Ревинская О.Г., Кравченко Н.С. Изучение моделей физических процессов и явлений на компьютере
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ɆɨɞɆ-07 Ⱦɜɢɠɟɧɢɟ ɢɧɟɪɬɧɨɝɨ ɬɟɥɚ ɜ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ ɩɨɥɟ
101
ɞɟɣɫɬɜɭɟɬ ɧɚ ɬɟɥɨ
2
m ɫ ɫɢɥɨɣ (
F
G
). ɍɱɢɬɵɜɚɹ ɨɛɳɢɣ ɜɢɞ ɰɟɧɬɪɚɥɶɧɨɣ
ɫɢɥɵ, ɡɚɩɢɲɟɦ ɜɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ (ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ) ɞɥɹ ɤɚɠɞɨ-
ɝɨ ɢɡ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɬɟɥ:
F
dt
rd
m
G
G
2
1
2
1
, F
dt
rd
m
G
G
2
2
2
2
,
ɝɞɟ
1
r
G
ɢ
2
r
G
– ɪɚɞɢɭɫ-ɜɟɤɬɨɪɵ ɜɡɚɢɦɨ-
ɞɟɣɫɬɜɭɸɳɢɯ ɬɟɥ ɜ ɩɪɨɢɡɜɨɥɶɧɨɣ ɞɟ-
ɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ (ɪɢɫ. 1),
ɩɪɢɱɟɦ
21
rrr
G
G
G
. Ɍɟɥɨ
1
m ɢɦɟɟɬ ɫɤɨ-
ɪɨɫɬɶ
1
v
G
, ɬɟɥɨ
2
m ɢɦɟɟɬ ɫɤɨɪɨɫɬɶ
2
v
G
.
Ɋɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɫɢɫɬɟɦɚ ɹɜɥɹ-
ɟɬɫɹ ɡɚɦɤɧɭɬɨɣ, ɬɚɤ ɤɚɤ ɧɚ ɧɟɟ ɧɟ ɞɟɣ-
ɫɬɜɭɸɬ ɞɪɭɝɢɟ ɫɢɥɵ, ɤɪɨɦɟ ɰɟɧɬɪɚɥɶ-
ɧɵɯ. ɐɟɧɬɪɚɥɶɧɚɹ ɫɢɥɚ
r
r
rFF
r
G
G
)(
ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɪɚɫɩɨɥɨɠɟɧɢɹ ɬɟɥ
r
G
. ɉɨɷɬɨɦɭ ɧɚ ɨɫ-
ɧɨɜɟ ɢɦɟɸɳɟɣɫɹ ɫɢɫɬɟɦɵ ɞɜɭɯ ɭɪɚɜɧɟɧɢɣ ɩɨɥɭɱɢɦ ɤɢɧɟɦɚɬɢɱɟɫɤɨɟ
ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɬɟɥ. Ⱦɥɹ ɷɬɨɝɨ ɩɨɞɟɥɢɦ ɤɚɠɞɨɟ ɢɡ
ɭɪɚɜɧɟɧɢɣ ɇɶɸɬɨɧɚ ɧɚ ɦɚɫɫɭ ɢ ɜɵɱɬɟɦ ɨɞɧɨ ɢɡ ɞɪɭɝɨɝɨ
r
r
rF
mm
dt
rrd
r
G
G
G
)(
11
)(
21
2
21
2
¸
¸
¹
·
¨
¨
©
§
;
r
r
rF
dt
rd
M
r
G
G
)(
2
2
.
ɝɞɟ
21
21
mm
mm
M
ɧɚɡɵɜɚɸɬ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɨɣ.
ɉɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɦɨɠɧɨ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɤɚɤ ɞɜɢɠɟɧɢɟ
ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ
M
ɜ ɩɨɥɟ ɰɟɧɬɪɚɥɶɧɨɣ ɫɢɥɵ
F
G
: ɬɟɥɨ ɦɚɫɫɨɣ
M
ɪɚɫɩɨɥɨɠɟɧɨ ɧɚ ɪɚɫɫɬɨɹɧɢɢ
r
G
ɨɬ ɢɫɬɨɱɧɢɤɚ ɩɨɥɹ ɢ ɞɜɢɠɟɬɫɹ ɫɨ ɫɤɨ-
ɪɨɫɬɶɸ
v
G
, ɪɚɜɧɨɣ ɫɤɨɪɨɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɬɟɥ
21
vvv
G
G
G
.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɚɞɚɱɚ ɨ ɞɜɢɠɟɧɢɢ ɞɜɭɯ ɬɟɥ ɜ ɮɢɡɢɤɟ ɫɜɨɞɢɬɫɹ ɤ
ɡɚɞɚɱɟ ɨ ɞɜɢɠɟɧɢɢ ɨɞɧɨɝɨ ɬɟɥɚ.
