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6.2. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
S
r
M
r
r
U
G
r
Ɂɚɞɚɱɚ 1. ȼ ɮɚɧɬɚɫɬɢɱɟɫɤɢɯ ɪɨɦɚɧɚɯ ɩɪɟɞɥɚɝɚɟɬɫɹ ɩɨɫɬɪɨɟɧɢɟ ɥɢɮɬɚ ɨɬ
Ɂɟɦɥɢ ɤ Ʌɭɧɟ. Ȼɭɞɟɦ ɟɝɨ ɫɱɢɬɚɬɶ ɨɞɧɨɪɨɞɧɵɦ ɰɢɥɢɧɞɪɨɦ ɦɚɫɫɨɣ m , ɩɥɨɳɚɞɶɸ
ɨɫɧɨɜɚɧɢɹ
ɢ ɜɵɫɨɬɨɣ h . Ɉɩɪɟɞɟɥɢɬɟ ɰɟɧɬɪ ɦɚɫɫ ɬɟɥɚ, ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɜ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ ɩɨɥɟ Ɂɟɦɥɢ ɧɚ ɩɨɥɸɫɟ, ɢ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɧɚ ɟɟ ɷɤɜɚɬɨɪɟ. Ɋɚɞɢɭɫ Ɂɟɦɥɢ ɩɪɢɧɹɬɶ ɪɚɜɧɵɦ R = 6500 ɤɦ, ɪɚɫɫɬɨɹɧɢɟ ɨɬ Ɂɟɦɥɢ ɞɨ Ʌɭɧɵ – ɜɵɫɨɬɚ ɰɢɥɢɧɞɪɚ 60 ɪɚɞɢɭɫɨɜ Ɂɟɦɥɢ. ɉɨɥɟɦ ɫɢɥɵ ɬɹɝɨɬɟɧɢɹ Ʌɭɧɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɩɨɥɟɦ ɬɹɝɨɬɟɧɢɹ Ɂɟɦɥɢ
ɢ ɜɪɚɳɟɧɢɟɦ Ʌɭɧɵ ɩɪɟɧɟɛɪɟɱɶ.
Ɋɟɲɟɧɢɟ:
ɉɨɫɤɨɥɶɤɭ ɩɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɰɢɥɢɧɞɪ ɨɞɧɨɪɨɞɧɵɣ, ɨɩɪɟɞɟɥɢɦ ɟɝɨ ɩɥɨɬ-
U
ɧɨɫɬɶ ɤɚɤ
.
hSm
ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɬɟɥɚ ɫ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɦɚɫɫɨɣ ɧɚɣɞɟɦ:
h
rdm
³
r
C
dm
³
Sdrr
U
³
0
m
2
Sh
U
m
mh
h
.
m
222
Ɇɵ ɩɨɥɭɱɢɥɢ ɜɩɨɥɧɟ ɪɚɡɭɦɧɵɣ ɪɟɡɭɥɶɬɚɬ, ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɨɞɧɨɪɨɞɧɨɝɨ ɰɢ-
ɥɢɧɞɪɚ ɪɚɫɩɨɥɨɠɟɧ ɜ ɬɨɱɤɟ ɩɨɥɨɜɢɧɵ ɟɝɨ ɜɵɫɨɬɵ, ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɰɟɧɬɪ ɦɚɫɫ
ɧɚɯɨɞɢɬɫɹ ɜ ɬɨɱɤɟ
30 195 000 19,5 10
rR ɤɦ ɦ
ɫ
7
.
Ɉɫɨɛɨ ɨɬɦɟɬɢɦ, ɱɬɨ ɜ ɨɞɧɨɪɨɞɧɨɦ ɩɨɥɟ ɬɹɠɟɫɬɢ ɢ ɩɪɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɦ
ɞɜɢɠɟɧɢɢ ɰɢɥɢɧɞɪɚ ɫ ɭɫɤɨɪɟɧɢɟɦ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɢ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɫɨɜɩɚɞɚɸɬ
ɫ ɰɟɧɬɪɨɦ ɦɚɫɫ ɬɟɥɚ. Ɉɞɧɚɤɨ ɜ ɧɟɨɞɧɨɪɨɞɧɵɯ ɩɨɥɹɯ ɷɬɨ ɧɟ ɬɚɤ.
ɇɚɣɞɟɦ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɬɟɥɚ ɧɚ ɩɨɥɸɫɟ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɰɢɥɢɧɞɪ ɦɨɠɧɨ ɪɚɡ-
ɛɢɬɶ ɧɚ ɷɥɟɦɟɧɬɚɪɧɵɟ ɫɥɨɢ, ɤɚɠɞɵɣ ɢɡ ɤɨɬɨɪɵɯ
Sd
dmU
ɫɢɥɨɜɨɝɨ ɩɨɥɹ ɦɚɫɫɨɣ
ɞɨ ɬɨɱɤɢ ɫɟɪɟɞɢɧɵ ɫɥɨɹ dr. ɐɟɧɬɪ ɬɹɠɟɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɬɨɱɤɚ ɩɪɢɥɨɠɟɧɢɹ
. ɇɚ ɬɚɤɭɸ ɦɚɫɫɭ ɞɟɣɫɬɜɭɟɬ ɝɪɚɜɢɬɚɰɢɨɧɧɚɹ ɫɢɥɚ ɫɨ ɫɬɨɪɨɧɵ ɰɟɧɬɪɚ
, ɜɟɥɢɱɢɧɚ ɤɨɬɨɪɨɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ:
Mdm
G
Ɂɞɟɫɶ
GdF
– ɝɪɚɜɢɬɚɰɢɨɧɧɚɹ ɩɨɫɬɨɹɧɧɚɹ,
ɢɦɟɟɬ ɜɵɫɨɬɭ dr, ɢ ɦɚɫɫɭ
SdrM
.
22
– ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɫɢɥɨɜɨɝɨ ɰɟɧɬɪɚ
ɪɟɡɭɥɶɬɢɪɭɸɳɟɣ ɫɢɥɵ ɬɹɠɟɫɬɢ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɬɟɥɚ. ɗɬɚ ɬɨɱɤɚ ɦɨɠɟɬ ɛɵɬɶ
ɨɩɪɟɞɟɥɟɧɚ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
hR
rdF
³
R
r
ɰɬ
.
hR
dF
³
R
ȼɵɱɢɫɥɢɦ ɨɬɞɟɥɶɧɨ ɱɢɫɥɢɬɟɥɶ ɢ ɡɧɚɦɟɧɚɬɟɥɶ ɷɬɨɣ ɮɨɪɦɭɥɵ. Ⱦɥɹ ɡɧɚɦɟɧɚ-
ɬɟɥɹ ɢɦɟɟɦ:
hR
GdF
Mdm
r
U
SdrM
G
³³³
R
22
r
1
hR
SGM
R
r
h
UU
SGM
.
