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13
ɋ
A
ȼ
ɋ
Ɋ
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 1
14
15
17
Ⱥ
30
F
30
ȼ
ɋ
60
A
16
D
Ɋ
ɋ
D
45
ȼ
A
ȼ
120
Ɋ
ȼ
A
F
50
ɋ
18
ɋ
60
F
ȼ
60
105
60
F
30
A
19
20
A
A
50
ɋ
ȼ
ȼ
D
50
11
D
Ɋ
ɋ
21
A
P
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 1
22
23
25
27
60
A
ȼ
105
F
ɋ
30
ɋ
A
50
Ⱥ
24
ȼ
75
ɋ
F
60
D
ȼ
ȼ
120
Ɋ
50
D
A
26
30
ȼ
ȼ
F
60
C
ɋ
F
50
45
ɋ
28
A
ɋ
ɋ
60
ȼ
ȼ
60
D
F
40
A
12
A
Ɋ
29
0
F
y
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 1
ɋ
3
120
ȼ
50
D
A
Ɋ
Ⱥ
30
45
ȼ
ɋ
Ɍɚɛɥɢɰɚ 1
ʋ ɩ/ɩ 1 2 3 4 5 6 7 8 9 10
F, P ɤɇ 9 3 5 6 4 7 10 12 2 8
ɉɪɢɦɟɪ ʋ 1
Ɂɚɞɚɱɚ ɧɚ ɨɩɪɟɞɟɥɟɧɢɟ ɪɟɚɤɰɢɣ ɧɟɜɟɫɨɦɵɯ ɫɬɟɪɠɧɟɣ.
Ƚɪɭɡ Ɋ = 80 ɤɇ ɩɨɞɧɢɦɚɟɬɫɹ ɥɟɛɟɞɤɨɣ ɩɪɢ ɩɨɦɨɳɢ ɤɚɧɚɬɚ, ɩɟɪɟɤɢɧɭɬɨɝɨ ɱɟɪɟɡ ɧɟɩɨɞɜɢɠɧɵɣ ɛɥɨɤ Ⱥ. Ɉɩɪɟɞɟɥɢɬɶ ɭɫɢɥɢɹ, ɜɨɡɧɢɤɚɸɳɢɟ ɜ ɫɬɟɪɠɧɹɯ Ⱥȼ ɢ Ⱥɋ, ɩɨɞɞɟɪɠɢɜɚɸɳɢɯ ɛɥɨɤ Ⱥ. Ɍɪɟɧɢɟɦ ɜ ɛɥɨɤɟ ɢ
ɟɝɨ ɪɚɡɦɟɪɚɦɢ ɩɪɟɧɟɛɪɟɱɶ.
ɍɫɢɥɢɹ ɜ ɫɬɟɪɠɧɹɯ ɪɚɜɧɵ ɩɨ ɦɨɞɭɥɸ ɪɟɚɤɰɢɹɦ ɫɬɟɪɠɧɟɣ ɢ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɢɦ ɧɚɩɪɚɜɥɟɧɵ, ɩɨɷɬɨɦɭ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɭɫɢɥɢɣ ɜ
S
ɢ
ɫɬɟɪɠɧɹɯ ɜɵɱɢɫɥɢɦ ɪɟɚɤɰɢɢ ɫɬɟɪɠɧɟɣ
S .
1
2
60
ȼ
30
Ⱥ
Ⱦ
60
S
1
T
x
30
ɋ
S
Ɋ
P
2
Ɋɢɫ. 9.
13
Ɋɟɲɚɟɦ ɡɚɞɚɱɭ ɚɧɚɥɢɬɢɱɟɫɤɢɦ ɫɩɨɫɨɛɨɦ.
P
T
Ɋɚɫɫɦɨɬɪɢɦ ɪɚɜɧɨɜɟɫɢɟ ɛɥɨɤɚ Ⱥ (ɪɢɫ. 9), ɩɪɟɞɫɬɚɜɢɜ ɟɝɨ ɜ ɜɢɞɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɫɜɨɛɨɞɧɨɝɨ ɬɟɥɚ, ɞɥɹ ɱɟɝɨ ɨɫɜɨɛɨɞɢɦ ɛɥɨɤ Ⱥ ɨɬ ɫɜɹɡɟɣ,
ɡɚɦɟɧɢɜ ɢɯ ɪɟɚɤɰɢɹɦɢ ɫɬɟɪɠɧɟɣ
ɱɟɬɵɪɟ ɫɢɥɵ: ɧɚɬɹɠɟɧɢɟ ɥɟɜɨɣ ɜɟɬɜɢ ɧɢɬɢ, ɪɚɜɧɨɟ
ɜɟɬɜɢ ɧɢɬɢ
S ɢ 2S , ɧɚɩɪɚɜɥɟɧɧɵɟ ɜɞɨɥɶ ɫɬɟɪɠɧɟɣ ɨɬ ɭɡɥɚ (ɢɫɬɢɧɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ
1
, ɪɚɜɧɨɟ ɩɨ ɜɟɥɢɱɢɧɟ ɬɚɤɠɟ Ɋ(Ɍ = Ɋ) ɢ ɪɟɚɤɰɢɢ ɫɬɟɪɠɧɟɣ
S ɢ 2S . Ɍɨɝɞɚ ɧɚ ɛɥɨɤ Ⱥ ɞɟɣɫɬɜɭɸɬ
1
, ɧɚɬɹɠɟɧɢɟ ɩɪɚɜɨɣ
ɪɟɚɤɰɢɣ ɫɬɟɪɠɧɟɣ ɭɬɨɱɧɹɟɬɫɹ ɩɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ).
ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ.
ɗɬɨ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɞɜɭɦɹ ɫɩɨɫɨɛɚɦɢ.
I ɫɩɨɫɨɛ.
Ɂɚ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɜɨɡɶɦɟɦ ɬɨɱɤɭ
ɬɚɥɶɧɨ ɜɞɨɥɶ ɥɢɧɢɢ ɞɟɣɫɬɜɢɹ ɫɢɥɵ
ɥɹɪɧɨ ɨɫɢ
ɏ.
Ⱦɥɹ ɩɥɨɫɤɨɣ ɫɢɫɬɟɦɵ ɫɯɨɞɹɳɢɯɫɹ ɫɢɥ ɫɨɫɬɚɜɥɹɟɦ ɞɜɚ ɭɪɚɜɧɟɧɢɹ
Ⱥ, ɨɫɶ ɏ ɧɚɩɪɚɜɢɦ ɝɨɪɢɡɨɧ-
T , ɨɫɶ ɍ ɩɪɨɜɨɞɢɦ ɩɟɪɩɟɧɞɢɤɭ-
ɪɚɜɧɨɜɟɫɢɹ:
FKX = 0; T – S1 ⋅ cos60° – S
⋅
cos30° = 0; (1)
2
FKɍ = 0; S1 ⋅ cos30° – S2 ⋅ cos60° – P = 0. (2)
Ɋɟɲɚɟɦ ɩɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɦɟɬɨɞɨɦ ɩɨɞɫɬɚɧɨɜɤɢ, ɬ. ɤ. ɤɚɠɞɨɟ
ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɩɨ ɞɜɟ ɧɟɢɡɜɟɫɬɧɵɟ ɜɟɥɢɱɢɧɵ.
ɂɡ ɭɪɚɜɧɟɧɢɹ (I) ɜɵɪɚɡɢɦ ɜɟɥɢɱɢɧɭ ɪɟɚɤɰɢɢ
S
:
1
S
=
1
2
60cos
. (3)
D
D
30cosST
⋅−
ɉɨɞɫɬɚɜɢɦ ɡɧɚɱɟɧɢɟ
S
ɜ ɭɪɚɜɧɟɧɢɟ (2) ɢ ɪɟɲɢɦ ɟɝɨ:
1
D
⋅−
30cosST
2
D
60cos
2
2
2
S
22
=
2
2
2
⋅−⋅
+
DD
=−⋅−⋅
;0P60cosS30cos
2
DD
60cosP30cosT
.
22
DD
60cos30cos
DDDD
=⋅−⋅−⋅−⋅
;060cosP60cosS30cosS30cosT
DDDD
⋅−⋅=+⋅
;60cosP30cosT)60cos30cos(S
14
ɉɨɞɫɬɚɜɢɦ ɱɢɫɥɨɜɵɟ ɡɧɚɱɟɧɢɹ ɜ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ:
y
80 866 80 0,5 80 (0,866 0,5)
S= = =29,3ɤɇ
2
––
⋅⋅⋅
31
+
44
1
.
ɂɡ ɜɵɪɚɠɟɧɢɹ (3) ɨɩɪɟɞɟɥɢɦ:
80 29,3 0,866
S = = 109,3 ɤɇ
1
⋅
–
0,5
.
