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Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
ɞɟɥɟɧɧɵɣɦɨɦɟɧɬ m
ɜɫɥɟɞɭɸɳɟɦɜɢɞɟ
ɬɨ ɜɬɨɪɨɟ ɭɫɥɨɜɢɟ ɢɡ (1.23) ɧɭɠɧɨ ɡɚɩɢɫɚɬɶ
x
2
w
w
x
m
w
2
x
.
D
ɋɜɨɛɨɞɧɵɣ ɤɪɚɣ. ȼ ɨɬɫɭɬɫɬɜɢɟ ɧɚ ɷɬɨɦɭɱɚɫɬɤɟɤɨɧɬɭɪɚ
ɜɧɟɲɧɢɯ ɫɢɥɨɜɵɯ ɮɚɤɬɨɪɨɜ ɩɨɝɨɧɧɵɯ ɧɚɝɪɭɡɨɤ ɜɫɟ ɩɨɝɨɧɧɵɟ
ɜɧɭɬɪɟɧɧɢɟɭɫɢɥɢɹɬɚɤɠɟɞɨɥɠɧɵɛɵɬɶɪɚɜɧɵɧɭɥɸɌɚɤɞɥɹɤɪɚɹ
y=b
(1.24)
HQM
y
y
by
.0;0;0
byby
Ɍɚɤɢɦɨɛɪɚɡɨɦ, ɜɦɟɫɬɨɞɜɭɯɧɟɨɛɯɨɞɢɦɵɯɭɫɥɨɜɢɣ ɡɞɟɫɶɩɨ
ɹɜɥɹɸɬɫɹ ɬɪɢ ɭɫɥɨɜɢɹ Ɉɞɧɚɤɨ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɩɪɢɛɥɢɠɟɧɧɨɣ
ɬɟɨɪɢɢɄɢɪɯɝɨɮɚ ɜɨɛɳɟɦɫɥɭɱɚɟɧɟɥɶɡɹɨɞɧɨɜɪɟɦɟɧɧɨɭɞɨɜɥɟɬɜɨ
ɪɢɬɶɞɜɭɦɩɨɫɥɟɞɧɢɦɭɫɥɨɜɢɹɦ >1@ɗɬɨ ɩɪɨɬɢɜɨɪɟɱɢɟ ɦɨɠ
ɧɨɭɫɬɪɚɧɢɬɶ ɡɚɦɟɧɢɜ ɩɨɩɟɪɟɱɧɭɸɫɢɥɭ Q
ɢɤɪɭɬɹɳɢɣɦɨɦɟɧɬ H
y
ɨɞɧɨɣ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɛɨɛɳɟɧɧɨɣ ɫɢɥɨɣ ɫɬɚɬɢɱɟɫɤɢ ɢɦ ɷɤɜɢɜɚɥɟɧɬ
ɧɨɣɢɨɬɜɟɱɚɸɳɟɣɩɪɨɝɢɛɭwɤɚɤɨɛɨɛɳɟɧɧɨɦɭɩɟɪɟɦɟɳɟɧɢɸ
(1.25)
,
QQV '
yyy
ɝɞɟ ¨Q
– ɞɨɩɨɥɧɢɬɟɥɶɧɚɹ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɫɬɚɬɢɱɟɫɤɢ ɷɤɜɢɜɚ
y
ɥɟɧɬɧɚɹɤɪɭɬɹɳɟɦɭɦɨɦɟɧɬɭ H. Ɍɨɝɞɚɤɚɤɢɜɩɪɟɞɵɞɭɳɢɯɫɥɭɱɚ
ɹɯ ɨɩɢɪɚɧɢɹ ɞɥɹ ɫɜɨɛɨɞɧɨɝɨ ɤɪɚɹ ɜɨɡɦɨɠɧɨ ɭɞɨɜɥɟɬɜɨɪɢɬɶ ɧɟ
ɬɪɟɦɚ ɬɨɥɶɤɨ ɞɜɭɦ ɫɢɥɨɜɵɦɭɫɥɨɜɢɹɦɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɬɨɥɶɤɨ
ɞɜɭɦɧɟɡɚɜɢɫɢɦɵɦɩɟɪɟɦɟɳɟɧɢɹɦɧɚɤɪɨɦɤɟ ɩɥɚɫɬɢɧɵ
Ɋɚɫɫɦɨɬɪɢɦ ɞɟɣɫɬɜɢɟ ɤɪɭɬɹɳɢɯ ɦɨɦɟɧɬɨɜ, ɪɚɫɩɪɟɞɟɥɟɧɧɵɯ
ɜɞɨɥɶɝɪɚɧɢ ȼɋ, ɧɚ ɫɜɨɛɨɞɧɨɦɤɪɚɸ y=b ɪɢɫ 1.6, ɚ). ɇɚɷɥɟɦɟɧ
ɬɚɪɧɨɦɭɱɚɫɬɤɟ dx ɞɟɣɫɬɜɭɟɬ ɤɪɭɬɹɳɢɣɦɨɦɟɧɬɪɚɜɧɵɣ H Âdx, ɤɨ
ɬɨɪɵɣɦɨɠɧɨɩɪɟɞɫɬɚɜɢɬɶɜɜɢɞɟɩɚɪɵɫɢɥ
xdHH
ɫɩɥɟɱɨɦ dx.
dx
ɇɚ ɫɨɫɟɞɧɟɦ ɭɱɚɫɬɤɟ dx ɷɬɚ ɩɚɪɚ ɫɢɥ ɩɨɥɭɱɢɬ ɩɪɢɪɚɳɟɧɢɟ dH
ɢɛɭɞɟɬɪɚɜɧɚ
w
H
ɫɬɚɤɢɦɠɟɩɥɟɱɨɦ dx ɪɢɫ 1.6, ɛ).
