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ɉɪɢɦɟɪ ʋ 3
F
y
Ɉɩɪɟɞɟɥɢɬɶ ɪɟɚɤɰɢɢ, ɜɨɡɧɢɤɚɸɳɢɟ ɜ ɩɪɹɦɨɥɢɧɟɣɧɨɣ ɛɚɥɤɟ,
ɨɞɢɧ ɤɨɧɟɰ ɤɨɬɨɪɨɣ ɠɟɫɬɤɨ ɡɚɞɟɥɚɧ ɜ ɬɨɱɤɟ Ⱥ.
Ɋȿɒȿɇɂȿ
Ɋɟɲɚɟɦ ɡɚɞɚɱɭ ɜ ɫɥɟɞɭɸɳɟɦ ɩɨɪɹɞɤɟ:
1.
Ɋɚɫɫɦɨɬɪɢɦ ɪɚɜɧɨɜɟɫɢɟ ɩɪɹɦɨɥɢɧɟɣɧɨɣ ɛɚɥɤɢ, ɠɟɫɬɤɨ ɡɚɞɟɥɚɧ-
ɧɨɣ ɨɞɧɢɦ ɤɨɧɰɨɦ ɜ ɫɬɟɧɤɭ (ɪɢɫ. 15).
ɂɡɨɛɪɚɡɢɦ ɫɯɟɦɭ ɤɨɧɫɬɪɭɤɰɢɢ ɫɨ ɜɫɟɦɢ ɡɚɞɚɧɧɵɦɢ ɫɢɥɚɦɢ.
2.
ɉɪɢɦɟɧɢɜ ɩɪɢɧɰɢɩ ɨɫɜɨɛɨɠɞɟɧɢɹ ɨɬ ɫɜɹɡɟɣ, ɩɨɤɚɠɟɦ ɫɨɫɬɚɜ-
3.
ɥɹɸɳɢɟ ɪɟɚɤɰɢɢ ɠɟɫɬɤɨɣ ɡɚɞɟɥɤɢ ɜ ɬɨɱɤɟ
– ɜɟɪɬɢɤɚɥɶɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ –
– ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ –
– ɪɟɚɤɬɢɜɧɵɣ ɦɨɦɟɧɬ –
4.
ɍɩɪɨɫɬɢɦ ɫɢɫɬɟɦɭ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɛɚɥɤɭ.
m .
A
V ;
A
Ⱥ:
H ;
A
V
A
m
A
F
H
A
2ɦ
60
F
y
ȼ
F
x
q
R
q
m
x
4ɦ 1ɦ
Ɋɢɫ. 15.
ɇɚɤɥɨɧɧɭɸ ɫɢɥɭ
ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ ɞɜɭɯ ɫɨɫɬɚɜɥɹɸɳɢɯ: ɜɟɪ-
ɬɢɤɚɥɶɧɨɣ ɢ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ, ɦɨɞɭɥɢ ɤɨɬɨɪɵɯ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɚɜɧɵ:
Fɭ = F ⋅ cos30° ɢ Fɯ = F ⋅ cos60°.
31
Ɋɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɭɸ ɧɚɝɪɭɡɤɭ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ
ɜɢɦ ɜ ɜɢɞɟ ɫɨɫɪɟɞɨɬɨɱɟɧɧɨɣ ɜɟɪɬɢɤɚɥɶɧɨɣ ɫɢɥɵ
q ⋅ 4, ɧɚɩɪɚɜɥɟɧɧɨɣ
q ɩɪɟɞɫɬɚ-
ɜɜɟɪɯ ɢ ɩɪɢɥɨɠɟɧɧɨɣ ɜ ɫɟɪɟɞɢɧɟ ɨɬɪɟɡɤɚ ɞɥɢɧɨɣ 4 ɦ.
5. Ɋɚɰɢɨɧɚɥɶɧɨɣ ɫɢɫɬɟɦɨɣ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ ɞɥɹ ɞɚɧɧɨɣ ɫɯɟ-
ɦɵ ɹɜɥɹɟɬɫɹ:
¦
°
¦
®
°
¦
¯
0m
=
A
0F
=
.