II ɫɩɨɫɨɛ.
x
S
1
A
S
2
60
30
T
ɉɪɨɜɟɞɟɦ ɨɞɧɭ ɢɡ ɨɫɟɣ, ɧɚɩɪɢɦɟɪ, ɨɫɶ
ɏ ɩɨ ɧɟɢɡɜɟɫɬɧɨɣ ɪɟɚɤɰɢɢ
ɝɞɚ, ɭɱɢɬɵɜɚɹ, ɱɬɨ ɫɬɟɪɠɧɢ
S (ɪɢɫ. 10). Ɍɨ-
2
Ⱥȼ ɢ Ⱥɋ ɜɡɚɢɦ-
ɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ, ɨɫɶ ɭ ɩɪɨɣɞɟɬ ɩɨ
S
ɧɚɩɪɚɜɥɟɧɢɸ ɪɟɚɤɰɢɢ
. ɋɨɫɬɚɜɢɦ ɭɪɚɜ-
1
ɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜ ɷɬɨɣ ɫɢɫɬɟɦɟ ɨɫɟɣ:
P
Ɋɢɫ. 10.
FKX = 0; – S2 + Ɍ ⋅ cos30° – Ɋ⋅ cos 60° = 0;
F
= 0; S1 – Ɋ ⋅ cos 30° – Ɍ ⋅ cos 60° = 0.
Kɍ
ɂɡ ɭɪɚɜɧɟɧɢɣ ɧɚɯɨɞɢɦ:
S2 = T ⋅ cos 30° – P ⋅ cos 60° = 80 ⋅ 0,866 – 80 ⋅ 0,5 = 29,3 ɤɇ;
S1 = Ɋ ⋅ cos 30° + Ɍ ⋅ cos 60° = 80 ⋅ 0,866 + 80 ⋅ 0,5 = 109,3 ɤɇ.
ɋɪɚɜɧɢɜɚɹ ɪɟɲɟɧɢɹ ɞɜɭɦɹ ɫɩɨɫɨɛɚɦɢ, ɡɚɤɥɸɱɚɟɦ, ɱɬɨ ɜɬɨɪɨɟ ɪɟɲɟɧɢɟ ɪɚɰɢɨɧɚɥɶɧɟɣ, ɬ. ɤ. ɩɪɢ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɜ ɤɚɠɞɨɦ
ɭɪɚɜɧɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɢɦɟɟɬɫɹ ɬɨɥɶɤɨ ɨɞɧɚ ɧɟɢɡɜɟɫɬɧɚɹ ɪɟɚɤɰɢɹ. Ɋɟɲɟɧɢɟ ɬɚɤɢɯ ɭɪɚɜɧɟɧɢɣ ɝɨɪɚɡɞɨ ɩɪɨɳɟ, ɱɟɦ ɜ ɩɟɪɜɨɦ ɫɥɭɱɚɟ.
ɉɨɥɭɱɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɪɟɚɤɰɢɣ ɫɜɹɡɟɣ ɩɨɥɨɠɢɬɟɥɶɧɵ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɟɪɜɨɧɚɱɚɥɶɧɨ, ɜɵɛɪɚɧɧɵɟ ɧɚɩɪɚɜɥɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ.
Ɉɛɚ ɫɬɟɪɠɧɹ
ɪɚɫɬɹɝɢɜɚɸɬɫɹ.
Ⱦɥɹ ɩɪɨɜɟɪɤɢ ɩɪɚɜɢɥɶɧɨɫɬɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɞɨɫɬɚɬɨɱɧɨ ɜɵɱɢɫɥɢɬɶ ɚɥɝɟɛɪɚɢɱɟɫɤɭɸ ɫɭɦɦɭ ɩɪɨɟɤɰɢɣ ɜɫɟɯ ɫɢɥ ɧɚ ɥɸɛɭɸ ɨɫɶ, ɤɨɬɨɪɚɹ
15
ɧɟ ɢɫɩɨɥɶɡɨɜɚɥɚɫɶ ɜ ɪɟɲɟɧɢɢ, ɧɨ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɭ
ɰɢɢ ɨɩɪɟɞɟɥɟɧɵ ɜɟɪɧɨ, ɬɨ ɜ ɪɟɡɭɥɶɬɚɬɟ ɜɵɱɢɫɥɟɧɢɣ ɩɨɥɭɱɢɬɫɹ
ɇɚɩɪɢɦɟɪ: ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ
1 ɫɩɨɫɨɛɨɦ, ɜɵɱɢɫɥɢɦ ɚɥɝɟɛɪɚɢ-
Ⱥ. ȿɫɥɢ ɪɟɚɤ-
0.
ɱɟɫɤɭɸ ɫɭɦɦɭ ɩɪɨɟɤɰɢɣ ɧɚ ɨɫɶ, ɫɨɜɩɚɞɚɸɳɭɸ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɫɢɥɵ
(ɪɢɫ. 10):
FKX = –S2 – Ɋ ⋅ cos 60° + Ɍ ⋅ cos 30° =
= –29,3 – 80 ⋅ 0,5 + 80 ⋅ 0,866 = –69,3 + 693 = 0.
Ɉɬɜɟɬ: S1 = 109,3 ɤɇ; S2 = 29,3 ɤɇ.