Ɋɚɫɫɦɨɬɪɢɦ, ɤɚɤ ɦɨɠɧɨ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ
ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɫɨɨɬɧɨɲɟɧɢɹɯ ɦɚɫɫ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɬɟɥ. ȿɫɥɢ ɦɚɫɫɚ
ɬɟɥɚ
2
m ɡɧɚɱɢɬɟɥɶɧɨ ɩɪɟɜɨɫɯɨɞɢɬ ɦɚɫɫɭ ɬɟɥɚ
1
m (
12
mm !! ), ɬɨ
x
y
z
r
r
1
r
2
m
1
m
2
v
1
v
2
Ɋɢɫ. 1
Ɇɟɯɚɧɢɤɚ
102
1
2
21
21
21
m
m
mm
mm
mm
M |
1
mM | .
ɉɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ ɩɪɟɨɛ-
ɪɚɡɭɟɬɫɹ ɜ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɦɚɫɫɨɣ
1
m ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɟɥɚ ɦɚɫ-
ɫɨɣ
2
m (ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ ɫɜɹɡɵɜɚɟɬɫɹ ɫ ɬɟɥɨɦ ɦɚɫɫɨɣ
2
m ):
r
r
rF
dt
rd
M
r
G
G
)(
2
2
r
r
rF
dt
rd
m
r
G
G
)(
2
2
1
.
Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɟɫɥɢ
21
mm !! , ɬɨ ɭɪɚɜɧɟɧɢɟ
ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɜ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ
ɬɟɥɚ ɦɚɫɫɨɣ
2
m ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɟɥɚ ɦɚɫɫɨɣ
1
m (ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ ɫɜɹɡɵɜɚ-
ɟɬɫɹ ɫ ɬɟɥɨɦ ɦɚɫɫɨɣ
1
m ). Ɍɨ ɟɫɬɶ ɬɟɥɨ ɛɨɥɶɲɟɣ ɦɚɫɫɵ ɜɵɫɬɭɩɚɟɬ ɜ ɤɚɱɟ-
ɫɬɜɟ ɢɫɬɨɱɧɢɤɚ ɩɨɥɹ, ɜ ɤɨɬɨɪɨɦ ɞɜɢɠɟɬɫɹ ɬɟɥɨ ɦɟɧɶɲɟɣ ɦɚɫɫɵ.
1.2. Ⱦɜɢɠɟɧɢɟ ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ ɜ ɩɨɥɟ
ɰɟɧɬɪɚɥɶɧɨɣ ɫɢɥɵ
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɢɦɟɟɦ ɞɜɢɠɟɧɢɟ ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ
M
,
ɨɩɢɫɵɜɚɟɦɨɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɦ ɭɪɚɜɧɟɧɢɟɦ
r
r
rF
dt
rd
M
r
GG
)(
2
2
,
ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɨɦ
r
G
ɢ ɫɤɨɪɨɫɬɶɸ
v
G
. ɇɚ ɬɟɥɨ ɩɪɢɜɟɞɟɧ-
ɧɨɣ ɦɚɫɫɵ ɞɟɣɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɚɹ ɫɢɥɚ
F
G
, ɹɜɥɹɸɳɚɹɫɹ ɰɟɧɬɪɚɥɶɧɨɣ.
ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ, ɤɚɤɢɦ ɛɭɞɟɬ ɞɜɢɠɟɧɢɟ ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ ɜ ɷɬɢɯ
ɭɫɥɨɜɢɹɯ.
1.
Ⱦɜɢɠɟɧɢɟ ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ ɜ ɩɨɥɟ ɰɟɧɬɪɚɥɶɧɨɣ ɫɢɥɵ
ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɢɦ, ɬɚɤ ɤɚɤ ɦɨɦɟɧɬ ɷɬɨɣ ɫɢɥɵ ɪɚɜɟɧ ɧɭɥɸ.
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɦɨɦɟɧɬ ɫɢɥɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɜɟɤɬɨɪɧɨɟ ɩɪɨɢɡ-
ɜɟɞɟɧɢɟ ɫɢɥɵ
F
G
ɧɚ ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɜ ɬɨɱɤɟ ɟɟ ɩɪɢɥɨɠɟɧɢɹ
r
G
. Ⱦɥɹ ɰɟɧ-
ɬɪɚɥɶɧɨɣ ɫɢɥɵ ɦɨɦɟɧɬ ɫɢɥɵ ɪɚɜɟɧ
0][
1
)(][ rr
r
rFFr
r
GG
G
G
.
2. ɋɨɝɥɚɫɧɨ ɜɬɨɪɨɦɭ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ ɞɟɣɫɬɜɢɟ ɫɢɥɵ ɩɪɢɜɨɞɢɬ ɤ
ɢɡɦɟɧɟɧɢɸ ɢɦɩɭɥɶɫɚ ɬɟɥɚ
dt
Pd
F
G
G
. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
0][ Fr
G
G
0][
dt
Pd
r
G
G
.
