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16) ɑɬɨ ɧɭɠɧɨ ɫɞɟɥɚɬɶ, ɱɬɨɛɵ ɭɜɟɥɢɱɢɬɶ ɪɟɡɭɥɶɬɢɪɭɸɳɭɸ ɫɤɨɪɨɫɬɶ ɪɚɤɟ-
ɬɵ ɩɪɢ ɬɨɦ ɠɟ ɤɨɥɢɱɟɫɬɜɟ ɜɵɛɪɨɲɟɧɧɵɯ ɝɚɡɨɜ?
1. ɍɦɟɧɶɲɢɬɶ ɜɪɟɦɹ ɪɚɡɝɨɧɚ ɪɚɤɟɬɵ.
2. ɍɜɟɥɢɱɢɬɶ ɜɪɟɦɹ ɪɚɡɝɨɧɚ ɪɚɤɟɬɵ.
3. ɍɦɟɧɶɲɢɬɶ ɨɛɳɭɸ ɧɚɱɚɥɶɧɭɸ ɦɚɫɫɭ ɪɚɤɟɬɵ.
4. ɍɜɟɥɢɱɢɬɶ ɨɛɳɭɸ ɧɚɱɚɥɶɧɭɸ ɦɚɫɫɭ ɪɚɤɟɬɵ.
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
17) Ʉɚɤ ɪɟɚɤɬɢɜɧɨɟ ɞɜɢɠɟɧɢɟ ɢɫɩɨɥɶɡɭɸɬ ɞɥɹ ɫɜɨɟɝɨ ɩɟɪɟɦɟɳɟɧɢɹ ɧɟɤɨɬɨ-
ɪɵɟ ɠɢɜɵɟ ɫɭɳɟɫɬɜɚ: ɤɚɥɶɦɚɪɵ, ɨɫɶɦɢɧɨɝɢ, ɤɚɪɚɤɚɬɢɰɵ, ɦɟɞɭɡɵ?
1. Ɋɟɡɤɨ ɢɡɦɟɧɹɸɬ ɮɨɪɦɭ.
2. ɋɨɜɟɪɲɚɸɬ ɦɨɳɧɵɟ ɞɜɢɠɟɧɢɹ ɳɭɩɚɥɶɰɚɦɢ.
3. ȼɫɚɫɵɜɚɸɬ, ɚ ɡɚɬɟɦ ɫ ɫɢɥɨɣ ɜɵɬɚɥɤɢɜɚɸɬ ɢɡ ɫɟɛɹ ɜɨɞɭ.
4. Ɂɚɤɨɧɵ ɪɟɚɤɬɢɜɧɨɝɨ ɞɜɢɠɟɧɢɹ ɧɟ ɢɫɩɨɥɶɡɭɸɬɫɹ.
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
18) ɉɨɱɟɦɭ ɜ ɦɨɦɟɧɬ ɜɤɥɸɱɟɧɢɹ ɜɨɞɵ ɤɨɧɟɰ ɪɟɡɢɧɨɜɨɝɨ ɲɥɚɧɝɚ, ɫɜɨɛɨɞɧɨ
ɥɟɠɚɳɟɝɨ ɧɚ Ɂɟɦɥɟ, ɫɨɜɟɪɲɚɟɬ ɪɟɡɤɨɟ ɞɜɢɠɟɧɢɟ ɜ ɫɬɨɪɨɧɭ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ
ɧɚɩɪɚɜɥɟɧɢɸ ɫɬɪɭɢ?
1. ȼɨɡɧɢɤɚɟɬ ɪɟɚɤɬɢɜɧɨɟ ɞɜɢɠɟɧɢɟ ɲɥɚɧɝɚ ɜɫɥɟɞɫɬɜɢɟ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ.
2. ɉɪɨɢɫɯɨɞɢɬ ɫɦɟɳɟɧɢɟ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɲɥɚɧɝɚ.
3. ȼ ɪɟɡɭɥɶɬɚɬɟ ɜɨɡɧɢɤɧɨɜɟɧɢɹ ɬɪɟɧɢɹ ɜɨɞɵ ɨ ɜɨɡɞɭɯ.
4. ȼ ɪɟɡɭɥɶɬɚɬɟ ɧɚɩɨɪɚ ɜɨɞɵ.
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ
ɩɪɟɞɥɨɠɟɧɨ.
141
9. ɈȻɔȿȾɂɇȿɇɇɕȿ ɄɈɇɌɊɈɅɖɇɕȿ ɌȿɋɌɕ
j
ɍɪɨɜɟɧɶ I
1. Ⱦɜɟ ɦɚɬɟɪɢɚɥɶɧɵɟ ɬɨɱɤɢ ɞɜɢɠɭɬɫɹ ɫɨɝɥɚɫɧɨ ɭɪɚɜɧɟɧɢɹɦ:
G
() (4 2 ) 2rt t t t i t
1
GG
23
(ɦ) ɢ
G
() (5 4 )rt t t t i
2
23
G
(ɦ).
ɇɚɣɬɢ ɫɤɨɪɨɫɬɢ ɷɬɢɯ ɬɨɱɟɤ ɜ ɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɢɯ ɭɫɤɨɪɟɧɢɹ ɪɚɜɧɵ.
2. ɉɭɬɶ, ɩɪɨɣɞɟɧɧɵɣ ɬɨɱɤɨɣ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ R = 4 ɦ, ɡɚɞɚɧ ɭɪɚɜ-
2
ɧɟɧɢɟɦ s(t) = 10 – 2t + t
(ɦ). ɇɚɣɬɢ ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ, ɧɨɪɦɚɥɶɧɨɟ ɢ ɩɨɥɧɨɟ ɭɫɤɨ-
ɪɟɧɢɹ ɬɨɱɤɢ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t = 2 ɫ.
3. ȼɟɫ ɬɟɥɚ ɦɚɫɫɨɣ ɬ = 100 ɤɝ, ɢɡɦɟɪɟɧɧɵɣ ɜ ɥɢɮɬɟ, ɞɜɢɠɭɳɟɦɫɹ ɜɧɢɡ, ɪɚ-
ɜɟɧ Ɋ = 1050 ɇ. Ʉɚɤɨɜ ɯɚɪɚɤɬɟɪ ɞɜɢɠɟɧɢɹ ɥɢɮɬɚ ɢ ɟɝɨ ɭɫɤɨɪɟɧɢɟ?
0
.
ɢ ɬ2.
1
4. ɇɟɩɨɞɜɢɠɧɨɟ ɚɬɨɦɧɨɟ ɹɞɪɨ ɪɚɡɥɟɬɚɟɬɫɹ ɧɚ ɞɜɚ ɨɫɤɨɥɤɚ ɦɚɫɫɚɦɢ ɬ
ɇɚɣɬɢ ɫɤɨɪɨɫɬɢ ɨɫɤɨɥɤɨɜ, ɟɫɥɢ ɢɯ ɨɛɳɚɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɪɚɜɧɚ Ɍ
5. Ɍɟɥɨ, ɭɩɚɜ ɫ ɧɟɤɨɬɨɪɨɣ ɜɵɫɨɬɵ, ɜ ɦɨɦɟɧɬ ɭɞɚɪɚ ɨ Ɂɟɦɥɸ ɨɛɥɚɞɚɟɬ ɢɦ-
ɩɭɥɶɫɨɦ ɪ = 100 ɤɝɦ/ɫ ɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɟɣ Ɍ = 500 Ⱦɠ. Ɉɩɪɟɞɟɥɢɬɶ ɦɚɫɫɭ
ɬɟɥɚ ɢ ɜɵɫɨɬɭ, ɫ ɤɨɬɨɪɨɣ ɨɧɨ ɭɩɚɥɨ.