2. ɄɈɇɌɊɈɅɖɇɕȿ ȼɈɉɊɈɋɕ
1.
ɑɬɨ ɬɚɤɨɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ, ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ?
ɑɬɨ ɬɚɤɨɟ ɧɟɫɜɨɛɨɞɧɨɟ ɬɟɥɨ?
2.
3.
ɑɬɨ ɧɚɡɵɜɚɟɬɫɹ ɫɜɹɡɶɸ?
4.
ɑɬɨ ɬɚɤɨɟ ɪɟɚɤɰɢɹ ɫɜɹɡɢ?
ɉɪɢɧɰɢɩ ɨɫɜɨɛɨɠɞɟɧɢɹ ɨɬ ɫɜɹɡɟɣ.
5.
6.
Ʉɚɤ ɧɚɩɪɚɜɥɟɧɵ ɪɟɚɤɰɢɢ ɝɥɚɞɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɠɟɫɬɤɨɝɨ
ɫɬɟɪɠɧɹ, ɝɢɛɤɨɣ ɧɢɬɢ?
ɑɬɨ ɬɚɤɨɟ ɩɥɨɫɤɚɹ ɫɢɫɬɟɦɚ ɫɯɨɞɹɳɢɯɫɹ ɫɢɥ (ɉɋɋɋ)?
7.
8.
ɍɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɉɋɋɋ ɜ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɮɨɪɦɟ.
9.
ɑɬɨ ɬɚɤɨɟ ɩɪɨɟɤɰɢɹ ɫɢɥɵ ɧɚ ɨɫɶ?
ɑɟɦɭ ɱɢɫɥɟɧɧɨ ɪɚɜɧɚ ɩɪɨɟɤɰɢɹ ɫɢɥɵ ɧɚ ɨɫɶ? ɉɪɚɜɢɥɨ ɡɧɚɤɨɜ.
10.
11.
ȼ ɤɚɤɨɦ ɫɥɭɱɚɟ ɩɪɨɟɤɰɢɹ ɫɢɥɵ ɧɚ ɨɫɶ ɪɚɜɧɚ ɧɭɥɸ, ɪɚɜɧɚ ɦɨɞɭ-
ɥɸ ɫɢɥɵ?
ɍɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɉɋɋɋ ɜ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɮɨɪɦɟ.
12.
S
2
16
ɉɊȺɄɌɂɑȿɋɄɈȿ ɁȺɇəɌɂȿ ʋ 2
F
0
Ɇ0(
F
Ɇ0(
F
Ɍɟɦɚ: Ɉɩɪɟɞɟɥɟɧɢɟ ɪɟɚɤɰɢɣ ɫɜɹɡɟɣ ɜ ɩɪɨɢɡɜɨɥɶɧɨɣ ɩɥɨɫɤɨɣ ɫɢ-
ɫɬɟɦɟ ɫɢɥ.
ɐɟɥɶ: Ɉɫɜɨɢɬɶ ɨɩɪɟɞɟɥɟɧɢɟ ɪɟɚɤɰɢɣ ɛɚɥɨɱɧɵɯ ɨɩɨɪ.
ȼɪɟɦɹ ɩɪɨɜɟɞɟɧɢɹ: 2 ɱɚɫɚ.
1. ɈɋɇɈȼɇɕȿ ɉɈɅɈɀȿɇɂə ɌȿɈɊɂɂ
ɉɪɨɢɡɜɨɥɶɧɨɣ ɩɥɨɫɤɨɣ ɫɢɫɬɟɦɨɣ ɫɢɥ ɧɚɡɵɜɚɟɬɫɹ ɬɚɤɚɹ ɫɢɫɬɟɦɚ
ɫɢɥ, ɥɢɧɢɢ ɞɟɣɫɬɜɢɹ ɤɨɬɨɪɵɯ ɪɚɫɩɨɥɨɠɟɧɵ ɜ ɨɞɧɨɣ ɩɥɨɫɤɨɫɬɢ ɢ ɧɟ ɩɟɪɟɫɟɤɚɸɬɫɹ ɜ ɨɞɧɨɣ ɬɨɱɤɟ.
Ɇɨɦɟɧɬɨɦ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɧɚɡɵɜɚɸɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɹ
ɦɨɞɭɥɹ ɫɢɥɵ ɧɚ ɟɟ ɩɥɟɱɨ.
ɉɥɟɱɨɦ ɧɚɡɵɜɚɟɬɫɹ ɞɥɢɧɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɚ, ɨɩɭɳɟɧɧɨɝɨ ɢɡ ɬɨɱɤɢ
ɧɚ ɥɢɧɢɸ ɞɟɣɫɬɜɢɹ ɫɢɥɵ, ɪɢɫ. 11.
Ɇɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɫɱɢɬɚɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɵɦ
ɫɢɥɚ ɫɬɪɟɦɢɬɫɹ ɜɪɚɳɚɬɶ ɫɜɨɟ ɩɥɟɱɨ ɩɨ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɟ (ɫɦ. ɪɢɫ. 11, ɛ)
ɢ ɨɬɪɢɰɚɬɟɥɶɧɵɦ – ɟɫɥɢ ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ (ɫɦ. ɪɢɫ. 11, ɚ).
Ɇɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɪɚɜɟɧ ɧɭɥɸ, ɟɫɥɢ ɥɢɧɢɹ ɞɟɣɫɬɜɢɹ ɫɢɥɵ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɷɬɭ ɬɨɱɤɭ.
0
h
.
h
, ɟɫɥɢ
.
) > 0
F
) < 0
ɚ) ɛ)
Ɋɢɫ. 11.
F
h
F
Ɋɢɫ. 12.
17
ɉɚɪɨɣ ɫɢɥ ɧɚɡɵɜɚɟɬɫɹ ɫɢɫɬɟɦɚ ɞɜɭɯ ɩɚɪɚɥɥɟɥɶɧɵɯ ɫɢɥ, ɪɚɜɧɵɯ ɩɨ
ɦɨɞɭɥɸ, ɢ ɧɚɩɪɚɜɥɟɧɧɵɯ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɫɬɨɪɨɧɵ (ɪɢɫ. 12).
ɉɚɪɚ ɫɢɥ ɨɤɚɡɵɜɚɟɬ ɧɚ ɬɟɥɨ ɜɪɚɳɚɬɟɥɶɧɨɟ ɞɟɣɫɬɜɢɟ, ɤɨɬɨɪɨɟ ɨɰɟɧɢɜɚɟɬɫɹ ɦɨɦɟɧɬɨɦ.
Ɇɨɦɟɧɬɨɦ ɩɚɪɵ ɫɢɥ ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟ ɨɞɧɨɣ ɢɡ ɫɢɥ ɩɚɪɵ
ɧɚ ɟɟ ɩɥɟɱɨ
()
hFF;Fm ⋅=
.
′
ɉɥɟɱɨɦ ɩɚɪɵ ɫɢɥ ɧɚɡɵɜɚɟɬɫɹ ɤɪɚɬɱɚɣɲɟɟ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɥɢɧɢɹɦɢ ɞɟɣɫɬɜɢɹ ɫɢɥ ɩɚɪɵ.
Ɇɨɦɟɧɬ ɩɚɪɵ ɫɢɥ ɫɱɢɬɚɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɵɦ, ɟɫɥɢ ɩɚɪɚ ɜɪɚɳɚɟɬ
ɫɜɨɟ ɩɥɟɱɨ (ɬɟɥɨ) ɩɨ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɟ, ɢ ɨɬɪɢɰɚɬɟɥɶɧɵɦ – ɟɫɥɢ ɩɪɨɬɢɜ
ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ.
ɋɜɨɣɫɬɜɚ ɩɚɪɵ ɫɢɥ:
– ɚɥɝɟɛɪɚɢɱɟɫɤɚɹ ɫɭɦɦɚ ɩɪɨɟɤɰɢɣ ɫɢɥ, ɨɛɪɚɡɭɸɳɢɯ ɩɚɪɭ, ɧɚ ɥɸ-
ɛɭɸ ɨɫɶ ɪɚɜɧɚ ɧɭɥɸ. ɉɨɷɬɨɦɭ ɜ ɭɪɚɜɧɟɧɢɢ ɩɪɨɟɤɰɢɣ ɫɢɥ ɧɚ ɤɚɤɭɸɥɢɛɨ ɨɫɶ ɧɟ ɡɚɩɢɫɵɜɚɸɬ ɦɨɦɟɧɬ ɩɚɪɵ ɫɢɥ;
– ɧɟ ɢɡɦɟɧɹɹ ɞɟɣɫɬɜɢɹ ɩɚɪɵ ɫɢɥ ɧɚ ɬɟɥɨ, ɟɟ ɦɨɠɧɨ ɩɟɪɟɦɟɳɚɬɶ ɜ
ɩɥɨɫɤɨɫɬɢ ɟɟ ɞɟɣɫɬɜɢɹ.
