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8.5. ɑɚɫɬɢɰɚ ɫɨ ɫɩɢɧɨɦ 2
/
1
s
ɢ ɦɚɝɧɢɬɧɵɦ ɦɨɦɟɧɬɨɦ ɧɚɯɨɞɢɬɫɹ ɜ
ɨɞɧɨɪɨɞɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ . ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫɩɢɧɨɜɨɟ
ɫɨɫɬɨɹɧɢɟ ɱɚɫɬɢɰɵ ɢɦɟɥɨ ɜɢɞ
0
P
B
p
T
n
T
F
2
sin
2
cos)0( ,
ɝɞɟ
n ɢ p – ɫɨɫɬɨɹɧɢɹ ɫɨ ɫɩɢɧɨɦ ɜɞɨɥɶ ɢ ɩɪɨɬɢɜ ɧɚɩɪɚɜɥɟɧɢɹ ,
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɇɚɣɬɢ:
B
ɚ) ɋɩɢɧɨɜɨɟ ɫɨɫɬɨɹɧɢɟ ɱɚɫɬɢɰɵ
)(tF ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
ɜɪɟɦɟɧɢ.
ɛ) Ɂɚɜɢɫɢɦɨɫɬɶ ɨɬ ɜɪɟɦɟɧɢ ɭɫɪɟɞɧɟɧɧɨɝɨ ɜɟɤɬɨɪɚ ɫɩɢɧɚ.
ɜ) ɋɬɚɰɢɨɧɚɪɧɵɟ ɫɩɢɧɨɜɵɟ ɫɨɫɬɨɹɧɢɹ ɱɚɫɬɢɰɵ.
8.6. Ⱦɥɹ ɫɢɫɬɟɦɵ ɢɡ ɞɜɭɯ ɱɚɫɬɢɰ ɫɨ ɫɩɢɧɨɦ 2/1
21
ss ɧɚɣɬɢ ɫɨɫɬɨɹɧɢɹ
z
SS, ɫ ɨɩɪɟɞɟɥɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɫɭɦɦɚɪɧɨɝɨ ɫɩɢɧɚ
S
ɢ ɩɪɨɟɤɰɢɢ
ɫɭɦɦɚɪɧɨɝɨ ɫɩɢɧɚ ; ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɛɚɡɢɫɧɵɦɢ ɫɨɫɬɨɹɧɢɹɦɢ ɜɢɞɚ
z
S
21
VV .
8.7. ɚ) Ⱦɥɹ ɷɥɟɤɬɪɨɧɚ ɜ ɫɨɫɬɨɹɧɢɹɯ ɫ ɪɚɡɥɢɱɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ
ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
l
ɩɟɪɟɱɢɫɥɢɬɶ ɜɨɡɦɨɠɧɵɟ ɫɨɫɬɨɹɧɢɹ ɫ ɨɩɪɟɞɟɥɟɧɧɵɦ
ɡɧɚɱɟɧɢɟɦ ɩɨɥɧɨɝɨ ɦɨɦɟɧɬɚ , ɩɪɟɞɫɬɚɜɥɹɸɳɟɝɨ ɫɨɛɨɣ ɤɜɚɧɬɨɜɨ-
ɦɟɯɚɧɢɱɟɫɤɭɸ ɫɭɦɦɭ ɨɪɛɢɬɚɥɶɧɨɝɨ ɢ ɫɩɢɧɨɜɨɝɨ ɦɨɦɟɧɬɨɜ ɷɥɟɤɬɪɨɧɚ. ɛ) ɇɚɣɬɢ
ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɨɩɟɪɚɬɨɪɚ ɷɧɟɪɝɢɢ «ɫɩɢɧ-ɨɪɛɢɬɚɥɶɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ»
ɜɢɞɚ , ɝɞɟ 0
j
sl
ˆ
ˆ
A !
A
– ɧɟɤɨɬɨɪɚɹ ɩɨɫɬɨɹɧɧɚɹ ɫ ɪɚɡɦɟɪɧɨɫɬɶɸ ɷɧɟɪɝɢɢ.
8.8. Ɉɩɟɪɚɬɨɪ ɷɧɟɪɝɢɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɚɝɧɢɬɧɵɯ ɦɨɦɟɧɬɨɜ ɷɥɟɤɬɪɨɧɚ ɢ
ɩɪɨɬɨɧɚ ɜ
s
1-ɫɨɫɬɨɹɧɢɢ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɶɤɨ ɫɩɢɧɨɜɵɦɢ
ɦɨɦɟɧɬɚɦɢ ɷɥɟɤɬɪɨɧɚ ɢ ɚɬɨɦɧɨɝɨ ɹɞɪɚ (ɬɚɤ ɤɚɤ ɨɪɛɢɬɚɥɶɧɵɣ ɦɨɦɟɧɬ ɷɥɟɤɬɪɨɧɚ
ɜ
s
-ɫɨɫɬɨɹɧɢɢ ɪɚɜɟɧ ɧɭɥɸ) ɢ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧ ɜ ɜɢɞɟ , ɝɞɟ
21
ˆˆ
ıı C 0!C –
ɩɨɫɬɨɹɧɧɚɹ ɫ ɪɚɡɦɟɪɧɨɫɬɶɸ ɷɧɟɪɝɢɢ, ɡɚɜɢɫɹɳɚɹ ɨɬ ɡɧɚɱɟɧɢɣ ɦɚɝɧɢɬɧɵɯ
ɦɨɦɟɧɬɨɜ ɷɥɟɤɬɪɨɧɚ ɢ ɹɞɪɚ, ɢ – ɦɚɬɪɢɰɵ ɉɚɭɥɢ, ɨɬɧɨɫɹɳɢɟɫɹ ɤ ɫɩɢɧɚɦ
ɷɥɟɤɬɪɨɧɚ ɢ ɹɞɪɚ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɇɚɣɬɢ ɞɥɹ ɭɤɚɡɚɧɧɨɝɨ ɨɩɟɪɚɬɨɪɚ
ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢ ɫɨɛɫɬɜɟɧɧɵɟ ɫɨɫɬɨɹɧɢɹ.
1
ˆ
ı
2
ˆ
ı
8.9.
