Ревинская О.Г., Кравченко Н.С. Изучение моделей физических процессов и явлений на компьютере
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ɆɨɞɆ-04 Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
41
³
³
³
³
³
³
U
U
dV
dVr
dV
dVr
dm
dmr
R
C
G
G
G
G
,
³³³
³³³
dxdydz
dxdydzr
R
C
G
G
.
Ɍɨɝɞɚ ɤɨɨɪɞɢɧɚɬɵ
C
x ,
C
y ,
C
z ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɨɩɪɟɞɟɥɹɸɬɫɹ
³³³
³³³
dxdydz
dxdydzx
x
C
,
³³³
³³³
dxdydz
dxdydzy
y
C
,
³³³
³³³
dxdydz
dxdydzz
z
C
.
Ⱦɥɹ ɬɟɥɚ, ɢɦɟɸɳɟɝɨ ɮɨɪɦɭ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɪɢɡɦɵ ɜɵɫɨɬɨɣ
h
,
ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ
dz ɦɨɠɧɨ ɜɵɩɨɥɧɢɬɶ ɧɟɡɚɜɢɫɢɦɨ. ɉɨɷɬɨɦɭ ɮɨɪɦɭ-
ɥɵ ɞɥɹ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɩɪɢɦɭɬ ɜɢɞ:
³³
³³
³³
³³
³³³
³³³
dxdy
dxdyx
dxdyh
dxdyxh
dxdydz
dxdyxdz
x
C
,
³³
³³
³³
³³
³³³
³³³
dxdy
dxdyy
dxdyh
dxdyyh
dxdydz
dxdyydz
y
C
,
2
2
2
h
dxdyh
dxdyh
dxdydz
dxdyzdz
z
C
³³
³³
³³³
³³³
.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ OZ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɛɭɞɟɬ ɧɚ-
ɯɨɞɢɬɶɫɹ ɧɚ ɨɞɢɧɚɤɨɜɨɦ ɪɚɫɫɬɨɹɧɢɢ
h/2 ɨɬ ɜɟɪɯɧɟɣ ɢ ɧɢɠɧɟɣ ɝɪɚɧɟɣ
ɩɪɢɡɦɵ:
2/hz
C
. Ɉɫɬɚɟɬɫɹ ɜɵɱɢɫɥɢɬɶ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɜ
ɩɥɨɫɤɨɫɬɢ XOY. Ⱦɥɹ ɷɬɨɝɨ ɧɟɨɛɯɨɞɢɦɨ ɜɵɱɢɫɥɢɬɶ ɪɹɞ ɞɜɨɣɧɵɯ ɢɧɬɟ-
ɝɪɚɥɨɜ:
³³
³³
dydx
dydxx
x
C
ɢ
³³
³³
dydx
dydxy
y
C
.
ɂɧɬɟɝɪɚɥɵ ɜ ɡɧɚɦɟɧɚɬɟɥɟ
³³
dxdy
ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɩɥɨɳɚɞɶ
ɜɟɪɯɧɟɣ (ɢɥɢ ɧɢɠɧɟɣ) ɝɪɚɧɢ, ɨɛɨɡɧɚɱɢɦ ɢɯ ɱɟɪɟɡ
³³
{ dxdyS
XY
.
Ɍɚɤ ɤɚɤ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɦɚɫɫ (ɢɧɟɪɰɢɢ) ɧɟɨɛɯɨɞɢɦɨ ɞɥɹ ɪɚɫɱɟ-
ɬɚ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ, ɫɥɟɞɭɟɬ ɜɵɱɢɫɥɹɬɶ ɤɨɨɪɞɢɧɚɬɵ
C
x ɢ
C
y ɜ ɬɨɣ ɠɟ
ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɱɬɨ ɢ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ (ɪɢɫ. 4). Ɍɨɝɞɚ ɩɨ ɬɟɨɪɟɦɟ
ɒɬɟɣɧɟɪɚ ɦɨɠɧɨ ɛɭɞɟɬ ɜɵɱɢɫɥɢɬɶ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ,
ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ (ɢɧɟɪɰɢɢ).
Ɋɚɫɫɦɨɬɪɢɦ ɧɟɫɤɨɥɶɤɨ ɩɪɢɦɟɪɨɜ.
Ɇɟɯɚɧɢɤɚ
42
ɐɟɧɬɪ ɢɧɟɪɰɢɢ ɬɪɟɭɝɨɥɶɧɨɣ ɩɪɢɡɦɵ, ɫɟɱɟɧɢɟɦ ɤɨɬɨɪɨɣ ɹɜɥɹɟɬɫɹ
ɩɪɹɦɨɭɝɨɥɶɧɵɣ ɬɪɟɭɝɨɥɶɧɢɤ
ȼ ɩɥɨɫɤɨɫɬɢ XOY ɮɢɝɭɪɚ ɨɝɪɚɧɢɱɟɧɚ ɩɪɹɦɵɦɢ
0
x
, 0 y ɢ
)(
1
ax
a
b
y
(ɪɢɫ. 6). Ɍɨɝɞɚ ɞɜɨɣɧɵɟ ɢɧɬɟɝɪɚɥɵ, ɧɟɨɛɯɨɞɢɦɵɟ ɞɥɹ
ɜɵɱɢɫɥɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ, ɩɪɢɦɭɬ ɜɢɞ:
³³³³
)(
00
1
xy
a
XY
dydxdydxS ;
³³³³
)(
00
1
xy
a
dyxdxdydxx ;
³³³³
)(
00
1
xy
a
ydydxdydxy .
ȼɵɱɢɫɥɢɦ:
³³
)(
00
1
xy
a
XY
dydxS =
)(
0
0
1
xy
a
ydx
³
=
³
a
dxax
a
b
0
=
¸
¹
·
¨
©
§
a
ax
a
b
0
2
)(
2
1
2
ab
.
³³
)(
00
1
xy
a
dyxdx =
)(
0
0
1
xy
a
yxdx
³
=
³
a
xdxax
a
b
0
=
=
¸
¹
·
¨
©
§
a
axx
a
b
0
23
2
1
3
1
6
2
ba
.
³³
)(
00
1
xy
a
ydydx =
)(
0
2
0
1
2
1
xy
a
ydx
¸
¹
·
¨
©
§
³
=
³
a
dxax
a
b
0
2
2
2
2
=
=
¸
¹
·
¨
©
§
a
ax
a
b
0
3
2
2
)(
3
1
2
6
2
ab
.
