨɦ ɜɫɟ ɨɫɬɚɥɶɧɵɟ ɬɨɱɤɢ ɞɜɢɠɭɬɫɹ ɜ ɩɥɨɫɤɨɫɬɹɯ, ɩɟɪɩɟɧɞɢɤɭɥɹɪ-

ɧɵɯ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɢ ɨɩɢɫɵɜɚɸɬ ɨɤɪɭɠɧɨɫɬɢ, ɰɟɧɬɪɵ ɤɨɬɨɪɵɯ ɥɟɠɚɬ ɧɚ ɷɬɨɣ ɨɫɢ.

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨɥɨɠɟɧɢɟ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɟɥ?

ɉɨɥɨɠɟɧɢɟ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɟɥɚ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɝɥɨɦ ɩɨɜɨɪɨɬɚ M , ɹɜɥɹɸɳɟɝɨɫɹ ɮɭɧɤɰɢɟɣ ɜɪɟɦɟɧɢ t .

Mf (t).

ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɭɪɚɜɧɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟ-

ɧɢɹ ɬɟɥɚ.

Ʉɚɤɚɹ ɜɟɥɢɱɢɧɚ ɧɚɡɵɜɚɟɬɫɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ?

ȼɟɥɢɱɢɧɚ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɛɵɫɬɪɨɬɭ ɢɡɦɟɧɟɧɢɹ ɭɝɥɚ ɩɨɜɨɪɨɬɚ M ɫ ɬɟ-

ɱɟɧɢɟɦ ɜɪɟɦɟɧɢ, ɧɚɡɵɜɚɟɬɫɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ ɬɟɥɚ

25

ZdM M . dt

Ʉɚɤɚɹ ɜɟɥɢɱɢɧɚ ɧɚɡɵɜɚɟɬɫɹ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ?

Ⱥɥɝɟɛɪɚɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɛɵɫɬɪɨɬɭ ɢɡɦɟɧɟɧɢɹ ɭɝɥɨ-

ɜɨɣ ɫɤɨɪɨɫɬɢ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ, ɧɚɡɵɜɚɟɬɫɹ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ ɬɟɥɚ

Ʉɚɤɨɟ ɜɪɚɳɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨɩɟɪɟɦɟɧɧɵɦ?

ȼɪɚɳɟɧɢɟ ɬɟɥɚ, ɩɪɢ ɤɨɬɨɪɨɦ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɩɨɫɬɨɹɧɧɨ H const ,

ɧɚɡɵɜɚɸɬ ɪɚɜɧɨɩɟɪɟɦɟɧɧɵɦ ɜɪɚɳɟɧɢɟɦ. ɉɪɢ ɷɬɨɦ ɟɫɥɢ ɚɛɫɨɥɸɬɧɚɹ ɜɟ-

ɥɢɱɢɧɚ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɭɜɟɥɢɱɢɜɚɟɬɫɹ, ɜɪɚɳɟɧɢɟ ɧɚɡɵɜɚɸɬ ɪɚɜɧɨɭɫ-

ɤɨɪɟɧɧɵɦ, ɚ ɟɫɥɢ ɭɦɟɧɶɲɚɟɬɫɹ — ɪɚɜɧɨɡɚɦɟɞɥɟɧɧɵɦ

M M0 Z0 t r H t2 . 2

Ʉɚɤɢɦɨɛɪɚɡɨɦɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹɜɪɚɳɟɧɢɟɬɜɺɪɞɨɝɨɬɟɥɚɜɟɤɬɨɪɚɦɢ Z ɢ H ?

ɋɨɫɬɨɹɧɢɟ ɞɜɢɠɟɧɢɹ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɯɚ-

ɪɚɤɬɟɪɢɡɭɟɬɫɹ ɜɟɤɬɨɪɨɦ Z, ɧɚɩɪɚɜɥɟɧɧɵɦ ɩɨ ɨɫɢ ɜɪɚɳɟɧɢɹ ɜ ɬɭ ɫɬɨɪɨ-

ɧɭ, ɨɬɤɭɞɚ ɜɪɚɳɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɩɪɨɢɫɯɨɞɹɳɢɦ ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ.

ɍɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɦɨɠɧɨ ɢɡɨɛɪɚɡɢɬɶ ɜ ɜɢɞɟ ɜɟɤɬɨɪɚ H Z , ɧɚɩɪɚɜɥɟɧɧɨɝɨ ɜɞɨɥɶ ɨɫɢ ɜɪɚɳɟɧɢɹ. ɉɪɢ ɷɬɨɦ ɧɚ-

ɩɪɚɜɥɟɧɢɟ H ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ Z, ɤɨɝɞɚ ɬɟɥɨ ɜɪɚɳɚɟɬɫɹ ɭɫɤɨ-

ɪɟɧɧɨ, ɢ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ Z, ɤɨɝɞɚ ɜɪɚɳɟɧɢɟ ɹɜɥɹɟɬɫɹ ɡɚɦɟɞɥɟɧɧɵɦ.

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɟɥɚ?

26

Ɇɨɞɭɥɶ ɜɪɚɳɚɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɪɚɜɟɧ ɩɪɨɢɡɜɟɞɟ-

ɧɢɸ ɤɪɚɬɱɚɣɲɟɝɨ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɬɨɱɤɢ ɞɨ ɨɫɢ ɜɪɚɳɟɧɢɹ ɧɚ ɭɝɥɨɜɭɸ ɫɤɨ-

ɪɨɫɬɶ ɬɟɥɚ:

v Z h.

ȼɟɤɬɨɪ ɜɪɚɳɚɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɢ v ɧɚɩɪɚɜɥɟɧ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɪɚɞɢɭɫɭ ɜ ɫɬɨɪɨɧɭ ɜɪɚɳɟɧɢɹ.

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ v ɩɪɢ ɩɨɦɨɳɢ ɮɨɪɦɭɥɵ ɗɣɥɟɪɚ?

ȼɪɚɳɚɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ ɪɚɜɧɚ ɜɟɤɬɨɪɧɨɦɭ ɩɪɨɢɡɜɟɞɟɧɢɸ ɜɟɤɬɨɪɚ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɬɟɥɚ ɧɚ ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɷɬɨɣ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ:

vZ ur .

Ʉɚɤ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ ɩɨ ɮɨɪɦɭ-

ɥɚɦ ɗɣɥɟɪɚ, ɟɫɥɢ ɨɫɶɸ ɜɪɚɳɟɧɢɹ ɹɜɥɹɟɬɫɹ ɨɫɶ z ? vx Z y , vy Z x , vz 0,

ɝɞɟ x, y - ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ.

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɜɺɪɞɨɝɨ ɬɟɥɚ?

ɍɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɪɚɜɧɨ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɧɨɪɦɚɥɶɧɨɝɨ ɢ ɤɚɫɚɬɟɥɶɧɨɝɨ ɭɫɤɨɪɟɧɢɣ:

aan aW .

Ʉɚɤ ɨɩɪɟɞɟɥɹɸɬɫɹ ɜɟɥɢɱɢɧɵ ɧɨɪɦɚɥɶɧɨɝɨ ɢ ɤɚɫɚɬɟɥɶɧɨɝɨ ɭɫɤɨɪɟɧɢɣ ɬɨɱɤɢ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɟɥɚ?