Ⱦɥɹ ɫɢɫɬɟɦɵ ɢɡ ɞɜɭɯ ɷɥɟɤɬɪɨɧɨɜ, ɧɚɯɨɞɹɳɢɯɫɹ ɜ ɫɨɫɬɨɹɧɢɢ ɫ
ɫɭɦɦɚɪɧɵɦ ɫɩɢɧɨɦ 0
S
, ɡɚɩɢɫɚɬɶ ɨɪɛɢɬɚɥɶɧɭɸ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ
ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɫɭɳɟɫɬɜɭɸɬ ɨɞɧɨɷɥɟɤɬɪɨɧɧɵɟ ɨɪɛɢɬɚɥɶɧɵɟ ɫɨɫɬɨɹɧɢɹ
ɢ ). ɍɤɚɡɚɧɢɹ: ɭɱɟɫɬɶ ɩɪɢɧɰɢɩ ɧɟɪɚɡɥɢɱɢɦɨɫɬɢ ɨɞɢɧɚɤɨɜɵɯ ɱɚɫɬɢɰ
),(
21
rr\
)(
r
p
M (r
q
M
80
(ɬɪɟɛɨɜɚɧɢɟ ɚɧɬɢɫɢɦɦɟɬɪɢɢ ɩɨɥɧɨɣ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɬɨɠɞɟɫɬɜɟɧɧɵɯ
ɮɟɪɦɢɨɧɨɜ); ɞɥɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɫɱɢɬɚɬɶ ɩɪɢɦɟɧɢɦɵɦ ɪɚɡɞɟɥɶɧɨɟ ɨɩɢɫɚɧɢɟ
ɨɪɛɢɬɚɥɶɧɨɝɨ ɢ ɫɩɢɧɨɜɨɝɨ ɫɨɫɬɨɹɧɢɣ.
8.10. ɉɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɫɢɫɬɟɦɟ ɢɡ ɞɜɭɯ ɷɥɟɤɬɪɨɧɨɜ ɫ ɨɞɧɨɱɚɫɬɢɱɧɵɦɢ
ɨɪɛɢɬɚɥɶɧɵɦɢ ɫɨɫɬɨɹɧɢɹɦɢ ɢ ))(
r
p
M (r
q
M ɡɚɩɢɫɚɬɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɪɟɞɧɟɝɨ
ɡɧɚɱɟɧɢɹ ɧɟ ɡɚɜɢɫɹɳɟɣ ɨɬ ɫɩɢɧɨɜ ɱɚɫɬɢɰ ɷɧɟɪɝɢɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
)(
21
rr U ɜ
ɞɜɭɯ ɫɥɭɱɚɹɯ: ɚ) ɫɭɦɦɚɪɧɵɣ ɫɩɢɧ ɫɢɫɬɟɦɵ 0
S
, ɛ) ɫɭɦɦɚɪɧɵɣ ɫɩɢɧ ɫɢɫɬɟɦɵ
1
S
.
Ɋɟɲɟɧɢɹ
8.1. Ɉɩɟɪɚɬɨɪ ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ ɧɚ ɨɫɶ
x
ɜ ɛɚɡɢɫɟ
z
n ,
z
p ɢɦɟɟɬ ɜɢɞ
ɦɚɬɪɢɰɵ
¸
¸
¹
·
¨
¨
©
§
V
02/1
2/10
ˆ
2
1
ˆ
xx
s . (1)
ɋɨɫɬɨɹɧɢɹ
F ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɪɨɟɤɰɢɟɣ ɫɩɢɧɚ ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ
ɭɪɚɜɧɟɧɢɸ
x
s
F F
xx
ss
ˆ
. (2)
Ɂɚɩɢɫɚɜ ɫɩɢɧɨɪ
F ɜ ɮɨɪɦɟ ɪɚɡɥɨɠɟɧɢɹ ɩɨ ɛɚɡɢɫɧɵɦ ɫɨɫɬɨɹɧɢɹɦ ɫ
ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɪɨɟɤɰɢɟɣ ɫɩɢɧɚ ,
z
s
zz
pFnF F
pn
, (3)
ɩɪɟɞɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ (2) ɜ ɜɢɞɟ ɫɢɫɬɟɦɵ ɨɞɧɨɪɨɞɧɵɯ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɞɥɹ
ɧɟɢɡɜɟɫɬɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɢ
n
F
p
F :
¯
®
FF
FF
pn
pn
0)2/1(
0)2/1(
x
x
s
s
(4)
Ɍɚɤɚɹ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ ɩɪɢ ɭɫɥɨɜɢɢ
0)1
ˆ
ˆ
(det
xx
ss ,
ɬɨ ɟɫɬɶ
04/1
2
x
s . (5)
81
Ɉɬɫɸɞɚ ɦɵ ɧɚɯɨɞɢɦ ɞɜɚ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɹ ɨɩɟɪɚɬɨɪɚ ; ɨɧɢ ɹɜɥɹɸɬɫɹ
ɞɨɩɭɫɬɢɦɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ ɧɚ ɨɫɶ
x
s
ˆ
x
:
2/1,2/1
x
s . (6)
Ʉɨɧɟɱɧɨ, ɷɬɨɬ ɪɟɡɭɥɶɬɚɬ ɡɚɪɚɧɟɟ ɨɱɟɜɢɞɟɧ, ɬɚɤ ɤɚɤ ɡɞɟɫɶ ɢɞɟɬ ɪɟɱɶ ɨ ɫɩɟɤɬɪɟ
ɡɧɚɱɟɧɢɣ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ( jjj ,...,1,)ɩɪɢ ɜɟɥɢɱɢɧɟ ɦɨɦɟɧɬɚ
2
/
1
s
j ; ɫɩɟɤɬɪ ɡɧɚɱɟɧɢɣ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɵɛɨɪɚ
ɧɚɩɪɚɜɥɟɧɢɹ, ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɤɨɬɨɪɨɦɭ ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɚɹ
ɩɪɨɟɤɰɢɹ.
ɉɨɞɫɬɚɜɢɜ ɜ (4) ɩɟɪɜɨɟ ɢɡ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ (6), ɩɨɥɭɱɢɦ:
pn
F F .
ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɧɨɪɦɢɪɨɜɚɧɧɨɟ ɧɚ ɟɞɢɧɢɰɭ ɫɩɢɧɨɜɨɟ ɫɨɫɬɨɹɧɢɟ
x
n{F ɫ
ɩɪɨɟɤɰɢɟɣ ɫɩɢɧɚ 2/1
x
s ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ:
zzx
pn n
2
1
2
1
. (7)
ȼɬɨɪɨɟ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ,2/1
x
s , ɩɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɜ (4) ɞɚɟɬ:
.ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɨɪɦɢɪɨɜɚɧɧɵɣ ɫɩɢɧɨɪ, ɩɪɢɧɚɞɥɟɠɚɳɢɣ ɷɬɨɦɭ
ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ, ɢɦɟɟɬ ɜɢɞ:
pn
F F
zzx
pn p
2
1
2
1
. (8)
Ɉɬɦɟɬɢɦ, ɱɬɨ ɫɩɢɧɨɪɵ (7) ɢ (8) ɜɡɚɢɦɧɨ ɨɪɬɨɝɨɧɚɥɶɧɵ ɢ ɦɨɝɭɬ ɛɵɬɶ
ɜɵɛɪɚɧɵ ɜ ɤɚɱɟɫɬɜɟ ɧɨɜɨɣ ɩɚɪɵ ɛɚɡɢɫɧɵɯ ɫɨɫɬɨɹɧɢɣ ɞɥɹ ɨɩɢɫɚɧɢɹ ɫɩɢɧɨɜɨɣ
ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ ɱɚɫɬɢɰɵ ɫɨ ɫɩɢɧɨɦ 2
/
1
s
. ȼ ɤɚɠɞɨɦ ɢɡ ɷɬɢɯ ɫɨɫɬɨɹɧɢɣ
ɜɟɪɨɹɬɧɨɫɬɶ ɨɛɧɚɪɭɠɟɧɢɹ ɥɸɛɨɝɨ ɢɡ ɡɧɚɱɟɧɢɣ 2/1r
z
s ɫɨɫɬɚɜɥɹɟɬ 50%.
ɉɨɞɨɛɧɭɸ ɫɜɹɡɶ ɦɟɠɞɭ ɞɜɭɦɹ ɩɚɪɚɦɢ ɛɚɡɢɫɧɵɯ ɫɨɫɬɨɹɧɢɣ ɦɵ ɨɛɫɭɠɞɚɥɢ ɜ
ɡɚɞɚɱɟ 2.5.
8.2. ɚ) ȼ ɛɚɡɢɫɟ, ɨɛɪɚɡɨɜɚɧɧɨɦ ɫɨɫɬɨɹɧɢɹɦɢ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɪɨɟɤɰɢɟɣ
ɫɩɢɧɚ , ɨɩɟɪɚɬɨɪ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɦɚɬɪɢɰɟɣ
z
s
y
s
ˆ
¸
¸
¹
·
¨
¨
©
§
V
02/
2/0
ˆ
2
1
ˆ
i
i
s
yy
. (1)
ɇɚɣɞɹ (ɤɚɤ ɜ ɡɚɞɚɱɟ 8.1) ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢ ɧɨɪɦɢɪɨɜɚɧɧɵɟ
ɫɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɵ ɨɩɟɪɚɬɨɪɚ (1), ɢɦɟɟɦ:
y
s
2/1,2/1
y
s ; (2)
82
zzy
i
pn n
22
1
,
zzy
i
pn p
22
1
. (3)
ɋɨɫɬɨɹɧɢɹ (3) ɦɨɠɧɨ ɬɚɤɠɟ ɩɨɥɭɱɢɬɶ ɫ ɩɨɦɨɳɶɸ ɩɨɜɨɪɨɬɨɜ ɛɚɡɢɫɧɨɝɨ ɫɩɢɧɨɪɚ
z
n ɧɚ ɭɝɥɵ 2
/
Sr ɜɨɤɪɭɝ ɨɫɢ
x
:
n
y
¸
¹
·
¨
©
§
S
x
R e
2
ˆ
z
n , p
y
¸
¹
·
¨
©
§
S
x
R e
2
ˆ
z
n . (4)
ɉɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɚɦɢ (7) ɢ (8) ɡɚɞɚɱɢ 8.1, ɧɚɯɨɞɢɦ ɜɟɪɨɹɬɧɨɫɬɢ
ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɣ
ɜ ɫɨɫɬɨɹɧɢɹɯ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɪɨɟɤɰɢɟɣ :
x
s
y
s
2/12/)1(
2
2
nn i
yx
, 2/12/)1(
2
2
np i
yx
2/12/)1(
2
2
pn i
yx
, 2/12/)1(
2
2
pp i
yx
. (5)
ȼɨɨɛɳɟ, ɩɪɢ ɫɪɚɜɧɟɧɢɢ ɞɜɭɯ ɫɩɢɧɨɜɵɯ ɫɨɫɬɨɹɧɢɣ (ɫɨ ɫɩɢɧɨɦ 1/2)
ɫɭɳɟɫɬɜɟɧɧɵɦ ɦɨɠɟɬ ɛɵɬɶ ɬɨɥɶɤɨ ɭɝɨɥ T ɦɟɠɞɭ ɧɚɩɪɚɜɥɟɧɢɹɦɢ ɨɛɨɢɯ ɫɩɢɧɨɜ,
ɩɨɬɨɦɭ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɞɨɫɬɚɬɨɱɧɨ ɩɨɥɚɝɚɬɶ, ɱɬɨ ɜɬɨɪɨɟ
ɫɨɫɬɨɹɧɢɟ ɩɨɥɭɱɟɧɨ ɩɨɜɨɪɨɬɨɦ ɩɟɪɜɨɝɨ ɫɨɫɬɨɹɧɢɹ ɧɚ ɭɝɨɥ
T
ɜɨɤɪɭɝ, ɧɚɩɪɢɦɟɪ,
ɨɫɢ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɦɟɟɬ ɜɢɞ:y
p
T
n
T
F
2
sin
2
cos , (6)
ɝɞɟ
n – ɩɟɪɜɨɟ ɫɨɫɬɨɹɧɢɟ, p – ɫɨɫɬɨɹɧɢɟ ɫ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦ ɧɚɩɪɚɜɥɟɧɢɟɦ
ɫɩɢɧɚ. Ɍɨɝɞɚ ɜɟɪɨɹɬɧɨɫɬɢ «ɫɨɛɵɬɢɣ» n ɢ ɜ ɫɨɫɬɨɹɧɢɢ (6) ɟɫɬɶp
2
cos
2
T
n
W ,
2
sin
2
T
p
W . (7)
ɇɚɣɞɟɧɧɵɟ ɜɵɲɟ ɡɧɚɱɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ (5) ɫɨɝɥɚɫɭɸɬɫɹ ɫ ɮɨɪɦɭɥɚɦɢ (7).