dx
x
w
H
ɉɪɢɬɚɤɨɦɩɪɟɞɫɬɚɜɥɟɧɢɢɞɟɣɫɬɜɢɟɤɪɭɬɹɳɢɯɦɨɦɟɧɬɨɜɦɨɠ
ɧɨɡɚɦɟɧɢɬɶɜɟɪɬɢɤɚɥɶɧɨɣ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɨɣɢɧɬɟɧɫɢɜɧɨ
w
ɫɬɶɸ
ɢ ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɦɢ ɫɢɥɚɦɢ H ɜɨɡɧɢɤɚɸɳɢɦɢ ɜɭɝɥɚɯ
xHw
B ɢ C ɪɢɫ 1.6, ɜ). ɋɭɦɦɢɪɭɹɷɬɭ ɧɚɝɪɭɡɤɭ ɫɨɝɥɚɫɧɨ ɫ ɩɨɩɟ
21
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
ɪɟɱɧɨɣɫɢɥɨɣ Q
, ɩɨɥɭɱɢɦɫɥɟɞɭɸɳɢɟɞɜɚɭɫɥɨɜɢɹɞɥɹɫɜɨɛɨɞɧɨɝɨ
y
ɤɪɚɹ y=b:
H
w
QM
yy
a)
B
ɛ)
ɜ)
H
B
Ɋɢɫ 1.6. ɉɪɟɞɫɬɚɜɥɟɧɢɟɤɪɭɬɹɳɢɯɦɨɦɟɧɬɨɜɧɚɝɪɚɧɢ ɩɥɚɫɬɢɧɵ
HÂdx
dx
dx dx
H
w
xHw
x
w
H
H
H
w
H
x
w
(1.26)
.0;0
byby
C
w
H
dx
w
x
dx
CB
H
C
x
x
x
ȼɵɪɚɡɢɜ ɭɫɢɥɢɹ ɱɟɪɟɡ ɮɭɧɤɰɢɸɩɪɨɝɢɛɨɜɡɚɩɢɲɟɦɝɪɚɧɢɱ
ɧɵɟɭɫɥɨɜɢɹ (1.26) ɜɬɚɤɨɦ ɜɢɞɟ
2
w
w
y
w
2
w
w
Q
2
2
w
2
§
w
w
w
¨
2
¨
y
yx
w
w
©
2
·
w
w
¸
Q
2
¸
x
w
¹
(1.27)
.0)2(;0
byby
ȼɫɥɭɱɚɟ ɠɟɟɫɥɢɫɜɨɛɨɞɧɵɣɤɪɚɣɛɭɞɟɬɡɚɞɚɧɧɚɝɪɚɧɹɯɩɥɚ
ɫɬɢɧɵ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɨɫɢ x ɩɨ ɚɧɚɥɨɝɢɢ ɫ ɝɪɚɧɢɱɧɵɟ
ɭɫɥɨɜɢɹɞɥɹ x=0, x=a ɡɚɩɢɲɭɬɫɹɬɚɤ:
2
w
w
x
w
2
w
w
Q
2
2
w
2
§
w
w
w
¨
2
¨
x
xy
w
w
©
2
·
w
w
¸
Q
2
¸
y
w
¹
(1.28)
.0)2(;0
,, 00
axax
Ʉɨɧɟɱɧɨ ɩɪɢ ɷɬɨɦ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɫɜɨɛɨɞɧɨɣ ɝɪɚɧɢ
ɛɭɞɭɬ ɭɞɨɜɥɟɬɜɨɪɹɬɶɫɹ ɩɪɢɛɥɢɠɟɧɧɨ ɇɨ ɧɚ ɨɫɧɨɜɚɧɢɢ ɩɪɢɧɰɢɩɚ
ɋɟɧ-ȼɟɧɚɧɚɡɚɦɟɧɚ ɩɨɩɟɪɟɱɧɨɣ ɫɢɥɵɢ ɤɪɭɬɹɳɟɝɨ ɦɨɦɟɧɬɚɫɬɚɬɢ
ɱɟɫɤɢ ɢɦ ɷɤɜɢɜɚɥɟɧɬɧɨɣ ɨɛɨɛɳɟɧɧɨɣ ɩɨɩɟɪɟɱɧɨɣ ɫɢɥɨɣ ɜɵɡɨɜɟɬ
ɥɢɲɶ ɦɟɫɬɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɜɛɥɢɡɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɫɜɨɛɨɞɧɨɝɨ
ɤɪɚɹɩɥɚɫɬɢɧɵ[1, 12].
22
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
1.3. Ɋɟɲɟɧɢɟɡɚɞɚɱɢɡɝɢɛɚɩɪɹɦɨɭɝɨɥɶɧɵɯ ɩɥɚɫɬɢɧ
ɗɥɟɦɟɧɬɚɪɧɵɟɫɥɭɱɚɢɢɡɝɢɛɚɩɥɚɫɬɢɧ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɪɚɫɱɟɬ ɬɨɧɤɢɯ ɭɩɪɭɝɢɯ ɩɥɚɫɬɢɧ ɫɜɨɞɢɬɫɹ
ɤɪɟɲɟɧɢɸɤɪɚɟɜɨɣɡɚɞɚɱɢɞɥɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ ɢɡɨ
ɝɧɭɬɨɣɩɨɜɟɪɯɧɨɫɬɢɩɥɚɫɬɢɧɵ
4
w
w
4
x
w
4
w
w
ww
4
q
w
w
4
22
y
yx
w
(1.29)
,2
D
ɝɞɟ w=w(x, y)–ɮɭɧɤɰɢɹɩɪɨɝɢɛɨɜ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢɩɥɚɫɬɢ
ɧɵ q=q(x, y)–ɪɚɫɩɪɟɞɟɥɟɧɧɚɹɩɨɜɟɪɯɧɨɫɬɧɚɹ ɧɚɝɪɭɡɤɚɩɟɪɩɟɧ
ɞɢɤɭɥɹɪɧɚɹɫɪɟɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ.