6. Ⱥɜɬɨɦɚɲɢɧɚ ɦɚɫɫɨɣ 2 ɬɨɧɧɵ ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɡɚ 6 ɫɟɤɭɧɞ, ɩɪɨɣɞɹ ɪɚɫ-
ɫɬɨɹɧɢɟ 30 ɦɟɬɪɨɜ. Ɉɩɪɟɞɟɥɢɬɶ ɫɢɥɭ ɬɨɪɦɨɠɟɧɢɹ.
7. Ɉɛɪɭɱ ɦɚɫɫɨɣ ɬ = 1 ɤɝ, ɪɚɞɢɭɫ
ɤɨɬɨɪɨɝɨ ɪɚɜɟɧ R = 60 ɫɦ, ɩɪɢɜɹɡɚɧ ɤ ɜɟ-
ɪɟɜɤɟ ɞɥɢɧɨɣ Ɛ = 40 ɫɦ. ȼɟɪɟɜɤɭ ɫ ɨɛɪɭɱɟɦ ɜɪɚɳɚɸɬ ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ. ɇɚɣɬɢ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɛɪɭɱɚ.
8. ɍɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɞɢɫɤɚ ɦɚɫɫɨɣ 20 ɤɝ ɢɡɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ
ɫɨɝɥɚɫɧɨ ɝɪɚɮɢɤɭ. ɇɚɣɬɢ ɪɚɞɢɭɫ ɷɬɨɝɨ ɞɢɫɤɚ, ɟɫɥɢ ɨɧ ɜɪɚɳɚɟɬɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ
ɤɚɫɚɬɟɥɶɧɨɣ ɫɢɥɵ F = 120 ɇ.
9. Ʉɨɥɟɫɨ ɪɚɞɢɭɫɨɦ
R = 30 ɫɦ ɢ ɦɚɫɫɨɣ ɬ = 3 ɤɝ ɫɤɚɬɵɜɚɟɬɫɹ ɫ ɜɟɪɲɢɧɵ
ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɞɥɢɧɨɣ Ɛ = 5 ɦ ɢ ɭɝɥɨɦ ɧɚɤɥɨɧɚ ɤ ɝɨɪɢɡɨɧɬɭ 30q. Ɉɩɪɟɞɟɥɢɬɶ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɤɨɥɟɫɚ, ɟɫɥɢ ɫɤɨɪɨɫɬɢ ɟɝɨ ɭ ɩɨɞɧɨɠɢɹ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɫɨɫɬɚɜɥɹɟɬ
X
= 3,13 ɦ/ɫ.
10. Ⱦɢɫɤ, ɦɚɫɫɚ ɤɨɬɨɪɨɝɨ ɪɚɜɧɚ 5 ɤɝ, ɪɚɜɧɨɦɟɪɧɨ ɜɪɚɳɚɟɬɫɹ ɩɨ ɢɧɟɪɰɢɢ ɜ ɝɨ-
ɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɜɨɤɪɭɝ ɨɫɢ ɫɢɦɦɟɬɪɢɢ. ɇɚ ɧɟɝɨ ɫɜɟɪɯɭ ɩɚɞɚɟɬ ɢ ɨɫɬɚɟɬɫɹ
ɥɟɠɚɬɶ ɧɚ ɪɚɫɫɬɨɹɧɢɢ ɪɚɜɧɨɦ ɩɨɥɨɜɢɧɟ ɪɚɞɢɭɫɚ ɦɚɥɨɟ ɬɟɥɨ ɦɚɫɫɨɣ 1 ɤɝ. ȼɨ ɫɤɨɥɶɤɨ ɪɚɡ ɢɡɦɟɧɢɬɫɹ ɱɚɫɬɨɬɚ ɜɪɚɳɟɧɢɹ ɞɢɫɤɚ?
11. ɇɚɣɬɢ ɝɢɞɪɨɫɬɚɬɢɱɟɫɤɨɟ ɞɚɜɥɟɧɢɟ ɜ ɨɡɟɪɟ ɧɚ ɝɥɭɛɢɧɟ 20
ɭɫɤɨɪɟɧɢɹ ɫɜɨɛɨɞɧɨɝɨ ɩɚɞɟɧɢɹ ɩɪɢɧɹɬɶ ɪɚɜɧɨɣ 10 ɦ/ɫ
3
ɧɹɬɶ ɪɚɜɧɨɣ 1000 ɤɝ/ɦ
.
2
. ɉɥɨɬɧɨɫɬɶ ɜɨɞɵ ɩɪɢ-
ɦ. ȼɟɥɢɱɢɧɭ
12. ɇɚɣɬɢ ɞɚɜɥɟɧɢɟ ɜ ɨɬɤɪɵɬɨɦ ɜɨɞɨɟɦɟ ɩɪɢ ɧɨɪɦɚɥɶɧɨɦ ɚɬɦɨɫɮɟɪɧɨɦ
3
ɞɚɜɥɟɧɢɢ ɧɚ ɝɥɭɛɢɧɟ 20 ɦ. ɉɥɨɬɧɨɫɬɶ ɜɨɞɵ ɩɪɢɧɹɬɶ ɪɚɜɧɨɣ 1000 ɤɝ/ɦ
.
142
ɍɪɨɜɟɧɶ II
G
1. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɡɚɞɚɧɨ ɭɪɚɜɧɟɧɢɟɦ
() (2 0,5 )rt t t i
Ɉɩɪɟɞɟɥɢɬɶ ɫɪɟɞɧɸɸ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ ɨɬ t
= 3 ɫ.
ɞɨ t
2
2
= 1 ɫ
1
G
2. Ɍɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɤɪɢɜɨɣ ɫ ɩɨɫɬɨɹɧɧɵɦ ɬɚɧɝɟɧɰɢɚɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
ɚ
= 0,5 ɦ/ɫ2. Ɉɩɪɟɞɟɥɢɬɶ ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɱɟɪɟɡ 4 ɫɟɤɭɧɞɵ ɨɬ ɧɚɱɚɥɚ
W
ɞɜɢɠɟɧɢɹ ɧɚ ɭɱɚɫɬɤɟ ɤɪɢɜɨɣ ɫ ɪɚɞɢɭɫɨɦ ɤɪɢɜɢɡɧɵ R = 8 ɦ.
3. Ɇɚɝɧɢɬ ɦɚɫɫɨɣ 50 ɝ ɩɪɢɥɢɩ ɤ ɠɟɥɟɡɧɨɣ ɜɟɪɬɢɤɚɥɶɧɨɣ ɫɬɟɧɤɟ. Ⱦɥɹ ɪɚɜ-
ɧɨɦɟɪɧɨɝɨ ɫɤɨɥɶɠɟɧɢɹ ɦɚɝɧɢɬɚ ɜɧɢɡ ɩɪɢɤɥɚɞɵɜɚɸɬ ɫɢɥɭ f = 2 ɇ. Ʉɚɤɭɸ ɫɢɥɭ
ɧɚɞɨ ɩɪɢɥɨɠɢɬɶ, ɱɬɨɛɵ ɦɚɝɧɢɬ ɧɚɱɚɥ ɫɤɨɥɶɡɢɬɶ ɜɜɟɪɯ?