8.3. ɚ) ɂɡ ɩɪɢɜɟɞɟɧɧɨɝɨ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɪɢɫɭɧɤɚ ɦɨɠɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɞɥɹ
ɩɟɪɟɜɨɞɚ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ ɜ ɧɨɜɨɟ ɩɨɥɨɠɟɧɢɟ zyx
~
,
~
,
~
, ɯɚɪɚɤɬɟɪɢɡɭɟɦɨɟ
ɭɝɥɚɦɢ , ɫɥɟɞɭɟɬ ɩɪɨɢɡɜɟɫɬɢ ɬɪɢ ɩɨɜɨɪɨɬɚ ɤɨɨɪɞɢɧɚɬɧɨɣ ɫɢɫɬɟɦɵ ɜɨɤɪɭɝ
ɩɟɪɜɨɧɚɱɚɥɶɧɵɯ ɨɫɟɣ
MTJ ,,
zy
x
,,:ɩɨɜɨɪɨɬ ɧɚ ɭɝɨɥ ɜɨɤɪɭɝ ɨɫɢ , ɡɚɬɟɦ – ɧɚ ɭɝɨɥJ z T
ɜɨɤɪɭɝ ɨɫɢ , ɢ, ɧɚɤɨɧɟɰ, – ɧɚ ɭɝɨɥy M ɫɧɨɜɚ ɜɨɤɪɭɝ ɨɫɢ . ɉɟɪɟɦɧɨɠɢɜ ɜ
z
83
ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɤɚɡɚɧɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶɸ ɜɪɚɳɟɧɢɣ ɫɩɢɧɨɪɧɵɟ ɦɚɬɪɢɰɵ
ɩɨɜɨɪɨɬɨɜ, ɩɨɥɭɱɢɦ ɢɫɤɨɦɭɸ ɦɚɬɪɢɰɭ
R
ˆ
:
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
©
§
TT
T
T
JTM
JMJM
JMJM
2
cos
2
sin
2
sin
2
cos
)(
ˆ
)(
ˆ
)(
ˆˆ
)(2/)(
2/)(2/)(
ii
ii
zyz
ee
ee
RRRR
eee . (1)
ɛ) Ɉɛɨɡɧɚɱɢɦ ɛɚɡɢɫɧɵɟ ɫɨɫɬɨɹɧɢɹ
2/1,2/1 r
z
ss ɤɚɤ
z
n ,
z
p .
ɗɥɟɦɟɧɬɵ ɩɟɪɜɨɝɨ ɫɬɨɥɛɰɚ ɦɚɬɪɢɰɵ (1) ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɤɨɦɩɨɧɟɧɬɵ
ɫɩɢɧɨɪɚ
z
R n
ˆ
:
)2/(cos
ˆ
2/)(
T nn
J
Mi
zz
eR ,
)2/(sin
ˆ
2/)(
T np
J
Mi
zz
eR . (2)
Ɍɚɤɨɣ ɫɩɢɧɨɪ, ɨɱɟɜɢɞɧɨ, ɞɨɥɠɟɧ ɨɩɢɫɵɜɚɬɶ ɫɩɢɧɨɜɨɟ ɫɨɫɬɨɹɧɢɟ
z
~
n ɫɨ
ɫɩɢɧɨɦ ɜɞɨɥɶ ɨɫɢ z
~
, ɩɨɫɤɨɥɶɤɭ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ɩɨɜɨɪɨɬ ɩɪɟɨɛɪɚɡɭɟɬ
ɧɚɩɪɚɜɥɟɧɢɟ ɨɫɢ ɜ ɧɚɩɪɚɜɥɟɧɢɟ
z z
~
. ȼ ɷɬɨɦ ɫɨɫɬɨɹɧɢɢ ɭɫɪɟɞɧɟɧɧɵɣ ɜɟɤɬɨɪ
ɫɩɢɧɚ ɛɭɞɟɬ ɢɦɟɬɶ ɧɚɩɪɚɜɥɟɧɢɟ ɨɫɢ z
~
. ɉɪɨɜɟɪɢɦ ɫɩɪɚɜɟɞɥɢɜɨɫɬɶ ɞɚɧɧɵɯ
ɭɬɜɟɪɠɞɟɧɢɣ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɵɦ ɜɵɱɢɫɥɟɧɢɟɦ ɤɨɦɩɨɧɟɧɬ ɭɫɪɟɞɧɟɧɧɨɝɨ ɜɟɤɬɨɪɚ
ɫɩɢɧɚ.
Ɍɚɤ ɤɚɤ ɜ ɤɜɚɧɬɨɜɨɣ ɦɟɯɚɧɢɤɟ ɞɜɚ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ, ɪɚɡɥɢɱɚɸɳɢɟɫɹ
ɥɢɲɶ ɮɚɡɨɜɵɦ ɦɧɨɠɢɬɟɥɟɦ, ɨɩɢɫɵɜɚɸɬ ɨɞɧɨ ɢ ɬɨ ɠɟ ɫɨɫɬɨɹɧɢɟ ɮɢɡɢɱɟɫɤɨɝɨ
ɨɛɴɟɤɬɚ, ɭɩɪɨɫɬɢɦ ɜɵɪɚɠɟɧɢɹ (2), ɪɚɫɫɦɚɬɪɢɜɚɹ ɜɦɟɫɬɨ
z
R n
ˆ
ɮɢɡɢɱɟɫɤɢ
ɷɤɜɢɜɚɥɟɧɬɧɵɣ ɫɩɢɧɨɪ
2/)(
J
M
F
i
e
z
R n
ˆ
. (3)
ȿɝɨ ɤɨɦɩɨɧɟɧɬɵ ɢɦɟɸɬ ɜɢɞ:
)2/(cos T F
n
, )2/(sin T F
M
p
i
e . (4)
ɉɪɨɢɡɜɟɞɹ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ (4) ɜɵɱɢɫɥɟɧɢɟ ɩɪɨɟɤɰɢɣ ɭɫɪɟɞɧɟɧɧɨɝɨ ɜɟɤɬɨɪɚ
ɫɩɢɧɚ ɩɨ ɮɨɪɦɭɥɚɦ (VIII) ɜɜɟɞɟɧɢɹ ɤ ɞɚɧɧɨɣ ɝɥɚɜɵ, ɩɨɥɭɱɢɦ
MT cossin
2
1
x
s , MT sinsin
2
1
y
s , T cos
2
1
z
s . (5)
ȼɢɞɧɨ, ɱɬɨ ɜɟɥɢɱɢɧɚ ɜɟɤɬɨɪɚ
s ɪɚɜɧɚ 2
/
1,ɚ ɧɚɩɪɚɜɥɟɧɢɟ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ
ɨɫɹɦ zy
x
,, ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɬɟɦɢ ɠɟ ɭɝɥɚɦɢ, ɱɬɨ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɨɫɢ z
~
.