ɤɯ
0F
=
ɤɭ
ɗɬɚ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɢɬɶ ɤɚɠɞɨɟ ɧɟɢɡɜɟɫɬɧɨɟ
ɢɡ ɬɪɟɯ (
mA; HA; V
) ɧɟɡɚɜɢɫɢɦɨ ɞɪɭɝ ɨɬ ɞɪɭɝɚ.
A
ɋɨɫɬɚɜɥɹɟɦ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɢ ɨɩɪɟɞɟɥɹɟɦ ɢɡ ɧɢɯ ɧɟɢɡɜɟɫɬ-
ɧɵɟ:
mA = 0; mȺ + F⋅ cos30° ⋅ 2 – q ⋅ 4 ⋅ 4 + m = 0,
m
+ 10 ⋅ 0,9 ⋅ 2 – 0,5 ⋅ 4 ⋅ 4 + 4 = 0,
Ⱥ
= –14 ɤɇ⋅ɦ.
m
A
= 0; HA + F ⋅ cos60° = 0,
F
ɤɯ
HA = –5 ɤɇ,
= 0; VȺ – F ⋅ cos30° + q ⋅ 4 = 0,
F
ɤɭ
– 10 ⋅ 0,9 + 0,5 ⋅ 4 = 0,
V
Ⱥ
= 7 ɤɇ.
V
Ⱥ
6. Ⱦɥɹ ɩɪɨɜɟɪɤɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɧɟɨɛɯɨɞɢɦɨ ɜɵɱɢɫɥɢɬɶ ɚɥɝɟɛɪɚɢɱɟɫɤɭɸ ɫɭɦɦɭ ɦɨɦɟɧɬɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ ɩɥɨɫɤɨɫɬɢ ɞɟɣɫɬɜɢɹ ɫɢɥ, ɧɚɩɪɢɦɟɪ ɬɨɱɤɢ
ȼ:
= mȺ + VȺ ⋅ 2 – q ⋅ 4 ⋅ 2 + m = –14 +7 ⋅ 2 – 0,5 ⋅ 4 ⋅ 2 + 4 = 0.
m
ȼ
Ⱦɟɣɫɬɜɢɬɟɥɶɧɵɟ ɧɚɩɪɚɜɥɟɧɢɹ –
ɧɵɦ ɧɚ ɱɟɪɬɟɠɟ, ɬ. ɤ. ɢɯ ɡɧɚɱɟɧɢɹ ɩɨɥɭɱɟɧɵ ɫɨ ɡɧɚɤɨɦ «
m
ɢ ɇ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɵ ɭɤɚɡɚɧ-
A
Ⱥ
–».
Ɉɬɜɟɬ: mA = –14 ɤɇ⋅ɦ; VȺ = 7 ɤɇ; ɇȺ = –5 ɤɇ.
32
ɏ
ɉɊȺɄɌɂɑȿɋɄɈȿ ɁȺɇəɌɂȿ ʋ 3
Ɍɟɦɚ: Ɉɩɪɟɞɟɥɟɧɢɟ ɩɨɥɨɠɟɧɢɹ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɩɥɨɫɤɢɯ ɫɟɱɟɧɢɣ
ɫɥɨɠɧɨɣ ɮɨɪɦɵ.
ɐɟɥɶ: Ɉɫɜɨɢɬɶ ɨɩɪɟɞɟɥɟɧɢɟ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɩɥɨɫɤɨɝɨ
ɫɟɱɟɧɢɹ ɫɥɨɠɧɨɣ ɮɨɪɦɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɡɚɞɚɧɧɵɯ ɨɫɟɣ.
ȼɪɟɦɹ ɩɪɨɜɟɞɟɧɢɹ: 2 ɱɚɫɚ.
1. ɈɋɇɈȼɇɕȿ ɉɈɅɈɀȿɇɂə ɌȿɈɊɂɂ
ɇɚ ɥɸɛɭɸ ɱɚɫɬɢɰɭ ɬɟɥɚ, ɧɚɯɨɞɹɳɟɝɨɫɹ ɜɛɥɢɡɢ ɡɟɦɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɞɟɣɫɬɜɭɟɬ ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɫɢɥɚ, ɧɚɡɵɜɚɟɦɚɹ ɫɢɥɨɣ
ɬɹɠɟɫɬɢ.
