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ȿɫɥɢ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɬɚɤ, ɱɬɨ ɟɟ ɫɤɨɪɨɫɬɶ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɩɨ
X
X
X
X
x
x
x
x
r
t
ɜɟɥɢɱɢɧɟ ɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ
G
, ɞɜɢɠɟɧɢɟ ɹɜɥɹɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɵɦ ɪɚɜ-
const
ɧɨɦɟɪɧɵɦ.
ɉɨ ɮɨɪɦɭɥɚɦ (2.2), (2.3) ɦɨɠɧɨ ɪɚɫɫɱɢɬɚɬɶ ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ ɜ ɥɸɛɨɣ ɡɚɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ. Ɉɞɧɚɤɨ ɧɚ ɩɪɚɤɬɢɤɟ ɱɚɫɬɨ ɜɨɡɧɢɤɚɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɜ
ɪɚɫɱɟɬɟ ɫɪɟɞɧɟɣ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ. ɂɦɟɧɧɨ ɷɬɚ ɫɤɨɪɨɫɬɶ
ɢɦɟɟɬɫɹ ɜ ɜɢɞɭ, ɤɨɝɞɚ ɝɨɜɨɪɹɬ, ɧɚɩɪɢɦɟɪ, ɨ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɚɜɬɨɦɨɛɢɥɹ, ɩɨɟɡɞɚ, ɜɟɥɨɫɢɩɟɞɢɫɬɚ ɢ ɬ. ɞ.
ȼɟɥɢɱɢɧɚ ɫɪɟɞɧɟɣ ɫɤɨɪɨɫɬɢ ȣ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɨɬɧɨɲɟɧɢɟɦ ɩɭɬɢ ¨S, ɩɪɨɣ-
ɫɪ
ɞɟɧɧɨɝɨ ɬɟɥɨɦ ɡɚ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ ¨t, ɤ ɷɬɨɦɭ ɩɪɨɦɟɠɭɬɤɭ:
S
'
X
. (2.4)
ɫɪ
t
'
ȼ ɪɟɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɫɤɨɪɨɫɬɶ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɩɪɢ ɞɜɢɠɟɧɢɢ ɢɡɦɟɧɹɟɬɫɹ. Ȼɵɫɬɪɨɬɭ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ ɯɚɪɚɤɬɟɪɢɡɭɸɬ ɜɟɤɬɨɪɧɨɣ ɜɟɥɢɱɢɧɨɣ, ɤɨɬɨɪɚɹ ɧɚɡɵɜɚɟɬɫɹ ɭɫɤɨɪɟɧɢɟɦ.
ɍɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɤɚɤ ɩɪɨɢɡɜɨɞɧɭɸ ɜɟɤ-
ɬɨɪɚ ɫɤɨɪɨɫɬɢ ɩɨ ɜɪɟɦɟɧɢ:
G
a
'
lim . (2.5)
t
o' 0
GG
d
dt
t
'
ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɢɡɜɟɫɬɧɚ ɡɚɜɢɫɢɦɨɫɬɶ ɦɨɞɭɥɹ ɫɤɨɪɨɫɬɢ ɨɬ ɜɪɟɦɟɧɢ ȣ(t), ɦɨɞɭɥɶ ɭɫɤɨɪɟɧɢɹ ɨɩɪɟɞɟɥɹɸɬ ɤɚɤ ɩɪɨɢɡɜɨɞɧɭɸ ɦɨɞɭɥɹ ɫɤɨɪɨɫɬɢ ɩɨ ɜɪɟɦɟɧɢ:
d
. (2.6)
a
dt
Ɇɨɞɭɥɶ ɭɫɤɨɪɟɧɢɹ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɨ ɩɪɨɟɤɰɢɹɦ ɜɟɤɬɨɪɚ ɭɫɤɨɪɟɧɢɹ
,,
aaa
ɤɚɤ ɩɪɨɢɡɜɨɞɧɵɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɩɪɨɟɤɰɢɣ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ
ɩɨ ɜɪɟɦɟɧɢ ɢɥɢ ɜɬɨɪɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɩɪɨɟɤɰɢɣ ɪɚɞɢɭɫ-
ɜɟɤɬɨɪɚ
ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
ɧɚ ɤɨɨɪɞɢɧɚɬɧɵɟ ɨɫɢ:
yz
222
aaaa
ɉɪɨɟɤɰɢɢ ɦɨɞɭɥɹ ɭɫɤɨɪɟɧɢɹ ɧɚ ɤɨɨɪɞɢɧɚɬɧɵɟ ɨɫɢ
G
ɩɨ ɜɪɟɦɟɧɢ:
ddx dyddz
XX
x z
;;;;;
aaaaaa
xxy y zz
dt dt dt dt dt dt
222
222
ȿɫɥɢ ɢɡɜɟɫɬɧɵ ɡɚɜɢɫɢɦɨɫɬɶ ɭɫɤɨɪɟɧɢɹ ɨɬ ɜɪɟɦɟɧɢ )(
d
X
y
G
X
, ɬɨ ɧɚɣɬɢ ɫɤɨɪɨɫɬɶ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
0
.
yz
,,
aaa
yz
aG ɢ ɫɤɨɪɨɫɬɶ
ɦɨɠɧɨ ɧɚɣɬɢ
,,
aaa
yz
.
ɦɨɠɧɨ ɩɨ ɮɨɪɦɭɥɟ:
GG
XX
t
G
adt
³
0
0
.