)(
hRR
81
Ɍɟɩɟɪɶ ɨɩɪɟɞɟɥɢɦ ɱɢɫɥɢɬɟɥɶ:
r
Z
hR
R
Mdm
rGdFr
r
hR
SrdrM
U
G
22
r
R
hR
dr
SGM
³³³³
r
R
UU
hR
SMG
.
)ln(
R
ȼ ɢɬɨɝɟ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ:
ɰɬ
hR
Rr
h
hR
)ln(
.
R
ɉɨɞɫɬɚɜɥɹɹ ɞɚɧɧɵɟ, ɢɦɟɟɦ:
7
ɦɤɦr
ɰɬ
10011,06,109)02,1ln(02,16500 | .
Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨɬ ɮɚɤɬ, ɱɬɨ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɥɟɠɢɬ ɧɚɦɧɨɝɨ ɧɢɠɟ,
ɱɟɦ ɰɟɧɬɪ ɦɚɫɫ ɫɢɫɬɟɦɵ, ɷɬɨ ɫɜɹɡɚɧɨ ɫ ɧɟɨɞɧɨɪɨɞɧɨɫɬɶɸ ɩɨɥɹ ɬɹɝɨɬɟɧɢɹ Ɂɟɦɥɢ.
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɧɢɠɧɢɟ ɫɥɨɢ ɰɢɥɢɧɞɪɚ ɩɪɢɬɹɝɢɜɚɸɬɫɹ ɤ Ɂɟɦɥɟ ɫ ɛɨɥɶɲɟɣ ɩɨ
ɜɟɥɢɱɢɧɟ ɫɢɥɨɣ, ɱɟɦ ɜɟɪɯɧɢɟ, ɩɨɫɤɨɥɶɤɭ ɪɚɫɫɬɨɹɧɢɟ ɨɬ
ɫɢɥɨɜɨɝɨ ɰɟɧɬɪɚ ɞɨ ɧɢɯ
ɦɟɧɶɲɟ, ɚ ɜɟɥɢɱɢɧɚ ɫɢɥɵ ɬɹɝɨɬɟɧɢɹ ɛɨɥɶɲɟ, ɱɟɦ ɞɨ ɜɟɪɯɧɢɯ.
Ɍɟɩɟɪɶ ɨɩɪɟɞɟɥɢɦ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɰɢɥɢɧɞɪɚ ɧɚ ɷɤɜɚɬɨɪɟ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɧɚ
ɤɚɠɞɵɣ ɜɵɞɟɥɟɧɧɵɣ ɫɥɨɣ ɜɟɥɢɱɢɧɨɣ d
ɞɟɣɫɬɜɭɟɬ ɰɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɚ ɢɧɟɪ-
ɰɢɢ:
22
rSdrrdmdF
ZUZ
,
ɝɞɟ
– ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ Ɂɟɦɥɢ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɰɢ-
ɥɢɧɞɪɚ ɫɨɝɥɚɫɧɨ ɫɨɨɬɧɨɲɟɧɢɸ:
hR
rdF
³
R
r
ɰɢ
.
hR
dF
³
R
ȼɧɨɜɶ ɧɚɣɞɟɦ ɡɧɚɦɟɧɚɬɟɥɶ ɩɪɟɞɥɨɠɟɧɧɨɝɨ ɜɵɪɚɠɟɧɢɹ:
hR
³³³
R
1
2
))((
1
ZUZUUZZ
2
ɢ ɱɢɫɥɢɬɟɥɶ:
hR
R
hR
³³³
R
1
ZUZUZ
3
332222
))((
RhRSdrrSrdmrdFr
2
ZU
S
3
Ɉɤɨɧɱɚɬɟɥɶɧɨ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɨɨɬɧɨɲɟɧɢɟɦ:
322
)33(32
hRhhR
r
ɰɢ
.
2
)2(
hRh
ɉɪɨɜɨɞɹ ɜɵɱɢɫɥɟɧɢɹ, ɢɦɟɟɦ:
223
2(3 3 )
Rh Rh h
r ɤɦ ɦ
|
ɰɢ
3
2
Rh h
(2 )
2 6,5 10 130 10
57
.
ɐɟɧɬɪ ɢɧɟɪɰɢɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɥɟɠɢɬ ɜɵɲɟ ɰɟɧɬɪɚ ɦɚɫɫ ɬɟɥɚ.
82
2222222
)2(
hRhSRhRSSdrrrdmdF
322
.
)33(
hhRRh
Ɂɚɞɚɱɚ 2. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɦɚɫɫɨɣ m
F
F
R
R
RgmR
ɠɭɳɟɦɫɹ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ a ɥɢɮɬɟ. ɇɚɣɞɢɬɟ ɫɢɥɭ ɧɚɬɹɠɟɧɢɹ ɧɢɬɢ ɜ
ɜɢɫɢɬ ɧɚ ɢɞɟɚɥɶɧɨɣ ɧɢɬɢ ɜ ɞɜɢ-
ɫɥɟɞɭɸɳɢɯ ɫɥɭɱɚɹɯ: ɚ) ɥɢɮɬ ɧɟɩɨɞɜɢɠɟɧ, ɛ) ɥɢɮɬ ɞɜɢɠɟɬɫɹ ɜɜɟɪɯ, ɜ) ɥɢɮɬ ɞɜɢɠɟɬɫɹ ɜɧɢɡ.
Ⱦɚɧɨ:
m
a
ɚ) Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ ɥɢɮɬɨɦ.
ȼ ɫɥɭɱɚɟ, ɟɫɥɢ ɥɢɮɬ ɧɟɩɨɞɜɢɠɟɧ, ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ ɧɢɬɢ
ɭɪɚɜɧɨɜɟɲɟɧɚ ɫɢɥɨɣ ɬɹɠɟɫɬɢ. ɉɨɷɬɨɦɭ ɜ ɷɬɨɦ ɫɥɭɱɚɟ
mgF
.