ɐɟɧɬɪɨɦ ɬɹɠɟɫɬɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɧɚɡɵɜɚɟɬɫɹ ɬɨɱɤɚ, ɧɟɢɡɦɟɧɧɨ ɫɜɹɡɚɧɧɚɹ ɫ ɷɬɢɦ ɬɟɥɨɦ, ɱɟɪɟɡ ɤɨɬɨɪɭɸ ɩɪɨɯɨɞɢɬ ɥɢɧɢɹ ɞɟɣɫɬɜɢɹ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ ɫɢɥ ɬɹɠɟɫɬɢ ɱɚɫɬɢɰ ɞɚɧɧɨɝɨ ɬɟɥɚ ɩɪɢ ɥɸɛɨɦ ɩɨɥɨɠɟɧɢɢ
ɬɟɥɚ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ.
Ʉɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ
¦¦¦
===
x ɭ z
ccc
ɯ
, ɭɤ, z
ɝɞɟ
– ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɟɤ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥ ɬɹɠɟɫɬɢ, ɱɚɫɬɢɰ ɬɟɥɚ;
ɤ
ɤ
G = Gɤ – ɫɢɥɚ ɬɹɠɟɫɬɢ ɬɟɥɚ; G
Ʉɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɨɞɧɨɪɨɞɧɨɝɨ ɬɟɥɚ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ
ɮɨɪɦɭɥɚɦ:
=
x
c
ɝɞɟ
V = Vɤ – ɨɛɴɟɦ ɬɟɥɚ; V
Ɇɚɲɢɧɨɫɬɪɨɢɬɟɥɸ ɜ ɩɪɚɤɬɢɱɟɫɤɨɣ ɞɟɹɬɟɥɶɧɨɫɬɢ ɩɪɢɯɨɞɢɬɫɹ
ɨɩɪɟɞɟɥɹɬɶ ɰɟɧɬɪ ɬɹɠɟɫɬɢ, ɝɥɚɜɧɵɦ ɨɛɪɚɡɨɦ, ɩɥɨɫɤɢɯ ɫɟɱɟɧɢɣ. Ȼɨɥɶɲɟɣ ɱɚɫɬɶɸ ɷɬɢ ɫɟɱɟɧɢɹ ɹɜɥɹɸɬɫɹ ɫɥɨɠɧɵɦɢ, ɫɨɫɬɨɹɳɢɦɢ ɢɡ ɩɪɨɫɬɵɯ
ɫɟɱɟɧɢɣ, ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɤɨɬɨɪɵɯ ɡɚɪɚɧɟɟ ɢɡɜɟɫɬɧɨ.
ȿɫɥɢ ɩɥɨɫɤɨɟ ɫɟɱɟɧɢɟ ɫɨɫɬɨɢɬ ɢɡ ɧɟɫɤɨɥɶɤɢɯ ɩɪɨɫɬɵɯ, ɬɨ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɨɩɪɟɞɟɥɹɸɬ, ɪɚɡɛɢɜɚɹ ɞɚɧɧɨɟ ɫɟɱɟɧɢɟ ɧɚ
Ʉɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɧɚɯɨɞɹɬ ɩɨ ɮɨɪɦɭɥɚɦ:
ɬɹɠɟɫɬɢ ɬɟɥɚ ɦɨɠɧɨ ɧɚɣɬɢ ɩɨ ɮɨɪɦɭɥɚɦ:
⋅⋅⋅
G
ɤɤ ɤɤ ɤɤ
;;,
GGG
⋅
ɏV
ɤɤ
V
– ɨɛɴɟɦ ɱɚɫɬɢɰɵ ɬɟɥɚ.
ɤ
G ɍ GZ
– ɫɢɥɚ ɬɹɠɟɫɬɢ ɱɚɫɬɢɰɵ ɬɟɥɚ.
ɤ
⋅
ɍɤV
ɤ
=
ɭ;
c
V
¦¦¦
=
z;
c
V
⋅
ZV
ɤɤ
,
ɩɪɨɫɬɵɟ.