84
Ɉɬɦɟɬɢɦ ɬɚɤɠɟ, ɱɬɨ ɤɜɚɞɪɚɬɵ ɦɨɞɭɥɟɣ ɜɟɥɢɱɢɧ (4) ɢɥɢ (2) ɢɦɟɸɬ ɫɦɵɫɥ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɜɟɪɨɹɬɧɨɫɬɟɣ; ɨɧɢ, ɤɚɤ ɢ ɞɨɥɠɧɨ ɛɵɬɶ, ɫɨɜɩɚɞɚɸɬ ɫ
ɜɟɪɨɹɬɧɨɫɬɹɦɢ (7) ɜ ɡɚɞɚɱɟ 8.2.
8.4. Ƚɚɦɢɥɶɬɨɧɢɚɧ ɉɚɭɥɢ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɫɭɦɦɨɣ ɞɜɭɯ ɨɩɟɪɚɬɨɪɨɜ:
ɫɩɨɪɛ
HHH
ˆˆˆ
, (1)
ɝɞɟ
ɨɪɛ
H
ˆ
)(
2
1
2
rA U
c
q
i
m
¸
¹
·
¨
©
§
= , (2)
ɫɩ
H
ˆ
Bs
ˆ
ˆ
0
P
s
Bμ
ɫɩ
ˆ
. (3)
ɉɭɫɬɶ ɜɟɤɬɨɪ ɫɩɢɧɨɜɨɝɨ ɫɨɫɬɨɹɧɢɹ
ɫɩ
F ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɜɨɥɧɨɜɨɦɭ
ɭɪɚɜɧɟɧɢɸ ɒɪɟɞɢɧɝɟɪɚ ɫɨ ɫɩɢɧɨɜɨɣ ɱɚɫɬɶɸ ɩɨɥɧɨɝɨ ɝɚɦɢɥɶɬɨɧɢɚɧɚ:
w
Fw
t
i
ɫɩ
=
ɫɩ
H
ˆ
ɫɩ
F . (4)
ȼ ɫɥɭɱɚɟ ɨɞɧɨɪɨɞɧɨɝɨ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ (ɦɨɠɟɬ ɛɵɬɶ ɡɚɜɢɫɹɳɟɝɨ ɨɬ ɜɪɟɦɟɧɢ)
ɭɪɚɜɧɟɧɢɟ (4) ɧɟ ɫɨɞɟɪɠɢɬ ɤɨɨɪɞɢɧɚɬ ɢ ɩɨɷɬɨɦɭ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ
B
r
ɫɩ
F ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɤɨɨɪɞɢɧɚɬɧɵɦ ɩɟɪɟɦɟɧɧɵɦ ɢ ɨɪɛɢɬɚɥɶɧɨɣ ɱɚɫɬɢ
ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɞɨɩɭɫɬɢɦɨ ɫɱɢɬɚɬɶ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɨɣ. ȼ ɫɜɨɸ
ɨɱɟɪɟɞɶ, ɜɟɤɬɨɪ ɨɪɛɢɬɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɱɚɫɬɢɰɵ
ɨɪɛ
H
ˆ
ɨɪɛ
\ , ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɣ
ɭɪɚɜɧɟɧɢɸ
w
\w
t
i
ɨɪɛ
=
ɨɪɛ
H
ˆ
ɨɪɛ
\ , (5)
ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɨɣ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɨɩɟɪɚɬɨɪɭ , ɬɨ ɟɫɬɶ
ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ
ɫɩ
H
ˆ
ɨɪɛ
\ ɤɚɤ ɜɟɥɢɱɢɧɭ, ɧɟ ɩɪɢɧɚɞɥɟɠɚɳɭɸ ɩɪɨɫɬɪɚɧɫɬɜɭ
ɫɩɢɧɨɜɵɯ ɫɨɫɬɨɹɧɢɣ. Ɍɨɝɞɚ ɪɟɡɭɥɶɬɚɬ ɞɟɣɫɬɜɢɹ ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɉɚɭɥɢ ɧɚ ɜɟɤɬɨɪ
ɫɨɫɬɨɹɧɢɹ ɜɢɞɚ
)
ɨɪɛ
\
ɫɩ
F (6)
ɡɚɩɢɲɟɬɫɹ ɜ ɮɨɪɦɟ
)H
ˆ
(
ɨɪɛ
H
ˆ
ɨɪɛ
\ )
ɫɩ
F
ɨɪɛ
\
ɫɩ
H
ˆ
ɫɩ
F . (7)
85
ɍɱɢɬɵɜɚɹ ɮɨɪɦɭɥɭ
t
w
w
(
ɨɪɛ
\
ɫɩ
F )
t
ɨɪɛ
w
\w
ɫɩ
F
ɨɪɛ
\
t
ɫɩ
w
Fw
(8)
ɢ ɪɚɜɟɧɫɬɜɚ (4), (5), (7), ɡɚɤɥɸɱɚɟɦ, ɱɬɨ ɫɨɫɬɨɹɧɢɟ (6) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɜɨɥɧɨɜɨɦɭ
ɭɪɚɜɧɟɧɢɸ ɒɪɟɞɢɧɝɟɪɚ ɫ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ ɉɚɭɥɢ:
)
w
)w
H
t
i
ˆ
= .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɩɢɫɚɧɢɟ ɫɨɫɬɨɹɧɢɣ ɱɚɫɬɢɰɵ ɜ ɮɨɪɦɟ (6) ɹɜɥɹɟɬɫɹ ɞɨɩɭɫɬɢɦɵɦ.