Ɇɨɞɭɥɶ ɧɨɪɦɚɥɶɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɪɚɜɟɧ ɩɪɨɢɡɜɟɞɟɧɢɸ ɤɪɚɬɱɚɣɲɟɝɨ ɪɚɫ-

ɫɬɨɹɧɢɹ (ɪɚɞɢɭɫɚ) ɨɬ ɬɨɱɤɢ ɞɨ ɨɫɢ ɜɪɚɳɟɧɢɹ ɧɚ ɤɜɚɞɪɚɬ ɩɨɥɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɬɟɥɚ:

27

Ɇɨɞɭɥɶ ɤɚɫɚɬɟɥɶɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɪɚɜɟɧ ɩɪɨɢɡɜɟɞɟɧɢɸ ɤɪɚɬɱɚɣɲɟɝɨ ɪɚɫ-

ɫɬɨɹɧɢɹ (ɪɚɞɢɭɫɚ) ɨɬ ɬɨɱɤɢ ɞɨ ɨɫɢ ɜɪɚɳɟɧɢɹ ɧɚ ɚɛɫɨɥɸɬɧɨɟ ɡɧɚɱɟɧɢɟ ɭɝ-

ɥɨɜɨɝɨ ɭɫɤɨɪɟɧɢɹ ɬɟɥɚ:

aW H h .

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɥɢɱɢɧɚ ɩɨɥɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɜɺɪ-

ɞɨɝɨ ɬɟɥɚ?

Ɇɨɞɭɥɶ ɩɨɥɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:

ɇɚɩɪɚɜɥɟɧɢɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɝɥɨɦ E , ɫɨɫɬɚɜɥɟɧɧɵɦ ɩɨɥɧɵɦ ɭɫɤɨɪɟɧɢɟɦ a ɫ ɪɚɞɢɭɫɨɦ ɨɤɪɭɠɧɨɫɬɢ, ɬɚɧɝɟɧɫ ɤɨɬɨɪɨɝɨ:

ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɭɫɤɨɪɟɧɢɟ aG ɧɚɩɪɚɜɥɟɧɨ ɩɨ ɪɚɞɢɭɫɭ ɤ ɰɟɧɬɪɭ ɨɤɪɭɠɧɨɫɬɢ,

ɨɩɢɫɵɜɚɟɦɨɣ ɬɨɱɤɨɣ.

Ʉɚɤ ɨɩɪɟɞɟɥɹɸɬɫɹ ɜɟɤɬɨɪɵ ɤɚɫɚɬɟɥɶɧɨɝɨ aW ɢ ɧɨɪɦɚɥɶɧɨɝɨ an ɭɫɤɨɪɟɧɢɣ?

Ʉɚɫɚɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɪɚɜɧɨ ɜɟɤ-

ɬɨɪɧɨɦɭ ɩɪɨɢɡɜɟɞɟɧɢɸ ɜɟɤɬɨɪɚ ɭɝɥɨɜɨɝɨ ɭɫɤɨɪɟɧɢɹ ɬɟɥɚ ɧɚ ɪɚɞɢɭɫ-

ɜɟɤɬɨɪ ɷɬɨɣ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ:

aW H ur .

ɇɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɪɚɜɧɨ ɜɟɤ-

ɬɨɪɧɨɦɭ ɩɪɨɢɡɜɟɞɟɧɢɸ ɜɟɤɬɨɪɚ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɬɟɥɚ ɧɚ ɜɪɚɳɚɬɟɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɷɬɨɣ ɬɨɱɤɢ:

28

an Z uv Z u Z ur .

Ʉɚɤ ɜɵɱɢɫɥɹɸɬɫɹ ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɬɟɥɚ, ɜɪɚɳɚɸɳɟɝɨɫɹ ɜɨɤɪɭɝ

ɨɫɢ z ?

ax H y Z2 x, ay Hx Z2 y, az 0.

Ʉɚɤɨɟ ɞɜɢɠɟɧɢɟ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɧɚɡɵɜɚɟɬɫɹ ɩɥɨɫɤɢɦ?

