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Ɋɟɲɟɧɢɟ:
G
G
G
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɧɚɪɹɞ ɢ Ɂɟɦɥɹ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɡɚɦɤɧɭɬɭɸ ɫɢɫɬɟɦɭ.
ɇɨ ɟɫɥɢ ɦɵ ɭɱɬɟɦ, ɱɬɨ ɞɥɢɬɟɥɶɧɨɫɬɶ ɜɡɪɵɜɚ ɱɪɟɡɜɵɱɚɣɧɨ ɦɚɥɚ, ɬɨ ɢɡɦɟɧɟɧɢɟɦ
ɢɦɩɭɥɶɫɚ ɫɧɚɪɹɞɚ ɡɚ ɜɪɟɦɹ ɜɡɪɵɜɚ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ. ɉɨɷɬɨɦɭ ɫɧɚɪɹɞ ɦɨɠɧɨ
ɫɱɢɬɚɬɶ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɨɣ (ɪɢɫ. 8.2), ɞɥɹ ɤɨɬɨɪɨɣ ɜɵɩɨɥɧɹɟɬɫɹ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ.
Ɋɢɫ. 8.2
11 2 2
,mV m V m V
ɝɞɟ
,
– ɦɚɫɫɵ ɨɫɤɨɥɤɨɜ,
m
m
1
2
G
G
,
– ɢɯ ɫɤɨɪɨɫɬɢ ɩɨɫɥɟ ɜɡɪɵɜɚ.
V
V
2
1
GGG
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɭɦɦɚɪɧɚɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɨɫɤɨɥɤɨɜ ɛɨɥɶɲɟ, ɱɟɦ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɧɚɪɹɞɚ. ɗɬɨ ɨɛɴɹɫɧɹɟɬɫɹ ɩɟɪɟɯɨɞɨɦ ɱɚɫɬɢ ɷɧɟɪɝɢɢ ɩɨɪɨɯɨɜɵɯ ɝɚɡɨɜ ɜ ɦɟɯɚɧɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ.
ɗɮɮɟɤɬɨɦ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦ, ɧɨ ɚɧɚɥɨɝɢɱɧɵɦ ɜɡɪɵɜɭ ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɚɛɫɨɥɸɬɧɨ ɧɟɭɩɪɭɝɢɣ ɭɞɚɪ.
Ⱥɛɫɨɥɸɬɧɨ ɧɟɭɩɪɭɝɢɦ ɭɞɚɪɨɦ ɧɚɡɵɜɚɟɬɫɹ ɬɚɤɨɟ
ɫɬɨɥɤɧɨɜɟɧɢɟ, ɩɨɫɥɟ ɤɨɬɨɪɨɝɨ ɫɤɨɪɨɫɬɢ ɬɟɥ ɨɤɚɡɵɜɚɸɬɫɹ ɨɞɢɧɚɤɨɜɵɦɢ. Ɋɚɫɫɦɨɬɪɢɦ ɫɬɨɥɤɧɨɜɟɧɢɟ ɞɜɭɯ ɬɟɥ ɦɚɫɫɚɦɢ
ɫɤɨɪɨɫɬɹɦɢ ɢ
G
G
, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɂɡ-ɡɚ ɬɨɝɨ, ɱɬɨ ɫɢɫɬɟɦɚ ɹɜɥɹɟɬɫɹ ɡɚɦɤɧɭ-
V
V
2
1
ɢ
m
ɉɭɫɬɶ ɷɬɢ ɬɟɥɚ ɞɜɢɝɚɸɬɫɹ ɫɨ
.m
1
2
ɬɨɣ, ɦɵ ɦɨɠɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ ɞɥɹ
ɧɚɯɨɠɞɟɧɢɹ ɫɤɨɪɨɫɬɢ ɬɟɥ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ:
GGG
)(
,
VmmVmVm
212211
ɨɬɤɭɞɚ ɥɟɝɤɨ ɧɚɣɬɢ ɫɤɨɪɨɫɬɶ ɫɢɫɬɟɦɵ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ:
VmVm
V
2211
.
)(
mm
21
ȼ ɱɚɫɬɧɨɫɬɢ, ɟɫɥɢ ɬɟɥɚ ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɵɟ ɦɚɫɫɵ ɢ ɥɟɬɹɬ ɧɚɜɫɬɪɟɱɭ ɞɪɭɝ
ɞɪɭɝɭ ɫ ɨɞɢɧɚɤɨɜɵɦɢ ɩɨ ɦɨɞɭɥɸ ɫɤɨɪɨɫɬɹɦɢ, ɬɨ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɢɯ ɫɤɨɪɨɫɬɶ ɛɭɞɟɬ ɪɚɜɧɚ ɧɭɥɸ. Ɇɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ:
mm
2
E
ɤ
2
21
V
2
,
121
ɚ ɞɨ ɫɬɨɥɤɧɨɜɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨɦ:
G
2
E
ɤ
11
1
2
VmVm
22
.
22
ɉɨɷɬɨɦɭ ɢɡɦɟɧɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɪɚɜɧɨ
mm
EE E V V VV
'
ɤɤ
21 12 12
12
mm
2( )
12
22
GG
2( ) .
Ɂɧɚɤ ɦɢɧɭɫ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɷɬɨɣ ɮɨɪɦɭɥɵ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɭɦɟɧɶɲɚɟɬɫɹ. ɑɚɫɬɶ ɷɧɟɪɝɢɢ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɞɪɭɝɢɟ ɜɢɞɵ ɷɧɟɪɝɢɢ
ɢɥɢ, ɤɚɤ ɝɨɜɨɪɹɬ, ɩɟɪɟɯɨɞɢɬ ɜ ɬɟɩɥɨ. ɉɪɟɨɛɪɚɡɭɟɦ ɩɪɚɜɭɸ ɱɚɫɬɶ
ɝɞɟ
mm
21
mm
)(2
21
u
– ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɬɟɥ,
12
mm
P
21
– ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ ɩɪɢɜɟɞɟɧɧɚɹ ɦɚɫɫɚ.
mm
21
2
2
2
1
)(2
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɨɤɚɡɚɧɚ ɫɥɟɞɭɸɳɚɹ ɬɟɨɪɟɦɚ.
1
2
GGGG
1
2
)(
2121
2
uVVVVVV
P P
,
12
2
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɜɵɞɟɥɹɸɳɟɟɫɹ ɩɪɢ ɚɛɫɨɥɸɬɧɨ ɧɟɭɩɪɭɝɨɦ ɭɞɚɪɟ, ɪɚɜɧɨ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɰɟɧɬɪɚ ɦɚɫɫ.