4. ɑɟɥɨɜɟɤ, ɫɬɨɹɳɢɣ ɧɚ ɥɶɞɭ, ɬɨɥɤɚɟɬ
ɫɚɧɢ ɦɚɫɫɨɣ 6 ɤɝ ɬɚɤ, ɱɬɨ ɨɧɢ ɨɬɤɚɬɵɜɚɸɬɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɟ 10 ɦɟɬɪɨɜ. ɇɚ ɤɚɤɨɟ ɪɚɫɫɬɨɹɧɢɟ ɩɪɢ ɷɬɨɦ ɨɬɤɚɬɢɬɫɹ ɱɟɥɨɜɟɤ, ɟɫɥɢ ɟɝɨ ɦɚɫɫɚ 60 ɤɝ?
5. Ɍɟɥɨ ɞɜɢɠɟɬɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ
G
2Ftij
GG
(ɇ). ɇɚɣɬɢ ɪɚɛɨɬɭ ɷɬɨɣ
ɫɢɥɵ, ɟɫɥɢ ɤɨɨɪɞɢɧɚɬɚ ɯ ɬɟɥɚ ɭɜɟɥɢɱɢɥɚɫɶ ɧɚ 'ɯ = 3 ɦ.
6. ɇɚ ɤɚɤɭɸ ɜɵɫɨɬɭ ɩɨɞɧɢɦɟɬɫɹ ɬɟɥɨ, ɫɤɨɥɶɡɹ ɜɜɟɪɯ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɫ ɭɝɥɨɦ
ɧɢɹ ɬɟɥɚ ɨ ɧɚɤɥɨɧɧɭɸ ɩɥɨɫɤɨɫɬɶ
D
= 45q, ɟɫɥɢ ɟɦɭ ɫɨɨɛɳɢɬɶ ɫɤɨɪɨɫɬɶ
P
= 0,2.
X
= 10 ɦ/ɫ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟ-
0
7. Ɉɞɧɨɪɨɞɧɵɣ ɞɢɫɤ ɦɚɫɫɨɣ 4 ɤɝ ɞɢɚɦɟɬɪɨɦ 60 ɫɦ ɜɪɚɳɚɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ, ɭɞɚɥɟɧɧɭɸ ɨɬ ɰɟɧɬɪɚ ɞɢɫɤɚ ɧɚ ɪɚɫɫɬɨɹɧɢɟ 20 ɫɦ.
Ɉɫɶ ɜɪɚɳɟɧɢɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɩɥɨɫɤɨɫɬɢ ɞɢɫɤɚ. ȼɨ ɫɤɨɥɶɤɨ ɪɚɡ ɢɡɦɟɧɢɬɫɹ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɟɝɨ ɜɪɚɳɟɧɢɹ, ɟɫɥɢ ɨɫɶ ɜɪɚɳɟɧɢɹ ɛɭɞɟɬ ɫɨɜɩɚɞɚɬɶ ɫ ɨɫɶɸ ɞɢɫɤɚ. ȼɪɚɳɚɸɳɢɣ ɦɨɦɟɧɬ ɫɢɥ ɫɱɢɬɚɬɶ ɩɨɫɬɨɹɧɧɵɦ.
8. Ɇɚɯɨɜɢɤ ɜ
ɳɢɣɫɹ ɫ ɱɚɫɬɨɬɨɣ 6 ɫ
ɜɢɞɟ ɞɢɫɤɚ ɦɚɫɫɨɣ ɬ = 10 ɤɝ ɢ ɪɚɞɢɭɫɨɦ R = 10 ɫɦ, ɜɪɚɳɚɸ-
–1
, ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɩɪɢ ɬɨɪɦɨɠɟɧɢɢ ɱɟɪɟɡ 5 ɫɟɤɭɧɞ. Ɉɩɪɟ-
ɞɟɥɢɬɶ ɬɨɪɦɨɡɹɳɢɣ ɦɨɦɟɧɬ.
9. Ⱦɢɫɤ ɧɚɱɢɧɚɟɬ ɜɪɚɳɚɬɶɫɹ ɜɨɤɪɭɝ ɫɜɨɟɣ ɨɫɢ ɫ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ
= 0,4 ɪɚɞ/ɫ
2
. Ɉɩɪɟɞɟɥɢɬɶ ɟɝɨ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɱɟɪɟɡ 10 ɫɟɤɭɧɞ ɨɬ ɧɚɱɚɥɚ
ɞɜɢɠɟɧɢɹ, ɟɫɥɢ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɩɪɢ ɷɬɨɦ ɞɨɫɬɢɝɚɟɬ ɜɟɥɢɱɢɧɵ L = 60 ɤɝɦ
10. Ɉɞɧɨɪɨɞɧɵɣ ɬɨɧɤɢɣ ɫɬɟɪɠɟɧɶ ɦɚɫɫɨɣ ɬ
ɢ ɞɥɢɧɨɣ Ɛ ɦɨɠɟɬ ɫɜɨɛɨɞɧɨ
1
2
/ɫ.
H
=
ɜɪɚɳɚɬɶɫɹ ɜɨɤɪɭɝ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɧɚ ɪɚɫɫɬɨɹɧɢɢ ɚ = Ɛ/4
ɨɬ ɜɟɪɯɧɟɝɨ ɤɨɧɰɚ ɫɬɟɪɠɧɹ, ɜ ɤɨɬɨɪɵɣ ɩɨɩɚɞɚɟɬ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɥɟɬɹɳɢɣ ɩɥɚɫɬɢɥɢɧɨɜɵɣ ɲɚɪɢɤ ɦɚɫɫɨɣ ɬ
ɫɨ ɫɤɨɪɨɫɬɶɸ X. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ
2
ɫɬɟɪɠɧɹ ɩɨɫɥɟ ɭɞɚɪɚ.
.
143
11. ɉɨɥɧɨɟ ɞɚɜɥɟɧɢɟ ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɬɪɭɛɟ ɪɚɜɧɨ 1500 ɉɚ, ɜɟɥɢɱɢɧɚ ɫɤɨɪɨɫɬɢ ɩɨɬɨɤɚ ɜɨɞɵ ɪɚɜɧɚ 1 ɦ/ɫ. ɇɚɣɬɢ ɫɬɚɬɢɱɟɫɤɨɟ ɞɚɜɥɟɧɢɟ ɜ ɬɪɭɛɟ. ɉɥɨɬɧɨɫɬɶ
3
ɜɨɞɵ ɩɪɢɧɹɬɶ ɪɚɜɧɨɣ 1000 ɤɝ/ɦ
.
