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5. ɆȿɏȺɇɂɄȺ ɌȼȿɊȾɈȽɈ ɌȿɅȺ
G
r
5.1. Ʉɪɚɬɤɢɟ ɫɜɟɞɟɧɢɹ ɢɡ ɬɟɨɪɢɢ
Ɍɜɟɪɞɨɟ ɬɟɥɨ ɦɚɫɫɨɣ m ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɫɢɫɬɟɦɭ ɦɚɬɟɪɢɚɥɶɧɵɯ
ɬɨɱɟɤ, ɜɡɚɢɦɧɨɟ ɪɚɫɩɨɥɨɠɟɧɢɟ ɤɨɬɨɪɵɯ ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɵɦ. ɉɨɥɨɠɟɧɢɟ
ɤɚɠɞɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ
ɜɟɤɬɨɪɨɦ
. ȼ ɫɥɭɱɚɟ ɫɥɨɠɧɨɝɨ ɩɨɫɬɭɩɚɬɟɥɶɧɨ-ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɬɜɟɪ-
r
i
ɡɚɞɚɟɬɫɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɪɚɞɢɭɫ-
m
i
ɞɨɝɨ ɬɟɥɚ ɨɤɚɡɵɜɚɟɬɫɹ ɫɭɳɟɫɬɜɭɟɬ ɬɨɱɤɚ, ɤɨɬɨɪɚɹ ɞɜɢɠɟɬɫɹ ɬɨɥɶɤɨ ɩɨɫɬɭɩɚɬɟɥɶɧɨ, ɨɧɚ ɧɚɡɵɜɚɟɬɫɹ ɰɟɧɬɪ ɦɚɫɫ ɬɜɟɪɞɨɝɨ ɬɟɥɚ.
ɐɟɧɬɪɨɦ ɦɚɫɫ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɧɚɡɵɜɚɟɬɫɹ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ, ɩɨɥɨɠɟɧɢɟ
ɤɨɬɨɪɨɣ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɚɞɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɨɦ:
mr
¦
ii
m
G
.
G
r
ɫ
ȼ ɫɥɭɱɚɟ ɧɟɩɪɟɪɵɜɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɦɚɫɫɵ ɰɟɧɬɪ ɦɚɫɫ ɜɵɱɢɫɥɹɸɬ ɩɨ
ɫɥɟɞɭɸɳɟɣ ɮɨɪɦɭɥɟ:
rdm
³
G
V
.
dm
³
V
Ɂɞɟɫɶ
r
c
– ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɰɟɧɬɪɚ ɜɵɞɟɥɟɧɧɨɣ ɷɥɟɦɟɧɬɚɪɧɨɣ ɦɚɫɫɵ ɬɟɥɚ dm,
ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɜɟɞɟɬɫɹ ɩɨ ɜɫɟɣ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɨɛɥɚɫɬɢ, ɡɚɧɢɦɚɟɦɨɣ ɬɟɥɨɦ.
Ɍɨɱɤɚ ɩɪɢɥɨɠɟɧɢɹ ɪɟɡɭɥɶɬɢɪɭɸɳɟɣ ɫɢɥɵ ɬɹɠɟɫɬɢ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɬɟɥɨ,
ɧɚɡɵɜɚɟɬɫɹ ɰɟɧɬɪɨɦ ɬɹɠɟɫɬɢ ɬɟɥɚ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɢ ɰɟɧɬɪ ɦɚɫɫ
ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɬɟɥɚ ɫɨɜɩɚɞɚɸɬ ɬɨɥɶɤɨ ɜ ɫɥɭɱɚɟ ɨɞɧɨɪɨɞɧɨɝɨ ɩɨɥɹ.
ɐɟɧɬɪ ɦɚɫɫ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɞɜɢɠɟɬɫɹ ɬɚɤ, ɤɚɤ ɞɜɢɝɚɥɚɫɶ ɛɵ ɦɚɬɟɪɢɚɥɶɧɚɹ
ɬɨɱɤɚ ɫ
ɫɢɥ:
ɦɚɫɫɨɣ, ɪɚɜɧɨɣ ɦɚɫɫɟ ɬɟɥɚ, ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɫɟɯ ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɟɥɭ
G
G
ma F
. (5.1)
¦
ɫ i
ɍɪɚɜɧɟɧɢɟ (5.1) ɞɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɭɫɬɚɧɨɜɢɬɶ ɞɜɢɠɟɧɢɟ ɰɟɧɬɪɚ ɦɚɫɫ ɬɜɟɪ-
ɞɨɝɨ ɬɟɥɚ, ɟɫɥɢ ɢɡɜɟɫɬɧɚ ɦɚɫɫɚ ɬɟɥɚ ɢ ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɧɟɝɨ ɫɢɥɵ.
ɉɪɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɤɨɝɞɚ ɜɫɟ ɬɨɱɤɢ ɬɟɥɚ ɩɨɥɭɱɚɸɬ ɡɚ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ ɪɚɜɧɵɟ ɩɨ ɜɟɥɢɱɢɧɟ ɢ ɧɚɩɪɚɜɥɟɧɢɸ ɩɟɪɟɦɟɳɟɧɢɹ, ɞɥɹ ɩɨɥɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɞɜɢɠɟɧɢɹ ɜɫɟɝɨ ɬɟɥɚ
ɞɨɫɬɚɬɨɱɧɨ
ɨɩɪɟɞɟɥɢɬɶ ɞɜɢɠɟɧɢɟ ɨɞɧɨɣ ɢɡ ɬɨɱɟɤ ɬɟɥɚ (ɧɚɩɪɢɦɟɪ, ɟɝɨ ɰɟɧɬɪɚ ɦɚɫɫ).
