Ревинская О.Г., Кравченко Н.С. Изучение моделей физических процессов и явлений на компьютере
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ɅȺȻɈɊȺɌɈɊɇȺə ɊȺȻɈɌȺ ʋ Ɇɨɞɗ-04
ɉɈ ɂɁɍɑȿɇɂɘ ɆɈȾȿɅȿɃ ɎɂɁɂɑȿɋɄɂɏ
ɉɊɈɐȿɋɋɈȼ ɂ əȼɅȿɇɂɃ ɇȺ ɄɈɆɉɖɘɌȿɊȿ
Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ
ɐɟɥɶ ɪɚɛɨɬɵ: ɢɡɭɱɟɧɢɟ ɩɨɜɟɞɟɧɢɹ ɨɞɧɨɣ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ
ɩɨɥɟ ɞɪɭɝɨɣ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ. ȼɵɹɜɥɟɧɢɟ ɜɡɚɢɦɨɫɜɹɡɢ ɩɚɪɚɦɟɬɪɨɜ
ɬɪɚɟɤɬɨɪɢɢ ɧɚɥɟɬɚɸɳɟɣ ɱɚɫɬɢɰɵ ɫ ɟɟ ɧɚɱɚɥɶɧɨɣ ɷɧɟɪɝɢɟɣ ɢ ɦɨɦɟɧɬɨɦ
ɢɦɩɭɥɶɫɚ.
1. Ɍɟɨɪɟɬɢɱɟɫɤɨɟ ɫɨɞɟɪɠɚɧɢɟ
ɋɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɦɚɬɟɪɢɚɥɶɧɭɸ ɬɨɱɤɭ, ɧɚɡɵɜɚɟɬɫɹ ɰɟɧ-
ɬɪɚɥɶɧɨɣ, ɟɫɥɢ ɨɧɚ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɪɚɫɫɬɨɹɧɢɹ ɞɨ ɢɫɬɨɱɧɢɤɚ ɩɨɥɹ
(ɫɢɥɵ) ɢ ɧɚɩɪɚɜɥɟɧɚ ɜɞɨɥɶ ɩɪɹɦɨɣ, ɫɨɟɞɢɧɹɸɳɟɣ ɦɚɬɟɪɢɚɥɶɧɭɸ ɬɨɱɤɭ ɫ
ɢɫɬɨɱɧɢɤɨɦ ɩɨɥɹ. ȼ ɨɛɳɟɦ ɜɢɞɟ ɰɟɧɬɪɚɥɶɧɚɹ ɫɢɥɚ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɚ
r
r
rFF
r
G
G
)( ,
ɝɞɟ
)(rF
r
– ɩɪɨɟɤɰɢɹ ɫɢɥɵ ɧɚ ɩɪɹɦɭɸ, ɫɨɟɞɢɧɹɸɳɭɸ ɦɚɬɟɪɢɚɥɶɧɭɸ
ɬɨɱɤɭ ɫ ɢɫɬɨɱɧɢɤɨɦ ɩɨɥɹ. Ɉɞɧɢɦ ɢɡ ɧɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɯ ɜɢɞɨɜ
ɰɟɧɬɪɚɥɶɧɵɯ ɫɢɥ ɹɜɥɹɟɬɫɹ ɫɢɥɚ, ɩɪɨɟɤɰɢɹ ɤɨɬɨɪɨɣ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨ-
ɧɚɥɶɧɚ ɤɜɚɞɪɚɬɭ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɢɫɬɨɱɧɢɤɚ ɩɨɥɹ ɞɨ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ
2
)(
r
rF
r
D
,
ɝɞɟ
D – ɤɨɧɫɬɚɧɬɚ, ɨɛɭɫɥɨɜɥɟɧɧɚɹ ɩɪɢɪɨɞɨɣ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ.