12. ȼ ɬɪɭɛɟ ɞɢɚɦɟɬɪɨɦ 6 ɫɦ ɫɤɨɪɨɫɬɶ ɩɨɬɨɤɚ 1 ɦ/ɫ. Ɋɚɫɫɱɢɬɚɬɶ ɜɟɥɢɱɢɧɭ
ɫɤɨɪɨɫɬɢ ɩɨɬɨɤɚ ɧɚ ɭɱɚɫɬɤɟ ɫɭɠɟɧɢɹ ɬɪɭɛɵ ɞɢɚɦɟɬɪɨɦ 3 ɫɦ.
ɍɪɨɜɟɧɶ III
1. Ɍɟɥɨ, ɛɪɨɲɟɧɧɨɟ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ, ɧɚɯɨɞɢɥɨɫɶ ɧɚ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɜɵɫɨɬɟ h = 8 ɦ ɞɜɚ ɪɚɡɚ ɫ ɢɧɬɟɪɜɚɥɨɦ 't = 2 ɫ. ɉɪɟɧɟɛɪɟɝɚɹ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ ɜɨɡɞɭɯɚ, ɜɵɱɢɫɥɢɬɶ ɧɚɱɚɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɛɪɨɲɟɧɧɨɝɨ ɬɟɥɚ.
2. Ɍɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ R = 2 ɦ ɬɚɤ, ɱɬɨ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɟɟ
M
ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ
(t) = 2t3. ɇɚɣɬɢ ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɜ
ɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɟɟ ɧɨɪɦɚɥɶɧɨɟ ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɹ ɪɚɜɧɵ.
3. ɋ ɤɚɤɢɦ ɭɫɤɨɪɟɧɢɟɦ ɛɭɞɟɬ ɫɤɨɥɶɡɢɬɶ ɬɟɥɨ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɫ ɭɝ-
ɥɨɦ ɧɚɤɥɨɧɚ 60q, ɟɫɥɢ ɩɪɢ ɭɝɥɟ ɧɚɤɥɨɧɚ 30q ɬɟɥɨ ɞɜɢɠɟɬɫɹ ɪɚɜɧɨɦɟɪɧɨ?
4. ɒɚɪɢɤ ɦɚɫɫɨɣ ɬ = 50 ɝ, ɩɚɞɚɹ ɫ ɜɵɫɨɬɵ h = 5 ɦ, ɭɩɪɭɝɨ ɨɬɫɤɚɤɢɜɚɟɬ ɨɬ
ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɞɫɬɚɜɤɢ. ɇɚɣɬɢ ɫɪɟɞɧɸɸ ɫɢɥɭ ɭɞɚɪɚ, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɩɨɞɫɬɚɜɤɭ, ɟɫɥɢ ɜɪɟɦɹ ɫɨɭɞɚɪɟɧɢɹ
'
t = 0,01 ɫ.
5. ɉɪɢ ɭɩɪɭɝɨɦ ɭɞɚɪɟ ɧɟɣɬɪɨɧɚ ɨ ɹɞɪɨ ɭɝɥɟɪɨɞɚ ɨɧ ɢɡɦɟɧɹɟɬ ɧɚɩɪɚɜɥɟɧɢɟ
ɫɤɨɪɨɫɬɢ ɧɚ 90q. Ɉɩɪɟɞɟɥɢɬɶ, ɜɨ ɫɤɨɥɶɤɨ ɪɚɡ ɢɡɦɟɧɢɬɫɹ ɷɧɟɪɝɢɹ ɧɟɣɬɪɨɧɚ ɜ ɪɟɡɭɥɶɬɚɬɟ ɭɞɚɪɚ, ɫɱɢɬɚɹ, ɱɬɨ ɦɚɫɫɚ ɹɞɪɚ ɭɝɥɟɪɨɞɚ ɜ 12 ɪɚɡ ɛɨɥɶɲɟ ɦɚɫɫɵ ɧɟɣɬɪɨɧɚ.
6. ȼɟɬɟɪ ɞɟɣɫɬɜɭɟɬ ɧɚ ɩɚɪɭɫ ɩɥɨɳɚɞɶɸ S ɫ ɫɢɥɨɣ F = A(
ɝɞɟ Ⱥ = const,
X
– ɫɤɨɪɨɫɬɶ ɜɟɬɪɚ,
ɜ
X
– ɫɤɨɪɨɫɬɶ ɥɨɞɤɢ. Ɉɩɪɟɞɟɥɢɬɶ, ɩɪɢ ɤɚɤɨɣ
ɥ
X
–
ɜ
X
)2,
ɥ
ɫɤɨɪɨɫɬɢ ɥɨɞɤɢ ɦɝɧɨɜɟɧɧɚɹ ɦɨɳɧɨɫɬɶ ɜɟɬɪɚ ɦɚɤɫɢɦɚɥɶɧɚ.
7. ɑɟɦɭ ɪɚɜɟɧ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɲɚɪɚ ɦɚɫɫɨɣ ɬ ɢ ɪɚɞɢɭɫɚ R, ɟɫɥɢ ɜɧɭɬɪɢ ɧɟɝɨ ɢɦɟɟɬɫɹ ɩɨɥɨɫɬɶ ɪɚɞɢɭɫɚ r, ɰɟɧɬɪ ɤɨɬɨɪɨɣ ɫɨɜɩɚɞɚɟɬ ɫ ɰɟɧɬɪɨɦ ɲɚɪɚ?
8. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɫ ɧɨɪɦɚɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
~ t4. ɉɪɢ ɷɬɨɦ ɦɨɦɟɧɬ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɨɱɤɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɜɪɚ-
ɚ
ɩ
ɳɟɧɢɹ, ɜɵɪɚɠɚɟɬɫɹ ɫɬɟɩɟɧɧɨɣ ɮɭɧɤɰɢɟɣ Ɇ ~ t
ɩ
. ɇɚɣɬɢ ɡɧɚɱɟɧɢɟ ɩ.
9. Ɉɞɧɨɪɨɞɧɵɣ ɫɬɟɪɠɟɧɶ ɞɥɢɧɨɣ Ɛ = 1 ɦ ɩɨɞɜɟɲɟɧ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ,
ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɜɟɪɯɧɢɣ ɤɨɧɟɰ. ɇɚ ɤɚɤɨɣ ɭɝɨɥ ɧɚɞɨ ɨɬɤɥɨɧɢɬɶ ɫɬɟɪɠɟɧɶ,
ɱɬɨɛɵ ɟɝɨ ɧɢɠɧɢɣ ɤɨɧɟɰ ɩɪɢ ɩɪɨɯɨɠɞɟɧɢɢ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɢɦɟɥ ɫɤɨ-
X
ɪɨɫɬɶ
= 5 ɦ/ɫ?
10. Ⱦɜɚ ɲɚɪɚ ɨɞɢɧɚɤɨɜɨɝɨ ɪɚɡɦɟɪɚ, ɢɡɝɨɬɨɜɥɟɧɧɵɟ ɢɡ ɦɚɬɟɪɢɚɥɨɜ, ɩɥɨɬɧɨɫɬɢ ɤɨɬɨɪɵɯ ɪɚɜɧɵ
U
ɢ
U
, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɜɪɚɳɚɸɬɫɹ ɧɟɡɚɜɢɫɢɦɨ ɞɪɭɝ ɨɬ ɞɪɭ-
1
2
ɝɚ ɜɨɤɪɭɝ ɨɛɳɟɣ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɢɯ ɰɟɧɬɪɵ ɫ ɭɝɥɨɜɵɦɢ
144
Z
ɢ
Z
ɫɤɨɪɨɫɬɹɦɢ
. ɋ ɤɚɤɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Z ɛɭɞɭɬ ɜɪɚɳɚɬɶɫɹ ɨɛɚ ɲɚɪɚ,
1
2
ɟɫɥɢ ɢɯ ɠɟɫɬɤɨ ɫɨɟɞɢɧɢɬɶ?
