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
351
5. Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. Ʉɚɤɢɟ ɩɨɥɹ ɧɚɡɵɜɚɸɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦɢ, ɨɞɧɨɪɨɞɧɵɦɢ?
2.
Ʉɚɤɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɨɩɢɫɵɜɚɟɬɫɹ ɞɜɢɠɟɧɢɟ ɡɚɪɹ-
ɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜɨ ɜɡɚɢɦɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ
ɦɚɝɧɢɬɧɨɦ ɩɨɥɹ?
3.
Ʉɚɤɨɣ ɤɪɢɜɨɣ ɨɩɢɫɵɜɚɟɬɫɹ ɞɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜɨ ɜɡɚɢɦ-
ɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ?
4.
Ʉɚɤ ɜɢɞ ɬɪɚɟɤɬɨɪɢɢ ɡɚɜɢɫɢɬ ɨɬ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨ-
ɫɬɶɸ ɱɚɫɬɢɰɵ, ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ ɢ ɢɧɞɭɤɰɢɢ ɦɚɝ-
ɧɢɬɧɨɝɨ ɩɨɥɹ?
5.
Ʉɚɤ ɡɚɜɢɫɢɬ ɩɟɪɢɨɞ ɜɪɚɳɟɧɢɹ ɱɚɫɬɢɰɵ ɨɬ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟ-
ɫɤɨɝɨ ɩɨɥɹ ɢ ɨɬ ɢɧɞɭɤɰɢɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ?
6.
Ʉɚɤ ɡɚɜɢɫɢɬ ɪɚɞɢɭɫ ɜɪɚɳɟɧɢɹ ɱɚɫɬɢɰɵ ɨɬ ɟɟ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ?
352
ɅȺȻɈɊȺɌɈɊɇȺə ɊȺȻɈɌȺ ʋ Ɇɨɞɗ-02
ɉɈ ɂɁɍɑȿɇɂɘ ɆɈȾȿɅȿɃ ɎɂɁɂɑȿɋɄɂɏ
ɉɊɈɐȿɋɋɈȼ ɂ əȼɅȿɇɂɃ ɇȺ ɄɈɆɉɖɘɌȿɊȿ
Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ
ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
ɐɟɥɶ ɪɚɛɨɬɵ: ɢɡɭɱɟɧɢɟ ɞɜɢɠɟɧɢɹ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɚɪɚɥ-
ɥɟɥɶɧɵɯ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ. Ɉɩɪɟɞɟɥɟɧɢɟ ɨɬɧɨɲɟɧɢɹ
ɡɚɪɹɞɚ ɱɚɫɬɢɰɵ ɤ ɟɟ ɦɚɫɫɟ.
1. Ɍɟɨɪɟɬɢɱɟɫɤɨɟ ɫɨɞɟɪɠɚɧɢɟ
ɂɡɭɱɟɧɢɟ ɞɜɢɠɟɧɢɹ ɡɚɪɹɠɟɧɧɵɯ ɱɚɫɬɢɰ ɜ ɫɬɚɰɢɨɧɚɪɧɵɯ ɨɞɧɨɪɨɞ-
ɧɵɯ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ ɩɪɟɞɫɬɚɜɥɹɟɬ ɛɨɥɶɲɨɣ ɢɧɬɟɪɟɫ,
ɤɚɤ ɫ ɬɟɨɪɟɬɢɱɟɫɤɨɣ, ɬɚɤ ɢ ɫ ɩɪɚɤɬɢɱɟɫɤɨɣ ɬɨɱɟɤ ɡɪɟɧɢɹ.
ɗɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ ɧɚɡɵɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɵɦ, ɟɫɥɢ ɧɚɩɪɹɠɟɧ-
ɧɨɫɬɶ ɩɨɥɹ
E
G
ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɩɪɨɫɬɪɚɧɫɬɜɚ ɨɞɢɧɚɤɨɜɚ ɤɚɤ ɩɨ ɦɨɞɭɥɸ,
ɬɚɤ ɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ. Ɇɚɝɧɢɬɧɨɟ ɩɨɥɟ ɧɚɡɵɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɵɦ, ɟɫɥɢ
ɢɧɞɭɤɰɢɹ ɩɨɥɹ
B
G
ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɩɪɨɫɬɪɚɧɫɬɜɚ ɨɞɢɧɚɤɨɜɚ ɤɚɤ ɩɨ ɦɨɞɭ-
ɥɸ, ɬɚɤ ɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ. ȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɩɨɥɹ ɧɚɡɵɜɚɸɬɫɹ ɧɟɨɞ-
ɧɨɪɨɞɧɵɦɢ.
ɗɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ ɧɚɡɵɜɚɟɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦ, ɟɫɥɢ ɜ ɥɸɛɨɣ
ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɧɚɩɪɹɠɟɧɧɨɫɬɶ ɩɨɥɹ
E
G
ɧɟ ɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɧɢ
ɩɨ ɦɨɞɭɥɸ, ɧɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ. Ɇɚɝɧɢɬɧɨɟ ɩɨɥɟ ɧɚɡɵɜɚɟɬɫɹ ɫɬɚɰɢɨ-
ɧɚɪɧɵɦ, ɟɫɥɢ ɜ ɥɸɛɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢɧɞɭɤɰɢɹ ɩɨɥɹ
B
G
ɧɟ ɦɟɧɹɟɬ-
ɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɧɢ ɩɨ ɦɨɞɭɥɸ, ɧɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ.
ȼ ɤɭɪɫɟ ɨɛɳɟɣ ɮɢɡɢɤɢ ɩɨɞɪɨɛɧɨ ɪɚɫɫɦɨɬɪɟɧɵ ɨɫɨɛɟɧɧɨɫɬɢ ɞɜɢ-
ɠɟɧɢɹ ɧɟɪɟɥɹɬɢɜɢɫɬɫɤɨɣ ɱɚɫɬɢɰɵ ɦɚɫɫɨɣ
m ɢ ɫ ɡɚɪɹɞɨɦ q ɨɬɞɟɥɶɧɨ ɜ
ɨɞɧɨɪɨɞɧɨɦ ɫɬɚɰɢɨɧɚɪɧɨɦ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɩɨɥɟ ɢ ɜ ɨɞɧɨɪɨɞɧɨɦ ɫɬɚɰɢɨ-
ɧɚɪɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ.
ȿɫɥɢ ɱɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɢ ɜ ɷɥɟɤɬɪɢɱɟɫɤɨɦ, ɢ ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ, ɧɚ
ɧɟɟ ɞɟɣɫɬɜɭɟɬ ɫɢɥɚ
EqF
ɷɥ
G
G
ɫɨ ɫɬɨɪɨɧɵ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ ɢ ɫɢɥɚ
][ BqF
ɦɚɝ
G
G
G
v ɫɨ ɫɬɨɪɨɧɵ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ (ɫɢɥɚ Ʌɨɪɟɧɰɚ). ɉɪɟɞɩɨɥɨ-
ɠɢɦ, ɱɬɨ ɜɥɢɹɧɢɟ ɞɪɭɝɢɯ ɫɢɥ (ɝɪɚɜɢɬɚɰɢɨɧɧɵɯ, ɫɢɥ ɬɪɟɧɢɹ) ɧɚ ɱɚɫɬɢɰɭ
ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɨ. Ɍɨɝɞɚ ɫɨɝɥɚɫɧɨ ɜɬɨɪɨɦɭ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ ɢɡɦɟɧɟ-
ɧɢɟ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ
v
G
m
ɨɛɭɫɥɨɜɥɟɧɨ ɞɟɣɫɬɜɢɟɦ ɫɢɥ
ɷɥ
F
G
ɢ
ɦɚɝ
F
G
Ɇɨɞɗ-02 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ
ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
353
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
ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
ȿɫɥɢ ɱɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬ-
ɧɨɦ ɩɨɥɹɯ (
BE
G
G
|| ), ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ XYZ ɭɞɨɛɧɨ ɜɵɛɪɚɬɶ ɫɥɟɞɭɸɳɢɦ
ɨɛɪɚɡɨɦ. ɇɚɩɪɚɜɢɦ ɨɫɶ OY ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɜɟɤɬɨɪɚ ɢɧɞɭɤɰɢɢ ɦɚɝ-
ɧɢɬɧɨɝɨ ɩɨɥɹ
B
G
ɢ ɧɚɩɪɚɜɥɟɧɢɹ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ
E
G
.
ɇɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɫɨɜɦɟɫɬɢɦ ɫ ɩɨɥɨɠɟɧɢɟɦ ɱɚɫɬɢɰɵ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨ-
ɦɟɧɬ ɜɪɟɦɟɧɢ (ɪɢɫ. 1). ȼ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɜɟɤɬɨɪ ɧɚɩɪɹ-
ɠɟɧɧɨɫɬɢ 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
y
,
0
zx
BB . Ʉɨɨɪɞɢɧɚɬɵ ɱɚɫɬɢɰɵ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɜɧɵ
ɧɭɥɸ:
0)0()0()0(
z
y
x
.