ȼ ɭɪɚɜɧɟɧɢɢ ɦɨɦɟɧɬɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɸɛɨɣ ɬɨɱɤɢ ɜ ɩɥɨɫɤɨɫɬɢ
ɞɟɣɫɬɜɢɹ ɩɚɪɵ ɫɢɥ, ɞɚɠɟ ɟɫɥɢ ɬɨɱɤɚ ɧɚɯɨɞɢɬɫɹ ɧɚ ɩɥɟɱɟ ɩɚɪɵ ɫɢɥ (ɥɢɧɢɢ, ɫɨɟɞɢɧɹɸɳɟɣ ɜɟɤɬɨɪɵ ɫɢɥ ɩɚɪɵ), ɧɟɨɛɯɨɞɢɦɨ ɡɚɩɢɫɚɬɶ ɦɨɦɟɧɬ
ɩɚɪɵ ɫɢɥ.
1.1. Ɋɚɜɧɨɜɟɫɢɟ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɩɨɞ ɞɟɣɫɬɜɢɟɦ
ɩɥɨɫɤɨɣ ɫɢɫɬɟɦɵ ɫɢɥ
Ⱦɥɹ ɪɚɜɧɨɜɟɫɢɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɩɥɨɫɤɨɣ ɫɢɫɬɟɦɵ ɫɢɥ, ɩɪɢɥɨɠɟɧ-
ɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ, ɧɟɨɛɯɨɞɢɦɨ ɢ ɞɨɫɬɚɬɨɱɧɨ, ɱɬɨɛɵ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ
′
R
ɢ ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ
Ɉ
, ɥɟɠɚɳɟɣ ɜ ɩɥɨɫɤɨɫɬɢ ɞɟɣɫɬɜɢɹ ɷɬɢɯ ɫɢɥ, ɛɵɥɢ ɪɚɜɧɵ ɧɭɥɸ, ɬ. ɟ.:
ȼ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɮɨɪɦɟ ɷɬɢ ɭɫɥɨɜɢɹ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɜ ɫɥɟɞɭɸ-
ɳɢɯ ɬɪɟɯ ɜɢɞɚɯ:
1.
=
¦
KX
¦
Kɍ
¦
M ɷɬɢɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ
0
′
;0F
=
;0F
=⋅
KA
Ɍɨɱɤɚ
.0)F(m
¦¦
Ⱥ ɜɵɛɢɪɚɟɬɫɹ ɩɪɨɢɡɜɨɥɶɧɨ.
====
0)F(mM;0FR
.
K00K
18
2.
3.
=
¦
¦
¦
KX
¦
¦
¦
;0)F(m
KA
=
;0)F(m
KB
=
.0)F(
=
;0)F(m
KA
=
;0)F(m
KB
=
.0)F(m
KC
Ɉɫɶ Ɉɏ, ɧɚ ɤɨɬɨɪɭɸ ɫɩɪɨɟɤɬɢɪɨɜɚɧɵ
ɫɢɥɵ, ɧɟ ɞɨɥɠɧɚ ɛɵɬɶ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɤ ɩɪɹɦɨɣ
ɉɪɨɢɡɜɨɥɶɧɵɟ ɬɨɱɤɢ
ɬɟɥɶɧɨ ɤɨɬɨɪɵɯ ɛɟɪɭɬɫɹ ɦɨɦɟɧɬɵ ɫɢɥ,
ɧɟ ɞɨɥɠɧɵ ɥɟɠɚɬɶ ɧɚ ɨɞɧɨɣ ɩɪɹɦɨɣ.
Ⱥȼ.
Ⱥ, ȼ, ɋ, ɨɬɧɨɫɢ-
ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ ɜɫɟ ɫɢɥɵ ɩɥɨɫɤɨɣ ɫɢɫɬɟɦɵ ɩɚɪɚɥɥɟɥɶɧɵ,
ɬɨ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɬɚɤɢɯ ɫɢɥ ɜɵɪɚɠɚɸɬɫɹ ɞɜɭɦɹ ɭɪɚɜɧɟɧɢɹɦɢ ɜ
ɫɥɟɞɭɸɳɢɯ ɞɜɭɯ ɜɢɞɚɯ:
1.
¦
KX
¦
;0)F(
==.0)F(m
KA
Ɉ
ɫɶ ɏ, ɧɚ ɤɨɬɨɪɭɸ ɫɩɪɨɟɤɬɢɪɨɜɚɧɵ ɫɢ-
ɥɵ, ɞɨɥɠɧɚ ɛɵɬɶ ɩɚɪɚɥɥɟɥɶɧɨɣ ɷɬɢɦ ɫɢɥɚɦ.
2.
¦
¦
;0)F(m
KA
==.0)F(m
KB
ɉɪɹɦɚɹ
ɥɚɦ.
