Механика.Методика решения задач

ɂɡɨɛɪɚɡɢɦ ɧɚ ɪɢɫɭɧɤɚɯ ɪɚɫɩɨɥɨɠɟɧɢɟ ɱɚɫɨɜ ɨɛɟɢɯ ɫɢɫɬɟɦ ɢ ɩɨɥɨɠɟɧɢɟ ɫɬɪɟɥɨɤ ɷɬɢɯ ɱɚɫɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɪɚɡɥɢɱɧɵɯ ɫɢɫɬɟɦ ɨɬ-

ɫɜɨɢɯ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ ɪɚɡɥɢɱɧɵ.

ɇɚ ɪɢɫ 5.5 ɪɚɫɩɨɥɨɠɟɧɢɟ ɱɚɫɨɜ ɢ ɩɨɥɨɠɟɧɢɟ ɢɯ ɫɬɪɟɥɨɤ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɫɥɭɱɚɸ, ɤɨɝɞɚ ɧɚɛɥɸɞɚɬɟɥɶ ɧɚɯɨɞɢɬɫɹ ɜ ɫɢɫɬɟɦɟ S, ɚ ɧɚ ɪɢɫ. 5.6 – ɤɨɝɞɚ ɧɚɛɥɸɞɚɬɟɥɶ ɧɚɯɨɞɢɬɫɹ ɜ ɫɢɫɬɟɦɟ S'.

V

Ɋɢɫ. 5.6

Ʉɚɤ ɜɢɞɢɦ, ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɟ ɢɧɬɟɪɜɚɥɵ ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɱɚɫɚɦɢ ɜ ɨɛɟɢɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ "ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ" ɧɚɛɥɸɞɚɬɟɥɟɣ, ɧɚɯɨɞɹɳɢɯɫɹ ɜ ɞɪɭɝɨɣ ɫɢɫɬɟɦɟ, ɭɦɟɧɶɲɚɟɬɫɹ ɜ J ɪɚɡ (ɫɦ. (5.35) ɢ (5.36)). ɉɪɢ ɷɬɨɦ ɩɨɤɚɡɚɧɢɹ ɱɚɫɨɜ ɥɢɧɟɣɧɨ ɡɚɜɢɫɹɬ ɨɬ ɢɯ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɜ ɫɜɨɢɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ. ȿɫɥɢ ɱɚɫɵ ɪɚɫɩɨɥɨ-

162 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ

ɠɟɧɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ ɫɜɨɟɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɫɤɨɪɨɫɬɢ ɟɟ ɞɜɢɠɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɬɨ ɨɧɢ ɨɬɫɬɚɸɬ ɨɬ ɱɚɫɨɜ, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɜ ɧɚɱɚɥɟ ɨɬɫɱɟɬɚ (ɫɦ. ɪɢɫ. 5.5 ɢ 5.6). ɂ ɧɚɨɛɨɪɨɬ, ɱɚɫɵ ɪɚɫɩɨɥɨɠɟɧɧɵɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ ɫɜɨɟɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɫɤɨɪɨɫɬɢ ɟɟ ɞɜɢɠɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɨɩɟɪɟɠɚɸɬ ɱɚɫɵ, ɪɚɫɩɨɥɨɠɟɧɧɵɟ ɜ ɧɚɱɚɥɟ ɨɬɫɱɟɬɚ (ɫɦ.

ɪɢɫ. 5.5 ɢ 5.6).

Ɉɰɟɧɢɦ ɦɚɤɫɢɦɚɥɶɧɨɟ ɪɚɡɥɢɱɢɟ ɜ ɩɨɤɚɡɚɧɢɹɯ ɱɚɫɨɜ, ɤɨɬɨɪɨɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɱɚɫɚɦ, ɪɚɫɩɨɥɨɠɟɧɧɵɦ ɧɚ ɦɚɤɫɢɦɚɥɶɧɨɦ ɪɚɫɫɬɨɹɧɢɢ

Ɂɚɞɚɱɚ 5.4

(ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɢɥɢ ɢɯ ɫɥɟɞɫɬɜɢɹ) Ɇɟɠɡɜɟɡɞɧɵɣ ɤɨɪɚɛɥɶ ɞɜɢɠɟɬɫɹ ɨɬ Ɂɟɦɥɢ ɤ ɡɜɟɡɞɟ, ɧɚɯɨɞɹ-