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
354
ɇɚɱɚɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɦɨɠɧɨ ɪɚɡɥɨɠɢɬɶ ɧɚ ɞɜɟ ɫɨɫɬɚɜ-
ɥɹɸɳɢɟ: ɩɚɪɚɥɥɟɥɶɧɭɸ ɩɨɥɸ
||
v
G
, ɪɚɜɧɭɸ ɩɨ ɦɨɞɭɥɸ
)0(
y
v
; ɩɟɪɩɟɧɞɢ-
ɤɭɥɹɪɧɭɸ ɩɨɥɹ
A
v
G
, ɦɨɞɭɥɶ ɤɨɬɨɪɨɣ ɪɚɜɟɧ
22
)0()0(
zx
vvv
A
.
Ɍɨɝɞɚ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ, ɨɩɢɫɵɜɚɸɳɚɹ ɞɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ
ɱɚɫɬɢɰɵ ɦɚɫɫɨɣ
m ɢ ɡɚɪɹɞɨɦ q ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝ-
ɧɢɬɧɨɦ ɩɨɥɹɯ, ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ XYZ ɩɪɢɦɟɬ ɜɢɞ
°
°
°
¯
°
°
°
®
.
,
,
B
m
q
dt
d
E
m
q
dt
d
B
m
q
dt
d
x
z
y
z
x
v
v
v
v
v
ɂɡ ɩɨɥɭɱɟɧɧɨɣ ɫɢɫɬɟɦɵ
ɭɪɚɜɧɟɧɢɣ ɜɢɞɧɨ, ɱɬɨ ɩɪɨɟɤɰɢɹ
ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ ɧɚ ɨɫɶ OY
y
v
ɢɡɦɟɧɹɟɬɫɹ ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɞɜɭɯ
ɞɪɭɝɢɯ ɩɪɨɟɤɰɢɣ. ɗɬɨ ɭɪɚɜɧɟɧɢɟ
ɦɨɠɧɨ ɪɟɲɚɬɶ ɨɬɞɟɥɶɧɨ. ɉɪɨɢɧ-
ɬɟɝɪɢɪɨɜɚɜ ɭɪɚɜɧɟɧɢɟ ɞɥɹ
y
v
, ɢ
ɭɱɢɬɵɜɚɹ, ɱɬɨ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨ-
ɦɟɧɬ ɜɪɟɦɟɧɢ ɱɚɫɬɢɰɚ ɢɦɟɥɚ
y-
ɤɨɦɩɨɧɟɧɬɵ ɫɤɨɪɨɫɬɢ, ɪɚɜɧɭɸ
)0(
y
v
, ɩɨɥɭɱɢɦ
Et
m
q
yy
)0(vv
.
Ɉɫɬɚɥɶɧɵɟ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɪɟɲɚɬɶ ɫɨɜɦɟɫɬɧɨ
°
°
¯
°
°
®
,
,
B
m
q
dt
d
B
m
q
dt
d
x
z
z
x
v
v
v
v
ɢɥɢ
°
°
¯
°
°
®
.
,
)/(
1
B
m
q
dt
d
dt
d
Bmq
x
z
z
x
v
v
v
v
ɑɬɨɛɵ ɪɟɲɢɬɶ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ, ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦ ɩɟɪɜɨɟ
ɭɪɚɜɧɟɧɢɟ ɩɨ ɜɪɟɦɟɧɢ ɢ ɩɨɞɫɬɚɜɢɦ ɜɨ ɜɬɨɪɨɟ
E
B
v
(
0
)
x
y
z
O
v
x
(0)
v
y
(0)
v
z
(0)
= v
~~
v
A
Ɋɢɫ. 1
Ɇɨɞɗ-02 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ
ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
355
B
m
q
dt
d
dt
d
Bmq
x
z
x
v
v
v
2
2
)/(
1
,
ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɞɥɹ ɨɞɧɨɣ ɧɟ-
ɢɡɜɟɫɬɧɨɣ
x
v
x
x
B
m
q
dt
d
v
v
2
2
2
¸
¹
·
¨
©
§
.