ɉɪɢ ɜɪɚɳɚɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ ɜɫɟ ɬɨɱɤɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɞɜɢɠɭɬɫɹ ɩɨ
ɨɤɪɭɠɧɨɫɬɹɦ, ɰɟɧɬɪɵ ɤɨɬɨɪɵɯ ɥɟɠɚɬ ɧɚ ɨɞɧɨɣ ɩɪɹɦɨɣ, ɧɚɡɵɜɚɟɦɨɣ ɨɫɶɸ ɜɪɚɳɟɧɢɹ. Ⱦɥɹ ɨɩɢɫɚɧɢɹ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɧɭɠɧɨ ɡɚɞɚɬɶ ɩɨɥɨɠɟɧɢɟ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɨɫɢ ɜɪɚɳɟɧɢɹ ɢ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɜ ɤɚɠɞɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
.
Ⱦɥɹ ɢɡɭɱɟɧɢɹ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɜɜɟɫɬɢ ɜ ɪɚɫɫɦɨɬɪɟɧɢɟ
ɫɥɟɞɭɸɳɢɟ ɮɢɡɢɱɟɫɤɢɟ ɜɟɥɢɱɢɧɵ: ɦɨɦɟɧɬ ɫɢɥɵ, ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ, ɦɨɦɟɧɬ
ɢɦɩɭɥɶɫɚ.
51
Ɇɨɦɟɧɬ ɫɢɥɵ
F
G
M
G
r
lFM
z
GGG
ɬɨɪɧɚɹ ɜɟɥɢɱɢɧɚ
G
ɝɞɟ
– ɪɚɞɢɭɫ-ɜɟɤɬɨɪ, ɩɪɨɜɟɞɟɧɧɵɣ ɢɡ ɬɨɱɤɢ Ɉ ɜ ɬɨɱɤɭ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ.
ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɢ Ɉ (ɪɢɫ. 5.1) – ɷɬɨ ɜɟɤ-
, ɨɩɪɟɞɟɥɹɟɦɚɹ ɜɵɪɚɠɟɧɢɟɦ:
G
G
G
>@
FrM
, , (5.2)
Ɋɢɫ. 5.1
ɂɡ ɨɩɪɟɞɟɥɟɧɢɹ ɫɥɟɞɭɟɬ, ɱɬɨ ɦɨɦɟɧɬ ɫɢɥɵ ɹɜɥɹɟɬɫɹ ɚɤɫɢɚɥɶɧɵɦ ɜɟɤɬɨɪɨɦ,
ɧɚɩɪɚɜɥɟɧɢɟ ɤɨɬɨɪɨɝɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɩɪɚɜɢɥɭ ɩɪɚɜɨɝɨ ɜɢɧɬɚ (ɧɚɩɪɚɜɥɟɧɢɟ
ɜɟɤɬɨɪɚ ɦɨɦɟɧɬɚ ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɜɪɚɳɟɧɢɹ ɜɢɧɬɚ, ɪɢɫɤɚ ɲɥɹɩɤɢ ɤɨɬɨɪɨɝɨ, ɩɨɜɨɪɚɱɢɜɚɟɬɫɹ ɨɬ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɤ ɜɟɤɬɨɪɭ ɫɢɥɵ ɩɨ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɟ
ɬɨ ɟɫɬɶ ɜ ɩɪɚɜɭɸ ɫɬɨɪɨɧɭ).
Ɇɨɞɭɥɶ ɜɟɤɬɨɪɚ ɦɨɦɟɧɬɚ ɫɢɥɵ ɪɚɜɟɧ
ɬɟɥɶɧɨ ɬɨɱɤɢ Ɉ. ɉɥɟɱɨɦ ɧɚɡɵɜɚɸɬ ɤɪɚɬɱɚɣɲɟɟ ɪɚɫɫɬɨɹɧɢɣ ɨɬ ɬɨɱɤɢ Ɉ ɞɨ ɥɢɧɢɢ
ɞɟɣɫɬɜɢɹ ɫɢɥɵ.
ȿɫɥɢ ɬɟɥɨ ɦɨɠɟɬ ɜɪɚɳɚɬɶɫɹ ɬɨɥɶɤɨ ɜɨɤɪɭɝ ɧɟɤɨɬɨɪɨɣ ɮɢɤɫɢɪɨɜɚɧɧɨɣ ɨɫɢ,
ɫɩɨɫɨɛɧɨɫɬɶ ɫɢɥɵ ɜɪɚɳɚɬɶ ɬɟɥɨ ɜɨɤɪɭɝ ɷɬɨɣ ɨɫɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɜɟɥɢɱɢɧɨɣ,
ɤɨɬɨɪɚɹ ɧɚɡɵɜɚɟɬɫɹ ɦɨɦɟɧɬɨɦ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ.
ɉɚɪɚɥɥɟɥɶɧɭɸ ɨɫɢ z ɫɨɫɬɚɜɥɹɸɳɭɸ ɦɨɦɟɧɬɚ ɫɢɥɵ (ɪɢɫ. 5.2)
ɬɨɱɤɢ Ɉ (ɥɟɠɚɳɟɣ ɧɚ ɨɫɢ) ɧɚɡɵɜɚɸɬ
FF
ɋɨɫɬɚɜɥɹɸɳɢɟ ɫɢɥɵ
ɉɨɜɨɪɨɬ ɜɨɤɪɭɝ ɨɫɢ z ɦɨɠɟɬ ɛɵɬɶ ɜɵɡɜɚɧ ɬɨɥɶɤɨ ɤɚɫɚɬɟɥɶɧɨɣ ɫɨɫɬɚɜɥɹɸ-
G
F
ɳɟɣ
ɛɨɥɶɲɟ ɟɟ ɩɥɟɱɨ R.
52
. ɉɪɢɱɟɦ ɷɬɚ ɫɨɫɬɚɜɥɹɸɳɚɹ ɬɟɦ ɭɫɩɟɲɧɟɟ ɨɫɭɳɟɫɬɜɢɬ ɩɨɜɨɪɨɬ, ɱɟɦ
W
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ:
,
//
ɦɨɦɟɧɬɨɦ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ:
>@
M
GG
ɧɟ ɦɨɝɭɬ ɜɵɡɜɚɬɶ ɜɪɚɳɟɧɢɹ ɜɨɤɪɭɝ ɨɫɢ z.