ɳɟɣɫɹ ɨɬ ɧɟɟ ɧɚ ɪɚɫɫɬɨɹɧɢɢ L = 3 ɫɜɟɬɨɜɵɯ ɝɨɞɚ, ɫɨ ɫɤɨɪɨɫɬɶɸ V = 5 106 ɦ/ɫ. Ⱦɨɫɬɢɝɧɭɜ ɡɜɟɡɞɵ, ɤɨɪɚɛɥɶ ɜɨɡɜɪɚɳɚɟɬɫɹ ɨɛɪɚɬɧɨ ɫ ɬɨɣ ɠɟ ɩɨ ɜɟɥɢɱɢɧɟ ɫɤɨɪɨɫɬɶɸ. ɇɚ ɤɚɤɨɟ ɜɪɟɦɹ 't ɱɚɫɵ ɧɚ ɤɨɪɚɛɥɟ ɨɬɫɬɚɧɭɬ ɨɬ ɡɟɦɧɵɯ ɱɚɫɨɜ ɩɨ ɜɨɡɜɪɚɳɟɧɢɢ ɤɨɪɚɛɥɹ ɧɚ Ɂɟɦɥɸ? ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɩɪɟɧɟɛɪɟɱɶ ɜɪɟɦɟɧɟɦ, ɡɚɬɪɚɱɟɧɧɵɦ ɧɚ ɪɚɡɝɨɧ ɢ ɬɨɪɦɨɠɟɧɢɟ ɪɚɤɟɬɵ.

Ɋɟɲɟɧɢɟ

I. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ S, ɫɜɹɡɚɧɧɚɹ ɫ Ɂɟɦɥɟɣ ɢ ɡɜɟɡɞɨɣ, ɹɜɥɹɟɬɫɹ ɢɧɟɪɰɢɚɥɶɧɨɣ. Ⱦɪɭɝɭɸ ɢɧɟɪɰɢɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ S' ɫɜɹɠɟɦ ɫ ɞɜɢɠɭɳɢɦɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ Ɂɟɦɥɢ ɦɟɠɡɜɟɡɞɧɵɦ ɤɨɪɚɛɥɟɦ. ɉɭɫɬɶ ɤɨɪɚɛɥɶ, ɚ ɡɧɚɱɢɬ ɢ ɫɢɫɬɟɦɚ S', ɞɜɢɠɭɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ V ɜɞɨɥɶ ɨɫɢ X ɫɢɫɬɟɦɵ S. Ɉɩɪɟɞɟɥɢɦ, ɤɚɤ ɨɛɵɱɧɨ, ɢɧɬɟɪɟɫɭɸɳɢɟ ɧɚɫ ɫɨɛɵɬɢɹ:

Ⱥ – ɦɟɠɡɜɟɡɞɧɵɣ ɤɨɪɚɛɥɶ ɧɚɱɚɥ ɞɜɢɝɚɬɶɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ V ɤ ɡɜɟɡɞɟ;

ȼ – ɤɨɪɚɛɥɶ ɞɨɥɟɬɟɥ ɞɨ ɡɜɟɡɞɵ; ɋ – ɦɟɠɡɜɟɡɞɧɵɣ ɤɨɪɚɛɥɶ ɜɨɡɜɪɚɬɢɥɫɹ ɨɛɪɚɬɧɨ ɧɚ Ɂɟɦɥɸ.

II.ɂɧɬɟɪɜɚɥɵ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ Ⱥ ɢ ȼ ɢ ɫɨɛɵɬɢɹɦɢ

ȼɢ ɋ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɪɚɜɧɵ

ɋ ɬɨɱɤɢ ɡɪɟɧɢɹ ɤɨɫɦɨɧɚɜɬɚ, ɧɚɯɨɞɹɳɟɝɨɫɹ ɜ ɤɨɪɚɛɥɟ, ɫɨɛɵɬɢɹ Ⱥ, ȼ ɢ ɫɨɛɵɬɢɹ ȼ, ɋ ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɤɨɪɚɛɥɟɦ, ɩɪɨɢɫɯɨɞɢɬ ɫɨɤɪɚɳɟɧɢɟ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ Ⱥ, ȼ ɢ ȼ, ɋ, ɬɨ ɟɫɬɶ ɧɚɛɥɸɞɚɟɬɫɹ "ɡɚɦɟɞɥɟɧɢɟɦ ɜɪɟɦɟɧɢ". Ɍɨɝɞɚ, ɫɨɝɥɚɫɧɨ (5.6), ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S':