11. ɇɚɣɬɢ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ, ɡɚ ɤɨɬɨɪɵɣ ɩɪɢ ɫɤɨɪɨɫɬɢ ɢɫɬɟɱɟɧɢɹ 10 ɦ/ɫ
ɢɡ ɩɨɠɚɪɧɨɝɨ ɲɥɚɧɝɚ ɞɢɚɦɟɬɪɨɦ 66 ɦɦ ɜɵɬɟɱɟɬ 2052 ɤɝ ɜɨɞɵ.
12. Ɉɩɪɟɞɟɥɢɬɶ ɯɚɪɚɤɬɟɪ ɞɜɢɠɟɧɢɹ ɜɨɞɵ ɜ ɬɪɭɛɟ ɞɢɚɦɟɬɪɨɦ 5 ɫɦ, ɟɫɥɢ ɫɤɨ-
ɪɨɫɬɶ ɩɨɬɨɤɚ ɪɚɜɧɚ 4 ɫɦ/ɫ. Ʉɨɷɮɮɢɰɢɟɧɬ ɞɢɧɚɦɢɱɟɫɤɨɣ ɜɹɡɤɨɫɬɢ ɜɨɞɵ ɩɪɢɧɹɬɶ
ɪɚɜɧɵɦ 0,001 ɉɚāɫ. ɉɥɨɬɧɨɫɬɶ
ɜɨɞɵ ɩɪɢɧɹɬɶ ɪɚɜɧɨɣ 1000 ɤɝ/ɦ3.
145
10. ɁȺȾȺɑɂ ɉɈ ɆȿɏȺɇɂɄȿ
V
U
U
Q
ɋ ɌȿɏɇɂɑȿɋɄɂɆ ɋɈȾȿɊɀȺɇɂȿɆ
10.1. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 1. Ɉɩɪɟɞɟɥɢɬɶ ɦɚɫɫɭ ɜɨɞɵ ɜ ɩɨɠɚɪɧɨɦ ɪɭɤɚɜɟ ɞɢɚɦɟɬɪɨɦ 51 ɦɦ
ɢ ɞɥɢɧɨɣ 20 ɦ.
Ⱦɚɧɨ: d = 51 ɦɦ; l = 20 ɦ.
ɇɚɣɬɢ: m.
Ɋɟɲɟɧɢɟ:
Ɇɚɫɫɚ ɜɨɞɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɮɨɪɦɭɥɵ
.
mU
ɉɥɨɬɧɨɫɬɶ ɜɨɞɵ ɩɪɢɧɢɦɚɟɦ
1000
ɤɝ/ɦ
3
.
Ɉɛɴɟɦ ɜɨɞɵ:
22
051,014,3
d
S
V
4
l
4
3
.ɦ04084,020
m = 1000ā0,04084 = 40,84 ɤɝ.
Ɉɬɜɟɬ: m = 40,84 ɤɝ.
Ɂɚɞɚɱɚ 2. Ɉɩɪɟɞɟɥɢɬɶ ɞɢɧɚɦɢɱɟɫɤɭɸ ɜɹɡɤɨɫɬɶ ɧɟɮɬɢ, ɟɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬ
ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɜɹɡɤɨɫɬɢ ɫɨɫɬɚɜɥɹɟɬ 0,624ā10
Ⱦɚɧɨ: Ȟ = 0,624ā10
ɇɚɣɬɢ:
K
.
-4 ɦ2
/ɫ; ȡ = 750 ɤɝ/ɦ3.
-4 ɦ2
/ɫ. ɉɥɨɬɧɨɫɬɶ ɧɟɮɬɢ 750 ɤɝ/ɦ3.
Ɋɟɲɟɧɢɟ:
Ⱦɢɧɚɦɢɱɟɫɤɚɹ ɜɹɡɤɨɫɬɶ ɧɟɮɬɢ:
K
= 0,624ā10
-4
ā750= 4,68ā10-2 ɉɚāɫ.
Ɉɬɜɟɬ:
K = 4,68ā10-2 ɉɚāɫ.
Ɂɚɞɚɱɚ 3. Ɉɩɪɟɞɟɥɢɬɶ ɪɚɫɯɨɞ ɜɨɞɵ (Q) ɢ ɫɪɟɞɧɸɸ ɫɤɨɪɨɫɬɶ ɜɵɯɨɞɚ ɜɨɞɵ
X
ɢɡ ɧɚɫɚɞɤɚ (
) ɩɨɠɚɪɧɨɝɨ ɫɬɜɨɥɚ ɞɢɚɦɟɬɪɨɦ 13 ɦɦ, ɟɫɥɢ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ
ɧ
ɜɨɞɵ ɩɨ ɪɭɤɚɜɭ ɞɢɚɦɟɬɪɨɦ 51 ɦɦ ɫɨɫɬɚɜɥɹɟɬ 2,2 ɦ/ɫ.
Ⱦɚɧɨ: d
ɇɚɣɬɢ: Q,
= 13 ɦɦ = 0,013 ɦ; dɪ = 51 ɦɦ = 0,051 ɦ;
ɧ
X
.
ɧ
X
= 2,2 ɦ/ɫ.
ɪ
Ɋɟɲɟɧɢɟ:
Ɉɩɪɟɞɟɥɹɟɦ ɩɥɨɳɚɞɶ ɠɢɜɨɝɨ ɫɟɱɟɧɢɹ ɩɨɠɚɪɧɨɝɨ ɪɭɤɚɜɚ.
= 0,785 d
S
ɪ
2
= 0,785ā0,0512 = 0,00204 ɦ2.
ɪ
Ɉɩɪɟɞɟɥɹɟɦ ɩɥɨɳɚɞɶ ɫɟɱɟɧɢɹ ɧɚɫɚɞɤɚ.
= 0,785 d
S
ɧ
2
= 0,785ā0,0132 = 0,000133 ɦ2.
ɧ
Ɉɩɪɟɞɟɥɹɟɦ ɪɚɫɯɨɞ ɜɨɞɵ.
X
= Q/S Q =
X
= 2,2ā0,002 = 0,0045 ɦ3/ɫ = 4,5 ɥ/ɫ.
ɪ Sɪ
Ɉɩɪɟɞɟɥɹɟɦ ɫɪɟɞɧɸɸ ɫɤɨɪɨɫɬɶ ɜɨɞɵ ɢɡ ɧɚɫɚɞɤɚ:
X
/
X
ɧ
= Sɪ /S
ɪ
ɧ
X
= (
X
ɧ
)/Sɧ = (2,2 ā 0,00204)/0,000133 = 33,74 ɦ/ɫ.
ɪ Sɪ
Ɉɬɜɟɬ: Q = 4,5 ɥ/ɫ;
X
= 33,74 ɦ/ɫ.