Ɂɚɞɚɱɚ 3. Ɉɩɢɲɢɬɟ ɚɛɫɨɥɸɬɧɨ ɭɩɪɭɝɢɣ ɭɞɚɪ.
Ɋɟɲɟɧɢɟ:
Ⱥɛɫɨɥɸɬɧɨ ɭɩɪɭɝɢɦ ɭɞɚɪɨɦ ɧɚɡɵɜɚɟɬɫɹ ɬɚɤɨɟ ɤɪɚɬɤɨɜɪɟɦɟɧɧɨɟ ɜɡɚɢɦɨɞɟɣ-
ɫɬɜɢɟ ɬɟɥ, ɩɨɫɥɟ ɤɨɬɨɪɨɝɨ ɜ ɨɛɨɢɯ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɬɟɥɚɯ ɧɟ ɨɫɬɚɟɬɫɹ ɧɢɤɚɤɢɯ ɞɟɮɨɪɦɚɰɢɣ. ɉɨɷɬɨɦɭ ɜɫɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ, ɤɨɬɨɪɨɣ ɨɛɥɚɞɚɥɢ ɬɟɥɚ ɞɨ
ɭɞɚɪɚ, ɩɨɫɥɟ ɭɞɚɪɚ ɜɧɨɜɶ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɫɬɚɥɤɢɜɚɸɳɢɯɫɹ ɬɟɥ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɨɞɧɨɝɨ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ
ɞɜɭɯ ɧɟɡɚɜɢɫɢɦɵɯ ɡɧɚɱɟɧɢɣ ɫɤɨɪɨɫɬɟɣ ɩɨɫɥɟ
ɭɞɚɪɚ ɭɠɟ ɧɟɞɨɫɬɚɬɨɱɧɨ. ɉɨɷɬɨɦɭ
ɧɟɨɛɯɨɞɢɦɨ ɡɚɩɢɫɚɬɶ ɞɜɚ ɭɪɚɜɧɟɧɢɹ – ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɢ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ:
2
G
ɝɞɟ , – ɫɤɨɪɨɫɬɢ ɲɚɪɨɜ ɞɨ ɭɞɚɪɚ,
G
V
V
1
2
G
G
c
c
, – ɢɯ ɫɤɨɪɨɫɬɢ ɩɨɫɥɟ ɭɞɚɪɚ.
V
V
1
2
11
2
22
c
2
c
11
,
VmVmVmVm
22112211
2
c
VmVmVmVm
22
,
2222
GGGG
c
Ɍɟɩɟɪɶ ɨɝɨɜɨɪɢɦ ɜɚɠɧɵɟ ɞɨɩɭɳɟɧɢɹ. Ȼɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɥɭɱɚɣ ɬɚɤ
ɧɚɡɵɜɚɟɦɨɝɨ
ɰɟɧɬɪɚɥɶɧɨɝɨ ɭɞɚɪɚ (ɢɥɢ ɥɨɛɨɜɨɝɨ ɫɬɨɥɤɧɨɜɟɧɢɹ). ɗɬɨ ɬɚɤɨɣ ɭɞɚɪ,
ɩɪɢ ɤɨɬɨɪɨɦ ɫɤɨɪɨɫɬɢ ɫɬɚɥɤɢɜɚɸɳɢɯɫɹ ɲɚɪɨɜ ɞɨ ɭɞɚɪɚ ɫɨɜɩɚɞɚɸɬ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɫ ɥɢɧɢɟɣ, ɫɨɟɞɢɧɹɸɳɟɣ ɰɟɧɬɪɵ ɲɚɪɨɜ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɪɢɜɟɞɟɧɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɩɪɢɧɢɦɚɟɬ ɜɢɞ:
122
c
c
c
c
,
)()(
VVmVVm
222111
c
c
.
))(())((
VVVVmVVVVm
2222211111
ɇɟɬɪɭɞɧɨ ɩɨɥɭɱɢɬɶ ɢ ɹɜɧɨɟ ɪɟɲɟɧɢɟ:
2)(
VmVmm
c
V
1
mm
22121
c
, .
V
21
2
2)(
VmVmm
11212
mm
21
Ɋɚɫɫɦɨɬɪɢɦ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ, ɢɦɟɸɳɢɣ ɩɪɚɤɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ.
ɉɭɫɬɶ ɨɞɢɧ ɲɚɪ ɞɨ ɭɞɚɪɚ ɩɨɤɨɢɬɫɹ, ɧɚɩɪɢɦɟɪ,
Vmm
)(
c
V
1
121
V
,
mm
21
Vm
2
c
2
11
mm
. Ɍɨɝɞɚ:
0
V
2
.
21
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɬɨɪɨɣ ɲɚɪ ɩɨɫɥɟ ɭɞɚɪɚ ɞɜɢɠɟɬɫɹ ɜ ɬɨɦ ɠɟ ɧɚɩɪɚɜɥɟɧɢɢ,
ɱɬɨ ɢ ɩɟɪɜɵɣ ɞɨ ɭɞɚɪɚ. ȼɟɥɢɱɢɧɚ ɨɬɧɨɲɟɧɢɹ
ɦɚɫɫ ɲɚɪɨɜ, ɱɬɨ ɩɪɨɞɟɦɨɧɫɬɪɢɪɨɜɚɧɨ ɧɚ ɪɢɫ. 8.3.
c
ɡɚɜɢɫɢɬ ɨɬ ɫɨɨɬɧɨɲɟɧɢɹ
)/(12VV
Ɋɢɫ. 8.3
ɇɚ ɪɢɫ. 8.4 ɩɪɟɞɫɬɚɜɥɟɧɚ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬɧɨɲɟɧɢɹ
ɦɚɫɫ.
Ɋɢɫ. 8.4
123
c
)/(11VV
ɨɬ ɨɬɧɨɲɟɧɢɹ
ȼ ɨɛɥɚɫɬɢ
V
V
M
ɧɢɸ, ɧɨ ɭɦɟɧɶɲɚɟɬɫɹ ɩɨ ɜɟɥɢɱɢɧɟ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɨɬɧɨɲɟɧɢɹ
ɲɚɪ ɩɪɢ ɷɬɨɦ ɞɜɢɠɟɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ ɛɨɥɶɲɟɣ, ɱɟɦ
, ɫɤɨɪɨɫɬɶ ɩɟɪɜɨɝɨ ɲɚɪɚ ɫɨɯɪɚɧɹɟɬɫɹ ɩɨ ɧɚɩɪɚɜɥɟ-
10
d mm
12
mm
m
. ɉɪɢ 1
V
1
2
m
1
ɩɟɪɜɵɣ ɲɚɪ
. ȼɬɨɪɨɣ
12
ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ, ɚ ɜɬɨɪɨɣ ɧɚɱɢɧɚɟɬ ɞɜɢɝɚɬɶɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ ɩɟɪɜɨɝɨ ɲɚɪɚ ɞɨ
m
2
1
ɫɬɨɥɤɧɨɜɟɧɢɹ. ɉɪɢ ɩɟɪɜɵɣ ɲɚɪ ɧɚɱɢɧɚɟɬ ɞɜɢɝɚɬɶɫɹ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ
!