ɆɨɞɆ-07 Ⱦɜɢɠɟɧɢɟ ɢɧɟɪɬɧɨɝɨ ɬɟɥɚ ɜ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ ɩɨɥɟ
103
0][][][][
dt
Pd
rP
dt
rd
dt
Pd
rPr
dt
d
G
G
G
G
G
G
G
G
.
ȼɟɤɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɢɦɩɭɥɶɫɚ ɬɟɥɚ ɧɚ ɟɝɨ ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɧɚ-
ɡɵɜɚɟɬɫɹ
ɦɨɦɟɧɬɨɦ ɢɦɩɭɥɶɫɚ
L
G
. ɂɡ ɩɨɥɭɱɟɧɧɨɝɨ ɜɵɪɚɠɟɧɢɹ ɫɥɟɞɭɟɬ,
ɱɬɨ
0
dt
Ld
G
–
ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɨɯɪɚɧɹɟɬɫɹ ɩɪɢ ɞɜɢɠɟɧɢɢ ɬɟɥɚ ɜ
ɩɨɥɟ ɰɟɧɬɪɚɥɶɧɵɯ ɫɢɥ
(
const][][ v
G
G
G
G
G
rMPrL
).
3. Ⱦɪɭɝɢɯ ɫɢɥ ɤɪɨɦɟ ɰɟɧɬɪɚɥɶɧɨɣ ɧɚ ɬɟɥɨ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ ɧɟ
ɞɟɣɫɬɜɭɟɬ, ɩɨɷɬɨɦɭ ɫɢɫɬɟɦɚ ɹɜɥɹɟɬɫɹ ɡɚɦɤɧɭɬɨɣ. Ⱦɥɹ ɧɟɟ
ɫɨɯɪɚɧɹɟɬɫɹ ɢ
ɩɨɥɧɚɹ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ
E
, ɜɤɥɸɱɚɸɳɚɹ ɤɢɧɟɬɢɱɟɫɤɭɸ
2
2
vM
T
ɢ ɩɨɬɟɧɰɢɚɥɶɧɭɸ
U
. ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ
U
ɪɚɜɧɚ ɪɚɛɨɬɟ
ɩɨ ɩɟɪɟɦɟɳɟɧɢɸ ɬɟɥɚ ɢɡ ɞɚɧɧɨɣ ɬɨɱɤɢ ɧɚ ɛɟɫɤɨɧɟɱɧɨɫɬɶ:
³³
ff
r
r
r
drrFrdFU )(
G
G
. ȿɫɥɢ ɫɢɥɚ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɤɜɚɞɪɚɬɭ
ɪɚɫɫɬɨɹɧɢɹ ɞɨ ɢɫɬɨɱɧɢɤɚ ɩɨɥɹ
2
)(
r
rF
r
D
, ɬɨ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ
r
dr
r
drrFU
rr
r
D
D
³³
ff
2
)( .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɥɧɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ ɜ ɩɨɥɟ
ɰɟɧɬɪɚɥɶɧɨɣ ɫɢɥɵ ɪɚɜɧɚ
const
2
2
D
r
M
E
v
.
1.3. ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ
ɜ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɞɟɣ-
ɫɬɜɢɟɦ ɬɨɥɶɤɨ ɰɟɧɬɪɚɥɶɧɨɣ ɫɢɥɵ, ɨɛɥɚɞɚɸɳɟɣ ɰɟɧɬɪɚɥɶɧɨɣ ɫɢɦɦɟɬɪɢɟɣ
r
r
rF
dt
rd
M
r
G
G
)(
2
2
.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢ ɞɜɢɠɟɧɢɟ ɬɟɥɚ ɨɛɥɚɞɚɟɬ ɰɟɧɬɪɚɥɶɧɨɣ ɫɢɦɦɟɬɪɢɟɣ.
Ʉɪɨɦɟ ɬɨɝɨ, ɤɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɞɜɢɠɟɧɢɟ ɬɟɥɚ ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɢɦ
(
const
L
G
). ɉɨɷɬɨɦɭ ɞɥɹ ɨɩɢɫɚɧɢɹ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɭɞɨɛɧɟɟ
ɩɟɪɟɣɬɢ ɜ ɩɨɥɹɪɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ (ɪɢɫ. 2). Ⱦɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ
Ɇɟɯɚɧɢɤɚ
104
x
ɢ y ɫɜɹɡɚɧɵ ɫ ɩɨɥɹɪɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ r ɢ
M
ɫɥɟɞɭɸɳɢɦɢ ɫɨɨɬɧɨ-
ɲɟɧɢɹɦɢ:
M cosr
x
, M sinry .