Ɋɟɲɟɧɢɟ:
ɛ) ɉɪɢ ɞɜɢɠɟɧɢɢ ɥɢɮɬɚ ɜɜɟɪɯ ɫ ɭɫɤɨɪɟɧɢɟɦ a ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɨɝɨ
ɧɚɛɥɸɞɚɬɟɥɹ ɧɚ Ɂɟɦɥɟ, ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɥɢɮɬɨɦ ɧɚ ɦɚɬɟɪɢɚɥɶɧɭɸ
ɬɨɱɤɭ, ɞɟɣɫɬɜɭɟɬ ɫɢɥɚ ɢɧɟɪɰɢɢ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ. ɉɨɷɬɨɦɭ
ɧɚɛɥɸɞɚɟɬɫɹ ɩɟɪɟɝɪɭɡɤɚ ɢ ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ ɪɚɜɧɚ:
)( agmF
,
ɜ) ɉɪɢ ɞɜɢɠɟɧɢɢ ɥɢɮɬɚ ɜɧɢɡ ɫ ɭɫɤɨɪɟɧɢɟɦ a ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɨɝɨ
ɧɚɛɥɸɞɚɬɟɥɹ ɧɚ Ɂɟɦɥɟ, ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɥɢɮɬɨɦ ɧɚ ɦɚɬɟɪɢɚɥɶɧɭɸ
ɬɨɱɤɭ, ɞɟɣɫɬɜɭɟɬ ɫɢɥɚ ɢɧɟɪɰɢɢ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ, ɩɨɷɬɨɦɭ ɫɢɥɚ
ɧɚɬɹɠɟɧɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ:
)( agmF
,
ɢ ɟɫɥɢ ɜɵɩɨɥɧɟɧɨ ɭɫɥɨɜɢɟ ,ag ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ ɧɢɬɢ ɨɤɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨɣ ɧɭɥɸ.
Ɂɚɞɚɱɚ 3. Ʌɟɬɱɢɤ ɫɨɜɟɪɲɚɟɬ ɦɟɪɬɜɭɸ ɩɟɬɥɸ ɪɚɞɢɭɫɨɦ 120
ɫ ɥɢɧɟɣɧɨɣ ɫɤɨɪɨɫɬɶɸ
X
ɦ
. Ɉɩɪɟɞɟɥɢɬɟ ɤɚɤɭɸ ɩɟɪɟɝɪɭɡɤɭ ɢɫɩɵɬɵɜɚɟɬ
60
ɫ
ɦ, ɞɜɢɝɚɹɫɶ
ɥɟɬɱɢɤ ɜ ɜɟɪɯɧɟɣ ɢ ɧɢɠɧɟɣ ɬɨɱɤɚɯ ɫɜɨɟɣ ɬɪɚɟɤɬɨɪɢɢ.
Ɋɟɲɟɧɢɟ:
ɂɫɩɨɥɶɡɭɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ ɥɟɬɱɢɤɨɦ. ȼ ɷɬɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɧɚ ɧɟɝɨ ɤɪɨɦɟ ɫɢɥɵ ɬɹɠɟɫɬɢ ,mg ɞɟɣɫɬɜɭɟɬ ɰɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ ɪɚɜ-
2
X
mF
ɧɚɹ ɩɨ ɜɟɥɢɱɢɧɟ
ɢ
.
Ⱦɥɹ ɜɟɪɯɧɟɣ ɬɨɱɤɢ ɬɪɚɟɤɬɨɪɢɢ ɫɢɥɵ ɬɹɠɟɫɬɢ ɢ ɰɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɵ ɢɧɟɪɰɢɢ ɧɚɩɪɚɜɥɟɧɵ ɜɞɨɥɶ ɨɞɧɨɣ ɩɪɹɦɨɣ ɚɧɬɢɩɚɪɚɥɥɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɭ, ɩɨɷɬɨɦɭ ɪɟɡɭɥɶɬɢɪɭɸɳɚɹ ɫɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɥɟɬɱɢɤɚ ɜ ɷɬɨɣ ɬɨɱɤɟ ɬɪɚɟɤɬɨɪɢɢ, ɪɚɜɧɚ:
22
XX
ɢ
mmgFmgF
R
gm
R
3600
10()(
120
mgmm
220)
.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɷɬɚ ɫɢɥɚ ɧɚɩɪɚɜɥɟɧɚ ɜɜɟɪɯ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɥɟɬɱɢɤ ɢɫɩɵɬɵɜɚɟɬ ɞɜɨɣɧɭɸ ɩɟɪɟɝɪɭɡɤɭ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɟ.
Ⱦɥɹ ɧɢɠɧɟɣ ɬɨɱɤɢ ɬɪɚɟɤɬɨɪɢɢ ɫɢɥɵ ɬɹɠɟɫɬɢ ɢ ɰɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɵ ɢɧɟɪɰɢɢ
ɜɧɨɜɶ ɧɚɩɪɚɜɥɟɧɵ ɜɞɨɥɶ ɨɞɧɨɣ ɩɪɹɦɨɣ ɩɚɪɚɥɥɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɭ, ɩɨɷɬɨɦɭ ɪɟɡɭɥɶɬɢɪɭɸɳɚɹ ɫɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɥɟɬɱɢɤɚ ɜ ɷɬɨɣ ɬɨɱɤɟ ɬɪɚɟɤɬɨɪɢɢ, ɪɚɜɧɚ:
22
ɢ
mmgFmgF
XX
3600
10()(
120
mgmm
440)
.
ɋɢɥɚ ɧɚɩɪɚɜɥɟɧɚ ɜɧɢɡ, ɢ ɥɟɬɱɢɤ ɢɫɩɵɬɵɜɚɟɬ ɱɟɬɵɪɟɯɤɪɚɬɧɭɸ ɩɟɪɟɝɪɭɡɤɭ
ɜ ɭɤɚɡɚɧɧɨɣ ɬɨɱɤɟ.
83
Ɂɚɞɚɱɚ 4. Ɉɞɧɨɪɨɞɧɵɣ ɰɢɥɢɧɞɪ ɦɚɫɫɵ m
R
g
ɠɟɥɨɛɚ, ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɛɨɤɨɜɵɦɢ ɫɬɟɧɤɚɦɢ ɤɨɬɨɪɨɝɨ ɪɚɜɧɨ ɪɚɞɢɭɫɭ
ɧɚɯɨɞɢɬɫɹ ɧɚ ɩɨɞɫɬɚɜɤɟ ɜ ɮɨɪɦɟ
ɰɢɥɢɧɞɪɚ. ɉɨɞɫɬɚɜɤɚ ɞɜɢɝɚɟɬɫɹ ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɫ ɭɫɤɨɪɟɧɢɟɦ a
(ɪɢɫ. 6.5). ɇɚɣɬɢ ɫɢɥɵ ɞɚɜɥɟɧɢɹ ɧɚ ɰɢɥɢɧɞɪ.