Ⱥȼ ɧɟ ɩɚɪɚɥɥɟɥɶɧɚ ɞɚɧɧɵɦ ɫɢ-
Ɂɚɞɚɱɢ ɧɚ ɪɚɜɧɨɜɟɫɢɟ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɧɚɯɨɞɹɳɟɝɨɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ
ɩɪɨɢɡɜɨɥɶɧɨɣ ɫɢɫɬɟɦɵ ɫɢɥ, ɪɟɤɨɦɟɧɞɭɟɬɫɹ ɪɟɲɚɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɩɨɪɹɞɤɟ:
1)
ɜɵɞɟɥɢɬɶ ɬɟɥɨ, ɪɚɜɧɨɜɟɫɢɟ ɤɨɬɨɪɨɝɨ ɫɥɟɞɭɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ;
2)
ɢɡɨɛɪɚɡɢɬɶ ɧɚ ɪɢɫɭɧɤɟ ɜɫɟ ɡɚɞɚɧɧɵɟ ɫɢɥɵ;
ɩɪɢɦɟɧɢɬɶ ɩɪɢɧɰɢɩ ɨɫɜɨɛɨɠɞɟɧɢɹ ɨɬ ɫɜɹɡɟɣ, ɬ. ɟ. ɦɵɫɥɟɧɧɨ ɨɬ-
3)
ɛɪɨɫɢɬɶ ɫɜɹɡɢ ɢ ɡɚɦɟɧɢɬɶ ɢɯ ɞɟɣɫɬɜɢɟ ɪɟɚɤɰɢɹɦɢ ɫɜɹɡɟɣ;
4)
ɜɵɛɪɚɬɶ ɧɚɩɪɚɜɥɟɧɢɟ ɞɟɤɚɪɬɨɜɵɯ ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ ɢ ɬɨɱɤɭ (ɢɥɢ
ɬɨɱɤɢ), ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɫɨɫɬɚɜɢɬɶ ɭɪɚɜɧɟɧɢɟ
ɦɨɦɟɧɬɨɜ;
5)
ɫɨɫɬɚɜɢɬɶ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ;
ɪɟɲɢɬɶ ɫɢɫɬɟɦɭ ɫɨɫɬɚɜɥɟɧɧɵɯ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ ɢ ɨɩɪɟɞɟ-
6)
ɥɢɬɶ ɧɟɢɡɜɟɫɬɧɵɟ ɜɟɥɢɱɢɧɵ.
ɑɬɨɛɵ ɩɨɥɭɱɢɬɶ ɛɨɥɟɟ ɩɪɨɫɬɵɟ ɪɟɲɟɧɢɹ ɩɪɢ ɫɨɫɬɚɜɥɟɧɢɢ ɭɪɚɜɧɟɧɢɣ, ɧɟɨɛɯɨɞɢɦɨ ɨɞɧɭ ɢɡ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ ɧɚɩɪɚɜɥɹɬɶ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɧɟɢɡɜɟɫɬɧɵɦ ɫɢɥɚɦ. ɋɨɫɬɚɜɥɹɹ ɭɪɚɜɧɟɧɢɟ ɦɨɦɟɧɬɨɜ, ɰɟɧɬɪɨɦ
ɦɨɦɟɧɬɨɜ ɥɭɱɲɟ ɜɵɛɪɚɬɶ ɬɨɱɤɭ, ɝɞɟ ɩɟɪɟɫɟɤɚɟɬɫɹ ɧɚɢɛɨɥɶɲɟɟ ɤɨɥɢɱɟɫɬɜɨ ɧɟɢɡɜɟɫɬɧɵɯ ɫɢɥ.
ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɦɨɦɟɧɬɨɜ ɢɧɨɝɞɚ ɭɞɨɛɧɨ ɪɚɡɥɨɠɢɬɶ ɞɚɧɧɭɸ
ɧɚɤɥɨɧɧɭɸ ɫɢɥɭ ɧɚ ɞɜɟ ɫɨɫɬɚɜɥɹɸɳɢɟ ɢ, ɩɨɥɶɡɭɹɫɶ ɬɟɨɪɟɦɨɣ ȼɚɪɢɧɶɨɧɚ, ɧɚɯɨɞɢɬɶ ɦɨɦɟɧɬ ɫɢɥɵ ɤɚɤ ɫɭɦɦɭ ɦɨɦɟɧɬɨɜ ɷɬɢɯ ɫɨɫɬɚɜɥɹɸɳɢɯ.