m
1
ɫɬɨɪɨɧɭ. ɉɪɢ ɨɱɟɧɶ ɛɨɥɶɲɨɦ ɡɧɚɱɟɧɢɢ ɷɬɨɝɨ ɨɬɧɨɲɟɧɢɹ, ɫɤɨɪɨɫɬɶ ɩɟɪɜɨɝɨ ɲɚɪɚ ɩɨ ɦɨɞɭɥɸ ɫɬɚɧɨɜɢɬɫɹ ɩɪɚɤɬɢɱɟɫɤɢ ɪɚɜɧɨɣ ɫɤɨɪɨɫɬɢ ɲɚɪɚ ɞɨ ɭɞɚɪɚ.
ȼɨ ɦɧɨɝɢɯ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱɚɯ ɜɨɡɧɢɤɚɟɬ ɫɢɬɭɚɰɢɹ, ɤɨɝɞɚ ɨɱɟɧɶ ɥɟɝɤɚɹ ɱɚɫɬɢɰɚ ɫɬɚɥɤɢɜɚɟɬɫɹ ɫ ɧɟɩɨɞɜɢɠɧɨɣ ɨɱɟɧɶ ɬɹɠɟɥɨɣ ɱɚɫɬɢɰɟɣ. Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɩɪɢɜɟɞɟɧɧɨɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ, ɥɟɝɤɚɹ ɱɚɫɬɢɰɚ ɩɪɢ ɰɟɧɬɪɚɥɶɧɨɦ ɭɞɚɪɟ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɞɜɢɠɟɬɫɹ ɜ ɨɛɪɚɬɧɭɸ ɫɬɨɪɨɧɭ, ɬ. ɟ. ɦɟɧɹɟɬ ɧɚɩɪɚɜɥɟɧɢɟ ɫɜɨɟɝɨ ɢɦɩɭɥɶɫɚ ɧɚ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɟ. ɂɡɦɟɧɟɧɢɟ ɢɦɩɭɥɶɫɚ ɷɬɨɣ ɥɟɝɤɨɣ ɱɚɫɬɢɰɵ ɪɚɜɧɨ:
G
c
' ,
GGG
2|
VmVmVmp
GG
ɩɨɫɤɨɥɶɤɭ
|c.
Ɋɢɫ. 8.5
ȼɚɠɧɵɦ ɚɫɩɟɤɬɨɦ ɷɬɨɣ ɡɚɞɚɱɢ ɹɜɥɹɟɬɫɹ ɩɟɪɟɞɚɱɚ ɷɧɟɪɝɢɢ ɩɪɢ ɫɬɨɥɤɧɨɜɟɧɢɢ. Ⱦɥɹ ɷɬɨɝɨ ɫɥɭɱɚɹ ɜɟɥɢɱɢɧɚ ɬɟɪɹɟɦɨɣ ɩɪɢ ɫɬɨɥɤɧɨɜɟɧɢɢ ɤɢɧɟɬɢɱɟɫɤɨɣ
ɷɧɟɪɝɢɢ ɪɚɜɧɚ:
2
c
4
V
V
m
E |'
mM
)1(
E
. ɇɚɩɪɢɦɟɪ, ɟɫɥɢ ɲɚɪɢɤ ɩɚɞɚ-
04ɤɤ
E
.
0
ɤɤɤ
22
)(
Mm
ɉɪɢ ɭɫɥɨɜɢɢ
E
'
m << M, ɩɨɥɭɱɢɦ:
22
c
VmmV
E
22
0
ɟɬ ɧɚ ɫɬɚɥɶɧɭɸ ɩɥɢɬɭ, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɭɸ ɫ Ɂɟɦɥɟɣ, ɬɨ ɞɨɥɹ ɩɟɪɟɞɚɜɚɟɦɨɣ ɷɧɟɪɝɢɢ ɛɭɞɟɬ ɢɫɱɟɡɚɸɳɟɟ ɦɚɥɚ. Ɉɞɧɚɤɨ ɜɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɜɫɬɪɟɱɚɸɬɫɹ ɫɢɬɭɚɰɢɢ,
ɤɨɝɞɚ ɥɟɝɤɢɟ ɱɚɫɬɢɰɵ ɢɫɩɵɬɵɜɚɸɬ ɦɧɨɝɨɤɪɚɬɧɵɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɫ ɬɹɠɟɥɵɦɢ ɧɟɩɨɞɜɢɠɧɵɦɢ ɱɚɫɬɢɰɚɦɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɩɵɬɚɜ ɛɨɥɶɲɨɟ ɱɢɫɥɨ ɫɬɨɥɤɧɨɜɟɧɢɣ,
ɥɟɝɤɚɹ ɱɚɫɬɢɰɚ, ɜ ɤɨɧɟɱɧɨɦ ɫɱɟɬɟ, ɩɨɬɟɪɹɟɬ ɫɜɨɸ ɤɢɧɟɬɢɱɟɫɤɭɸ
ɷɧɟɪɝɢɸ.
124
Ɂɚɞɚɱɚ 4. Ⱦɚɬɶ ɨɩɢɫɚɧɢɟ ɧɟɰɟɧɬɪɚɥɶɧɨɝɨ ɭɞɚɪɚ.
Ɋɟɲɟɧɢɟ:
ɐɟɧɬɪɚɥɶɧɵɣ ɭɞɚɪ ɩɪɨɢɫɯɨɞɢɬ ɨɱɟɧɶ ɪɟɞɤɨ. ɑɚɳɟ ɜɫɟɝɨ ɬɟɥɚ ɫɬɚɥɤɢɜɚɸɬɫɹ
ɬɚɤ, ɱɬɨ ɜɟɤɬɨɪɵ ɢɯ ɫɤɨɪɨɫɬɟɣ
G
, 2,1
V
i ɧɟ ɥɟɠɚɬ ɧɚ ɨɞɧɨɣ ɩɪɹɦɨɣ (ɪɢɫ. 8.6).