Ʉɪɚɟɜɨɣɡɚɞɚɱɟɣ ɧɚɡɵɜɚɟɬɫɹɡɚɞɚɱɚɧɚɯɨɠɞɟɧɢɹ ɬɚɤɨɝɨ ɪɟɲɟ
ɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɬɨɪɨɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɨɩɪɟ
ɞɟɥɟɧɧɵɦɭɫɥɨɜɢɹɦɧɚɝɪɚɧɢɰɚɯɤɪɚɹɯɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣɨɛɥɚɫɬɢ
Ɍɨɱɧɨɟɟɟɪɟɲɟɧɢɟ ɜɚɧɚɥɢɬɢɱɟɫɤɨɣɮɨɪɦɟɜɨɡɦɨɠɧɨ ɥɢɲɶ ɜɧɟɤɨ
ɬɨɪɵɯɱɚɫɬɧɵɯ ɫɥɭɱɚɹɯ ɝɟɨɦɟɬɪɢɢ ɩɥɚɫɬɢɧɵɟɟ ɧɚɝɪɭɠɟɧɢɹ ɢɭɫɥɨ
ɜɢɣɡɚɤɪɟɩɥɟɧɢɹȼɦɟɫɬɟɫɬɟɦɬɨɱɧɵɟɚɧɚɥɢɬɢɱɟɫɤɢɟɪɟɲɟɧɢɹɡɚɞɚɱ
ɹɜɥɹɸɬɫɹɫɜɨɟɨɛɪɚɡɧɵɦɷɬɚɥɨɧɨɦɫ ɤɨɬɨɪɵɦɦɨɠɧɨ ɫɪɚɜɧɢɜɚɬɶ ɪɟ
ɲɟɧɢɹɩɨɥɭɱɟɧɧɵɟɪɚɡɥɢɱɧɵɦɢɩɪɢɛɥɢɠɟɧɧɵɦɢɦɟɬɨɞɚɦɢ
Ɋɚɫɫɦɨɬɪɢɦɧɟɤɨɬɨɪɵɟ ɩɪɨɫɬɵɟ ɫɥɭɱɚɢɢɡɝɢɛɚ ɩɥɚɫɬɢɧ ɢɦɟ
ɸɳɢɟ ɜɚɠɧɨɟ ɡɧɚɱɟɧɢɟ ɞɥɹ ɩɨɧɢɦɚɧɢɹ ɨɫɨɛɟɧɧɨɫɬɟɣ ɢɯ ɪɚɛɨɬɵ
ɉɪɢɷɬɨɦɪɟɲɟɧɢɟɞɚɧɧɵɯɡɚɞɚɱɧɟɜɵɡɵɜɚɟɬɨɫɨɛɵɯɡɚɬɪɭɞɧɟɧɢɣ.
1. ɐɢɥɢɧɞɪɢɱɟɫɤɢɣɢɡɝɢɛɩɥɚɫɬɢɧɵ
ɉɪɟɞɫɬɚɜɢɦ ɫɟɛɟ ɛɟɫɤɨɧɟɱɧɨ ɞɥɢɧɧɭɸ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ y
ɩɥɚɫɬɢɧɭ ɪɢɫ 1.7)ɧɚɝɪɭɡɤɚ q ɧɚ ɤɨɬɨɪɭɸ ɩɨɫɬɨɹɧɧɚɜɞɨɥɶ ɷɬɨɣ
ɠɟɨɫɢɢɢɡɦɟɧɹɟɬɫɹɬɨɥɶɤɨɩɨ ɨɫɢ xɬ ɟ q=q(x). Ɉɱɟɜɢɞɧɨɱɬɨ
ɜɫɟɩɨɥɨɫɤɢɟɞɢɧɢɱɧɨɣɲɢɪɢɧɵɜɵɞɟɥɟɧɧɵɟɢɡɷɬɨɣɩɥɚɫɬɢɧɵɜɞɨɥɶ
ɨɫɢ xɛɭɞɭɬ ɢɡɝɢɛɚɬɶɫɹɨɞɢɧɚɤɨɜɨɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɰɟɥɨɦɩɥɚɫɬɢɧɚ
ɨɤɚɠɟɬɫɹɢɡɨɝɧɭɬɨɣɩɨɰɢɥɢɧɞɪɢɱɟɫɤɨɣɩɨɜɟɪɯɧɨɫɬɢ w=w(x).
ɉɨɫɤɨɥɶɤɭɮɭɧɤɰɢɹ ɩɪɨɝɢɛɨɜ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɧɟ ɡɚɜɢɫɢɬ ɨɬ y,
ɩɨɥɚɝɚɟɦ ɜ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ y ɪɚɜɧɵɦɢ ɧɭɥɸ ȼ ɪɟɡɭɥɶɬɚɬɟ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚɩɪɢɦɟɬɫɥɟɞɭɸɳɢɣɜɢɞ
4
xwd
4
xd
)()(
xq
(1.30)
.