Ɂɚɞɚɱɚ 5.5

(ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɢɥɢ ɢɯ ɫɥɟɞɫɬɜɢɹ)

ɋɬɟɪɠɟɧɶ, ɞɜɢɠɭɳɢɣɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ V = c/2 ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S, ɢɦɟɟɬ ɫɨɛɫɬɜɟɧɧɭɸ ɞɥɢɧɭ l0 = 1 ɦ. ȼ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S', ɫɜɹɡɚɧɧɨɣ ɫ ɞɜɢɠɭɳɢɦɫɹ ɫɬɟɪɠɧɟɦ, ɭɝɨɥ ɦɟɠɞɭ ɫɬɟɪɠɧɟɦ ɢ ɧɚ-

l0

M0

X X'

Ɋɢɫ. 5.7

Ɋɟɲɟɧɢɟ

I. ɉɨɫɤɨɥɶɤɭ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ, ɧɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟɥɢɬɶ ɞɥɢɧɭ ɫɬɟɪɠɧɹ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S, ɨɩɪɟɞɟɥɢɦ ɫɥɟɞɭɸɳɢɟ ɞɜɚ ɫɨɛɵɬɢɹ – Ⱥ ɢ ȼ. ɗɬɢ ɫɨɛɵɬɢɹ ɫɨɫɬɨɹɬ ɜ ɬɨɦ, ɱɬɨ ɨɞɧɨɜɪɟɦɟɧɧɨ ɢɡɦɟɪɟɧɵ ɩɨɥɨɠɟɧɢɹ ɞɜɭɯ ɤɨɧɰɨɜ ɫɬɟɪɠɧɹ ɜ ɫɢɫɬɟɦɟ S. ɉɭɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɷɬɢɯ ɫɨɛɵɬɢɣ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɪɚɜɧɵ ( x1 , y1 , t1 ) ɢ ( x2 , y2 , t2 ), ɩɪɢɱɟɦ t1 t2 . ȼ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S' ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɩɪɨɢɫɯɨɞɹɬ ɧɟ ɨɞɧɨɜɪɟɦɟɧɧɨ, ɢɯ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ

(x1c , y1c , t1c ) ɢ ( x2c , y2c , t2c ).

II. ɋɨɛɫɬɜɟɧɧɚɹ ɞɥɢɧɚ ɫɬɟɪɠɧɹ (ɞɥɢɧɚ ɫɬɟɪɠɧɹ ɜ ɧɟɩɨɞɜɢɠɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɝɨ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S') ɪɚɜɧɚ:

ɂɧɬɟɪɜɚɥɵ ǻxc ɢ ǻyc ɫɜɹɡɚɧɵ ɫ ɫɨɛɫɬɜɟɧɧɨɣ ɞɥɢɧɨɣ ɫɬɟɪɠɧɹ

l0 ɢ ɭɝɥɨɦ ɟɝɨ ɧɚɤɥɨɧɚ M0 ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S' ɫɥɟɞɭɸɳɢɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ:

ɉɨɫɤɨɥɶɤɭ ɫɨɛɵɬɢɹ A ɢ B ɩɪɨɢɫɯɨɞɹɬ ɨɞɧɨɜɪɟɦɟɧɧɨ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S, ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɞɥɢɧɵ ɫɬɟɪɠɧɹ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɫɥɟɞɫɬɜɢɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ – "ɫɨɤɪɚɳɟɧɢɟɦ ɞɥɢɧɵ" (ɫɦ. ɩ. 5.1.3). ɋɨɝɥɚɫɧɨ (5.7):

ɉɪɢ ɷɬɨɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ʌɨɪɟɧɰɚ (5.4) ɫɨɤɪɚɳɟɧɢɟ ɢɧɬɟɪɜɚɥɚ ǻy ɦɟɠɞɭ ɤɨɨɪɞɢɧɚɬɚɦɢ y1 ɢ y2 ɧɚɛɥɸ-

ɞɚɬɶɫɹ ɧɟ ɛɭɞɟɬ: ǻy ǻyc .