R
M
z
Fr
,
>@
FR
,
, ɝɞɟ l – ɩɥɟɱɨ ɫɢɥɵ ɨɬɧɨɫɢ-
ɨɬɧɨɫɢɬɟɥɶɧɨ
.
z
GGG
. (5.3)
W
,
Ɋɢɫ. 5.2
W
F
G
ȼɟɤɬɨɪɵ
ɜɡɚɢɦɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ, ɩɨɷɬɨɦɭ ɦɨɞɭɥɶ ɜɟɤɬɨɪɚ ɦɨ-
FRGG,
ɦɟɧɬɚ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɪɚɜɟɧ:
G
M
FR
z
,
W
ɝɞɟ R – ɩɥɟɱɨ ɤɚɫɚɬɟɥɶɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɫɢɥɵ ɢɥɢ ɪɚɞɢɭɫ ɨɤɪɭɠɧɨɫɬɢ ɜɪɚɳɟɧɢɹ ɬɨɱɤɢ, ɤ ɤɨɬɨɪɨɣ ɩɪɢɥɨɠɟɧɚ ɫɢɥɚ
.
ȿɫɥɢ ɤ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɟ ɩɪɢɥɨɠɟɧɨ ɧɟɫɤɨɥɶɤɨ ɫɢɥ ɜɨ ɜɪɚɳɚɬɟɥɶɧɨɦ
ɞɜɢɠɟɧɢɢ, ɬɨ ɦɨɦɟɧɬ ɪɟɡɭɥɶɬɢɪɭɸɳɟɣ ɫɢɥɵ ɪɚɜɟɧ ɜɟɤɬɨɪɧɨɣ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ
ɫɥɚɝɚɟɦɵɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ:
GGGG
.
M
M
M
z
z
z
21
M
...
zn
ɉɪɨɟɤɰɢɹ ɦɨɦɟɧɬɚ ɪɟɡɭɥɶɬɢɪɭɸɳɟɣ ɫɢɥɵ ɧɚ ɨɫɶ ɜɪɚɳɟɧɢɹ ɪɚɜɧɚ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɫɭɦɦɟ ɩɪɨɟɤɰɢɣ ɦɨɦɟɧɬɨɜ ɫɥɚɝɚɟɦɵɯ ɫɢɥ ɧɚ ɨɫɶ ɜɪɚɳɟɧɢɹ.
ɇɚ ɜɟɥɢɱɢɧɭ ɭɝɥɨɜɨɝɨ ɭɫɤɨɪɟɧɢɹ ɬɟɥɚ ɩɪɢ ɜɪɚɳɚɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ ɫɭɳɟɫɬɜɟɧɧɨɟ ɜɥɢɹɧɢɟ ɨɤɚɡɵɜɚɟɬ ɧɟ ɬɨɥɶɤɨ ɦɚɫɫɚ ɬɟɥɚ, ɧɨ ɢ ɟɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ.
ȼɟɥɢɱɢɧɚ, ɭɱɢɬɵɜɚɸɳɚɹ ɷɬɢ ɨɛɫɬɨɹɬɟɥɶɫɬɜɚ, ɧɚɡɵɜɚɟɬɫɹ
ɰɢɢ ɬɟɥɚ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ.
ɦɨɦɟɧɬɨɦ ɢɧɟɪ-
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z (ɬɜɟɪ-
ɞɨɝɨ ɬɟɥɚ) – ɷɬɨ ɮɢɡɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ, ɪɚɜɧɚɹ ɫɭɦɦɟ ɩɪɨɢɡɜɟɞɟɧɢɣ ɦɚɫɫ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ (
) ɧɚ ɤɜɚɞɪɚɬɵ ɢɯ ɪɚɫɫɬɨɹɧɢɣ ɨɬ ɨɫɢ z (
m
i
n
I
¦
z
i
2
m
iRi
1
):
R
i
, (5.4)
ɝɞɟ ɨɬɞɟɥɶɧɨ ɜɡɹɬɨɟ ɫɥɚɝɚɟɦɨɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ i-ɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z.
ɂɡ ɨɩɪɟɞɟɥɟɧɢɹ ɫɥɟɞɭɟɬ, ɱɬɨ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɹɜɥɹɟɬɫɹ ɜɟɥɢɱɢɧɨɣ ɚɞɞɢɬɢɜɧɨɣ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɪɚɜɟɧ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ
ɟɝɨ ɱɚɫɬɟɣ.
53
Ɋɚɫɱɟɬɧɵɟ ɮɨɪɦɭɥɵ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ ɨɞɧɨɪɨɞɧɵɯ ɫɢɦ-
I
I
I
r
X
X
ɦɟɬɪɢɱɧɵɯ ɬɟɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɬɟɥɚ, ɩɪɢɜɟɞɟɧɵ ɜ ɬɚɛɥ. 5.1.