19
R
A
ɑɬɨɛɵ ɧɚɣɬɢ ɦɨɦɟɧɬ ɨɬ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚɝɪɭɡɤɢ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ
q ɨɬɧɨɫɢɬɟɥɶɧɨ ɡɚɞɚɧɧɨɣ ɬɨɱɤɢ Ⱥ, ɧɟɨɛɯɨɞɢɦɨ:
– ɭɦɧɨɠɢɬɶ ɢɧɬɟɧɫɢɜɧɨɫɬɶ q ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚɝɪɭɡ-
ɤɢ ɧɚ ɞɥɢɧɭ ɭɱɚɫɬɤɚ
ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɭɸ
ɜ, ɧɚ ɤɨɬɨɪɨɦ ɪɚɫɩɪɟɞɟɥɟɧɚ ɷɬɚ ɧɚɝɪɭɡɤɚ (ɧɚɣɬɢ
R
), (ɪɢɫ. 13);
q
– ɭɦɧɨɠɢɬɶ ɩɨɥɭɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ qɜ ɧɚ ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɫɟɪɟɞɢɧɵ
ɭɱɚɫɬɤɚ
ɜ ɞɨ ɬɨɱɤɢ Ⱥ:
ɜ
·
§
A
¨
©
.
+⋅⋅=
ɚɜq)q(m
¸
2
¹
q
q
ȿɫɥɢ ɨɛɳɟɟ ɱɢɫɥɨ ɧɟɢɡɜɟɫɬɧɵɯ
ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɧɟ ɩɪɟɜɵɲɚɟɬ
ɱɢɫɥɚ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ, ɤɨɬɨ-
ɜ/2
ɜ/2
ɚɜ
Ɋɢɫ. 13.
ɪɵɟ ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ ɞɥɹ ɞɚɧɧɨɣ
ɫɢɫɬɟɦɵ ɫɢɥ, ɬɨ ɬɚɤɚɹ ɡɚɞɚɱɚ ɧɚɡɵɜɚɟɬɫɹ
ȿɫɥɢ ɠɟ ɨɛɳɟɟ ɱɢɫɥɨ ɧɟɢɡɜɟɫɬɧɵɯ
ɫɬɚɬɢɱɟɫɤɢ ɨɩɪɟɞɟɥɢɦɨɣ.
ɛɨɥɶɲɟ ɱɢɫɥɚ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ, ɬɨ ɡɚɞɚɱɚ ɧɚɡɵɜɚɟɬɫɹ
ɫɬɚɬɢɱɟ-
ɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɨɣ.
ȼ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɪɟɲɚɸɬ ɬɨɥɶɤɨ ɫɬɚɬɢɱɟɫɤɢ ɨɩɪɟɞɟɥɢɦɵɟ ɡɚɞɚɱɢ.
2. ɄɈɇɌɊɈɅɖɇɕȿ ȼɈɉɊɈɋɕ
1.
ɑɬɨ ɧɚɡɵɜɚɟɬɫɹ ɩɥɨɫɤɨɣ ɩɪɨɢɡɜɨɥɶɧɨɣ ɫɢɫɬɟɦɨɣ ɫɢɥ?
ɑɬɨ ɬɚɤɨɟ ɦɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ, ɩɥɟɱɨ ɫɢɥɵ?
2.
3.
ɉɪɚɜɢɥɨ ɡɧɚɤɨɜ ɞɥɹ ɦɨɦɟɧɬɚ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ.
4.
ȼ ɤɚɤɨɦ ɫɥɭɱɚɟ ɦɨɦɟɧɬ ɫɢɥɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɪɚɜɟɧ ɧɭɥɸ?
Ʉɚɤ ɧɚɣɬɢ ɦɨɦɟɧɬ ɨɬ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚɝɪɭɡɤɢ
5.
ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ (ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɞɟɣɫɬɜɢɣ)?
6.
ɑɬɨ ɬɚɤɨɟ ɩɚɪɚ ɫɢɥ?
Ʉɚɤɨɟ ɞɟɣɫɬɜɢɟ ɨɤɚɡɵɜɚɟɬ ɧɚ ɬɟɥɨ ɩɚɪɚ ɫɢɥ?
7.
8.
ɑɬɨ ɧɚɡɵɜɚɟɬɫɹ ɦɨɦɟɧɬɨɦ ɩɚɪɵ ɫɢɥ, ɩɥɟɱɨɦ ɩɚɪɵ ɫɢɥ?
9.
ɉɪɚɜɢɥɨ ɡɧɚɤɨɜ ɞɥɹ ɦɨɦɟɧɬɨɜ ɩɚɪɵ ɫɢɥ.
ɍɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɩɥɨɫɤɨɣ ɩɪɨɢɡɜɨɥɶɧɨɣ ɫɢɫɬɟɦɵ ɫɢɥ.
10.
Ɂɚɞɚɧɢɟ 2. Ɉɩɪɟɞɟɥɢɬɶ ɪɟɚɤɰɢɢ ɫɜɹɡɟɣ ɞɜɭɯɨɩɨɪɧɨɣ ɛɚɥɤɢ, ɞɚɧ-
ɧɵɟ ɜɡɹɬɶ ɢɡ ɬɚɛɥ. 2.
20
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