ɉɪɢɦɟɪɨɦ ɰɟɧɬɪɚɥɶɧɨɣ ɫɢɥɵ ɬɚɤɨɝɨ ɜɢɞɚ ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɫɢɥɚ ɤɭ-
ɥɨɧɨɜɫɤɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɞɜɭɯ ɱɚɫɬɢɰ ɫ ɡɚɪɹɞɚɦɢ
1
q ɢ
2
q , ɤɨɝɞɚ ɨɞɧɚ
ɢɡ ɧɢɯ ɜɵɫɬɭɩɚɟɬ ɜ ɤɚɱɟɫɬɜɟ ɢɫɬɨɱɧɢɤɚ ɩɨɥɹ ɞɥɹ ɞɪɭɝɨɣ
2
21
)(
r
qq
krF
r
.
1.1. Ɂɚɞɚɱɚ ɨ ɞɜɢɠɟɧɢɢ ɞɜɭɯ ɬɟɥ
ȼɡɚɢɦɨɞɟɣɫɬɜɢɟ ɞɜɭɯ ɡɚɪɹɠɟɧɧɵɯ ɬɟɥ ɨɬɧɨɫɢɬɫɹ ɜ ɮɢɡɢɤɟ ɤ ɡɚɞɚɱɟ
ɨ ɞɜɢɠɟɧɢɢ ɞɜɭɯ ɬɟɥ, ɤɨɬɨɪɭɸ ɦɨɠɧɨ ɫɜɟɫɬɢ ɤ ɡɚɞɚɱɟ ɨ ɞɜɢɠɟɧɢɢ ɨɞɧɨ-
ɝɨ ɬɟɥɚ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɫɫɵ ɜ ɩɨɥɟ ɰɟɧɬɪɚɥɶɧɵɯ ɫɢɥ.
Ɋɚɫɫɦɨɬɪɢɦ ɞɜɢɠɟɧɢɟ ɞɜɭɯ ɬɟɥ ɦɚɫɫɚɦɢ
1
m ɢ
2
m , ɫɢɥɵ ɜɡɚɢɦɨ-
ɞɟɣɫɬɜɢɹ ɤɨɬɨɪɵɯ ɹɜɥɹɸɬɫɹ ɰɟɧɬɪɚɥɶɧɵɦɢ. ɋɨɝɥɚɫɧɨ ɬɪɟɬɶɟɦɭ ɡɚɤɨɧɭ
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
392
ɇɶɸɬɨɧɚ, ɟɫɥɢ ɬɟɥɨ
2
m ɞɟɣɫɬɜɭɟɬ ɧɚ ɬɟɥɨ
1
m ɫ ɫɢɥɨɣ
F
G
, ɬɨ ɬɟɥɨ
1
m
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
GG
)(
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.
x
y
z
r
r
1
r
2
m
1
m
2
v
1
v
2
Ɋɢɫ. 1
Ɇɨɞɗ-04 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ
393
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
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. ȼ ɷɬɨɦ
ɫɥɭɱɚɟ ɬɟɥɨ, ɹɜɥɹɸɳɟɟɫɹ ɢɫɬɨɱɧɢɤɨɦ ɩɨɥɹ, ɧɚɡɵɜɚɸɬ
ɪɚɫɫɟɢɜɚɸɳɢɦ
ɰɟɧɬɪɨɦ
, ɚ ɛɨɥɟɟ ɥɟɝɤɨɟ ɬɟɥɨ – ɧɚɥɟɬɚɸɳɟɣ ɱɚɫɬɢɰɟɣ.
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
G
G
)(
2
2
ɢ ɯɚɪɚɤɬɟɪɢɡɭɟɦɨɟ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɨɦ
r
G
ɢ ɫɤɨɪɨɫɬɶɸ v
G
. ɇɚ ɬɟɥɨ ɩɪɢɜɟ-
ɞɟɧɧɨɣ ɦɚɫɫɵ ɞɟɣɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɚɹ ɫɢɥɚ
F
G
, ɹɜɥɹɸɳɚɹɫɹ ɰɟɧɬɪɚɥɶ-
ɧɨɣ. 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
G
G
G
G
.
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
394
2. ɋɨɝɥɚɫɧɨ ɜɬɨɪɨɦɭ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ ɞɟɣɫɬɜɢɟ ɫɢɥɵ ɩɪɢɜɨɞɢɬ ɤ
ɢɡɦɟɧɟɧɢɸ ɢɦɩɭɥɶɫɚ ɬɟɥɚ
dt
Pd
F
G
G
. ɉɨɫɤɨɥɶɤɭ ɞɜɢɠɟɧɢɟ ɹɜɥɹɟɬɫɹ ɩɥɨ-
ɫɤɢɦ, ɬɨ
0][ Fr
G
G
0][
dt
Pd
r
G
G
.