ɧ
146
Ɂɚɞɚɱɚ 4. Ɉɩɪɟɞɟɥɢɬɶ, ɩɪɢ ɤɚɤɨɦ ɪɚɫɯɨɞɟ ɜɨɞɵ ɩɨ ɩɨɠɚɪɧɨɦɭ ɪɭɤɚɜɭ ɞɢɚ-
tSV
X
t
sut
S
X
s
s
X
ɦɟɬɪɨɦ 66 ɦɦ ɪɟɠɢɦ ɞɜɢɠɟɧɢɹ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɥɚɦɢɧɚɪɧɵɦ. Ʉɨɷɮɮɢɰɢɟɧɬ ɤɢ-
-6 ɦ2
ɧɟɦɚɬɢɱɟɫɤɨɣ ɜɹɡɤɨɫɬɢ
Ⱦɚɧɨ: d = 66 ɦɦ; Re =2320;
= 1ā10
Q
= 1ā10
Q
/ɫ.
-6 ɦ2
/ɫ.
ɇɚɣɬɢ: Q.
Ɋɟɲɟɧɢɟ:
Ɉɩɪɟɞɟɥɢɦ ɫɪɟɞɧɸɸ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɜɨɞɵ, ɩɪɢ ɤɨɬɨɪɨɣ ɪɟɠɢɦ ɛɭɞɟɬ
ɥɚɦɢɧɚɪɧɵɦ.
Q
X
d
6
1012320Re
066,0
0,035 ɦ/ɫ.
Ɋɚɫɯɨɞ ɜɨɞɵ:
Q =
X
Sd2/4
14,3
035,0
4
42
ɦ3/ɫ.
102,1066,0
Ɉɬɜɟɬ: Q = 1,2 ā 10
-4 ɦ3
/ɫ.
Ɂɚɞɚɱɚ 5. ɇɚɣɬɢ ɤɨɷɮɮɢɰɢɟɧɬ ɩɨɥɟɡɧɨɝɨ ɞɟɣɫɬɜɢɹ ɜɨɞɨɦɟɬɧɨɝɨ ɞɜɢɝɚɬɟɥɹ
ɪɟɚɤɬɢɜɧɨɝɨ ɤɚɬɟɪɚ, ɟɫɥɢ ɩɥɨɳɚɞɶ ɜɯɨɞɧɨɝɨ ɨɬɜɟɪɫɬɢɹ ɞɜɢɝɚɬɟɥɹ S = 0,9 ɦ
ɚ ɜɵɯɨɞɧɨɝɨ ɨɬɜɟɪɫɬɢɹ s = 0,02 ɦ
2
.
Ɋɟɲɟɧɢɟ:
Ɂɚ ɜɪɟɦɹ
οݐ ɞɜɢɝɚɬɟɥɶ ɡɚɫɚɫɵɜɚɟɬ ɨɛɴɟɦ ɜɨɞɵ:
'
.
Ɋɚɛɨɬɚ Ⱥ ɞɜɢɝɚɬɟɥɹ ɪɚɜɧɚ ɢɡɦɟɧɟɧɢɸ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɜɨɞɵ. Ɉɛɴɟɦ
ɜɨɞɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɚɬɟɪɚ ɢɦɟɟɬ ɫɤɨɪɨɫɬɶ
1
ɤ
2
1
2
߭ ɢ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ:
1
322
XUXUXXU
2
.
tStSVE ' '
ȼɨɞɚ ɜɵɛɪɚɫɵɜɚɟɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ ɢ, ɢɦɟɟɬ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ:
1
ɤ
2
Ⱦɥɹ ɧɟɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɢ: ,
X
u
Ɍɨɝɞɚ ɮɨɪɦɭɥɚ ɞɥɹ ɪɚɫɱɟɬɚ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɜɵɛɪɚɫɵɜɚɟɦɨɣ ɜɨɞɵ
2/
.)(
utSE
UX
'
' '
S
.
s
ɩɪɢɦɟɬ ɜɢɞ:
3
1
S
/
E '
ɤ
2
3
.
t
XU
2
Ɋɚɛɨɬɚ ɪɚɜɧɚ ɢɡɦɟɧɟɧɢɸ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɜɨɞɵ:
23
sSS
/
ɤ
1
EEA '
ɤ
2
)(
3
.
t
XU
2
ɉɨɥɟɡɧɚɹ ɪɚɛɨɬɚ ɪɚɜɧɚ:
tFA
'
,
ɝɞɟ F
ɩɨɥɟɡ
– ɪɟɚɤɬɢɜɧɚɹ ɫɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɤɨɪɚɛɥɶ.
ɪɟɚɤ
ɪɟɚɤ
2
,
147
Ɋɟɚɤɬɢɜɧɚɹ ɫɢɥɚ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɪɚɜɧɚ ɢɡɦɟɧɟɧɢɸ ɢɦɩɭɥɶɫɚ ɜɨɞɵ
X
s
t
V
U
X
ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ:
ɪɟɚɤ
'
U
'
)()(
F
/
'
UX
mmu
t
'
t
t
t
'
t
2
§
S
2
¨
s
¨
©
ɩɨɥɟɡ
s
ɩɨɥɟɡ
2
Susu
·
¸
¸
¹
ɪɟɚɤ
3
XU
SA
s
'
U
t
XX
'
t
sS
22
SS
XUXX
s
,
tFA
'
)(
sS
.
t
'
22
X
Ssu
)(
,
Ʉɨɷɮɮɢɰɢɟɧɬ ɩɨɥɟɡɧɨɝɨ ɞɟɣɫɬɜɢɹ ɜɨɞɨɦɟɬɧɨɝɨ ɞɜɢɝɚɬɟɥɹ ɤɚɬɟɪɚ:
ɄɉȾ
A
ɩɨɥɟɡ
A
sS
3
S
ɄɉȾ 100 %.
)(
s
2s
Ss
t
/
1
'
2
sSS
2
Ɉɬɜɟɬ:
22
)(
3
'
,
t
XUXU
2s
ɄɉȾ 100 %.
Ss
Ɂɚɞɚɱɚ 6. ɇɚɫɨɫ ɩɨɞɚɟɬ ɨɛɴɟɦ V ɜɨɞɵ ɜ ɱɚɫ ɧɚ ɜɵɫɨɬɭ H ɩɨ ɬɪɭɛɟ ɞɢɚɦɟɬ-
ɪɨɦ d. Ʉɚɤɨɣ ɜɟɥɢɱɢɧɵ ɞɨɥɠɧɚ ɛɵɬɶ ɦɨɳɧɨɫɬɶ ɧɚɫɨɫɚ? Ɇɨɠɧɨ ɥɢ ɫ ɩɨɦɨɳɶɸ
ɧɚɫɨɫɚ ɦɟɧɶɲɟɣ ɦɨɳɧɨɫɬɢ ɩɨɞɚɜɚɬɶ ɨɛɴɟɦ V ɜɨɞɵ ɜ ɱɚɫ ɧɚ ɜɵɫɨɬɭ H?
Ɋɟɲɟɧɢɟ:
Ɂɚ ɜɪɟɦɹ
ɇɚɫɨɫ «ɝɨɧɢɬ» ɜɨɞɭ ɫɨ ɫɤɨɪɨɫɬɶɸ
οݐ ɧɚɫɨɫ ɩɨɞɚɟɬ ɧɚ ɜɵɫɨɬɭ ɇ ɦɚɫɫɭ ɜɨɞɵ:
m '
.