Ɍɨɝɞɚ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɜ ɩɥɨɫɤɨɫɬɢ XOY ɨɩɪɟɞɟɥɹɟɬɫɹ
ɤɨɨɪɞɢɧɚɬɚɦɢ:
3
a
x
C
ɢ
3
b
y
C
.
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɹɦɨɣ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ
ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɢ ɩɚɪɚɥɥɟɥɶɧɨɣ ɜɵɫɨɬɟ ɩɪɢɡɦɵ, ɩɨɥɭɱɢɦ ɢɡ ɬɟɨɪɟɦɵ
ɒɬɟɣɧɟɪɚ (ɢɡ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ, ɪɚɫɫɱɢɬɚɧɧɨɝɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɩɪɨ-
ɯɨɞɹɳɟɣ ɱɟɪɟɡ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ:
0
O
x , 0
O
y ):
)(
6
22222
CCT
T
CTC
yxmba
m
RmJJ
)(
18
)(
9
)(
6
222222
ba
m
ba
m
ba
m
J
TTT
C
.
ɆɨɞɆ-04 Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
43
ɐɟɧɬɪ ɢɧɟɪɰɢɢ ɫɟɝɦɟɧɬɚ ɤɪɭɝɨɜɨɝɨ ɞɢɫɤɚ ɬɨɥɳɢɧɨɣ h
ɍɱɢɬɵɜɚɹ ɮɨɪɦɭ ɬɟɥɚ, ɞɜɭɦɟɪɧɵɟ ɢɧɬɟɝɪɚɥɵ, ɧɟɨɛɯɨɞɢɦɵɟ ɞɥɹ
ɜɵɱɢɫɥɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ, ɭɞɨɛɧɟɟ ɛɪɚɬɶ ɜ ɩɨɥɹɪɧɵɯ ɤɨ-
ɨɪɞɢɧɚɬɚɯ:
³³
³³
M
M
drdr
drdrx
x
C
ɢ
³³
³³
M
M
drdr
drdry
y
C
,
ɝɞɟ
M cosr
x
ɢ
M sinry
.
Ɏɢɝɭɪɚ ɨɝɪɚɧɢɱɟɧɚ ɩɪɹɦɵɦɢ
0 M
,
2
S
M
, ɚ ɬɚɤɠɟ ɤɪɢɜɨɣ
ar
(ɪɢɫ. 7). Ɍɨɝɞɚ ɢɧɬɟɝɪɚɥɵ, ɧɟɨɛɯɨɞɢɦɵɟ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɩɨɥɨɠɟɧɢɹ
ɰɟɧɬɪɚ ɦɚɫɫ (ɢɧɟɪɰɢɢ), ɜɵɱɢɫɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
XY
S =
³³³³
S
M M
2/
00
drdrdrdr
a
;
³³³³
S
MM M
2/
00
2
cos ddrrdrdrx
a
;
³³³³
S
MM M
2/
00
2
sin ddrrdrdry
a
.
ȼɵɱɢɫɥɢɦ:
XY
S
=
³³
S
M
2/
00
drdr
a
=
2/
0
0
2
2
S
M
a
r
=
4
2
aS
.
³³
S
MM
2/
00
2
cos ddrr
a
=
2/
0
0
3
sin
3
S
M
a
r
=
3
3
a
.
³³
S
MM
2/
00
2
sin ddrr
a
=
2/
0
0
3
)cos(
3
S
M
a
r
=
3
3
a
.
Ɍɨɝɞɚ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɜ ɩɥɨɫɤɨɫɬɢ XOY ɨɩɪɟɞɟɥɹɟɬɫɹ
ɤɨɨɪɞɢɧɚɬɚɦɢ:
S
3
4a
x
C
ɢ
S
3
4a
y
C
.
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɹɦɨɣ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ
ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɢ ɩɚɪɚɥɥɟɥɶɧɨɣ ɜɵɫɨɬɟ ɩɪɢɡɦɵ, ɩɨɥɭɱɢɦ ɢɡ ɬɟɨɪɟɦɵ
ɒɬɟɣɧɟɪɚ (ɢɡ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ, ɪɚɫɫɱɢɬɚɧɧɨɝɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɩɪɨ-
ɯɨɞɹɳɟɣ ɱɟɪɟɡ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ:
0
O
x , 0
O
y ):
)(
2
222
CCT
T
CTC
yxma
m
RmJJ
Ɇɟɯɚɧɢɤɚ
44
22
2
2
2
2
)649(
189
16
2
2
a
m
a
m
a
m
J
TTT
C
S
S
S
.
4. Ɇɨɞɟɥɶ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɣ ɭɫɬɚɧɨɜɤɢ
ȼ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɫ ɩɨɦɨɳɶɸ ɫɪɟɞɫɬɜ ɤɨɦɩɶɸɬɟɪɧɨɣ ɝɪɚɮɢɤɢ ɦɨ-
ɞɟɥɢɪɭɟɬɫɹ ɜɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɜɨɡɧɢ-
ɤɚɸɳɟɟ ɜɫɥɟɞɫɬɜɢɟ ɭɩɪɭɝɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫ ɪɚɜɧɨɦɟɪɧɨ ɢ ɩɪɹɦɨɥɢ-
ɧɟɣɧɨ ɥɟɬɹɳɢɦ ɲɚɪɢɤɨɦ. ɋɢɥɚ ɬɹɠɟɫɬɢ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ
ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ, ɫɤɨɦɩɟɧɫɢɪɨɜɚɧɚ ɞɪɭɝɢɦɢ ɫɬɨɪɨɧɧɢɦɢ ɫɢɥɚɦɢ ɢ
ɧɟ ɨɤɚɡɵɜɚɟɬ ɜɥɢɹɧɢɹ ɧɢ ɧɚ ɞɜɢɠɟɧɢɟ ɬɟɥɚ, ɧɢ ɧɚ ɞɜɢɠɟɧɢɟ ɲɚɪɢɤɚ.