Ⱦɥɢɧɚ ɫɬɟɪɠɧɹ l ɨɩɪɟɞɟɥɹɟɬɫɹ ɟɝɨ ɩɪɨɟɤɰɢɹɦɢ ɧɚ ɨɫɢ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S:

III. ɉɨɞɫɬɚɜɢɜ ɜ ɮɨɪɦɭɥɭ (5.47) ɫɨɨɬɧɨɲɟɧɢɹ (5.45) ɢ (5.46),

Ɂɚɞɚɱɚ 5.6

(ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɢɥɢ ɢɯ ɫɥɟɞɫɬɜɢɹ) Ʉɨɫɦɢɱɟɫɤɢɣ ɤɨɪɚɛɥɶ ɥɟɬɢɬ ɫɨ ɫɤɨɪɨɫɬɶɸ V = 0,6ɫ ɨɬ ɨɞɧɨɝɨ

ɧɟɩɨɞɜɢɠɧɨɝɨ ɤɨɫɦɢɱɟɫɤɨɝɨ ɦɚɹɤɚ ɤ ɞɪɭɝɨɦɭ. ȼ ɬɨɬ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɨɧ ɧɚɯɨɞɢɬɫɹ ɩɨɫɟɪɟɞɢɧɟ ɦɟɠɞɭ ɦɚɹɤɚɦɢ, ɤɚɠɞɵɣ ɢɡ ɧɢɯ ɢɫɩɭɫɤɚɟɬ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɤɨɪɚɛɥɹ ɫɜɟɬɨɜɨɣ ɢɦɩɭɥɶɫ. ɇɚɣɬɢ, ɤɚɤɨɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ ɩɪɨɣɞɟɬ ɧɚ ɤɨɪɚɛɥɟ ɦɟɠɞɭ ɦɨɦɟɧɬɚɦɢ ɪɟɝɢɫɬɪɚɰɢɢ ɷɬɢɯ ɢɦɩɭɥɶɫɨɜ. Ɋɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɦɚɹɤɚɦɢ ɫɜɟɬ ɩɪɨɯɨɞɢɬ ɡɚ ɜɪɟɦɹ

W = 60 ɫɭɬ.

Ɋɟɲɟɧɢɟ

I. ɋɜɹɠɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ S' ɫ ɤɨɫɦɢɱɟɫɤɢɦ ɤɨɪɚɛɥɟɦ, ɚ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ S ɫ ɧɟɩɨɞɜɢɠɧɵɦɢ ɦɚɹɤɚɦɢ (ɪɢɫ. 5.8).

Ɋɢɫ. 5.8

ɉɭɫɬɶ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɫɨɫɬɨɹɬ ɜ ɬɨɦ, ɱɬɨ ɧɚ ɤɨɪɚɛɥɟ ɩɪɨɢɫɯɨɞɢɬ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɚɹ ɪɟɝɢɫɬɪɚɰɢɹ ɞɜɭɯ ɢɫɩɭɳɟɧɧɵɯ ɦɚɹɤɚɦɢ ɫɜɟɬɨɜɵɯ ɢɦɩɭɥɶɫɨɜ. ȼ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɫɧɚɱɚɥɚ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t1 ɜ ɬɨɱɤɟ ɫ ɤɨɨɪɞɢɧɚɬɨɣ x1 ɩɪɨɢɫɯɨɞɢɬ ɪɟɝɢɫɬɪɚɰɢɹ ɢɦɩɭɥɶɫɚ, ɢɫɩɭɳɟɧɧɨɝɨ ɦɚɹɤɨɦ ȼ. Ɋɟɝɢɫɬɪɚɰɢɹ ɢɦɩɭɥɶɫɚ, ɢɫɩɭɳɟɧɧɨɝɨ ɦɚɹɤɨɦ Ⱥ, ɩɪɨɢɫɯɨɞɢɬ ɜ ɩɨɫɥɟɞɭɸɳɢɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t2 ɜ ɬɨɱɤɟ ɫ ɤɨɨɪɞɢɧɚ-

ɬɨɣ x2.

II. Ⱦɥɹ ɧɚɛɥɸɞɚɬɟɥɹ, ɧɚɯɨɞɹɳɟɝɨɫɹ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S, ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɪɟɝɢɫɬɪɚɰɢɢ ɫɢɝɧɚɥɨɜ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɨɨɬɧɨɲɟɧɢɹɦɢ:

Ɍɨɝɞɚ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ Ⱥ ɢ ȼ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɛɭɞɟɬ ɪɚɜɟɧ:

Ɂɞɟɫɶ L – ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɤɨɫɦɢɱɟɫɤɢɦɢ ɦɚɹɤɚɦɢ, ɩɪɢɱɟɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ

L cW . (5.55)

ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɫɥɟɞɫɬɜɢɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ – "ɡɚɦɟɞɥɟɧɢɟɦ ɜɪɟɦɟɧɢ". ɉɨɫɤɨɥɶɤɭ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S' ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɩɪɨɢɫɯɨɞɹɬ ɜ ɨɞɧɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɬɨ, ɫɨɝɥɚɫɧɨ (5.6), ɞɨɥɠɧɨ ɧɚɛɥɸɞɚɬɶɫɹ ɫɨɤɪɚɳɟɧɢɟ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɦɢ ɫɨɛɵɬɢɹɦɢ ɜ ɷɬɨɣ ɫɢɫɬɟɦɟ:

ɞɟɥɢɦ ɢɫɤɨɦɵɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɦɨɦɟɧɬɚɦɢ ɪɟɝɢɫɬɪɚɰɢɢ ɫɜɟɬɨɜɵɯ ɢɦɩɭɥɶɫɨɜ ɧɚ ɤɨɪɚɛɥɟ:

ɉɨɞɫɬɚɜɥɹɹ ɜ (5.57) ɡɚɞɚɧɧɵɟ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɢ ɤɨɫɦɢɱɟɫɤɨɝɨ ɤɨɪɚɛɥɹ V = 0,6ɫ ɢ ɜɪɟɦɟɧɢ, ɧɟɨɛɯɨɞɢɦɨɝɨ ɞɥɹ ɩɪɨɯɨɠɞɟɧɢɹ ɫɜɟɬɚ ɦɟɠɞɭ ɦɚɹɤɚɦɢ W = 60 ɫɭɬ., ɩɨɥɭɱɚɟɦ:

ǻt' = 45 ɫɭɬ.

Ɂɚɞɚɱɚ 5.7

(ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɢɥɢ ɢɯ ɫɥɟɞɫɬɜɢɹ)

Ʉɨɪɚɛɥɶ, ɥɟɬɹɳɢɣ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɤ Ɂɟɦɥɟ, ɢɫɩɭɫɤɚɟɬ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɞɜɚ ɤɨɪɨɬɤɢɯ ɫɜɟɬɨɜɵɯ ɢɦɩɭɥɶɫɚ ɫ ɢɧɬɟɪɜɚɥɨɦ ɜɪɟɦɟɧɢ W1 = 1 ɦɢɧ. Ɉɬɪɚɠɟɧɧɵɣ ɨɬ Ɂɟɦɥɢ ɩɟɪɜɵɣ ɢɦɩɭɥɶɫ ɜɨɡɜɪɚɳɚɟɬɫɹ ɧɚ ɤɨɪɚɛɥɶ ɱɟɪɟɡ ɜɪɟɦɹ T = 1,5 ɦɟɫɹɰɚ. ɉɪɢ ɷɬɨɦ ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɩɪɢɧɹɬɵɦɢ ɫɢɝɧɚɥɚɦɢ ɫɨɫɬɚɜɥɹɟɬ W2 = 15 ɫ. ɉɪɨɦɟɠɭɬɤɢ ɜɪɟɦɟɧɢ W1, W2 ɢ T ɨɬɫɱɢɬɵɜɚɸɬɫɹ ɩɨ ɱɚɫɚɦ ɤɨɪɚɛɥɹ. ɇɚɣɬɢ ɫɤɨɪɨɫɬɶ ɤɨɪɚɛɥɹ ɢ ɜɪɟɦɹ TɁ, ɤɨɬɨɪɨɟ ɩɪɨɣɞɟɬ ɧɚ Ɂɟɦɥɟ ɨɬ ɦɨɦɟɧɬɚ ɪɟɝɢɫɬɪɚɰɢɢ ɡɟɦɧɵɦ ɧɚɛɥɸɞɚɬɟɥɟɦ ɩɟɪɜɨɝɨ ɫɜɟɬɨɜɨɝɨ ɢɦɩɭɥɶɫɚ ɞɨ ɩɪɢɥɟɬɚ ɤɨɪɚɛɥɹ.