ɢ ɫɨɜɟɪɲɚɟɬ ɪɚɛɨɬɭ, ɪɚɜɧɭɸ ɢɡɦɟɧɟ-
ɧɢɸ ɩɨɥɧɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɜɨɞɵ:
2
)(
tV
'
A '
XU
2
U
.)(
gHtV
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɳɧɨɫɬɶ ɧɚɫɨɫɚ:
A
N
;
N
t
'
ɇɚɣɞɟɦ ɫɤɨɪɨɫɬɶ ɩɨɞɚɱɢ ɜɨɞɵ. Ɂɚ ɜɪɟɦɹ
2
V
XU
2
U
VgH
.
οݐ ɱɟɪɟɡ ɩɨɩɟɪɟɱɧɨɟ ɫɟɱɟɧɢɟ ɬɪɭ-
ɛɵ ɩɥɨɳɚɞɶɸ S ɩɪɨɯɨɞɢɬ ɨɛɴɟɦ ɜɨɞɵ:
2
d
S
.
' '
ttV
X
4
ɂɡ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɚɣɞɟɦ ɮɨɪɦɭɥɭ ɞɥɹ ɪɚɫɱɟɬɚ ɫɤɨɪɨɫɬɢ ɩɨɞɚɱɢ ɜɨɞɵ:
V
4
.
X
2
d
S
Ⱦɥɹ ɪɚɫɱɟɬɚ ɦɨɳɧɨɫɬɢ ɧɚɫɨɫɚ, ɨɤɨɧɱɚɬɟɥɶɧɨ, ɩɨɥɭɱɚɟɦ ɮɨɪɦɭɥɭ:
3
U
32
V
N
S
42
d
U
VgH
.
148
ɂɡ ɩɨɥɭɱɟɧɧɨɝɨ ɜɵɪɚɠɟɧɢɹ ɫɥɟɞɭɸɬ ɜɵɜɨɞɵ:
H
N
U
ɱɟɦ ɛɨɥɶɲɟ ɞɢɚɦɟɬɪ d ɬɪɭɛɵ, ɬɟɦ ɦɟɧɶɲɟ ɧɟɨɛɯɨɞɢɦɚɹ ɦɨɳɧɨɫɬɶ ɧɚɫɨ-
1)
ɫɚ;
2)
ɜɟɥɢɱɢɧɭ ɦɨɳɧɨɫɬɢ N ɦɨɠɧɨ ɭɦɟɧɶɲɢɬɶ, ɧɟ ɢɡɦɟɧɹɹ ɞɢɚɦɟɬɪɚ ɬɪɭɛɵ:
ɟɫɥɢ ɬɪɭɛɭ ɨɛɪɟɡɚɬɶ ɧɚ ɜɵɫɨɬɟ
݄اܪ, ɬɨ ɜɨɞɚ ɛɭɞɟɬ ɜɵɥɟɬɚɬɶ ɢɡ ɬɪɭɛɵ
ɫ ɧɟɤɨɬɨɪɨɣ ɫɤɨɪɨɫɬɶɸ ɢ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɳɧɨɫɬɶ ɧɚɫɨɫɚ ɭɦɟɧɶɲɚɟɬɫɹ (ɩɪɟɧɟɛɪɟɝɚɟɦ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɟɣ) ɢ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
2
Vu
U
/
N
2
ɞɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɜɨɞɚ ɩɨɞɧɹɥɚɫɶ ɧɚ ɜɵɫɨɬɭ ɇ ɧɟɨɛɯɨɞɢɦɨ, ɱɬɨɛɵ ɟɟ ɫɤɨɪɨɫɬɶ
ɛɵɥɚ ɧɟ ɦɟɧɶɲɟ, ɱɟɦ
.2
min
gHu |
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɭɱɢɦ, ɱɬɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɧɚɫɨɫ
ɦɨɳɧɨɫɬɶɸ:
/
.
VgHNU
Ɉɬɜɟɬ: ɟɫɥɢ ɬɪɭɛɭ ɨɛɪɟɡɚɬɶ ɧɚ ɜɵɫɨɬɟ ݄اܪ, ɬɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɢɫ-
ɩɨɥɶɡɨɜɚɬɶ ɧɚɫɨɫ ɦɨɳɧɨɫɬɶɸ: .Vg
Ɂɚɞɚɱɚ 7. ɋɬɪɭɹ ɜɨɞɵ ɜ ɮɨɧɬɚɧɟ ɩɨɞɧɢɦɚɟɬɫɹ ɧɚ ɜɵɫɨɬɭ ɇ ɧɚɞ ɭɪɨɜɧɟɦ
ɜɵɯɨɞɧɨɣ ɬɪɭɛɵ ɧɚɫɨɫɚ. Ʉ ɷɬɨɣ ɜɵɯɨɞɧɨɣ ɬɪɭɛɟ ɩɨɞɫɨɟɞɢɧɹɸɬ ɜɟɪɬɢɤɚɥɶɧɭɸ
ɬɪɭɛɭ ɬɚɤɨɝɨ ɠɟ ɞɢɚɦɟɬɪɚ ɜɵɫɨɬɨɣ h (h < H). ȼɨ ɫɤɨɥɶɤɨ ɪɚɡ ɫɥɟɞɭɟɬ ɢɡɦɟɧɢɬɶ
ɦɨɳɧɨɫɬɶ ɧɚɫɨɫɚ ɩɨɫɥɟ ɩɨɞɫɨɟɞɢɧɟɧɢɹ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɬɪɭɛɵ, ɱɬɨɛɵ ɫɭɦɦɚɪɧɚɹ ɜɵɫɨɬɚ ɩɨɞɫɨɟɞɢɧɟɧɧɨɣ ɬɪɭɛɵ ɢ ɜɵɥɟɬɚɸɳɟɣ ɢɡ ɧɟɟ ɫɬɪɭɢ ɜɨɞɵ ɨɫɬɚɥɚɫɶ
ɪɚɜɧɨɣ ɇ? Ɍɪɟɧɢɟɦ ɜɨɞɵ ɨ ɫɬɟɧɤɢ ɬɪɭɛ ɩɪɟɧɟɛɪɟɱɶ.
Ɋɟɲɟɧɢɟ:
ɋɩɨɫɨɛ 1. Ɇɚɫɫɚ ɜɨɞɵ, ɩɨɫɬɭɩɚɸɳɚɹ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɢɡ ɧɚɫɨɫɚ ɩɪɢ
ɫɤɨɪɨɫɬɢ ɢɫɬɟɱɟɧɢɹ ȣ ɱɟɪɟɡ ɫɟɱɟɧɢɟ S:
m
0
t
XU
S
0
.
ɋɤɨɪɨɫɬɶ ɢɫɬɟɱɟɧɢɹ ɜɨɞɵ ɢɡ ɬɪɭɛɵ:
X
.22gH
Ɇɨɳɧɨɫɬɶ ɧɚɫɨɫɚ ɛɟɡ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɬɪɭɛɵ:
W
P
0
0ɤɨ
2
t
t
0
2
m
2
XXUX
S
)(
.
