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Сопротивление материалов. Часть 1. Учебное пособие

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ɋɟɱɟɧɢɟ ɯ = 3 ɦ (ɫɩɪɚɜɚ),

Qy = Rȼ = 5 ɤɇ,

Ɇz = 5 3 = 15 ɤɇɦ

(ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).

ɉɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɩɨ ɜɫɟɣ ɞɥɢɧɟ ɛɚɥɤɢ ɩɨɫɬɨɹɧɧɚ, ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɧɚ

ɭɱɚɫɬɤɚɯ Ⱥɋ ɢ ɋȼ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ ɢ ɜ ɫɟɱɟɧɢɢ

ɋ ɢɦɟɟɬ ɫɤɚ-

ɱɨɤ, ɪɚɜɧɵɣ ɩɨ ɜɟɥɢɱɢɧɟ ɞɟɣɫɬɜɭɸɳɟɦɭ ɜ ɷɬɨɦ ɫɟɱɟɧɢɢ ɦɨɦɟɧɬɭ.

ɗɩɸɪɵ Qy ɢ Mz

ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.3.8.

Ɂɚɞɚɱɚ 3.5

12ɤɇɦ

q(x)

18ɤɇ/ɦ

Ⱦɥɹ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɣ ɛɚɥɤɢ, ɢɡɨɛɪɚɠɟɧ-

B x

ɧɨɣ ɧɚ ɪɢɫ.3.9, ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ Qy ɢ Mz.

Ɋɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɧɚɝɪɭɡɤɢ, ɪɚɫɩɪɟɞɟ-

4 ɦ

2 ɦ

ɥɟɧɧɨɣ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ, ɪɚɜɧɚ

6 ɦ

1 18 6

54ɤɇ.

Ɉɩɪɟɞɟɥɢɦ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ

x0=3,65 ɦ

6ɆȺ = 0, 12 54 4 + 6Rȼ = 0,

Rȼ = 34 ɤɇ;

(ɤɇ)

6Ɇȼ = 0, 12 + 54 2 6RȺ = 0,

RȺ = 20 ɤɇ;

6Y = 0 (ɩɪɨɜɟɪɤɚ),

(ɤɇɦ)

54 20 34 = 54 54 = 0.

13,4

ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ Qy ɢ Mz

ɜ ɯɚɪɚɤɬɟɪ-

ɧɵɯ ɫɟɱɟɧɢɹɯ ɛɚɥɤɢ.

Ɋɢɫ.3.9

ɋɟɱɟɧɢɟ

ɯ = 0,

Qy = RȺ = 20 ɤɇ, Ɇz = 12 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨ-

ɥɨɤɧɚ).

Qy = Rȼ = 34 ɤɇ,

Ɇz = 0.

ɋɟɱɟɧɢɟ ɯ = 6 ɦ,

ɂɡ ɩɨɞɨɛɢɹ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ɧɚɯɨɞɢɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚ-

ɝɪɭɡɤɢ:

18 x

q(x)

3x .

ɋɨɝɥɚɫɧɨ ɡɚɜɢɫɢɦɨɫɬɹɦ (3.1) ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ ɤɜɚɞ-

ɪɚɬɧɨɣ ɩɚɪɚɛɨɥɵ

1 q(x)x 20

3 x2 ,

Qy (x) RA

ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ – ɩɨ ɡɚɤɨɧɭ ɤɭɛɢɱɟɫɤɨɣ ɩɚɪɚɛɨɥɵ

Ɇz (ɯ) 12 20ɯ

1 q(ɯ)ɯ

1 ɯ 12 20ɯ 1

ɯ3 .

Ɉɩɪɟɞɟɥɢɦ ɩɨɥɨɠɟɧɢɟ ɫɟɱɟɧɢɹ, ɝɞɟ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ

20 3

ɯ2

, ɯ = ɯ0 = 3,65 ɦ.

