Сопротивление материалов / Nesmeyanov - Soprotivleniye materialov. Nestandartniye zadachi i podkhodi k ikh resheniyu 2005

Ɋɢɫ.3.11

Ɋɢɫ.3.12

Ɉɛɨɡɧɚɱɢɜ ɫɢɥɭ ɜ ɩɪɭɠɢɧɟ ɱɟɪɟɡ R (ɜ ɬɨɱɤɟ B ɩɪɢɥɨɠɟɧɚ ɫɢɥɚ P3), ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɟ ɜɵɪɚɠɟɧɢɹ:

	'B= P3 /ɫɛ ,	'D= R/ɫɛ= (P3 – R)l3/(3EI),					'B – 'D='l1=P3 l1 /(ES).
Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɫɜɨɞɢɬɫɹ ɤ ɤɜɚɞɪɚɬɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɢ ɢɦɟɟɬ ɜɢɞ
	ɫɛ=( ( 4k 1)			– 1)/(2OȻ) ,
ɝɞɟ	OȻ= l3/(3EI),		k=OȻ /OɎ ,			OɎ=l1 /(ES).
ȼɨɡɜɪɚɳɚɹɫɶ ɤ ɪɢɫ.3.10, ɡɚɦɟɬɢɦ, ɱɬɨ ɢɫɤɨɦɚɹ ɠɟɫɬɤɨɫɬɶ ɫ ɫɭɦɦɢɪɭɟɬɫɹ ɢɡ
ɞɜɭɯ ɱɚɫɬɟɣ (ɚ) ɢ (ɛ):
	ɫ=ɫa+ɫɛ=1/OȻ+1/Oɛ=(				( 4k 1) +1)/(2OȻ).
Ɂɧɚɱɢɬ,
	u=P2/(2ɫ)=P2OȻ /(			( 4k 1) +1).
ɇɟɬɪɭɞɧɨ	ɡɚɦɟɬɢɬɶ,	ɱɬɨ		A	B	B	D	B l D
ɩɪɢ k=f	(ɤɚɤ ɩɪɢɧɹɬɨ
ɨɛɵɱɧɨ ɫɱɢɬɚɬɶ ɩɪɢ ɪɚɫɱɟ-			P2			P3	P3	c
ɬɚɯ ɩɟɪɟɦɟɳɟɧɢɣ ɜ ɪɚɦɚɯ)
ɢɡ ɩɨɥɭɱɟɧɧɨɝɨ ɪɟɡɭɥɶɬɚɬɚ
ɫɥɟɞɭɟɬ, ɱɬɨ ɫɦɟɳɟɧɢɟ u

ɪɚɜɧɨ ɧɭɥɸ. ɗɬɨ ɛɵɥɨ ɨɬɦɟɱɟɧɨ ɜɵɲɟ, ɢɫɯɨɞɹ ɢɡ

ɨɛɳɢɯ ɫɨɨɛɪɚɠɟɧɢɣ ɨ ɧɟɪɚɫɬɹɠɢɦɨɫɬɢ ɝɨɪɢɡɨɧɬɚɥɶɧɵɯ ɫɬɟɪɠɧɟɣ.

Ɋɟɲɟɧɢɟ ɡɚɞɚɱ ɫɭɳɟɫɬɜɟɧɧɨ ɭɩɪɨɳɚɟɬɫɹ, ɟɫɥɢ ɭɞɚɟɬɫɹ ɢɫɩɨɥɶɡɨɜɚɬɶ ɫɜɨɣɫɬɜɚ

			ɫɢɦɦɟɬɪɢɢ.
	P		ɉɭɫɬɶ ɮɟɪɦɚ ɢɡ ɲɟɫɬɢ ɨɞɢ-
			ɧɚɤɨɜɵɯ	ɫɢɦɦɟɬɪɢɱɧɨ ɪɚɫɩɨɥɨ-
	A	P	ɠɟɧɧɵɯ	ɫɬɟɪɠɧɟɣ (pɢɫ.3.13) ɧɚ-
			ɝɪɭɠɟɧɚ ɫɢɥɨɣ P. ɇɚɣɬɢ ɩɟɪɟɦɟ-
		PA	30° ɳɟɧɢɟ ɬɨɱɤɢ Ⱥ.
		PȻ	ɇɟɫɢɦɦɟɬɪɢɱɧɭɸ ɧɚɝɪɭɡɤɭ P
			ɦɨɠɧɨ ɪɚɡɥɨɠɢɬɶ ɧɚ ɫɢɦɦɟɬɪɢɱ-
	Ɋɢɫ.3.13	Ɋɢɫ.3.14	ɧɵɟ ɫɨɫɬɚɜɥɹɸɳɢɟ – ɜɟɪɬɢɤɚɥɶ-
			ɧɭɸ ɢ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɢ ɪɟɲɢɬɶ

ɞɜɟ ɡɚɞɚɱɢ ɨ ɠɟɫɬɤɨɫɬɹɯ ɩɪɢ ɪɚɡɧɵɯ ɜɢɞɚɯ ɜɨɡɞɟɣɫɬɜɢɹ. ɉɨɥɟɡɧɟɟ, ɨɞɧɚɤɨ, ɪɚɡɥɨɠɢɬɶ ɟɟ ɧɚ ɧɟɨɪɬɨɝɨɧɚɥɶɧɵɟ ɫɥɚɝɚɟɦɵɟ (pɢɫ.3.14): ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɠɟɫɬɤɨɫɬɢ ɜ ɧɚɩɪɚɜɥɟɧɢɹɯ ɫɢɥ PA ɢ PȻ ɨɞɢɧɚɤɨɜɵ (ɤɨɧɫɬɪɭɤɰɢɹ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɩɪɢ ɩɨɜɨɪɨɬɟ ɜɨɤɪɭɝ ɨɫɢ A ɧɚ ɭɝɨɥ 600). Ɉɞɧɚɤɨ ɡɚɞɚɱɚ ɭɩɪɨɫɬɢɬɫɹ, ɟɫɥɢ ɞɥɹ ɪɚɡɥɨɠɟɧɢɹ ɫɢɥɵ P ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɜɟ ɞɪɭɝɢɟ ɨɫɢ ɫɢɦɦɟɬɪɢɢ (pɢɫ.3.15). ɀɟɫɬɤɨɫɬɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɹɦ 1 ɢ 2 ɬɚɤɠɟ ɨɞɢɧɚɤɨɜɵ (ɫɢɦɦɟɬɪɢɹ ɤɨɧɫɬɪɭɤɰɢɢ ɩɪɢ ɩɨɜɨɪɨɬɟ ɧɚ 600), ɧɨ ɪɚɫɱɟɬɧɚɹ ɫɯɟɦɚ ɞɥɹ ɢɯ ɨɩɪɟɞɟɥɟɧɢɹ ɨɤɚɡɵɜɚɟɬɫɹ ɩɪɨɳɟ (pɢɫ.3.16). Ɂɚɞɚɱɚ ɡɟɪɤɚɥɶɧɨ ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɥɨɫɤɨɫɬɢ AB, ɢ ɜ ɫɬɟɪɠɧɹɯ, ɥɟɠɚɳɢɯ ɜ ɷɬɨɣ ɩɥɨɫɤɨ-

ɫɬɢ, ɭɫɢɥɢɹ ɨɬɫɭɬɫɬɜɭɸɬ. Ɍɚ ɠɟ ɤɨɫɚɹ ɫɢɦɦɟɬɪɢɹ ɩɨɡɜɨɥɹɟɬ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɢɫɤɨɦɚɹ ɠɟɫɬɤɨɫɬɶ ɜɞɜɨɟ ɜɵɲɟ, ɱɟɦ ɠɟɫɬɤɨɫɬɶ ɩɨɥɨɜɢɧɵ ɨɫɬɚɜɲɟɣɫɹ ɤɨɧɫɬɪɭɤɰɢɢ (ɪɢɫ.3.17). ɉɪɢɥɨɠɢɜ ɟɞɢɧɢɱɧɭɸ ɫɢɥɭ ɜɦɟɫɬɨ P1, ɧɚɣɞɟɦ ɩɨɞɚɬɥɢɜɨɫɬɶ (ɩɟɪɟɦɟɳɟɧɢɟ) ɢɡ ɢɧɬɟɝɪɚɥɚ Ɇɨɪɚ, «ɩɟɪɟɦɧɨɠɢɜ» ɷɩɸɪɭ N1 ɫɚɦɭ ɧɚ ɫɟɛɹ (ɭɫɢɥɢɹ ɜ ɫɬɟɪɠɧɹɯ ɪɚɜɧɵ 1/(2cos300)=1/ 3 )

		O=ɗN1uɗN1=2(1/ 3 1/	3 l/(ES))=2l/(3ES),
	ɨɬɤɭɞɚ ɢɫɤɨɦɚɹ ɠɟɫɬɤɨɫɬɶ ɮɟɪɦɵ ɫ=2/O=3ES/l.
	ȼɨɡɜɪɚɳɚɹɫɶ ɤ pɢɫ.3.15, ɢɫɩɨɥɶɡɭɹ ɩɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ, ɧɚɣɞɟɦ, ɱɬɨ ɜɟɤ-
		P		30°	ɬɨɪ	ɫɦɟɳɟɧɢɹ			ɛɭ-
					ɞɟɬ	ɫɤɥɚɞɵɜɚɬɶɫɹ
		1
	P				(ɩɨ ɩɪɚɜɢɥɚɦ ɫɥɨ-
		P2			ɠɟɧɢɹ		ɜɟɤɬɨɪɨɜ)
					ɢɡ	ɜɟɤɬɨɪɚ		ɜɞɨɥɶ
		A	B
	P1			1	P1,	ɪɚɜɧɨɝɨ |P1| /ɫ
					ɢ	ɜɟɤɬɨɪɚ		ɜɞɨɥɶ
	Ɋɢɫ.3.15	Ɋɢɫ.3.16		Ɋɢɫ.3.17	P2, ɪɚɜɧɨɝɨ |P2| /ɫ.
					ɋɨɨɬɧɨɲɟɧɢɟ				ɦɟ-
	ɠɞɭ ɫɥɚɝɚɟɦɵɦɢ ɬɨ ɠɟ, ɱɬɨ ɢ ɦɟɠɞɭ ɫɢɥɚɦɢ, ɡɧɚɱɢɬ, ɢ ɫɭɦɦɚ ɨɤɚɠɟɬɫɹ ɧɚɩɪɚɜɥɟɧ-
	ɧɨɣ ɩɨɞ ɭɝɥɨɦ D.
	Ɉɤɨɧɱɚɬɟɥɶɧɨ, ɞɥɹ ɜɟɤɬɨɪɨɜ ɫɦɟɳɟɧɢɹ ɢ ɫɢɥɵ ɫɩɪɚɜɟɞɥɢɜɨ ɜɵɪɚɠɟɧɢɟ

u= P/ɫ = Pl/(3ES).