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɚɧɚɥɨɝɢɱɧɨ ɭɪɚɜɧɟɧɢɸ, ɨɩɢɫɵɜɚɸɳɟɦɭ ɝɚɪɦɨɧɢɱɟɫɤɢɟ
ɤɨɥɟɛɚɧɢɹ. ɉɨɷɬɨɦɭ ɪɟɲɟɧɢɟ ɩɨɥɭɱɟɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɦɨɠɧɨ ɢɫɤɚɬɶ ɜ
ɜɢɞɟ
)sin( MZ tA
x
v , ɝɞɟ
A
,
Z
,
M
– ɢɫɤɨɦɵɟ ɤɨɧɫɬɚɧɬɵ. ɉɨɞɫɬɚɜɢɜ
x
v ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɢɦ
xx
x
B
m
q
tA
dt
d
vv
v
2
22
2
2
)sin(
¸
¹
·
¨
©
§
Z MZZ
.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
B
m
q
Z .
ɍɱɢɬɵɜɚɹ, ɱɬɨ
dt
d
dt
d
Bmq
xx
z
vv
v
Z
1
)/(
1
, ɩɨɥɭɱɢɦ
)sin( MZ tA
x
v ,
)cos( MZ tA
z
v .
Ɂɧɚɱɟɧɢɹ ɤɨɧɫɬɚɧɬ
A
ɢ
M
ɧɚɣɞɟɦ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ. ȼ ɦɨ-
ɦɟɧɬ ɜɪɟɦɟɧɢ
0 t ɱɚɫɬɢɰɚ ɢɦɟɥɚ ɫɤɨɪɨɫɬɢ )0(
x
v ɢ )0(
z
v :
M sin)0( A
x
v
ɢ
M cos)0( A
z
v
,
ɢɥɢ
)0(sin
x
A v M ɢ )0(cos
z
A v M .
Ɍɨɝɞɚ ɢɡ ɨɬɧɨɲɟɧɢɹ ɷɬɢɯ ɭɪɚɜɧɟɧɢɣ
)0(
)0(
tg
z
x
v
v
M
,
ɚ ɢɡ ɫɭɦɦɵ ɤɜɚɞɪɚɬɨɜ
222
2
)0()0(
A
vvv
zx
A .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɚɜɢɫɢɦɨɫɬɢ ɩɪɨɟɤɰɢɣ ɫɤɨɪɨɫɬɢ ɨɬ ɜɪɟɦɟɧɢ ɨɩɢ-
ɫɵɜɚɸɬɫɹ ɫɥɟɞɭɸɳɢɦɢ ɜɵɪɚɠɟɧɢɹɦɢ
)sin( MZ tA
x
v ,
Et
m
q
yy
)0(vv ,
)cos( MZ tA
z
v .
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
356
ɉɪɨɢɧɬɟɝɪɢɪɨɜɚɜ ɷɬɢ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɩɪɨɟɤɰɢɣ ɫɤɨɪɨɫɬɢ, ɩɨɥɭɱɢɦ
ɡɚɜɢɫɢɦɨɫɬɶ ɤɨɨɪɞɢɧɚɬ
x
, y ɢ
z
ɨɬ ɜɪɟɦɟɧɢ
t
)cos()( MZ
Z
t
A
Ctx
x
,
2
2
1
)0()0()(
Et
m
q
tyty
y
v ,
)sin()( MZ
Z
t
A
Ctz
z
.
ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɱɚɫɬɢɰɚ ɧɚɯɨɞɢɥɚɫɶ ɜ ɧɚɱɚɥɟ ɤɨɨɪ-
ɞɢɧɚɬ
0)0(
x
, 0)0( y , 0)0(
z
. ɉɨɷɬɨɦɭ
)0(
1
cos
zx
A
C
v
Z
M
Z
ɢ
)0(
1
sin
xz
A
C
v
Z
M
Z
.
ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɟ
Z
A
R
. Ɍɨɝɞɚ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɦɨɠ-
ɧɨ ɡɚɩɢɫɚɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ
)cos()0(
1
)( MZ
Z
tRtx
z
v ,
2
2
1
)0()(
Et
m
q
tty
y
v ,
)sin()0(
1
)( MZ
Z
tRtz
x
v .
1.2. Ⱥɧɚɥɢɡ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ
Ⱦɥɹ ɚɧɚɥɢɡɚ ɯɚɪɚɤɬɟɪɚ ɬɪɚɟɤɬɨɪɢɢ ɩɪɨɜɟɞɟɦ ɫɥɟɞɭɸɳɢɟ ɩɪɟɨɛɪɚ-
ɡɨɜɚɧɢɹ. Ɂɚɤɨɧɵ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜɞɨɥɶ ɨɫɟɣ OX ɢ OZ
)cos( MZ tRCx
x
,
)sin( MZ tRCz
z
.
ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ
)cos( MZ tRCx
x
,
)sin( MZ tRCz
z
.
Ɉɛɚ ɭɪɚɜɧɟɧɢɹ ɜɨɡɜɟɞɟɦ ɜ ɤɜɚɞɪɚɬ ɢ ɫɥɨɠɢɦ
))(sin)((cos
222
22
MZMZ ttRCzCx
zx
,
2
22
RCzCx
zx
ɢɥɢ
2
22
Rzzxx
RR
,
ɝɞɟ
xR
Cx ,
zR
Cz .
Ɇɨɞɗ-02 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ
ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
357
ɉɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɨɤɪɭɠɧɨɫɬɢ ɫ ɰɟɧ-
ɬɪɨɦ ɜ ɬɨɱɤɟ (
x
R
, z
R
) ɢ ɪɚɞɢɭɫɨɦ R ɜ ɩɥɨɫɤɨɫɬɢ XOZ.
Ɉɞɧɨɜɪɟɦɟɧɧɨ ɱɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɪɚɜɧɨɭɫɤɨɪɟɧɧɨ ɜɞɨɥɶ ɨɫɢ OY:
2
2
1
)0()(
Et
m
q
tty
y
v
.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɭɩɟɪɩɨɡɢɰɢɸ
ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɜ ɩɥɨɫɤɨɫɬɢ XOZ ɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ
ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ OY.
ɉɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ:
ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
E
m
q
a
ɩɨɫɬ
2
1
ɢ ɪɚɜɧɨɦɟɪɧɨ ɦɟɧɹɸɳɟɣɫɹ ɫɤɨɪɨɫɬɶɸ
Et
m
q
yyɩɨɫɬ
{ )0(vvv .
ȼɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ:
ɩɨɫɬɨɹɧɧɨɣ ɱɚɫɬɨɬɨɣ
B
m
q
Z
,
ɪɚɞɢɭɫɨɦ
22
)0()0(
1
zx
R vv
Z
,
ɧɚɱɚɥɶɧɨɣ ɮɚɡɨɣ ɜɪɚɳɟɧɢɹ
M, ɞɥɹ ɤɨɬɨɪɨɣ
)0(
)0(
tg
z
x
v
v
M
,
ɢ ɩɨɥɨɠɟɧɢɟɦ ɰɟɧɬɪɚ ɜɪɚɳɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢ-
ɧɚɬ XOZ:
)0(
1
zx
C v
Z
,
)0(
1
xz
C v
Z
.
Ʌɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɪɚɜɧɚ
A
Z vvvv
22
)0()0(
zxɜɪ
R ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɧɟɲɧɢɯ ɩɨɥɟɣ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ
ɩɨɥɹɯ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɞɜɢɠɟɧɢɟ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɜ ɩɥɨɫɤɨɫɬɢ XOZ,
ɰɟɧɬɪ ɤɨɬɨɪɨɣ ɪɚɜɧɨɭɫɤɨɪɟɧɧɨ ɞɜɢɠɟɬɫɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ OY (ɪɢɫ. 2).
Ɍɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɬɪɟɯɦɟɪɧɭɸ ɤɪɢɜɭɸ,
ɤɨɬɨɪɚɹ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢ ɨɩɢɫɚɧɚ ɩɨɥɭɱɟɧɧɵɦɢ ɡɚɤɨɧɚɦɢ ɞɜɢɠɟɧɢɹ, ɢ
ɧɚɡɵɜɚɟɬɫɹ
ɜɢɧɬɨɜɨɣ ɥɢɧɢɟɣ.
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
358
ɇɚɩɪɹɠɟɧɧɨɫɬɶ
ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ
E
ɢ
ɢɧɞɭɤɰɢɹ ɦɚɝɧɢɬɧɨɝɨ ɩɨ-
ɥɹ
B
ɨɤɚɡɵɜɚɸɬ ɧɟɡɚɜɢ-
ɫɢɦɨɟ ɜɥɢɹɧɢɟ ɧɚ ɯɚɪɚɤ-
ɬɟɪ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ.
