Механика.Методика решения задач

ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ J, ɩɨɩɚɞɚɸɳɟɝɨ ɜ ɭɯɨ ɱɟɥɨɜɟɤɚ. ɋɱɢɬɚɬɶ ɩɥɨɳɚɞɶ ɭɯɚ, ɨɪɢɟɧɬɢɪɨɜɚɧɧɨɝɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɧɚɩɪɚɜɥɟɧɢɸ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ, s = 4 ɫɦ2. ɉɥɨɬɧɨɫɬɶ ɜɨɡɞɭɯɚ ɜ ɨɬɫɭɬɫɬɜɢɟ ɜɨɥɧɵ U = 1,3 ɤɝ/ɦ3, ɫɤɨɪɨɫɬɶ ɡɜɭɤɚ ɜ ɜɨɡɞɭɯɟ c = 334 ɦ/ɫ.

Ɋɟɲɟɧɢɟ

I. Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɢɫɩɨɥɶɡɭɟɦ ɞɟɤɚɪɬɨɜɭ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ. ȼ ɤɚɱɟɫɬɜɟ ɦɨɞɟɥɢ ɜɨɥɧɵ ɫɦɟɳɟɧɢɣ ɜɵɛɟɪɟɦ ɩɥɨɫɤɭɸ ɝɚɪɦɨɧɢɱɟɫɤɭɸ ɜɨɥɧɭ, ɪɚɫɩɪɨɫɬɪɚɧɹɸɳɭɸɫɹ ɜɞɨɥɶ ɨɫɢ X ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. ɉɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɫɱɢɬɚɟɦ ɚɞɢɚɛɚɬɢɱɟɫɤɢɦ.

II. ɋɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ J, ɩɚɞɚɸɳɟɝɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɨɜɟɪɯɧɨɫɬɢ ɭɯɚ ɩɥɨɳɚɞɶɸ s, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɨɣ

(9.48) ɪɚɜɧɨ:

ɬɭɞɭ ɜɨɥɧɵ ɞɚɜɥɟɧɢɣ ǻP0 , ɡɚɞɚɧɧɭɸ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ.

ɉɪɢ ɚɞɢɚɛɚɬɢɱɟɫɤɨɦ ɩɪɨɰɟɫɫɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɜ ɝɚɡɟ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɢɡɦɟɧɟɧɢɹ ɩɥɨɬɧɨɫɬɢ ɫɪɟɞɵ ɢ ɞɚɜɥɟɧɢɹ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ (ɫɦ. (9.33) ɢ (9.36)):

ɝɞɟ P ɢ U – ɞɚɜɥɟɧɢɟ ɢ ɩɥɨɬɧɨɫɬɶ ɜɨɡɞɭɯɚ ɜ ɨɬɫɭɬɫɬɜɢɟ ɜɨɥɧɵ.

Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɚɦɩɥɢɬɭɞɵ ɡɜɭɤɨɜɨɝɨ ɞɚɜɥɟɧɢɹ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɬɚɤɠɟ ɮɨɪɦɭɥɨɣ (9.31) ɞɥɹ ɫɤɨɪɨɫɬɢ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɜ ɜɨɡɞɭ-

ɦɟɧɟɧɢɟ ɞɚɜɥɟɧɢɹ ɜɨɡɞɭɯɚ, ɜɵɡɜɚɧɧɨɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟɦ ɜ ɧɟɦ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ:

ɥɹɪɧɨ ɩɨɜɟɪɯɧɨɫɬɢ ɭɯɚ ɩɥɨɳɚɞɶɸ s, ɩɨɥɭɱɚɟɦ ɩɨɞɫɬɚɧɨɜɤɨɣ ɜ (9.98) ɨɛɴɟɦɧɨɣ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɜɨɥɧɵ (9.99) ɫ ɭɱɟɬɨɦ (9.106):

ɉɨɞɫɬɚɜɢɜ ɜ (9.107) ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ, ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ:

J = 4.6 10–5 ȼɬ.