Ɋɟɲɟɧɢɟ:
ɉɟɪɟɣɞɟɦ ɜ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ ɞɜɢɠɭɳɢɦɫɹ ɬɟɥɨɦ (ɪɢɫ. 6.6).
ɇɚ ɰɟɧɬɪ ɟɝɨ ɦɚɫɫ ɜ ɰɟɧɬɪɟ ɰɢɥɢɧɞɪɚ ɧɚ ɧɟɝɨ ɞɟɣɫɬɜɭɟɬ ɫɢɥɚ ɢɧɟɪɰɢɢ )( ma .
Ʉ ɷɬɨɣ ɠɟ ɬɨɱɤɟ ɩɪɢɥɨɠɟɧɚ ɫɢɥɚ ɬɹɠɟɫɬɢ ().mg ɋɢɥɵ
ɪɟɚɤɰɢɢ ɨɩɨɪ, ɪɚɜɧɵɟ ɩɨ
ɜɟɥɢɱɢɧɟ ɢɫɤɨɦɵɦ ɫɢɥɚɦ ɞɚɜɥɟɧɢɹ, ɩɪɢɥɨɠɟɧɵ ɜ ɬɨɱɤɚɯ ɤɚɫɚɧɢɹ ɰɢɥɢɧɞɪɚ ɢ
ɠɟɥɨɛɚ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɤɚɫɚɬɟɥɶɧɵɦ ɜ ɷɬɢɯ ɬɨɱɤɚɯ, ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɚɩɪɚɜɥɟɧɵ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɪɚɞɢɭɫɚ ɰɢɥɢɧɞɪɚ, ɨɛɨɡɧɚɱɢɦ ɢɯ
.
,NN
12
Ɋɢɫ. 6.5 Ɋɢɫ. 6.6
Ɋɚɫɫɦɨɬɪɢɦ ɜɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɫɬɟɪɠɧɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɟɜɨɣ ɬɨɱɤɢ
ɤɚɫɚɧɢɹ. Ɂɞɟɫɶ ɦɨɦɟɧɬɵ ɫɢɥ )( ma ɢ
ɫɬɪɟɥɤɢ, ɚ ɦɨɦɟɧɬ ɫɢɥɵ ɬɹɠɟɫɬɢ m
ɜɪɚɳɚɸɬ ɰɢɥɢɧɞɪ ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ
N
2
ɩɨ ɱɚɫɨɜɨɣ. Ɂɚɩɢɲɟɦ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɜɵɛɪɚɧɧɨɣ ɬɨɱɤɢ:
lmglNlma
.
mgNma
3
RRl
)
2
R
l
,
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ
mg
2
Ɂɞɟɫɶ
ma
2
2
S
cos( RRl
6
3
)
,
N
2
2
sin(
S
3
ɩɥɟɱɢ ɫɢɥ. Ɍɟɩɟɪɶ ɧɟɬɪɭɞɧɨ ɨɩɪɟɞɟɥɢɬɶ ɢɫɤɨɦɨɟ ɜɵɪɚɠɟɧɢɟ ɫɢɥɵ ɪɟɚɤɰɢɢ
ɜ ɩɪɚɜɨɣ ɬɨɱɤɟ ɤɚɫɚɧɢɹ:
g
(
mN
2
)
a
.
3
Ⱥɧɚɥɨɝɢɱɧɵɟ ɜɵɱɢɫɥɟɧɢɹ ɞɥɹ ɜɪɚɳɟɧɢɹ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɚɜɨɣ ɬɨɱɤɢ ɤɚɫɚ-
ɧɢɹ, ɩɨɡɜɨɥɹɸɬ ɩɨɥɭɱɢɬɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɢɥɵ ɪɟɚɤɰɢɢ ɧɚ ɥɟɜɭɸ ɬɨɱɤɭ ɤɚɫɚɧɢɹ
g
ɰɢɥɢɧɞɪɚ, ɤɨɬɨɪɨɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
(
mN
1
)
a
.
3
84
Ɂɚɞɚɱɚ 5. Ƚɪɭɡ ɦɚɫɫɨɣ l = 1 ɤɝ ɩɨɞɜɟɲɟɧ ɧɚ ɧɢɬɢ ɞɥɢɧɨɣ l = 1 ɦ ɢ ɨɩɢɫɵɜɚ-
g
R
D
l
R
Z
l
ɟɬ ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɨɤɪɭɠɧɨɫɬɶ ɫ ɩɨɫɬɨɹɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ
ɫSZ. Ɉɩɪɟɞɟɥɢɬɟ ɫɢɥɭ ɭɩɪɭɝɨɫɬɢ ɧɢɬɢ ɢ ɭɝɨɥ, ɤɨɬɨɪɵɣ ɨɛɪɚɡɭɟɬ ɧɢɬɶ
12
ɫ ɜɟɪɬɢɤɚɥɶɸ, ɟɫɥɢ ɬɨɱɤɚ ɩɨɞɜɟɫɚ ɞɜɢɠɟɬɫɹ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ ɫ ɭɫɤɨɪɟɧɢ-
ɟɦ
ɦ
1
ɚ .
2
ɫ
Ɋɟɲɟɧɢɟ:
ɉɟɪɟɣɞɟɦ ɜ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ ɝɪɭɡɨɦ. ɇɚ ɧɟɝɨ, ɨɱɟɜɢɞɧɨ, ɞɟɣ-
ɫɬɜɭɟɬ ɫɢɥɚ ɬɹɠɟɫɬɢ )(m
, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ, ɫɢɥɚ ɢɧɟɪɰɢɢ, ɫɜɹ-
ɡɚɧɧɚɹ ɫ ɞɜɢɠɟɧɢɟɦ ɬɨɱɤɢ ɩɨɞɜɟɫɚ, ɬɚɤɠɟ ɧɚɩɪɚɜɥɟɧɚ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɜ ɷɬɨɣ
ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ ɢ ɪɚɜɧɚ ɩɨ ɜɟɥɢɱɢɧɟ ().ma
ɐɟɧɬɪɨɛɟɠɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ ɧɚɩɪɚɜɥɟɧɚ ɨɬ ɰɟɧɬɪɚ ɨɤɪɭɠɧɨɫɬɢ, ɩɨ ɤɨɬɨɪɨɣ
ɞɜɢɠɟɬɫɹ ɝɪɭɡ, ɢ ɪɚɜɧɚ ɩɨ ɜɟɥɢɱɢɧɟ
Z
. Ɂɞɟɫɶ
)(2Rm
– ɪɚɞɢɭɫ ɨɤɪɭɠɧɨɫɬɢ, ɩɨ
ɤɨɬɨɪɨɣ ɞɜɢɠɟɬɫɹ ɝɪɭɡ. ɗɬɢ ɫɢɥɵ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɞɪɭɝ ɞɪɭɝɭ, ɩɨɷɬɨɦɭ ɢɯ
ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɪɚɜɧɚ
222
. ɂɦɟɧɧɨ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ
)()( Rmmamg
Z
ɭɤɚɡɚɧɧɵɯ ɫɢɥ ɢ ɭɪɚɜɧɨɜɟɲɢɜɚɟɬɫɹ ɫɢɥɨɣ ɭɩɪɭɝɨɫɬɢ ɧɢɬɢ:
ɭ
.