4.ɄɈɇɋɌɊɍɄɐɂɂ ɋ ȺȻɋɈɅɘɌɇɈ ɀȿɋɌɄɂɆɂ ɗɅȿɆȿɇɌȺɆɂ

ȼ ɢɧɠɟɧɟɪɧɨɣ ɩɪɚɤɬɢɤɟ ɱɚɫɬɨ ɜɫɬɪɟɱɚɸɬɫɹ ɤɨɧɫɬɪɭɤɰɢɢ, ɜ ɤɨɬɨɪɵɯ ɠɟɫɬɤɨɫɬɶ ɨɞɧɢɯ ɷɥɟɦɟɧɬɨɜ (ɧɚɩɪɢɦɟɪ, ɨɩɨɪ) ɡɧɚɱɢɬɟɥɶɧɨ ɛɨɥɶɲɟ ɠɟɫɬɤɨɫɬɢ ɞɪɭɝɢɯ. ȼ ɬɚɤɢɯ ɫɥɭɱɚɹɯ ɩɟɪɜɵɟ ɷɥɟɦɟɧɬɵ ɭɞɨɛɧɨ ɫɱɢɬɚɬɶ ɚɛɫɨɥɸɬɧɨ ɠɟɫɬɤɢɦɢ. ɗɬɨ ɞɨɩɭɳɟɧɢɟ ɧɟ ɩɪɢɜɨɞɢɬ ɤ ɡɧɚɱɢɬɟɥɶɧɵɦ ɨɲɢɛɤɚɦ, ɧɨ ɦɨɠɟɬ ɫɭɳɟɫɬɜɟɧɧɨ ɭɩɪɨɫɬɢɬɶ ɪɟɲɟɧɢɟ. Ɍɪɭɞɧɨɫɬɶ ɜ ɬɚɤɢɯ ɡɚɞɚɱɚɯ ɫɜɹɡɚɧɚ ɫ ɧɟɤɨɬɨɪɨɣ ɧɟɨɛɵɱɧɨɫɬɶɸ ɜɨɡɧɢɤɚɸɳɢɯ ɜ ɤɨɧɬɚɤɬɟ ɫ ɚɛɫɨɥɸɬɧɨ ɠɟɫɬɤɢɦ ɬɟɥɨɦ ɫɢɥ.

Ɂɚɞɚɱɚ 1 (ɪɢɫ.4.1). Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɫɢɥɭ Ɋ, ɤɨɬɨɪɚɹ ɩɨɡɜɨɥɢɬ ɜɫɬɚɜɢɬɶ ɫɬɟɪɠɟɧɶ ɜ ɤɪɢɜɨɥɢɧɟɣɧɨɟ ɨɬɜɟɪɫɬɢɟ ɚɛɫɨɥɸɬɧɨ ɠɟɫɬɤɨɝɨ ɬɟɥɚ (ɬɪɟɧɢɟɦ ɩɪɟɧɟɛɪɟɱɶ). ɋɬɟɪɠɟɧɶ ɩɨɫɬɨɹɧɧɨɣ ɠɟɫɬɤɨɫɬɢ EI ɢɦɟɟɬ ɞɥɢɧɭ L.

Ɋɟɲɟɧɢɟ. Ɋɚɫɫɦɨɬɪɢɦ ɫɨɫɬɨɹɧɢɟ ɫɬɟɪɠɧɹ, ɤɨɝɞɚ ɨɧ ɜɨɲɟɥ ɜ ɨɬɜɟɪɫɬɢɟ ɧɚ ɞɥɢɧɭ l (ɪɢɫ.4.1). ȿɝɨ ɩɪɹɦɚɹ ɱɚɫɬɶ ɫɠɚɬɚ, ɨɫɬɚɥɶɧɚɹ – ɢɡɨɝɧɭɬɚ ɢ ɢɦɟɟɬ ɩɨɫɬɨɹɧɧɭɸ ɤɪɢɜɢɡɧɭ F = 1/R; ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɨɧɚ ɢɫɩɵɬɵɜɚɟɬ ɱɢɫɬɵɣ ɢɡɝɢɛ ɦɨɦɟɧɬɨɦ EIF=EI/R. ɇɟɬɪɭɞɧɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɪɟɚɤɰɢɹɦɢ ɚɛɫɨɥɸɬɧɨ ɠɟɫɬɤɨɣ ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɨɩɨɪɵ ɹɜɥɹɸɬɫɹ ɞɜɟ ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɟ ɩɚɪɵ ɫɢɥ ɫ ɭɤɚɡɚɧɧɵɦ ɦɨɦɟɧɬɨɦ – ɧɚ ɤɨɧɰɟ ɫɬɟɪɠɧɹ ɢ ɧɚ ɜɯɨɞɟ ɜ ɨɬɜɟɪɫɬɢɟ.

ɉɨɫɤɨɥɶɤɭ ɩɥɟɱɨ ɤɚɠɞɨɣ ɩɚɪɵ ɧɟ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɧɭɥɟɜɨɝɨ, ɡɧɚɱɟɧɢɹ ɫɢɥ ɛɟɫɤɨɧɟɱɧɵ. ȼ ɷɬɨɦ ɫɨɫɬɨɢɬ ɭɩɨɦɹɧɭɬɚɹ ɫɬɪɚɧɧɨɫɬɶ ɪɟɚɤɰɢɢ ɠɟɫɬɤɨɣ ɨɩɨɪɵ, ɥɨɝɢɱɧɨ ɜɵɬɟɤɚɸɳɚɹ ɢɡ ɨɛɵɱɧɵɯ ɞɨɩɭɳɟɧɢɣ ɪɚɫɱɟɬɧɨɣ ɫɯɟɦɵ.

ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɞɟɮɨɪɦɚɰɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ

U=³F 2EIdz/2=EIl/(2R2)

(4.1)

(ɷɧɟɪɝɢɟɣ ɫɠɚɬɢɹ ɫɬɟɪɠɧɹ ɩɪɟɧɟɛɪɟɝɚɟɦ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɷɧɟɪɝɢɟɣ ɢɡɝɢɛɚ). ɉɪɢ ɢɡɦɟɧɟɧɢɢ ɞɥɢɧɵ l ɧɚ ɦɚɥɭɸ ɜɟɥɢɱɢɧɭ ' ɷɧɟɪɝɢɹ U ɜɨɡɪɚɫɬɚɟɬ ɧɚ ɜɟɥɢɱɢɧɭ (EI/(2R2))' ɢ ɞɨɩɨɥɧɢɬɟɥɶɧɚɹ ɪɚɛɨɬɚ ɫɢɥɵ P ɪɚɜɧɚ P' . ɂɡ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɫɥɟɞɭɟɬ ɪɚɜɟɧɫɬɜɨ ɷɬɢɯ ɜɟɥɢɱɢɧ, ɨɬɤɭɞɚ

P= EI/(2R2).

(4.2)

ɋɢɥɚ P ɧɟ ɡɚɜɢɫɢɬ ɨɬ l, ɩɨɤɚ l L. ɇɚ ɪɢɫ.4.2 ɩɨɤɚɡɚɧɵ ɝɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɢ P ɢ U

l	P U	U	ɨɬ ɞɥɢɧɵ l. Ʉɚɤ ɜɢɞɢɦ, ɡɚɞɚ-
			ɱɚ ɨɬɧɨɫɢɬɫɹ ɤ		ɝɟɨɦɟɬɪɢ-
P	P		ɱɟɫɤɢ ɧɟɥɢɧɟɣɧɵɦ		(ɤɪɚɟɜɵɟ
			ɭɫɥɨɜɢɹ ɢɡɦɟɧɹɸɬɫɹ ɜ ɩɪɨ-
			ɰɟɫɫɟ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ), ɩɨ-
R		l	ɷɬɨɦɭ ɡɞɟɫɶ ɨɬɫɭɬɫɬɜɭɟɬ ɩɪɢ-
			ɜɵɱɧɨɟ	ɤɜɚɞɪɚɬɢɱɧɨɟ		ɜɨɡɪɚ-
Ɋɢɫ.4.1	Ɋɢɫ.4.2		ɫɬɚɧɢɟ	ɩɨɬɟɧɰɢɚɥɶɧɨɣ		ɷɧɟɪ-
			ɝɢɢ ɢ ɥɢɧɟɣɧɨɟ ɜɨɡɪɚɫɬɚɧɢɟ

ɧɚɝɪɭɡɤɢ. ȼ ɦɨɦɟɧɬ, ɤɨɝɞɚ l=L, ɫɢɥɚ ɩɚɞɚɟɬ ɞɨ ɧɭɥɹ, ɢ ɫɬɟɪɠɟɧɶ ɜɵɥɟɬɚɟɬ ɢɡ ɞɪɭɝɨɝɨ ɤɨɧɰɚ ɨɬɜɟɪɫɬɢɹ, ɩɪɟɜɪɚɳɚɹ ɧɚɤɨɩɥɟɧɧɭɸ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɜ ɤɢɧɟɬɢɱɟɫɤɭɸ.

Ɂɚɞɚɱɚ 2 (ɪɢɫ. 4.3). Ȼɟɫɤɨɧɟɱɧɨ ɞɥɢɧɧɵɣ ɜɟɫɨɦɵɣ ɫɬɟɪɠɟɧɶ (ɩɨɝɨɧɧɵɣ ɜɟɫ q) ɥɟɠɢɬ ɧɚ ɠɟɫɬɤɨɦ ɨɫɧɨɜɚɧɢɢ. Ɉɩɪɟɞɟɥɢɬɶ ɜɵɫɨɬɭ ɩɨɞɴɟɦɚ h ɩɪɢ ɧɚɝɪɭɠɟɧɢɢ ɫɢɥɨɣ P.