Ɍɚɛɥɢɰɚ 5.1
ɇɚɡɜɚɧɢɟ ɬɟɥɚ Ɋɚɫɱɟɬɧɚɹ ɮɨɪɦɭɥɚ
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ,
Ɍɟɥɨ – ɨɛɪɭɱ ɢɥɢ ɬɨɧɤɨɫɬɟɧɧɵɣ
ɰɢɥɢɧɞɪ ɦɚɫɫɨɣ m ɢ ɪɚɞɢɭɫɨɦ R
Ɍɟɥɨ – ɞɢɫɤ ɢɥɢ ɫɩɥɨɲɧɨɣ ɰɢɥɢɧɞɪ
ɦɚɫɫɨɣ m ɢ ɪɚɞɢɭɫɨɦ R
Ɍɟɥɨ – ɲɚɪ ɦɚɫɫɨɣ m ɪɚɞɢɭɫɚ R
Ɍɟɥɨ – ɬɨɧɤɢɣ ɞɢɫɤ (ɬɨɥɳɢɧɚ ɞɢɫɤɚ
ɦɧɨɝɨ ɦɟɧɶɲɟ ɟɝɨ ɪɚɞɢɭɫɚ) ɦɚɫɫɨɣ m
ɢ ɪɚɞɢɭɫɚ R
Ɍɟɥɨ – ɬɨɧɤɢɣ ɞɥɢɧɧɵɣ ɫɬɟɪɠɟɧɶ
ɫ ɫɟɱɟɧɢɟɦ ɥɸɛɨɣ ɮɨɪɦɵ ɦɚɫɫɨɣ m
ɢ ɞɥɢɧɨɣ l
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɩɥɨɫɤɨɫɬɢ ɨɛɪɭɱɚ
ɢ ɫɨɜɩɚɞɚɸɳɟɣ ɫ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɨɫɶɸ ɰɢɥɢɧɞɪɚ:
z
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ,
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɩɥɨɫɤɨɫɬɢ ɞɢɫɤɚ
ɢ ɫɨɜɩɚɞɚɸɳɟɣ ɫ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɨɫɶɸ ɰɢɥɢɧɞɪɚ:
z
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɲɚɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ,
ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɲɚɪɚ:
ImR
z
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɫɨɜɩɚɞɚɸɳɟɣ
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ,
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɤ ɫɬɟɪɠɧɸ ɢ ɩɪɨɯɨɞɹɳɟɣ
ɫ ɞɢɚɦɟɬɪɨɦ ɞɢɫɤɚ:
z
ɱɟɪɟɡ ɟɝɨ ɫɟɪɟɞɢɧɭ:
z
2
5
1
4
12
mR
mRI
1
2
2
2
2
2
mR
2
ml
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ I ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɢɡɜɨɥɶɧɨɣ ɨɫɢ (ɪɢɫ. 5.3) ɪɚɜɟɧ ɫɭɦɦɟ
ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ I
ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɚɪɚɥɥɟɥɶɧɨɣ ɞɚɧɧɨɣ ɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟ-
0
ɪɟɡ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɬɟɥɚ, ɢ ɩɪɨɢɡɜɟɞɟɧɢɹ ɦɚɫɫɵ ɬɟɥɚ m ɧɚ ɤɜɚɞɪɚɬ ɪɚɫɫɬɨɹɧɢɹ r
ɦɟɠɞɭ ɨɫɹɦɢ:
II mr
0
Ⱦɚɧɧɚɹ ɮɨɪɦɭɥɚ ɹɜɥɹɟɬɫɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɜɵɪɚɠɟɧɢɟɦ
. Ɍɟɨɪɟɦɚ ɒɬɟɣɧɟɪɚ ɫɜɨɞɢɬ ɜɵɱɢɫɥɟɧɢɟ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɧɟɪɚ
2
. (5.5)
ɬɟɨɪɟɦɵ ɒɬɟɣ-
ɩɪɨɢɡɜɨɥɶɧɨɣ ɨɫɢ ɤ ɜɵɱɢɫɥɟɧɢɸ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɬɟɥɚ.
Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ Ɉ ɪɚɜɟɧ:
G
– ɪɚɞɢɭɫ-ɜɟɤɬɨɪ, ɩɪɨɜɟɞɟɧɧɵɣ ɢɡ ɬɨɱɤɢ Ɉ ɜ ɬɭ ɬɨɱɤɭ ɩɪɨɫɬɪɚɧɫɬɜɚ,
ɝɞɟ
ɜ ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɬɫɹ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ (ɪɢɫ. 5.4);
ɫɤɨɪɨɫɬɶɸ
G
G
– ɢɦɩɭɥɶɫ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ m, ɤɨɬɨɪɚɹ ɞɜɢɠɟɬɫɹ ɫɨ
mp
G
.
G
GG
>@
prL
, , (5.6)
54
plL
D
r
l
z
G
>@>
@
WXW
GGGGG
z
R
G
X
G
Ɋɢɫ. 5.3 Ɋɢɫ. 5.4
Ɇɨɞɭɥɶ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
,
ɝɞɟ
ɧɚɩɪɚɜɥɟɧɢɟ ɤɨɬɨɪɨɝɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɩɪɚɜɢɥɭ ɩɪɚɜɨɝɨ ɜɢɧɬɚ.
ɨɫɢ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ
ɛɵɬɶ ɨɩɪɟɞɟɥɟɧɚ ɩɨ ɮɨɪɦɭɥɟ:
sin
– ɩɥɟɱɨ ɜɟɤɬɨɪɚ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ.
Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɹɜɥɹɟɬɫɹ ɚɤɫɢɚɥɶɧɵɦ ɜɟɤɬɨɪɨɦ,
Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z – ɷɬɨ ɫɨɫɬɚɜɥɹɸɳɚɹ
L
Ɇɨɠɧɨ ɞɨɤɚɡɚɬɶ, ɱɬɨ ɫɨɫɬɚɜɥɹɸɳɚɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɩɨ ɨɫɢ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ Ɉ, ɥɟɠɚɳɟɣ ɧɚ ɨɫɢ z (ɪɢɫ. 5.5):
G
L
z
Ɋɢɫ. 5.5
GG
>@
. (5.7)
,
pr
z
G
L
ɩɨ ɷɬɨɣ
G
ɦɨɠɟɬ
L
z
L
ɝɞɟ
55
– ɫɨɫɬɚɜɥɹɸɳɚɹ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚɹ ɤ ɨɫɢ z;
– ɤɚɫɚɬɟɥɶɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɫɤɨɪɨɫɬɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ m.
W
,, RmpR
,
Ɇɨɞɭɥɶ ɫɨɫɬɚɜɥɹɸɳɟɣ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɩɨ ɨɫɢ ɪɚɜɟɧ:
G
Z
Z
I
L
Z
zIz
z
Z
z
H
G
L
z
Rm
X
,
W
ɝɞɟ R – ɪɚɞɢɭɫ ɨɤɪɭɠɧɨɫɬɢ, ɩɨ ɤɨɬɨɪɨɣ ɜɪɚɳɚɟɬɫɹ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɦɚɫɫɨɣ m
ɜɨɤɪɭɝ ɨɫɢ z.