Ɂɚɞɚɱɚ 9.5

Ɍɨɱɟɱɧɵɣ ɢɡɨɬɪɨɩɧɨ ɢɡɥɭɱɚɸɳɢɣ ɢɫɬɨɱɧɢɤ ɡɜɭɤɚ S ɧɚɯɨɞɢɬɫɹ ɧɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɟ ɤ ɩɥɨɫɤɨɫɬɢ ɤɨɥɶɰɚ, ɩɪɨɯɨɞɹɳɟɦ ɱɟɪɟɡ ɟɝɨ ɰɟɧɬɪ P (ɫɦ. ɪɢɫ. 9.9).

Ɋɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɬɨɱɤɨɣ P ɢ ɢɫɬɨɱɧɢɤɨɦ S ɪɚɜɧɨ L = 1 ɦ, ɪɚɞɢɭɫ ɤɨɥɶɰɚ – R = 0,5 ɦ. ɇɚɣɬɢ ɫɪɟɞɧɢɣ ɩɨɬɨɤ ɷɧɟɪɝɢɢ ɱɟɪɟɡ ɩɥɨɫɤɭɸ ɩɨɜɟɪɯɧɨɫɬɶ, ɨɝɪɚɧɢɱɟɧɧɭɸ ɤɨɥɶɰɨɦ, ɟɫɥɢ ɜ ɬɨɱɤɟ P ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ I0 = 30 ɦɤȼɬ/ɦ2. Ɂɚɬɭɯɚɧɢɟɦ ɜɨɥɧ ɩɪɟɧɟɛɪɟɱɶ.

Ɋɟɲɟɧɢɟ

I. Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɢɫɩɨɥɶɡɭɟɦ ɩɨɥɹɪɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɫ ɰɟɧɬɪɨɦ ɜ ɬɨɱɤɟ P, ɹɜɥɹɸɳɟɣɫɹ ɰɟɧɬɪɨɦ ɤɨɥɶɰɚ.

ɉɨɫɤɨɥɶɤɭ ɢɫɬɨɱɧɢɤ ɡɜɭɤɚ S ɹɜɥɹɟɬɫɹ ɬɨɱɟɱɧɵɦ ɢ ɢɡɨɬɪɨɩɧɨ ɢɡɥɭɱɚɸɳɢɦ, ɬɨ ɨɧ ɜɨɡɛɭɠɞɚɟɬ ɫɮɟɪɢɱɟɫɤɭɸ ɜɨɥɧɭ, ɚɦɩɥɢɬɭɞɚ ɤɨɬɨɪɨɣ ɢɡɦɟɧɹɟɬɫɹ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɪɚɫɫɬɨɹɧɢɸ ɨɬ ɢɫɬɨɱɧɢɤɚ (ɫɦ. (9.13)), ɚ ɡɧɚɱɢɬ, ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɤɜɚɞɪɚɬɭ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɢɫɬɨɱɧɢɤɚ ɞɨ ɬɨɱɤɢ ɧɚɛɥɸɞɟɧɢɹ.

II. Ɋɚɡɨɛɶɟɦ ɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɧɚ ɤɨɧɰɟɧɬɪɢɱɟɫɤɢɟ ɤɨɥɶɰɟɜɵɟ ɡɨɧɵ, ɡɚɤɥɸɱɟɧɧɵɟ ɦɟɠɞɭ ɨɤɪɭɠɧɨɫɬɹɦɢ ɫ ɪɚɞɢɭɫɚɦɢ U ɢ U+dU (0 d U d R) ɫ ɰɟɧɬɪɚɦɢ ɜ ɬɨɱɤɟ P.

ɉɨɫɤɨɥɶɤɭ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɫɮɟɪɢɱɟɫɤɨɣ ɜɨɥɧɵ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɤɜɚɞɪɚɬɭ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɢɫɬɨɱɧɢɤɚ ɞɨ ɬɨɱɤɢ ɧɚɛɥɸɞɟɧɢɹ, ɬɨ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɜɨɥɧɵ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɤɨɥɶɰɟɜɭɸ ɡɨɧɭ ɪɚɞɢɭɫɨɦ U (ɫɦ. ɪɢɫ. 9.9), ɪɚɜɧɚ:

ɉɨɬɨɤ ɷɧɟɪɝɢɢ dJ ɱɟɪɟɡ ɜɵɞɟɥɟɧɧɭɸ ɮɢɡɢɱɟɫɤɢ ɛɟɫɤɨɧɟɱɧɨ ɬɨɧɤɭɸ ɤɨɥɶɰɟɜɭɸ ɡɨɧɭ (ɫɦ. ɪɢɫ. 9.9) ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɜɨɥɧɵ (ɫɦ. ɩ. 9.1.5), ɪɚɜɟɧ:

III. ɂɫɤɨɦɵɣ ɩɨɬɨɤ ɷɧɟɪɝɢɢ J ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ, ɨɝɪɚɧɢɱɟɧɧɭɸ ɤɨɥɶɰɨɦ ɪɚɞɢɭɫɨɦ R, ɨɩɪɟɞɟɥɢɦ, ɢɧɬɟɝɪɢɪɭɹ (9.109) ɫ ɭɱɟ-

ɬɨɦ (9.108):

354 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ

ɉɨɞɫɬɚɜɢɜ ɜ (9.110) ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɪɚɫɫɬɨɹɧɢɹ L ɦɟɠɞɭ ɬɨɱɤɨɣ P ɢ ɢɫɬɨɱɧɢɤɨɦ S, ɪɚɞɢɭɫɚ ɤɨɥɶɰɚ R ɢ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ I0 ɜ ɬɨɱɤɟ P, ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ:

J = 20 ɦɤȼɬ.

Ɂɚɞɚɱɚ 9.6

ɇɚ ɨɫɢ X ɧɚɯɨɞɹɬɫɹ ɩɪɢɟɦɧɢɤ D ɢ ɢɫɬɨɱɧɢɤ S ɡɜɭɤɨɜɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɜɨɥɧ ɫ ɱɚɫɬɨɬɨɣ Qs = 2000 Ƚɰ. ɂɫɬɨɱɧɢɤ ɭɫɬɚɧɨɜɥɟɧ ɧɚ ɬɟɥɟɠɤɟ, ɫɨɜɟɪɲɚɸɳɟɣ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɤɨɥɟɛɚɧɢɹ ɜɞɨɥɶ ɷɬɨɣ ɨɫɢ ɫ ɭɝɥɨɜɨɣ ɱɚɫɬɨɬɨɣ Z ɢ ɚɦɩɥɢɬɭɞɨɣ Ⱥ = 50 ɫɦ. ɋɤɨɪɨɫɬɶ ɡɜɭɤɚ ɫ =340 ɦ/ɫ. ɉɪɢ ɤɚɤɨɦ ɡɧɚɱɟɧɢɢ Z ɲɢɪɢɧɚ ɱɚɫɬɨɬɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ɡɜɭɤɚ, ɜɨɫɩɪɢɧɢɦɚɟɦɨɝɨ ɧɟɩɨɞɜɢɠɧɵɦ ɩɪɢɟɦɧɢɤɨɦ, ɛɭɞɟɬ ɫɨɫɬɚɜɥɹɬɶ 'Q = 20 Ƚɰ?

Ɋɟɲɟɧɢɟ

I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɨɫɶ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɥɟɧɚ ɨɬ ɢɫɬɨɱɧɢɤɚ ɤ ɩɪɢɟɦɧɢɤɭ (ɫɦ. ɪɢɫ. 6.1). ɉɨɫɤɨɥɶɤɭ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɧɟ ɨɝɨɜɚɪɢ-

Ⱦɨɩɥɟɪɚ (ɫɦ. ɩ. 9.1.6).

II. ɒɢɪɢɧɚ ɱɚɫɬɨɬɧɨɝɨ ɢɧɬɟɪɜɚɥɚ 'Q ɡɜɭɤɨɜɵɯ ɜɨɥɧ, ɜɨɫɩɪɢɧɢɦɚɟɦɵɯ ɩɪɢɟɦɧɢɤɨɦ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɡɧɨɫɬɶɸ ɦɚɤɫɢɦɚɥɶɧɨɣ ɢ ɦɢɧɢɦɚɥɶɧɨɣ ɱɚɫɬɨɬ ɷɬɢɯ ɜɨɥɧ.

ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɢɫɬɨɱɧɢɤ ɩɪɢɛɥɢɠɚɟɬɫɹ ɤ ɩɪɢɟɦɧɢɤɭ, ɦɚɤɫɢ-

ɝɞɟ Xs0 – ɚɦɩɥɢɬɭɞɚ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɢɫɬɨɱɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ

ɫɪɟɞɵ, c – ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ ɜ ɫɪɟɞɟ.