2
ɉɨɫɥɟ ɭɫɬɚɧɨɜɤɢ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɬɪɭɛɵ ɢɡɦɟɧɢɬɫɹ ɫɤɨɪɨɫɬɶ ɢɫɬɟɱɟɧɢɹ ɜɨɞɵ:
2
).(2
hHgu
Ɇɨɳɧɨɫɬɶ ɧɚɫɨɫɚ ɩɪɢ ɧɚɥɢɱɢɢ ɬɪɭɛɵ:
W
P
t
,ɤFu
ɝɞɟ F – ɫɢɥɚ ɞɚɜɥɟɧɢɹ ɫɬɪɭɢ ɩɪɢ ɩɨɞɴɟɦɟ ɧɚ ɜɵɫɨɬɭ h:
.0ghSFU
149
Ɉɤɨɧɱɚɬɟɥɶɧɨ ɞɥɹ ɪɚɫɱɟɬɚ ɢɫɤɨɦɨɣ ɦɨɳɧɨɫɬɢ ɧɚɫɨɫɚ ɩɨɥɭɱɚɟɦ:
2
)(
usu
U
0
P
P
P
0
U
2
X
H
,
ughS
0
hHu
.
ɋɩɨɫɨɛ 2. ɇɚɩɨɪ ɜɨɞɵ ɜ ɞɜɭɯ ɫɥɭɱɚɹɯ ɨɞɢɧɚɤɨɜ, ɬɚɤ ɤɚɤ ɨɞɢɧɚɤɨɜɚ ɩɨɥɧɚɹ
ɜɵɫɨɬɚ ɮɨɧɬɚɧɨɜ.
Ɍɨɝɞɚ ɞɥɹ ɩɟɪɜɨɧɚɱɚɥɶɧɨɣ ɢ ɤɨɧɟɱɧɨɣ ɦɨɳɧɨɫɬɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ:
)(2
hHg
X
0
FuPFP
P
X
0
u
P
;;
2
gH
hH
.
H
Ɉɬɜɟɬ:
P
P
0
hH
.
H
10.2. Ɂɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ
ɍɪɨɜɟɧɶ 1
1. ɇɟɤɨɬɨɪɚɹ ɝɢɞɪɨɷɥɟɤɬɪɨɫɬɚɧɰɢɹ ɜ ɝɨɞ ɜɵɪɚɛɚɬɵɜɚɟɬ 23,5 ɦɥɪɞ ɤȼɬ/ɱ
ɷɥɟɤɬɪɨɷɧɟɪɝɢɢ. ɋɤɨɥɶɤɨ ɜɨɞɵ ɞɨɥɠɧɨ ɩɪɨɯɨɞɢɬɶ ɡɚ ɝɨɞ ɱɟɪɟɡ ɝɢɞɪɨɬɭɪɛɢɧɵ
ɫɬɚɧɰɢɢ? ȼɵɫɨɬɚ ɩɥɨɬɢɧɵ h = 222 ɦ. ɋɱɢɬɚɬɶ, ɱɬɨ ɜɫɹ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ
ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɷɥɟɤɬɪɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ.
Ɉɬɜɟɬ: 3,8 ā 10
10 ɦ3
2. ɋɬɚɥɶɧɨɣ ɤɚɧɚɬ, ɜɵɞɟɪɠɢɜɚɸɳɢɣ ɜɟɫ ɧɟɩɨɞɜɢɠɧɨɣ ɤɚɛɢɧɵ ɥɢɮɬɚ, ɢɦɟɟɬ
ɞɢɚɦɟɬɪ 8 ɦɦ. Ʌɢɮɬ ɜɨ ɜɪɟɦɹ ɪɚɛɨɬɵ ɩɪɢ ɜɧɟɡɚɩɧɨɣ ɨɫɬɚɧɨɜɤɟ ɩɨɥɭɱɚɟɬ ɭɫɤɨɪɟɧɢɟ 8 g. Ɉɩɪɟɞɟɥɢɬɟ, ɫɤɨɥɶɤɨ ɬɚɤɢɯ ɤɚɧɚɬɨɜ ɫɥɟɞɭɟɬ ɫɨɟɞɢɧɢɬɶ ɩɚɪɚɥɥɟɥɶɧɨ
ɩɪɢ ɭɫɬɚɧɨɜɤɟ ɥɢɮɬɚ.
Ɉɬɜɟɬ: 9.
3. Ⱦɢɚɦɟɬɪ ɬɪɭɛɵ ɧɟɮɬɟɦɚɝɢɫɬɪɚɥɢ ɋɚɦɨɬɥɨɪ – ɍɫɬɶ-Ȼɚɥɵɤ – Ʉɭɪɝɚɧ – ɍɮɚ –
Ⱥɥɶɦɟɬɶɟɜɫɤ ɫɨɫɬɚɜɥɹɟɬ 1220 ɦɦ. Ʉɚɤɨɟ ɤɨɥɢɱɟɫɬɜɨ ɧɟɮɬɢ ɩɪɨɯɨɞɢɬ
ɜ ɝɨɞ ɩɨ ɧɟɮɬɟɩɪɨɜɨɞɭ, ɟɫɥɢ ɠɢɞɤɨɫɬɶ ɬɟɱɟɬ ɫɨ ɫɤɨɪɨɫɬɶɸ 1 ɦ/ɫ?
Ɉɬɜɟɬ: # 37 000 000 ɦ
3
4. ɉɪɢ ɫɬɪɨɢɬɟɥɶɫɬɜɟ ɢɪɪɢɝɚɰɢɨɧɧɵɯ ɫɨɨɪɭɠɟɧɢɣ ɭɤɚɬɵɜɚɸɬ ɝɪɭɧɬ. Ʉɚɤɨ-
ɜɚ ɨɛɴɟɦɧɚɹ ɩɥɨɬɧɨɫɬɶ ɝɪɭɧɬɚ ɩɨɫɥɟ ɭɩɥɨɬɧɟɧɢɹ, ɟɫɥɢ ɨɧ ɨɫɟɞɚɟɬ ɧɚ 10 ɫɦ,
ɚ ɩɟɪɜɨɧɚɱɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɫɨɫɬɚɜɥɹɥɚ 1400 ɤɝ/ɦ
3
?
Ɉɬɜɟɬ: 1560 ɤɝ/ɦ
3
5. ɉɟɧɨɫɬɟɤɥɨ ɩɨɥɭɱɚɸɬ ɜɫɩɟɧɢɜɚɧɢɟɦ ɫɬɟɤɥɚ ɜ ɩɪɨɰɟɫɫɟ ɜɚɪɤɢ, ɜɜɨɞɹ ɝɚ-
ɡɨɨɛɪɚɡɭɸɳɢɟ ɜɟɳɟɫɬɜɚ. Ʉɚɤɭɸ ɱɚɫɬɶ ɨɛɴɟɦɚ ɩɟɧɨɫɬɟɤɥɚ (ɜ %) ɡɚɧɢɦɚɸɬ ɝɚɡɵ,
ɟɫɥɢ ɟɝɨ ɩɥɨɬɧɨɫɬɶ 200 ɤɝ/ɦ
3
?
Ɉɬɜɟɬ: 92 %.
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