Ⱦɥɹ ɨɞɧɨɪɨɞɧɨɝɨ ɬɟɥɚ, ɫɢɦɦɟɬɪɢɱɧɨɝɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɦɨ-
L
ɦɟɧɬ ɢɦɩɭɥɶɫɚ
ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɫ ɜɟɤɬɨɪɨɦ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ
ɝɞɟ I – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ.
ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ Ɉ, ɥɟɠɚɳɟɣ ɧɚ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɫɨɜɩɚɞɚɟɬ
G
G
ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
G
, (5.8)
ɉɪɨɟɤɰɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɬɟɥɚ ɧɚ ɨɫɶ z, ɫɨɜɩɚɞɚɸɳɭɸ ɫ ɨɫɶɸ ɜɪɚɳɟ-
ɧɢɹ, ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
L
, (5.9)
ɝɞɟ
– ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ;
I
– ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɬɟɥɚ.
ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɫɨɨɬɧɨɲɟɧɢɟ (5.9) ɫɩɪɚɜɟɞɥɢɜɨ ɞɥɹ ɥɸɛɨɝɨ ɬɟɥɚ, ɚ
ɫɨɨɬɧɨɲɟɧɢɟ (5.8) ɢɦɟɟɬ ɦɟɫɬɨ ɬɨɥɶɤɨ ɜ ɫɥɭɱɚɟ ɜɪɚɳɟɧɢɹ ɬɟɥɚ ɜɨɤɪɭɝ ɨɫɢ ɫɢɦɦɟɬɪɢɢ ɢ ɞɥɹ ɧɟɫɢɦɦɟɬɪɢɱɧɨɝɨ ɬɟɥɚ, ɜɪɚɳɚɸɳɟɝɨɫɹ ɜɨɤɪɭɝ ɨɞɧɨɣ ɢɡ ɫɜɨɢɯ
ɝɥɚɜɧɵɯ ɨɫɟɣ.
Ɉɫɧɨɜɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɢɧɚɦɢɤɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɶ z, ɫɨɜɩɚɞɚɸɳɭɸ ɫ ɨɫɶɸ ɜɪɚɳɟɧɢɹ, ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ:
n
ɜɧɟɲ
¦
i
, (5.10)
zi
1
ɝɞɟ
H
– ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ (ɨɫɢ);
I
IM
z
– ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɬɟɥɚ;
ɜɧɟɲ
M
– ɩɪɨɟɤɰɢɹ ɦɨɦɟɧɬɚ i-ɨɣ ɜɧɟɲɧɟɣ ɫɢɥɵ ɧɚ ɨɫɶ z, ɫɨɜɩɚɞɚɸɳɭɸ ɫ
zi
ɨɫɶɸ ɜɪɚɳɟɧɢɹ;
n
ɜɧɟɲ
M
– ɚɥɝɟɛɪɚɢɱɟɫɤɚɹ ɫɭɦɦɚ ɩɪɨɟɤɰɢɣ ɦɨɦɟɧɬɨɜ ɜɫɟɯ n ɜɧɟɲɧɢɯ
zi
1
i¦
ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɨ.
ȿɫɥɢ ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɢɯ ɫɢɥ ɧɚ ɬɜɟɪɞɨɟ ɬɟɥɨ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɢɥɢ
ɫɭɦɦɚɪɧɵɣ ɦɨɦɟɧɬ ɜɧɟɲɧɢɯ ɫɢɥ ɪɚɜɟɧ ɧɭɥɸ, ɬɨ ɩɪɚɜɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (5.10)
ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ. ȼ ɷɬɢɯ ɭɫɥɨɜɢɹɯ ɫɨɫɬɚɜɥɹɸɳɚɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɬɜɟɪɞɨɝɨ
ɬɟɥɚ ɩɨ ɨɫɢ ɫɨɯɪɚɧɹɟɬɫɹ:
Z
d
I
z
d
HH
dt
I
z
dt
I
ZZ
z
Lconst
z
.;;0;;0 const
(5.11)
56
Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɦɨɠɟɬ ɛɵɬɶ ɫɮɨɪɦɭɥɢɪɨɜɚɧ ɫɥɟ-
I
ɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ
ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɵɦ.
ɂɡ ɭɪɚɜɧɟɧɢɹ (5.11) ɫɥɟɞɭɟɬ, ɱɬɨ ɩɪɢ ɪɚɜɟɧɫɬɜɟ ɧɭɥɸ ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ
ɦɨɦɟɧɬɚ ɜɫɟɯ ɜɧɟɲɧɢɯ ɫɢɥ ɬɟɥɨ ɜɪɚɳɚɟɬɫɹ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ. ȿɫɥɢ ɜ
ɷɬɢɯ ɭɫɥɨɜɢɹɯ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɦɨɠɟɬ ɢɡɦɟɧɹɬɶɫɹ ɜɫɥɟɞɫɬɜɢɟ ɢɡɦɟɧɟɧɢɹ
ɜɡɚɢɦɧɨɝɨ ɪɚɫɩɨɥɨɠɟɧɢɹ ɟɝɨ ɨɬɞɟɥɶɧɵɯ ɱɚɫɬɟɣ ɨɬ ɡɧɚɱɟɧɢɹ
ɞɨ
I
1z
, ɬɨ ɢɡɦɟ-
I
2z
ɧɟɧɢɟ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɜɥɟɱɟɬ ɡɚ ɫɨɛɨɣ ɢɡɦɟɧɟɧɢɟ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɨɬ ɜɟɥɢ-
Z
ɱɢɧɵ
Z
ɞɨ
1
. Ɉɞɧɚɤɨ ɷɬɢ ɢɡɦɟɧɟɧɢɹ ɞɨɥɠɧɵ ɛɵɬɶ ɬɚɤɢɦɢ, ɱɬɨɛɵ ɩɪɨɢɡɜɟɞɟ-
2
ɧɢɟ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɧɚ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɨɫɬɚɥɨɫɶ ɩɨɫɬɨɹɧɧɵɦ:
II
ZZ
11 2 2zz
. (5.12)
ȼɧɭɬɪɟɧɧɢɟ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ ɦɟɠɞɭ ɬɟɥɚɦɢ ɫɢɫɬɟɦɵ, ɦɨɝɭɬ ɜɵɡɜɚɬɶ ɢɡɦɟɧɟɧɢɹ ɦɨɦɟɧɬɨɜ ɢɦɩɭɥɶɫɚ ɨɬɞɟɥɶɧɵɯ ɱɚɫɬɟɣ ɫɢɫɬɟɦɵ. Ɉɞɧɚɤɨ ɷɬɢ ɢɡɦɟɧɟɧɢɹ
ɛɭɞɭɬ ɜɫɟɝɞɚ ɬɚɤɨɜɵ, ɱɬɨ ɫɭɦɦɚɪɧɵɣ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɤɚɤ ɰɟɥɨɝɨ
ɨɫɬɚɟɬɫɹ ɛɟɡ ɢɡɦɟɧɟɧɢɣ. ɉɨɥɧɵɣ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɦɨɠɟɬ ɢɡɦɟɧɹɬɶɫɹ
ɬɨɥɶɤɨ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɢɯ ɫɢɥ.