11
ȼɵɱɢɫɥɟɧɢɟ ɞɥɢɧɵ ɩɭɬɢ. ɂɡ ɮɨɪɦɭɥɵ (2.3) ɷɥɟɦɟɧɬɚɪɧɵɣ ɩɭɬɶ, ɩɪɨɣɞɟɧ-
tdS
X
t
X
X
X
ɧɵɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ: d
.
Ɍɨɝɞɚ ɩɭɬɶ, ɩɪɨɣɞɟɧɧɵɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ ɨɬ ɧɚɱɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ, ɪɚɜɧɨɝɨ ɧɭɥɸ, ɞɨ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t ɦɨɠɧɨ ɧɚɣɬɢ ɩɭɬɟɦ
ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ:
t
X
Sdt
³
. (2.7)
0
Ƚɪɚɮɢɱɟɫɤɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɮɨɪɦɭɥɵ (2.7) ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ. 2.3.
Ɋɢɫ. 2.3
ȿɫɥɢ ɢɡɜɟɫɬɟɧ ɝɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɫɤɨɪɨɫɬɢ ɨɬ ɜɪɟɦɟɧɢ ȣ(t), ɡɚɞɚɧɵ
ɧɚɱɚɥɶɧɵɣ ɢ ɤɨɧɟɱɧɵɣ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ
()tt
ɠɭɬɨɤ ɜɪɟɦɟɧɢ
ɱɢɫɥɟɧɧɨ ɪɚɜɟɧ ɩɥɨɳɚɞɢ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɮɢɝɭɪɵ ɩɨɞ
21
,tt
, ɬɨ ɩɪɨɣɞɟɧɧɵɣ ɩɭɬɶ ɡɚ ɩɪɨɦɟ-
12
ɝɪɚɮɢɤɨɦ ɫɤɨɪɨɫɬɢ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɩɪɹɦɵɦɢ ɜɨɫɫɬɚɧɨɜɥɟɧɧɵɦɢ ɢɡ ɦɨɦɟɧɬɨɜ
,tt
ɜɪɟɦɟɧɢ
12
.
ɉɪɹɦɨɥɢɧɟɣɧɨɟ ɪɚɜɧɨɩɟɪɟɦɟɧɧɨɟ ɞɜɢɠɟɧɢɟ
ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɬɚɤ, ɱɬɨ ɟɟ ɭɫɤɨɪɟɧɢɟ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɢ ɩɨ ɜɟɥɢɱɢɧɟ cons
G
a
, ɞɜɢɠɟɧɢɟ ɹɜɥɹɟɬɫɹ ɩɪɹɦɨɥɢ-
ɧɟɣɧɵɦ ɪɚɜɧɨɩɟɪɟɦɟɧɧɵɦ. Ɍɚɤɨɟ ɩɪɹɦɨɥɢɧɟɣɧɨɟ ɞɜɢɠɟɧɢɟ ɦɨɠɟɬ ɛɵɬɶ ɪɚɜɧɨ-
ɭɫɤɨɪɟɧɧɵɦ ɢ ɪɚɜɧɨɡɚɦɟɞɥɟɧɧɵɦ.
ɍɫɤɨɪɟɧɢɟ ɩɪɢ ɪɚɜɧɨɩɟɪɟɦɟɧɧɨɦ ɞɜɢɠɟɧɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
GG
XX
G
a
G
ɝɞɟ
– ɫɤɨɪɨɫɬɶ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t;
G
X
– ɫɤɨɪɨɫɬɶ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ;
0
0
,
t
t – ɜɪɟɦɹ, ɩɪɨɲɟɞɲɟɟ ɫ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
ɪɚɜɧɨɭɫɤɨɪɟɧɧɨɦ, ɩɪɹɦɨɥɢɧɟɣɧɨɦ ɞɜɢɠɟɧɢɢ ɧɚɩɪɚɜɥɟɧɢɟ ɜɟɤɬɨɪɚ
ɉɪɢ
ɫɤɨɪɨɫɬɢ ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɜɟɤɬɨɪɚ ɭɫɤɨɪɟɧɢɹ, ɢ ɩɪɨɟɤɰɢɹ ɫɤɨɪɨɫɬɢ ɧɚ
ɧɚɩɪɚɜɥɟɧɢɟ ɤɨɨɪɞɢɧɚɬɧɨɣ ɨɫɢ (ɧɚɩɪɢɦɟɪ, ɏ) ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
.
ta
0
12
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɨɣ (2.7) ɜ ɩɪɟɞɟɥɚɯ ɨɬ ɧɭɥɹ ɞɨ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɦɨ-
X
X
M
M
ɦɟɧɬɚ ɜɪɟɦɟɧɢ t ɧɚɣɞɟɦ ɮɨɪɦɭɥɭ ɞɥɹ ɪɚɫɱɟɬɚ ɩɪɨɣɞɟɧɧɨɝɨ ɩɭɬɢ ɩɪɢ ɪɚɜɧɨɭɫɤɨɪɟɧɧɨɦ, ɩɪɹɦɨɥɢɧɟɣɧɨɦ ɞɜɢɠɟɧɢɢ:
tt
Sdt atdtSt
XX X
³³
00
ɉɪɢ
ɪɚɜɧɨɡɚɦɟɞɥɟɧɧɨɦ ɩɪɹɦɨɥɢɧɟɣɧɨɦ ɞɜɢɠɟɧɢɢ ɧɚɩɪɚɜɥɟɧɢɟ ɜɟɤɬɨɪɚ
();
00
at
2
2
.