Ɋɟɲɟɧɢɟ

I.Ɉɩɪɟɞɟɥɢɦ ɢɧɬɟɪɟɫɭɸɳɢɟ ɧɚɫ ɫɨɛɵɬɢɹ:

Ⱥ– ɢɫɩɭɫɤɚɧɢɟ ɤɨɪɚɛɥɟɦ ɩɟɪɜɨɝɨ ɫɜɟɬɨɜɨɝɨ ɢɦɩɭɥɶɫɚ; ȼ – ɢɫɩɭɫɤɚɧɢɟ ɤɨɪɚɛɥɟɦ ɜɬɨɪɨɝɨ ɫɜɟɬɨɜɨɝɨ ɢɦɩɭɥɶɫɚ;

Ⱥ1 – ɨɬɪɚɠɟɧɢɟ ɩɟɪɜɨɝɨ ɢɦɩɭɥɶɫɚ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ Ɂɟɦɥɢ; ȼ1 – ɨɬɪɚɠɟɧɢɟ ɜɬɨɪɨɝɨ ɢɦɩɭɥɶɫɚ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ Ɂɟɦɥɢ; Ⱥ2 – ɪɟɝɢɫɬɪɚɰɢɹ ɩɟɪɜɨɝɨ ɢɦɩɭɥɶɫɚ ɤɨɪɚɛɥɟɦ; ȼ2 – ɪɟɝɢɫɬɪɚɰɢɹ ɜɬɨɪɨɝɨ ɢɦɩɭɥɶɫɚ ɤɨɪɚɛɥɟɦ;

ɋ – ɩɪɢɥɟɬ ɤɨɪɚɛɥɹ ɧɚ Ɂɟɦɥɸ.

ɇɚ ɪɢɫ. 5.9 ɫɯɟɦɚɬɢɱɧɨ (ɛɟɡ ɫɨɛɥɸɞɟɧɢɹ ɦɚɫɲɬɚɛɚ) ɢɡɨɛɪɚɠɟɧɚ ɜɪɟɦɟɧɧɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɭɤɚɡɚɧɧɵɯ ɫɨɛɵɬɢɣ ɜ ɫɢɫɬɟɦɚɯ ɨɬɱɟɬɚ, ɫɜɹɡɚɧɧɵɯ ɫ Ɂɟɦɥɟɣ (ɜɟɪɯɧɹɹ ɧɚ ɪɢɫɭɧɤɟ ɨɫɶ ɜɪɟɦɟɧɢ t) ɢ ɫ ɤɨɪɚɛɥɟɦ (ɧɢɠɧɹɹ ɨɫɶ t').

T

Ɋɢɫ. 5.9

ɋɨɛɵɬɢɹ, ɩɪɨɢɡɨɲɟɞɲɢɟ ɧɚ ɤɨɪɚɛɥɟ, ɨɛɨɡɧɚɱɟɧɵ ɧɟɡɚɤɪɚɲɟɧɧɵɦɢ ɤɪɭɠɤɚɦɢ, ɚ ɫɨɛɵɬɢɹ, ɩɪɨɢɡɨɲɟɞɲɢɟ ɧɚ Ɂɟɦɥɟ, – ɡɚɤɪɚɲɟɧɧɵɦɢ ɤɪɭɠɤɚɦɢ. ɇɚ ɪɢɫɭɧɤɟ ɢɡɨɛɪɚɠɟɧɵ ɬɚɤɠɟ ɢɧɬɟɪɜɚɥɵ ɜɪɟɦɟɧɢ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ, ɡɚɞɚɧɧɵɟ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɩɨ ɱɚɫɚɦ ɤɨɪɚɛɥɹ, – W1, W2 ɢ T. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɫɥɟɞɫɬɜɢɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ – ɬɚɤ ɧɚɡɵɜɚɟɦɵɦ "ɡɚɦɟɞɥɟɧɢɟɦ ɜɪɟɦɟɧɢ" (ɫɦ. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ, ɮɨɪɦɭɥɚ (5.6)) – ɢɧɬɟɪɜɚɥɵ ɜɪɟɦɟɧɢ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɬɟɦ ɠɟ ɩɚɪɚɦ ɫɨɛɵɬɢɣ, ɢɡɦɟɪɟɧɧɵɟ ɩɨ ɡɟɦɧɵɦ ɱɚɫɚɦ, ɭɜɟɥɢɱɢɜɚɸɬɫɹ ɜ J ɪɚɡ (ɫɦ. ɪɢɫ. 5.9).

II. ɉɨɫɤɨɥɶɤɭ ɩɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɡɚɞɚɧ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ Ɍ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ Ⱥ ɢ Ⱥ2 ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɤɨɪɚɛɥɟɦ, ɡɚɩɢɲɟɦ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ ǻtA ɦɟɠɞɭ ɬɟɦɢ ɠɟ ɫɨɛɵɬɢɹɦɢ ɜ ɫɢɫ-

ɝɞɟ LA – ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ Ɂɟɦɥɟɣ ɢ ɤɨɪɚɛɥɟɦ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɩɪɨɢɡɨɲɥɨ ɫɨɛɵɬɢɟ Ⱥ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ Ɂɟɦɥɟɣ.

ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɢɫɩɨɥɶɡɭɹ ɫɥɟɞɫɬɜɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ʌɨɪɟɧɰɚ, "ɡɚɦɟɞɥɟɧɢɟ ɜɪɟɦɟɧɢ", ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ:

ɍɪɚɜɧɟɧɢɟ, ɚɧɚɥɨɝɢɱɧɨɟ (5.58), ɡɚɩɢɲɟɦ ɞɥɹ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ ǻtB ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ B ɢ B2 ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ

ɝɞɟ LB – ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ Ɂɟɦɥɟɣ ɢ ɤɨɪɚɛɥɟɦ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɩɪɨɢɡɨɲɥɨ ɫɨɛɵɬɢɟ B ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ Ɂɟɦɥɟɣ.

Ɂɚɩɢɲɟɦ ɩɪɨɣɞɟɧɧɵɣ ɤɨɪɚɛɥɟɦ ɩɭɬɶ ɫ ɦɨɦɟɧɬɚ ɢɫɩɭɫɤɚɧɢɹ ɩɟɪɜɨɝɨ ɫɜɟɬɨɜɨɝɨ ɢɦɩɭɥɶɫɚ ɞɨ ɦɨɦɟɧɬɚ ɢɫɩɭɫɤɚɧɢɹ ɜɬɨɪɨɝɨ ɫɜɟɬɨɜɨɝɨ ɢɦɩɭɥɶɫɚ (ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɣ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɫɨɛɵɬɢɹɦɢ A ɢ B) ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ Ɂɟɦɥɟɣ (ɡɚ ɜɪɟɦɹ JW1 ):

ɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɤɨɪɚɛɥɹ V ɢ ɜɪɟɦɹ TɁ, ɤɨɬɨɪɨɟ ɩɪɨɣɞɟɬ ɧɚ Ɂɟɦɥɟ ɨɬ ɦɨɦɟɧɬɚ ɪɟɝɢɫɬɪɚɰɢɢ ɡɟɦɧɵɦ ɧɚɛɥɸɞɚɬɟɥɟɦ ɩɟɪɜɨɝɨ ɫɜɟɬɨɜɨɝɨ ɢɦɩɭɥɶɫɚ ɞɨ ɩɪɢɥɟɬɚ ɤɨɪɚɛɥɹ:

ɉɨɞɫɬɚɜɥɹɹ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢɧɬɟɪɜɚɥɨɜ ɜɪɟɦɟɧɢ, ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ, ɩɨɥɭɱɢɦ:

V 0,6c , TɁ #1 ɦɟɫɹɰ .

Ɂɚɞɚɱɚ 5.8

(ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ ɢɥɢ ɢɯ ɫɥɟɞɫɬɜɢɹ)

Ⱦɜɚ ɡɜɟɡɞɨɥɟɬɚ ɫ ɜɵɤɥɸɱɟɧɧɵɦɢ ɞɜɢɝɚɬɟɥɹɦɢ ɞɜɢɠɭɬɫɹ ɧɚɜɫɬɪɟɱɭ ɞɪɭɝ ɞɪɭɝɭ (ɫɦ. ɪɢɫ. 5.10). ɇɚ ɧɨɫɭ ɢ ɧɚ ɤɨɪɦɟ ɩɟɪɜɨɝɨ ɡɜɟɡɞɨɥɟɬɚ ɩɟɪɢɨɞɢɱɟɫɤɢ, ɤɚɠɞɵɟ W1 = 1 ɫ, ɩɨ ɱɚɫɚɦ ɷɬɨɝɨ ɡɜɟɡɞɨɥɟɬɚ ɨɞɧɨɜɪɟɦɟɧɧɨ ɡɚɠɢɝɚɸɬɫɹ ɫɢɝɧɚɥɶɧɵɟ ɨɝɧɢ. ɇɚ ɜɬɨɪɨɦ ɡɜɟɡɞɨɥɟɬɟ ɤɚɠɞɵɟ W2 = 0,5 ɫ ɧɚɛɥɸɞɚɸɬ ɞɜɟ ɜɫɩɵɲɤɢ ɫ ɢɧɬɟɪɜɚɥɨɦ ɜɪɟɦɟɧɢ 'W = 1 ɦɤɫ. ɇɚɣɬɢ ɫɨɛɫɬɜɟɧɧɭɸ ɞɥɢɧɭ l0 ɩɟɪɜɨɝɨ ɡɜɟɡɞɨɥɟɬɚ ɢ ɫɤɨɪɨɫɬɶ U ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɡɜɟɡɞɨɥɟɬɨɜ.