i
Ɋɢɫ. 8.6
ȼ ɷɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ ɭɞɚɪ ɚɛɫɨɥɸɬɧɨ ɭɩɪɭɝɢɣ, ɦɵ ɞɨɥɠɧɵ ɩɨɫɬɭɩɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. ɉɪɢ ɫɬɨɥɤɧɨɜɟɧɢɢ ɫɨɯɪɚɧɹɟɬɫɹ ɤɨɥɢɱɟɫɬɜɨ ɞɜɢɠɟɧɢɹ, ɩɨɷɬɨɦɭ:
ɝɞɟ
GG GG
mV m V mV m V
11 2 2 11 2 2
,
– ɦɚɫɫɵ ɲɚɪɢɤɨɜ,
m
m
1
2
G
G
c
, 2,1
,
V
V
i – ɢɯ ɫɤɨɪɨɫɬɢ ɞɨ ɢ ɩɨɫɥɟ ɭɞɚɪɚ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
i
i
cc
,
ɉɨɫɤɨɥɶɤɭ ɤɨɥɢɱɟɫɬɜɨ ɞɜɢɠɟɧɢɹ – ɜɟɥɢɱɢɧɚ ɜɟɤɬɨɪɧɚɹ, ɬɨ ɦɨɠɧɨ ɪɚɡɥɨɠɢɬɶ ɢɦɩɭɥɶɫɵ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɧɨɪɦɚɥɶɧɭɸ ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɭɸ ɨɫɢ:
Ɂɞɟɫɶ
, 2,1 i – ɫɨɫɬɚɜɥɹɸɳɢɟ ɜɟɤɬɨɪɨɜ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɥɢɧɢɢ, ɫɨɟɞɢ-
V
in
ɧɹɸɳɟɣ ɰɟɧɬɪɵ ɲɚɪɨɜ, ɚ
mV +mV =mV +m V
11 22 11 22
nnnn
mV +mV = mV +mV
11IJ 22IJ 11IJ 22IJ
– ɫɨɫɬɚɜɥɹɸɳɢɟ ɜɟɤɬɨɪɨɜ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɩɟɪɩɟɧ-
V
W
i
ɞɢɤɭɥɹɪɧɨɦ ɷɬɨɣ ɥɢɧɢɢ. Ɍɟɩɟɪɶ ɭɱɬɟɦ, ɱɬɨ
cc
cc
,
.
n
222
ɢ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ
VVV
W
ɷɧɟɪɝɢɢ ɩɪɢɦɟɬ ɜɢɞ:
2
2
2
2
2
)(
1
11
2
2
)(
2
22
2
2
c
c
)(
1
11
2
2
c
c
VVmVVmVVmVVm
)(
WWWW
nnnn
2
22
2
.
Ɇɵ ɩɨɥɭɱɢɥɢ ɫɢɫɬɟɦɭ ɬɪɟɯ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɱɟɬɵɪɟɯ ɧɟɢɡɜɟɫɬɧɵɯ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɟɨɛɯɨɞɢɦɨ ɫɞɟɥɚɬɶ ɭɬɨɱɧɟɧɢɹ. ɇɚɩɢɫɚɜ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ, ɦɵ ɩɪɟɞɩɨɥɨɠɢɥɢ, ɱɬɨ ɲɚɪɵ ɢɞɟɚɥɶɧɨ ɝɥɚɞɤɢɟ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ
ɫɢɥɵ ɬɪɟɧɢɹ ɦɟɠɞɭ ɫɬɚɥɤɢɜɚɸɳɢɦɢɫɹ ɲɚɪɚɦɢ ɨɬɫɭɬɫɬɜɭɸɬ. ɋɥɟɞɫɬɜɢɟɦ ɷɬɨɝɨ
ɹɜɥɹɟɬɫɹ ɧɟɢɡɦɟɧɧɨɫɬɶ ɬɚɧɝɟɧɰɢɚɥɶɧɵɯ ɫɨɫɬɚɜɥɹɸɳɢɯ ɢɯ ɫɤɨɪɨɫɬɟɣ
c
. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ ɧɟɰɟɧɬɪɚɥɶɧɨɦ ɚɛɫɨɥɸɬɧɨ ɭɩɪɭɝɨɦ ɭɞɚɪɟ ɢɞɟ-
VV
WW
22
VVc
WW
11
ɚɥɶɧɨ ɝɥɚɞɤɢɯ ɲɚɪɨɜ ɧɨɪɦɚɥɶɧɵɟ ɫɨɫɬɚɜɥɹɸɳɢɟ ɫɤɨɪɨɫɬɟɣ ɜɟɞɭɬ ɫɟɛɹ ɬɚɤ ɠɟ,
ɤɚɤ ɢ ɩɪɢ ɰɟɧɬɪɚɥɶɧɨɦ ɭɞɚɪɟ, ɚ ɬɚɧɝɟɧɰɢɚɥɶɧɵɟ ɫɨɫɬɚɜɥɹɸɳɢɟ ɧɟ ɢɡɦɟɧɹɸɬɫɹ.
,
125
Ɋɚɫɫɦɨɬɪɢɦ ɨɞɧɨ ɢɡ ɜɚɠɧɵɯ ɫɥɟɞɫɬɜɢɣ, ɜɵɬɟɤɚɸɳɢɯ ɢɡ ɷɬɨɝɨ. ɉɭɫɬɶ ɲɚ-
D
D
E
ɪɢɤ ɩɚɞɚɟɬ ɧɚ ɦɚɫɫɢɜɧɭɸ ɧɟɩɨɞɜɢɠɧɭɸ ɩɥɢɬɭ ɩɨɞ ɭɝɥɨɦ
. ɉɨɞ ɤɚɤɢɦ ɭɝɥɨɦ
ɲɚɪɢɤ ɨɬɪɚɡɢɬɫɹ ɨɬ ɧɟɟ (ɫɦ. ɪɢɫ. 8.7).
Ɋɢɫ. 8.7
Ȼɭɞɟɦ ɩɨɥɚɝɚɬɶ ɭɞɚɪ ɚɛɫɨɥɸɬɧɨ ɭɩɪɭɝɢɦ. ɍɝɥɵ ɩɚɞɟɧɢɹ ɢ ɨɬɪɚɠɟɧɢɹ ɨɬɫɱɢɬɵɜɚɸɬɫɹ ɨɬ ɩɟɪɩɟɧɞɢɤɭɥɹɪɚ ɤ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɢɬɵ. ɂɡ ɪɢɫ. 8.7 ɫɥɟɞɭɟɬ, ɱɬɨ:
ɉɨɫɤɨɥɶɤɭ ɩɪɢ ɬɚɤɨɦ ɭɞɚɪɟ
, ɧɟɩɨɞɜɢɠɧɨɣ
mm
21
ɟɬɫɹ ɨɬ ɫɬɟɧɤɢ ɩɨɞ ɭɝɥɨɦ
V
1
W
D
, ɚ
tg
V
n
1
ɩɥɢɬɵ, ɢɦɟɟɦ
0
V
2
. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɭɝɨɥ ɨɬɪɚɠɟɧɢɹ ɱɚɫɬɢɰɵ ɨɬ
tg
VV
E
c
WW
11
V
V
,
c
1
W
.