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
Ɇɨɞɗ-04 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ
395
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
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 ɫɤɨɪɨɫɬɢ )0,,( yx
G
v . Ɇɨɦɟɧɬ ɢɦ-
ɩɭɥɶɫɚ
][ v
G
G
G
rML
ɜ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬɚɯ ɡɚɩɢɫɵɜɚɟɬɫɹ ɱɟɪɟɡ ɨɩɪɟ-
ɞɟɥɢɬɟɥɶ 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
const
2
M
MrL .
Ⱥɧɚɥɨɝɢɱɧɨ ɩɪɟɨɛɪɚɡɭɟɦ ɤɢɧɟɬɢɱɟɫɤɭɸ ɢ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪ-
ɝɢɸ. Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ:
Y
X
r
v
M
O
y
x
M
Ɋɢɫ. 2
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
396
)(
22
22
2
yx
MM
T
v
)cossin2cossin
cossin2sincos(
2
22222
22222
MMMMMM
MMMMMM
rrrr
rrrr
M
)(
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 (ɰɟɧɬɪɨɦ ɪɚɫɫɟɹɧɢɹ).
Ɇɨɞɗ-04 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ
397
ɗɧɟɪɝɢɸ ɫɢɫɬɟɦɵ ɜ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ
r
M
r
L
r
M
E
D
2
2
2
2
2
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 (1/r ), ɨɛ-
ɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ ɪɚɫɫɬɨɹɧɢɸ ɞɨ ɪɚɫɫɟɢɜɚɸɳɟɝɨ ɰɟɧɬɪɚ (ɢɫɬɨɱ-
ɧɢɤɚ ɩɨɥɹ), ɜ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɫɬɟɩɟɧɹɯ. ɉɨɷɬɨɦɭ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɪɟɲɟ-
ɧɢɹ 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
,
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
398
2
2
2
2
2
M
[
[
d
d
M
L
r
.
ɉɨɞɫɬɚɜɥɹɹ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɜ
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ, ɩɨɥɭɱɢɦ
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
:
Ɇɨɞɗ-04 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ
399
)sin())sin((
)1)cos((
minmin
2
min
MMH MMMH
MMH
Mp
Lp
r
.
Ɍɨɝɞɚ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ
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
).
ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ ɫɥɭɱɚɣ, ɤɨɝɞɚ
H > 0 ɢ p > 0. ɍɱɢɬɵɜɚɹ, ɱɬɨ ɤɨɫɢ-
ɧɭɫ ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ ɨɬ –1 ɞɨ +1, ɪɚɫɫɬɨɹɧɢɟ
r ɞɨ ɪɚɫɫɟɢɜɚɸɳɟɝɨ
ɰɟɧɬɪɚ ɛɭɞɟɬ ɦɢɧɢɦɚɥɶɧɵɦ ɩɪɢ ɭɝɥɟ
min
M M . ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
min
M –
ɷɬɨ ɭɝɨɥ, ɩɪɢ ɤɨɬɨɪɨɦ ɧɚɥɟɬɚɸɳɚɹ ɱɚɫɬɢɰɚ ɩɨɞɥɟɬɚɟɬ ɤ ɪɚɫɫɟɢɜɚɸɳɟɦɭ
ɰɟɧɬɪɭ ɧɚ ɦɢɧɢɦɚɥɶɧɨɟ ɪɚɫɫɬɨɹɧɢɟ
1
min
H
p
r
.