Ɋɟɲɟɧɢɟ. Ɂɚɞɚɱɚ ɫɢɦɦɟɬɪɢɱɧɚ; ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɩɪɚɜɭɸ ɩɨɥɨɜɢɧɭ ɫɬɟɪɠɧɹ (zt 0). Ɉɫɧɨɜɚɧɢɟ ɫɱɢɬɚɟɦ ɚɛɫɨɥɸɬɧɨ ɠɟɫɬɤɢɦ. ɉɨɥɚɝɚɟɦ (ɞɚɥɟɟ ɷɬɨ ɦɨɠɧɨ ɩɪɨɜɟɪɢɬɶ), ɱɬɨ ɫɬɟɪɠɟɧɶ ɤɨɧɬɚɤɬɢɪɭɟɬ ɫ ɨɩɨɪɨɣ ɧɚ ɜɫɟɦ ɢɧɬɟɪɜɚɥɟ z t l/2 – ɩɪɚɜɟɟ ɬɨɱɤɢ A. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɨɝɢɛ, ɩɨɜɨɪɨɬ, ɤɪɢɜɢɡɧɚ ɫɬɟɪɠɧɹ F ɢ, ɨɬɫɸɞɚ, ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɩɪɢ zt l/2 ɪɚɜɧɵ ɧɭɥɸ.

Ɋɚɫɫɦɨɬɪɢɦ ɨɫɬɚɥɶɧɭɸ ɱɚɫɬɶ ɫɬɟɪɠɧɹ (zd l/2). Ⱦɥɹ ɧɟɟ ɞɨɥɠɧɵ ɜɵɩɨɥɧɹɬɶɫɹ

ɫɥɟɞɭɸɳɢɟ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ (ɪɢɫ.4.4). ɇɚ ɥɟɜɨɦ ɤɪɚɸ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ (ɩɨ ɭɫɥɨ- y y

	P	M( 0)	P/2
			q
A	A			z
	h		A
	z
l/2	l/2		R	r
	Ɋɢɫ.4.3		Ɋɢɫ.4.4

ɜɢɹɦ ɫɢɦɦɟɬɪɢɢ) ɪɚɜɧɚ P/2, ɩɪɨɝɢɛ X(0)=h, ɩɨɜɨɪɨɬ dX/dz ɪɚɜɟɧ ɧɭɥɸ ɢ, ɩɨɜɢɞɢɦɨɦɭ, ɢɦɟɟɬɫɹ ɧɟɧɭɥɟɜɨɣ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ M(0). ɇɚ ɩɪɚɜɨɦ ɤɪɚɸ (z=l/2)

z = l/2

ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɪɚɜɟɧ ɧɭɥɸ, ɬɚɤ ɤɚɤ ɩɪɚɜɟɟ ɬɨɱɤɢ A ɫɬɟɪɠɟɧɶ ɧɟ ɢɡɝɢɛɚɟɬɫɹ, ɚ ɨɞɧɨɫɬɨɪɨɧɧɹɹ ɨɩɨɪɚ, ɜ ɨɬɥɢɱɢɟ ɨɬ ɞɜɭɫɬɨɪɨɧɧɟɣ, ɤɚɤ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɪɢɦɟɪɟ, ɧɟ ɦɨɠɟɬ ɫɨɡɞɚɬɶ ɫɨɫɪɟɞɨɬɨɱɟɧɧɨɣ ɩɚɪɵ ɫɢɥ.

Ɂɚɬɨ ɷɬɚ ɨɩɨɪɚ ɦɨɠɟɬ ɞɚɬɶ ɞɪɭɝɭɸ ɪɟɚɤɰɢɸ: ɫɨɫɪɟɞɨɬɨɱɟɧɧɭɸ ɫɢɥɭ R, ɧɚɩɪɚɜɥɟɧɧɭɸ ɜɜɟɪɯ. ɗɬɨ ɤɚɠɟɬɫɹ ɧɟɩɪɚɜɞɨɩɨɞɨɛɧɵɦ, ɩɨɫɤɨɥɶɤɭ ɤɨɧɬɚɤɬ ɫɬɟɪɠɧɹ ɫ ɨɩɨɪɨɣ ɩɪɨɢɫɯɨɞɢɬ ɩɨ ɥɢɧɢɢ ɢ ɪɟɚɤɰɢɹ, ɤɚɡɚɥɨɫɶ ɛɵ, ɦɨɠɟɬ ɛɵɬɶ ɬɨɥɶɤɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ, ɨɞɧɚɤɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɭɫɥɨɜɢɹ (X(l/2)=0, dX/dz(l/2)=0) ɢ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɧɟ ɨɫɬɚɜɥɹɸɬ ɞɪɭɝɨɣ ɜɨɡɦɨɠɧɨɫɬɢ.

ȿɫɥɢ ɛɵ ɨɩɨɪɚ ɛɵɥɚ ɩɨɞɚɬɥɢɜɨɣ, ɪɟɚɤɰɢɹ ɨɩɨɪɵ ɦɨɝɥɚ ɛɵɬɶ ɥɢɲɶ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɩɨ ɡɚɤɨɧɭ, ɫɜɹɡɚɧɧɨɦɭ ɫ ɟɟ ɨɫɚɞɤɨɣ. ɑɟɦ ɜɵɲɟ ɠɟɫɬɤɨɫɬɶ ɨɩɨɪɵ, ɬɟɦ ɛɨɥɟɟ ɪɟɡɤɨ ɦɨɠɟɬ ɢɡɦɟɧɹɬɶɫɹ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɨɩɨɪɧɨɣ ɪɟɚɤɰɢɢ; ɜ ɩɪɟɞɟɥɟ ɫɨɫɪɟɞɨɬɨɱɟɧɧɚɹ R ɧɟ ɬɨɥɶɤɨ ɜɨɡɦɨɠɧɚ, ɧɨ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɧɟɨɛɯɨɞɢɦɚ: ɢɧɚɱɟ ɩɨɜɨɪɨɬ ɫɟ-

q ɱɟɧɢɹ ɧɟ ɦɨɝ ɛɵ ɛɵɬɶ ɧɭɥɟɜɵɦ. ɍɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ (6Y= 0) ɢɦɟɟɬ ɜɢɞ

		R		(ql – P)/2 – R= 0.		(4.3)
	l/2
			Ⱦɥɹ ɡɚɩɢɫɢ ɟɳɟ ɨɞɧɨɝɨ ɭɫɥɨɜɢɹ ɭɞɨɛɧɨ ɜɢɞɨɢɡɦɟ-
	Ɋɢɫ.4.5
			ɧɢɬɶ ɫɯɟɦɭ ɡɚɞɚɱɢ (pɢɫ.4.5), ɩɨɥɚɝɚɹ ɧɟɩɨɞɜɢɠɧɵɦ
ɥɟɜɵɣ ɤɪɚɣ. Ɍɨɝɞɚ ɩɪɨɝɢɛ ɩɪɚɜɨɝɨ ɤɪɚɹ ɪɚɜɟɧ h, ɟɝɨ ɩɨɜɨɪɨɬ ɪɚɜɟɧ ɧɭɥɸ. Ⱦɜɟ ɩɨ-
ɫɥɟɞɧɢɟ ɜɟɥɢɱɢɧɵ ɧɚɯɨɞɹɬɫɹ ɢɡ ɢɧɬɟɝɪɚɥɚ Ɇɨɪɚ, ɨɬɤɭɞɚ, ɫ ɭɱɟɬɨɦ (4.3), ɫɥɟɞɭɟɬ:
	l=3P/(2q), h=4.39 10 – 3P4/(q3EI).
ȿɳɟ ɛɨɥɟɟ ɫɬɪɚɧɧɵɣ ɪɟɡɭɥɶɬɚɬ ɩɨɥɭɱɚɟɦ ɜ ɫɥɟɞɭɸɳɟɣ ɡɚɞɚɱɟ.
ɉɪɢɦɟɪ 3. Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɧɚɩɪɹɠɟɧɢɹ ɜ ɩɨɩɟɪɟɱɧɵɯ ɫɟɱɟɧɢɹɯ ɫɬɟɪɠɧɹ
(pɢɫ.4.6), ɩɪɢɠɢɦɚɟɦɨɝɨ ɧɚɝɪɭɡɤɨɣ q(M) ɤ ɚɛɫɨɥɸɬɧɨ ɠɟɫɬɤɨɦɭ ɰɢɥɢɧɞɪɢɱɟɫɤɨɦɭ
ɨɫɧɨɜɚɧɢɸ ɪɚɞɢɭɫɚ R. Ɍɪɟɧɢɟ ɦɟɠɞɭ ɫɨɩɪɢɤɚɫɚɸɳɢɦɢɫɹ ɩɨɜɟɪɯɧɨɫɬɹɦɢ ɨɬɫɭɬɫɬ-
ɜɭɟɬ, ɫɬɟɪɠɟɧɶ ɤɚɫɚɟɬɫɹ ɨɩɨɪɵ ɩɨ ɜɫɟɣ ɞɥɢɧɟ. ɇɚɝɪɭɡɤɚ q ɹɜɥɹɟɬɫɹ ɫɬɪɨɝɨ ɪɚɞɢ-
1	q( )		q( )	ɚɥɶɧɨɣ.	ɉɨɩɟɪɟɱɧɨɣ	ɫɠɢɦɚɟɦɨɫɬɶɸ
		M		ɫɬɟɪɠɧɹ ɩɪɟɧɟɛɪɟɝɚɟɦ.
	d		2	Ɋɟɲɟɧɢɟ. ɇɚ ɢɡɨɝɧɭɬɵɣ ɫɬɟɪɠɟɧɶ
		1
1		N		ɞɟɣɫɬɜɭɸɬ ɫɬɪɨɝɨ ɪɚɞɢɚɥɶɧɵɟ ɜɧɟɲɧɢɟ
			r	ɫɢɥɵ: ɚɤɬɢɜɧɵɟ q(M),		ɧɚɩɪɚɜɥɟɧɧɵɟ ɤ
		1	( )	ɰɟɧɬɪɭ	ɰɢɥɢɧɞɪɚ, ɢ	ɪɟɚɤɰɢɢ ɨɩɨɪɵ
			2
	R			r(M), ɧɚɩɪɚɜɥɟɧɧɵɟ ɨɬ ɰɟɧɬɪɚ. ɉɨɞ
			R
				ɞɟɣɫɬɜɢɟɦ ɷɬɢɯ ɫɢɥ ɫɬɟɪɠɟɧɶ ɩɪɢɠɚɬ ɤ
Ɉ		Ɉ		ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɢ, ɫɥɟɞɨ-
				ɜɚɬɟɥɶɧɨ, ɢɦɟɟɬ ɩɨɫɬɨɹɧɧɭɸ ɤɪɢɜɢɡɧɭ
Ɋɢɫ.4.6		Ɋɢɫ.4.7		(ɪɚɞɢɭɫ ɟɟ U=R+d/2). Ɂɧɚɱɢɬ, ɜ ɫɬɟɪɠ-
ɧɟ ɞɟɣɫɬɜɭɟɬ ɩɨɫɬɨɹɧɧɵɣ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ M=EI/(R+d/2) (ɱɢɫɬɵɣ ɢɡɝɢɛ). ɗɬɨ
ɜɨɡɦɨɠɧɨ ɥɢɲɶ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɩɨ ɤɨɧɰɚɦ ɫɬɟɪɠɧɹ ɩɪɢɥɨɠɟɧɵ ɜɧɟɲɧɢɟ ɫɨɫɪɟɞɨ-
ɬɨɱɟɧɧɵɟ ɩɚɪɵ ɫɢɥ ɫ ɦɨɦɟɧɬɨɦ M. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤɚɤɨɣ ɛɵ ɧɢ ɛɵɥɚ ɪɚɫɩɪɟɞɟ-
ɥɟɧɧɚɹ ɧɚɝɪɭɡɤɚ ɩɨ ɞɥɢɧɟ ɫɬɟɪɠɧɹ, ɨɧɚ ɭɪɚɜɧɨɜɟɲɢɜɚɟɬɫɹ ɬɚɤɨɣ ɠɟ ɪɚɫɩɪɟɞɟɥɟɧ-
ɧɨɣ ɪɟɚɤɰɢɟɣ ɫɨ ɫɬɨɪɨɧɵ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɨɩɨɪɵ. ɇɨ ɧɚ ɤɨɧɰɚɯ ɫɬɟɪɠɧɹ ɢɧɬɟɧ-
ɫɢɜɧɨɫɬɢ ɜɧɟɲɧɟɣ ɧɚɝɪɭɡɤɢ q(M) ɢ ɪɟɚɤɰɢɢ r(M) ɫɬɪɟɦɹɬɫɹ ɤ ɛɟɫɤɨɧɟɱɧɨɫɬɢ ɫ ɛɟɫ-