ɂɡɦɟɧɟɧɢɟ ɢɧɞɭɤ-
ɰɢɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ
B
ɨɤɚɡɵɜɚɟɬ ɜɥɢɹɧɢɟ ɬɨɥɶɤɨ
ɧɚ ɜɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟ-
ɧɢɟ ɱɚɫɬɢɰɵ, ɦɟɧɹɹ ɟɝɨ
ɱɚɫɬɨɬɭ
Z ɢ ɪɚɞɢɭɫ
R
.
ɂɡɦɟɧɟɧɢɟ ɧɚɩɪɹɠɟɧɧɨ-
ɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ
E
ɨɤɚɡɵɜɚɟɬ ɜɥɢɹɧɢɟ
ɬɨɥɶɤɨ ɧɚ ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ, ɦɟɧɹɹ ɭɫɤɨɪɟɧɢɟ
ɩɨɫɬ
a . ȿɫɥɢ ɩɪɨ-
ɟɤɰɢɹ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ
EE
y
ɢ ɡɚɪɹɞ ɱɚɫɬɢɰɵ q
ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɵɟ ɡɧɚɤɢ, ɬɨ ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɛɭɞɟɬ ɩɪɨɢɫɯɨ-
ɞɢɬɶ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
0!
ɩɨɫɬ
a . Y-ɤɨɦɩɨɧɟɧɬɚ ɫɤɨɪɨɫɬɢ
ɛɭɞɟɬ ɜɨɡɪɚɫɬɚɬɶ ɫɨ ɜɪɟɦɟɧɟɦ. Ɋɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɜɢɬɤɚɦɢ ɜɢɧɬɨɜɨɣ ɥɢ-
ɧɢɢ ɛɭɞɟɬ ɫɬɚɧɨɜɢɬɶɫɹ ɜɫɟ ɛɨɥɶɲɟ ɢ ɛɨɥɶɲɟ – ɜɢɬɤɢ «ɪɚɫɬɹɝɢɜɚɸɬɫɹ».
ȿɫɥɢ ɩɪɨɟɤɰɢɹ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ
EE
y
ɢ ɡɚɪɹɞ ɱɚɫ-
ɬɢɰɵ
q ɢɦɟɸɬ ɪɚɡɧɵɟ ɡɧɚɤɢ, ɬɨ ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɛɭɞɟɬ ɩɪɨɢɫ-
ɯɨɞɢɬɶ ɫ ɨɬɪɢɰɚɬɟɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
0
ɩɨɫɬ
a
. Y-ɤɨɦɩɨɧɟɧɬɚ ɫɤɨɪɨɫɬɢ
ɫɧɚɱɚɥɚ ɛɭɞɟɬ ɭɛɵɜɚɬɶ ɞɨ ɧɭɥɹ, ɚ ɡɚɬɟɦ ɬɟɥɨ ɧɚɱɧɟɬ ɞɜɢɝɚɬɶɫɹ ɜ ɩɪɨɬɢ-
ɜɨɩɨɥɨɠɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ. Ɋɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɜɢɬɤɚɦɢ ɜɢɧɬɨɜɨɣ ɥɢɧɢɢ
ɛɭɞɟɬ ɫɧɚɱɚɥɚ ɭɦɟɧɶɲɚɬɶɫɹ (ɜɢɬɤɢ «ɫɠɢɦɚɸɬɫɹ»), ɚ ɡɚɬɟɦ ɧɚɱɧɟɬ ɭɜɟɥɢ-
ɱɢɜɚɬɶɫɹ (ɜɢɬɤɢ «ɪɚɫɬɹɝɢɜɚɸɬɫɹ») ɩɪɢ ɞɜɢɠɟɧɢɢ ɜ ɨɛɪɚɬɧɭɸ ɫɬɨɪɨɧɭ.