33
A
A
ɏA
y
⋅
ɤɤ
x
=
c
ɭ;
=
c
ɍ
⋅
A
¦¦
ɤɤ
;
Ⱥ
– ɩɥɨɳɚɞɶ ɩɪɨɫɬɵɯ ɫɟɱɟɧɢɣ; Ⱥ = Ⱥɤ – ɩɥɨɳɚɞɶ ɜɫɟɝɨ ɩɥɨɫɤɨɝɨ
ɝɞɟ
ɤ
ɫɟɱɟɧɢɹ;
ɯ
, ɭ
Ʉ
ɯ
, ɭɫ – ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɩɥɨɫɤɨɝɨ ɫɟɱɟɧɢɹ;
ɫ
– ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɨɬɞɟɥɶɧɵɯ ɱɚɫɬɟɣ ɫɟɱɟɧɢɹ.
Ʉ
ɉɪɢ ɷɬɨɦ ɱɢɫɥɨ ɫɥɚɝɚɟɦɵɯ ɜ ɤɚɠɞɨɦ ɢɡ ɱɢɫɥɢɬɟɥɟɣ ɛɭɞɟɬ ɪɚɜɧɨ
ɱɢɫɥɭ ɩɪɨɫɬɵɯ ɫɟɱɟɧɢɣ, ɧɚ ɤɨɬɨɪɵɟ ɪɚɡɛɢɬɨ ɞɚɧɧɨɟ ɫɥɨɠɧɨɟ ɫɟɱɟɧɢɟ.
ȿɫɥɢ ɜ ɫɟɱɟɧɢɢ ɢɦɟɸɬɫɹ ɨɬɜɟɪɫɬɢɹ (ɜɵɪɟɡɵ), ɬɨ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ
ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɩɨɥɶɡɭɸɬɫɹ ɬɟɦɢ ɠɟ ɮɨɪɦɭɥɚɦɢ, ɱɬɨ ɢ ɞɥɹ ɫɩɥɨɲɧɵɯ
ɫɟɱɟɧɢɣ, ɧɨ ɬɨɥɶɤɨ ɩɥɨɳɚɞɢ ɜɵɪɟɡɚɧɧɵɯ
ɱɚɫɬɟɣ ɫɱɢɬɚɸɬ ɨɬɪɢɰɚɬɟɥɶɧɵɦɢ.
ȿɫɥɢ ɫɟɱɟɧɢɟ ɢɦɟɟɬ ɩɥɨɫɤɨɫɬɶ, ɨɫɶ ɢɥɢ ɰɟɧɬɪ ɫɢɦɦɟɬɪɢɢ, ɬɨ ɰɟɧɬɪ
ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ ɧɚɯɨɞɢɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜ ɩɥɨɫɤɨɫɬɢ ɧɚ ɨɫɢ ɢɥɢ ɜ
ɰɟɧɬɪɟ ɫɢɦɦɟɬɪɢɢ.
Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɪɚɫɱɟɬɨɜ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɥɨɠɧɵɯ ɫɟɱɟɧɢɣ ɢɫɩɨɥɶɡɭɸɬ ɩɨɧɹɬɢɟ ɫɬɚɬɢɱɟɫɤɨɝɨ ɦɨɦɟɧɬɚ.
ɋɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɫɟɱɟɧɢɹ ɩɪɨɫɬɨɣ ɮɨɪɦɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
:
Sx = yc ⋅ A,
ɭɫ – ɤɨɨɪɞɢɧɚɬɚ ɭ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ (ɤɪɚɬɱɚɣɲɟɟ ɪɚɫɫɬɨɹɧɢɟ
ɝɞɟ
ɨɬ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ ɞɨ ɨɫɢ
ɯ, ɪɢɫ. 16); Ⱥ – ɩɥɨɳɚɞɶ ɫɟɱɟɧɢɹ.
ɯ
ɫ
ɫ
ɫ
ɭ
x
Ɋɢɫ. 16.
Ⱥɧɚɥɨɝɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ
ɭ:
= ɯc ⋅ A.
S
ɭ
34
ȿɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ ɫɬɚɬɢɱɟɫɤɨɝɨ ɦɨɦɟɧɬɚ [ɦ
S
x
y
x
x
x
x
x
3
, ɫɦ3, ɦɦ3].
ɋɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɨɠɢɬɟɥɶɧɵɦ, ɨɬɪɢɰɚɬɟɥɶɧɵɦ
ɢ ɪɚɜɧɵɦ ɧɭɥɸ (ɪɢɫ. 17).
>0
x
x
S = 0
S <0
x
Ɋɢɫ. 17.
ɋɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɫɟɱɟɧɢɹ ɫɥɨɠɧɨɣ ɮɨɪɦɵ ɪɚɜɟɧ ɚɥɝɟɛɪɚɢɱɟ-
ɫɤɨɣ ɫɭɦɦɟ ɫɬɚɬɢɱɟɫɤɢɯ ɦɨɦɟɧɬɨɜ ɟɝɨ ɩɪɨɫɬɵɯ ɱɚɫɬɟɣ (ɪɢɫ 18, ɚ, ɛ):
n
=
SS .
¦
ixx
1i
=
1
2
x
1
2
x
ɚ) ɛ)
S
= S
1
+ S
2x
= S
– S
S
1
2
Ɋɢɫ. 18.
ɋɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɫɟɱɟɧɢɹ, ɢɦɟɸɳɟɝɨ ɨɫɶ ɫɢɦɦɟɬɪɢɢ, ɪɚɜɟɧ
ɧɭɥɸ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɫɢɦɦɟɬɪɢɢ
Sy = 0 (ɫɦ. ɪɢɫ. 18, ɛ).
ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱ ɧɚ ɨɩɪɟɞɟɥɟɧɢɟ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ ɫɥɨɠ-
ɧɨɣ ɮɨɪɦɵ ɪɟɤɨɦɟɧɞɭɟɬɫɹ ɩɪɢɞɟɪɠɢɜɚɬɶɫɹ ɫɥɟɞɭɸɳɟɝɨ ɩɨɪɹɞɤɚ:
1.
Ɋɚɡɛɢɬɶ ɫɥɨɠɧɨɟ ɫɟɱɟɧɢɟ ɧɚ ɩɪɨɫɬɵɟ ɫɟɱɟɧɢɹ, ɩɨɥɨɠɟɧɢɟ ɰɟɧ-
ɬɪɨɜ ɬɹɠɟɫɬɢ ɤɨɬɨɪɵɯ ɢɡɜɟɫɬɧɨ.
Ɉɩɪɟɞɟɥɢɬɶ ɜɟɥɢɱɢɧɵ, ɜɯɨɞɹɳɢɟ ɜ ɮɨɪɦɭɥɵ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ
2.
ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ. Ɋɟɤɨɦɟɧɞɭɟɬɫɹ ɫɨɫɬɚɜɢɬɶ ɬɚɛɥɢɰɭ, ɜɧɨɫɹ ɜ ɧɟɟ ɧɨɦɟɪɚ ɩɪɨɫɬɵɯ ɱɚɫɬɟɣ, ɪɚɡɦɟɪɵ ɩɥɨɳɚɞɟɣ, ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɨɜ ɬɹɠɟɫɬɢ
ɩɪɨɫɬɵɯ ɫɟɱɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɵɯ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ.
35
3.
BCȾ
E
A
B
A
Ⱦ
B
Ⱦ
A
R
R
Ɉɩɪɟɞɟɥɢɬɶ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ ɫɥɨɠɧɨɣ ɮɨɪ-
ɦɵ ɢ ɩɨɤɚɡɚɬɶ ɢɯ ɧɚ ɪɢɫɭɧɤɟ.
