Ревинская О.Г., Кравченко Н.С. Изучение моделей физических процессов и явлений на компьютере
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Ɇɨɞɗ-01 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ
ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
331
ɦɚɝɷɥ
FFm
dt
d
G
G
G
)( v .
ɍɱɢɬɵɜɚɹ ɧɟɪɟɥɹɬɢɜɢɫɬɫɤɢɣ ɯɚɪɚɤɬɟɪ ɞɜɢɠɟɧɢɹ, ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ
ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
][ BqEq
dt
d
m
G
G
G
G
v
v
.
ȼ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ OX, OY, OZ
ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɨɩɢɫɵɜɚɟɬɫɹ ɫɢɫɬɟɦɨɣ ɭɪɚɜɧɟɧɢɣ
°
°
°
¯
°
°
°
®
.
,
,
xyyxz
z
zxxzy
y
yzzyx
x
BB
m
q
E
m
q
dt
d
BB
m
q
E
m
q
dt
d
BB
m
q
E
m
q
dt
d
vv
v
vv
v
vv
v
Ɍɚɤ ɤɚɤ ɢ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɢ ɦɚɝɧɢɬɧɨɟ ɩɨɥɹ ɫɬɚɰɢɨɧɚɪɧɵ, ɤɚɠɞɨɟ
ɢɯ ɧɢɯ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɟɞɢɧɫɬɜɟɧɧɵɦ ɧɚɩɪɚɜɥɟɧɢɟɦ ɜɟɤɬɨɪɨɜ
E
G
ɢ B
G
,
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɉɨɥɭɱɟɧɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɦɨɠɟɬ ɛɵɬɶ ɫɭɳɟɫɬɜɟɧ-
ɧɨ ɭɩɪɨɳɟɧɚ ɫɩɟɰɢɚɥɶɧɵɦ ɜɵɛɨɪɨɦ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ XYZ, ɜ ɤɨɬɨɪɨɣ
ɧɚɩɪɚɜɥɟɧɢɟ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɢɧɞɭɤɰɢɢ
ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɟɣ ɫɨɜɩɚɞɚɸɬ ɫ ɧɚɩɪɚɜɥɟɧɢɹɦɢ ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ.
1.1. ɍɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜɨ ɜɡɚɢɦɧɨ
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
ȿɫɥɢ ɱɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɜɨ ɜɡɚɢɦɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɷɥɟɤɬɪɢɱɟ-
ɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ (
B
E
G
G
A
), ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ XYZ ɭɞɨɛɧɨ ɜɵ-
ɛɪɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. ɇɚɩɪɚɜɢɦ ɨɫɶ OZ ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɜɟɤɬɨ-
ɪɚ ɢɧɞɭɤɰɢɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ
B
G
, ɚ ɨɫɶ OY – ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɧɚɩɪɹ-
ɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ
E
G
. ɇɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɫɨɜɦɟɫɬɢɦ ɫ ɩɨɥɨ-
ɠɟɧɢɟɦ ɱɚɫɬɢɰɵ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ (ɪɢɫ. 1). ȼ ɜɵɛɪɚɧɧɨɣ
ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɜɟɤɬɨɪ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ
)0,,0( EE
G
ɢɦɟɟɬ ɬɨɥɶɤɨ ɨɞɧɭ ɨɬɥɢɱɧɭɸ ɨɬ ɧɭɥɹ ɩɪɨɟɤɰɢɸ:
EE
y
,
0
zx
EE
. ȼɟɤɬɨɪ ɦɚɝɧɢɬɧɨɣ ɢɧɞɭɤɰɢɢ
),0,0( BB
G
ɢɦɟɟɬ ɬɨɠɟ ɬɨɥɶɤɨ
ɨɞɧɭ ɨɬɥɢɱɧɭɸ ɨɬ ɧɭɥɹ ɩɪɨɟɤɰɢɸ:
BB
z
, 0
yx
BB . Ʉɨɨɪɞɢɧɚɬɵ
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
332
ɱɚɫɬɢɰɵ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɜɧɵ ɧɭɥɸ:
0)0()0()0(
z
y
x
.