ɉɟɪɟɩɢɲɟɦ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ
ɷɧɟɪɝɢɢ ɢ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ
ɢɦɩɭɥɶɫɚ ɜ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ. Ⱦɥɹ
ɷɬɨɝɨ ɫɧɚɱɚɥɚ ɡɚɩɢɲɟɦ ɤɨɦɩɨɧɟɧɬɵ
ɫɤɨɪɨɫɬɢ:
MMM sincos rr
x
,
MMM cossin rry
.
ȼ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬɚɯ ɜɟɤɬɨɪ
r
G
ɩɪɢ ɞɜɢɠɟɧɢɢ ɜ ɩɥɨɫɤɨɫɬɢ XOY ɢɦɟɥ
ɤɨɨɪɞɢɧɚɬɵ
)0,,( y
x
r
G
, ɚ ɜɟɤɬɨɪ ɫɤɨ-
ɪɨɫɬɢ
)0,,( yx
G
v . Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ
][ v
G
G
G
rML ɜ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬɚɯ ɡɚɩɢɫɵɜɚɟɬɫɹ ɱɟɪɟɡ ɨɩɪɟɞɟɥɢɬɟɥɶ
ɜɢɞɚ
l
l
l
[] 0 ( )
0
ijk
L
M r M x y M xy xy k
xy
G
G
G
v
.
ȼɟɤɬɨɪ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɧɚɩɪɚɜɥɟɧ ɬɨɥɶɤɨ ɜɞɨɥɶ ɨɪɬɚ
l
k
, ɩɨɷɬɨ-
ɦɭ
)( y
x
y
x
M
L
. ɉɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɜ ɩɨ-
ɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ, ɩɨɞɫɬɚɜɢɜ ɜ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɜɵ-
ɪɚɠɟɧɢɹ ɜ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ:
MMMMMMMMMM )sinsincossincoscossincos(
rrrrrrrr
M
L
M MMM
2222
)sin(cos MrMr
Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ
constMrL M
2
.
Ⱥɧɚɥɨɝɢɱɧɨ ɩɪɟɨɛɪɚɡɭɟɦ ɤɢɧɟɬɢɱɟɫɤɭɸ ɢ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪ-
ɝɢɸ. Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ:
)(
22
22
2
yx
MM
T
v
)cossin2cossin
cossin2sincos(
2
22222
22222
MMMMMM
MMMMMM
rrrr
rrrr
M
Y
X
r
v
M
O
y
x
M
Ɋɢɫ. 2
ɆɨɞɆ-07 Ⱦɜɢɠɟɧɢɟ ɢɧɟɪɬɧɨɝɨ ɬɟɥɚ ɜ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ ɩɨɥɟ
105
)(
2
222
M
rr
M
T .
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɨɜɨɤɭɩɧɨɫɬɶ ɩɨɫɬɭɩɚ-
ɬɟɥɶɧɨɝɨ
2
2
r
M
(ɜɞɨɥɶ ɪɚɞɢɭɫɚ) ɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ
2222
22
Z M r
M
r
M
, ɝɞɟ
M Z
– ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ.
ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ
r
U
D
.
Ɍɨɝɞɚ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ
const)(
2
222
D
M
r
rr
M
E
.
ɂɡ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɜɵɪɚɡɢɦ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ
M
ɢ ɩɨɞɫɬɚɜɢɦ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ:
2
M
r
L
M
const)(
2
22
2
2
D
r
r
M
L
r
M
E
ɢɥɢ
const
2
2
2
2
2
D
r
M
r
L
r
M
E
.
ɉɪɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɩɨ ɜɪɟɦɟɧɢ
02
2
23
2
D
r
r
r
M
r
L
rr
M
ɢɥɢ
23
2
r
M
r
L
rM
D
.
ɉɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ
ɜɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸ-
ɬɨɧɚ ɜ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
: ɬɟɥɨ ɩɪɢɨɛɪɟɬɚɟɬ ɪɚɞɢɚɥɶɧɨɟ
ɭɫɤɨɪɟɧɢɟ
r
ɜɫɥɟɞɫɬɜɢɟ ɞɟɣɫɬɜɢɹ ɞɜɭɯ ɫɢɥ: ɰɟɧɬɪɨɛɟɠɧɨɣ (
3
2
M
r
L
) ɢ ɰɟɧ-
ɬɪɚɥɶɧɨɣ (
2
r
D
). ɇɚɥɢɱɢɟ ɰɟɧɬɪɨɛɟɠɧɨɣ ɫɢɥɵ ɝɨɜɨɪɢɬ ɨ ɬɨɦ, ɱɬɨ ɜɵ-
ɛɪɚɧɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ ɹɜɥɹɟɬɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ. Ɉɧɚ ɞɜɢɠɟɬɫɹ ɭɫɤɨ-
ɪɟɧɧɨ ɜɦɟɫɬɟ ɫ ɢɫɬɨɱɧɢɤɨɦ ɩɨɥɹ.