ɤɨɧɟɱɧɨ ɦɚɥɵɦ ɫɞɜɢɝɨɦ ɩɨ ɭɝɥɭ. ɗɬɨɬ ɫɞɜɢɝ ɢ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɢɬɶ ɞɜɟ ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɟ ɩɚɪɵ ɫɢɥ ɫ ɦɨɦɟɧɬɨɦ M. ɇɨɪɦɚɥɶɧɚɹ ɫɢɥɚ ɨɬɫɭɬɫɬɜɭɟɬ, ɩɨɫɤɨɥɶɤɭ ɜɧɟɲɧɹɹ ɧɚɝɪɭɡɤɚ ɷɤɜɢɜɚɥɟɧɬɧɚ ɷɬɢɦ ɞɜɭɦ ɩɚɪɚɦ ɫɢɥ.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɫɟɱɟɧɢɢ 1–1 ɞɟɣɫɬɜɭɸɬ ɧɨɪɦɚɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ, ɨɩɪɟɞɟɥɹɟɦɵɟ ɜɵɪɚɠɟɧɢɟɦ

V =Ey/(R+0.5d),

ɤɨɨɪɞɢɧɚɬɚ y ɨɬɫɱɢɬɵɜɚɟɬɫɹ ɨɬ ɰɟɧɬɪɚɥɶɧɨɣ ɨɫɢ ɫɟɱɟɧɢɹ ɫɬɟɪɠɧɹ. ȼɨɡɧɢɤɧɨɜɟɧɢɟ ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɯ ɩɚɪ ɫɢɥ ɩɨ ɤɪɚɹɦ ɷɬɨɝɨ ɫɬɟɪɠɧɹ ɫɜɹɡɚɧɨ ɫ

ɩɪɢɧɹɬɵɦɢ ɞɨɩɭɳɟɧɢɹɦɢ ɢ ɜɟɫɶɦɚ ɧɟɨɱɟɜɢɞɧɨ. ɇɚɩɪɢɦɟɪ, ɜ ɪɚɛɨɬɚɯ [1,2], ɝɞɟ ɛɵɥɚ ɪɚɫɫɦɨɬɪɟɧɚ ɚɧɚɥɨɝɢɱɧɚɹ ɡɚɞɚɱɚ, ɧɚɥɢɱɢɟ ɩɚɪ ɫɢɥ ɧɟ ɛɵɥɨ ɡɚɦɟɱɟɧɨ ɢ ɜ ɪɟɡɭɥɶɬɚɬɟ, ɩɨɫɥɟ ɪɚɫɫɦɨɬɪɟɧɢɹ ɦɨɦɟɧɬɚ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɱɚɫɬɶ ɫɬɟɪɠɧɹ ɩɪɚɜɟɟ ɫɟɱɟɧɢɹ 1–1 (pɢɫ.4.7), ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɤɪɢɜɢɡɧɵ O (ɦɨɦɟɧɬɵ ɨɬ ɪɚɞɢɚɥɶɧɨɣ ɧɚɝɪɭɡɤɢ ɢ ɪɟɚɤɰɢɢ ɡɞɟɫɶ ɫɱɢɬɚɸɬɫɹ ɧɭɥɟɜɵɦɢ ɜɫɥɟɞɫɬɜɢɟ ɢɯ ɪɚɞɢɚɥɶɧɨɫɬɢ) ɩɨɥɭɱɟɧɨ, ɱɬɨ ɜ ɫɟɱɟɧɢɢ 1–1 ɞɨɥɠɧɚ ɞɟɣɫɬɜɨɜɚɬɶ ɭɪɚɜɧɨɜɟɲɢɜɚɸɳɚɹ ɫɠɢɦɚɸɳɚɹ ɧɨɪɦɚɥɶɧɚɹ ɫɢɥɚ N= – M/U (U=R+d/2).

Ɋɟɲɢɬɟ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ

Ɂɚɞɚɱɚ 1 (pɢɫ.4.8). Ʉɪɢɜɨɥɢɧɟɣɧɵɣ ɫɬɟɪɠɟɧɶ ɤɜɚɞɪɚɬɧɨɝɨ (huh) ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫ ɪɚɞɢɭɫɨɦ ɤɪɢɜɢɡɧɵ R >> h ɡɚɠɢɦɚɟɬɫɹ ɦɟɠɞɭ ɞɜɭɦɹ ɚɛɫɨɥɸɬɧɨ ɠɟɫɬɤɢɦɢ ɢ ɝɥɚɞɤɢɦɢ (ɬɪɟɧɢɟ ɨɬɫɭɬɫɬɜɭɟɬ) ɩɨɜɟɪɯɧɨɫɬɹɦɢ. ɇɚɝɪɭɠɟɧɢɟ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɞɨ ɩɨɥɧɨɝɨ ɜɵɩɪɹɦɥɟɧɢɹ ɫɬɟɪɠɧɹ. Ɍɪɟɛɭɟɬɫɹ ɩɨɫɬɪɨɢɬɶ ɷɩɸɪɵ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥɨɜɵɯ ɮɚɤɬɨɪɨɜ.

Ɂɚɞɚɱɚ 2. ɋɬɚɥɶɧɚɹ ɩɨɥɨɫɚ ɫɬɹɝɢɜɚɟɬɫɹ ɜɨɤɪɭɝ ɚɛɫɨɥɸɬɧɨ ɠɟɫɬɤɨɝɨ ɰɢɥɢɧɞɪɚ ɪɚɞɢɭɫɚ R (ɪɢɫ.4.9). Ɉɩɪɟɞɟɥɢɬɶ ɜɟɥɢɱɢɧɭ ɫɢɥ, ɜɨɡɧɢɤɚɸɳɢɯ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɤɨɧɬɚɤɬɚ ɞɜɭɯ ɬɟɥ (ɩɪɢɧɹɬɶ ɲɢɪɢɧɭ ɩɨɥɨɫɵ b=1).

Ɂɚɞɚɱɚ 3. ɉɪɹɦɨɣ ɫɬɟɪɠɟɧɶ ɤɪɭɝɥɨɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ (d=1 ɦɦ), ɜɵɩɨɥɧɟɧɧɵɣ ɢɡ ɢɞɟɚɥɶɧɨ ɩɥɚɫɬɢɱɟɫɤɨɝɨ ɦɚɬɟɪɢɚɥɚ (VT=200 Mɉɚ), ɢɡɨɝɧɭɬ ɢ ɛɟɡ ɡɚɡɨɪɚ ɜɫɬɚɜɥɟɧ ɜ ɚɛɫɨɥɸɬɧɨ ɝɥɚɞɤɢɣ ɩɚɡ ɤɪɭɝɥɨɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɨ ɫɪɟɞɧɢɦ

h

R	S
h	2 R

P	ɉɥɨɬɧɵɣ	h	d
	ɫɬɵɤ
P		R	R
			Ɉ
Ɋɢɫ.4.8	Ɋɢɫ.4.9		Ɋɢɫ.4.10

ɪɚɞɢɭɫɨɦ R=4 ɦ (pɢɫ.4.10). Ɉɩɪɟɞɟɥɢɬɶ ɜɟɥɢɱɢɧɭ ɩɥɚɫɬɢɱɟɫɤɢɯ ɞɟɮɨɪɦɚɰɢɣ. Ɇɨɞɭɥɶ ɭɩɪɭɝɨɫɬɢ ɦɚɬɟɪɢɚɥɚ ɫɬɟɪɠɧɹ E=2 105 Ɇɉɚ.