c
n
1
c
, ɬɨ ɜ ɭɫɥɨɜɢɹɯ ɦɚɫɫɢɜɧɨɣ
VV
WW
22
c
V=V
. ɉɨɷɬɨɦɭ ɲɚɪɢɤ ɨɬɪɚɠɚ-
11
nn
ɫɬɟɧɤɢ ɪɚɜɟɧ ɭɝɥɭ ɟɟ ɩɚɞɟɧɢɹ. ȼ ɫɥɭɱɚɟ ɲɟɪɨɯɨɜɚɬɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɚɫɚɬɟɥɶɧɚɹ
ɫɨɫɬɚɜɥɹɸɳɚɹ ɫɤɨɪɨɫɬɢ ɦɨɠɟɬ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɨɭɞɚɪɟɧɢɹ ɢɡɦɟɧɢɬɶɫɹ, ɢ ɭɝɨɥ ɨɬɪɚɠɟɧɢɹ ɭɠɟ ɧɟ ɛɭɞɟɬ ɪɚɜɟɧ ɭɝɥɭ ɩɚɞɟɧɢɹ.
Ɂɚɞɚɱɚ 5. Ɋɚɫɫɦɨɬɪɟɬɶ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ, ɛɪɨɲɟɧɧɨɣ ɩɨɞ ɭɝɥɨɦ ɤ ɝɨɪɢ-
ɡɨɧɬɭ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ ɟɟ ɞɜɢɠɟɧɢɹ ɜ ɰɟɧɬɪɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ (ɪɢɫ. 8.8).
Ɋɢɫ. 8.8
126
Ɋɟɲɟɧɢɟ:
r
L
V
R
D
g
M
Ɋɚɫɫɦɨɬɪɢɦ ɦɚɬɟɪɢɚɥɶɧɭɸ ɬɨɱɤɭ, ɦɚɫɫɨɣ
ɬɪɚɥɶɧɚɹ ɫɢɥɚ
G
G
)( , ɝɞɟ
rrfF
G
– ɪɚɞɢɭɫ-ɜɟɤɬɨɪ, ɩɪɨɜɟɞɟɧɧɵɣ ɢɡ ɧɚɱɚɥɚ ɨɬɫɱɟ-
m, ɧɚ ɤɨɬɨɪɭɸ ɞɟɣɫɬɜɭɟɬ ɰɟɧ-
ɬɚ, ɜ ɤɨɬɨɪɨɦ ɧɚɯɨɞɢɬɫɹ ɫɢɥɨɜɨɣ ɰɟɧɬɪ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ, ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɦɨɦɟɧɬ
ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɫɢɥɨɜɨɣ ɰɟɧɬɪ ɪɚɜɟɧ ɧɭɥɸ
KK
KKK
ªº
,,()0MrF rrfr
¬¼
>@
Ɍɨɝɞɚ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɫɢɥɨɜɨɣ ɰɟɧɬɪ, ɫɨɯɪɚɧɹɟɬɫɹ
.
G
const
. ɉɨɫɤɨɥɶɤɭ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ – ɜɟɥɢɱɢɧɚ
ɜɟɤɬɨɪɧɚɹ, ɩɨɥɭɱɟɧɧɨɟ ɪɚɜɟɧɫɬɜɨ ɨɡɧɚɱɚɟɬ ɫɨɯɪɚɧɟɧɢɟ ɷɬɨɣ ɜɟɥɢɱɢɧɵ ɤɚɤ ɩɨ
ɧɚɩɪɚɜɥɟɧɢɸ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɬɚɤ ɢ ɩɨ ɟɟ ɚɛɫɨɥɸɬɧɨɦɭ ɡɧɚɱɟɧɢɸ.
ɉɨɫɤɨɥɶɤɭ ɧɚ ɱɚɫɬɢɰɭ ɞɟɣɫɬɜɭɟɬ ɫɢɥɚ ɝɪɚɜɢɬɚɰɢɢ, ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɫɢɥɨɜɨɣ ɰɟɧɬɪ, ɫɨɯɪɚɧɹɟɬɫɹ. Ⱥɛɫɨɥɸɬɧɨɟ ɡɧɚɱɟɧɢɟ ɜɟɥɢɱɢɧɵ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɜ ɦɨɦɟɧɬ ɟɟ ɛɪɨɫɚɧɢɹ
S
·
ɝɞɟ
§
D cos
sin
¸
¨
2
¹
©
– ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ ɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɬɨɱɤɟ ɬɪɚɟɤɬɨɪɢɢ,
D
RmVRmVL . Ɍɨɝɞɚ ɞɥɹ ɥɸɛɨɣ ɬɨɱɤɢ ɬɪɚɟɤɬɨɪɢɢ:
00
,
E D sin)(cos
0
hRmVRmVL
ɪɚɜɧɨ
– ɪɚɞɢɭɫ Ɂɟɦɥɢ,
ȕ – ɭɝɨɥ ɦɟɠɞɭ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɨɦ ɢ ɫɤɨɪɨɫɬɶɸ,
h – ɜɵɫɨɬɚ ɩɨɞɴɟɦɚ ɬɟɥɚ ɧɚɞ ɩɨɜɟɪɯɧɨɫɬɶɸ Ɂɟɦɥɢ. ȼ ɱɚɫɬɧɨɫɬɢ, ɞɥɹ
ɧɚɢɜɵɫɲɟɣ ɬɨɱɤɢ ɬɪɚɟɤɬɨɪɢɢ ɢɦɟɟɬ ɦɟɫɬɨ ɫɥɟɞɭɸɳɟɟ ɫɨɨɬɧɨɲɟɧɢɟ:
.
)(cos
0
hRVRV
ȿɫɥɢ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɷɬɭ ɠɟ ɡɚɞɚɱɭ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ ɧɚɛɥɸɞɚɬɟɥɹ, ɧɚɯɨɞɹɳɟɝɨɫɹ ɜ ɬɨɱɤɟ ɛɪɨɫɚɧɢɹ, ɬɨ ɜ ɷɬɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɞɟɣɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɚɹ ɫɢɥɚ –
ɫɢɥɚ ɬɹɠɟɫɬɢ
G
, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɜ ɥɸɛɨɣ ɬɨɱɤɟ ɬɪɚɟɤɬɨɪɢɢ.
m
ɉɨɷɬɨɦɭ ɜ ɷɬɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɞɟɣɫɬɜɭɟɬ ɦɨɦɟɧɬ ɫɢɥɵ ɬɹɠɟɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ, ɢ ɪɚɜɧɵɣ ɩɨ ɜɟɥɢɱɢɧɟ ,
ɯ – ɤɨɨɪɞɢɧɚɬɚ ɩɨ ɨɫɢ ɚɛɫɰɢɫɫ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɷɬɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ
ɝɞɟ
mgx
ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ ɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɢɥɢ ɧɟ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɵɦ. ɇɟɬɪɭɞɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɟɝɨ ɜɟɥɢɱɢɧɚ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ ɤɜɚɞɪɚɬɢɱɧɨ:
2
gt
mVL
0
2
cos
.