ɗɧɟɪɝɢɸ ɫɢɫɬɟɦɵ ɜ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ
r
M
r
L
r
M
E
D
2
2
2
2
2
Ɇɟɯɚɧɢɤɚ
106
ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɫɨɜɨɤɭɩɧɨɫɬɶ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɪɚɞɢɚɥɶ-
ɧɨɝɨ ɞɜɢɠɟɧɢɹ
2
2
r
M
, ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɰɟɧɬɪɚɥɶɧɨɝɨ ɩɨɥɹ
r
D
ɢ
ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ (ɰɟɧɬɪɨɛɟɠɧɨɣ ɷɧɟɪ-
ɝɢɢ)
2
2
2
M
r
L
. ȼɟɥɢɱɢɧɭ, ɪɚɜɧɭɸ
r
M
r
L
rɎ
D
2
2
2
)(
, ɧɚɡɵɜɚɸɬ
ɷɮɮɟɤɬɢɜ-
ɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɟɣ
ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ.
1.4. Ɍɪɚɟɤɬɨɪɢɹ ɬɟɥɚ ɜ ɰɟɧɬɪɚɥɶɧɨɦ ɩɨɥɟ
ȼ ɩɨɥɭɱɟɧɧɨɦ ɭɪɚɜɧɟɧɢɢ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ (ɜɬɨ-
ɪɨɦ ɡɚɤɨɧɟ ɇɶɸɬɨɧɚ)
23
2
r
M
r
L
rM
D
ɜɫɟ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɬɟɥɨ, ɡɚɜɢɫɹɬ ɬɨɥɶɤɨ ɨɬ ɜɟɥɢɱɢɧɵ, ɨɛɪɚɬɧɨ
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ ɪɚɫɫɬɨɹɧɢɸ
r ɞɨ ɢɫɬɨɱɧɢɤɚ ɩɨɥɹ. ɉɨɷɬɨɦɭ ɞɥɹ ɧɚ-
ɯɨɠɞɟɧɢɹ ɪɟɲɟɧɢɹ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɭɞɨɛɧɨ ɫɞɟɥɚɬɶ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɵɯ
)(
1
)(
M[
tr
.
Ɍɨɝɞɚ ɩɨɥɭɱɟɧɧɨɟ ɪɚɧɟɟ ɜɵɪɚɠɟɧɢɟ
2
M
r
L
M
ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɤɚɤ
2
2
[ M
M
L
M
r
L
.
ȼɵɪɚɡɢɦ ɩɟɪɜɭɸ ɢ ɜɬɨɪɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ ɪɚɫɫɬɨɹɧɢɹ
r ɜ ɧɨɜɵɯ
ɩɟɪɟɦɟɧɧɵɯ
M
[
[
[
M
[
M
[
M
M
[
[
[
[
{
d
d
M
L
d
d
dt
d
d
d
dt
d
dt
dr
r
2
2222
1111
,
M
[
d
d
M
L
r
.
2
2
2
2
2
M
[
[
M
¸
¸
¹
·
¨
¨
©
§
M
[
M
¸
¸
¹
·
¨
¨
©
§
M
[
d
d
M
L
dt
d
d
d
d
d
M
L
d
d
dt
d
M
L
r
,
2
2
2
2
2
M
[
[
d
d
M
L
r
.
ɉɨɞɫɬɚɜɥɹɹ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɜ
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ, ɩɨɥɭɱɢɦ
ɆɨɞɆ-07 Ⱦɜɢɠɟɧɢɟ ɢɧɟɪɬɧɨɝɨ ɬɟɥɚ ɜ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ ɩɨɥɟ
107
23
2
r
M
r
L
rM
D
23
2
2
2
2
2
2
D[[
M
[
[
M
L
d
d
M
L
M
,
22
2
L
M
d
d D
[
M
[
ɢɥɢ
0
22
2
¸
¹
·
¨
©
§
D
[
M
[
L
M
d
d
.
Ɂɚɦɟɧɚ ɩɟɪɟɦɟɧɧɵɯ
2
)()(
L
MD
M[ MF
ɩɨɡɜɨɥɹɟɬ ɫɜɟɫɬɢ ɩɨɥɭɱɟɧ-
ɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤ ɫɬɚɧɞɚɪɬɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ ɫ
ɱɚɫɬɨɬɨɣ ɪɚɜɧɨɣ ɟɞɢɧɢɰɟ
0
2
2
F
M
F
d
d
.