5. ɆȿɌɈȾ ɋɂɅ ɂ ɆȿɒȺɘɓɂȿ ɋȼəɁɂ

ɋɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɵɟ ɡɚɞɚɱɢ ɩɪɟɞɫɬɚɜɥɹɸɬ ɜɚɠɧɭɸ ɝɪɭɩɩɭ ɡɚɞɚɱ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ ɢ ɞɨɫɬɚɜɥɹɸɬ ɫɬɭɞɟɧɬɚɦ ɦɧɨɝɨ ɧɟɩɪɢɹɬɧɨɫɬɟɣ. ɇɢɠɟ ɩɪɟɞɥɨɠɟɧɚ ɨɪɢɝɢɧɚɥɶɧɚɹ ɦɟɬɨɞɢɤɚ ɪɟɚɥɢɡɚɰɢɢ ɦɟɬɨɞɚ ɫɢɥ, ɨɬɥɢɱɚɸɳɚɹɫɹ ɨɬ ɫɬɚɧɞɚɪɬɧɨɣ ɛɨɥɟɟ ɫɜɨɛɨɞɧɵɦ ɜɵɛɨɪɨɦ ɨɫɧɨɜɧɨɣ ɫɢɫɬɟɦɵ. ɉɨɫɥɟɞɧɹɹ ɧɟ ɞɨɥɠɧɚ ɛɵɬɶ ɤɢɧɟɦɚɬɢɱɟɫɤɢ ɧɟɢɡɦɟɧɹɟɦɨɣ, ɯɨɬɹ ɢ ɦɨɠɟɬ ɛɵɬɶ ɬɚɤɨɣ. ɗɬɚ ɦɟɬɨɞɢɤɚ ɟɫɬɟɫɬɜɟɧɧɵɦ ɨɛɪɚɡɨɦ ɪɚɡɜɢɜɚɟɬ ɦɟɬɨɞ ɫɟɱɟɧɢɣ ɜ ɫɬɚɬɢɱɟɫɤɢ ɨɩɪɟɞɟɥɢɦɵɯ ɡɚɞɚɱɚɯ ɧɚ ɡɚɞɚɱɢ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɵɟ ɢ ɢɦɟɟɬ, ɧɚ ɧɚɲ ɜɡɝɥɹɞ, ɪɹɞ ɩɪɟɢɦɭɳɟɫɬɜ ɩɟɪɟɞ ɬɪɚɞɢɰɢɨɧɧɨɣ.

5.1. ɉɨɧɹɬɢɟ "ɦɟɲɚɸɳɢɟ ɫɜɹɡɢ"

ȼɫɩɨɦɧɢɦ ɬɟɯɧɢɤɭ ɨɩɪɟɞɟɥɟɧɢɹ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥɨɜɵɯ ɮɚɤɬɨɪɨɜ (Ɏ) ɜ ɫɬɚɬɢɱɟɫɤɢ ɨɩɪɟɞɟɥɢɦɨɣ ɡɚɞɚɱɟ. Ⱦɥɹ ɷɬɨɝɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɦɟɬɨɞ ɫɟɱɟɧɢɣ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɨɬɨɪɵɦ ɧɟɨɛɯɨɞɢɦɨ:

xɪɚɡɪɟɡɚɬɶ ɫɬɟɪɠɟɧɶ;

xɨɬɛɪɨɫɢɬɶ ɨɞɧɭ ɢɡ ɱɚɫɬɟɣ ɜɦɟɫɬɟ ɫɨ ɫɜɹɡɹɦɢ ɢ ɞɟɣɫɬɜɭɸɳɢɦɢ ɧɚ ɧɟɟ ɫɢɥɚ-

ɦɢ;

xɡɚɦɟɧɢɬɶ ɟɟ ɞɟɣɫɬɜɢɟ ɜɧɭɬɪɟɧɧɢɦɢ ɫɢɥɨɜɵɦɢ ɮɚɤɬɨɪɚɦɢ ;

xɭɪɚɜɧɨɜɟɫɢɬɶ, ɬɨ ɟɫɬɶ ɡɚɩɢɫɚɬɶ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɨɫɬɚɜɲɟɣɫɹ ɱɚɫɬɢ, ɢɡ ɪɟɲɟɧɢɹ ɤɨɬɨɪɵɯ ɧɚɯɨɞɹɬɫɹ ɡɧɚɱɟɧɢɹ Ɏ.

ɋɬɭɞɟɧɬɵ ɢɧɨɝɞɚ ɤɨɞɢɪɭɸɬ ɷɬɭ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɥɢɬɟɪɚɦɢ ɊɈɁɍ.

Ɉɬɦɟɬɢɦ, ɱɬɨ ɨɫɬɚɜɲɚɹɫɹ ɱɚɫɬɶ ɬɟɥɚ ɧɟ ɞɨɥɠɧɚ ɢɦɟɬɶ ɫɜɹɡɟɣ ɫ "ɡɟɦɥɟɣ", ɢɧɚɱɟ ɡɚɩɢɫɚɬɶ ɧɟɨɛɯɨɞɢɦɵɟ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɧɟ ɭɞɚɫɬɫɹ. ɉɨɷɬɨɦɭ, ɧɟ ɜɫɟɝɞɚ ɨɬɱɟɬɥɢɜɨ ɷɬɨ ɮɨɪɦɭɥɢɪɭɹ, ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɩɪɨɢɡɜɨɞɹɬ ɜ ɞɜɚ ɷɬɚɩɚ:

x ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɣ (ɭɫɬɪɚɧɟɧɢɟ ɩɨɦɟɯ – ɫɧɹɬɢɟ ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ ɢ ɨɩɪɟɞɟɥɟɧɢɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɨɩɨɪɧɵɯ ɪɟɚɤɰɢɣ; ɡɞɟɫɶ ɢɫɩɨɥɶɡɭɟɬɫɹ ɬɨɬ ɠɟ ɦɟɬɨɞ ɊɈɁɍ);

x ɨɫɧɨɜɧɨɣ (ɨɩɪɟɞɟɥɟɧɢɟ Ɏ – ɩɨ ɧɚɡɜɚɧɧɨɣ ɜɵɲɟ ɫɯɟɦɟ).

Ʉɨɧɫɬɪɭɤɰɢɸ, ɩɨɥɭɱɟɧɧɭɸ ɢɡ ɡɚɞɚɧɧɨɣ ɡɚɞɚɱɢ ɩɨɫɥɟ ɫɧɹɬɢɹ ɜɫɟɯ ɧɚɝɪɭɡɨɤ ɢ ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ, ɧɚɡɨɜɟɦ, ɩɨ ɢɡɜɟɫɬɧɨɣ ɚɧɚɥɨɝɢɢ, ɨɫɧɨɜɧɨɣ ɫɢɫɬɟɦɨɣ ("Ɉ"). Ɂɚɞɚɱɭ, ɩɨɥɭɱɟɧɧɭɸ ɢɡ ɨɫɧɨɜɧɨɣ ɫɢɫɬɟɦɵ ɩɭɬɟɦ ɜɨɡɜɪɚɳɟɧɢɹ ɜɫɟɯ ɜɧɟɲɧɢɯ ɜɨɡɞɟɣɫɬɜɢɣ ɢ ɪɟɚɤɰɢɣ (ɜɨɡɦɨɠɧɨ, ɟɳɟ ɧɟ ɧɚɣɞɟɧɧɵɯ) ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ, ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɷɤɜɢɜɚɥɟɧɬɧɨɣ ("ɗ"). ɉɪɟɞɜɚɪɢɬɟɥɶɧɵɣ ɷɬɚɩ ɜɤɥɸɱɚɟɬ ɩɨɥɭɱɟɧɢɟ ɨɫɧɨɜɧɨɣ, ɡɚɬɟɦ ɷɤɜɢɜɚɥɟɧɬɧɨɣ ɫɢɫɬɟɦɵ, ɤɢɧɟɦɚɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ ɨɫɧɨɜɧɨɣ ɫɢɫɬɟɦɵ, ɡɚɩɢɫɶ ɧɚ ɨɫɧɨɜɟ ɩɨɫɥɟɞɧɟɝɨ ɭɫɥɨɜɢɣ ɪɚɜɧɨɜɟɫɢɹ ɢ ɪɟɲɟɧɢɟ ɷɬɢɯ ɭɪɚɜɧɟɧɢɣ – ɞɥɹ ɩɨɫɥɟɞɭɸɳɟɝɨ ɩɨɫɬɪɨɟɧɢɹ ɷɩɸɪ ɜ ɨɫɧɨɜɧɨɦ ɷɬɚɩɟ ɦɟɬɨɞɚ ɫɟɱɟɧɢɣ.

ɂɡ ɫɤɚɡɚɧɧɨɝɨ ɫɥɟɞɭɟɬ, ɱɬɨ "ɦɟɲɚɸɳɢɦɢ" ɹɜɥɹɸɬɫɹ ɫɜɹɡɢ, ɧɟ ɩɨɡɜɨɥɹɸɳɢɟ ɜ ɨɫɧɨɜɧɨɦ ɷɬɚɩɟ ɩɨɥɭɱɚɬɶ ɩɨɫɥɟ ɥɸɛɨɝɨ ɧɟɨɛɯɨɞɢɦɨɝɨ ɪɚɡɪɟɡɚ ɧɟ ɫɜɹɡɚɧɧɭɸ ɫ ɡɟɦɥɟɣ ɨɫɬɚɜɲɭɸɫɹ ɱɚɫɬɶ.