ɉɪɨɫɬɵɟ ɫɟɱɟɧɢɹ
ɐɟɧɬɪ ɬɹɠɟɫɬɢ ɩɥɨɳɚɞɢ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ
ɧɚɯɨɞɢɬɫɹ ɧɚ ɩɟɪɟɫɟɱɟɧɢɢ
ɟɝɨ ɞɢɚɝɨɧɚɥɟɣ
ɐɟɧɬɪ ɬɹɠɟɫɬɢ ɩɥɨɳɚɞɢ ɬɪɟɭɝɨɥɶɧɢɤɚ
C
ɧɚɯɨɞɢɬɫɹ ɜ ɬɨɱɤɟ ɩɟɪɟɫɟɱɟɧɢɹ ɦɟɞɢɚɧ ɧɚ
ɪɚɫɫɬɨɹɧɢɢ 1/3 ɦɟɞɢɚɧɵ, ɫɱɢɬɚɹ ɨɬ ɫɬɨɪɨɧɵ
ɬɪɟɭɝɨɥɶɧɢɤɚ
ɐɟɧɬɪ ɬɹɠɟɫɬɢ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ
ɬɪɟɭɝɨɥɶɧɢɤɚ ɥɟɠɢɬ ɧɚ ɩɟɪɟɫɟɱɟɧɢɢ
C
ɩɪɹɦɵɯ, ɩɚɪɚɥɥɟɥɶɧɵɯ ɤɚɬɟɬɚɦ
ɬɪɟɭɝɨɥɶɧɢɤɚ ɢ ɨɬɫɬɚɸɳɢɯ ɨɬ ɧɢɯ
ɧɚ ɪɚɫɫɬɨɹɧɢɢ 1/3 ɞɥɢɧɵ ɤɚɬɟɬɚ,
ɫɱɢɬɚɹ ɨɬ ɩɪɹɦɨɝɨ ɭɝɥɚ
ɐɟɧɬɪ ɬɹɠɟɫɬɢ ɩɥɨɳɚɞɢ ɩɨɥɭɤɪɭɝɚ
c
3
R ɧɚɯɨɞɢɬɫɹ
x ɧɚ ɪɚɫɫɬɨɹɧɢɢ:
R4
≈=
π
R424,0
ɫ ɪɚɞɢɭɫɨɦ
C
0
X
c
x
ɧɚ ɨɫɢ ɫɢɦɦɟɬɪɢɢ
X
Ɍɚɛɥɢɰɚ 3
ɐɟɧɬɪ ɬɹɠɟɫɬɢ ɩɥɨɳɚɞɢ ɤɪɭɝɨɜɨɝɨ ɫɟɤɬɨɪɚ
ɥɟɠɢɬ ɧɚ ɟɝɨ ɨɫɢ ɫɢɦɦɟɬɪɢɢ ɢ ɨɬɫɬɨɢɬ
ɨɬ ɰɟɧɬɪɚ ɤɪɭɝɚ
α
0
x
α
ɝɞɟ
α
– ɩɨɥɨɜɢɧɚ ɰɟɧɬɪɚɥɶɧɨɝɨ ɭɝɥɚ
0 ɧɚ ɪɚɫɫɬɨɹɧɢɢ, ɪɚɜɧɨɦ:
α
⋅
sinR2
=
X
c
,
α
3
ɜ ɪɚɞɢɚɧɚɯ
36
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɧɚ ɨɩɪɟɞɟɥɟɧɢɟ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɟɱɟ-
y
y
ɧɢɹ ɫɥɨɠɧɨɣ ɮɨɪɦɵ, ɫ ɩɪɢɦɟɧɟɧɢɟɦ ɫɬɚɬɢɱɟɫɤɢɯ ɦɨɦɟɧɬɨɜ, ɦɨɠɧɨ
ɨɫɭɳɟɫɬɜɢɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɩɨɪɹɞɤɟ:
1.
Ɋɚɡɛɢɬɶ ɫɟɱɟɧɢɟ ɧɚ ɩɪɨɫɬɵɟ ɱɚɫɬɢ.
ȼɵɱɢɫɥɢɬɶ ɩɥɨɳɚɞɢ ɩɪɨɫɬɵɯ ɱɚɫɬɟɣ ɫɟɱɟɧɢɹ ɩɨ ɫɨɨɬɜɟɬɫɬɜɭɸ-
2.
ɳɢɦ ɮɨɪɦɭɥɚɦ.
3.
ȼɵɱɢɫɥɢɬɶ ɩɥɨɳɚɞɶ ɜɫɟɝɨ ɫɟɱɟɧɢɹ.
Ɉɛɨɡɧɚɱɢɬɶ ɰɟɧɬɪɵ ɬɹɠɟɫɬɢ ɩɪɨɫɬɵɯ ɱɚɫɬɟɣ: ɋ
4.
5.