	ȼ ɫɟɱɟɧɢɢ ɯ0 ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɢɦɟɟɬ ɷɤɫɬɪɟɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ
	Mmax		Ɇz (3,65)		12 20 3,65				1	3 3,65 3,65 13,65 13,4 ɤɇɦ
									2	3
										(ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).
	ɗɩɸɪɵ Qy ɢ Ɇz				ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.3.9.
	Ɂɚɞɚɱɚ 3.6								Ⱦɥɹ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɣ ɛɚɥɤɢ ɫ ɤɨɧɫɨ-
	12ɤɇ		36ɤɇ	6ɤɇ/ɦ					Ⱦɥɹ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɣ ɛɚɥɤɢ ɫ ɤɨɧɫɨ-
	12ɤɇ		36ɤɇ	6ɤɇ/ɦ				ɥɶɸ, ɢɡɨɛɪɚɠɟɧɧɨɣ ɧɚ ɪɢɫ.3.10, ɩɨɫɬɪɨɢɦ
A					B		x	ɷɩɸɪɵ Qy ɢ Ɇz. ɇɚɣɞɟɦ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ.
A								6ɆȺ = 0, 12 2 36 4 6 2 7 6Rȼ = 0,
			RB
RA
	2 ɦ 2				2 ɦ					Rȼ = 42 ɤɇ;
	2 ɦ 2		ɦ 2 ɦ		2 ɦ					Rȼ = 42 ɤɇ;
y								6Ɇȼ = 0, 12 4 36 2 6 2 1 6RȺ = 0,
y			_		30					RȺ = 18 ɤɇ;
			_		30					RȺ = 18 ɤɇ;
								6Y = 0 (ɩɪɨɜɟɪɤɚ),
	+								12 + 36 + 6 2 18 42 = 60 60 = 0.
18	+	6	12				(ɤɇ)		ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ Qy		ɢ Mz ɜ ɯɚɪɚɤ-
			12					ɬɟɪɧɵɯ ɫɟɱɟɧɢɹɯ ɛɚɥɤɢ.
			12				M		ɋɟɱɟɧɢɟ ɯ = 0, Qy = RȺ = 18 ɤɇ, Ɇz = 0.
			+				M		ɋɟɱɟɧɢɟ ɯ = 0, Qy = RȺ = 18 ɤɇ, Ɇz = 0.
			+				(ɤɇɦ)		ɋɟɱɟɧɢɟ ɯ = 2 ɦ (ɫɥɟɜɚ),		Qy = 18 ɤɇ,
	36									Ɇz = 18 2 = 36 ɤɇɦ
	36		48							(ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).
			Ɋɢɫ.3.10
	ɋɟɱɟɧɢɟ ɯ = 2 ɦ (ɫɩɪɚɜɚ),							Qy = 18 12 = 6 ɤɇ, Ɇz = 36 ɤɇɦ.
	ɋɟɱɟɧɢɟ ɯ = 4 ɦ (ɫɥɟɜɚ),							Qy = 6 ɤɇ, Ɇz = 18 4 12 2 = 48 ɤɇɦ
										(ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).
	ɋɟɱɟɧɢɟ ɯ = 6 ɦ (ɫɩɪɚɜɚ),							Qy = 6 2 = 12 ɤɇ, Ɇz = – 6 2 1 = – 12 ɤɇɦ
										(ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ).
	ɋɟɱɟɧɢɟ ɯ = 6 ɦ (ɫɥɟɜɚ),							Qy = 12 42 = 30 ɤɇ, Ɇz = – 12 ɤɇɦ.
	ɋɟɱɟɧɢɟ ɯ = 8 ɦ,							Qy = 0,		Ɇz = 0.
	ɇɚ ɤɨɧɫɨɥɶɧɨɣ ɱɚɫɬɢ ɛɚɥɤɢ, ɝɞɟ ɢɦɟɟɬɫɹ ɪɚɫɩɪɟɞɟɥɟɧɧɚɹ ɧɚɝɪɭɡɤɚ, ɩɨɩɟ-
ɪɟɱɧɚɹ ɫɢɥɚ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ, ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ – ɩɨ
ɡɚɤɨɧɭ ɤɜɚɞɪɚɬɧɨɣ ɩɚɪɚɛɨɥɵ ɫ ɜɵɩɭɤɥɨɫɬɶɸ, ɨɛɪɚɳɟɧɧɨɣ ɜ ɫɬɨɪɨɧɭ ɞɟɣɫɬɜɢɹ
ɧɚɝɪɭɡɤɢ. ɇɚ ɨɫɬɚɥɶɧɵɯ ɭɱɚɫɬɤɚɯ ɩɨɩɟɪɟɱɧɚɹ ɧɚɝɪɭɡɤɚ ɨɬɫɭɬɫɬɜɭɟɬ, ɩɨɷɬɨɦɭ
ɩɨɩɟɪɟɱɧɚɹ			ɫɢɥɚ ɩɨɫɬɨɹɧɧɚ, ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ
ɡɚɤɨɧɭ. ȼ ɫɟɱɟɧɢɹɯ, ɝɞɟ ɞɟɣɫɬɜɭɸɬ ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɟ ɫɢɥɵ ɢɥɢ ɨɩɨɪɧɵɟ
ɪɟɚɤɰɢɢ,		ɧɚ ɷɩɸɪɟ		Qy		ɢɦɟɸɬɫɹ ɫɤɚɱɤɢ, ɪɚɜɧɵɟ ɩɨ ɜɟɥɢɱɢɧɟ ɞɟɣɫɬɜɭɸɳɢɦ
ɫɢɥɚɦ, ɚ ɧɚ ɷɩɸɪɟ Ɇz					ɢɦɟɸɬɫɹ ɬɨɱɤɢ ɢɡɥɨɦɚ.
	ɗɩɸɪɵ Qy ɢ Mz				ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.3.10.
32

Ɂɚɞɚɱɚ 3.7						12ɤɇ/ɦ
Ⱦɥɹ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɣ ɛɚɥɤɢ ɫ ɤɨɧɫɨ-				ɤɇɦ		12ɤɇ/ɦ				15ɤɇ
ɥɹɦɢ (ɪɢɫ.3.11) ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ Qy			ɢ Ɇz.	E	A	D			B	C	x
ɇɚɯɨɞɢɦ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ.				E		D		R	B	C	x
ɇɚɯɨɞɢɦ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ.				y	1ɦ RȺ 3 ɦ
6ɆȺ = 0, 6 12 3 1,5 15 6
6ɆȺ = 0, 6 12 3 1,5 15 6							2	ɦ B 1ɦ
+ 5Rȼ = 0,	Rȼ = 30 ɤɇ;					ɯ0	_ 15
6Ɇȼ = 0, 6 + 12 3 3,5 15 1						+
– 5RȺ = 0,	RȺ = 21ɤɇ;				21	+		15		15 (ɤɇ)
6Y = 0 (ɩɪɨɜɟɪɤɚ),								15	_
12 3 + 15 21 30 = 51 51 = 0.									_		M
12 3 + 15 21 30 = 51 51 = 0.				6		+					M
ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ Qy		ɢ Ɇz	ɜ ɯɚɪɚɤ-	6		+	15				(ɤɇɦ)
ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ Qy		ɢ Ɇz	ɜ ɯɚɪɚɤ-								(ɤɇɦ)
ɬɟɪɧɵɯ ɫɟɱɟɧɢɹɯ ɛɚɥɤɢ.
ɬɟɪɧɵɯ ɫɟɱɟɧɢɹɯ ɛɚɥɤɢ.						24,4
						24,4
ɋɟɱɟɧɢɟ ɯ = 0, Qy = 0,		Ɇz = 6 ɤɇɦ				Ɋɢɫ.3.11
ɋɟɱɟɧɢɟ ɯ = 0, Qy = 0,		Ɇz = 6 ɤɇɦ
(ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).