ɇɚɱɚɥɶɧɚɹ
ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɲɚɪɢɤɚ ɦɨɠɟɬ ɢɡɦɟɧɹɬɶɫɹ ɜ ɞɢɚɩɚ-
ɡɨɧɟ ɨɬ 40 ɞɨ 80 ɫɦ/ɫ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɜɪɟɦɟɧɢ, ɡɚ ɤɨɬɨɪɨɟ ɬɟɥɨ ɫɨɜɟɪ-
ɲɢɬ ɡɚɞɚɧɧɨɟ ɱɢɫɥɨ ɨɛɨɪɨɬɨɜ, ɢɫɩɨɥɶɡɭɟɬɫɹ ɫɟɤɭɧɞɨɦɟɪ, ɫɩɨɫɨɛɧɵɣ ɢɡ-
ɦɟɪɹɬɶ ɜɪɟɦɹ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ 1 ɦɢɥɥɢɫɟɤɭɧɞɵ. ɋɟɤɭɧɞɨɦɟɪ ɦɨɠɟɬ ɛɵɬɶ
ɧɚɫɬɪɨɟɧ ɞɥɹ ɢɡɦɟɪɟɧɢɹ ɜɪɟɦɟɧɢ ɪɚɡɥɢɱɧɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɨɛɨɪɨɬɨɜ ɬɟɥɚ.
ɉɪɢ ɭɤɚɡɚɧɧɵɯ ɭɫɥɨɜɢɹɯ ɩɨɝɪɟɲɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ
ɬɟɥɚ ɜ ɷɤɫɩɟɪɢɦɟɧɬɟ ɧɟ ɩɪɟɜɵɲɚɟɬ 0,7-1%.
Ɋɚɛɨɬɚ ɜɵɩɨɥɧɹɟɬɫɹ ɧɚ IBM-ɫɨɜɦɟɫɬɢɦɨɦ ɩɟɪɫɨɧɚɥɶɧɨɦ ɤɨɦɩɶɸ-
ɬɟɪɟ ɜ ɜɢɞɟ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ Windows-ɩɪɢɥɨɠɟɧɢɹ. Ⱦɥɹ ɭɞɨɛɫɬɜɚ ɜɵ-
ɆɨɞɆ-04 Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
45
ɩɨɥɧɟɧɢɹ ɪɚɛɨɬɵ ɜ ɩɪɨɝɪɚɦɦɟ ɩɪɟɞɭɫɦɨɬɪɟɧɵ ɬɪɢ ɪɚɡɞɟɥɚ: ɤɪɚɬɤɨɟ ɨɩɢ-
ɫɚɧɢɟ ɪɚɛɨɬɵ; ɩɨɪɹɞɨɤ ɜɵɩɨɥɧɟɧɢɹ ɪɚɛɨɬɵ ɢ ɷɤɫɩɟɪɢɦɟɧɬ. ɉɟɪɟɤɥɸɱɟ-
ɧɢɟ ɦɟɠɞɭ ɪɚɡɞɟɥɚɦɢ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɤɧɨɩɨɤ «ɏɨɞ ɪɚɛɨɬɵ»
ɢ «ɗɤɫɩɟɪɢɦɟɧɬ». ɇɚɠɚɬɢɟ ɷɬɢɯ ɤɧɨɩɨɤ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɤɨɧɬɟɤɫɬɚ ɪɚ-
ɛɨɬɵ ɩɪɨɝɪɚɦɦɵ ɩɪɢɜɨɞɢɬ ɥɢɛɨ ɤ ɜɵɡɨɜɭ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɪɚɡɞɟɥɨɜ,
ɥɢɛɨ ɤ ɜɨɡɜɪɚɳɟɧɢɸ ɜ ɪɚɡɞɟɥ ɨɩɢɫɚɧɢɹ
.
ȼɚɪɢɚɧɬɵ ɜɵɩɨɥɧɟɧɢɹ ɪɚɛɨɬɵ
(ɬɟɥɨ ɢɦɟɟɬ ɮɨɪɦɭ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɪɢɡɦɵ)
ȼɚɪɢɚɧɬ 1.
ɋɟɱɟɧɢɟ ɩɪɢɡɦɵ:
ɩɪɹɦɨɭɝɨɥɶɧɢɤ
ȼɚɪɢɚɧɬ 2.
ɋɟɱɟɧɢɟ ɩɪɢɡɦɵ:
ɤɜɚɞɪɚɬ
ȼɚɪɢɚɧɬ 3.
ɋɟɱɟɧɢɟ ɩɪɢɡɦɵ:
ɩɪɹɦɨɭɝɨɥɶɧɵɣ
ɬɪɟɭɝɨɥɶɧɢɤ
O
y
x
a
b
y
1
= b
r
h
a
b
O
O’
– b
y
2
= – b
b
O
y
x
a
a
y
1
=a
r
h
a
O
O’
a
O
y
x
a
b
y
1
= –
a
b
(x – a)
r
h
a
b
O
O
’
ȼɚɪɢɚɧɬ 4.
ɋɟɱɟɧɢɟ ɩɪɢɡɦɵ:
ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ
ɬɪɟɭɝɨɥɶɧɢɤ
ȼɚɪɢɚɧɬ 5.
ɋɟɱɟɧɢɟ ɩɪɢɡɦɵ:
ɩɚɪɚɥɥɟɥɨɝɪɚɦɦ
ȼɚɪɢɚɧɬ 6.
ɋɟɱɟɧɢɟ ɩɪɢɡɦɵ:
ɪɨɦɛ
O
y
x
a
b
y
1
= –
a
b
(x – a)
r
h a
b
O
O’
– b
y
2
=
a
b
(x – a)
b
O
y
x
a
b
y
2
= –
a
b
x
r
h
a
b
O
O’
y
1
= –
a
b
(x – a)
– b
O
y
x
a = b
3
2b
y
2
= –
a
b
x
r
ha
b
O
O’
y
1
= –
a
b
(x – 2a)
–b
b
b
Ɇɟɯɚɧɢɤɚ
46
Ɋɚɡɞɟɥ ɩɪɨɝɪɚɦɦɵ «ɗɤɫɩɟɪɢɦɟɧɬ» ɫɨ-
ɞɟɪɠɢɬ ɩɚɧɟɥɶ ɢɧɫɬɪɭɦɟɧɬɨɜ ɫ ɤɧɨɩɤɚɦɢ ɞɥɹ
ɜɵɛɨɪɚ ɬɟɥɚ, ɫɱɟɬɱɢɤ ɞɥɹ ɢɡɦɟɧɟɧɢɹ ɧɚɱɚɥɶɧɨɣ
ɫɤɨɪɨɫɬɢ ɲɚɪɢɤɚ, ɛɟɝɭɧɤɢ ɩɪɨɤɪɭɬɤɢ ɞɥɹ ɢɡ-
ɦɟɧɟɧɢɹ ɩɨɥɨɠɟɧɢɹ ɨɫɢ ɜɪɚɳɟɧɢɹ ɩɨ ɝɨɪɢɡɨɧ-
ɬɚɥɢ ɢ ɩɨ ɜɟɪɬɢɤɚɥɢ, ɧɚɫɬɪɚɢɜɚɟɦɵɣ ɫɟɤɭɧɞɨ-
ɦɟɪ ɞɥɹ ɢɡɦɟɪɟɧɢɹ ɜɪɟɦɟɧɢ ɜɪɚɳɟɧɢɹ, ɚ ɬɚɤɠɟ
ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɟ ɤɧɨɩɤɢ, ɩɨɡɜɨɥɹɸɳɢɟ
ɭɩɪɚɜɥɹɬɶ ɷɤɫɩɟɪɢɦɟɧɬɨɦ.