ɉɥɨɫɤɨɩɚɪɚɥɥɟɥɶɧɵɦ (ɢɥɢ ɩɥɨɫɤɢɦ) ɞɜɢɠɟɧɢɟɦ ɬɜɺɪɞɨɝɨ ɬɟɥɚ ɧɚɡɵɜɚɟɬ-

ɫɹ ɬɚɤɨɟ ɞɜɢɠɟɧɢɟ, ɩɪɢ ɤɨɬɨɪɨɦ ɜɫɟ ɬɨɱɤɢ ɬɟɥɚ ɞɜɢɠɭɬɫɹ ɩɚɪɚɥɥɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɨɣ (ɨɫɧɨɜɧɨɣ) ɩɥɨɫɤɨɫɬɢ.

ɇɚɡɨɜɢɬɟ ɨɫɧɨɜɧɵɟ ɜɢɞɵ ɞɜɢɠɟɧɢɹ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ.

ɉɨɥɨɠɟɧɢɟ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɪɟɦɹ ɩɚɪɚɦɟɬɪɚɦɢ: ɤɨɨɪɞɢ-

ɧɚɬɚɦɢ ɩɨɥɸɫɚ xc f1(t), yc f2 (t) ɢ ɭɝɥɨɦ ɩɨɜɨɪɨɬɚ ɨɬɧɨɫɢɬɟɥɶɧɨ

ɩɨɥɸɫɚ Mc f3 (t).

Ɉɫɧɨɜɧɵɦɢ ɜɢɞɚɦɢ ɞɜɢɠɟɧɢɹ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɹɜɥɹɸɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɜɦɟɫɬɟ ɫ ɩɨɥɸɫɨɦ ɫ ɢ ɜɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɥɸɫɚ. ɉɪɢɱɺɦ, ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɡɚɜɢɫɢɬ ɨɬ ɜɵɛɨɪɚ ɩɨɥɸɫɚ, ɚ

ɜɪɚɳɚɬɟɥɶɧɨɟ ɨɬ ɜɵɛɨɪɚ ɩɨɥɸɫɚ ɧɟ ɡɚɜɢɫɢɬ.

Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɤɨɪɨɫɬɶ ɥɸɛɨɣ ɬɨɱɤɢ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɩɪɢ ɪɚɡɥɨɠɟɧɢɹ

ɩɥɨɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɧɚ ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ ɢ ɜɪɚɳɚɬɟɥɶɧɨɟ?

ɋɤɨɪɨɫɬɶ ɥɸɛɨɣ ɬɨɱɤɢ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɪɚɜɧɚ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɫɤɨɪɨɫɬɢ ɩɨɥɸɫɚ vc ɢ ɜɪɚɳɚɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɷɬɨɣ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɥɸɫɚ vMC :

vM vC vMC .

ȼɟɤɬɨɪ vMC Z uMC ɩɟɪɩɟɧɞɢɤɭɥɹɪɟɧ ɩɪɹɦɨɣ (MC ), ɫɨɟɞɢɧɹɸɳɟɣ ɬɨɱɤɭ M ɫ ɩɨɥɸɫɨɦ C .

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ɋɤɨɪɨɫɬɶ ɬɨɱɤɢ ɢɡɨɛɪɚɠɚɟɬɫɹ ɞɢɚɝɨɧɚɥɶɸ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ, ɩɨɫɬɪɨɟɧ-

ɧɨɝɨ ɩɪɢ ɬɨɱɤɟ ɧɚ ɫɤɨɪɨɫɬɢ ɩɨɥɸɫɚ, ɩɟɪɟɧɟɫɺɧɧɨɣ ɜ ɬɨɱɤɭ, ɢ ɜɪɚɳɚɬɟɥɶ-

ɧɨɣ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɜɨɤɪɭɝ ɩɨɥɸɫɚ.

ɇɚɡɨɜɢɬɟ ɫɥɟɞɫɬɜɢɹ ɢɡ ɬɟɨɪɟɦɵ ɨ ɫɤɨɪɨɫɬɹɯ ɬɨɱɟɤ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ.

ɋɥɟɞɫɬɜɢɟ 1. ɉɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɟɣ ɬɨɱɟɤ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɧɚ ɩɪɹɦɭɸ ɢɯ ɫɨɟɞɢɧɹɸɳɭɸ ɪɚɜɧɵ ɦɟɠɞɭ ɫɨɛɨɣ.