D
23
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
y
y
y
1
Ɋɢɫ ɐɢɥɢɧɞɪɢɱɟɫɤɢɣɢɡɝɢɛɩɥɚɫɬɢɧɵ
0
a
q
x
0
Ɂɞɟɫɶ ɢɫɩɨɥɶɡɭɟɬɫɹɨɛɵɤɧɨɜɟɧɧɚɹɚɧɟɱɚɫɬɧɚɹɩɪɨɢɡɜɨɞɧɚɹ
ɩɨɫɤɨɥɶɤɭ w ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɨɞɧɨɝɨ ɚɪɝɭɦɟɧɬɚ x ɍɪɚɜɧɟɧɢɟ
(1.30) ɨɩɢɫɵɜɚɸɳɟɟ ɰɢɥɢɧɞɪɢɱɟɫɤɢɣ ɢɡɝɢɛ ɩɥɚɫɬɢɧɵ ɫɨɜɩɚɞɚɟɬ
ɫɭɪɚɜɧɟɧɢɟɦɢɡɝɢɛɚɛɚɥɤɢɭɤɨɬɨɪɨɣɠɟɫɬɤɨɫɬɶɫɟɱɟɧɢɹɧɚɢɡɝɢɛ
EI ɪɚɜɧɚɰɢɥɢɧɞɪɢɱɟɫɤɨɣɠɟɫɬɤɨɫɬɢɩɥɚɫɬɢɧɵ D (EI = D). Ɉɬɫɸɞɚ
ɜɟɥɢɱɢɧɚ D ɩɨɥɭɱɢɥɚɧɚɢɦɟɧɨɜɚɧɢɟɰɢɥɢɧɞɪɢɱɟɫɤɨɣɠɟɫɬɤɨɫɬɢ.
ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɰɢɥɢɧɞɪɢɱɟ
ɫɤɨɝɨɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵɩɪɢɥɸɛɨɦɡɚɤɨɧɟɩɨɩɟɪɟɱɧɨɣɧɚɝɪɭɡɤɢq (x)
ɧɟɜɵɡɵɜɚɟɬɨɫɨɛɵɯɫɥɨɠɧɨɫɬɟɣɩɨɷɬɨɦɭɡɚɞɚɱɚɪɟɲɚɟɬɫɹɬɨɱɧɨ
ɉɭɫɬɶɧɚɩɪɢɦɟɪq=q
x/a (ɫɦɪɢɫ 1.7), ɬɨɝɞɚ ɩɨɫɥɟɱɟɬɵɪɟɯ
0
ɤɪɚɬɧɨɝɨɢɧɬɟɝɪɢɪɨɜɚɧɢɹɭɪɚɜɧɟɧɢɹɩɨɥɭɱɢɦ
5
120
xq
0
(1.31)
,
Da
2
321
, ɋ2, ɋ3, ɋ4– ɱɟɬɵɪɟɩɨɫɬɨɹɧɧɵɟɢɧɬɟɝɪɢɪɨɜɚɧɢɹɨɩɪɟɞɟɥɹɟ
ɝɞɟɋ
1
3
xCxCxCCw
4
ɦɵɟ ɢɡ ɱɟɬɵɪɟɯ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɩɨ ɞɜɚ ɭɫɥɨɜɢɹ ɞɥɹ ɡɚɞɟɥɚɧ
ɧɨɝɨɤɪɚɹ x=0 ɢɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɝɨɤɪɚɹ x=a):
2
w
w
w
w
x
w
00
w
w
2
w
x
(1.32)
.0;0;0;0
a
xaxxx
ɉɨɞɫɬɚɜɢɜɩɪɨɝɢɛɵ w (1.31) ɢ ɢɯɩɪɨɢɡɜɨɞɧɵɟ ɜ ɝɪɚɧɢɱɧɵɟɭɫɥɨ
ɜɢɹɢɪɟɲɢɜɫɢɫɬɟɦɭɢɡɱɟɬɵɪɟɯɚɥɝɟɛɪɚɢɱɟɫɤɢɯɭɪɚɜɧɟɧɢɣ
ɧɚɯɨɞɢɦɋ
= ɋ2= 0,
1
2
aq
7
0
C
240
,
4
D
C
3
aq
9
0
. ȼɪɟɡɭɥɶɬɚɬɟɩɨɥɭ
D
240
ɱɢɦɫɥɟɞɭɸɳɟɟɜɵɪɚɠɟɧɢɟɞɥɹɩɪɨɝɢɛɨɜ:
4
2
§
aq
0
240
¨
¨
D
©
xw
)(
x
x
2
a
a
24
5
3
3
·
x
(1.33)
¸
.297
5
¸
a
¹
ɉɪɢ ɰɢɥɢɧɞɪɢɱɟɫɤɨɦ ɢɡɝɢɛɟ ɤɨɝɞɚ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ y ɪɚɜɧɵ
y
y
ɧɭɥɸɜɵɪɚɠɟɧɢɹ ɞɥɹɦɨɦɟɧɬɨɜɩɪɢɦɭɬɜɢɞ
2
w
w
HɆ
DɆ
2
x
w
2
w
w
DɆ
Q
2
w
x
xyx
(1.34)
Q
.0;;
ɋɭɱɟɬɨɦɢɦɟɟɦɫɥɟɞɭɸɳɭɸɡɚɜɢɫɢɦɨɫɬɶ ɞɥɹɦɨɦɟɧɬɚ:
2
§
aq
0
120
¨
¨
©
xM
)(
x
x
a
3
·
x
¸
.20277
3
¸
a
¹