222
)()( RmmamgF
Z
ȼ ɢɬɨɝɟ, ɬɚɧɝɟɧɫ ɭɝɥɚ, ɤɨɬɨɪɵɣ ɫɨɫɬɚɜɥɹɟɬ ɧɢɬɶ ɫ ɜɟɪɬɢɤɚɥɶɸ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɨɬɧɨɲɟɧɢɟɦ ɦɨɞɭɥɹ ɰɟɧɬɪɨɛɟɠɧɨɣ ɫɢɥɵ ɤ ɜɟɥɢɱɢɧɟ ɪɟɡɭɥɶɬɢɪɭɸɳɟɣ ɫɢɥɵ
ɬɹɠɟɫɬɢ ɢ ɫɢɥɵ ɢɧɟɪɰɢɢ, ɫɜɹɡɚɧɧɨɣ ɫ ɞɜɢɠɟɧɢɟɦ ɫ ɭɫɤɨɪɟɧɢɟɦ ɬɨɱɤɢ ɩɨɞɜɟɫɚ, ɢ
ɪɚɜɟɧ:
22
tg
D
Rm
agm
R
ZZ
)(
.
ag
Ɋɚɞɢɭɫ ɨɤɪɭɠɧɨɫɬɢ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɧɚɱɟɧɢɟɦ ɧɚɣɞɟɧɧɨɝɨ
2
DZ
sin
l
ɭɝɥɚ ɢ ɞɥɢɧɨɣ ɧɢɬɢ, ɤɚɤ
ag
D
cos
Ⱥ ɡɧɚɱɟɧɢɟ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ:
11
22
S
FmgmamR ɇ
q|
ɭ
279,0
|
14
()()(11)(4sin74)417
sin
. ɉɨɷɬɨɦɭ ɢɦɟɟɦ
74D|q
, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
222 22 2
ZS
.
D
tg
ɢɥɢ
ag
.
6.3. Ɂɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ
1. ȼ ɥɢɮɬɟ, ɩɨɞɧɢɦɚɸɳɟɦɫɹ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ ,a ɫɨɜɟɪɲɚɟɬ ɤɨɥɟ-
ɛɚɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ. Ɉɩɪɟɞɟɥɢɬɟ ɩɟɪɢɨɞ ɟɝɨ ɤɨɥɟɛɚɧɢɣ.
,a
2. ȼ ɥɢɮɬɟ, ɞɜɢɠɭɳɟɦɫɹ ɜɧɢɡ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
ɤɨɥɟɛɥɟɬɫɹ ɦɚ-
ɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ. Ɉɩɪɟɞɟɥɢɬɟ ɩɟɪɢɨɞ ɟɝɨ ɤɨɥɟɛɚɧɢɣ.
3. ȼ ɩɨɟɡɞɟ, ɞɜɢɠɭɳɟɦɫɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɫ ɭɫɤɨɪɟɧɢɟɦ
,a
ɧɚɯɨɞɢɬɫɹ ɦɚɬɟ-
ɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ. ɇɚɣɞɢɬɟ ɩɟɪɢɨɞ ɟɝɨ ɤɨɥɟɛɚɧɢɣ.
85
4. ȼ ɜɚɝɨɧɟ ɩɨɟɡɞɚ ɜɢɫɢɬ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ. ȼ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ
D
P
D
P
D
P
D
ɜɪɟɦɟɧɢ ɩɨɟɡɞ ɬɪɨɝɚɟɬɫɹ ɢ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
ɜɫɥɟɞɫɬɜɢɟ ɱɟɝɨ ɦɚɹɬɧɢɤ ɨɬɤɥɨɧɹɟɬɫɹ. Ʉɚɤɨɜ ɭɝɨɥ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɨɬɤɥɨɧɟɧɢɹ
ɦɚɹɬɧɢɤɚ?
,a
5. ȼ ɥɢɮɬɟ, ɩɨɞɧɢɦɚɸɳɟɦɫɹ ɫ ɭɫɤɨɪɟɧɢɟɦ
ɫɬɨɢɬ ɹɳɢɤ ɫ ɩɟɫɤɨɦ. Ʉɚ-
ɦɟɧɶ ɦɚɫɫɨɣ m , ɧɚɯɨɞɹɳɢɣɫɹ ɧɚ ɜɵɫɨɬɟ h ɧɚɞ ɩɨɜɟɪɯɧɨɫɬɶɸ ɩɟɫɤɚ, ɧɚɱɢɧɚɟɬ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɢɮɬɚ ɩɚɞɚɬɶ ɛɟɡ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ. Ʉɚɤɨɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ
ɜɵɞɟɥɢɬɫɹ ɩɪɢ ɭɞɚɪɟ ɤɚɦɧɹ ɨ ɩɟɫɨɤ?
6. ȼ ɩɨɟɡɞɟ, ɞɜɢɠɭɳɟɦɫɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɫ ɭɫɤɨɪɟɧɢɟɦ
,a
ɫɬɨɢɬ ɹɳɢɤ
ɫ ɩɟɫɤɨɦ. Ʉɚɦɟɧɶ ɦɚɫɫɨɣ m , ɧɚɯɨɞɹɳɢɣɫɹ ɧɚ ɜɵɫɨɬɟ h ɧɚɞ ɩɨɜɟɪɯɧɨɫɬɶɸ ɩɟɫɤɚ, ɧɚɱɢɧɚɟɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɟɡɞɚ ɩɚɞɚɬɶ ɛɟɡ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ. Ʉɚɤɨɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ ɜɵɞɟɥɢɬɫɹ ɩɪɢ ɭɞɚɪɟ ɤɚɦɧɹ ɨ ɩɟɫɨɤ?