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ, ɜɪɚɳɚɸɳɟɝɨɫɹ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ,
ɪɚɜɧɚ:
2
I
Z
z
Ɍ
, (5.13)
2
ɝɞɟ
– ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ;
z
Ȧ – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɬɟɥɚ.
ȼ ɡɚɜɟɪɲɟɧɢɢ ɤɪɚɬɤɨɝɨ ɨɛɡɨɪɚ ɬɟɨɪɢɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɫɞɟɥɚɬɶ ɪɹɞ
ɜɚɠɧɵɯ ɜɵɜɨɞɨɜ:
1) ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɹɜɥɹɟɬɫɹ ɦɟɪɨɣ ɢɧɟɪɬɧɨɫɬɢ ɬɟɥɚ ɩɪɢ ɜɪɚɳɚɬɟɥɶ-
ɧɨɦ ɞɜɢɠɟɧɢɢ ɢ ɜ ɷɬɨɦ ɫɦɵɫɥɟ – ɚɧɚɥɨɝɨɦ ɦɚɫɫɵ ɩɪɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɦ ɞɜɢɠɟ-
ɧɢɢ ɬɟɥɚ;
2) ɞɥɹ ɨɫɦɵɫɥɟɧɢɹ ɨɫɧɨɜɧɵɯ ɩɨɧɹɬɢɣ ɦɟɯɚɧɢɤɢ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɫɨɩɨɫɬɚɜɢɬɶ
ɢ ɩɪɨɚɧɚɥɢɡɢɪɨɜɚɬɶ ɮɨɪɦɭɥɵ ɦɟɯɚɧɢɤɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫ ɚɧɚɥɨɝɢɱɧɵɦɢ ɮɨɪɦɭɥɚɦɢ ɦɟɯɚɧɢɤɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫ ɩɨɦɨɳɶɸ ɬɚɛɥ. 5.2;
3) ɩɨɧɹɬɢɹ ɦɨɦɟɧɬɚ ɫɢɥɵ ɢ
ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ, ɧɟɫɦɨɬɪɹ ɧɚ ɬɨ, ɱɬɨ ɜɜɨɞɹɬɫɹ
ɧɚ ɨɫɧɨɜɟ ɪɚɫɫɦɨɬɪɟɧɢɹ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ, ɫɭɳɟɫɬɜɭɸɬ ɛɟɡɨɬɧɨɫɢɬɟɥɶɧɨ ɤ ɜɪɚɳɟɧɢɸ. Ɍɚɤ, ɧɚɩɪɢɦɟɪ, ɥɸɛɨɟ ɬɟɥɨ, ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɬɨɝɨ, ɜɪɚɳɚɟɬɫɹ ɨɧɨ
ɢɥɢ ɩɨɤɨɢɬɫɹ, ɨɛɥɚɞɚɟɬ ɨɩɪɟɞɟɥɟɧɧɵɦ ɦɨɦɟɧɬɨɦ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɸɛɨɣ
ɨɫɢ, ɩɨɞɨɛɧɨ ɬɨɦɭ, ɤɚɤ ɬɟɥɨ ɨɛɥɚɞɚɟɬ ɦɚɫɫɨɣ ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɫɨɫɬɨɹɧɢɹ ɫɜɨɟɝɨ
ɞɜɢɠɟɧɢɹ. Ɇɨɦɟɧɬ ɫɢɥɵ ɬɚɤɠɟ ɫɭɳɟɫɬɜɭɟɬ
ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɬɨɝɨ, ɜɪɚɳɚɟɬɫɹ ɬɟɥɨ
ɜɨɤɪɭɝ ɨɫɢ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɛɟɪɟɬɫɹ ɦɨɦɟɧɬ, ɢɥɢ ɩɨɤɨɢɬɫɹ. ȼ ɩɨɫɥɟɞɧɟɦ
ɫɥɭɱɚɟ ɦɨɦɟɧɬ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɥɵ ɭɪɚɜɧɨɜɟɲɢɜɚɟɬɫɹ ɦɨɦɟɧɬɚɦɢ ɞɪɭɝɢɯ
ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɨ.
Ʌɸɛɨɟ ɞɜɢɠɟɧɢɟ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɨ ɤɚɤ ɫɭɩɟɪɩɨɡɢ-
ɰɢɹ ɞɜɭɯ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɜɢɞɨɜ ɞɜɢɠɟɧɢɹ: ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ.