Ɍɨɝɞɚ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ, ɨɩɢɫɵɜɚɸɳɚɹ ɞɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ
ɱɚɫɬɢɰɵ ɦɚɫɫɨɣ
m
ɢ ɡɚɪɹɞɨɦ
q
ɜɨ ɜɡɚɢɦɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɷɥɟɤɬɪɢ-
ɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ, ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ XYZ ɩɪɢ-
ɦɟɬ ɜɢɞ
°
°
°
¯
°
°
°
®
.0
,
,
dt
d
B
m
q
E
m
q
dt
d
B
m
q
dt
d
z
x
y
y
x
v
v
v
v
v
ɂɡ ɭɪɚɜɧɟɧɢɹ ɞɥɹ
z
v ɜɢɞ-
ɧɨ, ɱɬɨ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɜɞɨɥɶ
ɧɚɩɪɚɜɥɟɧɢɹ OZ ɧɟ ɢɡɦɟɧɹɟɬɫɹ
ɫɨ ɜɪɟɦɟɧɟɦ. ȿɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ
ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ Z-ɤɨɦɩɨɧɟɧɬɚ
ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ
)0(
z
v ɛɵɥɚ
ɪɚɜɧɚ ɧɭɥɸ, ɬɨ ɞɜɢɠɟɧɢɟ ɛɭɞɟɬ
ɩɪɨɢɫɯɨɞɢɬɶ ɜ ɩɥɨɫɤɨɫɬɢ XOY.
ȿɫɥɢ ɜ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɟ ɨɬ-
ɫɱɟɬɚ ɱɚɫɬɢɰɚ ɢɦɟɟɬ ɨɬɥɢɱɧɭɸ
ɨɬ ɧɭɥɹ Z-ɤɨɦɩɨɧɟɧɬɭ ɫɤɨɪɨɫɬɢ
ɱɚɫɬɢɰɵ
)0(
z
v , ɦɨɠɧɨ ɩɟɪɟɣɬɢ ɤ
ɞɪɭɝɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɞɜɢɠɭɳɟɣɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɟɪ-
ɜɨɣ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ, ɪɚɜɧɨɣ
)0(
z
v
G
. Ɍɨɝɞɚ ɜ ɧɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨ-
ɨɪɞɢɧɚɬ ɱɚɫɬɢɰɚ ɛɭɞɟɬ ɢɦɟɬɶ ɫɤɨɪɨɫɬɶ ɜ ɧɚɩɪɚɜɥɟɧɢɢ OZ
)0(
z
v ɪɚɜɧɭɸ
ɧɭɥɸ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜɨ ɜɡɚɢɦɧɨ ɩɟɪ-
ɩɟɧɞɢɤɭɥɹɪɧɵɯ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɢɦ.
Ⱦɥɹ ɞɚɥɶɧɟɣɲɟɝɨ ɢɡɭɱɟɧɢɹ ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ
ɜɵɛɪɚɧɚ ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨ Z-ɤɨɦɩɨɧɟɧɬɚ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ
0)0(
z
v ɪɚɜɧɚ ɧɭɥɸ. Ɍɨɝɞɚ ɬɪɚɟɤɬɨɪɢɹ ɱɚɫɬɢɰɵ ɛɭɞɟɬ ɥɟɠɚɬɶ ɜ ɩɥɨɫ-
ɤɨɫɬɢ XOY ɢ ɨɩɢɫɵɜɚɬɶɫɹ ɫɢɫɬɟɦɨɣ ɭɪɚɜɧɟɧɢɣ
E
B
v
(
0
)
x
y
z
O
v
x
(0)
v
y
(0)
v
z
(0)
Ɋɢɫ. 1
Ɇɨɞɗ-01 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ
ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
333
°
°
¯
°
°
®
,
,
B
m
q
E
m
q
dt
d
B
m
q
dt
d
x
y
y
x
v
v
v
v
ɢɥɢ
°
°
¯
°
°
®
.