2. ɑɢɫɬɵɣɢɡɝɢɛɩɥɚɫɬɢɧɵ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɹɦɨɭɝɨɥɶɧɭɸ ɩɥɚɫɬɢɧɭ ɫɜɨɛɨɞɧɭɸɨɬ ɡɚɤɪɟɩ
ɥɟɧɢɣ ɢ ɧɚɝɪɭɠɟɧɧɭɸ ɩɨ ɤɨɧɬɭɪɭ ɪɚɫɩɪɟɞɟɥɟɧɧɵɦɢ ɦɨɦɟɧɬɚɦɢ
ɩɨɫɬɨɹɧɧɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɢm
= const ɢ m
1
= const ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
2
ɪɢɫ 1.8, ɚ). ɇɚɱɚɥɨɤɨɨɪɞɢɧɚɬɩɨɦɟɫɬɢɦɜɰɟɧɬɪɟɩɥɚɫɬɢɧɵ
ɚ)
z
m
2
Ɋɢɫ 1.8. ɑɢɫɬɵɣ ɢɡɝɢɛɩɥɚɫɬɢɧɵ
x
m
1
ɛ)
z
ɉɨɫɤɨɥɶɤɭɩɨɩɟɪɟɱɧɚɹɧɚɝɪɭɡɤɚɨɬɫɭɬɫɬɜɭɟɬɢɦɟɟɦɨɞɧɨɪɨɞ
ɧɨɟɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚ
4
w
w
4
w
x
4
w
w
ww
yx
4
w
w
22
4
w
y
(1.35)
,02
ɪɟɲɟɧɢɟɤɨɬɨɪɨɝɨɛɭɞɟɦɢɫɤɚɬɶɜɜɢɞɟ
2
(1.36)
.)(5,0
yCxCw
2
ɢ ɋ2ɧɚɣɞɟɦ ɢɡ ɝɪɚɧɢɱɧɵɯ
1
ɉɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɋ
ɭɫɥɨɜɢɣM
= m1ɢMy= m
x
2
1
ɉɨɞɫɬɚɜɢɜɜɩɨɥɭɱɢɦ
2
25
x
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
2
§
w
w
¨
DɆ
x
y
¨
x
w
©
2
§
w
¨
DɆ
¨
y
w
©
2
·
w
w
¸
Q
2
w
2
2
y
w
2
w
w
Q
2
x
w
¸
¹
·
¸
¸
¹
mCCD
;
Q
121
DH
mCCD
Q
;
212
2
w
w
Q
.0)1(
yx
ww
ɂɡɪɟɲɟɧɢɹɩɟɪɜɵɯɞɜɭɯɭɪɚɜɧɟɧɢɣɧɚɯɨɞɢɦ
Q
mm
C
1
D
12
C
;
2
2
)1(
Q
Q
mm
D
21
.
2
)1(
Q
Ɍɨɝɞɚ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɪɨɝɢɛɨɜ ɬ ɟ ɭɪɚɜɧɟɧɢɟ ɢɡɨɝɧɭɬɨɣ
ɩɨɜɟɪɯɧɨɫɬɢɩɥɚɫɬɢɧɵɪɢɫ 1.8, ɛ), ɡɚɩɢɲɟɬɫɹɜɜɢɞɟ
),(
D
2
Q
1
ymmxmm
yxw
ª
«
¬
)1(2
2
12
QQ
21
2
º
(1.37)
.)()(
»
¼
ȼɨɜɫɟɯɫɟɱɟɧɢɹɯɩɥɚɫɬɢɧɵɩɚɪɚɥɥɟɥɶɧɵɯ ɨɫɹɦx ɢ y, ɞɟɣ
ɫɬɜɭɸɬɬɨɥɶɤɨɩɨɫɬɨɹɧɧɵɟɢɡɝɢɛɚɸɳɢɟɦɨɦɟɧɬɵM
Ⱦɪɭɝɢɟ ɜɧɭɬɪɟɧɧɢɟɭɫɢɥɢɹɨɬɫɭɬɫɬɜɭɸɬ: H = Q
Ɋɚɫɫɦɨɬɪɢɦɧɟɫɤɨɥɶɤɨɱɚɫɬɧɵɯɫɥɭɱɚɟɜɉɭɫɬɶm
m
),(
yxw
D
)1(2
Q
yx
x
= Qy= 0.
x
22
.
1
= m1ɢMy= m
= m2= mɬɨɝɞɚ
2
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɩɚɪɚɛɨɥɨɢɞɚ ɜɪɚɳɟɧɢɹ ɂɡɨɝɧɭɬɚɹ ɩɨɜɟɪɯɧɨɫɬɶ
ɜɷɬɨɦ ɫɥɭɱɚɟɩɪɟɞɫɬɚɜɥɹɟɬ ɱɚɫɬɶɫɮɟɪɵ ɬɚɤɤɚɤ ɪɚɞɢɭɫɵɤɪɢɜɢɡ
ɧɵɨɞɢɧɚɤɨɜɵɜɨɜɫɟɬɨɱɤɚɯɩɥɚɫɬɢɧɵ
ȼɨɡɶɦɟɦɞɪɭɝɨɣ ɱɚɫɬɧɵɣɫɥɭɱɚɣ m
= m; m
1
ɍɪɚɜɧɟɧɢɟ
2
ɩɪɢɦɟɬɫɥɟɞɭɸɳɢɣɜɢɞ
),(
yxw Q
m
D
2
)1(2
Q
22
.
yx
.