ɋɟɱɟɧɢɟ ɯ = 1 ɦ		(ɫɥɟɜɚ),	Qy = 0, Ɇz = 6		ɤɇɦ.
ɋɟɱɟɧɢɟ ɯ = 1 ɦ		(ɫɩɪɚɜɚ),	Qy = RȺ = 21	ɤɇ,	Ɇz = 6	ɤɇɦ.
ɋɟɱɟɧɢɟ ɯ = 4	ɦ, Qy = 21 12 3 = 15 ɤɇ,				Ɇz = 6	+ 21 3 12 3 1,5 =
			= 15 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).
ɋɟɱɟɧɢɟ ɯ = 6	ɦ (ɫɩɪɚɜɚ),		Qy = 15 ɤɇ,		Ɇz = 15 1 = 15 ɤɇɦ
				(ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ).
ɋɟɱɟɧɢɟ ɯ = 6	ɦ (ɫɥɟɜɚ),		Qy = 15 30 = 15 ɤɇ, Ɇz = 15 ɤɇɦ.
ɋɟɱɟɧɢɟ ɯ = 7	ɦ,		Qy = 15 ɤɇ,	Ɇz = 0.

ɇɚ ɭɱɚɫɬɤɟ ɫ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚɝɪɭɡɤɨɣ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ ɫɨ ɫɦɟɧɨɣ ɡɧɚɤɚ ɫ ɩɥɸɫɚ ɧɚ ɦɢɧɭɫ. ɂɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɧɚ ɷɬɨɦ ɭɱɚɫɬɤɟ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ ɤɜɚɞɪɚɬɧɨɣ ɩɚɪɚɛɨɥɵ ɢ ɢɦɟɟɬ ɷɤɫɬɪɟɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɜ ɫɟɱɟɧɢɢ, ɝɞɟ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɪɚɜɧɚ ɧɭɥɸ. ɂɡ ɩɨɞɨɛɢɹ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ɧɚ ɷɩɸɪɟ Qy ɨɩɪɟɞɟɥɹɟɦ ɤɨɨɪɞɢɧɚɬɭ ɫɟɱɟɧɢɹ ɯ0 , ɝɞɟ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ, ɢ ɞɥɹ ɷɬɨɝɨ ɫɟɱɟɧɢɹ ɜɵɱɢɫɥɹɟɦ ɷɤɫɬɪɟɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ.

	ɯ0 1	21	,	ɯ0 = 2,75 ɦ ;
	4 ɯ0	15	,	ɯ0 = 2,75 ɦ ;
	4 ɯ0	15
Mmax Ɇz (2,75)		6 21		1,75	12	1,752	24,4 ɤɇɦ
Mmax Ɇz (2,75)		6 21		1,75	12	2	24,4 ɤɇɦ
						2

(ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).

ɇɚ ɭɱɚɫɬɤɟ ȿȺ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɪɚɜɧɚ ɧɭɥɸ, ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɢɦɟɟɬ ɩɨɫɬɨɹɧɧɨɟ ɡɧɚɱɟɧɢɟ. ɇɚ ɭɱɚɫɬɤɚɯ Dȼ ɢ ȼɋ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɩɨɫɬɨɹɧɧɚ, ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ. ȼ ɫɟɱɟɧɢɹɯ Ⱥ ɢ ȼ ɧɚ

ɷɩɸɪɟ Qy ɢɦɟɸɬɫɹ ɫɤɚɱɤɢ.

ɗɩɸɪɵ Qy ɢ Mz ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.3.11.

Ɂɚɞɚɱɚ 3.8
Ⱦɥɹ ɛɚɥɤɢ ɫ ɩɪɨɦɟɠɭɬɨɱɧɵɦ ɲɚɪɧɢɪɨɦ, ɢɡɨɛɪɚɠɟɧɧɨɣ ɧɚ ɪɢɫ.3.12, ɩɨ-
ɫɬɪɨɢɦ ɷɩɸɪɵ Qy ɢ Mz.
Ȼɚɥɤɚ ɹɜɥɹɟɬɫɹ ɫɬɚɬɢɱɟɫɤɢ ɨɩɪɟɞɟɥɢɦɨɣ, ɩɨɫɤɨɥɶɤɭ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɬɪɟɯ
ɨɩɨɪɧɵɯ ɪɟɚɤɰɢɣ RȺ, Rȼ ɢ RD				ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɢ
ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ 6Ɇɋ				= 0	ɞɥɹ ɥɟɜɨɣ ɢɥɢ ɩɪɚɜɨɣ ɱɚɫɬɢ ɛɚɥɤɢ.
Ɋɚɫɱɟɬ ɩɪɨɜɟɞɟɦ ɫ ɩɨɦɨɳɶɸ ɬɚɤ ɧɚɡɵɜɚɟɦɨɣ ɩɨɷɬɚɠɧɨɣ ɫɯɟɦɵ. Ɋɚɡɪɟɠɟɦ
ɦɵɫɥɟɧɧɨ ɛɚɥɤɭ ɩɨ ɩɪɨɦɟɠɭɬɨɱɧɨɦɭ ɲɚɪɧɢɪɭ ɋ. Ȼɚɥɤɚ CD ɧɟ ɦɨɠɟɬ ɪɚɛɨɬɚɬɶ
ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ ɢ ɨɩɢɪɚɟɬɫɹ ɧɚ ɧɟɫɭɳɭɸ ɛɚɥɤɭ Ⱥɋ.
12ɤɇ/ɦ			18ɤɇ			ȼɧɚɱɚɥɟ ɩɪɨɢɡɜɟɞɺɦ ɪɚɫɱɺɬ ɧɟ-
		C		D		ɫɨɦɨɣ ɛɚɥɤɢ	CD, ɢɦɟɸɳɟɣ ɭɫɥɨɜ-
A	B		E
					x	ɧɭɸ ɲɚɪɧɢɪɧɭɸ ɨɩɨɪɭ ɜ ɫɟɱɟɧɢɢ
						ɋ.
3 ɦ	1 ɦ	1 ɦ	2 ɦ
						Ɉɩɪɟɞɟɥɹɟɦ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ.
y		ɇɟɫɨɦɚɹ ɛɚɥɤɚ				6Ɇɋ = 0,	18 1 3RD = 0,
ɇɟɫɭɳɚɹ ɛɚɥɤɚ
							RD = 6 ɤɇ;

18 ɤɇ

6ɆD = 0,

18 2 3RC = 0,

RC = 12 ɤɇ;

6Y = 0 (ɩɪɨɜɟɪɤɚ),

18 12 6 = 18 18 = 0.