,
)/(
1
B
m
q
E
m
q
dt
d
dt
d
Bmq
x
y
y
x
v
v
v
v
ɑɬɨɛɵ ɪɟɲɢɬɶ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ, ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦ ɩɟɪɜɨɟ
ɭɪɚɜɧɟɧɢɟ ɩɨ ɜɪɟɦɟɧɢ ɢ ɩɨɞɫɬɚɜɢɦ ɜɨ ɜɬɨɪɨɟ
B
m
q
E
m
q
dt
d
dt
d
Bmq
x
y
x
v
v
v
2
2
)/(
1
,
ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɞɥɹ ɨɞɧɨɣ ɧɟ-
ɢɡɜɟɫɬɧɨɣ
x
v
BE
m
q
B
m
q
dt
d
x
x
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
22
2
2
v
v
ɢɥɢ
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
B
E
B
m
q
dt
d
x
x
v
v
2
2
2
.
ȼɜɟɞɟɦ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɵɯ
B
E
u
xx
v . Ɍɨɝɞɚ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ
ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ
x
x
uB
m
q
dt
ud
2
2
2
¸
¹
·
¨
©
§
. ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɚɧɚɥɨɝɢɱɧɨ
ɭɪɚɜɧɟɧɢɸ, ɨɩɢɫɵɜɚɸɳɟɦɭ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɤɨɥɟɛɚɧɢɹ. ɉɨɷɬɨɦɭ ɪɟɲɟ-
ɧɢɟ ɦɨɠɧɨ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
)sin( TZ tAu
x
, ɝɞɟ
A
, Z, T – ɢɫɤɨɦɵɟ ɤɨɧ-
ɫɬɚɧɬɵ. ɉɨɞɫɬɚɜɢɜ
x
u ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɢɦ
xx
x
uB
m
q
utA
dt
ud
2
22
2
2
)sin(
¸
¹
·
¨
©
§
Z TZZ
.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
B
m
q
Z .
ȼɟɪɧɭɜɲɢɫɶ ɤ ɩɟɪɟɦɟɧɧɨɣ
B
E
u
xx
v ɢ ɭɱɢɬɵɜɚɹ, ɱɬɨ
dt
d
dt
d
Bmq
xx
y
vv
v
Z
1
)/(
1
, ɩɨɥɭɱɢɦ
B
E
tA
x
TZ )sin(v
,
)cos( TZ tA
y
v .
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
334
Ɂɧɚɱɟɧɢɹ ɤɨɧɫɬɚɧɬ
A
ɢ T ɧɚɣɞɟɦ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ. ȼ ɦɨ-
ɦɟɧɬ ɜɪɟɦɟɧɢ
0 t ɱɚɫɬɢɰɚ ɢɦɟɥɚ ɫɤɨɪɨɫɬɢ )0(
x
v ɢ
)0(
y
v
:
B
E
A
x
T sin)0(v ɢ T cos)0( A
y
v ,
ɢɥɢ
B
E
A
x
T )0(sin v ɢ
)0(cos
y
A v T
.
Ɍɨɝɞɚ ɢɡ ɨɬɧɨɲɟɧɢɹ ɷɬɢɯ ɭɪɚɜɧɟɧɢɣ
)0(
)0(
tg
y
x
B
E
v
v
T
,
ɚ ɢɡ ɫɭɦɦɵ ɤɜɚɞɪɚɬɨɜ
2
2
2
)0()0(
yx
B
E
A vv
¸
¹
·
¨
©
§
.
ɉɪɨɢɧɬɟɝɪɢɪɨɜɚɜ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɫɤɨɪɨɫɬɢ, ɩɨɥɭɱɢɦ ɡɚɜɢɫɢɦɨɫɬɶ
ɤɨɨɪɞɢɧɚɬ
x
ɢ
y
ɨɬ ɜɪɟɦɟɧɢ
t
t
B
E
t
A
Ctx
x
TZ
Z
)cos()( ,
)sin()( TZ
Z
t
A
Cty
y
.
ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɱɚɫɬɢɰɚ ɧɚɯɨɞɢɥɚɫɶ ɜ ɧɚɱɚɥɟ ɤɨɨɪ-
ɞɢɧɚɬ
0)0(
x
,
0)0( y
,
0)0(
z
. ɉɨɷɬɨɦɭ
)0(
1
cos
yx
A
C v
Z
T
Z
ɢ
¸
¹
·
¨
©
§
Z
T
Z
B
EA
C
xy
)0(
1
sin v
.
ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɟ
Z
A
R
. Ɍɨɝɞɚ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɦɨɠ-
ɧɨ ɡɚɩɢɫɚɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ
t
B
E
tRtx
y
TZ
Z
)cos()0(
1
)( v ,
)sin()0(
1
)( TZ
¸
¹
·
¨
©
§
Z
tR
B
E
ty
x
v ,
0)( t
z
.
1.2. Ⱥɧɚɥɢɡ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ
Ⱦɥɹ ɚɧɚɥɢɡɚ ɯɚɪɚɤɬɟɪɚ ɬɪɚɟɤɬɨɪɢɢ ɩɪɨɜɟɞɟɦ ɫɥɟɞɭɸɳɢɟ ɩɪɟɨɛɪɚ-
ɡɨɜɚɧɢɹ. ɉɨɥɭɱɟɧɧɵɟ ɡɚɤɨɧɵ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ
Ɇɨɞɗ-01 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ
ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
335
t
B
E
tRCx
x
TZ )cos(
,
)sin( TZ tRCy
y
.
ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ
)cos( TZ
¸
¹
·
¨
©
§
tRt
B
E
Cx
x
,
)sin( TZ tRCy
y
.
Ɉɛɚ ɭɪɚɜɧɟɧɢɹ ɜɨɡɜɟɞɟɦ ɜ ɤɜɚɞɪɚɬ ɢ ɫɥɨɠɢɦ
))(sin)((cos
222
2
2
TZTZ
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
ttRCyt
B
E
Cx
yx
,
2
2
2
RCyt
B
E
Cx
yx
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
ɢɥɢ
2
22
Ryyxx
RR
,
ɝɞɟ
t
B
E
Cx
xR
,
yR
Cy .
ɉɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ
ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɨɤɪɭɠ-
ɧɨɫɬɢ ɫ ɰɟɧɬɪɨɦ ɜ ɬɨɱɤɟ
(x
R
, y
R
) ɢ ɪɚɞɢɭɫɨɦ R. ɂɡ ɹɜ-
ɧɨɝɨ ɜɢɞɚ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ
ɨɤɪɭɠɧɨɫɬɢ (x
R
, y
R
) ɫɥɟɞɭɟɬ,
ɱɬɨ ɰɟɧɬɪ ɨɤɪɭɠɧɨɫɬɢ ɪɚɜ-
ɧɨɦɟɪɧɨ ɞɜɢɠɟɬɫɹ ɜɞɨɥɶ ɨɫɢ
OX (ɪɢɫ. 2).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɜɢɠɟ-
ɧɢɟ ɱɚɫɬɢɰɵ ɩɪɟɞɫɬɚɜɥɹɟɬ
ɫɭɩɟɪɩɨɡɢɰɢɸ ɜɪɚɳɚɬɟɥɶɧɨ-
ɝɨ ɞɜɢɠɟɧɢɹ ɜ ɩɥɨɫɤɨɫɬɢ XOY ɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɜ ɧɚɩɪɚɜɥɟ-
ɧɢɢ ɨɫɢ OX.
ɉɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ:
ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ
B
E
ɩɨɫɬ
v .
ȼɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ:
ɩɨɫɬɨɹɧɧɨɣ ɱɚɫɬɨɬɨɣ
B
m
q
Z
,
R
x
y
C
x
C
y
T
v(0)
v
ɩɨɫɬ
Ɋɢɫ. 2
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
336
ɪɚɞɢɭɫɨɦ
2
2
)0()0(
1
yx
B
E
R vv
¸
¹
·
¨
©
§
Z
,
ɧɚɱɚɥɶɧɨɣ ɮɚɡɨɣ ɜɪɚɳɟɧɢɹ
T , ɞɥɹ ɤɨɬɨɪɨɣ
)0(
)0(
tg
y
x
B
E
v
v
T
,
ɢ ɩɨɥɨɠɟɧɢɟɦ ɰɟɧɬɪɚ ɜɪɚɳɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢ-
ɧɚɬ XOY:
)0(
1
yx
C v
Z
,
¸
¹
·
¨
©
§
Z
B
E
C
xy
)0(
1
v .
Ʌɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɪɚɜɧɚ
A
B
E
R
yxɜɪ
¸
¹
·
¨
©
§
Z
2
2
)0()0( vvv
.
Ɍɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɞɜɭɦɟɪɧɭɸ
ɤɪɢɜɭɸ, ɤɨɬɨɪɚɹ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢ ɨɩɢɫɚɧɚ ɩɨɥɭɱɟɧɧɵɦɢ ɡɚɤɨɧɚɦɢ ɞɜɢ-
ɠɟɧɢɹ, ɢ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɧɚɡɵɜɚɟɬɫɹ
ɬɪɨɯɨɢɞɨɣ.
ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ
E
ɢ ɢɧɞɭɤɰɢɢ
ɦɚɝɧɢɬɧɨɝɨ
B
ɩɨɥɟɣ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɫɤɨɪɨɫɬɹɦɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ
ɩɨɫɬ
v ɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ
ɜɪ
v ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɢɱɧɵɦ
(ɪɢɫ. 3).
Ɋɚɫɫɦɨɬɪɢɦ ɜɨɡɦɨɠɧɵɟ ɫɥɭɱɚɢ.
ȿɫɥɢ ɫɤɨɪɨɫɬɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ
ɩɨɫɬ
v ɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ
ɜɪ
v
ɞɜɢ-
ɠɟɧɢɹ ɪɚɜɧɵ
ɜɪɩɨɫɬ
vv , ɬɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɰɢɤɥɨɢɞɨɣ
(ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɬɪɨɯɨɢɞɵ) (ɪɢɫ. 3ɚ). Ɍɚɤɨɟ ɞɜɢɠɟɧɢɟ ɜɨɡɦɨɠɧɨ ɧɚɛɥɸ-
ɞɚɬɶ, ɧɚɩɪɢɦɟɪ, ɤɨɝɞɚ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɪɚɜɧɚ ɧɭɥɸ:
0)0(
x
v ,
0)0(
y
v
2
2
)0()0(
yx
B
E
B
E
vv
¸
¹
·
¨
©
§
ɜɪɩɨɫɬ
vv
.
ȿɫɥɢ ɫɤɨɪɨɫɬɶ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ
ɩɨɫɬ
v ɞɜɢɠɟɧɢɹ ɦɟɧɶɲɟ ɫɤɨɪɨɫɬɢ
ɜɪɚɳɚɬɟɥɶɧɨɝɨ
ɜɪ
v ɞɜɢɠɟɧɢɹ
ɜɪɩɨɫɬ
vv , ɬɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɧɚɡɵɜɚ-
ɟɬɫɹ
ɬɪɨɯɨɢɞɨɣ ɢ ɢɦɟɟɬ ɩɟɬɥɢ (ɪɢɫ. 3ɛ).
ɜɪɩɨɫɬ
vv
2
2
)0()0(
yx
B
E
B
E
vv
¸
¹
·
¨
©
§
.
Ɍɚɤɨɟ ɞɜɢɠɟɧɢɟ ɜɨɡɦɨɠɧɨ, ɧɚɩɪɢɦɟɪ, ɤɨɝɞɚ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ
ɱɚɫɬɢɰɵ ɧɚɩɪɚɜɥɟɧɚ ɜɞɨɥɶ ɨɫɢ OY (
0)0(
x
v ,
0)0( z
y
v
) ɢɥɢ ɧɚɱɚɥɶɧɚɹ
Ɇɨɞɗ-01 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ
ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
337
ɫɤɨɪɨɫɬɶ ɢɦɟɟɬ ɧɚɩɪɚɜɥɟɧɢɟ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɟ ɨɫɢ OX (
0)0(
x
v ,
0)0(
y
v ).
ȿɫɥɢ ɫɤɨɪɨɫɬɶ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ
ɩɨɫɬ
v ɞɜɢɠɟɧɢɹ ɛɨɥɶɲɟ ɫɤɨɪɨɫɬɢ
ɜɪɚɳɚɬɟɥɶɧɨɝɨ
ɜɪ
v ɞɜɢɠɟɧɢɹ
ɜɪɩɨɫɬ
vv ! , ɬɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɬɚɤɠɟ
ɧɚɡɵɜɚɟɬɫɹ
ɬɪɨɯɨɢɞɨɣ, ɧɨ ɩɟɬɟɥɶ ɧɟ ɢɦɟɟɬ (ɪɢɫ. 3ɜ).
x
y
O
a
x
y
O
ɛ
x
y
O
ɜ
x
y
O
ɝ
Ɋɢɫ. 3
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
338
ɜɪɩɨɫɬ
vv !