ɉɪɢɦɟɪ 1. Ɂɚɞɚɱɚ ɧɚ ɪɢɫ.5.1 ɜɩɨɥɧɟ ɝɨɬɨɜɚ ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɨɫɧɨɜɧɨɝɨ ɷɬɚɩɚ ɦɟɬɨɞɚ ɫɟɱɟɧɢɣ: ɩɪɨɜɟɞɹ ɥɸɛɨɟ ɫɟɱɟɧɢɟ, ɨɬɛɪɨɫɢɦ ɥɟɜɭɸ ɱɚɫɬɶ ɢ ɨɫɬɚɜɢɦ ɩɪɚɜɭɸ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ ɧɟɬ ɢ ɞɚɧɧɚɹ ɫɯɟɦɚ ɨɞɧɨɜɪɟɦɟɧɧɨ ɩɪɟɞɫɬɚɜɥɹɟɬ «ɷɤɜɢɜɚɥɟɧɬɧɭɸ» ɫɢɫɬɟɦɭ. ɇɨ ɜ ɡɚɞɚɱɟ ɧɚ ɪɢɫ.5.2ɚ ɷɬɨɝɨ

Ɋɢɫ.5.3

ɫɞɟɥɚɬɶ ɫɪɚɡɭ ɧɟɥɶɡɹ. ɇɟɨɛɯɨɞɢɦɨ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɧɚɣɬɢ ɥɢɛɨ ɥɟɜɭɸ ɨɩɨɪɧɭɸ ɪɟɚɤɰɢɸ (ɪɢɫ.5.2ɛ), ɥɢɛɨ ɩɪɚɜɭɸ (ɪɢɫ.5.2ɜ). Ɂɚɦɟɬɢɦ ɫɪɚɡɭ, ɱɬɨ ɦɨɠɧɨ ɨɬɛɪɨɫɢɬɶ ɨɛɟ ɫɜɹɡɢ: ɢɧɨɝɞɚ ɷɬɨ ɭɞɨɛɧɟɟ ɞɥɹ ɩɨɫɥɟɞɭɸɳɟɝɨ ɩɨɫɬɪɨɟɧɢɹ ɷɩɸɪ Ɏ. ɂ ɧɚɨɛɨɪɨɬ:

ɜ ɡɚɞɚɱɟ ɧɚ ɪɢɫ.5.3

P	P		P	P	ɫɜɹɡɢ ɧɟ ɦɟɲɚɸɬ
					ɩɨɫɬɪɨɢɬɶ	ɷɩɸɪɭ
		A			ɢɡɝɢɛɚɸɳɟɝɨ	ɦɨ-
Ɋɢɫ.5.1	ɚ)		ɛ)	ɜ)	ȼ ɦɟɧɬɚ M ɢ ɡɚɬɟɦ –
			Ɋɢɫ.5.2		ɩɨɩɟɪɟɱɧɨɣ	ɫɢɥɵ

PQ=dM/dz. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɧɹɬɢɟ ɦɟɲɚɸɳɟɣ ɫɜɹ-

ɡɢ ɜɟɫɶɦɚ ɫɜɨɛɨɞɧɨ. ɇɚɩɪɢɦɟɪ, ɞɥɹ ɡɚɞɚɱɢ ɪɢɫ.5.3 ɧɚ ɪɢɫ.5.4 ɩɨɤɚɡɚɧ ɪɹɞ ɷɤɜɢɜɚɥɟɧɬɧɵɯ ɡɚɞɚɱ (ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ ɡɞɟɫɶ ɢ ɞɚɥɟɟ ɨɛɨɡɧɚɱɚɸɬɫɹ ɥɚɬɢɧɫɤɢɦɢ ɛɭɤɜɚɦɢ). ɉɨɫɥɟɞɧɢɣ ɜɚɪɢɚɧɬ ɧɟɭɞɨɛɟɧ, ɧɨ ɜɨɡ-

Pɦɨɠɟɧ; ɨɧ ɩɨɤɚɡɵɜɚɸɬ ɛɨɥɶɲɭɸ ɫɜɨɛɨɞɭ ɩɨɧɹɬɢɹ

			ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ.
	A		Ɇɟɲɚɸɳɢɟ ɫɜɹɡɢ ɦɨɝɭɬ ɛɵɬɶ ɧɟ ɬɨɥɶɤɨ ɜɧɟɲ-
B		P	ɧɢɦɢ (ɫɜɹɡɵɜɚɸɳɢɦɢ ɫ "ɡɟɦɥɟɣ"), ɧɨ ɢ ɜɧɭɬɪɟɧɧɢ-
			ɦɢ.
			ɉɪɢɦɟɪ 2. ȼ ɡɚɞɚɱɟ ɧɚ ɪɢɫ.5.5ɚ ɨɬɛɪɚɫɵɜɚɧɢɟ
C		P	ɜɧɟɲɧɢɯ ɫɜɹɡɟɣ ɟɳɟ ɧɟ ɩɨɞɝɨɬɚɜɥɢɜɚɟɬ ɨɫɧɨɜɧɨɣ
B			ɷɬɚɩ ɦɟɬɨɞɚ ɫɟɱɟɧɢɣ: ɧɟɨɛɯɨɞɢɦɨ "ɪɚɡɨɦɤɧɭɬɶ" ɪɚ-
			ɦɭ. ɇɚ ɪɢɫ.5.5ɛ ɩɨɤɚɡɚɧɚ ɫɢɫɬɟɦɚ "ɗ" ɩɪɢ ɜɵɛɨɪɟ
C	A		ɦɢɧɢɦɚɥɶɧɨɝɨ ɱɢɫɥɚ ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ; ɧɚ ɪɢɫ.5.5ɜ
		F P	– ɛɨɥɟɟ ɭɞɨɛɧɚɹ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɷɩɸɪ, ɧɨ ɬɪɟɛɭɸ-
B	F		ɳɚɹ ɛɨɥɟɟ ɝɪɨɦɨɡɞɤɨɝɨ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɷɬɚɩɚ ɨɩ-
			ɪɟɞɟɥɟɧɢɹ ɪɟɚɤɰɢɣ ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ. Ɂɚɦɟɬɢɦ, ɱɬɨ
	E		ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ (ɤɚɤ ɢ ɜ ɧɟɤɨɬɨɪɵɯ ɞɪɭɝɢɯ) ɢ ɩɪɟɞ-
C	G	D	ɜɚɪɢɬɟɥɶɧɵɣ ɷɬɚɩ ɭɞɨɛɧɨ ɪɚɡɛɢɬɶ ɧɚ ɞɜɚ: ɜɧɚɱɚɥɟ
	Ɋɢɫ.5.4		ɨɬɛɪɨɫɢɬɶ ɬɨɥɶɤɨ ɜɧɟɲɧɸɸ ɫɜɹɡɶ ɢ ɧɚɣɬɢ ɪɟɚɤɰɢɸ
			A (ɪɢɫ.5.6, A=P), ɚ ɡɚɬɟɦ ɫɧɹɬɶ ɜɧɭɬɪɟɧɧɢɟ ɦɟ-

ɲɚɸɳɢɟ ɫɜɹɡɢ. ȼɚɪɢɚɧɬɵ ɬɚɤɢɯ ɷɤɜɢɜɚɥɟɧɬɧɵɯ ɫɢɫɬɟɦ ɩɨɤɚɡɚɧɵ ɧɚ ɪɢɫ.5.5ɛ,ɜ;

P		P+C	B		P+C B
		B			B
					ɜ)	D
	ɚ)	C		ɛ)	C
					E
					E	D
				A

A

Ɋɢɫ. 5.5

ɜɨɡɦɨɠɧɨ ɟɳɟ ɛɨɥɟɟ ɞɪɨɛɧɨɟ ɞɟɥɟɧɢɟ – ɞɥɹ ɬɟɯ, ɤɬɨ ɧɟ ɨɲɢɛɚɟɬɫɹ ɜ ɡɚɩɢɫɢ ɢ ɪɟɲɟɧɢɢ ɭɫɥɨɜɢɣ ɪɚɜɧɨɜɟɫɢɹ, ɧɨ ɧɟ ɥɸɛɢɬ ɫɬɪɨɢɬɶ ɷɩɸɪɵ Ɏ ɜ ɝɪɨɦɨɡɞɤɢɯ ɪɚɦɚɯ.

Ⱦɨɛɚɜɢɦ ɞɜɟ ɪɟɤɨɦɟɧɞɚɰɢɢ, ɨɛɥɟɝɱɚɸɳɢɟ ɪɟɲɟɧɢɟ:

	P		1. ɉɪɢ ɪɚɫɱɟɬɚɯ ɮɟɪɦ (ɬɨ ɟɫɬɶ ɫɬɟɪɠɧɟɜɵɯ ɤɨɧɫɬɪɭɤɰɢɣ,
	l		ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɵɯ ɪɚɛɨɬɚɸɬ ɜ ɭɫɥɨɜɢɹɯ ɪɚɫɬɹɠɟɧɢɹ-ɫɠɚɬɢɹ)
		2l	ɪɟɤɨɦɟɧɞɭɟɬɫɹ ɩɨɥɭɱɚɬɶ ɫɢɫɬɟɦɭ "ɗ" ɫɥɟɞɭɸɳɢɦ ɫɬɚɧɞɚɪɬ-
	l l		ɧɵɦ ɫɩɨɫɨɛɨɦ: ɪɚɡɪɟɡɚɬɶ ɜɫɟ ɫɬɟɪɠɧɢ. Ɍɨɝɞɚ ɩɪɟɞɜɚɪɢɬɟɥɶ-
		l	ɧɵɣ ɷɬɚɩ ɹɜɥɹɟɬɫɹ ɨɞɧɨɜɪɟɦɟɧɧɨ ɨɫɧɨɜɧɵɦ, ɩɨɫɤɨɥɶɤɭ ɪɟɚɤ-
			ɰɢɹɦɢ ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ ɹɜɥɹɸɬɫɹ ɢɫɤɨɦɵɟ ɡɧɚɱɟɧɢɹ ɧɨɪ-

	A	ɦɚɥɶɧɵɯ ɫɢɥ (ɪɢɫ.5.7).
		Ⱦɨɛɚɜɢɦ: ɱɬɨɛɵ ɧɟ ɛɵɥɨ ɨɲɢɛɨɤ ɜ ɡɧɚɤɚɯ, ɜɫɟ ɪɟɚɤɰɢɢ Ni
	Ɋɢɫ.5.6
		(i – ɧɨɦɟɪ ɫɬɟɪɠɧɹ) ɧɟɨɛɯɨɞɢɦɨ ɧɚɩɪɚɜɥɹɬɶ ɜ ɫɬɨɪɨɧɭ ɪɚɫɬɹ-

ɠɟɧɢɹ, ɞɚɠɟ ɟɫɥɢ ɨɱɟɜɢɞɧɨ, ɱɬɨ ɫɬɟɪɠɟɧɶ ɪɚɛɨɬɚɟɬ ɧɚ ɫɠɚɬɢɟ. Ɋɟɲɟɧɢɟ ɞɚɟɬ ɧɟ ɬɨɥɶɤɨ ɚɛɫɨɥɸɬɧɵɟ ɡɧɚɱɟɧɢɹ, ɧɨ ɢ ɡɧɚɤɢ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ Ni.