D
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɨɥɭɱɢɥɢ ɞɜɚ ɫɭɳɟɫɬɜɟɧɧɨ ɪɚɡɧɵɯ ɪɟɡɭɥɶɬɚɬɚ.
ȼɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ ɦɵ ɩɪɟɧɟɛɪɟɝɥɢ ɰɟɧɬɪɚɥɶɧɵɦ ɯɚɪɚɤɬɟɪɨɦ ɫɢɥɵ ɝɪɚɜɢɬɚɰɢɢ,
ɩɨɥɚɝɚɹ, ɱɬɨ ɨɧɚ ɜɟɪɬɢɤɚɥɶɧɨ ɧɚɩɪɚɜɥɟɧɚ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ. ɂɦɟɧɧɨ ɷɬɨ ɨɛɫɬɨɹɬɟɥɶɫɬɜɨ ɢ ɩɪɢɜɨɞɢɬ ɤ ɪɚɫɯɨɠɞɟɧɢɸ ɪɟɡɭɥɶɬɚɬɨɜ. ɉɨɷɬɨɦɭ ɩɪɢ ɪɟɲɟɧɢɢ ɧɟɨɛɯɨɞɢɦɨ ɫɥɟɞɢɬɶ ɤɚɤ ɡɚ ɜɵɛɨɪɨɦ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɬɚɤ ɢ ɡɚ ɢɫɩɨɥɶɡɭɟɦɵɦɢ ɩɪɢɛɥɢɠɟɧɢɹɦɢ.
127
Ɂɚɞɚɱɚ 6. ɋɬɟɪɠɟɧɶ ɞɥɢɧɨɣ l ɢ ɦɚɫɫɨɣ M ɦɨɠɟɬ ɜɪɚɳɚɬɶɫɹ ɜɨɤɪɭɝ ɧɟɩɨ-
Z
ɞɜɢɠɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɜɟɪɯɧɸɸ ɬɨɱɤɭ. ȼ ɫɬɟɪɠɟɧɶ ɧɚ ɪɚɫɫɬɨɹɧɢɢ
r ɨɬ ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɩɨɩɚɞɚɟɬ ɥɟɬɹɳɚɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ V ɩɭɥɹ
ɦɚɫɫɨɣ
m ɢ ɡɚɫɬɪɟɜɚɟɬ ɜ ɧɟɦ. ɇɚ ɤɚɤɨɣ ɭɝɨɥ ɨɧ ɨɬɤɥɨɧɢɬɫɹ ɨɬ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɩɨ-
ɥɨɠɟɧɢɹ?
Ɋɟɲɟɧɢɟ:
ɉɪɟɠɞɟ ɜɫɟɝɨ, ɩɪɨɚɧɚɥɢɡɢɪɭɟɦ ɮɢɡɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ. Ɉɧɚ ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ
ɬɟɥ – ɫɬɟɪɠɧɹ ɢ ɩɭɥɢ. ɉɭɥɸ ɦɨɠɧɨ ɩɪɢɧɹɬɶ ɡɚ ɦɚɬɟɪɢɚɥɶɧɭɸ ɬɨɱɤɭ, ɧɨ ɫɬɟɪɠɟɧɶ, ɟɫɬɟɫɬɜɟɧɧɨ, ɬɚɤɨɜɵɦ ɧɟ ɹɜɥɹɟɬɫɹ. Ɋɚɫɫɦɚɬɪɢɜɚɟɦɨɟ ɹɜɥɟɧɢɟ ɡɚɤɥɸɱɚɟɬɫɹ
ɜɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ ɷɬɢɯ ɬɟɥ. ɗɬɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɹɜɥɹɟɬɫɹ ɚɛɫɨɥɸɬɧɨ ɧɟɭɩɪɭɝɢɦ ɭɞɚɪɨɦ, ɢ
ɩɨɫɥɟ ɩɨɩɚɞɚɧɢɹ ɩɭɥɢ ɬɟɥɚ ɞɜɢɠɭɬɫɹ ɫ ɨɞɢɧɚɤɨɜɵɦɢ ɫɤɨɪɨɫɬɹɦɢ.
ȿɫɬɟɫɬɜɟɧɧɨ, ɱɬɨ ɜ ɷɬɨɦ ɩɪɨɰɟɫɫɟ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɧɟ ɫɨɯɪɚɧɹɟɬɫɹ. ɏɚɪɚɤɬɟɪ ɫɢɥ, ɜɨɡɧɢɤɚɸɳɢɯ ɩɪɢ ɷɬɨɦ, ɧɟɢɡɜɟɫɬɟɧ, ɩɨɷɬɨɦɭ ɪɟɲɢɬɶ ɞɚɧɧɭɸ ɡɚɞɚɱɭ
ɞɢɧɚɦɢɱɟɫɤɢɦ ɦɟɬɨɞɨɦ ɧɟɜɨɡɦɨɠɧɨ ɢ ɫɥɟɞɭɟɬ ɨɛɪɚɬɢɬɶɫɹ ɤ ɡɚɤɨɧɚɦ ɫɨɯɪɚɧɟɧɢɹ.