ɫɤɨɪɨɫɬɢ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɧɚɩɪɚɜɥɟɧɢɸ ɜɟɤɬɨɪɚ ɭɫɤɨɪɟɧɢɹ ɢ ɩɪɨɟɤɰɢɹ ɫɤɨɪɨɫɬɢ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɤɨɨɪɞɢɧɚɬɧɨɣ ɨɫɢ (ɧɚɩɪɢɦɟɪ, ɏ) ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
.
ta
0
Ⱥɧɚɥɨɝɢɱɧɨ ɧɚɣɞɟɦ ɮɨɪɦɭɥɭ ɞɥɹ ɪɚɫɱɟɬɚ ɩɪɨɣɞɟɧɧɨɝɨ ɩɭɬɢ ɩɪɢ ɪɚɜɧɨɡɚɦɟɞɥɟɧɧɨɦ, ɩɪɹɦɨɥɢɧɟɣɧɨɦ ɞɜɢɠɟɧɢɢ:
tt
Sdt atdtSt
XX X
³³
00
();
00
at
2
2
.
Ƚɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɢ ɭɫɤɨɪɟɧɢɹ, ɫɤɨɪɨɫɬɢ ɢ ɤɨɨɪɞɢɧɚɬɵ ɬɟɥɚ ɨɬ ɜɪɟɦɟɧɢ
ɩɪɢ ɪɚɜɧɨɩɟɪɟɦɟɧɧɨɦ ɞɜɢɠɟɧɢɢ ɩɪɟɞɫɬɚɜɥɟɧɵ ɧɚ ɪɢɫ. 2.4.
Ɋɢɫ. 2.4
ɍɫɤɨɪɟɧɢɟ ɩɪɢ ɤɪɢɜɨɥɢɧɟɣɧɨɦ ɞɜɢɠɟɧɢɢ
Ʉɪɢɜɢɡɧɚ ɩɥɨɫɤɨɣ ɥɢɧɢɢ ɜ ɤɚɤɨɣ-ɥɢɛɨ ɟɟ ɬɨɱɤɟ ɪɚɜɧɚ ɤɪɢɜɢɡɧɟ ɨɤɪɭɠɧɨɫɬɢ,
ɫɥɢɜɚɸɳɟɣɫɹ ɜ ɞɚɧɧɨɦ ɦɟɫɬɟ ɫ ɤɪɢɜɨɣ ɧɚ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɦ ɟɟ ɭɱɚɫɬɤɟ (ɪɢɫ. 2.5).
Ⱥɧɚɥɢɬɢɱɟɫɤɢ
ɤɪɢɜɢɡɧɚ ɤɪɢɜɨɣ ɋ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ:
d
'
lim , (2.8)
C
t
o' 0
dS
S
'
ɝɞɟ ¨ij – ɭɝɨɥ ɦɟɠɞɭ ɤɚɫɚɬɟɥɶɧɵɦɢ ɤ ɤɪɢɜɨɣ ɜ ɬɨɱɤɚɯ, ɨɬɫɬɨɹɳɢɯ ɞɪɭɝ ɨɬ ɞɪɭɝɚ
ɧɚ ¨S.
13
Ɋɢɫ. 2.5
X
W
Ʉɪɢɜɢɡɧɚ ɩɥɨɫɤɨɣ ɥɢɧɢɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɫɤɨɪɨɫɬɶɸ ɢɡɦɟɧɟɧɢɹ ɧɚɩɪɚɜɥɟɧɢɹ ɤɪɢɜɨɣ, ɬ. ɟ. ɫɤɨɪɨɫɬɶɸ ɩɨɜɨɪɨɬɚ ɤɚɫɚɬɟɥɶɧɨɣ ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ ɜɞɨɥɶ ɤɪɢɜɨɣ. ȼɟɥɢɱɢɧɚ, ɨɛɪɚɬɧɚɹ ɤɪɢɜɢɡɧɟ ɬɪɚɟɤɬɨɪɢɢ ɋ, ɪɚɜɧɚ
ɪɚɞɢɭɫɭ ɤɪɢɜɢɡɧɵ ɬɪɚ-
ɟɤɬɨɪɢɢ R:
R1 .
C
ȼ ɫɥɭɱɚɟ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɨɩɪɟɞɟɥɟɧɧɵɣ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɫɨɜɩɚɞɚɟɬ ɫ ɪɚɞɢɭɫɨɦ ɨɤɪɭɠɧɨɫɬɢ.
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɩɪɢ ɞɜɢɠɟɧɢɢ ɬɨɱɤɢ ɩɨ ɩɥɨɫɤɨɣ ɤɪɢɜɨɣ ɟɟ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɢ ɩɨ ɜɟɥɢɱɢɧɟ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ
ɬɨɱɤɢ ɛɭɞɟɬ ɢɦɟɬɶ ɞɜɟ ɫɨɫɬɚɜɥɹɸɳɢɟ: ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ
ɪɢɡɭɸɳɟɟ ɛɵɫɬɪɨɬɭ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ ɩɨ ɦɨɞɭɥɸ, ɢ ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟ-
G
a
, ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɟ ɛɵɫɬɪɨɬɭ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ.