P	P
	"ɗ"
N2	N4
	N3
N1	N4

2P
Ɋɢɫ. 5.7	2P

2. ȿɫɥɢ ɡɚɞɚɱɚ (ɤɨɧɫɬɪɭɤɰɢɹ ɢ ɜɨɡɞɟɣɫɬɜɢɟ) ɫɢɦɦɟɬɪɢɱɧɚ, ɬɨ "ɗ" ɫɥɟɞɭɟɬ ɜɵɛɢɪɚɬɶ ɬɚɤɠɟ ɫɢɦɦɟɬɪɢɱɧɨɣ; ɟɫɥɢ ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɚ, ɬɨ ɫɥɟɞɭɟɬ ɫɨɯɪɚɧɹɬɶ ɤɨɫɭɸ ɫɢɦɦɟɬɪɢɸ.

ɉɪɢɦɟɪ 3. Ɋɚɦɚ, ɩɨɤɚɡɚɧɧɚɹ ɧɚ ɪɢɫ.5.8, ɫɢɦɦɟɬɪɢɱɧɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɪɚɳɟɧɢɹ ɜɨɤɪɭɝ ɬɨɱɤɢ C ɧɚ ɭɝɨɥ S. ɇɚɝɪɭɡɤɚ ɩɪɢ ɷɬɨɦ ɦɟɧɹɟɬ ɧɚɩɪɚɜɥɟɧɢɟ, ɡɧɚɱɢɬ ɡɚɞɚɱɚ

ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɚ.		ɋɥɟɞɨɜɚɬɟɥɶɧɨ,	ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɵ ɢ		ɨɩɨɪɧɵɟ	ɪɟɚɤɰɢɢ
	Ⱥ			Ⱥ	(ɪɢɫ.5.8ɛ), ɨɬɤɭ-
					ɞɚ ɥɟɝɤɨ		ɭɫɬɚ-
		ȼ		ȼ	ɧɨɜɢɬɶ,		ɱɬɨ
					A=P/2,		B=0.
	ɋ	Ɋ ɋ		Ɋ ɋ
Ɋ					Ɉɬɦɟɬɢɦ,		ɱɬɨ
				Ⱥ	ɟɫɥɢ ɩɪɢ ɜɵɛɨɪɟ
					ɷɤɜɢɜɚɥɟɧɬɧɨɣ
			ȼ		ɫɢɫɬɟɦɵ		ɧɚɪɭ-
	ɚ)	ɛ)		ɜ)	ɲɢɬɶ	ɭɫɥɨɜɢɟ
					ɫɢɦɦɟɬɪɢɢ		(ɧɚ
		Ɋɢɫ. 5.8
					ɪɢɫ.5.8ɜ		ɦɟ-

ɲɚɸɳɢɟ ɫɜɹɡɢ ɫɧɹɬɵ, ɢ ɜ ɷɬɨɦ ɫɦɵɫɥɟ ɫɢɫɬɟɦɚ "ɗ" ɞɨɩɭɫɬɢɦɚ), ɬɨ ɡɚɞɚɱɚ ɨɤɚɠɟɬɫɹ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɨɣ ɢ ɟɟ ɪɟɲɟɧɢɟ ɩɨɬɪɟɛɭɟɬ ɨɫɨɛɨɝɨ ɦɟɬɨɞɚ (ɧɚɩɪɢɦɟɪ, ɦɟɬɨɞɚ ɫɢɥ), ɱɬɨɛɵ ɩɨɥɭɱɢɬɶ ɜ ɢɬɨɝɟ ɬɨɬ ɠɟ ɨɬɜɟɬ.

ɋɜɨɛɨɞɧɵɣ ɜɵɛɨɪ ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ ɬɪɟɛɭɟɬ ɨɫɨɛɟɧɧɨ ɜɧɢɦɚɬɟɥɶɧɨɝɨ ɨɬɧɨɲɟɧɢɹ ɤ ɜɵɛɨɪɭ ɭɫɥɨɜɢɣ ɪɚɜɧɨɜɟɫɢɹ. ȿɫɥɢ ɨɬɛɪɨɫɢɬɶ ɜɫɟ ɫɜɹɡɢ, ɬɨ ɦɨɠɧɨ ɡɚɩɢɫɵɜɚɬɶ ɛɟɡ ɪɚɡɞɭɦɢɣ ɨɛɵɱɧɵɟ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ (ɫɭɦɦɵ ɩɪɨɟɤɰɢɣ ɫɢɥ, ɦɨɦɟɧɬɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɢɡɜɨɥɶɧɵɯ ɨɫɟɣ), ɧɨ ɟɫɥɢ ɨɝɪɚɧɢɱɢɬɶɫɹ ɧɟɨɛɯɨɞɢɦɵɦ ɦɢɧɢɦɭɦɨɦ, ɬɨ ɫɨɜɟɪɲɟɧɧɨ ɨɛɹɡɚɬɟɥɟɧ ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɣ ɤɢɧɟɦɚɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ ɦɟɯɚɧɢɡɦɚ, ɩɨɥɭɱɚɸɳɟɝɨɫɹ ɩɨɫɥɟ ɫɧɹɬɢɹ ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ: ɧɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟɥɢɬɶ ɤɨɥɢɱɟɫɬɜɨ ɢ ɤɚɱɟɫɬɜɨ ɩɨɥɭɱɟɧɧɵɯ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ. ɋɥɟɞɭɟɬ ɩɨɦɧɢɬɶ, ɱɬɨ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ – ɷɬɨ ɜɫɟɝɞɚ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɦɟɯɚɧɢɡɦɚ, ɬɨ ɟɫɬɶ ɤɨɧɫɬɪɭɤɰɢɢ, ɢɦɟɸɳɟɣ ɜɨɡɦɨɠɧɨɫɬɶ ɩɟɪɟɦɟɳɚɬɶɫɹ, ɧɨ ɧɟ ɩɟɪɟɦɟɳɚɸɳɟɣɫɹ ɢɦɟɧɧɨ ɜɫɥɟɞɫɬɜɢɟ ɪɚɜɟɧɫɬɜɚ ɧɭɥɸ ɞɜɢɠɭɳɟɣ ɧɚɝɪɭɡɤɢ. ȼ ɫɢɦɦɟɬɪɢɱɧɨɣ ɡɚɞɚɱɟ ɩɪɢ ɫɢɦɦɟɬɪɢɱɧɨɣ "ɗ" ɫɥɟɞɭɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɫɢɦɦɟɬɪɢɱɧɵɟ ɞɜɢɠɟɧɢɹ (ɢ ɡɚɩɢɫɵɜɚɬɶ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ), ɜ ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɨɣ – ɬɨɥɶɤɨ ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɵɟ.

ȼ ɩɨɫɥɟɞɧɟɣ ɡɚɞɚɱɟ (ɪɢɫ.5.8) ɫɢɦɦɟɬɪɢɱɧɵɦ ɞɜɢɠɟɧɢɟɦ "ɗ" (ɪɢɫ.5.8ɛ) ɹɜɥɹɟɬɫɹ ɬɨɥɶɤɨ ɩɨɜɨɪɨɬ ɜɨɤɪɭɝ ɬɨɱɤɢ ɋ; ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɵɦ – ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɜ ɥɸɛɨɦ ɧɚɩɪɚɜɥɟɧɢɢ. ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɵɟ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ – ɪɚɜɟɧɫɬɜɨ ɧɭɥɸ ɫɭɦɦɵ ɩɪɨɟɤɰɢɣ ɫɢɥ ɧɚ ɞɜɟ ɨɫɢ.

5.2. ɗɤɜɢɜɚɥɟɧɬɧɚɹ ɫɢɫɬɟɦɚ ɜ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɵɯ ɡɚɞɚɱɚɯ

Ɇɨɠɟɬ ɩɨɤɚɡɚɬɶɫɹ, ɱɬɨ ɩɨɧɹɬɢɟ ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ ɞɨɫɬɚɬɨɱɧɨ ɨɱɟɜɢɞɧɨ ɢ ɧɟ ɡɚɫɥɭɠɢɜɚɟɬ ɫɬɨɥɶɤɢɯ ɫɥɨɜ (ɨɧɨ ɩɪɚɤɬɢɱɟɫɤɢ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜɫɟɝɞɚ, ɬɨɥɶɤɨ, ɦɨɠɟɬ ɛɵɬɶ, ɛɟɡ ɨɬɱɟɬɥɢɜɨɣ ɮɨɪɦɭɥɢɪɨɜɤɢ). ɗɬɢ ɫɥɨɜɚ ɛɵɥɢ ɩɨɞɝɨɬɨɜɤɨɣ ɤ ɬɨɦɭ, ɱɬɨɛɵ ɭɛɟɞɢɬɶ ɱɢɬɚɬɟɥɹ ɧɟ ɦɟɧɹɬɶ ɬɟɯɧɢɤɢ ɪɚɫɱɟɬɚ ɩɪɢ ɩɟɪɟɯɨɞɟ ɤ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɵɦ ɫɢɫɬɟɦɚɦ. ɉɪɢɧɢɦɚɟɦɨɟ ɨɛɵɱɧɨ ɬɪɟɛɨɜɚɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɧɟɢɡɦɟɧɹɟɦɨɫɬɢ ɷɤɜɢɜɚɥɟɧɬɧɨɣ ɫɢɫɬɟɦɵ ɜ ɦɟɬɨɞɟ ɫɢɥ – ɧɟ ɧɟɨɛɯɨɞɢɦɨɫɬɶ, ɚ ɩɪɨɫɬɨ ɬɪɚɞɢɰɢɹ. Ƚɨɪɚɡɞɨ ɭɞɨɛɧɟɟ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɷɤɜɢɜɚɥɟɧɬɧɨɣ ɡɚɞɚɱɢ ɫɧɢɦɚɬɶ ɧɟ "ɥɢɲɧɢɟ", ɚ ɦɟɲɚɸɳɢɟ ɫɜɹɡɢ. ȼɨɩɪɨɫ ɨ ɫɬɟɩɟɧɢ ɫɬɚɬɢɱɟɫɤɨɣ ɧɟɨɩɪɟɞɟɥɢɦɨɫɬɢ ɨɬɤɥɚɞɵɜɚɟɬɫɹ ɩɪɢ ɷɬɨɦ ɞɨ ɬɨɝɨ ɦɨɦɟɧɬɚ, ɤɨɝɞɚ ɷɬɨ ɛɭɞɟɬ ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɧɟɨɛɯɨɞɢɦɨ (ɜ ɡɚɞɚɱɟ ɧɚ ɪɢɫ.5.8 ɷɬɨɝɨ ɧɟ ɩɨɧɚɞɨɛɢɥɨɫɶ ɜɨɨɛɳɟ; ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɟɟ ɢɥɢ ɧɟ ɫɱɢɬɚɬɶ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɨɣ).