ɤɇ/ɦ

ȼɵɩɨɥɧɢɦ ɪɚɫɱɟɬ ɧɟɫɭɳɟɣ ɛɚɥ-

12ɤɇ

ɤɢ. ȼɥɢɹɧɢɟ ɧɟɫɨɦɨɣ ɛɚɥɤɢ ɋD ɧɚ

ɧɟɫɭɳɭɸ ɛɚɥɤɭ

Ⱥɋ ɯɚɪɚɤɬɟɪɢɡɭ-

ɟɬɫɹ

ɞɟɣɫɬɜɢɟɦ

ɫɢɥɵ

12 ɤɇ,

ɢɦɟɸɳɟɣ

ɧɚɩɪɚɜɥɟɧɢɟ,

ɩɪɨɬɢɜɨ-

ɩɨɥɨɠɧɨɟ

ɧɚɩɪɚɜɥɟɧɢɸ

ɭɫɥɨɜɧɨɣ

ɯ0

ɨɩɨɪɧɨɣ ɪɟɚɤɰɢɢ RC.

6ɆȺ

= 0,

12 3 1,5 12 4

(ɤɇ)

6Ɇȼ

+ 3Rȼ = 0, Rȼ = 34 ɤɇ;

= 0,

12 3 1,5 12 1

– 3RA = 0, RA = 14 ɤɇ;

6Y = 0 (ɩɪɨɜɟɪɤɚ),

12 3 12 34 14 = 48 48 = 0.

(ɤɇɦ)

ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ

Qy ɢ Mz

8,17

ɜ ɯɚɪɚɤɬɟɪɧɵɯ ɫɟɱɟɧɢɹɯ ɛɚɥɤɢ.

ɋɟɱɟɧɢɟ ɯ = 0, Qy = RA =

Ɋɢɫ.3.12

= 14 ɤɇ,

Ɇz = 0.

ɋɟɱɟɧɢɟ ɯ = 3 ɦ (ɫɩɪɚɜɚ),

Qy = 12 ɤɇ,

Ɇz = 12 1 = – 12 ɤɇɦ

(ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ).

ɋɟɱɟɧɢɟ ɯ = 3 ɦ (ɫɥɟɜɚ),

Qy = 12 – 34 = – 22 ɤɇ,

Ɇz = 12 ɤɇɦ.

ɋɟɱɟɧɢɟ ɯ = 4 ɦ,

Qy = 12 ɤɇ,

Ɇz = 0.

ɋɟɱɟɧɢɟ ɯ = 5	ɦ (ɫɩɪɚɜɚ),	Qy = 6 ɤɇ,	Ɇz = 6 2 = 12 ɤɇɦ
ɋɟɱɟɧɢɟ ɯ = 5			(ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).
ɋɟɱɟɧɢɟ ɯ = 5	ɦ (ɫɥɟɜɚ),	Qy = 12 ɤɇ, Ɇz = 12 1 = 12 ɤɇɦ.
ɋɟɱɟɧɢɟ ɯ = 7	ɦ,	Qy = 6 ɤɇ,	Ɇz = 0.

ɂɡ ɩɨɞɨɛɢɹ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ɧɚ ɷɩɸɪɟ Qy ɨɩɪɟɞɟɥɹɟɦ ɤɨɨɪɞɢɧɚɬɭ ɫɟɱɟɧɢɹ ɯ0, ɢ ɞɥɹ ɷɬɨɝɨ ɫɟɱɟɧɢɹ ɜɵɱɢɫɥɹɟɦ ɷɤɫɬɪɟɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ.

	ɯ0	14	,		ɯ0 = 1,17 ɦ ;
	3 ɯ0	22	,		ɯ0 = 1,17 ɦ ;
	3 ɯ0	22
Mmax Ɇz (1,17 ) 14 1,17		12		1,17	2
Mmax Ɇz (1,17 ) 14 1,17		12		2	8,17 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).
				2

ɇɚ ɭɱɚɫɬɤɚɯ ȼȿ ɢ ȿD ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɩɨɫɬɨɹɧɧɚ, ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ. ȼ ɫɟɱɟɧɢɹɯ ȼ ɢ ȿ ɧɚ ɷɩɸɪɟ ɩɨɩɟɪɟɱɧɵɯ ɫɢɥ

ɢɦɟɸɬɫɹ ɫɤɚɱɤɢ, ɚ ɧɚ ɷɩɸɪɟ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ ɢɦɟɸɬɫɹ ɬɨɱɤɢ ɢɡɥɨɦɚ. ɗɩɸɪɵ Qy ɢ Ɇz ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.3.12.

Ɂɚɞɚɱɚ 3.9

Ⱦɥɹ ɛɚɥɤɢ ɫ ɧɚɤɥɨɧɧɵɦ ɭɱɚɫɬɤɨɦ, ɢɡɨɛɪɚɠɟɧɧɨɣ ɧɚ ɪɢɫ.3.13, ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ N, Q ɢ M.

Ɉɩɪɟɞɟɥɢɦ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ.

6X = 0, ɇɋ = 0;

6Ɇȼ = 0, 18 4 2 12 2 4Rɋ = 0,

Rɋ = 30 ɤɇ;

6Ɇɋ = 0, 12 6 18 4 2 4Rȼ = 0,

Rȼ = 54 ɤɇ;

6Y = 0 (ɩɪɨɜɟɪɤɚ),

12 18 4 54 30 = 84 84 = 0.

ɉɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɩɪɨɞɨɥɶɧɵɯ ɢ ɩɨɩɟɪɟɱɧɵɯ ɫɢɥ N ɢ Q ɜ ɩɪɟɞɟɥɚɯ ɧɚɤɥɨɧɧɨɝɨ ɭɱɚɫɬɤɚ ɧɚɞɨ ɫɨɫɬɚɜɢɬɶ ɩɪɨɟɤɰɢɢ ɧɚɝɪɭɡɨɤ ɢ ɪɟɚɤɰɢɣ ɧɚ ɨɫɶ ɫɬɟɪɠɧɹ ɢ ɧɚ ɧɨɪɦɚɥɶ ɤ ɨɫɢ.

ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ N, Q ɢ M ɜ ɯɚɪɚɤɬɟɪɧɵɯ ɫɟɱɟɧɢɹɯ ɛɚɥɤɢ.

ɋɟɱɟɧɢɟ ɯ = 0, N = 0, Q = 12 ɤɇ, Ɇ = 0.

ɋɟɱɟɧɢɟ ɯ = 2 ɦ (ɫɥɟɜɚ), N = 0,

Q = 12 ɤɇ, Ɇ = 12 2 = 24 ɤɇɦ

(ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ). ɋɟɱɟɧɢɟ ɯ = 2 ɦ (ɫɩɪɚɜɚ),

18ɤɇ/ɦ

12ɤɇ

A	B	30ɨ
		x
y	2 ɦ RB
		4 ɦ

		+		15 N
		+		(ɤɇ)
	_			(ɤɇ)
21	_		26
21		_	26
		_
	ɯ0	6 – ɯ0		(ɤɇ)
12 _		6 – ɯ0
12 _	+
	+
	36,4			M
24_ 24		+		(ɤɇɦ)
		+
		25
		25
	Ɋɢɫ.3.13

N 54 12 sin30$ 21 ɤɇ (ɫɠɚɬɢɟ),	Q 54 12 cos30$ 36,4 ɤɇ,
Ɇ = 24 ɤɇɦ.

ɋɟɱɟɧɢɟ ɯ = 6 ɦ,	N	30sin30$	15 ɤɇ, (ɪɚɫɬɹɠɟɧɢɟ).
	Q	30cos30$		26 ɤɇ,	Ɇ = 0.
ɂɡ ɩɨɞɨɛɢɹ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ɧɚ ɷɩɸɪɟ			Q	ɨɩɪɟɞɟɥɹɟɦ ɤɨɨɪɞɢɧɚɬɭ ɯ0, ɝɞɟ

ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ, ɢ ɞɥɹ ɷɬɨɝɨ ɫɟɱɟɧɢɹ ɜɵɱɢɫɥɹɟɦ ɷɤɫɬɪɟɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ.

	ɯ0 2	36,4 ,	ɯ	0	= 4,33 ɦ ,		6 ɯ 1,67 ɦ ;

	6 ɯ0	26					0

Mmax Ɇ 4,33		30	1,67		18	1,672	25 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).
						2

ɗɩɸɪɵ N, Q ɢ M ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ. 3.13.

Ɂɚɞɚɱɚ 3.10

Ⱦɥɹ ɤɨɧɫɨɥɶɧɨɝɨ ɥɨɦɚɧɨɝɨ ɫɬɟɪɠɧɹ, ɢɡɨɛɪɚɠɟɧɧɨɝɨ ɧɚ ɪɢɫ.3.14,ɚ, ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ N, Q ɢ M. ɉɪɟɞɜɚɪɢɬɟɥɶɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɨɩɨɪɧɵɯ ɪɟɚɤɰɢɣ ɜ ɡɚɞɟɥɤɟ ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ.

ɚ)

18 ɤɇ 1ɦ

24ɤɇ/ɦ

2ɦ

2 ɦ

ɛ)

(ɤɇ)

ɜ)	18			ɝ)	66		ɞ)
					66
							ɉɪɨɜɟɪɤɚ ɪɚɜɧɨɜɟɫɢɹ
							ɉɪɨɜɟɪɤɚ ɪɚɜɧɨɜɟɫɢɹ
	+		18		M		ɠɺɫɬɤɨɝɨ ɭɡɥɚ ɋ
			18					48ɤɇ
								48ɤɇ
48	(ɤɇ)	_			(ɤɇɦ)		18ɤɇ	12ɤɇɦ
48							18ɤɇ	12ɤɇɦ
	+			12		36	ɋ	18ɤɇ
	+			12	12	36	ɋ	48ɤɇ
					12			48ɤɇ
					Ɋɢɫ.3.14

Ɉɩɪɟɞɟɥɹɟɦ ɜɧɭɬɪɟɧɧɢɟ ɭɫɢɥɢɹ N, Q ɢ M ɜ ɯɚɪɚɤɬɟɪɧɵɯ ɫɟɱɟɧɢɹɯ ɫɬɟɪɠɧɟɣ, ɧɚɱɢɧɚɹ ɫɨ ɫɜɨɛɨɞɧɨɝɨ ɤɨɧɰɚ.

ɋɬɟɪɠɟɧɶ Ⱥȼ	NȺ = Nȼ = 0, QȺ = Qȼ = 18 ɤɇ, ɆȺ = 0,
ɋɬɟɪɠɟɧɶ ȼɋ	Ɇȼ = 18 2 = 36 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɩɪɚɜɵɟ ɜɨɥɨɤɧɚ).
ɋɬɟɪɠɟɧɶ ȼɋ	Nȼ = Nɋ = 18 ɤɇ (ɫɠɚɬɢɟ), Qȼ = 0, Qɋ = 24 2 = 48 ɤɇ,
	Ɇȼ = 36 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ),
ɋɬɟɪɠɟɧɶ ɋD	Ɇɋ = 24 2 1 36 = 12 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ).
ɋɬɟɪɠɟɧɶ ɋD	NC = ND = 2 24 = 48 ɤɇ (ɪɚɫɬɹɠɟɧɢɟ), QC = QD = 18 ɤɇ,
	Ɇɋ = 12 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɩɪɚɜɵɟ ɜɨɥɨɤɧɚ),
	ɆD = 24 2 1 18 1 = 66 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɩɪɚɜɵɟ ɜɨɥɨɤɧɚ).

ɗɩɸɪɵ N, Q ɢ M ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.3.14,ɛ,ɜ,ɝ. Ɉɩɨɪɧɵɟ ɪɟɚɤɰɢɢ ɜ ɡɚɞɟɥɤɟ ɪɚɜɧɵ: HD = 18 ɤɇ, RD = 48 ɤɇ, MD = 66 ɤɇɦ.