5. ɉɨɪɹɞɨɤ ɜɵɩɨɥɧɟɧɢɹ ɪɚɛɨɬɵ
5.1. Ʉɪɚɬɤɨɟ ɨɩɢɫɚɧɢɟ ɯɨɞɚ ɪɚɛɨɬɵ
1. ȼɵɛɟɪɢɬɟ ɬɟɥɨ (ɩɨ ɭɤɚɡɚɧɢɸ ɩɪɟɩɨɞɚɜɚɬɟɥɹ).
2. ɍɫɬɚɧɨɜɢɬɟ ɦɢɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ X-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟ-
ɧɢɹ.
3. ɍɫɬɚɧɨɜɢɬɟ ɦɢɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ Y-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟ-
ɧɢɹ.
4. ɇɚɫɬɪɨɣɬɟ ɫɟɤɭɧɞɨɦɟɪ ɧɚ ɢɡɦɟɪɟɧɢɟ ɜɪɟɦɟɧɢ ɩɹɬɢ ɨɛɨɪɨɬɨɜ ɬɟ-
ɥɚ.
5. Ɂɚɞɚɣɬɟ ɦɢɧɢɦɚɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɲɚɪɢɤɚ. ȼɵɩɨɥɧɢɬɟ ɷɤɫɩɟɪɢ-
ɦɟɧɬ.
6. ɉɨɜɬɨɪɢɬɟ ɷɤɫɩɟɪɢɦɟɧɬ ɞɥɹ ɬɪɟɯ ɫɤɨɪɨɫɬɟɣ ɲɚɪɢɤɚ.
7. ɉɨɜɬɨɪɢɬɟ ɨɩɵɬɵ ɞɥɹ ɲɟɫɬɢ ɡɧɚɱɟɧɢɣ X-ɤɨɨɪɞɢɧɚɬɵ
ɨɫɢ ɜɪɚ-
ɳɟɧɢɹ.
8. ɉɨɜɬɨɪɢɬɟ ɨɩɵɬɵ ɞɥɹ ɲɟɫɬɢ ɡɧɚɱɟɧɢɣ Y-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚ-
ɳɟɧɢɹ.
9. Ⱦɥɹ ɤɚɠɞɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ ɜɵɱɢɫɥɢɬɟ ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ ɢɧɟɪ-
ɰɢɢ.
10. Ⱦɥɹ ɤɚɠɞɨɝɨ ɩɨɥɨɠɟɧɢɹ ɨɫɢ ɜɵɱɢɫɥɢɬɟ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɦɨ-
ɦɟɧɬɚ ɢɧɟɪɰɢɢ.
11. ɉɨɫɬɪɨɣɬɟ ɝɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɢ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɨɬ Y-
ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟɧɢɹ.
12. Ɉɩɪɟɞɟɥɢɬɟ ɩɨɥɨɠɟɧɢɟ ɦɢɧɢɦɭɦɨɜ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɢɡ ɝɪɚ-
ɮɢɤɨɜ.
ȼɚɪɢɚɧɬ 7.
ɋɟɱɟɧɢɟ ɩɪɢɡɦɵ:
ɩɨɥɭɤɪɭɝ
O
y
x
a
a
r= a
r
h
a
a
O
O’
M
ɆɨɞɆ-04 Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
47
13. ɉɨɫɬɪɨɣɬɟ ɝɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɢ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɨɬ X-
ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟɧɢɹ.
14. Ɉɩɪɟɞɟɥɢɬɟ ɩɨɥɨɠɟɧɢɟ ɦɢɧɢɦɭɦɨɜ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɢɡ ɝɪɚ-
ɮɢɤɨɜ.
15. Ɉɩɪɟɞɟɥɢɬɟ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɢɧɟɪ-
ɰɢɢ.
16. ȼɵɱɢɫɥɢɬɟ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɡɧɚɱɟɧɢɹ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɬɟɥɚ ɢ
ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɢɧɟɪɰɢɢ.
17. ɋɞɟɥɚɣɬɟ ɜɵɜɨɞɵ.
5.2. ɉɨɞɪɨɛɧɨɟ ɨɩɢɫɚɧɢɟ ɯɨɞɚ ɪɚɛɨɬɵ
ɉɪɢ ɜɵɩɨɥɧɟɧɢɢ ɪɚɛɨɬɵ ɪɟɤɨɦɟɧɞɭɟɬɫɹ ɫɥɟɞɭɸɳɚɹ ɩɨɫɥɟɞɨɜɚ-
ɬɟɥɶɧɨɫɬɶ ɞɟɣɫɬɜɢɣ:
1. ɉɚɧɟɥɶ ɢɧɫɬɪɭɦɟɧɬɨɜ «Ɍɟɥɨ» ɫɨɞɟɪɠɢɬ ɪɚɫɤɪɵɜɚɸɳɭɸɫɹ ɩɚɧɟɥɶ ɤɧɨɩɨɤ
«ɋɟɱɟɧɢɟ», ɩɨɡɜɨɥɹɸɳɭɸ ɜɵɛɪɚɬɶ ɫɟɱɟɧɢɟ ɬɟɥɚ, ɤɨɬɨɪɨɟ ɛɭɞɟɬ ɭɱɚɫɬɜɨɜɚɬɶ ɜ ɷɤɫɩɟ-
ɪɢɦɟɧɬɟ. ȼɫɟ ɬɟɥɚ ɹɜɥɹɸɬɫɹ ɩɪɹɦɨɭɝɨɥɶɧɵɦɢ ɩɪɢɡɦɚɦɢ ɫ ɩɨɩɟɪɟɱɧɵɦɢ ɫɟɱɟɧɢɹɦɢ ɜ
ɜɢɞɟ: ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ, ɤɜɚɞɪɚɬɚ, ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ, ɪɚɜɧɨɛɟɞɪɟɧɧɨɝɨ
ɬɪɟɭɝɨɥɶɧɢɤɚ, ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ, ɪɨɦɛɚ ɢ ɩɨɥɭɤɪɭɝɚ. ȼɵɛɟɪɢɬɟ ɫɟɱɟɧɢɟ ɬɟɥɚ, ɫ ɤɨ-
ɬɨɪɵɦ ɛɭɞɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɷɤɫɩɟɪɢɦɟɧɬ (ɩɨ ɭɤɚɡɚɧɢɸ ɩɪɟɩɨɞɚɜɚɬɟɥɹ). Ⱦɥɹ ɜɵɛɪɚɧ-
ɧɨɝɨ ɬɟɥɚ ɪɹɞɨɦ ɫ ɩɚɧɟɥɶɸ ɢɧɫɬɪɭɦɟɧɬɨɜ ɚɜɬɨɦɚɬɢɱɟɫɤɢ ɭɤɚɡɵɜɚɸɬɫɹ ɟɝɨ ɦɚɫɫɚ, ɥɢ-
ɧɟɣɧɵɟ ɪɚɡɦɟɪɵ, ɚ ɬɚɤɠɟ ɱɟɪɬɟɠ (ɫɯɟɦɚ), ɧɟɨɛɯɨɞɢɦɵɟ ɞɥɹ ɪɚɫɱɟɬɚ ɬɟɨɪɟɬɢɱɟɫɤɢɯ
ɡɧɚɱɟɧɢɣ ɩɨɥɨɠɟɧɢɹ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɢ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ.