57
Ɍɚɛɥɢɰɚ 5.2
X
Z
H
I
X
Z
F
G
M
Z
X
r
Z
X
ɉɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ȼɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ
G
– ɥɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ
G
a
– ɥɢɧɟɣɧɨɟ ɭɫɤɨɪɟɧɢɟ
m – ɦɚɫɫɚ
G
ɍɪɚɜɧɟɧɢɟ ɞɢɧɚɦɢɤɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ
ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɦɚɫɫɨɣ m ɜ ɩɪɨɟɤɰɢɹɯ
ɧɚ ɨɫɶ z (2 ɡɚɤɨɧ ɇɶɸɬɨɧɚ):
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ:
G
– ɢɦɩɭɥɶɫ
mp
– ɫɢɥɚ
n
ma F
¦
z
1
i
2
X
m
Ɍ
2
zi
ɜɧɟɲ
L – ɩɪɨɟɤɰɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ
z
G
– ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ
G
– ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ
– ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ
z
zI
ɧɚ ɨɫɶ z
G
– ɦɨɦɟɧɬ ɫɢɥɵ
Ɉɫɧɨɜɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɢɧɚɦɢɤɢ
ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ
ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɶ z:
n
IM
Z
z
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ:
Ɍ
ɜɧɟɲ
¦
i
I
z
2
zi
1
2
Z
ɉɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ ɬɟɥɚ – ɷɬɨ ɞɜɢɠɟɧɢɟ, ɩɪɢ ɤɨɬɨɪɨɦ ɜɫɟ ɬɨɱɤɢ ɬɟɥɚ ɩɟ-
ɪɟɦɟɳɚɸɬɫɹ ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɥɨɫɤɨɫɬɹɯ. ɉɪɢɦɟɪɨɦ ɩɥɨɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɦɨɠɟɬ
ɫɥɭɠɢɬɶ ɤɚɱɟɧɢɟ ɰɢɥɢɧɞɪɚ ɩɨ ɩɥɨɫɤɨɫɬɢ (ɪɢɫ. 5.6).
Ɋɢɫ. 5.6
ɉɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ ɹɜɥɹɟɬɫɹ ɱɚɫɬɧɵɦ ɫɥɭɱɚɟɦ ɫɥɨɠɧɨɝɨ ɞɜɢɠɟɧɢɹ ɬɟɥɚ.
ȿɫɥɢ ɜɵɛɪɚɬɶ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɫɥɨɠɧɨɟ ɞɜɢɠɟɧɢɟ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɧɟɩɨɞɜɢɠɧɨɣ, ɬɨ ɞɜɢɠɟɧɢɟ ɬɟɥɚ ɦɨɠɧɨ
ɜɢɬɶ ɤɚɤ ɜɪɚɳɟɧɢɟ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ
G
ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɤɨɬɨɪɚɹ ɞɜɢɠɟɬɫɹ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɨɣ ɫɢɫɬɟɦɵ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ
Ɉɛɭɫɥɨɜɥɟɧɧɚɹ ɜɪɚɳɟɧɢɟɦ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɥɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ Ɉ
ɜɟɱɚɸɳɚɹ ɪɚɞɢɭɫɭ-ɜɟɤɬɨɪɭ
G
, ɩɪɨɜɟɞɟɧɧɨɦɭ ɤ ɧɟɣ ɢɡ ɬɨɱɤɢ Ɉ, ɪɚɜɧɚ
G
.
0
ɩɪɟɞɫɬɚ-
/
, ɨɬ-
GG
G
>@
r
,
.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɤɨɪɨɫɬɶ ɷɬɨɣ ɬɨɱɤɢ ɩɪɢ ɫɥɨɠɧɨɦ ɞɜɢɠɟɧɢɢ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ:
GG G
XX Z
G
. (5.14)
, r
>@
0
58
ɗɥɟɦɟɧɬɚɪɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɩɪɢ ɩɥɨɫɤɨɦ ɞɜɢɠɟɧɢɢ ɜɫɟɝɞɚ
X
ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɩɨɜɨɪɨɬ ɜɨɤɪɭɝ ɧɟɤɨɬɨɪɨɣ ɨɫɢ, ɧɚɡɵɜɚɟɦɨɣ ɦɝɧɨɜɟɧɧɨɣ
ɨɫɶɸ ɜɪɚɳɟɧɢɹ. ɗɬɚ ɨɫɶ ɦɨɠɟɬ ɥɟɠɚɬɶ ɤɚɤ ɜ ɩɪɟɞɟɥɚɯ ɬɟɥɚ, ɬɚɤ ɢ ɜɧɟ ɟɝɨ ɩɪɟɞɟɥɨɜ. ȼ ɫɥɭɱɚɟ ɤɚɬɹɳɟɝɨɫɹ ɩɨ ɩɥɨɫɤɨɫɬɢ ɰɢɥɢɧɞɪɚ ɦɝɧɨɜɟɧɧɚɹ ɨɫɶ Ɉ
/
ɫɨɜɩɚɞɚɟɬ ɫ
ɥɢɧɢɟɣ ɤɚɫɚɧɢɹ ɰɢɥɢɧɞɪɚ ɫ ɩɥɨɫɤɨɫɬɶɸ. ɉɪɢ ɤɚɱɟɧɢɢ ɰɢɥɢɧɞɪɚ ɦɝɧɨɜɟɧɧɚɹ ɨɫɶ
ɩɟɪɟɦɟɳɚɟɬɫɹ ɤɚɤ ɩɨ ɩɥɨɫɤɨɫɬɢ (ɬ. ɟ. ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ), ɬɚɤ ɢ ɩɨ ɩɨɜɟɪɯɧɨɫɬɢ ɰɢɥɢɧɞɪɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɪɹɞ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɜɪɚɳɟɧɢɣ ɜɨɤɪɭɝ
ɦɝɧɨɜɟɧɧɵɯ ɨɫɟɣ.