ɉɪɢ ɞɜɢɠɟɧɢɢ ɬɟɥɟɠɤɢ ɩɨ ɝɚɪɦɨɧɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ ɚɦɩɥɢɬɭɞɚ ɟɟ ɫɤɨɪɨɫɬɢ, ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢ ɚɦɩɥɢɬɭɞɚ ɫɤɨɪɨɫɬɢ ɢɫɬɨɱɧɢɤɚ,

III. ɂɫɤɨɦɚɹ ɲɢɪɢɧɚ ɱɚɫɬɨɬɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ɡɜɭɤɚ 'Q, ɜɨɫɩɪɢɧɢɦɚɟɦɨɝɨ ɧɟɩɨɞɜɢɠɧɵɦ ɩɪɢɟɦɧɢɤɨɦ, ɫɨɝɥɚɫɧɨ (9.111) ɢ (9.113) ɪɚɜɧɚ:

ɂɫɩɨɥɶɡɭɹ ɜɡɚɢɦɨɫɜɹɡɶ ɚɦɩɥɢɬɭɞɵ ɫɤɨɪɨɫɬɢ ɢɫɬɨɱɧɢɤɚ ɫ ɚɦɩɥɢɬɭɞɨɣ ɟɝɨ ɤɨɥɟɛɚɧɢɣ ɜɦɟɫɬɟ ɫ ɬɟɥɟɠɤɨɣ (9.112), ɩɨɥɭɱɚɟɦ:

ɉɨɫɤɨɥɶɤɭ ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ ɹɜɥɹɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɜɟɥɢɱɢɧɨɣ, ɬɨ ɞɥɹ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ ɬɟɥɟɠɤɢ ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ:

ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ǻQ Qs , ɫɥɟɞɨɜɚɬɟɥɶɧɨ (9.117) ɦɨɠɧɨ ɭɩɪɨɫɬɢɬɶ, ɨɝɪɚɧɢɱɢɜɚɹɫɶ ɜ ɪɚɡɥɨɠɟɧɢɢ ɤɜɚɞɪɚɬɧɨɝɨ

ɉɨɞɫɬɚɧɨɜɤɚ ɱɢɫɥɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɡɚɞɚɧɧɵɯ ɜ ɡɚɞɚɱɟ, ɞɚɟɬ:

Z # 3,4 ɪɚɞ/ɫ .

Ɂɚɞɚɱɚ 9.7

ȼ ɭɩɪɭɝɨɣ ɨɞɧɨɪɨɞɧɨɣ ɫɪɟɞɟ ɫ ɩɥɨɬɧɨɫɬɶɸ U ɪɚɫɩɪɨɫɬɪɚɧɹɸɬɫɹ ɞɜɟ ɩɥɨɫɤɢɟ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɩɪɨɞɨɥɶɧɵɟ ɜɨɥɧɵ ɫɦɟɳɟɧɢɣ ɫɨ ɫɤɨɪɨɫɬɶɸ c, ɨɞɢɧɚɤɨɜɵɦɢ ɚɦɩɥɢɬɭɞɚɦɢ a ɢ ɱɚɫɬɨɬɚɦɢ Z , ɨɞɧɚ – ɜɞɨɥɶ ɨɫɢ X, ɞɪɭɝɚɹ – ɜɞɨɥɶ ɨɫɢ Y ɧɟɤɨɬɨɪɨɣ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. ɇɚɣɬɢ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ ɜɨɥɧɨɜɨɝɨ ɩɨɥɹ ɜɞɨɥɶ ɩɪɹɦɨɣ y = x ɜ ɩɥɨɫɤɨɫɬɢ XY, ɫɱɢɬɚɹ ɨɞɢɧɚɤɨɜɵɦɢ ɧɚɱɚɥɶɧɵɟ ɮɚɡɵ ɤɨɥɟɛɚɧɢɣ ɱɚɫɬɢɰ ɫɪɟɞɵ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ, ɨɛɭɫɥɨɜɥɟɧɧɵɯ ɤɚɠɞɨɣ ɜɨɥɧɨɣ ɜ ɨɬɞɟɥɶɧɨɫɬɢ.