Ɋɟɲɟɧɢɟɦ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɹɜɥɹɟɬɫɹ ɝɚɪɦɨɧɢɱɟɫɤɚɹ ɡɚɜɢɫɢ-
ɦɨɫɬɶ
F
ɨɬ
M
ɫ ɚɦɩɥɢɬɭɞɨɣ
A
ɢ ɧɟɤɨɬɨɪɨɣ ɧɚɱɚɥɶɧɨɣ ɮɚɡɨɣ
min
M :
)cos(
min
MM F A .
ȼɨɡɜɪɚɳɚɹɫɶ ɤ ɩɟɪɟɦɟɧɧɵɦ
[
ɢ
r
, ɩɨɥɭɱɢɦ ɢɫɤɨɦɨɟ ɪɟɲɟɧɢɟ ɜ
ɜɢɞɟ
)cos(
min
2
MM
D
[ F A
L
M
2
min
)cos(
1
L
M
A
r
D
MM
.
¸
¸
¹
·
¨
¨
©
§
MM
D
D
1)cos(
1
min
2
2
M
AL
L
M
r
.
ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɹ
M
L
p
D
2
ɢ
M
AL
D
H
2
. Ɍɨɝɞɚ ɪɟɲɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ
1)cos(
min
MMH
p
r
.
Ɉɩɪɟɞɟɥɢɬɶ ɤɨɧɫɬɚɧɬɭ
H ɦɨɠɧɨ, ɩɨɞɫɬɚɜɢɜ r ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ
ɷɧɟɪɝɢɢ
r
M
r
L
r
M
E
D
2
2
2
2
2
.
ȼɨɡɶɦɟɦ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ
r
ɢ ɩɨɞɫɬɚɜɢɦ ɜ ɧɟɟ
2
M
r
L
M
:
)sin())sin((
)1)cos((
minmin
2
min
MMH MMMH
MMH
Mp
Lp
r
.
Ɍɨɝɞɚ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ
Ɇɟɯɚɧɢɤɚ
108
MMHMMH
2
min
2
2
min
22
22
2
)1)cos((
2
)(sin
2
Mp
L
pM
LM
E
)1)cos((
min
MMH
D
p
ȼɵɩɨɥɧɢɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɢ ɩɨɞɫɬɚɜɢɜ
M
L
p
D
2
,
ɩɨɥɭɱɢɦ
E
L
M
H
D
)1(
2
2
2
2
.
Ɉɬɫɸɞɚ
1
2
2
2
D
r H E
M
L
.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ
ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨ-
ɪɢɢ
ɢɦɟɟɬ ɜɢɞ
1)cos(
min
MMH
p
r
, ɝɞɟ
M
L
p
D
2
, E
M
L
2
2
2
1
D
r H
.
ɗɬɨ
ɭɪɚɜɧɟɧɢɟ ɤɨɧɢɱɟɫɤɢɯ ɫɟɱɟɧɢɣ, ɡɚɩɢɫɚɧɧɨɟ ɜ ɩɨɥɹɪɧɵɯ ɤɨ-
ɨɪɞɢɧɚɬɚɯ. Ʉɨɧɫɬɚɧɬɚ
p ɧɚɡɵɜɚɟɬɫɹ ɮɨɤɚɥɶɧɵɦ ɩɚɪɚɦɟɬɪɨɦ ɤɪɢɜɨɣ, ɚ
ɤɨɧɫɬɚɧɬɚ
H – ɷɤɫɰɟɧɬɪɢɫɢɬɟɬɨɦ. ɂɡ ɤɭɪɫɚ ɝɟɨɦɟɬɪɢɢ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɜ
ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɷɤɫɰɟɧɬɪɢɫɢɬɟɬɚ ɤɨɧɢɱɟɫɤɢɟ ɫɟɱɟɧɢɹ ɦɨɝɭɬ ɩɪɢɧɢɦɚɬɶ
ɮɨɪɦɭ ɤɪɭɝɚ (
0 H
), ɷɥɥɢɩɫɚ (
1H
), ɩɚɪɚɛɨɥɵ (
1 H
) ɢɥɢ ɝɢɩɟɪɛɨɥɵ
(
1!H
).
Ɋɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ
H – ɜɟɥɢɱɢɧɚ ɩɨɥɨɠɢ-
ɬɟɥɶɧɚɹ
0tH . ɍɱɢɬɵɜɚɹ, ɱɬɨ ɤɨɫɢɧɭɫ ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ ɨɬ –1 ɞɨ +1,
ɪɚɫɫɬɨɹɧɢɟ
r ɞɨ ɢɫɬɨɱɧɢɤɚ ɩɨɥɹ ɛɭɞɟɬ ɦɢɧɢɦɚɥɶɧɵɦ ɩɪɢ ɭɝɥɟ
min
M M .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
min
M – ɷɬɨ ɭɝɨɥ, ɩɪɢ ɤɨɬɨɪɨɦ ɧɚɥɟɬɚɸɳɚɹ ɱɚɫɬɢɰɚ ɩɨɞ-
ɥɟɬɚɟɬ ɤ ɢɫɬɨɱɧɢɤɭ ɩɨɥɹ ɧɚ ɦɢɧɢɦɚɥɶɧɨɟ ɪɚɫɫɬɨɹɧɢɟ
1
min
H
p
r
.