ȿɫɥɢ ɧɚɩɪɹɠɟɧɧɨɫɬɶ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ
E
ɢ ɢɧɞɭɤɰɢɹ ɦɚɝɧɢɬ-
ɧɨɝɨ ɩɨɥɹ
B
ɨɬɥɢɱɧɵ ɨɬ ɧɭɥɹ, ɞɜɢɠɟɧɢɟ ɹɜɥɹɟɬɫɹ ɬɪɟɯɦɟɪɧɵɦ. ɑɚɫɬɢɰɚ
ɛɟɡ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ
0)0( v
G
ɛɭɞɟɬ ɞɜɢɝɚɬɶɫɹ ɪɚɜɧɨɭɫɤɨɪɟɧɧɨ ɜɞɨɥɶ
ɨɫɢ OY. ȿɫɥɢ ɱɚɫɬɢɰɚ ɢɦɟɟɬ ɨɬɥɢɱɧɭɸ ɨɬ ɧɭɥɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɭɸ ɫɨ-
ɫɬɚɜɥɹɸɳɭɸ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ (
0z
A
v ), ɨɧɚ ɛɭɞɟɬ ɞɜɢɝɚɬɶɫɹ ɩɨ ɜɢɧ-
ɬɨɜɨɣ ɥɢɧɢɢ. Ɋɚɞɢɭɫ ɜɢɧɬɨɜɨɣ ɥɢɧɢɢ ɡɚɜɢɫɢɬ ɨɬ ɦɨɞɭɥɹ ɩɟɪɩɟɧɞɢɤɭ-
ɥɹɪɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɢ ɨɬ ɢɧɞɭɤɰɢɢ ɦɚɝɧɢɬɧɨɝɨ
ɩɨɥɹ.
R
C
x
C
z
M
v
ɩɨɫɬ
v
(
0
)
x
y
z
v
ɩɨɫɬ
Ɋɢɫ. 2
Ɇɨɞɗ-02 Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ
ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ
359
2. Ɋɚɛɨɱɢɟ ɮɨɪɦɭɥɵ
Ⱦɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɹɜɥɹɟɬɫɹ ɩɟɪɢɨɞɢɱɟɫɤɢɦ ɫ ɩɟɪɢɨɞɨɦ
Bmq
T
)/(
22 S
Z
S
.
ȿɫɥɢ ɩɚɪɚɥɥɟɥɶɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɫɤɨɪɨɫɬɢ ɪɚɜɧɚ ɧɭɥɸ
(
0)0(
y
v ), ɬɨ y-ɤɨɨɪɞɢɧɚɬɚ ɱɚɫɬɢɰɵ ɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ
2
2
1
)(
Et
m
q
ty
.
Ɂɚ ɤɚɠɞɵɣ ɩɨɥɭɩɟɪɢɨɞ ɜɪɚɳɟɧɢɹ
2/
T
ɱɚɫɬɢɰɚ ɫɨɜɟɪɲɚɟɬ ɩɨɥɭɨɛɨɪɨɬ ɢ
ɫɦɟɳɚɟɬɫɹ ɩɨ ɨɫɢ OY ɧɚ ɧɟɤɨɬɨɪɨɟ ɪɚɫɫɬɨɹɧɢɟ. Ɂɚ
n-ɵɣ ɩɨɥɭɩɟɪɢɨɞ
(ɩɨɥɭ ɜɢɬɨɤ)
y-ɤɨɨɪɞɢɧɚɬɚ ɢɡɦɟɧɢɬɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɟ
2
)12(
8
1
2
)1(
2
TnE
m
qT
ny
T
nyd
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
ɢɥɢ
)12(
)/(
2
2
2
S
n
Bmq
E
d
,
ɩɪɨɣɞɹ ɩɪɢ ɷɬɨɦ ɩɨɥɜɢɬɨɤ ɜɢɧɬɨɜɨɣ ɥɢɧɢɢ. Ɋɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɬɨɱɤɚɦɢ
ɜɢɧɬɨɜɨɣ ɥɢɧɢɢ, ɢɦɟɸɳɢɦɢ ɨɞɢɧɚɤɨɜɭɸ ɮɚɡɭ ɜɪɚɳɟɧɢɹ, ɧɚɡɵɜɚɟɬɫɹ
ɲɚɝɨɦ ɜɢɧɬɨɜɨɣ ɥɢɧɢɢ. ɒɚɝ ɜɢɧɬɨɜɨɣ ɥɢɧɢɢ – ɷɬɨ ɩɪɢɪɚɳɟɧɢɟ y-
ɤɨɨɪɞɢɧɚɬɵ ɱɚɫɬɢɰɵ ɡɚ ɩɨɥɧɵɣ ɩɟɪɢɨɞ ɜɪɚɳɟɧɢɹ
T
. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
d
– ɞɥɢɧɚ ɩɨɥɭɲɚɝɚ ɜɢɧɬɨɜɨɣ ɥɢɧɢɢ. ȿɫɥɢ ɧɚɩɪɹɠɟɧɧɨɫɬɶ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ
ɩɨɥɹ ɨɬɥɢɱɧɚ ɨɬ ɧɭɥɹ, ɬɨ ɞɥɢɧɚ ɩɨɥɭɲɚɝɚ ɜɢɧɬɨɜɨɣ ɥɢɧɢɢ ɪɚɡɥɢɱɧɚ ɞɥɹ
ɪɚɡɧɵɯ ɩɨɥɭɜɢɬɤɨɜ (ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɧɨɦɟɪɭ ɜɢɬɤɚ
12 n
). Ɂɧɚɹ
ɧɨɦɟɪ n ɩɨɥɭɜɢɬɤɚ ɢ ɢɡɦɟɪɢɜ ɞɥɢɧɭ ɩɨɥɭɲɚɝɚ
d , ɦɨɠɧɨ ɧɚɣɬɢ ɨɬɧɨɲɟ-
ɧɢɟ ɡɚɪɹɞɚ ɱɚɫɬɢɰɵ ɤ ɟɟ ɦɚɫɫɟ
)12(
2
2
2
S
n
Bd
E
m
q
.