7. Ʉ ɩɨɬɨɥɤɭ ɤɚɸɬɵ ɪɚɜɧɨɦɟɪɧɨ ɢɞɭɳɟɝɨ ɬɟɩɥɨɯɨɞɚ ɩɨɞɜɟɲɟɧ ɲɚɪ. Ʉɚɤ
ɢɡɦɟɧɢɬɫɹ ɩɨɥɨɠɟɧɢɟ ɲɚɪɚ, ɟɫɥɢ ɬɟɩɥɨɯɨɞ:
ɚ) ɧɚɱɧɟɬ ɭɜɟɥɢɱɢɜɚɬɶ ɫɤɨɪɨɫɬɶ
, ɧɟ ɦɟɧɹɹ ɧɚɩɪɚɜɥɟɧɢɟ ɞɜɢɠɟɧɢɹ;
ɛ) ɩɨɜɟɪɧɟɬ ɜ ɫɬɨɪɨɧɭ;
ɜ) ɜɧɟɡɚɩɧɨ ɨɫɬɚɧɨɜɢɬɫɹ?
8. ɇɚ ɷɤɜɚɬɨɪɟ ɪɟɲɢɥɢ ɩɨɫɬɪɨɢɬɶ ɛɚɲɧɸ, ɱɬɨɛɵ ɧɚ ɜɟɪɯɧɟɦ ɷɬɚɠɟ ɥɸɞɢ
ɧɚɯɨɞɢɥɢɫɶ ɜ ɫɨɫɬɨɹɧɢɢ ɧɟɜɟɫɨɦɨɫɬɢ. Ɉɰɟɧɢɬɟ ɜɵɫɨɬɭ ɩɨɞɨɛɧɨɣ ɛɚɲɧɢ.
9. ɇɚ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɫ ɭɝɥɨɦ ɧɚɤɥɨɧɚ
ɫɨɤ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɦɟɠɞɭ ɛɪɭɫɤɨɦ ɢ ɩɥɨɫɤɨɫɬɶɸ
ɧɚɯɨɞɢɬɫɹ ɧɟɛɨɥɶɲɨɣ ɛɪɭ-
ɇɚɤɥɨɧɧɚɹ ɩɥɨɫɤɨɫɬɶ ɞɜɢɠɟɬɫɹ ɫ ɭɫɤɨɪɟɧɢɟɦ a ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɭɤɚɡɚɧɧɨɦ ɧɚ
ɪɢɫɭɧɤɟ. ɉɪɢ ɤɚɤɨɦ ɦɢɧɢɦɚɥɶɧɨɦ ɭɫɤɨɪɟɧɢɢ ɛɪɭɫɨɤ ɧɚɱɧɟɬ ɫɤɨɥɶɡɢɬɶ ɩɨ ɩɥɨɫɤɨɫɬɢ?
10. ɇɚ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɫ ɭɝɥɨɦ ɧɚɤɥɨɧɚ
ɫɨɤ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɦɟɠɞɭ ɛɪɭɫɤɨɦ ɢ ɩɥɨɫɤɨɫɬɶɸ
ɧɚɯɨɞɢɬɫɹ ɧɟɛɨɥɶɲɨɣ ɛɪɭ-
ɇɚɤɥɨɧɧɚɹ ɩɥɨɫɤɨɫɬɶ ɞɜɢɠɟɬɫɹ ɫ ɭɫɤɨɪɟɧɢɟɦ a ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɭɤɚɡɚɧɧɨɦ ɧɚ
ɪɢɫɭɧɤɟ 6.7. ɉɪɢ ɤɚɤɨɦ ɦɢɧɢɦɚɥɶɧɨɦ ɭɫɤɨɪɟɧɢɢ ɛɪɭɫɨɤ ɧɚɱɧɟɬ ɫɤɨɥɶɡɢɬɶ ɩɨ
ɩɥɨɫɤɨɫɬɢ?
11. ɇɚ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɫ ɭɝɥɨɦ ɧɚɤɥɨɧɚ
ɫɨɤ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɦɟɠɞɭ ɛɪɭɫɤɨɦ ɢ ɩɥɨɫɤɨɫɬɶɸ
ɧɚɯɨɞɢɬɫɹ ɧɟɛɨɥɶɲɨɣ ɛɪɭ-
ɇɚɤɥɨɧɧɚɹ ɩɥɨɫɤɨɫɬɶ ɞɜɢɠɟɬɫɹ ɫ ɭɫɤɨɪɟɧɢɟɦ a ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɭɤɚɡɚɧɧɨɦ ɧɚ
ɪɢɫɭɧɤɟ 6.8. ɉɪɢ ɤɚɤɨɦ ɦɢɧɢɦɚɥɶɧɨɦ ɭɫɤɨɪɟɧɢɢ ɛɪɭɫɨɤ ɧɚɱɧɟɬ ɫɤɨɥɶɡɢɬɶ ɩɨ
ɩɥɨɫɤɨɫɬɢ?
,a
.
.
.
Ɋɢɫ. 6.7 Ɋɢɫ. 6.8
12. Ɍɟɥɟɠɤɚ, ɧɚ ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɬɫɹ ɫɨɫɭɞ ɫ ɠɢɞɤɨɫɬɶɸ, ɞɜɢɠɟɬɫɹ ɜ ɝɨɪɢ-
ɡɨɧɬɚɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɫ ɭɫɤɨɪɟɧɢɟɦ a . Ɉɩɪɟɞɟɥɢɬɟ ɭɝɨɥ ɧɚɤɥɨɧɚ
ɩɨɜɟɪɯ-
ɧɨɫɬɢ ɠɢɞɤɨɫɬɢ ɤ ɝɨɪɢɡɨɧɬɚɥɢ, ɫɱɢɬɚɹ ɩɨɥɨɠɟɧɢɟ ɠɢɞɤɨɫɬɢ ɜ ɫɨɫɭɞɟ ɭɫɬɚɧɨɜɢɜɲɢɦɫɹ.
86
13. ɉɭɫɬɶ ɢɦɟɟɬɫɹ ɩɥɚɧɟɬɚ, ɩɥɨɬɧɨɫɬɶ ɜɟɳɟɫɬɜɚ ɤɨɬɨɪɨɣ ɪɚɜɧɚ
U
F
G
F
G
. ɂɡɜɟɫɬ-
ɧɨ, ɱɬɨ ɬɟɥɨ ɧɚ ɷɤɜɚɬɨɪɟ ɜɟɫɢɬ ɜ n ɪɚɡ ɦɟɧɶɲɟ, ɱɟɦ ɧɚ ɩɨɥɸɫɟ. ɉɨ ɷɬɢɦ ɞɚɧɧɵɦ
ɨɩɪɟɞɟɥɢɬɟ ɩɟɪɢɨɞ ɨɛɪɚɳɟɧɢɹ ɩɥɚɧɟɬɵ ɜɨɤɪɭɝ ɫɜɨɟɣ ɨɫɢ.