ɋɢɫɬɟɦɚ (ɬɟɥɨ + ɩɭɥɹ) ɹɜɥɹɟɬɫɹ ɧɟɡɚɦɤɧɭɬɨɣ. ɇɟɡɚɦɤɧɭɬɨɫɬɶ ɫɢɫɬɟɦɵ ɨɛɭɫɥɨɜɥɟɧɚ ɧɚɥɢɱɢɟɦ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ
ɩɨɩɚɞɚɧɢɢ ɩɭɥɢ ɷɬɢ ɫɢɥɵ
. ɇɨ ɷɬɢ ɫɢɥɵ ɞɟɣɫɬɜɭɸɬ ɢɦɟɧɧɨ ɜ ɬɨɱɤɟ ɩɨɞɜɟɫɚ, ɩɨɷɬɨɦɭ ɢɯ ɦɨɦɟɧɬ ɨɬ-
ɟɬɫɹ
ɢɡɦɟɧɹɸɬɫɹ, ɚ ɡɧɚɱɢɬ, ɢɦɩɭɥɶɫ ɫɢɫɬɟɦɵ ɧɟ ɫɨɯɪɚɧɹ-
ɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɬɨɱɤɢ ɪɚɜɟɧ ɧɭɥɸ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ ɩɨɩɚɞɚɧɢɢ ɩɭɥɢ
ɧɹɟɬɫɹ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ
Ɂɞɟɫɶ
ɢ
I
– ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɫɬɟɪɠɧɹ ɢ ɩɭɥɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚ-
I
1
2
ɧɚ ɬɟɥɨ ɜ ɬɨɱɤɟ ɩɨɞɜɟɫɚ. ɉɨɷɬɨɦɭ ɩɪɢ
ɫɨɯɪɚ-
, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɩɨɞɜɟɫɚ:
.
IImVr
21
ɳɟɧɢɹ, ɚ Z – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ. ɉɨɫɥɟ ɡɚɫɬɪɟɜɚɧɢɹ ɩɭɥɢ ɫɢɫɬɟɦɚ ɩɨɥɭɱɚɟɬ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ, ɡɚ ɫɱɟɬ ɱɟɝɨ ɢ ɧɚɱɢɧɚɟɬ ɩɨɞɧɢɦɚɬɶɫɹ ɧɚ ɧɟɤɨɬɨɪɭɸ ɜɵɫɨɬɭ. ȼ ɩɪɨɰɟɫɫɟ ɩɨɞɴɟɦɚ ɫɢɫɬɟɦɵ, ɟɫɥɢ ɩɪɟɧɟɛɪɟɱɶ ɫɢɥɚɦɢ ɬɪɟɧɢɹ, ɜɵɩɨɥɧɹɟɬɫɹ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ:
,
EEE
ɩɤk
ɝɞɟ
0
ɢ
E
– ɤɢɧɟɬɢɱɟɫɤɚɹ ɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
E
ɤ
ɩ
ɉɪɢ ɩɨɞɴɟɦɟ ɫɢɫɬɟɦɵ ɧɚ ɦɚɤɫɢɦɚɥɶɧɭɸ ɜɵɫɨɬɭ ɜɫɹ ɧɚɱɚɥɶɧɚɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɩɟɪɟɯɨɞɢɬ ɜ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɦɚɫɫ
ɜ ɧɚɱɚɥɶɧɨɦ ɫɨɫɬɨɹɧɢɢ:
2
II
Z
21
2
glMmghMm
.cos1
M
ɉɨɞɫɬɚɜɥɹɹ ɜ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɡɧɚɱɟɧɢɟ Z, ɩɨɥɭɱɚɟɦ:
22
rmV
1cos
M
2
.
glII
21
128
Ɂɚɞɚɱɚ 7. ɑɚɫɬɢɰɚ ɦɚɫɫɨɣ m ɡɚɤɪɟɩɥɟɧɚ ɧɚ ɧɢɬɢ ɢ ɜɪɚɳɚɟɬɫɹ (ɫɦ. ɪɢɫ. 8.9).
Ʉɨɝɞɚ ɞɥɢɧɚ ɧɢɬɢ ɪɚɜɧɚ
, ɥɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɪɚɜɧɚ
r
0
. ɑɟɦɭ ɪɚɜɧɚ
V
0
ɪɚɛɨɬɚ, ɫɨɜɟɪɲɚɟɦɚɹ ɩɪɢ ɭɤɨɪɚɱɢɜɚɧɢɢ ɧɢɬɢ?
Ɋɢɫ. 8.9
Ɋɟɲɟɧɢɟ:
ɋɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɱɚɫɬɢɰɭ ɫɨ ɫɬɨɪɨɧɵ ɧɢɬɢ, ɧɚɩɪɚɜɥɟɧɚ ɩɨ ɪɚɞɢɭɫɭ.
ɉɨɷɬɨɦɭ ɜ ɩɪɨɰɟɫɫɟ ɭɤɨɪɚɱɢɜɚɧɢɹ ɧɢɬɢ ɦɨɦɟɧɬ ɷɬɨɣ ɫɢɥɵ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ
ɜɪɚɳɟɧɢɹ, ɪɚɜɟɧ ɧɭɥɸ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɡɚɤɨɧɨɦ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ:
.
mVrrmV
00
ɋɨɜɟɪɲɚɟɦɚɹ ɪɚɛɨɬɚ ɪɚɜɧɚ ɢɡɦɟɧɟɧɢɸ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ.
Ɍɨɝɞɚ:
2
·
§
1
2
2
1
2
0
2
r
2
0
¸
¨
mVVVmA
0
¨
©
.
1
2
¸
r
¹
Ɂɚɞɚɱɚ 8. ɒɚɪɢɤ ɦɚɫɫɨɣ m ɫɤɨɥɶɡɢɬ ɛɟɡ ɬɪɟɧɢɹ ɩɨ ɜɧɭɬɪɟɧɧɟɣ ɩɨɜɟɪɯɧɨ-
ɫɬɢ ɤɨɧɭɫɚ (ɪɢɫ. 8.10). Ɍɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ – ɧɚɤɥɨɧɟɧɧɵɣ ɤ ɩɨɜɟɪɯɧɨɫɬɢ
h
Ɂɟɦɥɢ ɷɥɥɢɩɫ. ɂɡɜɟɫɬɧɵ ɜɵɫɨɬɵ
ɩɨɞɴɟɦɚ. ɇɚɣɬɢ ɫɤɨɪɨɫɬɢ ɲɚɪɢɤɚ
ɢ h2 ɜ ɬɨɱɤɚɯ ɧɚɢɦɟɧɶɲɟɝɨ ɢ ɧɚɢɛɨɥɶɲɟɝɨ
1
V
ɢ V2 ɜ ɷɬɢɯ ɬɨɱɤɚɯ.