2
2
)0()0(
yx
B
E
B
E
vv
¸
¹
·
¨
©
§
!
.
Ɍɚɤɨɟ ɞɜɢɠɟɧɢɟ ɜɨɡɦɨɠɧɨ, ɧɚɩɪɢɦɟɪ, ɤɨɝɞɚ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ
ɱɚɫɬɢɰɵ ɢɦɟɟɬ ɧɚɩɪɚɜɥɟɧɢɟ, ɫɨɜɩɚɞɚɸɳɟɟ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɩɨɫɬɭɩɚ-
ɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ (ɜɞɨɥɶ ɨɫɢ OX
0)0(
y
v ), ɚ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɧɟɡɧɚ-
ɱɢɬɟɥɶɧɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɦɨɞɭɥɹ ɫɤɨɪɨɫɬɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ
B
E
x
|)0(v , ɧɨ ɧɟ ɪɚɜɟɧ ɟɦɭ.
ȿɫɥɢ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɢɦɟɟɬ ɬɨ ɠɟ ɧɚɩɪɚɜɥɟɧɢɟ, ɱɬɨ ɢ
ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ (ɜɞɨɥɶ ɨɫɢ OX
0)0(
y
v
), ɚ ɟɟ ɦɨɞɭɥɶ ɪɚɜɟɧ
ɦɨɞɭɥɸ ɫɤɨɪɨɫɬɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ
B
E
x
)0(v
, ɬɨ ɪɚɞɢɭɫ ɜɪɚ-
ɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ. ɑɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ
ɪɚɜɧɨɦɟɪ-
ɧɨ ɢ ɩɪɹɦɨɥɢɧɟɣɧɨ
ɜɞɨɥɶ ɨɫɢ OX (ɪɢɫ. 3ɝ).
2. Ɋɚɛɨɱɢɟ ɮɨɪɦɭɥɵ
Ⱦɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɹɜɥɹɟɬɫɹ ɩɟɪɢɨɞɢɱɟɫɤɢɦ ɫ ɩɟɪɢɨɞɨɦ
Bmq
T
)/(
22 S
Z
S
. ȼ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ OX ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɨɩɢɫɵɜɚ-
ɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ
t
B
E
tRCtx
x
TZ )cos()(. Ɂɚ ɩɟɪɢɨɞ ɜɪɚɳɟɧɢɹ
T
ɱɚɫɬɢɰɚ ɫɨɜɟɪɲɚɟɬ ɩɨɥɧɵɣ ɨɛɨɪɨɬ ɢ ɫɦɟɳɚɟɬɫɹ ɩɨ ɨɫɢ OX ɧɚ ɪɚɫɫɬɨɹ-
ɧɢɟ
T
B
E
x '
, ɩɪɨɣɞɹ ɩɪɢ ɷɬɨɦ ɰɟɥɵɣ ɜɢɬɨɤ ɬɪɨɯɨɢɞɵ (ɰɢɤɥɨɢɞɵ). Ɂɚ
n
ɩɟɪɢɨɞɨɜ ɱɚɫɬɢɰɚ ɜɞɨɥɶ ɨɫɢ OX ɫɦɟɫɬɢɬɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɟ
nT
B
E
x ' ,
ɩɪɢ ɷɬɨɦ
y
-ɤɨɨɪɞɢɧɚɬɚ ɱɚɫɬɢɰɵ ɨɤɚɠɟɬɫɹ ɬɚɤɨɣ ɠɟ, ɤɚɤ
n
ɩɟɪɢɨɞɨɜ ɧɚ-
ɡɚɞ. ɂɡɦɟɪɢɜ ɞɥɢɧɭ
n
ɰɟɥɵɯ ɜɢɬɤɨɜ
x
'
ɬɪɨɯɨɢɞɵ (ɰɢɤɥɨɢɞɵ) ɜɞɨɥɶ ɨɫɢ
OX, ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɟɪɢɨɞ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ
En
Bx
T
'
, ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɱɚɫɬɨɬɭ Z ɢ ɨɬɧɨɲɟɧɢɟ ɡɚɪɹɞɚ ɱɚɫɬɢɰɵ ɤ ɟɟ
ɦɚɫɫɟ
BTm
q S
2
(ɭɞɟɥɶɧɵɣ ɡɚɪɹɞ).