ɉɨɜɟɪɯɧɨɫɬɶ ɨɩɢɫɵɜɚɟɦɚɹ ɷɬɢɦ ɭɪɚɜɧɟɧɢɟɦ ɢɦɟɟɬ ɫɟɞɥɨɨɛɪɚɡ
ɧɭɸ ɮɨɪɦɭ ɢ ɧɚɡɵɜɚɟɬɫɹ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɦ ɩɚɪɚɛɨɥɨɢɞɨɦ ɝɨɪɢ
ɡɨɧɬɚɥɹɦɢɟɟɹɜɥɹɸɬɫɹɝɢɩɟɪɛɨɥɵ Ȼɥɚɝɨɞɚɪɹɜɥɢɹɧɢɸɤɨɷɮɮɢɰɢ
ɟɧɬɚ ɉɭɚɫɫɨɧɚ ɩɥɚɫɬɢɧɚ ɢɡɝɢɛɚɟɬɫɹ ɧɟ ɬɨɥɶɤɨ ɜ ɩɥɨɫɤɨɫɬɢ
ɞɟɣɫɬɜɢɹ ɦɨɦɟɧɬɚ M
= m ɧɨ ɩɨɥɭɱɚɟɬ ɢɨɛɪɚɬɧɵɣ ɜɵɝɢɛɜ ɩɟɪ
x
ɩɟɧɞɢɤɭɥɹɪɧɨɣɩɥɨɫɤɨɫɬɢ
ɇɚɤɨɧɟɰɩɪɢɦɟɦ m
= m; m2=–m ȼɷɬɨɦ ɫɥɭɱɚɟɭɪɚɜɧɟɧɢɟ
1
ɩɪɨɝɢɛɨɜɬɚɤɠɟɢɦɟɟɬɜɢɞ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɝɨ ɩɚɪɚɛɨɥɨɢɞɚ:
26
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
m
yxw
),(
D
Q
)1(2
22
.
yx
ɇɟɬɪɭɞɧɨ ɩɨɤɚɡɚɬɶ ɱɬɨ ɩɪɢ ɬɚɤɨɦ ɧɚɝɪɭɠɟɧɢɢ ɜ ɤɨɫɵɯ ɫɟɱɟɧɢɹɯ
ɩɥɚɫɬɢɧɵɧɚɤɥɨɧɟɧɧɵɯɤɨɫɹɦx ɢy ɧɚɭɝɨɥĮ ɩɥɚɫɬɢɧɚ ɢɫ
ɩɵɬɵɜɚɟɬɞɟɮɨɪɦɚɰɢɸɱɢɫɬɨɝɨɤɪɭɱɟɧɢɹɬ ɟ H
= –m, M
Į
Į
= 0.
Ɋɟɲɟɧɢɟɩɪɹɦɨɣɢɨɛɪɚɬɧɨɣɡɚɞɚɱɢɡɝɢɛɚɩɥɚɫɬɢɧ
ɉɪɹɦɚɹ ɡɚɞɚɱɚ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸ
ɳɢɦ ɨɛɪɚɡɨɦ: ɡɚɞɚɧɵ ɧɚɝɪɭɡɤɚ ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɬɪɟɛɭɟɬɫɹ
ɨɩɪɟɞɟɥɢɬɶ ɩɪɨɝɢɛɵ ɩɥɚɫɬɢɧɵ ɢ ɞɟɣɫɬɜɭɸɳɢɟ ɜ ɧɟɣ ɜɧɭɬɪɟɧɧɢɟ
ɭɫɢɥɢɹɊɟɲɟɧɢɟɩɪɹɦɨɣɡɚɞɚɱɢɡɚɬɪɭɞɧɟɧɨɧɟɨɛɯɨɞɢɦɨɫɬɶɸɢɧɬɟ
ɝɪɢɪɨɜɚɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢɡɝɢɛɚ ɢ ɝɥɚɜɧɨɟ
ɭɞɨɜɥɟɬɜɨɪɹɬɶɪɚɡɧɨɨɛɪɚɡɧɵɦɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦ
Ɋɚɫɫɦɨɬɪɢɦ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɭɸ ɩɨ ɤɨɧɬɭɪɭ ɩɪɹɦɨɭɝɨɥɶɧɭɸ
ɩɥɚɫɬɢɧɭ ɧɚɝɪɭɠɟɧɧɭɸ ɩɨɩɟɪɟɱɧɨɣ ɧɚɝɪɭɡɤɨɣ ɂɧɬɟɧɫɢɜɧɨɫɬɶ
ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚɝɪɭɡɤɢ ɡɚɞɚɞɢɦ ɜ ɜɢɞɟ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ
y
ɮɭɧɤɰɢɢ
sinsin
xqqSS
0
a
ɪɢɫ 1.9). Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ – ɮɭɧɤɰɢɸ
b
ɩɪɨɝɢɛɨɜɩɥɚɫɬɢɧɵwɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸɡɚɞɚɧɧɨɣɧɚɝɪɭɡɤɟɢɝɪɚ
ɧɢɱɧɵɦɭɫɥɨɜɢɹɦɛɭɞɟɦɢɫɤɚɬɶɜɚɧɚɥɨɝɢɱɧɨɦ ɜɢɞɟ
q
0
x
b/2
b/2
a/2 a/2
y
Ɋɢɫ 1.9ɉɪɹɦɨɭɝɨɥɶɧɚɹɲɚɪɧɢɪɧɨɨɩɟɪɬɚɹɩɥɚɫɬɢɧɚ
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ ɡɚɩɢɲɟɬɫɹ
ɜɬɚɤɨɦɜɢɞɟ
4
w SS
w
4
x
w
4
w
w
yx
ww
4
w
22
y
w
27
0
4
a
D
q
w
y
x
(1.38)
.sinsin2
b
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɪɚɧɢɱɧɵɟɭɫɥɨɜɢɹɞɥɹɲɚɪɧɢɪɧɨɨɩɟɪɬɨɣɩɨɤɨɧɬɭɪɭɩɥɚɫɬɢɧɵ
2
w
w
w
2
x
w
w
2
w
w
2
y
w
(1.39)
.0;0;0;0
,,,,
0000
bybyaxax
Ɋɟɲɟɧɢɟɭɪɚɜɧɟɧɢɹɡɚɞɚɞɢɦɜɮɨɪɦɟɟɝɨɩɪɚɜɨɣɱɚɫɬɢ:
y
xCwSS
a
(1.40)
.sinsin
b
Ɏɭɧɤɰɢɹ ɩɪɨɝɢɛɨɜ ɡɚɩɢɫɚɧɧɚɹ ɜ ɜɢɞɟ (1.40) ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɯɚɪɚɤɬɟɪɭ ɞɟɮɨɪɦɚɰɢɢ ɩɥɚɫɬɢɧɵ ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ 1.39).