2. ɉɨɥɡɭɧɨɤ ɩɪɨɤɪɭɬɤɢ «X» ɩɨɡɜɨɥɹɟɬ ɢɡɦɟɧɹɬɶ X-ɤɨɨɪɞɢɧɚɬɭ ɨɫɢ ɜɪɚɳɟɧɢɹ,
ɪɚɫɩɨɥɚɝɚɹ ɟɟ ɤɚɤ ɜɧɭɬɪɢ ɬɟɥɚ, ɬɚɤ ɢ ɜɧɟ ɟɝɨ. ɍɫɬɚɧɨɜɢɬɟ ɩɨɥɡɭɧɨɤ ɩɪɨɤɪɭɬɤɢ ɬɚɤ,
ɱɬɨɛɵ X-ɤɨɨɪɞɢɧɚɬɚ ɨɫɢ ɛɵɥɚ ɪɚɜɧɚ ɧɭɥɸ.
3. ɉɨɥɡɭɧɨɤ ɩɪɨɤɪɭɬɤɢ «Y» ɩɨɡɜɨɥɹɟɬ ɢɡɦɟɧɹɬɶ Y-ɤɨɨɪɞɢɧɚɬɭ ɨɫɢ ɜɪɚɳɟɧɢɹ,
ɪɚɫɩɨɥɚɝɚɹ ɟɟ ɤɚɤ ɜɧɭɬɪɢ ɬɟɥɚ, ɬɚɤ ɢ ɜɧɟ ɟɝɨ. ɍɫɬɚɧɨɜɢɬɟ ɩɨɥɡɭɧɨɤ ɩɪɨɤɪɭɬɤɢ ɬɚɤ,
ɱɬɨɛɵ Y-ɤɨɨɪɞɢɧɚɬɚ ɨɫɢ ɩɪɢɧɹɥɚ ɛɵ ɦɢɧɢɦɚɥɶɧɨɟ (ɧɢɠɧɟɟ) ɡɧɚɱɟɧɢɟ.
4. ȼɫɬɪɨɟɧɧɵɣ ɫɟɤɭɧɞɨɦɟɪ ɦɨɠɟɬ ɢɡɦɟɪɹɬɶ ɜɪɟɦɹ ɪɚɡɥɢɱɧɨɝɨ ɤɨɥɢɱɟɫɬɜɚ
ɨɛɨɪɨɬɨɜ. ɇɚ ɩɚɧɟɥɢ ɫɟɤɭɧɞɨɦɟɪɚ ɪɚɫɩɨɥɨɠɟɧ
ɫɱɟɬɱɢɤ, ɩɨɡɜɨɥɹɸɳɢɣ ɢɡɦɟɧɹɬɶ ɤɨ-
ɥɢɱɟɫɬɜɨ ɨɛɨɪɨɬɨɜ. ɍɫɬɚɧɨɜɢɬɟ ɫɱɟɬɱɢɤ ɬɚɤ, ɱɬɨɛɵ ɫɟɤɭɧɞɨɦɟɪ ɢɡɦɟɪɹɥ ɦɚɤɫɢɦɚɥɶ-
ɧɨ ɜɨɡɦɨɠɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɨɛɨɪɨɬɨɜ ɬɟɥɚ.
5. ɇɚ ɩɚɧɟɥɢ ɢɧɫɬɪɭɦɟɧɬɨɜ «ɒɚɪɢɤ» ɪɚɫɩɨɥɨɠɟɧ ɫɱɟɬɱɢɤ «ɇɚɱɚɥɶɧɚɹ ɫɤɨ-
ɪɨɫɬɶ», ɩɨɡɜɨɥɹɸɳɢɣ ɢɡɦɟɧɹɬɶ ɫɤɨɪɨɫɬɶ ɲɚɪɢɤɚ ɜ ɞɢɚɩɚɡɨɧɟ ɨɬ 40 ɞɨ 80 ɫɦ/ɫ. Ɂɚ-
ɞɚɣɬɟ ɦɢɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ ɲɚɪɢɤɚ.
ɇɚɠɦɢɬɟ ɤɧɨɩɤɭ «ɇɚɱɚɬɶ ɷɤɫɩɟɪɢɦɟɧɬ» – ɜ ɡɚɞɚɧɧɵɯ ɭɫɥɨɜɢɹɯ ɧɚɱɧɟɬ ɞɜɢ-
ɝɚɬɶɫɹ ɲɚɪɢɤ. ɉɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɲɚɪɢɤ ɩɨɥɟɬɢɬ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɫɬɨɪɨɧɭ, ɚ
ɬɟɥɨ ɧɚɱɧɟɬ ɜɪɚɳɚɬɶɫɹ – ɚɜɬɨɦɚɬɢɱɟɫɤɢ ɜɤɥɸɱɢɬɫɹ ɫɟɤɭɧɞɨɦɟɪ. ɗɤɫɩɟɪɢɦɟɧɬ ɡɚɤɨɧ-
ɱɢɬɫɹ, ɤɨɝɞɚ ɬɟɥɨ ɜɵɩɨɥɧɢɬ ɡɚɞɚɧɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɨɛɨɪɨɬɨɜ.
6. ɋɤɨɪɨɫɬɶ ɲɚɪɢɤɚ ɦɨɠɧɨ ɢɡɦɟɧɹɬɶ ɜ ɞɢɚɩɚɡɨɧɟ ɨɬ 40 ɞɨ 80 ɫɦ/ɫ. Ɋɚɡɞɟɥɢɬɟ
ɞɢɚɩɚɡɨɧ ɧɚ ɪɚɜɧɵɟ ɱɚɫɬɢ (ɧɚɩɪɢɦɟɪ, ɱɟɪɟɡ 20 ɫɦ/ɫ: 40, 60, 80 ɫɦ/ɫ). ɉɨɜɬɨɪɢɬɟ ɷɤɫ-
ɩɟɪɢɦɟɧɬ ɞɥɹ ɬɪɟɯ ɫɤɨɪɨɫɬɟɣ ɲɚɪɢɤɚ.
Ɇɟɯɚɧɢɤɚ
48
ȼɊȿɆə ȼɊȺɓȿɇɂə ɂ ɄɈɅɂɑȿɋɌȼɈ ɈȻɈɊɈɌɈȼ ɁȺɉɂɒɂɌȿ ȼ ɌȺȻ-
Ʌɂɐɍ 1.
7. Ⱦɢɚɩɚɡɨɧ ɢɡɦɟɧɟɧɢɹ X-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟɧɢɹ ɡɚɜɢɫɢɬ ɨɬ ɝɨɪɢɡɨɧɬɚɥɶ-
ɧɵɯ ɪɚɡɦɟɪɨɜ ɬɟɥɚ. Ɋɚɡɞɟɥɢɬɟ ɞɢɚɩɚɡɨɧ ɢɡɦɟɧɟɧɢɹ X-ɤɨɨɪɞɢɧɚɬɵ ɧɚ 5 ɪɚɜɧɵɯ ɱɚɫ-
ɬɟɣ. Ⱦɥɹ ɲɟɫɬɢ ɩɨɥɭɱɟɧɧɵɯ ɩɨɥɨɠɟɧɢɣ ɨɫɢ ɜɪɚɳɟɧɢɹ ɩɨ ɝɨɪɢɡɨɧɬɚɥɢ ɢ ɮɢɤɫɢɪɨ-
ɜɚɧɧɨɝɨ ɡɧɚɱɟɧɢɹ Y-ɤɨɨɪɞɢɧɚɬɵ ɩɨɜɬɨɪɢɬɟ ɨɩɵɬ ɞɥɹ ɬɪɟɯ ɫɤɨɪɨɫɬɟɣ ɲɚɪɢɤɚ.
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɬɨɱɧɨ ɡɚɞɚɬɶ ɜɵɛɪɚɧɧɨɟ ɡɧɚɱɟɧɢɟ X-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟ-
ɧɢɹ ɢɫɩɨɥɶɡɭɣɬɟ ɫɤɪɨɥɥɢɧɝ ɦɵɲɢ. Ɍɨɱɧɨɟ ɩɨɥɨɠɟɧɢɟ ɨɫɢ ɜɪɚɳɟɧɢɟ ɚɜɬɨɦɚɬɢɱɟɫɤɢ
ɭɤɚɡɵɜɚɟɬɫɹ ɪɹɞɨɦ ɫ ɨɛɥɚɫɬɶɸ ɷɤɫɩɟɪɢɦɟɧɬɚ ɜ ɜɢɞɟ: «Ʉɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟɧɢɹ:
X = ** ɫɦ Y = ** ɫɦ».
8. Ⱦɢɚɩɚɡɨɧ ɢɡɦɟɧɟɧɢɹ Y-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟɧɢɹ ɡɚɜɢɫɢɬ ɨɬ ɜɟɪɬɢɤɚɥɶɧɵɯ
ɪɚɡɦɟɪɨɜ ɬɟɥɚ (ɞɥɹ ɪɚɡɧɵɯ ɬɟɥ ɜɟɪɬɢɤɚɥɶɧɵɟ ɪɚɡɦɟɪɵ ɦɨɝɭɬ ɪɚɡɥɢɱɚɬɶɫɹ ɜ ɪɚɡɥɢɱ-
ɧɵɯ ɭɱɚɫɬɤɚɯ
ɬɟɥɚ). Ɋɚɡɞɟɥɢɬɟ ɞɢɚɩɚɡɨɧ ɢɡɦɟɧɟɧɢɹ Y-ɤɨɨɪɞɢɧɚɬɵ ɧɚ 5 ɪɚɜɧɵɯ ɱɚɫ-
ɬɟɣ.
Ⱦɥɹ ɤɚɠɞɨɝɨ ɢɡ ɲɟɫɬɢ ɩɨɥɭɱɟɧɧɵɯ ɩɨɥɨɠɟɧɢɣ ɨɫɢ ɜɪɚɳɟɧɢɹ ɩɨ ɜɟɪɬɢɤɚɥɢ
ɩɨɜɬɨɪɢɬɟ ɨɩɵɬ ɞɥɹ ɲɟɫɬɢ ɡɧɚɱɟɧɢɣ X-ɤɨɨɪɞɢɧɚɬɵ ɢ ɬɪɟɯ ɫɤɨɪɨɫɬɟɣ ɲɚɪɢɤɚ.
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɬɨɱɧɨ ɡɚɞɚɬɶ ɜɵɛɪɚɧɧɨɟ ɡɧɚɱɟɧɢɟ Y-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟ-
ɧɢɹ ɢɫɩɨɥɶɡɭɣɬɟ ɫɤɪɨɥɥɢɧɝ ɦɵɲɢ. Ɍɨɱɧɨɟ ɩɨɥɨɠɟɧɢɟ ɨɫɢ ɜɪɚɳɟɧɢɟ ɚɜɬɨɦɚɬɢɱɟɫɤɢ
ɭɤɚɡɵɜɚɟɬɫɹ ɪɹɞɨɦ ɫ ɨɛɥɚɫɬɶɸ ɷɤɫɩɟɪɢɦɟɧɬɚ ɜ ɜɢɞɟ: «Ʉɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟɧɢɹ:
X = ** ɫɦ Y = ** ɫɦ».