Ɍɨɱɤɚ ɬɪɚɟɤɬɨɪɢɢ ɧɚɥɟɬɚɸɳɟɣ ɱɚɫɬɢɰɵ, ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɪɚɫɫɟɢ-
ɜɚɸɳɟɝɨ ɰɟɧɬɪɚ ɞɨ ɤɨɬɨɪɨɣ ɦɢɧɢɦɚɥɶɧɨ, ɧɚɡɵɜɚɟɬɫɹ
ɩɟɪɢɝɟɟɦ. ȼ ɩɟɪɢ-
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
400
ɝɟɟ ɪɚɞɢɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ
)sin(
min
MMH
Mp
L
r
ɨɛɪɚɳɚɟɬɫɹ ɜ
ɧɨɥɶ, ɚ ɭɝɥɨɜɚɹ
2
M
r
L
M Z
ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɚɹ
M
r
L
r Z
M
v
ɫɤɨɪɨɫɬɢ
ɦɚɤɫɢɦɚɥɶɧɵ.
Ⱥɧɚɥɢɡ ɬɪɚɟɤɬɨɪɢɢ ɩɪɢ ɞɪɭɝɢɯ ɡɧɚɱɟɧɢɹɯ
H ɢ p ɜɵɩɨɥɧɹɟɬɫɹ ɚɧɚ-
ɥɨɝɢɱɧɨ.
1.5. Ɍɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ
ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ
ȼ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ ɫɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɡɚɪɹɠɟɧɧɭɸ ɱɚɫɬɢɰɭ
1
q
ɫɨ ɫɬɨɪɨɧɵ ɞɪɭɝɨɣ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ
2
q
, ɹɜɥɹɟɬɫɹ ɰɟɧɬɪɚɥɶɧɨɣ ɢ
ɪɚɜɧɚ
2
21
)(
r
qq
krF
r
, ɝɞɟ
0
4
1
SH
k
,
0
H =8,8510
–12
Ɏ/ɦ.
Ʉɨɧɫɬɚɧɬɚ
D
ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɩɪɢɦɟɬ ɜɢɞ
21
qkq D
ɢ ɦɨɠɟɬ ɛɵɬɶ ɤɚɤ ɩɨɥɨɠɢɬɟɥɶɧɨɣ (
0!D
) ɜ ɫɥɭɱɚɟ ɨɬɬɚɥɤɢɜɚɧɢɹ ɨɞɧɨ-
ɢɦɟɧɧɵɯ ɡɚɪɹɞɨɜ, ɬɚɤ ɢ ɨɬɪɢɰɚɬɟɥɶɧɨɣ (
0D ) ɜ ɫɥɭɱɚɟ ɩɪɢɬɹɠɟɧɢɹ ɡɚ-
ɪɹɞɨɜ, ɢɦɟɸɳɢɯ ɪɚɡɧɵɟ ɡɧɚɤɢ.
Ɋɚɫɫɦɨɬɪɢɦ, ɤɚɤ
ɛɭɞɟɬ ɞɜɢɝɚɬɶɫɹ ɡɚɪɹ-
ɠɟɧɧɚɹ ɱɚɫɬɢɰɚ ɦɚɫ-
ɫɨɣ
1
m ɢ ɡɚɪɹɞɨɦ
1
q ɜ
ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ, ɟɫ-
ɥɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨ-
ɦɟɧɬ ɜɪɟɦɟɧɢ ɨɧɚ
ɢɦɟɥɚ ɫɤɨɪɨɫɬɶ
0
v
G
ɢ
ɧɚɯɨɞɢɥɚɫɶ ɧɚ ɪɚɫ-
ɫɬɨɹɧɢɢ
0
r
ɨɬ ɪɚɫɫɟɢ-
ɜɚɸɳɟɝɨ ɰɟɧɬɪɚ ɦɚɫ-
ɫɨɣ
2
m ɢ ɡɚɪɹɞɨɦ
2
q
(
12
mm !! ). Ʉɚɤ ɛɵɥɨ
ɩɨɤɚɡɚɧɨ ɪɚɧɟɟ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ ɫɜɹɡɵɜɚɟɬɫɹ ɫ ɬɟɥɨɦ
ɛɨɥɶɲɟɣ ɦɚɫɫɵ –
2
m
(ɪɢɫ. 3):
1
m
– ɧɚɥɟɬɚɸɳɚɹ ɱɚɫɬɢɰɚ,
2
m
– ɪɚɫɫɟɢ-
b
r
0
v
0
m
1
m
2
M
0
T
x
0
XO
Ɋɢɫ. 3