Ɋɟɲɟɧɢɟ

I. ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɡɚɞɚɧɚ ɞɟɤɚɪɬɨɜɚ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ, ɜɞɨɥɶ ɨɫɟɣ X ɢ Y ɤɨɬɨɪɨɣ ɪɚɫɩɪɨɫɬɪɚɧɹɸɬɫɹ ɞɜɟ ɩɥɨɫɤɢɟ ɩɪɨɞɨɥɶɧɵɟ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɜɨɥɧɵ.

II. Ɂɚɩɢɲɟɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɡɚɤɨɧɵ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɛɟɝɭɳɢɯ ɩɥɨɫɤɢɯ ɩɪɨɞɨɥɶɧɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɜɨɥɧ

ɧɨɜɨɟ ɱɢɫɥɨ ɞɥɹ ɨɛɟɢɯ ɜɨɥɧ, M0 ɧɚɱɚɥɶɧɵɟ ɮɚɡɵ ɤɨɥɟɛɚɧɢɣ ɱɚɫ-

ɬɢɰ ɫɪɟɞɵ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ, ɨɛɭɫɥɨɜɥɟɧɧɵɯ ɤɚɠɞɨɣ ɜɨɥɧɨɣ ɜ ɨɬɞɟɥɶɧɨɫɬɢ.

Ɉɩɪɟɞɟɥɢɦ ɚɦɩɥɢɬɭɞɭ A ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ ɜɨɥɧɨɜɨɝɨ ɩɨɥɹ

ɇɚɩɪɚɜɢɦ ɜɫɩɨɦɨɝɚɬɟɥɶɧɭɸ ɨɫɶ ī ɜɞɨɥɶ ɩɪɹɦɨɣ y = x ɜ ɩɥɨɫɤɨɫɬɢ XY ɢ ɨɛɨɡɧɚɱɢɦ ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɜɞɨɥɶ ɷɬɨɝɨ ɧɚɩɪɚɜ-

ɋɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ ɜɨɥɧɨɜɨɝɨ ɜɨɡɦɭɳɟɧɢɹ ɜɞɨɥɶ ɩɪɹɦɨɣ y = x ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (9.48) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ:

ɡɧɚɱɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ ɜɨɥɧɨɜɨɝɨ ɩɨɥɹ ɜɞɨɥɶ ɩɪɹɦɨɣ y = x ɜ ɩɥɨɫɤɨɫɬɢ XY:

ɪɢɫ. 9.11). ɂɫɬɨɱɧɢɤ ɢ ɩɪɢɟɦɧɢɤ ɧɟɩɨɞɜɢɠɧɵ ɢ ɪɚɫɩɨɥɨɠɟɧɵ ɧɚ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɧɨɪɦɚɥɢ ɤ ɨɬɪɚɠɚɬɟɥɸ, ɤɨɬɨɪɵɣ ɭɞɚɥɹɟɬɫɹ ɨɬ ɢɫɬɨɱɧɢɤɚ ɫɨ ɫɤɨɪɨɫɬɶɸ u = 6 ɫɦ/ɫ. ɋɤɨɪɨɫɬɶ ɡɜɭɤɚ ɫ = 340 ɦ/ɫ. ɇɚɣɬɢ ɱɚɫɬɨɬɭ ɛɢɟɧɢɣ, ɪɟɝɢɫɬɪɢɪɭɟɦɵɯ ɩɪɢɟɦɧɢɤɨɦ.

358 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ

X

Ɋɢɫ. 9.11

Ɋɟɲɟɧɢɟ

I. ȼɵɛɟɪɟɦ ɧɚɩɪɚɜɥɟɧɢɟ ɨɫɢ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ, ɫɨɜɩɚɞɚɸɳɢɦ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɞɜɢɠɟɧɢɹ ɨɬɪɚɠɚɬɟɥɹ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 9.11. ɉɪɢɟɦɧɢɤ ɪɟɝɢɫɬɪɢɪɭɟɬ ɫɭɩɟɪɩɨɡɢɰɢɸ ɞɜɭɯ ɡɜɭɤɨɜɵɯ ɜɨɥɧ: ɢɫɩɭɳɟɧɧɨɣ ɢɫɬɨɱɧɢɤɨɦ ɢ ɨɬɪɚɠɟɧɧɨɣ ɨɬ ɞɜɢɠɭɳɟɝɨɫɹ ɨɬɪɚɠɚɬɟɥɹ.

II. ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ (9.51), ɫɜɹɡɵɜɚɸɳɟɣ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ ɞɜɢɠɭɳɢɯɫɹ ɢɫɬɨɱɧɢɤɚ ɢ ɩɪɢɟɦɧɢɤɚ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ:

ɬɟɥɶɧɨ ɫɪɟɞɵ, c – ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ ɜ ɫɪɟɞɟ, Qs – ɱɚɫɬɨɬɚ ɢɡɥɭɱɚɟɦɨɣ ɢɫɬɨɱɧɢɤɨɦ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ, Qd – ɱɚɫɬɨɬɚ ɜɨɥ-

ɧɵ, ɤɨɬɨɪɭɸ ɪɟɝɢɫɬɪɢɪɭɟɬ ɞɟɬɟɤɬɨɪ. ȼɫɟ ɫɤɨɪɨɫɬɢ ɧɚɩɪɚɜɥɟɧɵ ɜ ɨɞɧɭ ɫɬɨɪɨɧɭ.

ɑɚɫɬɨɬɚ ɜɨɥɧɵ, ɤɨɬɨɪɭɸ ɡɚɮɢɤɫɢɪɨɜɚɥ ɛɵ ɞɟɬɟɤɬɨɪ, ɧɚɯɨɞɹɳɢɣɫɹ ɧɚ ɞɜɢɠɭɳɟɦɫɹ ɨɬɪɚɠɚɬɟɥɟ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ

ɋɱɢɬɚɹ, ɱɬɨ ɨɬɪɚɠɚɬɟɥɶ ɧɟ ɦɟɧɹɟɬ ɱɚɫɬɨɬɭ ɜɨɥɧɵ ɩɪɢ ɨɬɪɚɠɟɧɢɢ, ɡɚɩɢɲɟɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɱɚɫɬɨɬɵ ɜɨɥɧɵ, ɨɬɪɚɠɟɧɧɨɣ ɨɬ ɨɬɪɚɠɚɬɟɥɹ ɢ ɡɚɪɟɝɢɫɬɪɢɪɨɜɚɧɧɨɣ ɧɟɩɨɞɜɢɠɧɵɦ ɩɪɢɟɦɧɢɤɨɦ ɫɨɝɥɚɫ-

ɧɨ (9.128):

Ɂɧɚɤ ɩɥɸɫ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɮɨɪɦɭɥɵ (9.130) ɨɛɭɫɥɨɜɥɟɧ ɬɟɦ, ɱɬɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɤɨɪɨɫɬɶ ɨɬɪɚɠɚɬɟɥɹ (ɢɫɬɨɱɧɢɤɚ ɨɬɪɚɠɟɧɧɨɣ

ɜɨɥɧɵ) ɢ ɫɤɨɪɨɫɬɶ ɨɬɪɚɠɟɧɧɨɣ ɜɨɥɧɵ ɧɚɩɪɚɜɥɟɧɵ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɫɬɨɪɨɧɵ.

Ⱦɥɹ ɱɚɫɬɨɬɵ ɛɢɟɧɢɣ (ɫɦ. ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ 8.10 ɜ Ƚɥɚɜɟ 8), ɜɨɡɧɢɤɚɸɳɢɯ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɭɩɟɪɩɨɡɢɰɢɢ ɜɨɥɧɵ ɫ ɱɚɫɬɨɬɨɣ Q0 , ɢɫɩɭ-

ɳɟɧɧɨɣ ɧɟɩɨɞɜɢɠɧɵɦ ɢɫɬɨɱɧɢɤɨɦ, ɢ ɜɨɥɧɵ ɫ ɱɚɫɬɨɬɨɣ Q 2 , ɨɬɪɚ-

ɉɨɞɫɬɚɜɥɹɹ ɜ (9.132) ɡɚɞɚɧɧɵɟ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɩɨɥɭɱɢɦ ɡɧɚɱɟɧɢɟ ɱɚɫɬɨɬɵ ɛɢɟɧɢɣ, ɡɚɪɟɝɢɫɬɪɢɪɨɜɚɧɧɵɯ ɩɪɢɟɦɧɢɤɨɦ:

Qɛɢɟɧ 0,6 Ƚɰ .