ɦ
14. ɉɨɟɡɞ ɞɜɢɠɟɬɫɹ ɪɚɜɧɨɦɟɪɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ
X
ɪɚɞɢɭɫɨɦ ɦR 200 . ȼ ɟɝɨ ɜɚɝɨɧɟ ɜɡɜɟɲɢɜɚɸɬ ɝɪɭɡ ɦɚɫɫɨɣ ɤɝm 5
ɩɨ ɡɚɤɪɭɝɥɟɧɢɸ
20
ɫ
ɫ ɩɨɦɨɳɶɸ
ɞɢɧɚɦɨɦɟɬɪɚ. Ɉɩɪɟɞɟɥɢɬɟ ɪɟɡɭɥɶɬɚɬ ɜɡɜɟɲɢɜɚɧɢɹ.
6.4. ɇɟɢɧɟɪɰɢɚɥɶɧɵɟ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
Ɍɟɫɬɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ɡɧɚɧɢɣ
ɍɪɨɜɟɧɶ 1
ɇɚ ɪɢɫɭɧɤɟ 6.9 ɢɡɨɛɪɚɠɟɧɨ ɞɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ m
ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ
ɨɞɧɨɜɪɟɦɟɧɧɨ ɜ ɞɜɭɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ
)(YOXS
ɢ
cccc
.
)( XOYS
Ɋɢɫ. 6.9
1)
ȿɫɥɢ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ
(ɂɋɈ), ɬɨ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ
S
ɹɜɥɹɟɬɫɹ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɨɣ ɨɬɫɱɟɬɚ
S
c
ɹɜɥɹɟɬɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɨɣ ɨɬɫɱɟɬɚ, ɟɫɥɢ
ɨɧɚ…
ɇɚɯɨɞɢɬɫɹ ɜ ɫɨɫɬɨɹɧɢɢ ɩɨɤɨɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S.
1.
2.
Ⱦɜɢɠɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S.
3.
Ⱦɜɢɠɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢ-
ɫɬɟɦɵ ɨɬɫɱɟɬɚ S.
Ⱦɜɢɠɟɬɫɹ ɫ ɭɫɤɨɪɟɧɢɟɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S.
4.
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
2) ɍɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ m ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ
ɫɢɬɟɥɶɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ (
Ⱥɛɫɨɥɸɬɧɵɦ ɭɫɤɨɪɟɧɢɟɦ (a
1.
ɉɟɪɟɧɨɫɧɵɦ ɭɫɤɨɪɟɧɢɟɦ (
2.
G
).
aG).
ɩɟɪ
ɧɚ ɪɢɫ. 6.9) ɧɚɡɵɜɚɟɬɫɹ…
)(YOXS
ɨɬɧɨ-
87
Ɉɬɧɨɫɢɬɟɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ (
G
F
G
G
G
F
G
F
G
3.
ɇɨɪɦɚɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ (
4.
ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
5.
G
).
a
ɨɬɧ
).
a
n
3) ɍɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ m ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ
c
ɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ
ɧɚ ɪɢɫ. 6.9) ɧɚɡɵɜɚɟɬɫɹ…
1. Ⱥɛɫɨɥɸɬɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
2. ɉɟɪɟɧɨɫɧɵɦ ɭɫɤɨɪɟɧɢɟɦ (
3. Ɉɬɧɨɫɢɬɟɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ (
4. ɇɨɪɦɚɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ (
)(O
G
( a
).
aG).
ɩɟɪ
G
).
a
ɨɬɧ
(
).
a
n
4) ɍɫɤɨɪɟɧɢɟ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ
ɬɟɥɶɧɨ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ )(O ɧɚɡɵɜɚɟɬɫɹ…
1. Ⱥɛɫɨɥɸɬɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
2. ɉɟɪɟɧɨɫɧɵɦ ɭɫɤɨɪɟɧɢɟɦ (
3. Ɉɬɧɨɫɢɬɟɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ (
4. ɇɨɪɦɚɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
G
( a
aG).
ɩɟɪ
(
a
n
).
).
G
).
a
ɨɬɧ
5) ȼ ɤɚɠɞɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɭɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ m
ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ
1.
2.
3.
4.
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɪɚɜɧɨ…
GGG
aaa
.
ɨɬɧɩɟɪ
GGG
aaa
.
ɨɬɧɩɟɪ
GGG
aaa
.
ɩɟɪɨɬɧ
GGG
aaa
.
ɩɟɪɨɬɧ
6) ȼ ɤɚɠɞɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɭɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ m
ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ
1.
2.
3.
4.
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
7) Ɇɨɞɭɥɶ ɩɟɪɟɧɨɫɧɨɝɨ ɭɫɤɨɪɟɧɢɹ (
1. ɋɢɫɬɟɦɚ ɨɬɫɱɟɬɚ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɪɚɜɧɨ…
GGG
aaa
.
ɨɬɧɩɟɪ
GGG
aaa
.
ɨɬɧɩɟɪ
GGG
aaa
.
ɩɟɪɨɬɧ
GGG
aaa
.
ɩɟɪɨɬɧ
aG) ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɵɦ ɧɭɥɸ, ɟɫɥɢ…
ɩɟɪ
cccc
ɧɚɯɨɞɢɬɫɹ ɜ ɫɨɫɬɨɹɧɢɢ ɩɨɤɨɹ ɨɬɧɨɫɢɬɟɥɶ-
)( XOYS
ɨɬɧɨ-
c
ɨɬɧɨɫɢ-
)(O
cccc
)( XOYS
ɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ
88
.
)(YOXS
2. ɋɢɫɬɟɦɚ ɨɬɫɱɟɬɚ
G
cccc
ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ
)( XOYS
ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ
3. ɋɢɫɬɟɦɚ ɨɬɫɱɟɬɚ
cccc
ɧɚɯɨɞɢɬɫɹ ɜ ɫɨɫɬɨɹɧɢɢ ɩɨɤɨɹ ɢɥɢ ɞɜɢ-
)( XOYS
.
)(YOXS
ɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ
4. ɋɢɫɬɟɦɚ ɨɬɫɱɟɬɚ
ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
.