ɋɥɟɞɫɬɜɢɟ 2. Ʉɨɧɰɵ ɫɤɨɪɨɫɬɟɣ ɬɨɱɟɤ ɧɟɢɡɦɟɧɹɟɦɨɝɨ ɨɬɪɟɡɤɚ ɥɟɠɚɬ ɧɚ ɨɞɧɨɣ ɩɪɹɦɨɣ ɢ ɞɟɥɹɬ ɷɬɭ ɩɪɹɦɭɸ ɧɚ ɱɚɫɬɢ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵɟ ɪɚɫɫɬɨɹ-

ɧɢɹɦ ɦɟɠɞɭ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɬɨɱɤɚɦɢ.

Ʉɚɤɭɸ ɬɨɱɤɭ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɧɚɡɵɜɚɸɬ ɦɝɧɨɜɟɧɧɵɦ ɰɟɧɬɪɨɦ ɫɤɨɪɨɫɬɟɣ?

Ɍɨɱɤɚ P ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ, ɫɤɨɪɨɫɬɶ ɤɨɬɨɪɨɣ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɜɧɚ ɧɭɥɸ, ɧɚɡɵɜɚɟɬɫɹ ɦɝɧɨɜɟɧɧɵɦ ɰɟɧɬɪɨɦ ɫɤɨɪɨɫɬɟɣ (Ɇɐɋ). Ɇɝɧɨ-

ɜɟɧɧɵɣ ɰɟɧɬɪ ɫɤɨɪɨɫɬɟɣ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɧɚɯɨɞɢɬɫɹ ɧɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɟ

ɤ ɧɚɩɪɚɜɥɟɧɢɸ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ vɆ , ɧɚ ɪɚɫɫɬɨɹɧɢɢ ɨɬ ɬɨɱɤɢ, ɪɚɜɧɨɦ vM .

Z

Ʉɚɤ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɤɨɪɨɫɬɢ ɬɨɱɟɤ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɩɪɢ ɩɨɦɨɳɢ ɦɝɧɨɜɟɧ-

ɧɨɝɨ ɰɟɧɬɪɚ ɫɤɨɪɨɫɬɟɣ?

ɋɤɨɪɨɫɬɶ ɥɸɛɨɣ ɬɨɱɤɢ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɩɪɟɞ-

ɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɜɪɚɳɚɬɟɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɷɬɨɣ ɬɨɱɤɢ ɜɨɤɪɭɝ ɦɝɧɨɜɟɧɧɨɝɨ ɰɟɧɬɪɚ ɫɤɨɪɨɫɬɟɣ:

vM PM Z, vM A PM ,

ɬ. ɟ. ɫɤɨɪɨɫɬɶ ɥɸɛɨɣ ɬɨɱɤɢ ɩɥɨɫɤɨɣ ɮɢɝɭɪɵ ɢɦɟɟɬ ɦɨɞɭɥɶ, ɪɚɜɧɵɣ ɩɪɨ-

ɢɡɜɟɞɟɧɢɸ ɦɝɧɨɜɟɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɮɢɝɭɪɵ ɧɚ ɞɥɢɧɭ ɨɬɪɟɡɤɚ, ɫɨ-

ɟɞɢɧɹɸɳɟɝɨ ɬɨɱɤɭ ɫ ɦɝɧɨɜɟɧɧɵɦ ɰɟɧɬɪɨɦ ɫɤɨɪɨɫɬɟɣ, ɢ ɧɚɩɪɚɜɥɟɧɚ ɩɟɪ-

ɩɟɧɞɢɤɭɥɹɪɧɨ ɷɬɨɦɭ ɨɬɪɟɡɤɭ ɜ ɫɬɨɪɨɧɭ ɜɪɚɳɟɧɢɹ ɮɢɝɭɪɵ.

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