U

l0

Ɋɢɫ. 5.10

Ɋɟɲɟɧɢɟ

I. ɋɜɹɠɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ S ɫ ɩɟɪɜɵɦ ɡɜɟɡɞɨɥɟɬɨɦ, ɬɨɝɞɚ ɜɬɨɪɨɣ ɡɜɟɡɞɨɥɟɬ, ɫ ɤɨɬɨɪɵɦ ɫɜɹɠɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ S', ɛɭɞɟɬ ɞɜɢɝɚɬɶɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S ɫɨ ɫɤɨɪɨɫɬɶɸ U (ɪɢɫ. 5.10).

Ɉɩɪɟɞɟɥɢɦ ɢɧɬɟɪɟɫɭɸɳɢɟ ɧɚɫ ɫɨɛɵɬɢɹ:

Ⱥ ɢ ȼ – ɞɜɟ ɛɥɢɠɚɣɲɢɟ ɩɨ ɜɪɟɦɟɧɢ ɜɫɩɵɲɤɢ, ɩɪɨɢɫɯɨɞɹɳɢɟ ɧɚ ɧɨɫɭ ɩɟɪɜɨɝɨ ɡɜɟɡɞɨɥɟɬɚ;

Ⱥ1 ɢ ȼ1 – ɪɟɝɢɫɬɪɚɰɢɹ ɷɬɢɯ ɜɫɩɵɲɟɤ ɧɚ ɜɬɨɪɨɦ ɡɜɟɡɞɨɥɟɬɟ; C – ɜɫɩɵɲɤɚ ɧɚ ɤɨɪɦɟ ɩɟɪɜɨɝɨ ɡɜɟɡɞɨɥɟɬɚ, ɤɨɬɨɪɚɹ ɩɪɨ-

ɢɡɨɲɥɚ ɨɞɧɨɜɪɟɦɟɧɧɨ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɫɨ ɜɫɩɵɲɤɨɣ ɧɚ ɧɨɫɭ ɷɬɨɝɨ ɡɜɟɡɞɨɥɟɬɚ (ɫɨɛɵɬɢɟ Ⱥ);

ɋ1 – ɪɟɝɢɫɬɪɚɰɢɹ ɧɚ ɜɬɨɪɨɦ ɡɜɟɡɞɨɥɟɬɟ ɜɫɩɵɲɤɢ, ɩɪɨɢɡɨɲɟɞɲɟɣ ɧɚ ɤɨɪɦɟ ɩɟɪɜɨɝɨ ɡɜɟɡɞɨɥɟɬɚ.

ɇɚ ɪɢɫ. 5.11 ɫɯɟɦɚɬɢɱɧɨ ɢɡɨɛɪɚɠɟɧɚ ɜɪɟɦɟɧɧɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɭɤɚɡɚɧɧɵɯ ɫɨɛɵɬɢɣ ɜ ɫɢɫɬɟɦɚɯ ɨɬɱɟɬɚ, ɫɜɹɡɚɧɧɵɯ ɫ ɩɟɪɜɵɦ (ɜɟɪɯɧɹɹ ɧɚ ɪɢɫɭɧɤɟ ɨɫɶ ɜɪɟɦɟɧɢ t) ɢ ɫɨ ɜɬɨɪɵɦ ɡɜɟɡɞɨɥɟɬɨɦ (ɧɢɠɧɹɹ ɨɫɶ t').

ɋɨɛɵɬɢɹ, ɩɪɨɢɡɨɲɟɞɲɢɟ ɧɚ ɩɟɪɜɨɦ ɡɜɟɡɞɨɥɟɬɟ, ɨɛɨɡɧɚɱɟɧɵ ɧɟɡɚɤɪɚɲɟɧɧɵɦɢ ɤɪɭɠɤɚɦɢ, ɚ ɫɨɛɵɬɢɹ, ɩɪɨɢɡɨɲɟɞɲɢɟ ɧɚ ɜɬɨɪɨɦ ɡɜɟɡɞɨɥɟɬɟ, – ɡɚɤɪɚɲɟɧɧɵɦɢ ɤɪɭɠɤɚɦɢ.