ɧɢɟ
n
G
a
, ɯɚɪɚɤɬɟ-
W
Ɍɚɧɝɟɧɰɢɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
G
ɝɞɟ
G
a
W
G
– ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ, ɧɚɩɪɚɜɥɟɧɧɵɣ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɬɪɚɟɤɬɨɪɢɢ ɜ ɞɚɧ-
'
W
'
'
X
§
limlim , (2.9)
¨
©
·
W
¸
o'o'
tt 00
'
tt
¹
GG
d
X
W
dt
ɧɨɣ ɟɟ ɬɨɱɤɟ ɜ ɫɬɨɪɨɧɭ ɞɜɢɠɟɧɢɹ ɢ ɬɨɠɞɟɫɬɜɟɧɧɵɣ ɟɞɢɧɢɱɧɨɦɭ ɜɟɤɬɨɪɭ ɫɤɨɪɨɫɬɢ (ɪɢɫ. 2.6).
Ɋɢɫ. 2.6
ɇɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ ɬɪɚɟɤɬɨɪɢɢ:
G
2
XX
GG
an
n
'
n
lim
0
t
'o
tR
'
, (2.10)
14
ɝɞɟ ȣ – ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ;
R
M
M
M
R – ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ;
nG– ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɧɨɪɦɚɥɢ ɤ ɬɪɚɟɤɬɨɪɢɢ ɜ ɞɚɧɧɨɣ ɟɟ ɬɨɱɤɟ (ɪɢɫ. 2.6).
ȼɟɤɬɨɪ ɩɨɥɧɨɝɨ ɭɫɤɨɪɟɧɢɹ aG ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧ ɜ ɜɢɞɟ ɫɭɦɦɵ ɞɜɭɯ
G
G
a
a
ɢ
ɜɟɤɬɨɪɨɜ
W
ɪɢɢ, ɚ ɜɬɨɪɨɣ (
, ɨɞɢɧ ɢɡ ɤɨɬɨɪɵɯ (
n
G
a
) ɩɟɪɩɟɧɞɢɤɭɥɹɪɟɧ ɜɟɤɬɨɪɭ ɫɤɨɪɨɫɬɢ ɢ ɧɚɩɪɚɜɥɟɧ ɤ ɰɟɧɬɪɭ
n
ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ (ɪɢɫ. 2.6).
ȿɫɥɢ ɫɤɨɪɨɫɬɶ ɪɚɫɬɟɬ ɩɨ ɜɟɥɢɱɢɧɟ, ɬɨ ɜɟɤɬɨɪ
ɠɟɧɢɹ, ɟɫɥɢ ɫɤɨɪɨɫɬɶ ɭɛɵɜɚɟɬ ɩɨ ɜɟɥɢɱɢɧɟ, ɬɨ ɜɟɤɬɨɪ
G
a
) ɧɚɩɪɚɜɥɟɧ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɬɪɚɟɤɬɨ-
W
G
a
ɧɚɩɪɚɜɥɟɧ ɜ ɫɬɨɪɨɧɭ ɞɜɢ-
W
G
a
ɧɚɩɪɚɜɥɟɧ ɜ ɫɬɨɪɨɧɭ,
W
ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɧɚɩɪɚɜɥɟɧɢɸ ɞɜɢɠɟɧɢɹ.
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɦɨɞɭɥɶ ɭɫɤɨɪɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɩɪɢ ɞɜɢɠɟɧɢɢ ɩɨ
ɩɥɨɫɤɨɣ ɤɪɢɜɨɣ ɪɚɜɟɧ:
2
2
§·
22
aaa
n
XX
¨¸
W
©¹
2
d
§·
.
¨¸
dt
©¹
ȿɫɥɢ ɧɚɩɪɚɜɥɟɧɢɟ ɫɤɨɪɨɫɬɢ ɧɟ ɢɡɦɟɧɹɟɬɫɹ, ɞɜɢɠɟɧɢɟ ɹɜɥɹɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɵɦ. Ʉɪɢɜɢɡɧɚ ɩɪɹɦɨɣ ɥɢɧɢɢ ɪɚɜɧɚ ɧɭɥɸ (ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ R ɪɚɜɟɧ ɛɟɫɤɨɧɟɱɧɨɫɬɢ). Ɍɨɝɞɚ ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɛɭɞɟɬ ɪɚɜɧɨ ɧɭɥɸ, ɚ ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ
ɪɚɜɧɨ ɬɚɧɝɟɧɰɢɚɥɶɧɨɦɭ ɭɫɤɨɪɟɧɢɸ (
GG
aa
W
).
Ʉɢɧɟɦɚɬɢɤɚ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ
ȼɫɟ ɬɨɱɤɢ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɜɪɚɳɚɸɳɟɝɨɫɹ ɜɨɤɪɭɝ ɧɟɤɨɬɨɪɨɣ
ɨɫɢ ɈɈ, ɞɜɢɠɭɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɹɦ, ɰɟɧɬɪɵ ɤɨɬɨɪɵɯ ɥɟɠɚɬ ɧɚ ɨɫɢ ɜɪɚɳɟɧɢɹ.
Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ ɤɚɠɞɨɣ ɬɨɱɤɢ (ɜɟɤɬɨɪ, ɩɪɨɜɟɞɟɧɧɵɣ ɢɡ ɰɟɧɬɪɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ
ɨɤɪɭɠɧɨɫɬɢ ɜ ɞɚɧɧɭɸ ɬɨɱɤɭ) ɩɨɜɨɪɚɱɢɜɚɟɬɫɹ ɡɚ ɜɪɟɦɹ ¨t ɧɚ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɭɝɨɥ
¨ij – ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɬɜɟɪɞɨɝɨ ɬɟɥɚ.