Ɍɨɱɤɚ ɬɪɚɟɤɬɨɪɢɢ ɧɚɥɟɬɚɸɳɟɣ ɱɚɫɬɢɰɵ, ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɢɫɬɨɱɧɢɤɚ
ɩɨɥɹ ɞɨ ɤɨɬɨɪɨɣ ɦɢɧɢɦɚɥɶɧɨ, ɧɚɡɵɜɚɟɬɫɹ
ɩɟɪɢɝɟɟɦ. ȼ ɩɟɪɢɝɟɟ ɪɚɞɢɚɥɶ-
ɧɚɹ ɫɤɨɪɨɫɬɶ ɬɟɥɚ
)sin(
min
MMH
Mp
L
r
ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ (ɫɤɨɪɨɫɬɶ v
G
ɆɨɞɆ-07 Ⱦɜɢɠɟɧɢɟ ɢɧɟɪɬɧɨɝɨ ɬɟɥɚ ɜ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ ɩɨɥɟ
109
ɧɚɩɪɚɜɥɟɧɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɭ
r
G
), ɚ ɭɝɥɨɜɚɹ
2
M
r
L
M Z
ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɚɹ
M
r
L
r Z
M
v ɫɤɨɪɨɫɬɢ ɦɚɤɫɢɦɚɥɶɧɵ.
ȿɫɥɢ ɬɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɡɚɦɤɧɭɬɚ (ɷɥɥɢɩɫ), ɬɨ ɤɪɨɦɟ ɩɟɪɢɝɟɹ,
ɨɧɚ ɢɦɟɟɬ ɢ
ɚɩɨɝɟɣ – ɬɨɱɤɭ ɧɚɢɛɨɥɶɲɟɝɨ ɭɞɚɥɟɧɢɹ ɨɬ ɢɫɬɨɱɧɢɤɚ ɩɨɥɹ
1
||
max
H
p
r
.
Ɍɨɱɤɟ ɚɩɨɝɟɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɭɝɨɥ, ɨɬɥɢɱɚɸɳɢɣɫɹ ɨɬ
min
M ɧɚ S,
SrM M
min
. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɚɩɨɝɟɟ ɪɚɞɢɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɬɟɥɚ
r
ɬɚɤɠɟ
ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ (ɫɤɨɪɨɫɬɶ
v
G
ɧɚɩɪɚɜɥɟɧɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɪɚɞɢɭɫ-
ɜɟɤɬɨɪɭ
r
G
), ɚ ɭɝɥɨɜɚɹ
2
M
r
L
M Z
ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɚɹ
M
r
L
r Z
M
v
ɫɤɨɪɨɫɬɢ ɦɢɧɢɦɚɥɶɧɵ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɭɝɨɥ
min
M ɡɚɞɚɟɬ ɧɚɩɪɚɜɥɟɧɢɟ ɩɪɹɦɨɣ, ɫɨɟɞɢɧɹɸ-
ɳɟɣ ɩɟɪɢɝɟɣ ɢ ɚɩɨɝɟɣ.
ɉɪɢ ɨɬɪɢɰɚɬɟɥɶɧɨɦ ɷɤɫɰɟɧɬɪɢɫɢɬɟɬɟ
0dH , ɚɩɨɝɟɣ ɢ ɩɟɪɢɝɟɣ ɦɟ-
ɧɹɸɬɫɹ ɦɟɫɬɚɦɢ
1
min
H
p
r
,
H
1
||
max
p
r
.
1.5. Ɍɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɜ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ ɩɨɥɟ
ȼ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ
ɩɨɥɟ ɫɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ
ɬɟɥɨ ɦɚɫɫɨɣ
1
m ɫɨ ɫɬɨɪɨɧɵ
ɞɪɭɝɨɝɨ ɬɟɥɚ ɦɚɫɫɨɣ
2
m , ɹɜ-
ɥɹɟɬɫɹ ɰɟɧɬɪɚɥɶɧɨɣ ɢ ɪɚɜɧɚ
2
21
)(
r
mm
GrF
r
, ɝɞɟ
G
=6,67210
–11
ɇɦ
2
/ɤɝ
2
.