ɉɟɪɟɣɞɹ ɧɚ ɹɡɵɤ ɜɨɥɧɨɜɵɯ ɮɭɧɤɰɢɣ, ɦɨɠɧɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ ɜ ɭɤɚɡɚɧɧɨɦ
ɨɩɢɫɚɧɢɢ ɤɨɨɪɞɢɧɚɬɧɵɟ ɢ ɫɩɢɧɨɜɵɟ ɩɟɪɟɦɟɧɧɵɟ ɩɨɥɧɨɫɬɶɸ ɪɚɡɞɟɥɟɧɵ. ɋ ɷɬɨɣ
ɰɟɥɶɸ ɜɜɟɞɟɦ ɜ ɪɚɫɫɦɨɬɪɟɧɢɟ ɛɚɡɢɫɧɵɟ ɫɨɫɬɨɹɧɢɹ ɱɚɫɬɢɰɵ
[ ,
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɩɪɟɞɟɥɟɧɧɵɦ ɤɨɨɪɞɢɧɚɬɚɦ ɢ ɨɩɪɟɞɟɥɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ
ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ :
r
V{
z
s
V [ r . (9)
ɋɤɚɥɹɪɧɨ ɭɦɧɨɠɢɜ (6) ɧɚ (9), ɩɨɥɭɱɢɦ:
)()()( ttt
ɫɩɨɪɛ
FV\ )[ r , (10)
ɢɥɢ
)(),(),( ttt
V
F\ [) r . (11)
Ɂɞɟɫɶ ɨɡɧɚɱɚɟɬ ɫɨɜɨɤɭɩɧɨɫɬɶ ɩɟɪɟɦɟɧɧɵɯ[V,
r , ɬɚɤ ɱɬɨ ),(
t
[) ɩɪɟɞɫɬɚɜɥɹɟɬ
ɫɨɛɨɣ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ, ɤɜɚɞɪɚɬɨɦ ɦɨɞɭɥɹ ɤɨɬɨɪɨɣ ɨɩɪɟɞɟɥɹɟɬɫɹ
ɨɞɧɨɜɪɟɦɟɧɧɨ ɜɟɪɨɹɬɧɨɫɬɶ ɡɧɚɱɟɧɢɣ ɢ ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ ɱɚɫɬɢɰɵ
r V.
ȼɟɪɨɹɬɧɨɫɬɶ ɧɚɣɬɢ ɱɚɫɬɢɰɭ ɜ ɷɥɟɦɟɧɬɟ ɨɛɴɟɦɚ d
V
ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ ɢ ɩɪɢ
ɷɬɨɦ ɨɛɧɚɪɭɠɢɬɶ ɭ ɱɚɫɬɢɰɵ ɡɧɚɱɟɧɢɟ ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ
r
V
z
s ɟɫɬɶ
2
,
),( tdVW [)
Vr
.
ɉɨɫɤɨɥɶɤɭ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ (11) ɢɦɟɟɬ ɜɢɞ ɩɪɨɢɡɜɟɞɟɧɢɹ ɨɪɛɢɬɚɥɶɧɨɣ ɢ
ɫɩɢɧɨɜɨɣ ɮɭɧɤɰɢɣ, ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɤɨɨɪɞɢɧɚɬ
2
),( tr\
ɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ
ɜɟɪɨɹɬɧɨɫɬɢ
2
)(t
V
F
ɞɥɹ ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɧɟ ɫɜɹɡɚɧɵ ɞɪɭɝ ɫ
ɞɪɭɝɨɦ.
ɉɪɢ ɢɡ ɭɪɚɜɧɟɧɢɹ (4) ɫɥɟɞɭɟɬ, ɱɬɨ ɫɩɢɧɨɜɚɹ ɱɚɫɬɶ ɜɟɤɬɨɪɚ
ɫɨɫɬɨɹɧɢɹ ɧɟ ɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɢ ɟɟ ɪɨɥɶ ɦɨɠɟɬ ɜɵɩɨɥɧɹɬɶ, ɜ ɱɚɫɬɧɨɫɬɢ,
ɥɸɛɨɟ ɢɡ ɛɚɡɢɫɧɵɯ ɫɩɢɧɨɜɵɯ ɫɨɫɬɨɹɧɢɣ
0
B
V . ɉɭɫɬɶ ɞɥɹ ɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɪɟɱɶ
86
ɢɞɟɬ ɨ ɱɚɫɬɢɰɟ ɫɨ ɫɩɢɧɨɦ 2
/
1,ɧɚɩɪɢɦɟɪ, – ɨɛ ɷɥɟɤɬɪɨɧɟ ɜ ɚɬɨɦɟ. ȼ ɤɚɱɟɫɬɜɟ
ɨɪɛɢɬɚɥɶɧɵɯ ɫɨɫɬɨɹɧɢɣ ɨɛɵɱɧɨ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɟ ɫɨɫɬɨɹɧɢɹ
ɷɥɟɤɬɪɨɧɚ
n (ɷɬɨ ɩɨɧɹɬɢɟ ɦɨɠɟɬ ɛɵɬɶ ɫ ɧɟɤɨɬɨɪɨɣ ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ
ɨɩɪɟɞɟɥɟɧɨ ɞɚɠɟ ɜ ɦɧɨɝɨɷɥɟɤɬɪɨɧɧɨɦ ɚɬɨɦɟ, ɧɟɫɦɨɬɪɹ ɧɚ ɫɭɳɟɫɬɜɟɧɧɭɸ ɪɨɥɶ
ɤɭɥɨɧɨɜɫɤɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɟɠɞɭ ɜɫɟɦɢ ɷɥɟɤɬɪɨɧɚɦɢ); – ɫɨɜɨɤɭɩɧɨɫɬɶ
ɤɜɚɧɬɨɜɵɯ ɱɢɫɟɥ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɛɟɡ ɭɱɟɬɚ ɫɩɢɧɚ. Ɍɨɝɞɚ ɩɨɥɧɵɣ
ɜɟɤɬɨɪ ɫɨɫɬɨɹɧɢɹ ɱɚɫɬɢɰɵ, ɜɵɫɬɭɩɚɸɳɢɣ ɜ ɞɚɧɧɨɦ ɤɨɧɬɟɤɫɬɟ ɤɚɤ ɜɟɤɬɨɪ
ɛɚɡɢɫɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɷɥɟɤɬɪɨɧɚ, ɡɚɩɢɲɟɬɫɹ ɫɨɝɥɚɫɧɨ (6) ɜ ɜɢɞɟ
n
)
V,n
Vn .