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɟɨɛɯɨɞɢɦɨ ɜɵɩɨɥɧɢɬɶ ɷɤɫɩɟɪɢɦɟɧɬ ɞɥɹ 36 ɩɨɥɨɠɟɧɢɣ ɨɫɢ
ɜɪɚɳɟɧɢɹ (ɞɥɹ ɤɚɠɞɨɣ ɢɡ 6 Y-ɤɨɨɪɞɢɧɚɬ ɜɵɛɢɪɚɸɬ 6 X-ɤɨɨɪɞɢɧɚɬ) ɢ ɬɪɟɯ ɡɧɚɱɟɧɢɣ
ɫɤɨɪɨɫɬɢ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ.
ȼɧɢɦɚɧɢɟ! Ɍɨɱɤɚ ɭɞɚɪɚ ɲɚɪɢɤɚ ɨ ɬɟɥɨ ɜɨ ɜɫɟɯ ɨɩɵɬɚɯ ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɨɣ.
ɉɨɷɬɨɦɭ ɩɪɢ ɢɡɦɟɧɟɧɢɢ Y-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɦɟɧɹɟɬɫɹ ɩɥɟɱɨ ɫɢɥɵ (ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɨɫɢ
ɞɨ ɩɪɹɦɨɣ, ɜɞɨɥɶ ɤɨɬɨɪɨɣ ɞɜɢɠɟɬɫɹ ɲɚɪɢɤ).
ɁȺɉɂɒɂɌȿ ɉɅȿɑɈ ɋɂɅɕ ȼ ɌȺȻɅɂɐɍ 1.
9. Ⱦɥɹ ɤɚɠɞɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ ɜɵɱɢɫɥɢɬɟ ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ. ɋɨɝɥɚɫ-
ɧɨ ɪɚɛɨɱɟɣ ɮɨɪɦɭɥɟ ɞɥɹ ɷɬɨɝɨ ɧɟɨɛɯɨɞɢɦɨ ɡɧɚɬɶ: ɦɚɫɫɭ ɢ ɧɚɱɚɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɲɚ-
ɪɢɤɚ, ɩɥɟɱɨ ɫɢɥɵ ɢ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ.
10. Ⱦɥɹ ɤɚɠɞɨɝɨ ɩɨɥɨɠɟɧɢɹ ɨɫɢ ɜɵɱɢɫɥɢɬɟ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ
ɦɨɦɟɧɬɚ ɢɧɟɪ-
ɰɢɢ <J> ɩɨ ɡɧɚɱɟɧɢɹɦ, ɩɨɥɭɱɟɧɧɵɦ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɣ ɫɤɨɪɨɫɬɢ ɲɚɪɢɤɚ.
ɁȺɉɂɒɂɌȿ ɉɈɅɍɑȿɇɇɕȿ ɁɇȺɑȿɇɂə ȼ ɌȺȻɅɂɐɍ 1.
Ⱦɥɹ ɭɞɨɛɫɬɜɚ ɩɨɫɬɪɨɟɧɢɹ ɝɪɚɮɢɤɨɜ ɡɚɩɨɥɧɢɬɟ ɬɚɛɥɢɰɭ 2 ɞɚɧɧɵɦɢ ɢɡ ɬɚɛɥɢ-
ɰɵ 1.
11. ɉɨɫɬɪɨɣɬɟ ɝɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɢ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɨɬ Y-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ
ɜɪɚɳɟɧɢɹ ɩɪɢ X = const. ɉɨɥɭɱɢɬɟ ɫɟɦɟɣɫɬɜɨ ɤɪɢɜɵɯ, ɦɢɧɢɦɭɦɵ ɤɨɬɨɪɵɯ ɥɟɠɚɬ ɧɚ
ɨɞɧɨɣ ɩɪɹɦɨɣ. ɋɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ
ɒɬɟɣɧɟɪɚ ɤɚɠɞɵɣ ɝɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɦɨɦɟɧɬɚ
ɢɧɟɪɰɢɢ ɨɬ Y-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɤɜɚɞɪɚɬɢɱɧɵɣ ɜɢɞ.
12. Ɇɢɧɢɦɭɦɵ ɜɫɟɯ ɲɟɫɬɢ ɤɪɢɜɵɯ ɥɟɠɚɬ ɧɚ ɨɞɧɨɣ ɩɪɹɦɨɣ. ɉɪɨɜɟɞɢɬɟ ɷɬɭ
ɩɪɹɦɭɸ ɞɨ ɩɟɪɟɫɟɱɟɧɢɹ ɫ ɨɫɶɸ ɤɨɨɪɞɢɧɚɬ, ɜɞɨɥɶ ɤɨɬɨɪɨɣ ɨɬɤɥɚɞɵɜɚɥɢ Y-
ɤɨɨɪɞɢɧɚɬɭ ɨɫɢ ɜɪɚɳɟɧɢɹ – ɩɨɥɭɱɢɬɟ Y-ɤɨɨɪɞɢɧɚɬɭ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ Y
C
.
13. ɉɨɫɬɪɨɣɬɟ ɝɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɢ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɨɬ X-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ
ɜɪɚɳɟɧɢɹ ɩɪɢ Y = const. ɉɨɥɭɱɢɬɟ ɫɟɦɟɣɫɬɜɨ ɤɪɢɜɵɯ, ɦɢɧɢɦɭɦɵ ɤɨɬɨɪɵɯ ɥɟɠɚɬ ɧɚ
ɆɨɞɆ-04 Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
49
ɨɞɧɨɣ ɩɪɹɦɨɣ. ɋɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ ɒɬɟɣɧɟɪɚ ɤɚɠɞɵɣ ɝɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɦɨɦɟɧɬɚ
ɢɧɟɪɰɢɢ ɨɬ X-ɤɨɨɪɞɢɧɚɬɵ ɨɫɢ ɜɪɚɳɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɤɜɚɞɪɚɬɢɱɧɵɣ ɜɢɞ.
14. Ɇɢɧɢɦɭɦɵ ɜɫɟɯ ɲɟɫɬɢ ɤɪɢɜɵɯ ɥɟɠɚɬ ɧɚ ɨɞɧɨɣ ɩɪɹɦɨɣ. ɉɪɨɜɟɞɢɬɟ ɷɬɭ
ɩɪɹɦɭɸ ɞɨ ɩɟɪɟɫɟɱɟɧɢɹ ɫ ɨɫɶɸ ɤɨɨɪɞɢɧɚɬ, ɜɞɨɥɶ ɤɨɬɨɪɨɣ ɨɬɤɥɚɞɵɜɚɥɢ X-
ɤɨɨɪɞɢɧɚɬɭ ɨɫɢ ɜɪɚɳɟɧɢɹ – ɩɨɥɭɱɢɬɟ X-ɤɨɨɪɞɢɧɚɬɭ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ X
C
.