Ɂɚɞɚɱɚ 9.9

ɋɬɚɥɶɧɚɹ ɫɬɪɭɧɚ ɞɥɢɧɨɣ L = 110 ɫɦ, ɩɥɨɬɧɨɫɬɶɸ U = 7,8 ɝ/ɫɦ3 ɢ ɞɢɚɦɟɬɪɨɦ d = 1 ɦɦ ɧɚɬɹɧɭɬɚ ɦɟɠɞɭ ɩɨɥɸɫɚɦɢ ɷɥɟɤɬɪɨɦɚɝɧɢɬɚ. ɉɪɢ ɩɪɨɩɭɫɤɚɧɢɢ ɩɨ ɫɬɪɭɧɟ ɩɟɪɟɦɟɧɧɨɝɨ ɬɨɤɚ ɱɚɫɬɨɬɨɣ Q = 256 Ƚɰ ɜ ɧɟɣ ɜɨɡɛɭɠɞɚɟɬɫɹ ɭɩɪɭɝɚɹ ɩɨɩɟɪɟɱɧɚɹ ɜɨɥɧɚ, ɩɪɢɱɟɦ ɧɚ ɞɥɢɧɟ ɫɬɪɭɧɵ "ɭɤɥɚɞɵɜɚɟɬɫɹ" n = 5 ɩɨɥɭɜɨɥɧ. ɇɚɣɬɢ ɫɢɥɭ ɧɚɬɹɠɟɧɢɹ ɫɬɪɭɧɵ.

Ɋɟɲɟɧɢɟ

I. Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɢɡɦɟɧɟɧɢɟ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɫɬɪɭɧɵ, ɜɵɡɜɚɧɧɨɟ ɭɩɪɭɝɨɣ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɨɣ, ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɨ (ɫɦ. ɩ. 9.1.4.Ȼ). Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɨɫɶ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɢɦ ɜɞɨɥɶ ɫɬɪɭɧɵ (ɫɦ. ɪɢɫ. 9.12).

Ɋɢɫ. 9.12

II. Ɂɚɩɢɲɟɦ ɜɡɚɢɦɨɫɜɹɡɶ ɫɤɨɪɨɫɬɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɭɩɪɭɝɢɯ ɩɨɩɟɪɟɱɧɵɯ ɜɨɥɧ ɜ ɫɬɪɭɧɟ ɢ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɫɬɪɭɧɵ (ɫɦ. (9.27) ɜ

ɩ. 9.1.4.ȼ)

Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɢɫɤɨɦɨɣ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɫɬɪɭɧɵ ɧɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɭɩɪɭɝɢɯ ɩɨɩɟɪɟɱɧɵɯ ɜɨɥɧ ɜ ɫɬɪɭɧɟ (ɫɦ. (9.133)).

ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɧɚ ɞɥɢɧɟ ɫɬɪɭɧɵ "ɭɤɥɚɞɵɜɚɟɬɫɹ" n ɩɨɥɭ-

ɉɨɞɫɬɚɜɥɹɹ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɜɯɨɞɹɳɢɯ ɜ (9.137), ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ:

T # 77,7 ɇ .

Ɂɚɞɚɱɚ 9.10

ɇɚɣɬɢ ɱɚɫɬɨɬɵ Qn , ɧɚ ɤɨɬɨɪɵɯ ɛɭɞɟɬ ɪɟɡɨɧɢɪɨɜɚɬɶ ɬɪɭɛɚ

ɞɥɢɧɨɣ L = 1,7 ɦ, ɡɚɤɪɵɬɚɹ ɫ ɨɞɧɨɝɨ ɤɨɧɰɚ, ɟɫɥɢ ɫɤɨɪɨɫɬɶ ɡɜɭɤɚ ɜ ɜɨɡɞɭɯɟ ɪɚɜɧɚ c = 340 ɦ/ɫ.

Ɋɟɲɟɧɢɟ

I.ɑɚɫɬɨɬɵ, ɧɚ ɤɨɬɨɪɵɯ ɛɭɞɟɬ ɪɟɡɨɧɢɪɨɜɚɬɶ ɬɪɭɛɚ, ɫɨɜɩɚɞɚɸɬ

ɫɱɚɫɬɨɬɚɦɢ ɧɨɪɦɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ ɱɚɫɬɢɰ ɜɨɡɞɭɯɚ ɜ ɬɪɭɛɟ, ɨɛɪɚɡɭɸɳɢɯ ɫɬɨɹɱɢɟ ɜɨɥɧɵ.