ȿɫɥɢ ɫɤɚɥɹɪɧɨ ɭɦɧɨɠɢɬɶ ɨɛɟ ɫɬɨɪɨɧɵ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɧɚ
r , ɬɨ ɩɨɥɭɱɢɦ
ɫɢɦɜɨɥɢɱɟɫɤɨɟ ɜɵɪɚɠɟɧɢɟ
)
V
)(
,
r
n
V
\ )(r
n
,
ɨɛɨɡɧɚɱɚɸɳɟɟ ɫɬɚɰɢɨɧɚɪɧɨɟ ɫɨɫɬɨɹɧɢɟ ɷɥɟɤɬɪɨɧɚ ɫ ɨɪɛɢɬɚɥɶɧɨɣ ɜɨɥɧɨɜɨɣ
ɮɭɧɤɰɢɟɣ )ɢ ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɪɨɟɤɰɢɟɣ ɫɩɢɧɚ(
r
n
\V. Ɍɚɤɢɟ ɜɵɪɚɠɟɧɢɹ ɩɪɢ
2
/
1,2
/
1 V ɜ ɥɢɬɟɪɚɬɭɪɟ ɢɧɨɝɞɚ ɡɚɩɢɫɵɜɚɸɬɫɹ ɤɚɤ
D\ )(
r
n
, E\ )(r
n
.
ɋɥɟɞɭɟɬ ɨɞɧɚɤɨ ɩɨɦɧɢɬɶ, ɱɬɨ ɩɪɢ ɷɬɨɦ ɜɟɥɢɱɢɧɵ D ɢ ɧɟ ɩɪɢɧɢɦɚɸɬ
ɱɢɫɥɨɜɵɯ ɡɧɚɱɟɧɢɣ, ɚ ɥɢɲɶ ɫɢɦɜɨɥɢɡɢɪɭɸɬ ɫɩɢɧɨɜɵɟ ɫɨɫɬɨɹɧɢɹ
E
n ɢ p .
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ, ɩɪɢ ɧɚɥɢɱɢɢ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ,
ɪɟɲɟɧɢɟ
)(
rB
)(t) ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɒɪɟɞɢɧɝɟɪɚ ɫ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ ɉɚɭɥɢ
ɬɚɤɠɟ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɨ ɩɨ ɛɚɡɢɫɧɵɦ ɫɨɫɬɨɹɧɢɹɦ (9):
)(t) V)
¦¦
V
V
r
r
r
)(
,
t .
ɇɨ ɚɦɩɥɢɬɭɞɚ ɜɟɪɨɹɬɧɨɫɬɢ )ɢɥɢ, ɱɬɨ ɩɪɚɤɬɢɱɟɫɤɢ ɬɨ ɠɟ ɫɚɦɨɟ, ɜɨɥɧɨɜɚɹ
ɮɭɧɤɰɢɹ )
(
,
t
V
)
r
;,(
t
V) r ɭɠɟ ɧɟ ɫɜɨɞɢɬɫɹ ɤ ɩɪɨɢɡɜɟɞɟɧɢɸ (11), ɩɨɫɤɨɥɶɤɭ ɢɦɟɸɳɚɹɫɹ
ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɤɨɪɪɟɥɹɰɢɹ ɡɧɚɱɟɧɢɣ ɢ
r V ɩɪɨɹɜɥɹɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɪɚɡɥɢɱɧɵɦ
ɡɧɚɱɟɧɢɹɦ ɨɬɜɟɱɚɸɬ ɪɚɡɧɵɟ ɮɨɪɦɵ ɡɚɜɢɫɢɦɨɫɬɢ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɨɬ .
Ɇɨɠɧɨ ɫɤɚɡɚɬɶ, ɱɬɨ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɱɚɫɬɢɰɵ ɫɨ ɫɩɢɧɨɦ
V
r
s
ɢɦɟɟɬ 12
s
ɤɨɦɩɨɧɟɧɬ:
);,(
t
s
r) , );1,(
t
s
) r , … , );,(
t
s
) r .
ɉɪɢɦɟɪ ɩɨɞɨɛɧɨɣ ɤɨɪɪɟɥɹɰɢɢ ɞɚɟɬ ɤɚɪɬɢɧɚ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɨɩɵɬɭ
ɒɬɟɪɧɚ–Ƚɟɪɥɚɯɚ ɫ ɱɚɫɬɢɰɚɦɢ ɫɩɢɧɚ 2
/
1.ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɞɨ
87
ɜɯɨɠɞɟɧɢɹ ɜɨɥɧɨɜɨɝɨ ɩɚɤɟɬɚ ɜ ɩɪɢɛɨɪ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦ ɦɚɝɧɢɬɧɵɦ ɩɨɥɟɦ, ɟɳɟ
ɞɨɩɭɫɬɢɦɨ ɫɱɢɬɚɬɶ ɤɨɨɪɞɢɧɚɬɧɭɸ ɱɚɫɬɶ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɜɟɥɢɱɢɧɨɣ, ɧɟ
ɡɚɜɢɫɹɳɟɣ ɨɬ ɫɩɢɧɨɜɨɝɨ ɫɨɫɬɨɹɧɢɹ ɱɚɫɬɢɰɵ. Ɉɞɧɚɤɨ ɡɚɬɟɦ ɜ ɪɟɡɭɥɶɬɚɬɟ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦ ɦɚɝɧɢɬɧɵɦ ɩɨɥɟɦ ɨɛɪɚɡɭɸɬɫɹ ɞɜɚ
ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ ɪɚɡɧɟɫɟɧɧɵɯ «ɨɛɥɚɤɚ ɜɟɪɨɹɬɧɨɫɬɢ»: ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ
ɡɧɚɱɟɧɢɹ ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ 2
/
1r V ɱɚɫɬɢɰɚ ɛɭɞɟɬ ɨɛɧɚɪɭɠɢɜɚɬɶɫɹ ɥɢɛɨ ɜ
ɨɞɧɨɣ ɥɢɛɨ ɜ ɞɪɭɝɨɣ ɨɛɥɚɫɬɢ
r -ɩɪɨɫɬɪɚɧɫɬɜɚ.