Ɇɨɞɗ-01 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ
ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
339
ȼ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ OY ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɨɩɢɫɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢ-
ɟɦ
)sin()( TZ tRCty
y
ɢ ɹɜɥɹɟɬɫɹ ɨɝɪɚɧɢɱɟɧɧɵɦ. Ʉɨɨɪɞɢɧɚɬɚ y
ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ ɨɬ
RCy
y
min
(ɤɨɝɞɚ 1sin ) ɞɨ RCy
y
max
(ɤɨɝɞɚ
1sin ). ɂɡɦɟɪɢɜ
min
y ɢ
max
y , ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɪɚɞɢɭɫ ɜɪɚɳɟ-
ɧɢɹ
)(
minmax
2
1
yyR
.
Ɂɚɜɢɫɢɦɨɫɬɶ ɪɚɞɢɭɫɚ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɨɬ ɧɚ-
ɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɧɨɫɢɬ ɫɥɨɠɧɵɣ ɯɚɪɚɤɬɟɪ
2
2
)0()0(
1
yx
B
E
R vv
¸
¹
·
¨
©
§
Z
. Ɉɞɧɚɤɨ, ɜɨɡɦɨɠɧɵ ɞɜɚ ɫɥɭɱɚɹ, ɤɨɝɞɚ ɷɬɚ
ɡɚɜɢɫɢɦɨɫɬɶ ɡɧɚɱɢɬɟɥɶɧɨ ɭɩɪɨɳɚɟɬɫɹ:
x ɫɤɨɪɨɫɬɶ ɧɚɩɪɚɜɥɟɧɚ ɜɞɨɥɶ ɨɫɢ OX (
0)0(
y
v
):
B
E
R
x
Z
)0(
1
v ;
x ɫɤɨɪɨɫɬɶ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ OX
B
E
x
)0(v
( 0)0( z
y
v ): )0(
1
y
R v
Z
;
ȼ ɨɛɨɢɯ ɫɥɭɱɚɹɯ ɡɚɜɢɫɢɦɨɫɬɢ ɪɚɞɢɭɫɚ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɨɬ
ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ
))0((
x
RR v ɢ ))0((
y
RR v ɧɨɫɹɬ ɤɭɫɨɱ-
ɧɨ-ɥɢɧɟɣɧɵɣ ɯɚɪɚɤɬɟɪ. Ɇɨɞɭɥɶ ɬɚɧɝɟɧɫɚ ɭɝɥɚ ɧɚɤɥɨɧɚ ɤɚɠɞɨɝɨ ɨɬɪɟɡɤɚ
ɩɪɹɦɨɣ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ
Z:
Z
D
1
tg
. ɑɬɨ ɬɚɤɠɟ ɩɨɡɜɨɥɹɟɬ ɨɩ-
ɪɟɞɟɥɢɬɶ ɨɬɧɨɲɟɧɢɟ ɡɚɪɹɞɚ ɱɚɫɬɢɰɵ ɤ ɟɟ ɦɚɫɫɟ
m
q
.
3. Ɇɨɞɟɥɶ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɣ ɭɫɬɚɧɨɜɤɢ
ȼ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɫ ɩɨɦɨɳɶɸ ɫɪɟɞɫɬɜ ɤɨɦɩɶɸɬɟɪɧɨɣ ɝɪɚɮɢɤɢ ɦɨ-
ɞɟɥɢɪɭɟɬɫɹ ɞɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜɨ ɜɡɚɢɦɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪ-
ɧɵɯ ɨɞɧɨɪɨɞɧɵɯ ɫɬɚɰɢɨɧɚɪɧɵɯ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ. ɇɚ-
ɩɪɹɠɟɧɧɨɫɬɶ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ ɦɨɠɧɨ ɢɡɦɟɧɹɬɶ ɜ ɢɧɬɟɪɜɚɥɟ ɨɬ –200
ɞɨ 200 ȼ/ɫɦ, ɢɧɞɭɤɰɢɸ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ – ɜ ɢɧɬɟɪɜɚɥɟ ɨɬ 0 ɞɨ 200 ɦɌɥ.
ɇɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɡɚɞɚɟɬɫɹ ɩɭɬɟɦ ɢɡɦɟɧɟɧɢɹ ɟɟ ɩɪɨɟɤɰɢɣ
ɧɚ
ɨɫɢ OX ɢ OY. ɉɪɨɟɤɰɢɹ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɧɚ ɨɫɶ OZ ɜɫɟɝɞɚ ɫɱɢɬɚɟɬɫɹ
ɪɚɜɧɨɣ ɧɭɥɸ. Ɉɫɶ OZ ɢ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɧɚɩɪɚɜɥɟɧɵ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ
ɩɥɨɫɤɨɫɬɢ ɷɤɫɩɟɪɢɦɟɧɬɚ.
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
340
ɑɚɫɬɢɰɚ ɧɚɱɢɧɚɟɬ ɫɜɨɟ ɞɜɢɠɟɧɢɟ ɢɡ ɧɚɱɚɥɚ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ,
ɪɚɫɩɨɥɨɠɟɧɧɨɝɨ ɜ ɰɟɧɬɪɟ ɨɛɥɚɫɬɢ ɷɤɫɩɟɪɢɦɟɧɬɚ. ȼɨ ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ
ɱɚɫɬɢɰɚ ɨɫɬɚɜɥɹɟɬ ɫɥɟɞ ɧɚ ɩɥɨɫɤɨɫɬɢ ɷɤɫɩɟɪɢɦɟɧɬɚ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɩɪɨ-
ɜɨɞɢɬɶ ɢɡɦɟɪɟɧɢɹ. Ⱦɥɹ ɩɨɜɵɲɟɧɢɹ ɬɨɱɧɨɫɬɢ ɢɡɦɟɪɟɧɢɣ ɜ ɪɚɛɨɬɟ ɫɭɳɟ-
ɫɬɜɭɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɭɜɟɥɢɱɟɧɢɹ ɦɚɫɲɬɚɛɚ ɨɬɨɛɪɚɠɟɧɢɹ ɨɛɥɚɫɬɢ ɷɤɫɩɟ-
ɪɢɦɟɧɬɚ.
Ⱦɥɹ ɢɡɦɟɪɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɱɚɫɬɢɰɵ ɜ ɪɚɛɨɬɟ ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɜɟ ɥɢ-
ɧɟɣɤɢ, ɰɟɧɚ ɞɟɥɟɧɢɹ ɤɨɬɨɪɵɯ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɦɚɫɲɬɚɛɚ ɢɡɨɛɪɚɠɟɧɢɹ
ɦɨɠɟɬ ɢɡɦɟɧɹɬɶɫɹ ɨɬ 1 ɞɨ 0,2 ɫɦ. ɉɪɢ ɭɤɚɡɚɧɧɵɯ ɭɫɥɨɜɢɹɯ ɩɨɝɪɟɲɧɨɫɬɶ
ɨɩɪɟɞɟɥɟɧɢɹ ɨɬɧɨɲɟɧɢɹ ɡɚɪɹɞɚ ɱɚɫɬɢɰɵ ɤ ɟɟ ɦɚɫɫɟ ɜ ɷɤɫɩɟɪɢɦɟɧɬɟ ɧɟ
ɩɪɟɜɵɲɚɟɬ 0,3–0,9%.
Ɋɚɛɨɬɚ ɜɵɩɨɥɧɹɟɬɫɹ ɧɚ IBM-ɫɨɜɦɟɫɬɢɦɨɦ ɩɟɪɫɨɧɚɥɶɧɨɦ ɤɨɦɩɶɸ-
ɬɟɪɟ ɜ ɜɢɞɟ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ Windows-ɩɪɢɥɨɠɟɧɢɹ. Ⱦɥɹ
ɭɞɨɛɫɬɜɚ ɜɵ-
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.