Ɉɩɪɟɞɟɥɢɬɶ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɨɜ ɬɹɠɟɫɬɢ ɩɪɨɫɬɵɯ ɱɚɫɬɟɣ ɨɬɧɨ-
, ɋ2, ɋ3,….
1
ɫɢɬɟɥɶɧɨ ɡɚɞɚɧɧɵɯ ɨɫɟɣ.
ȼɵɱɢɫɥɢɬɶ ɫɬɚɬɢɱɟɫɤɢɟ ɦɨɦɟɧɬɵ ɩɪɨɫɬɵɯ ɫɟɱɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶ-
6.
ɧɨ ɨɫɢ
ɏ.
ȼɵɱɢɫɥɢɬɶ ɫɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ
7.
ɨɫɢ
ɏ.
8.
ȼɵɱɢɫɥɢɬɶ ɫɬɚɬɢɱɟɫɤɢɟ ɦɨɦɟɧɬɵ ɩɪɨɫɬɵɯ ɫɟɱɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶ-
ɍ.
ɧɨ ɨɫɢ
9.
ȼɵɱɢɫɥɢɬɶ ɫɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɨɫɢ
ɍ.
Ɉɩɪɟɞɟɥɢɬɶ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ (ɬɨɱɤɢ ɋ), ɜɫɟɝɨ ɫɟɱɟ-
10.
ɧɢɹ.
11.
ɉɨɤɚɡɚɬɶ ɧɚ ɫɟɱɟɧɢɢ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ.
ȼ ɩɪɢɜɟɞɟɧɧɨɣ ɬɚɛɥ. 3 ɞɚɧɵ ɫɜɟɞɟɧɢɹ ɨ ɩɨɥɨɠɟɧɢɢ ɰɟɧɬɪɨɜ ɬɹɠɟ-
ɫɬɢ ɧɟɤɨɬɨɪɵɯ ɩɪɨɫɬɵɯ ɫɟɱɟɧɢɣ.
Ɂɚɞɚɧɢɟ 4. Ɉɩɪɟɞɟɥɢɬɶ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɡɚɞɚɧɧɨɝɨ ɫɟ-
ɱɟɧɢɹ. ɇɟɨɛɯɨɞɢɦɵɟ ɪɚɡɦɟɪɵ ɜɡɹɬɶ ɜ ɬɚɛɥ. 4.
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 4
1
2
ca b
c
b
x0
R
a
bb
R
bb
x
aa
37
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 4
R
y
y
y
y
9
0
y
y
3
4
R
0
5
c
45
0
7
c
0
b
b
a
x
a
a
a
0
6
a
a
x
b
a
b
b
R
c
c
b
x
8
R
R
aya
0
ac
a
45
0
x
c
R
x
a
x
b
a
y
10
a
c
b
c
R
bc
x
45
0
R
x
acb
38
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 4
y
y
y
y
11
12
R
b
b
a
x
b
ab
r=c
14
0
x
a
x
R
a
r=
13
R
0
b
a
a
a
a
4
0
a + b
15
R
0
16
a
y
y
b
c
R
x
c
c
R
a
b
a
c
cc
r=
2
b
a
x
0
a
a
x
39
Ɍɚɛɥɢɰɚ 4
ɪ
ʋ
ɜɚ
ɢɚɧɬɚ
1
2 4 12 15 20
3
4 6 4 10 1,5
5
6 6 12 21 4
7
8 10 20 24 8
9
10 9 4 5 15
11
12 6 10 18 5
13
14 46 36 10 6
15
16 14 6 9 18
17
18 29 17 8 6
19
20 6 7 10 15
21
22 15 9 3 2
23
24 6 7,5 4 15
25
26 9 10 5 14
27
28 21 24 20 10
29
30 12 5 18 7
31
32 4,5 6 3 16
ʋ
ɫɯɟɦɵ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
ɚ (ɫɦ) b (ɫɦ) ɫ (ɫɦ) R (ɫɦ)
2 8 6 14
9 5 14 2
4 7 14 3
6 14 15 4
12 5 6 20
8 15 25 6
34 24 8 4
8 4 5 10
15 9 6 4
4 5 8 10
6 4 2 1
5 6 3 12
12 16 8 20
12 15 10 6
9 4 12 5
3 4 2 12
40
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