8.5. ɚ) ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɪɚɡɞɟɥɶɧɵɦ ɨɩɢɫɚɧɢɟɦ ɨɪɛɢɬɚɥɶɧɵɯ ɢ ɫɩɢɧɨɜɵɯ
ɩɟɪɟɦɟɧɧɵɯ, ɨɛɫɭɠɞɚɜɲɢɦɫɹ ɜ ɡɚɞɚɱɟ 8.4. Ⱦɥɹ ɱɚɫɬɢɰɵ ɫɨ ɫɩɢɧɨɦ 2
/
1
s
ɢɦɟɟɦ 2
/
ˆ
ˆ
ıs
, ɬɚɤ ɱɬɨ ɫɩɢɧɨɜɚɹ ɱɚɫɬɶ ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɟɫɬɶ
0
ˆ
P
ɫɩ
Bı
ˆ
. (1)H
ɇɚɩɪɚɜɢɦ ɨɫɶ ɜɞɨɥɶ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ; ɬɨɝɞɚ
z
¸
¸
¹
·
¨
¨
©
§
P
P
VP
B
B
BH
zɫɩ
0
0
0
0
0
ˆ
ˆ
. (2)
Ɋɟɲɟɧɢɟ
)(tF ɭɪɚɜɧɟɧɢɹ
w
Fw
t
t
i
)(
=
ɫɩ
H
ˆ
)(tF (3)
ɫ ɧɟ ɡɚɜɢɫɹɳɢɦ ɨɬ ɜɪɟɦɟɧɢ ɦɚɝɧɢɬɧɵɦ ɩɨɥɟɦ ɫɥɟɞɭɟɬ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
)0()(
/
ˆ
F F
=tHi
ɫɩ
et . (4)
ɉɨɞɫɬɚɜɢɜ ɜ (4) ɹɜɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɧɚɱɚɥɶɧɨɝɨ ɫɩɢɧɨɜɨɝɨ ɫɨɫɬɨɹɧɢɹ,
p
T
n
T
F
2
sin
2
cos)0( , (5)
ɩɨɥɭɱɢɦ
2
cos)(
T
F
t n
=/
ˆ
tHi
ɫɩ
e
2
sin
T
p
=/
ˆ
tHi
ɫɩ
e . (6)
ɉɨɫɤɨɥɶɤɭ ɛɚɡɢɫɧɵɟ ɫɩɢɧɨɜɵɟ ɫɨɫɬɨɹɧɢɹ
n ɢ p ɹɜɥɹɸɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɞɥɹ
ɨɩɟɪɚɬɨɪɚ (2),
nP n BH
ɫɩ 0
ˆ
, pP p BH
ɫɩ 0
ˆ
, (7)
ɬɨ
88
n
=/
ˆ
tHi
ɫɩ
e n
P =/
0
tBi
e , p
=/
ˆ
tHi
ɫɩ
e p
P =/
0
tBi
e . (8)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
2
cos)(
T
F
t n
P =/
0
tBi
e
2
sin
T
p
P =/
0
tBi
e . (9)
ɛ) ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɫɪɟɞɧɢɯ ɡɧɚɱɟɧɢɣ ɩɪɨɟɤɰɢɣ ɫɩɢɧɚ ɦɨɠɧɨ
ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɤɚɤ ɜɟɤɬɨɪɨɦ ɫɨɫɬɨɹɧɢɹ (9), ɬɚɤ ɢ ɥɸɛɵɦ ɜɟɤɬɨɪɨɦ ɫɨɫɬɨɹɧɢɹ,
ɨɬɥɢɱɚɸɳɢɦɫɹ ɨɬ (9) ɩɪɨɢɡɜɨɥɶɧɵɦ ɮɚɡɨɜɵɦ ɦɧɨɠɢɬɟɥɟɦ. ȼ ɱɚɫɬɧɨɫɬɢ,
ɭɞɨɛɧɨ ɜɡɹɬɶ ɜɟɤɬɨɪ ɫɨɫɬɨɹɧɢɹ
)(
/
0
te
tBi
F
P =
; ɬɚɤɨɣ ɫɩɢɧɨɪ ɨɛɥɚɞɚɟɬ
ɤɨɦɩɨɧɟɧɬɚɦɢ
)2/(cos
T F
n
, )2/(sin
)(
T F
M
p
ti
e , (10)
ɝɞɟ
t
B
t
=
0
2
)(
P
M
. (11)
ɋɨɝɥɚɫɧɨ ɪɟɡɭɥɶɬɚɬɭ ɡɚɞɚɱɢ 8.3 ɜ ɭɤɚɡɚɧɧɨɦ ɡɞɟɫɶ ɫɩɢɧɨɜɨɦ ɫɨɫɬɨɹɧɢɢ
ɧɚɩɪɚɜɥɟɧɢɟ ɭɫɪɟɞɧɟɧɧɨɝɨ ɜɟɤɬɨɪɚ ɫɩɢɧɚ
)(ts ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɨɫɹɦ zy
x
,,
ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɝɥɚɦɢ
T ɢ )(
t
M . ɉɨɫɤɨɥɶɤɭ ɭɝɨɥ )(
t
M ɢɡɦɟɧɹɟɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ
=
B
d
t
d
0
2P
M
Z
, (12)
ɦɵ ɡɚɤɥɸɱɚɟɦ, ɱɬɨ ɜɟɤɬɨɪ
)(ts ɩɪɟɰɟɫɫɢɪɭɟɬ (ɜɪɚɳɚɟɬɫɹ) ɜɨɤɪɭɝ ɧɚɩɪɚɜɥɟɧɢɹ
ɫ ɱɚɫɬɨɬɨɣ (12). ɉɪɢ ɷɬɨɦ ɭɝɨɥ ɦɟɠɞɭ ɧɚɩɪɚɜɥɟɧɢɹɦɢB
)(ts ɢ ɢɦɟɟɬ
ɩɨɫɬɨɹɧɧɨɟ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ
T; ɨɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ.
B
ɉɪɢ 0ɫɩɢɧɨɜɨɟ ɫɨɫɬɨɹɧɢɟ (9) ɹɜɥɹɟɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦ, ɷɬɨ ɟɫɬɶ
ɫɨɫɬɨɹɧɢɟ ɫɨ ɫɩɢɧɨɦ ɜɞɨɥɶ . ȿɦɭ ɨɬɜɟɱɚɟɬ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɨɩɟɪɚɬɨɪɚ
T
B
ɫɩ
H
ˆ
B
0
P
H
n
. (13)
ɉɪɢ ɫɩɢɧɨɜɨɟ ɫɨɫɬɨɹɧɢɟ ɱɚɫɬɢɰɵ ɬɚɤɠɟ ɨɤɚɡɵɜɚɟɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦ – ɷɬɨ
ɛɭɞɟɬ ɫɨɫɬɨɹɧɢɟ ɫɨ ɫɩɢɧɨɦ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ,
ɩɪɢɧɚɞɥɟɠɚɳɟɟ ɭɪɨɜɧɸ ɷɧɟɪɝɢɢ
S T
B
B
0
P H
p
. (14)
ɑɚɫɬɨɬɚ ɩɪɟɰɟɫɫɢɢ ɫɩɢɧɚ (12) ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɚɡɧɨɫɬɢ ɡɧɚɱɟɧɢɣ ɷɧɟɪɝɢɢ (13) ɢ
(14).
89