)(YOXS
cccc
ɪɚɜɧɨɦɟɪɧɨ ɜɪɚɳɚɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ
)( XOYS
.
)(YOXS
8) ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬ-
ɫɱɟɬɚ (ɂɋɈ) ɢɦɟɟɬ ɜɢɞ…
G
G
amF
1.
G
2.
G
3.
G
4.
5. ɩɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
ɝɞɟ
,
G
amF
;
, ɝɞɟ
;
ɩɟɪ
G
amF
ɨɬɧ
G
amF
n
a
GGG
aaa
– ɚɛɫɨɥɸɬɧɨɟ ɭɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ;
ɨɬɧɩɟɪ
– ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ;
n
9) ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬ-
ɫɱɟɬɚ ɢɦɟɟɬ ɜɢɞ…
G
amF
G
G
G
, ɝɞɟ
G
G
amF
G
amF
ɝɞɟ
,
ɩɟɪ
n
;
G
G
amFam
G
a
1.
2.
3.
4.
5. ɩɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
GGG
aaa
– ɚɛɫɨɥɸɬɧɨɟ ɭɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ;
ɨɬɧɩɟɪ
;
ɩɟɪɨɬɧ
–
ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ;
n
10) Ⱥɧɚɥɢɬɢɱɟɫɤɚɹ ɮɨɪɦɚ ɡɚɩɢɫɢ II ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ ɜ ɇɂɋɈ, ɤɨɬɨɪɚɹ ɞɜɢ-
ɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ, ɢɦɟɟɬ ɜɢɞ…
G
1.
G
2.
G
3.
G
4.
5. ɩɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
GG
FFam
ɢɨɬɧ
GG
FFam
ɢɩɟɪ
GG
FFam
ɝɞɟ
,
ɢ
G
Fam
ɝɞɟ
F
,
ɢ
G
ɝɞɟ
F
,
ɝɞɟ
,
G
ɢ
– ɫɢɥɚ ɢɧɟɪɰɢɢ;
ɢ
G
– ɫɢɥɚ ɢɧɟɪɰɢɢ;
F
ɢ
G
F
– ɫɢɥɚ ɢɧɟɪɰɢɢ;
ɢ
– ɫɢɥɚ ɢɧɟɪɰɢɢ;
11) ɋɢɥɭ, ɤɨɬɨɪɚɹ ɜɨɡɧɢɤɚɟɬ ɜ ɪɟɡɭɥɶɬɚɬɟ ɭɫɤɨɪɟɧɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ,
ɧɚɡɵɜɚɸɬ:
1. ɐɟɧɬɪɚɥɶɧɨɣ ɫɢɥɨɣ.
2. Ƚɪɚɜɢɬɚɰɢɨɧɧɨɣ ɫɢɥɵ.
3. ȼɟɫɨɦ ɬɟɥɚ.
4. ɋɢɥɨɣ ɢɧɟɪɰɢɢ.
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
89
12) ɋɢɥɵ ɢɧɟɪɰɢɢ…
1. ɇɟ ɢɦɟɸɬ ɩɪɨɬɢɜɨɞɟɣɫɬɜɭɸɳɟɣ ɫɢɥɵ.
2. ɇɟ ɹɜɥɹɸɬɫɹ ɪɟɡɭɥɶɬɚɬɨɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɬɟɥ.
3. ɉɨɹɜɥɹɸɬɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɭɫɤɨɪɟɧɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
4. əɜɥɹɸɬɫɹ ɫɢɥɚɦɢ ɩɪɢɪɨɞɵ.
5. ɇɟ ɹɜɥɹɸɬɫɹ ɪɟɡɭɥɶɬɚɬɨɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɬɟɥ, ɧɟ ɢɦɟɸɬ ɩɪɨɬɢɜɨɞɟɣ-
ɫɬɜɭɸɳɟɣ ɫɢɥɵ, ɩɨɹɜɥɹɸɬɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɭɫɤɨɪɟɧɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
13) ɋɢɥɚ ɢɧɟɪɰɢɢ ɜ ɇɂɋɈ, ɤɨɬɨɪɚɹ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ ɫ ɭɫɤɨɪɟɧɢɟɦ,
ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
G
G
amF
1.
G
2.
G
3.
G
4.
ɢ
5. ɩɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
;
ɩɟɪɢ
G
amF
;
ɩɟɪɢ
G
amF
;
ɨɬɧɢ
G
amF
;
14) ȼ ɇɂɋɈ ɧɟɜɨɡɦɨɠɧɨ ɫɨɡɞɚɬɶ ɚɧɚɥɢɬɢɱɟɫɤɭɸ ɮɨɪɦɭ ɡɚɩɢɫɢ…
1. ɉɟɪɜɨɝɨ ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ.
2. ȼɬɨɪɨɝɨ ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ.
3. Ɍɪɟɬɶɟɝɨ ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ.
4. Ɂɚɤɨɧɚ ɜɫɟɦɢɪɧɨɝɨ ɬɹɝɨɬɟɧɢɹ.
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
15) ɇɚ ɪɢɫɭɧɤɟ 6.10 ɫɢɥɚ ɢɧɟɪɰɢɢ…
1. ɇɟ ɞɟɣɫɬɜɭɟɬ ɧɚ ɩɨɞɜɟɲɟɧɧɵɣ ɲɚɪɢɤ ɜ ɫɥɭɱɚɹɯ (ɚ) ɢ (ɛ).
2. ɇɟ ɞɟɣɫɬɜɭɟɬ ɧɚ ɩɨɞɜɟɲɟɧɧɵɣ ɲɚɪɢɤ ɜ ɫɥɭɱɚɟ (ɛ).
3. Ⱦɟɣɫɬɜɭɟɬ ɧɚ ɩɨɞɜɟɲɟɧɧɵɣ ɲɚɪɢɤ ɜ ɫɥɭɱɚɟ (ɚ).
4. ɇɟ ɞɟɣɫɬɜɭɟɬ ɧɚ ɩɨɞɜɟɲɟɧɧɵɣ ɲɚɪɢɤ ɜ ɫɥɭɱɚɟ (ɚ) ɢ ɞɟɣɫɬɜɭɟɬ ɧɚ ɩɨɞ-
ɜɟɲɟɧɧɵɣ ɲɚɪɢɤ ɜ
5. ɉɪɚɜɢɥɶɧɨɝɨ ɨɬɜɟɬɚ ɧɟ ɩɪɟɞɥɨɠɟɧɨ.
ɚ ɛ
ɫɥɭɱɚɟ (ɛ):
Ɋɢɫ. 6.10
G
90
G
amF
.
ɉɢ
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