Ɉɩɪɟɞɟɥɢɦ ɩɨɫɬɨɹɧɧɭɸ C ɪɚɜɧɭɸ ɚɦɩɥɢɬɭɞɟ ɩɪɨɝɢɛɚ ɜ ɰɟɧɬɪɟ
ɩɥɚɫɬɢɧɵ. ɉɨɫɥɟɩɨɞɫɬɚɧɨɜɤɢɮɭɧɤɰɢɢɩɪɨɝɢɛɚ1.40) ɜɭɪɚɜɧɟɧɢɟ
(1.38ɢɫɨɤɪɚɳɟɧɢɹɧɚɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟɦɧɨɠɢɬɟɥɢɩɨɥɭɱɚɟɦ
2
2
2
§
S
¨
ɋ
¨
©
·
S
¸
2
2
¸
ba
¹
q
D
0
,
ɨɬɫɸɞɚ
wɋ
0
44
baq
0
.
2224
)(
S
baD
ȼ ɪɟɡɭɥɶɬɚɬɟ ɭɪɚɜɧɟɧɢɟ ɢɡɨɝɧɭɬɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ
ɩɪɢɦɟɬɜɢɞ
44
baq
w
0
2224
S
baD
)(
SS
y
x
a
.sinsin
b
ȿɫɥɢɢɧɬɟɧɫɢɜɧɨɫɬɶɪɚɫɩɪɟɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɢɡɚɞɚɟɬɫɹɜɜɢɞɟ
2
2
2
qqSS sinsin
0
xm
a
w
yn
, ɬɨ
b
0
q
0
S
§
m
4
Dqɋ
¨
a
©
2
22224
bnamD
·
n
S
2
ɢɮɭɧɤɰɢɹɩɪɨɝɢɛɚ
¸
2
b
¹
SS
xm
a
yn
.sinsin
b
(1.41)
ɂɦɟɹɮɭɧɤɰɢɸɩɪɨɝɢɛɚ, ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɜɫɟ ɤɨɦɩɨɧɟɧɬɵ
ɧɚɩɪɹɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨɫɨɫɬɨɹɧɢɹɜɥɸɛɨɣɬɨɱɤɟɩɥɚɫɬɢ
ɧɵ
ɉɨɫɬɚɧɨɜɤɢ ɨɛɪɚɬɧɨɣɡɚɞɚɱɢɢɡɝɢɛɚɩɥɚɫɬɢɧɵ ɛɨɥɟɟɪɚɡɧɨ
ɨɛɪɚɡɧɵ Ɂɞɟɫɶ ɦɨɝɭɬ ɛɵɬɶ ɡɚɞɚɧɵ ɥɢɛɨ ɭɪɚɜɧɟɧɢɟ ɭɩɪɭɝɨɣ ɩɨ
ɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ ɥɢɛɨɪɚɫɩɪɟɞɟɥɟɧɢɟɧɚɩɪɹɠɟɧɢɣ ɢɥɢ ɞɟɮɨɪ
ɦɚɰɢɣ Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɜɫɟ ɨɫɬɚɥɶɧɵɟ ɧɟɢɡɜɟɫɬɧɵɟ ɤɨɦɩɨɧɟɧɬɵ
ɨɩɪɟɞɟɥɹɸɳɢɟɧɚɩɪɹɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɟɫɨɫɬɨɹɧɢɟɩɥɚɫɬɢɧɵ
28
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɹɦɨɭɝɨɥɶ
ɧɭɸ ɩɥɚɫɬɢɧɭ ɪɢɫ Ɂɚɞɚ
0
x
b
ɞɢɦɫɹ ɭɪɚɜɧɟɧɢɟɦ ɢɡɨɝɧɭɬɨɣ
ɫɪɟɞɢɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢ
ɧɵɜɜɢɞɟ
CwSS
x
a
y
3
sinsin
b
C = const; a=2b; Ȟ
,
y
Ɋɢɫ ɉɪɹɦɨɭɝɨɥɶɧɚɹ
a
ɩɥɚɫɬɢɧɚ
Ɍɪɟɛɭɟɬɫɹ
1) ɭɫɬɚɧɨɜɢɬɶ ɤɚɤɢɦɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɡɚ
ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟɭɩɪɭɝɨɣɩɨɜɟɪɯɧɨɫɬɢw (x, y);
2) ɨɩɪɟɞɟɥɢɬɶɩɨɫɬɨɹɧɧɵɣɤɨɷɮɮɢɰɢɟɧɬɋɢɫɩɨɥɶɡɭɹɞɢɮɮɟ
ɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ;
3) ɫɨɫɬɚɜɢɬɶ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ M
, M
x
ɢɤɪɭɬɹɳɟɝɨɦɨɦɟɧɬɚH;
4) ɩɨɫɬɪɨɢɬɶɷɩɸɪɵɦɨɦɟɧɬɨɜɜɞɨɥɹɯɨɬ
ɢbɜɫɟɱɟɧɢɹɯ,
q
0
ɩɚɪɚɥɥɟɥɶɧɵɯɨɫɹɦx ɢy.