ȼɵɪɟɡɚɟɦ ɠɟɫɬɤɢɣ ɭɡɟɥ ɋ ɢ ɩɪɨɜɟɪɹɟɦ ɟɝɨ ɪɚɜɧɨɜɟɫɢɟ ɩɨɞ ɞɟɣɫɬɜɢɟɦ

ɭɫɢɥɢɣ ɜ ɫɬɟɪɠɧɹɯ, ɫɯɨɞɹɳɢɯɫɹ ɜ ɭɡɥɟ (ɪɢɫ.3.14,ɞ). ɍɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ 6X = = 0, 6Y = 0, 6Ɇɋ = 0 ɜɵɩɨɥɧɹɸɬɫɹ.

Ɂɚɞɚɱɚ 3.11

Ⱦɥɹ ɫɬɟɪɠɧɹ ɫ ɤɪɢɜɨɥɢɧɟɣɧɵɦ ɭɱɚɫɬɤɨɦ ɜ ɜɢɞɟ ɩɨɥɭɨɤɪɭɠɧɨɫɬɢ (ɪɢɫ.3.15,ɚ) ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ ɜɧɭɬɪɟɧɧɢɯ ɭɫɢɥɢɣ N, Q ɢ M.

3	ɦ

HA A

MA RA

			ɛ)					ɜ) 3	N (ɤɇ)
			ɛ)		n	M Q	t	ɜ) 3	3	5,2
					N	M Q	t	5,2_		5,2
6	ɤɇ	6	ɤɇ		N	Ʉ		5,2_		+
6	ɤɇ					Ʉ

C		B			Μ			6		6
				RcosΜ

ɝ)	(ɤɇ)		ɞ)	M (ɤɇɦ)
ɝ)	6		ɞ)
5,2	6	5,2
5,2	+	5,2		9	9
				9	9
3		3		15,6	15,6
		_			18
	6	_		18	18
				18	18

			Ɋɢɫ.3.15

ɍɫɬɚɧɨɜɢɦ ɡɚɤɨɧɵ ɢɡɦɟɧɟɧɢɹ ɜɧɭɬɪɟɧɧɢɯ ɭɫɢɥɢɣ ɧɚ ɤɪɢɜɨɥɢɧɟɣɧɨɦ ɭɱɚɫɬɤɟ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɭɝɥɚ Μ. ɉɪɢɪɚɜɧɢɜɚɹ ɤ ɧɭɥɸ ɫɭɦɦɭ ɩɪɨɟɤɰɢɣ ɜɫɟɯ ɫɢɥ ɧɚ ɧɨɪɦɚɥɶ n ɢ ɧɚ ɤɚɫɚɬɟɥɶɧɭɸ t ɤ ɫɟɱɟɧɢɸ, ɚ ɬɚɤɠɟ ɫɭɦɦɭ ɦɨɦɟɧɬɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ Ʉ (ɪɢɫ.3.15,ɛ), ɩɨɥɭɱɢɦ

6n = 0, N 6cosΜ = 0, 6t = 0, Q 6sinΜ = 0, 6ɆK = 0, Ɇ 6RcosΜ = 0,

N = 6cosΜ ;

Q= 6sinΜ ;

Ɇ= 18cosΜ .

ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜɵɱɢɫɥɹɟɦ			Ɇ = 18 ɤɇɦ;
Μ = 0,	N = 6 ɤɇ,	Q = 0,	Ɇ = 18 ɤɇɦ;
Μ = 30θ,	N = 5,2 ɤɇ,	Q = 3 ɤɇ,	Ɇ = 15,6 ɤɇɦ;
Μ = 60θ,	N = 3 ɤɇ,	Q = 5,2 ɤɇ,	Ɇ = 9 ɤɇɦ;
Μ = 90θ,	N = 0,	Q = 6 ɤɇ,	Ɇ = 0;
Μ = 120θ,	N = 3 ɤɇ,	Q = 5,2 ɤɇ,	Ɇ = 9 ɤɇɦ;
Μ = 150θ,	N = 5,2 ɤɇ,	Q = 3 ɤɇ,	Ɇ = 15,6 ɤɇɦ;
Μ = 180θ,	N = 6 ɤɇ,	Q = 0,	Ɇ = 18 ɤɇɦ.
ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ N, Q ɢ M ɜ ɯɚɪɚɤɬɟɪɧɵɯ ɫɟɱɟɧɢɹɯ ɝɨɪɢɡɨɧɬɚɥɶɧɨɝɨ
ɫɬɟɪɠɧɹ ȼɋ.
ɋɟɱɟɧɢɟ ɋ :	N = 0, Q = 6 ɤɇ,	Ɇ = 0.
ɋɟɱɟɧɢɟ ȼ :	N = 0, Q = 6 ɤɇ,	Ɇ = 6 3 = 18 ɤɇɦ
		(ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ).

Ɉɬɤɥɚɞɵɜɚɹ ɜɵɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɫɟɱɟɧɢɹɯ ɩɟɪɩɟɧɞɢ-

ɤɭɥɹɪɧɨ ɤ ɨɫɢ ɫɬɟɪɠɧɹ ɢ ɫɨɟɞɢɧɹɹ ɩɨɥɭɱɟɧɧɵɟ ɬɨɱɤɢ, ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ ɜɧɭɬɪɟɧɧɢɯ ɭɫɢɥɢɣ N, Q ɢ M. ɗɬɢ ɷɩɸɪɵ ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.3.15,ɜ,ɝ,ɞ.

Ɉɩɨɪɧɵɟ ɪɟɚɤɰɢɢ ɜ ɡɚɞɟɥɤɟ ɪɚɜɧɵ: RA = 6 ɤɇ, HA = 0, ɆȺ = 18 ɤɇɦ.