15. Ⱦɥɹ ɜɫɟɯ ɩɨɥɨɠɟɧɢɣ ɨɫɢ, ɞɥɹ ɤɨɬɨɪɵɯ ɛɵɥ ɜɵɩɨɥɧɟɧ ɷɤɫɩɟɪɢɦɟɧɬ, ɩɨ
ɬɟɨɪɟɦɟ ɒɬɟɣɧɟɪɚ ɨɩɪɟɞɟɥɢɬɟ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ
2
CTC
RmJJ ! .
ȿɫɥɢ ɜ ɧɟɤɨɬɨɪɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɩɨɥɨɠɟɧɢɟ ɬɨɱɤɢ, ɱɟɪɟɡ ɤɨɬɨɪɭɸ ɩɪɨ-
ɯɨɞɢɬ ɨɫɶ ɜɪɚɳɟɧɢɹ, ɡɚɞɚɟɬɫɹ ɤɨɨɪɞɢɧɚɬɚɦɢ
O
X ɢ
O
Y , ɚ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ
– ɤɨɨɪɞɢɧɚɬɚɦɢ
C
X ɢ
C
Y , ɬɨɝɞɚ ɤɜɚɞɪɚɬ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ ɞɨ ɰɟɧɬɪɚ
ɢɧɟɪɰɢɢ ɪɚɜɟɧ
222
)()(
COCOC
YYXXR
.
Ⱦɥɹ ɤɚɠɞɨɝɨ ɨɩɵɬɚ ɜɵɱɢɫɥɢɬɟ ɤɜɚɞɪɚɬ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ ɞɨ ɰɟɧɬɪɚ
ɢɧɟɪɰɢɢ ɢ ɪɚɫɫɱɢɬɚɣɬɟ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ.
ɁȺɉɂɒɂɌȿ ɊȿɁɍɅɖɌȺɌɕ ȼ ɌȺȻɅɂɐɍ 3.
ɉɨ ɩɨɥɭɱɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ ɜɵɱɢɫɥɢɬɟ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ
ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ.
16. ȼɵɱɢɫɥɢɬɟ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɡɧɚɱɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɬɟɥɚ ɢ ɦɨ-
ɦɟɧɬɚ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɢɧɟɪɰɢɢ.
Ⱦɥɹ ɷɬɨɝɨ ɜɵɱɢɫɥɢɬɟ ɢɧɬɟɝɪɚɥɵ
³³
³³
dydx
dydxx
x
C
ɢ
³³
³³
dydx
dydxy
y
C
, ɚ ɬɚɤɠɟ
³³
U
)(
)(
22
0
1
2
)(
xf
xf
a
dyyxdxhJ
– ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ. Ɍɟɨɪɟ-
ɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ
ɢɧɟɪɰɢɢ, ɜɵɱɢɫɥɢɬɟ ɩɨ ɬɟɨɪɟɦɟ ɒɬɟɣɧɟɪɚ:
)(
222
CCTCTC
yxmJRmJJ (ɩɨ-
ɥɨɠɟɧɢɟ ɨɫɢ ɜɪɚɳɟɧɢɹ:
0
O
x
,
0
O
y
).
17. ɋɞɟɥɚɣɬɟ ɜɵɜɨɞɵ:
ɇɚɪɢɫɭɣɬɟ ɫɯɟɦɭ ɬɟɥɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɩɥɨɫɤɨɫɬɶ XOY. Ɉɬɦɟɬɶɬɟ ɧɚ ɷɬɨɣ ɫɯɟɦɟ
ɜɫɟ ɩɨɥɨɠɟɧɢɹ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɞɥɹ ɤɨɬɨɪɵɯ ɛɵɥɢ ɜɵɩɨɥɧɟɧɵ ɨɩɵɬɵ, ɢ ɩɨɥɨɠɟɧɢɟ
ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ.
əɜɥɹɟɬɫɹ ɥɢ ɜɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɪɚɜɧɨɦɟɪɧɵɦ
ɜ ɨɬɫɭɬɫɬɜɢɟ ɫɢɥ ɬɪɟɧɢɹ?
Ʉɚɤ ɡɚɜɢɫɢɬ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɨɬ ɫɤɨɪɨɫɬɢ ɲɚɪɢɤɚ, ɩɪɢɜɨɞɹɳɟɝɨ
ɬɟɥɨ ɜ ɞɜɢɠɟɧɢɟ
?
Ʉɚɤ ɡɚɜɢɫɢɬ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɨɬ ɩɨɥɨɠɟɧɢɹ ɨɫɢ ɜɪɚɳɟɧɢɹ?
Ʉɚɤ ɡɚɜɢɫɢɬ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬ ɩɥɟɱɚ ɫɢɥɵ?
ɋɪɚɜɧɢɬɟ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ
ɢɧɟɪɰɢɢ.
ɋɪɚɜɧɢɬɟ ɬɟɨɪɟɬɢɱɟɫɤɨɟ ɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɟ ɡɧɚɱɟɧɢɹ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɬɟ-
ɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ.
Ɇɟɯɚɧɢɤɚ
50
Ɍɚɛɥɢɰɚ 1
X-ɤɨɨɪɞɢɧɚɬɚ ɨɫɢ ɜɪɚɳɟ-
ɧɢɹ x
O
, ɫɦ
Y-ɤɨɨɪɞɢɧɚɬɚ ɨɫɢ ɜɪɚɳɟ-
ɧɢɹ y
O
, ɫɦ
ɉɥɟɱɨ ɫɢɥɵ
l = y
ɭɞ
– y
O
, ɫɦ
Ʉɨɥɢɱɟɫɬɜɨ ɨɛɨɪɨɬɨɜ N
ɇɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɲɚ-
ɪɢɤɚ v, ɫɦ/ɫ
ȼɪɟɦɹ ɜɪɚɳɟɧɢɹ t, ɫ
ɉɟɪɢɨɞ ɜɪɚɳɟɧɢɹ T, ɫ
ɍɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟ-
ɧɢɹ Z, ɪɚɞ/ɫ
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ J,
ɤɝ
ɫɦ
2
ɋɪɟɞɧɢɣ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ
< J >, ɤɝ
ɫɦ
2
Ɍɚɛɥɢɰɚ 2
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɢɡɜɨɥɶɧɨɣ ɨɫɢ < J >, ɤɝ
ɫɦ
2
x
O
, ɫɦ
y
O
, ɫɦ