Ʉɨɧɫɬɚɧɬɚ
D ɜ ɷɬɨɦ
ɫɥɭɱɚɟ ɩɪɢɦɟɬ ɜɢɞ
21
mmG D
ɢ ɦɨɠɟɬ ɛɵɬɶ ɬɨɥɶɤɨ ɨɬɪɢ-
ɰɚɬɟɥɶɧɨɣ (
0D
).
r
A
v
A
m
1
m
2
M
min
A
Ɋɢɫ. 3
Ɇɟɯɚɧɢɤɚ
110
Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɩɪɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ
p ɢ H ɬɪɚɟɤɬɨɪɢɹ ɞɜɢ-
ɠɟɧɢɹ ɬɟɥɚ ɢɦɟɟɬ ɯɨɬɹ ɛɵ ɨɞɧɭ ɬɨɱɤɭ, ɜ ɤɨɬɨɪɨɣ ɫɤɨɪɨɫɬɶ ɩɟɪɩɟɧɞɢɤɭ-
ɥɹɪɧɚ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɭ. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ
Ⱥ ɬɨɱɤɭ ɬɪɚɟɤɬɨɪɢɢ, ɜ ɤɨɬɨɪɨɣ
ɫɤɨɪɨɫɬɶ
A
v
G
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɭ
A
r
G
(ɪɢɫ. 3).
Ɍɨɝɞɚ ɜ ɬɨɱɤɟ
Ⱥ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɦɨɠɧɨ ɡɚɩɢ-
ɫɚɬɶ
][
AA
rML v
G
G
G
AAAA
MrMrL vv
S
2
sin ,
AA
MrL v .
ɗɧɟɪɝɢɹ ɜ ɬɨɱɤɟ
Ⱥ
A
A
r
M
E
D
2
2
v
.
ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ, ɡɚɩɢɫɚɧɧɨɟ ɜ ɩɨɥɹɪɧɵɯ
ɤɨɨɪɞɢɧɚɬɚɯ:
1)cos(
min
MMH
p
r
, ɝɞɟ
M
L
p
D
2
, E
M
L
2
2
2
1
D
H
.
ȼ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ ɪɚɫɫɬɨɹɧɢɟ
r
ɫɱɢɬɚɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɵɦ.
ɉɪɢ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ ɩɪɢɬɹɠɟɧɢɢ (
0D ) ɮɨɤɚɥɶɧɵɣ ɩɚɪɚɦɟɬɪ
ɬɪɚɟɤɬɨɪɢɢ
p
ɨɬɪɢɰɚɬɟɥɟɧ (
0p
), ɚ ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ
H
ɜ ɡɚɜɢɫɢɦɨɫɬɢ
ɨɬ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ
E
ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɥɸɛɵɟ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɡɧɚɱɟ-
ɧɢɹ. ɉɨɷɬɨɦɭ ɦɨɠɧɨ ɧɚɛɥɸɞɚɬɶ ɪɚɡɥɢɱɧɵɟ ɤɨɧɢɱɟɫɤɢɟ ɫɟɱɟɧɢɹ. Ʉɚɤ
ɜɢɞɧɨ ɢɡ ɮɨɪɦɭɥɵ, ɫɜɹɡɵɜɚɸɳɟɣ ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ ɢ ɷɧɟɪɝɢɸ, ɱɬɨɛɵ ɧɚ-
ɛɥɸɞɚɬɶ ɞɜɢɠɟɧɢɟ ɫ ɷɤɫɰɟɧɬɪɢɫɢɬɟɬɨɦ
10 Hd , ɧɟɨɛɯɨɞɢɦɨ, ɱɬɨɛɵ
ɷɧɟɪɝɢɹ
E
ɛɵɥɚ ɨɬɪɢɰɚɬɟɥɶɧɨɣ. ȿɫɥɢ ɷɧɟɪɝɢɹ ɪɚɜɧɚ ɧɭɥɸ, ɛɭɞɟɦ ɧɚ-
ɛɥɸɞɚɬɶ ɞɜɢɠɟɧɢɟ ɩɨ ɩɚɪɚɛɨɥɟ.
Ⱥɧɚɥɢɡ ɬɪɚɟɤɬɨɪɢɢ ɬɟɥɚ ɜ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ ɩɨɥɟ ɦɨɠɧɨ ɬɚɤɠɟ
ɨɫɭɳɟɫɬɜɥɹɬɶ ɧɚ ɨɫɧɨɜɟ ɫɨɩɨɫɬɚɜɥɟɧɢɹ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ ɧɚɥɟɬɚɸɳɟɣ
ɱɚɫɬɢɰɵ
)(
2
2
rɎr
M
E
ɢ ɷɮɮɟɤɬɢɜɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ )(rɎ ,
ɜɜɟɞɟɧɧɨɣ ɜ ɪɚɡɞɟɥɟ 1.3.
Ɋɚɫɫɦɨɬɪɢɦ ɷɮɮɟɤɬɢɜɧɭɸ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ
r
M
r
L
rɎ
D
2
2
2
)(
.