Ɂɚɞɚɱɚ 3.12

Ⱦɥɹ ɪɚɦɵ ɫ ɲɚɪɧɢɪɧɵɦɢ ɨɩɨɪɚɦɢ (ɪɢɫ.3.16,ɚ) ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ N, Q

ɢ M. ɚ)

18ɤɇ/ɦ		12ɤɇɦ
HA A
	C	D	ɦ	x
RA	E	12ɤɇ1,5
y	E		ɦ
			ɦ
		B	1,5
		B	1,5
3 ɦ	RB 1ɦ

ɛ)	(ɤɇ)
25 +	(ɤɇ)
25 +	_
ɯ0	_	+
	29	12

ɜ)	N (ɤɇ)	ɝ)			ɞ)	(ɤɇɦ)
12	N (ɤɇ)	ɉɪɨɜɟɪɤɚ ɪɚɜɧɨɜɟɫɢɹ			M	(ɤɇɦ)
12	_	ɠɺɫɬɤɨɝɨ ɭɡɥɚ ɋ				6	18
	_	ɠɺɫɬɤɨɝɨ ɭɡɥɚ ɋ
	_	12ɤɇ	29ɤɇ	12ɤɇɦ	17,36		12
		12ɤɇ
		6ɤɇɦ ɋ
		6ɤɇɦ ɋ		12ɤɇ
	29	18ɤɇɦ 29ɤɇ

Ɋɢɫ.3.16

Ɉɩɪɟɞɟɥɢɦ ɜɟɥɢɱɢɧɵ ɨɩɨɪɧɵɯ ɪɟɚɤɰɢɣ.

6X = 0,

ɇȺ 12 = 0, ɇȺ = 12 ɤɇ;

6ɆȺ = 0, 18 3 1,5 12 12 1,5 3Rȼ = 0,

Rȼ = 29 ɤɇ;

6Ɇȼ = 0,

18 3 1,5 12 12 1,5 12 3 3RȺ = 0, RȺ = 25 ɤɇ;

6Y = 0 (ɩɪɨɜɟɪɤɚ),

18 3 – 25 – 29 = 54 – 54 = 0.

ȼɵɱɢɫɥɹɟɦ ɜɧɭɬɪɟɧɧɢɟ ɭɫɢɥɢɹ ɜ ɯɚɪɚɤɬɟɪɧɵɯ ɫɟɱɟɧɢɹɯ ɤɚɠɞɨɝɨ ɭɱɚɫɬɤɚ

ɪɚɦɵ.

ɋɬɟɪɠɟɧɶ ȺD

ɋɟɱɟɧɢɟ Ⱥ:

N = 12 ɤɇ,

Q = 25 ɤɇ,

Ɇ = 0.

ɋɟɱɟɧɢɟ ɋ (ɫɥɟɜɚ):

N = 12 ɤɇ, Q = 25 18 3 = 29 ɤɇ,

Ɇ = 25 3 18 3 1,5 = 6 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ).

ɋɟɱɟɧɢɟ ɋ (ɫɩɪɚɜɚ): N = 0,

Q = 0,

Ɇ = 12 ɤɇɦ

(ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).

ɋɟɱɟɧɢɟ D:

N = 0,

Q = 0,

Ɇ = 12 ɤɇɦ.

ɋɬɟɪɠɟɧɶ ȼɋ

ɋɟɱɟɧɢɟ ȼ:

N = 29 ɤɇ,

Q = 0,

Ɇ = 0.

ɋɟɱɟɧɢɟ ȿ (ɫɧɢɡɭ):

N = 29 ɤɇ,

Q = 0,

Ɇ = 0.

ɋɟɱɟɧɢɟ ȿ (ɫɜɟɪɯɭ):

N = 29 ɤɇ,

Q = 12 ɤɇ, Ɇ = 0.

ɋɟɱɟɧɢɟ ɋ:

N = 29 ɤɇ,

Q = 12 ɤɇ, Ɇ = 12 1,5 = 18 ɤɇɦ

(ɪɚɫɬɹɧɭɬɵ ɩɪɚɜɵɟ ɜɨɥɨɤɧɚ).

ɂɡ ɩɨɞɨɛɢɹ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ɧɚ ɷɩɸɪɟ

Q (ɪɢɫ.3.16,ɛ) ɨɩɪɟɞɟɥɹɟɦ ɤɨɨɪɞɢɧɚ-

ɬɭ ɯ0, ɝɞɟ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ, ɢ ɞɥɹ ɷɬɨɝɨ ɫɟɱɟɧɢɹ ɜɵɱɢɫɥɹɟɦ

ɷɤɫɬɪɟɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ.

ɯ0

25 ,

ɯ0 = 1,39 ɦ ;

3 ɯ0

Mmax

Ɇ 1,39

25 1,39

1,392

17,36 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).

ɗɩɸɪɵ N, Q ɢ M ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.3.16,ɛ,ɜ,ɞ.

ȼɵɪɟɠɟɦ ɦɵɫɥɟɧɧɨ ɭɡɟɥ

ɢ ɩɪɨɜɟɪɢɦ ɟɝɨ ɪɚɜɧɨɜɟɫɢɟ ɩɨɞ ɞɟɣɫɬɜɢɟɦ

ɜɧɭɬɪɟɧɧɢɯ ɭɫɢɥɢɣ ɜ ɫɬɟɪɠɧɹɯ, ɫɯɨɞɹɳɢɯɫɹ ɜ ɭɡɥɟ (ɪɢɫ.3.16,ɝ). ɇɟɬɪɭɞɧɨ ɜɢ-

ɞɟɬɶ, ɱɬɨ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ 6X = 0,

6Y = 0,

6Ɇ = 0

ɜɵɩɨɥɧɹɸɬɫɹ.

Ɂɚɞɚɱɚ 3.13

Ⱦɥɹ ɤɨɧɫɨɥɶɧɨɣ ɛɚɥɤɢ (ɪɢɫ.3.17,ɚ)

ɢɡɨɛɪɚɠɟɧɚ ɷɩɸɪɚ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ

(ɪɢɫ.3.17,ɛ). Ɉɩɪɟɞɟɥɢɦ ɧɚɝɪɭɡɤɭ, ɞɟɣɫɬ-

ɚ/2

ɚ=2ɦ

ɜɭɸɳɭɸ ɧɚ ɛɚɥɤɭ, ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ ɢ ɩɨ-

ɫɬɪɨɢɦ ɷɩɸɪɭ ɩɨɩɟɪɟɱɧɵɯ ɫɢɥ.