1
Ɋɢɫ. 8.10
129
Ɋɟɲɟɧɢɟ:
g
K
GGG
G
ɑɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɩɨ ɷɥɥɢɩɫɭ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɟɡɭɥɶɬɢɪɭɸɳɚɹ ɫɢɥɵ ɬɹɠɟ-
m
ɫɬɢ
ɢ ɫɢɥɵ ɪɟɚɤɰɢɹ ɨɩɨɪɵ
G
F
ɧɚɩɪɚɜɥɟɧɚ ɜ ɫɬɨɪɨɧɭ ɨɞɧɨɝɨ ɢɡ ɟɝɨ ɮɨɤɭɫɨɜ.
p
Ɋɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɫɢɬɭɚɰɢɹ ɚɧɚɥɨɝɢɱɧɚ ɞɜɢɠɟɧɢɸ ɜ ɰɟɧɬɪɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ,
ɝɞɟ ɫɢɥɨɜɨɣ ɰɟɧɬɪ ɤɚɤ ɪɚɡ ɢ ɧɚɯɨɞɢɬɫɹ ɜ ɮɨɤɭɫɟ ɬɪɚɟɤɬɨɪɢɢ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
ɜɵɩɨɥɧɹɟɬɫɹ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ. ɉɨɷɬɨɦɭ ɞɥɹ ɞɜɭɯ ɜɵɞɟɥɟɧɧɵɯ ɬɨɱɟɤ ɧɚɢɛɨɥɶɲɟɝɨ ɢ ɧɚɢɦɟɧɶɲɟɝɨ ɩɨɞɴɟɦɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɟɪɲɢɧɵ ɤɨɧɭɫɚ
ɢɦɟɟɦ:
,
,, rVmrVm
2211
ɝɞɟ
>@> @
GG
rr
– ɪɚɞɢɭɫ-ɜɟɤɬɨɪɵ ɨɬ ɜɟɪɲɢɧɵ ɤɨɧɭɫɚ ɤ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɬɨɱɤɚɦ ɬɪɚ-
2,1
ɟɤɬɨɪɢɢ. Ɋɚɫɤɪɨɟɦ ɜɟɤɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ:
,
hVhV
2211
ɝɞɟ
– ɜɵɫɨɬɵ ɞɨ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɬɨɱɟɤ.
, hh
21
ɉɨɫɤɨɥɶɤɭ ɞɜɢɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɤɨɧɫɟɪɜɚɬɢɜɧɵɯ ɫɢɥ, ɬɨ
ɜɵɩɨɥɧɹɟɬɫɹ ɢ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ:
1
2
1
2
1
2
mghmVmghmV
.
21
2
2
Ɍɟɩɟɪɶ ɨɤɨɧɱɚɬɟɥɶɧɨ ɨɩɪɟɞɟɥɢɦ ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɟɣ:
2
hV
g
21
,
hh
21
2
g
hV
12
.
hh
21
Ɋɚɫɫɦɨɬɪɟɧɧɵɟ ɩɪɢɦɟɪɵ ɩɨɤɚɡɵɜɚɸɬ ɩɪɨɞɭɤɬɢɜɧɨɫɬɶ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɡɚɤɨɧɨɜ ɫɨɯɪɚɧɟɧɢɹ ɜ ɬɟɯ ɫɢɬɭɚɰɢɹɯ, ɤɨɝɞɚ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɨɞɯɨɞɚ
ɜɟɫɶɦɚ ɡɚɬɪɭɞɧɢɬɟɥɶɧɨ ɢɥɢ ɧɟɜɨɡɦɨɠɧɨ. ȼ ɬɨ ɠɟ ɜɪɟɦɹ ɨɱɟɜɢɞɟɧ ɢ ɧɟɞɨɫɬɚɬɨɤ
ɬɚɤɨɝɨ ɩɨɞɯɨɞɚ. Ɉɧ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɧɟɜɨɡɦɨɠɧɨɫɬɢ ɞɟɬɚɥɶɧɨɝɨ ɨɩɢɫɚɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɝɨ ɩɨɜɟɞɟɧɢɹ ɫɢɫɬɟɦɵ, ɤɨɬɨɪɨɟ ɜɨɡɦɨɠɧɨ ɜ ɪɚɦɤɚɯ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɨɞɯɨɞɚ.
1. Ɇɚɯɨɜɨɟ ɤɨɥɟɫɨ ɧɚɱɢɧɚɟɬ ɜɪɚɳɚɬɶɫɹ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢ-
ɟɦ 0,5 ɪɚɞ/ɫ
8.3. Ɂɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ
2
ɢ ɱɟɪɟɡ 15 ɫɟɤɭɧɞ ɩɨɫɥɟ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ ɩɪɢɨɛɪɟɬɚɟɬ ɦɨɦɟɧɬ ɢɦ-
ɩɭɥɶɫɚ 73,5 ɤɝāɦ²/ɫ. ɇɚɣɬɢ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɤɨɥɟɫɚ ɱɟɪɟɡ 20 ɫɟɤɭɧɞ ɩɨɫɥɟ
ɧɚɱɚɥɚ ɜɪɚɳɟɧɢɹ.
ɑɟɥɨɜɟɤ ɜɟɫɨɦ 600 ɇ ɫɬɨɢɬ ɧɚ ɤɪɚɸ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɚɬɮɨɪɦɵ ɦɚɫ-
2.
ɫɨɣ 100 ɤɝ, ɤɨɬɨɪɚɹ ɜɪɚɳɚɟɬɫɹ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ
ɰɟɧɬɪ ɩɥɚɬɮɨɪɦɵ, ɞɟɥɚɹ 10 ɨɛɨɪɨɬɨɜ ɜ ɦɢɧɭɬɭ. ɇɚɣɬɢ ɱɚɫɬɨɬɭ ɜɪɚɳɟɧɢɹ ɩɥɚɬɮɨɪɦɵ ɩɨɫɥɟ ɩɟɪɟɯɨɞɚ ɱɟɥɨɜɟɤɚ ɨɬ ɤɪɚɹ ɩɥɚɬɮɨɪɦɵ ɤ ɟɟ ɰɟɧɬɪɭ? ɋɱɢɬɚɬɶ ɩɥɚɬɮɨɪɦɭ ɨɞɧɨɪɨɞɧɵɦ ɤɪɭɝɥɵɦ ɞɢɫɤɨɦ, ɚ ɱɟɥɨɜɟɤɚ – ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ.
ɋɬɚɥɶɧɨɣ ɲɚɪɢɤ ɦɚɫɫɨɣ 20 ɝ ɩɚɞɚɟɬ ɫ ɜɵɫɨɬɵ 1 ɦ ɧɚ ɫɬɚɥɶɧɭɸ ɩɥɢɬɭ
3.
ɢ ɨɬɫɤɚɤɢɜɚɟɬ ɨɬ ɧɟɟ ɧɚ ɜɵɫɨɬɭ 81 ɫɦ. ɇɚɣɬɢ: 1) ɢɦɩɭɥɶɫ ɫɢɥɵ, ɩɨɥɭɱɟɧɧɨɣ
ɩɥɢɬɨɣ ɡɚ ɜɪɟɦɹ ɭɞɚɪɚ; 2) ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɤɨɬɨɪɨɟ ɜɵɞɟɥɢɥɨɫɶ ɩɪɢ ɭɞɚɪɟ.
130
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