Ɋɟɲɟɧɢɟ
ɋɨɫɬɚɜɢɦɜɵɪɚɠɟɧɢɹ ɞɥɹɭɝɥɨɜɩɨɜɨɪɨɬɚ.
ȼɧɚɩɪɚɜɥɟɧɢɢɨɫɢ x:
T
x
Ɍɚɤɤɚɤɩɪɢx=0 ɢɦɟɟɦ
w
w
C
x
w
w
x
a
a
2
w
w
0;0
2
w
x
sincos
ɧɨ
y
SSS
3
.
b
ɫɥɟɞɨɜɚɬɟɥɶɧɨ
0zT
x
ɩɨɷɬɨɣɝɪɚɧɢ ɩɥɚɫɬɢɧɚɲɚɪɧɢɪɧɨ ɨɩɟɪɬɚ ɉɪɢx=aɬɚɤɠɟ ɢɦɟɟɦ
2
w
w
,
0;0
w
2
w
x
ɬ ɟɢɷɬɚ ɝɪɚɧɶ ɩɥɚɫɬɢɧɵ ɲɚɪɧɢɪɧɨɨɩɟɪɬɚ
0zT
x
ȼɧɚɩɪɚɜɥɟɧɢɢɨɫɢ y:
SSS
3
x
y
cossin
.
b
ɬɨɩɨɷɬɨɣɝɪɚɧɢɩɥɚɫɬɢ
0zT
Ɍɚɤɤɚɤɩɪɢy=0
w
T
y
w
w
3
w
C
y
2
w
w
2
w
y
a
b
,
0;0
y
ɧɚ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɚ ɉɪɢ y=bɩɨ ɬɟɦ ɠɟ ɩɪɢɱɢɧɚɦ ɩɥɚɫɬɢɧɚ
ɬɚɤɠɟ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɚ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨ ɜɫɟɦ ɫɜɨɢɦ ɝɪɚɧɹɦ
ɩɥɚɫɬɢɧɚɹɜɥɹɟɬɫɹɲɚɪɧɢɪɧɨɨɩɟɪɬɨɣ ɪɢɫ 1.11).
29
y
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
2. ɋɨɫɬɚɜɢɦɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɨɝɧɭɬɨɣɫɪɟɞɢɧ
ɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ ȼɵɱɢɫɥɢɦ ɩɪɨɢɡɜɨɞɧɵɟ ɨɬ ɮɭɧɤɰɢɢ
ɩɪɨɝɢɛɚw(x, y)Ⱦɥɹɫɨɤɪɚɳɟɧɢɹɡɚɩɢɫɢ ɜɜɟɞɟɦɨɛɨɡɧɚɱɟɧɢɹ:
xXS
;
a
4
w
IV
w
x
w
x
yYS
3
;
b
w
;
4
w
IV
w
y
w
2
w
w
cc
w
x
4
w
4
y
;
2
w
x
IV
;
w
xy
2
w
w
cc
w
y
w
;
2
y
w
4
w
.
22
yx
ww
Ɍɨɝɞɚ ɩɨɥɭɱɢɦ
2
S
cc
Cw
x
IV
w
x
2
a
4
S
C
4
a
sinsin
sinsin
IV
w
C
xy
YX
YX
9
;
w C
;
4
S
22
ba
cc
y
IV
y
2
S
9
Cw
2
b
4
81
S
4
b
.
YX
sinsin
;
YX
sinsin
YX
sinsin
;
ɉɪɢ ɷɬɨɦ ɮɭɧɤɰɢɸ ɩɨɩɟɪɟɱɧɨɣ ɧɚɝɪɭɡɤɢ ɛɭɞɟɦ ɡɚɞɚɜɚɬɶ
ɜɚɧɚɥɨɝɢɱɧɨɦɜɢɞɟ:
3
qYXqq
x
00
a
y
SS
sinsinsinsin
.
b
ɉɨɞɫɬɚɜɢɦ ɡɧɚɱɟɧɢɹɩɪɨɢɡɜɨɞɧɵɯɢɜɵɪɚɠɟɧɢɟɞɥɹɧɚɝɪɭɡɤɢ
ɜɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟ ɢɡɝɢɛɚ (1.29):
4
§
YXC sinsin
sinsin
¨
©
Ɉɬɫɸɞɚɩɨɞɫɬɚɜɢɜa=2bɩɨɥɭɱɢɦ
ȼɵɪɚɠɟɧɢɹɞɥɹ ɦɨɦɟɧɬɨɜ M
cc
cc
Q
yxx
cc
cc
Q
xyy
2
sinsin
2
sinsin
30
81181
S
bbaa
C
, Myɢ H ɛɭɞɭɬɫɥɟɞɭɸɳɢɦɢ:
x
YXCDwwDɆ
YXCDwwDɆ
q
·
0
¸
4224
D
¹
4
bq
0
§
¨
©
§
¨
©
D
S
91
·
QS
¸
22
ba
¹
19
·
QS
¸
22
ab
¹
.
YX
bq
0
4
831756,85
2
bq
0
286
2
bq
0
5,93
4
.
D
;sinsin
YX
;sinsin
YX
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