Ɉɱɟɧɶ ɦɚɥɵɟ ɩɨɜɨɪɨɬɵ, ɩɪɢ ɤɨɬɨɪɵɯ ɩɭɬɶ, ɩɪɨɯɨɞɢɦɵɣ ɥɸɛɨɣ ɬɨɱɤɨɣ ɬɟɥɚ
ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɩɪɹɦɨɥɢɧɟɣɧɵɦ, ɦɨɝɭɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɤɚɤ ɜɟɤɬɨɪɵ
ሬ
ο߮
. ɇɚɩɪɚɜɥɟɧɢɟ ɷɬɢɯ ɜɟɤɬɨɪɨɜ ɫɜɹɡɵɜɚɸɬ ɩɪɚɜɢɥɨɦ ɩɪɚɜɨɝɨ ɜɢɧɬɚ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɜɪɚɳɟɧɢɹ ɬɟɥɚ. ȼɟɤɬɨɪɵ ɬɚɤɨɝɨ ɬɢɩɚ, ɧɚɩɪɚɜɥɟɧɢɟ ɤɨɬɨɪɵɯ ɫɜɹɡɵɜɚɟɬɫɹ
ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɜɪɚɳɟɧɢɹ (ɢɥɢ ɨɛɯɨɞɚ), ɧɚɡɵɜɚɸɬ ɚɤɫɢɚɥɶɧɵɦɢ ɜɟɤɬɨɪɚɦɢ.
ɍɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ ɬɟɥɚ ɧɚɡɵɜɚɟɬɫɹ ɜɟɤɬɨɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɤɨɬɨɪɚɹ ɨɩɪɟɞɟ-
ɥɹɟɬɫɹ ɤɚɤ ɩɪɨɢɡɜɨɞɧɚɹ ɜɟɤɬɨɪɚ ɭɝɥɚ ɩɨɜɨɪɨɬɚ ɩɨ ɜɪɟɦɟɧɢ:
G
Z
'
lim . (2.11)
t
o' 0
'
GG
d
dt
t
ȼɟɤɬɨɪ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɹɜɥɹɟɬɫɹ ɚɤɫɢɚɥɶɧɵɦ ɜɟɤɬɨɪɨɦ, ɚ ɟɝɨ ɧɚɩɪɚɜɥɟ-
ɧɢɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɩɪɚɜɢɥɭ ɩɪɚɜɨɝɨ ɜɢɧɬɚ (ɪɢɫ. 2.7).
Ɇɨɞɭɥɶ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɪɚɜɟɧ:
d
Z
.
dt
ȼɟɤɬɨɪ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɦɨɠɟɬ ɢɡɦɟɧɹɬɶɫɹ ɤɚɤ ɩɨ ɜɟɥɢɱɢɧɟ, ɬɚɤ ɢ ɩɨ
ɧɚɩɪɚɜɥɟɧɢɸ (ɡɚ ɫɱɟɬ ɩɨɜɨɪɨɬɚ ɨɫɢ ɜɪɚɳɟɧɢɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ). Ȼɵɫɬɪɨɬɚ ɢɡɦɟ-
15
ɧɟɧɢɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɜɟɤɬɨɪɧɨɣ ɜɟɥɢɱɢɧɨɣ, ɤɨɬɨɪɚɹ ɧɚɡɵ-
Z
Z
Z
GGG
Z
X
R
Z
X
ɜɚɟɬɫɹ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ.
Ɋɢɫ. 2.7
ɍɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɩɪɨɢɡɜɨɞɧɚɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɩɨ
ɜɪɟɦɟɧɢ:
G
H
'
lim . (2.12)
t
o' 0
'
GG
d
dt
t
ȿɫɥɢ ɧɚɩɪɚɜɥɟɧɢɟ ɨɫɢ ɜɪɚɳɟɧɢɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɵɦ, ɢɡ-
ɜɟɫɬɧɚ ɡɚɜɢɫɢɦɨɫɬɶ ɦɨɞɭɥɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɨɬ ɜɪɟɦɟɧɢ Ȧ(t), ɬɨ
ɜɨɝɨ ɭɫɤɨɪɟɧɢɹ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɩɪɨɢɡɜɨɞɧɚɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɩɨ ɜɪɟɦɟɧɢ:
d
H
.
dt
ɦɨɞɭɥɶ ɭɝɥɨ-
ȼ ɷɬɨɣ ɮɨɪɦɭɥɟ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ – ɷɬɨ ɚɥɝɟɛɪɚɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ, ɤɨɬɨɪɚɹ
ɩɨɥɨɠɢɬɟɥɶɧɚ, ɟɫɥɢ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɪɚɫɬɟɬ ɩɨ ɜɟɥɢɱɢɧɟ (ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɜɟɤɬɨ-
GG
,
ZH
ɪɵ
ɭɦɟɧɶɲɚɟɬɫɹ (ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɧɚɩɪɚɜɥɟɧɢɹ ɜɟɤɬɨɪɨɜ
ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɨɟ ɧɚɩɪɚɜɥɟɧɢɟ), ɢ ɨɬɪɢɰɚɬɟɥɶɧɚ, ɟɫɥɢ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ
GG
,
ZH
ɩɪɨɬɢɜɨɩɨɥɨɠɧɵ).
Ʌɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ ɬɟɥɚ ɩɪɢ ɜɪɚɳɟɧɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɤɬɨɪɧɵɦ
ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɬɟɥɚ ɢ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɬɨɱɤɢ:
>@
. (2.13)
r
,
Ɇɨɞɭɥɶ ɥɢɧɟɣɧɨɣ ɫɤɨɪɨɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
,
ɝɞɟ R – ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɬɨɱɤɢ ɞɨ ɨɫɢ ɜɪɚɳɟɧɢɹ (ɪɚɞɢɭɫ ɨɤɪɭɠɧɨɫɬɢ, ɩɨ ɤɨɬɨɪɨɣ
ɜɪɚɳɚɟɬɫɹ ɬɨɱɤɚ ɬɜɟɪɞɨɝɨ ɬɟɥɚ).
ɇɚɩɪɚɜɥɟɧɢɟ ɜɟɤɬɨɪɚ ɥɢɧɟɣɧɨɣ ɫɤɨɪɨɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɩɪɚɜɢɥɭ ɩɪɚɜɨɝɨ
ɜɢɧɬɚ (ɪɢɫ. 2.8). Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɦɨɞɭɥɹ ɧɨɪɦɚɥɶɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɬɨɱɟɤ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɟɥɚ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (2.10) ɢ ɫ ɭɱɟɬɨɦ ɜɵɪɚɠɟɧɢɹ (2.13) ɩɨɥɭɱɢɦ
ɮɨɪɦɭɥɭ:
2
aR
Z
n
.
16
Ɋɢɫ. 2.8
t
Z
G
M
Z
Z
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɬɚɧɝɟɧɰɢɚɥɶɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɬɨɱɟɤ ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɟɥɚ
ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (2.9) ɢ ɫ ɭɱɟɬɨɦ ɜɵɪɚɠɟɧɢɹ (2.13) ɩɨɥɭɱɢɦ ɮɨɪɦɭɥɭ:
aR
H
W
.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ ɢ ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɹ ɪɚɫɬɭɬ ɥɢɧɟɣɧɨ ɫ
ɪɚɫɫɬɨɹɧɢɟɦ ɬɨɱɤɢ ɞɨ ɨɫɢ ɜɪɚɳɟɧɢɹ R.
Ɇɨɞɭɥɶ ɥɢɧɟɣɧɨɝɨ (ɩɨɥɧɨɝɨ) ɭɫɤɨɪɟɧɢɹ ɬɨɱɟɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
22 2 2 2
aaa R R
n
()()
ZH
W
.
Ɋɚɜɧɨɦɟɪɧɨɟ ɜɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ
ȼɪɚɳɟɧɢɟ ɫ ɩɨɫɬɨɹɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ
ɧɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨ-
cons
ɦɟɪɧɵɦ. ɍɝɨɥ ɩɨɜɨɪɨɬɚ ɨɬ ɧɚɱɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ, ɪɚɜɧɨɝɨ ɧɭɥɸ, ɞɨ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t ɪɚɜɟɧ:
tt
dt dt t t
ZZ ZMZ
³³
00
;
.
Ʉ ɨɫɧɨɜɧɵɦ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦ ɪɚɜɧɨɦɟɪɧɨɝɨ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɨɬɧɨɫɹɬɫɹ ɩɟɪɢɨɞ ɜɪɚɳɟɧɢɹ Ɍ (ɜɪɟɦɹ ɨɞɧɨɝɨ ɩɨɥɧɨɝɨ ɨɛɨɪɨɬɚ) ɢ ɱɚɫɬɨɬɚ ɜɪɚɳɟɧɢɹ n (ɱɢɫɥɨ
ɨɛɨɪɨɬɨɜ ɡɚ ɨɞɧɭ ɫɟɤɭɧɞɭ).
ɉɟɪɢɨɞ ɜɪɚɳɟɧɢɹ ɦɨɠɧɨ ɪɚɫɫɱɢɬɚɬɶ ɩɨ ɮɨɪɦɭɥɚɦ:
S
t
; T
T ,
N
2
ɝɞɟ t – ɜɪɟɦɹ, ɡɚ ɤɨɬɨɪɨɟ ɬɟɥɨɦ ɫɨɜɟɪɲɟɧɨ N ɩɨɥɧɵɯ ɨɛɨɪɨɬɨɜ ɩɪɢ ɜɪɚɳɟɧɢɢ ɫ
ɩɨɫɬɨɹɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Ȧ.
ɑɚɫɬɨɬɚ ɜɪɚɳɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɜɵɪɚɠɟɧɢɣ:
N
n
;
t
1
n
.
T
ɍɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ Ȧ ɫɜɹɡɚɧɚ ɫ ɱɚɫɬɨɬɨɣ ɜɪɚɳɟɧɢɹ n ɫɨɨɬɧɨɲɟɧɢɟɦ:
n
S
2
.
17
Ɋɚɜɧɨɩɟɪɟɦɟɧɧɨɟ ɜɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ
t
H
G
Z
Z
HZZ
Z
HZZ
ȼɪɚɳɟɧɢɟ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ
ɧɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨ-
cons
ɩɟɪɟɦɟɧɧɵɦ, ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɛɵɬɶ ɪɚɜɧɨɭɫɤɨɪɟɧɧɵɦ ɢ ɪɚɜɧɨɡɚɦɟɞɥɟɧɧɵɦ.
ɍɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɩɪɢ ɪɚɜɧɨɩɟɪɟɦɟɧɧɨɦ ɞɜɢɠɟɧɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
ɪɚɜɧɨɭɫɤɨɪɟɧɧɨɦ ɜɪɚɳɟɧɢɢ ɦɨɞɭɥɶ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɪɚɫɬɟɬ ɨɬ ɡɧɚ-
ɉɪɢ
ɱɟɧɢɹ ɧɚɱɚɥɶɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ
G
H
Z
GG
.
0
t
ɢ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɦɨɠɟɬ ɛɵɬɶ
0
ɪɚɫɫɱɢɬɚɧ ɩɨ ɮɨɪɦɭɥɟ:
.
t
0
ɍɝɨɥ ɩɨɜɨɪɨɬɚ ɨɬ ɧɚɱɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ ɪɚɜɧɨɝɨ ɧɭɥɸ ɞɨ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
MMZ ZH MZ
ɪɚɜɧɨɡɚɦɟɞɥɟɧɧɨɦ ɜɪɚɳɟɧɢɢ ɦɨɞɭɥɶ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɭɦɟɧɶɲɚ-
ɉɪɢ
tt
ddt tdt t
³³ ³
t
00
();
00
ɟɬɫɹ ɨɬ ɡɧɚɱɟɧɢɹ ɧɚɱɚɥɶɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ
ɢ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
0
2
H
t
.
2
ɦɨɠɟɬ ɛɵɬɶ ɪɚɫɫɱɢɬɚɧ ɩɨ ɮɨɪɦɭɥɟ:
.
t
0
ɍɝɨɥ ɩɨɜɨɪɨɬɚ ɨɬ ɧɚɱɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ, ɪɚɜɧɨɝɨ ɧɭɥɸ, ɞɨ ɩɪɨɢɡ-
ɜɨɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
tt
MMZ ZH MZ
ddt tdt t
³³ ³
00
t
();
00
ɉɪɢ ɪɚɜɧɨɭɫɤɨɪɟɧɧɨɦ ɜɪɚɳɟɧɢɢ ɧɚɩɪɚɜɥɟɧɢɹ ɜɟɤɬɨɪɨɜ
ɚ ɩɪɢ ɪɚɜɧɨɡɚɦɟɞɥɟɧɧɨɦ ɜɪɚɳɟɧɢɢ ɧɚɩɪɚɜɥɟɧɢɹ ɜɟɤɬɨɪɨɜ
2
H
t
.
2
GG
ZH
,
GG
,
ZH
ɹɜɥɹɸɬɫɹ ɩɪɨ-
ɫɨɜɩɚɞɚɸɬ,
ɬɢɜɨɩɨɥɨɠɧɵɦɢ.
Ʉɥɸɱɟɜɵɟ ɩɨɧɹɬɢɹ ɢ ɬɟɪɦɢɧɵ
Ⱥɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ
Ⱦɜɢɠɟɧɢɟ
Ⱦɜɢɠɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɟ
Ⱦɜɢɠɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɟ ɪɚɜɧɨɦɟɪɧɨɟ
Ⱦɜɢɠɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɟ ɪɚɜɧɨɩɟɪɟɦɟɧɧɨɟ
Ⱦɜɢɠɟɧɢɟ ɤɪɢɜɨɥɢɧɟɣɧɨɟ
Ⱦɜɢɠɟɧɢɟ ɩɪɹɦɨɥɢɧɟɣɧɨɟ ɪɚɜɧɨɦɟɪɧɨɟ
Ⱦɜɢɠɟɧɢɟ ɩɪɹɦɨɥɢɧɟɣɧɨɟ ɪɚɜɧɨɩɟɪɟɦɟɧɧɨɟ
Ⱦɥɢɧɚ ɩɭɬɢ
Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ
ɉɟɪɟɦɟɳɟɧɢɟ
Ɋɚɞɢɭɫ ɤɪɢɜɢɡɧɵ
Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ
ɋɢɫɬɟɦɚ ɨɬɫɱɟɬɚ
ɋɤɨɪɨɫɬɶ
ɋɤɨɪɨɫɬɶ ɥɢɧɟɣɧɚɹ
ɋɤɨɪɨɫɬɶ ɫɪɟɞɧɹɹ
ɋɤɨɪɨɫɬɶ ɭɝɥɨɜɚɹ
Ɍɪɚɟɤɬɨɪɢɹ
ɍɫɤɨɪɟɧɢɟ
ɍɫɤɨɪɟɧɢɟ ɥɢɧɟɣɧɨɟ
ɍɫɤɨɪɟɧɢɟ ɧɨɪɦɚɥɶɧɨɟ
ɍɫɤɨɪɟɧɢɟ ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ
ɍɫɤɨɪɟɧɢɟ ɭɝɥɨɜɨɟ
18
Ɏɨɪɦɭɥɵ ɢ ɟɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ ɛɚɡɨɜɵɯ
GGG
G
X
X
X
ɦ
M
M
M
R
Z
X
ɭ
Z
Z
Z
ɦ
ɦ
R
ɦ
ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɜ ɫɢɫɬɟɦɟ ɋɂ
ɇɚɡɜɚɧɢɟ Ɏɨɪɦɭɥɚ
1. Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ
(ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ
ɤɨɨɪɞɢɧɚɬ)
Ɇɨɞɭɥɶ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ
2. ɋɤɨɪɨɫɬɶ
Ɇɨɞɭɥɶ ɫɤɨɪɨɫɬɢ
3. Ⱦɥɢɧɚ ɩɭɬɢ
4. ɋɤɨɪɨɫɬɶ ɫɪɟɞɧɹɹ
5. ɍɫɤɨɪɟɧɢɟ
Ɇɨɞɭɥɶ ɭɫɤɨɪɟɧɢɹ
6. ɋɤɨɪɨɫɬɶ ɭɝɥɨɜɚɹ
7. ɋɤɨɪɨɫɬɶ ɥɢɧɟɣɧɚɹ
Ɇɨɞɭɥɶ ɥɢɧɟɣɧɨɣ
ɫɤɨɪɨɫɬɢ
8. ɍɫɤɨɪɟɧɢɟ ɭɝɥɨɜɨɟ
Ɇɨɞɭɥɶ ɭɝɥɨɜɨɝɨ
ɫɤɨɪɟɧɢɹ
9. ɍɫɤɨɪɟɧɢɟ
ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ
10. ɍɫɤɨɪɟɧɢɟ
ɧɨɪɦɚɥɶɧɨɟ
11. ɍɫɤɨɪɟɧɢɟ
ɥɢɧɟɣɧɨɟ
GGG
lim lim
a
W
tt
'o 'o
GG
an
aaa
kzjyixr
222
rxyz
G
'
lim
X
t
o' 0
'
G
X
G
a
G
Z
S
'
lim
t
o' 0
t
'
t
X
³
0
'
X
ɫɪ
'
'
lim
t
o' 0
t
'
a
'
lim
t
o' 0
t
'
d
Z
dt
GG
, r
XZ
>@
G
H
''
00
''
n
22
n
'
lim
t
o' 0
t
'
d
H
G
XXX
dt
§·
W
¨¸
ttdt
©¹
G
XX
'
n
lim
'o
t
0
'
tR
§·
¨¸
W
©¹
GG
rdtr
dt
dS
dt
dtS
S
t
GG
d
dt
d
dt
GG
d
dt
G
GG
d
dt
d
WW
2
2
2
vdv
2
§·
¨¸
dt
©¹
ȿɞɢɧɢɰɚ
ɢɡɦɟɪɟɧɢɹ
ɜ ɫɢɫɬɟɦɟ ɋɂ
ɦɟɬɪ (ɦ)
ɦɟɬɪ
ɫɟɤɭɧɞɚ
ɦ
·
§
¸
¨
ɫ
¹
©
ɦɟɬɪ (ɦ)
ɦɟɬɪ
ɫɟɤɭɧɞɚ
ɪɚɞɢɚɧ ɪɚɞ
ɫɟɤɭɧɞɚ ɫ
ɦ
§
¨
ɫ
©
2
ɫ
§·
¨¸
©¹
ɪɚɞ
ɫ
ɫ
ɫ
ɫ
ɦ
§
¨
ɫ
©
2
2
2
2
ɦɟɬɪ
ɫɟɤɭɧɞɚ
·
¸
¹
·
¸
¹
19
2.2. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
t
G
GGH
Ɂɚɞɚɱɚ 1. ɇɚɩɢɫɚɬɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɬɟɥɚ, ɟɫɥɢ ɨɧɨ ɛɪɨɲɟɧɨ
ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɫ ɜɵɫɨɬɵ ɇ = 15 ɦ ɫɨ ɫɤɨɪɨɫɬɶɸ
X
= 3 ɦ/ɫ. ɇɚɣɬɢ ɩɟɪɟɦɟɳɟɧɢɟ
0
ɬɟɥɚ ɡɚ ɩɟɪɜɭɸ ɫɟɤɭɧɞɭ ɞɜɢɠɟɧɢɹ.
Ⱦɚɧɨ:
ɇ = 15 ɦ
X
= 3 ɦ/ɫ
0
= 1 ɫ
1
G
() ?
rt
?
r
'
1
1. ȼɜɟɞɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ: ɜ ɤɚɱɟɫɬɜɟ ɬɟɥɚ ɨɬɫɱɟ-
ɬɚ ɩɪɢɦɟɦ Ɂɟɦɥɸ, ɨɫɢ ɯ ɢ y ɧɚɩɪɚɜɢɦ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ,
ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɢ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ (ɪɢɫ. 2.9) ɢ
ɧɚɱɧɟɦ ɨɬɫɱɟɬ ɜɪɟɦɟɧɢ ɜ ɦɨɦɟɧɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
Ɋɟɲɟɧɢɟ:
Ɋɢɫ. 2.9
2. ȼ ɨɛɳɟɦ ɜɢɞɟ ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɬɟɥɚ ɢɦɟɟɬ ɜɢɞ:
.kzjyixr
3. Ɍɟɥɨ ɞɜɢɠɟɬɫɹ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ ɚ = g. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɤɨɨɪɞɢɧɚ-
ɬɵ ɬɟɥɚ ɢɡɦɟɧɹɸɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɩɨ ɡɚɤɨɧɭ:
2
at
x
2
2
at
y
2
;
X
= 0; ɚɯ = 0; ɚy = –g ɢ ɜɵɪɚɠɟɧɢɹ
0y
ȼ ɧɚɲɟɣ ɡɚɞɚɱɟ ɯ
xt x t
() ;
°
°
®
°
yt y t
() .
°
¯
= 0; y0 = ɇ;
0
X
00
x
X
00
y
X
=
X
0ɯ
0
ɩɪɢɧɢɦɚɸɬ ɜɢɞ:
() 0 0; () ;
xt t xt t
°°
®®
() 0 () .
yt H yt H
°°
¯¯
XX
00
22
ɢɥɢ
gt gt
22
20
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