ɛ)

ɇɚ ɭɱɚɫɬɤɟ ȼɋ ɷɩɸɪɚ Ɇz

ɢɦɟɟɬ ɜɢɞ

ɧɚɤɥɨɧɧɨɣ ɩɪɹɦɨɣ. Ɋɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ

ɜɨɥɨɤɧɚ ɛɚɥɤɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɤ ɤɨɧɰɭ

Mmax

(ɤɇɦ)

ɛɚɥɤɢ ɩɪɢɥɨɠɟɧɚ ɫɨɫɪɟɞɨɬɨɱɟɧɧɚɹ ɫɢɥɚ,

ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɧɢɡ ɢ ɪɚɜɧɚɹ

Ɋɢɫ.3.17

Ɋ1	Ɇȼ	20	10ɤɇ .
	ɚ	2

ɇɚ ɭɱɚɫɬɤɟ Ⱥȼ ɷɩɸɪɚ Ɇz ɢɦɟɟɬ ɜɢɞ ɤɜɚɞɪɚɬɧɨɣ ɩɚɪɚɛɨɥɵ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɚ ɷɬɨɦ ɭɱɚɫɬɤɟ ɧɚ ɛɚɥɤɭ ɞɟɣɫɬɜɭɟɬ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɚɹ ɧɚɝɪɭɡɤɚ q,

ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɧɢɡ. ȼɟɥɢɱɢɧɭ	q		ɨɩɪɟɞɟɥɢɦ ɩɨ ɚɛɫɨɥɸɬɧɵɦ ɡɧɚɱɟɧɢɹɦ ɢɡɝɢ-
ɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ ɆȺ, Ɇȼ ɢ ɆD			ɢ ɩɨ ɜɟɥɢɱɢɧɟ “ɫɬɪɟɥɤɢ” f qa2/8 ɤɜɚɞɪɚɬ-
ɧɨɣ ɩɚɪɚɛɨɥɵ ɜ ɫɟɪɟɞɢɧɟ ɭɱɚɫɬɤɚ Ⱥȼ.
ɋɨɝɥɚɫɧɨ ɫɜɨɣɫɬɜɭ ɫɪɟɞɧɟɣ ɥɢɧɢɢ ɬɪɚɩɟɰɢɢ, ɢɦɟɟɦ
f ɆD		ɆȺ Ɇȼ		.

			2

Ɉɬɫɸɞɚ ɩɨɥɭɱɢɦ

4 ɆȺ Ɇ

ȼ 8ɆD

ɚ2

ɆȺ= 60ɤɇɦ

q = 40ɤɇ/ɦ Ɋ1=10ɤɇ

RA= 60ɤɇ

Ɋ2= 30ɤɇ

_ 20

ɛ)

(ɤɇ)

Ɋɢɫ.3.18

	4 60 20 8 20	40ɤɇ/ɦ.
	4	40ɤɇ/ɦ.
	4

ȼ ɫɟɱɟɧɢɢ ȼ ɧɚ ɷɩɸɪɟ Ɇz ɢɦɟɟɬɫɹ ɢɡɥɨɦ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɷɬɨɦ ɫɟɱɟɧɢɢ ɩɪɢɥɨɠɟɧɚ ɫɨɫɪɟɞɨɬɨɱɟɧɧɚɹ ɫɢɥɚ Ɋ2, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɜɟɪɯ. ȼɟɥɢɱɢɧɭ ɫɢɥɵ ɧɚɣɞɟɦ ɩɨ ɡɧɚɱɟɧɢɸ ɢɡɝɢɛɚɸɳɟɝɨ

ɦɨɦɟɧɬɚ ɜ ɡɚɞɟɥɤɟ

ɆȺ = 60 ɤɇɦ:

Ɇ		Ɋ	2ɚ Ɋ ɚ				qa2
Ɇ	Ⱥ	Ɋ	2ɚ Ɋ ɚ
	Ⱥ	1		2			2
				40		22	2
10 2		2 2Ɋ		40		22		60ɤɇɦ.
10 2		2 2Ɋ						60ɤɇɦ.
			2		2
					2

Ɉɬɫɸɞɚ ɩɨɥɭɱɢɦ Ɋ2 = 30 ɤɇ.

ɇɚ ɪɢɫ.3.18,ɚ ɩɨɤɚɡɚɧɵ ɧɚɝɪɭɡɤɢ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɛɚɥɤɭ, ɢ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ. ɗɩɸɪɚ ɩɨɩɟɪɟɱɧɵɯ ɫɢɥ ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ.3.18,ɛ.

Ɂɚɞɚɱɚ 3.14

Ⱦɥɹ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɣ ɛɚɥɤɢ ɫ ɤɨɧɫɨɥɶɸ (ɪɢɫ.3.19,ɚ) ɢɡɨɛɪɚɠɟɧɚ ɷɩɸɪɚ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ (ɪɢɫ.3.19,ɛ). Ɉɩɪɟɞɟɥɢɦ ɧɚɝɪɭɡɤɭ, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɛɚɥɤɭ, ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ ɢ ɩɨɫɬɪɨɢɦ ɷɩɸɪɭ ɩɨɩɟɪɟɱɧɵɯ ɫɢɥ.

ɇɚ ɤɨɧɫɨɥɶɧɨɣ ɱɚɫɬɢ

ɋD ɛɚɥɤɢ

ɚ)

ɞɟɣɫɬɜɭɟɬ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɚɹ

ɚ=2 ɦ

ɧɚɝɪɭɡɤɚ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɧɢɡ, ɤɨɬɨɪɭɸ

ɚ=2 ɦ

ɨɩɪɟɞɟɥɢɦ ɩɨ ɜɟɥɢɱɢɧɟ ɢɡɝɢɛɚɸɳɟɝɨ

ɛ)

ɦɨɦɟɧɬɚ ɜ ɫɟɱɟɧɢɢ ɋ:

2 20

Ɇɋ

10ɤɇ/ɦ .

(ɤɇɦ)

ȼ ɫɟɱɟɧɢɢ

ɤ ɛɚɥɤɟ ɩɪɢɥɨɠɟɧ

ɦɨɦɟɧɬ

Ɇ = 12 ɤɇɦ (ɪɢɫ.3.19,ɚ), ɜɵ-

40	Ɋɢɫ.3.19

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