ȼ ɩɪɨɟɤɰɢɢ ɧɚ ɩɥɨɫɤɨɫɬɶ XOZ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɩɪɟɞɫɬɚɜɥɹɟɬ
ɫɨɛɨɣ ɞɜɢɠɟɧɢɟ ɩɨ ɤɪɭɝɭ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɹɜɥɹɟɬɫɹ ɨɝɪɚɧɢɱɟɧɧɵɦ. Ʉɨ-
ɨɪɞɢɧɚɬɚ
)cos()( MZ tRCtx
x
ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ ɨɬ RCx
x
min
(ɤɨɝɞɚ
1cos ) ɞɨ RCx
x
max
(ɤɨɝɞɚ 1cos ). ɂɡɦɟɪɢɜ
min
x ɢ
max
x ,
ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɪɚɞɢɭɫ ɜɪɚɳɟɧɢɹ
)(
minmax
2
1
xxR . Ʉɨɨɪɞɢɧɚɬɚ
)sin()( MZ tRCtz
z
ɬɨɠɟ ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ ɨɬ RCz
z
min
(ɤɨ-
ɝɞɚ
1sin
) ɞɨ
RCz
z
max
(ɤɨɝɞɚ
1sin
). ɉɨɷɬɨɦɭ ɪɚɞɢɭɫ ɜɪɚɳɚ-
ɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɦɨɠɧɨ ɬɚɤɠɟ ɨɩɪɟɞɟɥɢɬɶ ɢ ɩɨ ɡɧɚɱɟɧɢɹɦ
min
z ɢ
max
z
: )(
minmax
2
1
zzR .
ɗɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ
360
Ɂɚɜɢɫɢɦɨɫɬɶ ɪɚɞɢɭɫɚ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɨɬ ɩɟɪ-
ɩɟɧɞɢɤɭɥɹɪɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɧɨɫɢɬ ɥɢɧɟɣɧɵɣ ɯɚ-
ɪɚɤɬɟɪ
Z
Z
A
v
vv
22
)0()0(
1
zx
R (ɨɬ ɩɚɪɚɥɥɟɥɶɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ
ɫɤɨɪɨɫɬɢ ɪɚɞɢɭɫ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɧɟ ɡɚɜɢɫɢɬ). Ɍɚɧɝɟɧɫ ɭɝɥɚ ɧɚ-
ɤɥɨɧɚ ɩɪɹɦɨɣ
)(
A
vRR ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ Z:
Z
D
1
tg
. ɑɬɨ
ɬɚɤɠɟ ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɢɬɶ 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 ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ ɨɞɧɨɪɨɞɧɵɯ
ɫɬɚɰɢɨɧɚɪɧɵɯ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɹɯ. ɇɚɩɪɹɠɟɧɧɨɫɬɶ ɷɥɟɤ-
ɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ ɦɨɠɧɨ ɢɡɦɟɧɹɬɶ ɜ ɢɧɬɟɪɜɚɥɟ ɨɬ –200 ɞɨ 200 ȼ/ɫɦ, ɢɧ-
ɞɭɤɰɢɸ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ – ɜ ɢɧɬɟɪɜɚɥɟ ɨɬ 0 ɞɨ 200 ɦɌɥ. ɇɚɱɚɥɶɧɚɹ ɫɤɨ-
ɪɨɫɬɶ ɱɚɫɬɢɰɵ ɡɚɞɚɟɬɫɹ ɩɭɬɟɦ ɢɡɦɟɧɟɧɢɹ ɟɟ
ɩɪɨɟɤɰɢɣ ɧɚ ɨɫɢ OX, OY ɢ