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɩɪɢ ɩɥɨɫɤɨɦ ɞɜɢɠɟɧɢɢ ɹɜɥɹɟɬɫɹ ɚɞɞɢɬɢɜ-
ɧɨɣ ɜɟɥɢɱɢɧɨɣ ɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɭɦɦɭ ɤɢɧɟɬɢɱɟɫɤɢɯ ɷɧɟɪɝɢɣ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ
ɞɜɢɠɟɧɢɹ ɫɨ ɫɤɨɪɨɫɬɶɸ, ɪɚɜɧɨɣ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɢ ɷɧɟɪɝɢɢ
ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɜɨɤɪɭɝ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɬɟɥɚ:
22
mI
XZ
Ɍ
ɫɫ
22
, (5.15)
ɝɞɟ m – ɦɚɫɫɚ ɬɟɥɚ;
– ɫɤɨɪɨɫɬɶ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ;
c
– ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɢɧɟɪɰɢɢ;
I
c
Ȧ – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ.
Ʉɥɸɱɟɜɵɟ ɩɨɧɹɬɢɹ ɢ ɬɟɪɦɢɧɵ
Ⱦɜɢɠɟɧɢɟ ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ
Ⱦɜɢɠɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɟ
Ⱦɜɢɠɟɧɢɟ ɩɥɨɫɤɨɟ
Ɇɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ
Ɇɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ
Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɦɚɬɟɪɢɚɥɶɧɨɣ
Ɉɫɧɨɜɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɢɧɚɦɢɤɢ
ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
ɩɪɢ ɜɪɚɳɚɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɩɪɢ
ɩɥɨɫɤɨɦ ɞɜɢɠɟɧɢɢ
ɐɟɧɬɪ ɢɧɟɪɰɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
ɬɨɱɤɢ
Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ
59
Ɏɨɪɦɭɥɵ ɢ ɟɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ ɜ ɫɢɫɬɟɦɟ ɋɂ
ɭ
>
@
>
@
G
G
G
G
G
Z
ɨɫɧɨɜɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ
ɇɚɡɜɚɧɢɟ Ɏɨɪɦɭɥɚ
1. Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ, ɡɚɞɚɸɳɢɣ
ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɦɚɫɫ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
ɫ ɞɢɫɤɪɟɬɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɦɚɫɫɵ
2. Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ, ɡɚɞɚɸɳɢɣ
ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɦɚɫɫ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
ɫ ɧɟɩɪɟɪɵɜɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɦɚɫɫɵ
3. Ɇɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ
ɥɶ ɦɨɦɟɧɬɚ ɫɢɥɵ
ɦɨɞ
4. Ɇɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ
ɦɨɞɭɥɶ ɦɨɦɟɧɬɚ ɫɢɥɵ
5. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ Z
6. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɢɡɜɨɥɶɧɨɣ ɨɫɢ
(ɬɟɨɪɟɦɚ ɒɬɟɣɧɟɪɚ)
7. Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɦɚɬɟɪɢɚɥɶɧɨɣ
ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɢɡɜɨɥɶɧɨɣ
ɬɨɱɤɢ ɦɨɞɭɥɶ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ
8. Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɨɞɧɨɪɨɞɧɨɝɨ
ɫɢɦɦɟɬɪɢɱɧɨɝɨ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ
9. ɉɪɨɟɤɰɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ
ɧɚ ɨɫɶ ɜɪɚɳɟɧɢɹ
10. Ɉɫɧɨɜɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɢɧɚɦɢɤɢ
ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ
(ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɜɪɚɳɟɧɢɹ)
11. Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ
12. Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ
ɩɪɢ ɜɪɚɳɚɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ
13. ɋɤɨɪɨɫɬɶ ɬɟɥɚ ɩɪɢ ɩɥɨɫɤɨɦ
14. Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ
ɢɦɩɭɥɶɫɚ
ɞɜɢɠɟɧɢɢ
ɩɪɢ ɩɥɨɫɤɨɦ ɞɜɢɠɟɧɢɢ
G
m
¦
G
r
c
G
r
c
G
G
M
G
M
ImR
zi
II mr
L
z
IM
H
z
II
ZZ
11 2 2zz
Ɍ
GG G
XX Z
mI
Ɍ
iri
m
rdm
³
V
z
z
i
XZ
22
dm
³
V
G
G
FrM
;
lFM
G
G
Fr
z
FR
W
n
¦
0
i
I
0
22
cc
2
i
1
2
>@
prL
plL
Z
IL
zI
n
ɜɧɟɲ
¦
zi
1
2
Z
z
2
G
, r
>@
ȿɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ
ɜ ɫɢɫɬɟɦɟ ɋɂ
[m] = [ɤɝ];
[r] = [ɦ]
[dm] = [ɤɝ];
[r] = [ɦ]
[M] = [ɇāɦ];
[l] = [ɦ];
[F] = [ɇ]
[M] = [ɇāɦ];
[F] = [ɇ];
[R] = [ɦ]
[I] = [ɤɝāɦ²];
[m] = [ɤɝ];
[R] = [ɦ]
[I] = [ɤɝāɦ²];
[m] = [ɤɝ];
[c] = [ɦ]
[L] = [Ⱦɠāɫ];
[p] = [ɇāc];
[l] = [ɦ]
[L] = [Ⱦɠāɫ];
[I] = [ɤɝāɦ²];
[Ȧ] = [ɪɚɞ/ɫ]
[L] = [Ⱦɠāɫ];
[I] = [ɤɝāɦ²];
[Ȧ] = [ɪɚɞ/ɫ]
[I] = [ɤɝāɦ²];
[İ] = [ɪɚɞ/ɫ²];
[M] = [ɇāɦ]
[I] = [ɤɝāɦ²];
[Ȧ] = [ɪɚɞ/ɫ]
[Ɍ] = [Ⱦɠ];
[I] = [ɤɝāɦ²];
[Ȧ] = [ɪɚɞ/ɫ]
[ȣ] = [ɦ/ɫ];
[Ȧ] = [ɪɚɞ/ɫ]
[Ɍ] = [Ⱦɠ];
[m] = [ɤɝ];
[ȣ] = [ɦ/ɫ];
[I] = [ɤɝāɦ²];
[Ȧ] = [ɪɚɞ/ɫ]
60
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