ɂɬɚɤ, ɩɨ ɫɯɟɦɟ, ɨɩɢɫɚɧɧɨɣ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɭɧɤɬɟ, ɜɵɩɨɥɧɹɟɦ ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɣ ɷɬɚɩ ɦɟɬɨɞɚ ɫɟɱɟɧɢɣ:

xɨɬɛɪɚɫɵɜɚɟɦ ɦɟɲɚɸɳɢɟ ɫɜɹɡɢ,

xɡɚɦɟɧɹɟɦ ɢɯ ɪɟɚɤɰɢɹɦɢ,

xɩɪɨɢɡɜɨɞɢɦ ɤɢɧɟɦɚɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ ɩɨɥɭɱɟɧɧɨɝɨ ɦɟɯɚɧɢɡɦɚ,

xɡɚɩɢɫɵɜɚɟɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ.

Ⱦɚɥɟɟ ɜɨɡɦɨɠɧɵ ɫɥɟɞɭɸɳɢɟ ɫɢɬɭɚɰɢɢ:

ɚ). Ɋɟɚɤɰɢɢ ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɭɫɥɨɜɢɣ ɪɚɜɧɨɜɟɫɢɹ ɨɞɧɨɡɧɚɱɧɨ (ɜɵɪɚɠɚɸɬɫɹ ɱɟɪɟɡ ɩɚɪɚɦɟɬɪɵ ɜɧɟɲɧɟɣ ɧɚɝɪɭɡɤɢ).

ɛ). Ⱦɥɹ ɢɯ ɨɩɪɟɞɟɥɟɧɢɹ ɧɟɯɜɚɬɚɟɬ ɟɳɟ k ɭɫɥɨɜɢɣ. Ɍɨɝɞɚ ɜɜɨɞɹɬ k ɩɪɨɢɡɜɨɥɶɧɵɯ ɜɟɥɢɱɢɧ x1 .. xk, ɧɚɡɵɜɚɟɦɵɯ ɧɟɢɡɜɟɫɬɧɵɦɢ ɦɟɬɨɞɚ ɫɢɥ. Ɋɟɚɤɰɢɢ ɫɜɹɡɟɣ ɜɵɪɚɠɚɸɬ ɱɟɪɟɡ ɩɚɪɚɦɟɬɪɵ ɜɧɟɲɧɟɣ ɧɚɝɪɭɡɤɢ ɢ ɧɟɢɡɜɟɫɬɧɵɟ xi. ɑɢɫɥɨ ɧɟɢɡɜɟɫɬɧɵɯ

k ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɫɬɟɩɟɧɶɸ ɫɬɚɬɢɱɟɫɤɨɣ ɧɟɨɩɪɟɞɟɥɢɦɨɫɬɢ. Ɉɧɨ ɧɟ ɜɫɟɝɞɚ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɪɢɧɹɬɨɦɭ ɜ ɭɱɟɛɧɢɤɚɯ ɬɟɪɦɢɧɭ.

ɜ). ɍɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɧɟ ɜɵɩɨɥɧɹɸɬɫɹ ɧɢ ɩɪɢ ɤɚɤɢɯ ɡɧɚɱɟɧɢɹɯ ɪɟɚɤɰɢɣ ɦɟɲɚɸɳɢɯ ɫɜɹɡɟɣ; ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ ɧɟ ɢɦɟɟɬ ɪɟɲɟɧɢɹ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɦɵ ɢɦɟɟɦ ɞɟɥɨ ɫ ɦɟɯɚɧɢɡɦɨɦ, ɧɟ ɧɚɯɨɞɹɳɢɦɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ ɩɪɢ ɞɚɧɧɨɣ ɧɚɝɪɭɡɤɟ. ɗɬɨ – ɞɢɧɚɦɢɱɟɫɤɚɹ ɡɚɞɚɱɚ.

ɝ). ɍɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜɵɩɨɥɧɹɸɬɫɹ ɬɨɥɶɤɨ ɩɪɢ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɜɧɟɲɧɢɯ ɫɢɥ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɦɵ ɢɦɟɟɦ ɞɟɥɨ ɫ ɦɟɯɚɧɢɡɦɨɦ, ɧɚɯɨɞɹɳɢɦɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ.

ɞ). ɍɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜɵɩɨɥɧɹɸɬɫɹ ɬɨɥɶɤɨ ɩɪɢ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɜɧɟɲɧɢɯ ɫɢɥ, ɧɨ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɪɟɚɤɰɢɢ ɧɟɯɜɚɬɚɟɬ k ɭɫɥɨɜɢɣ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɦɵ ɢɦɟɟɦ ɞɟɥɨ ɫɨ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɵɦ (ɫɬɟɩɟɧɶ ɧɟɨɩɪɟɞɟɥɢɦɨɫɬɢ k) ɦɟɯɚɧɢɡɦɨɦ, ɧɚɯɨɞɹɳɢɦɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ.

ɋɥɭɱɚɣ ɚ) ɪɚɫɫɦɚɬɪɢɜɚɥɫɹ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɭɧɤɬɟ (ɫɦ., ɧɚɩɪɢɦɟɪ, ɡɚɞɚɱɭ ɧɚ ɪɢɫ.5.8).

ɋɥɭɱɚɣ ɛ) ɜɨɡɧɢɤɚɟɬ ɜ ɬɨɣ ɠɟ ɪɚɦɟ, ɟɫɥɢ ɟɟ ɧɚɝɪɭɡɢɬɶ ɫɢɦɦɟɬɪɢɱɧɨ (ɪɢɫ.5.9).

l		Ⱥ	l			Ɂɞɟɫɶ ɟɞɢɧɫɬɜɟɧɧɨɟ ɫɢɦɦɟɬɪɢɱɧɨɟ
	l			l		ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ – ɫɭɦɦɚ ɦɨɦɟɧ-
Ɇ		ȼ	Ɇ			ɬɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ C – ɜɤɥɸɱɚ-
	ɋ				ɋ	ɟɬ ɞɜɟ ɧɟɢɡɜɟɫɬɧɵɟ Mc=M+2Bl –
	l			l	ȼ	2Al= 0 ɢ ɢɦɟɟɬ ɛɟɫɤɨɧɟɱɧɨɟ ɦɧɨɠɟ-
					Ⱥ	ɫɬɜɨ ɪɟɲɟɧɢɣ, ɧɚɩɪɢɦɟɪ, A=X1, B=
	l				l	X1 –	–M/(2l), ɢɥɢ A=M/l+X1,
		Ɋɢɫ. 5.9				B=	=M/(2l)+X1.
							ȼɨɡɦɨɠɧɵ ɦɧɨɝɢɟ ɞɪɭɝɢɟ ɪɟ-

ɲɟɧɢɹ, ɤɚɠɞɨɟ ɢɡ ɤɨɬɨɪɵɯ ɞɨɥɠɧɨ ɜɤɥɸɱɚɬɶ ɧɟɤɨɬɨɪɭɸ (ɫɜɨɸ) ɩɪɨɢɡɜɨɥɶɧɭɸ ɩɟɪɟɦɟɧɧɭɸ X1.

ɋɢɬɭɚɰɢɹ ɜ) ɜɨɡɧɢɤɚɟɬ, ɧɚɩɪɢɦɟɪ, ɜ ɮɟɪɦɟ, ɩɪɢɜɟɞɟɧɧɨɣ ɧɚ ɪɢɫ.5.10 (ɜ ɮɟɪɦɟ ɞɟɜɹɬɶ, ɚ ɧɟ ɞɜɟɧɚɞɰɚɬɶ ɫɬɟɪɠɧɟɣ!). ɋ ɭɱɟɬɨɦ ɫɢɦɦɟɬɪɢɢ ɢ ɪɟɤɨɦɟɧɞɚɰɢɢ ɞɥɹ ɮɟɪɦ (ɪɚɡɪɟɡɚɬɶ ɜ "ɗ" ɜɫɟ ɫɬɟɪɠɧɢ) ɩɨɥɭɱɚɟɦ "ɗ" ɜ ɜɢɞɟ ɩɚɪɵ ɮɪɚɝɦɟɧɬɨɜ (ɪɢɫ.5.10ɛ), ɩɨɜɬɨɪɹɸɳɢɯɫɹ ɬɪɢɠɞɵ. ɇɢ ɩɪɢ ɤɚɤɨɦ ɡɧɚɱɟɧɢɢ ɫɢɥɵ P (ɤɪɨɦɟ P= 0) ɪɚɜɧɨɜɟɫɢɟ ɨɛɨɢɯ ɮɪɚɝɦɟɧɬɨɜ ɧɟ ɞɨɫɬɢɠɢɦɨ, ɱɬɨ ɫɥɟɞɭɟɬ ɢɡ ɞɜɭɯ ɭɫɥɨɜɢɣ ɪɚɜɧɨɜɟɫɢɹ:

N1+N2=0,	(5.1)
N1+N2+P= 0.	(5.2)

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɚɧɧɚɹ ɮɟɪɦɚ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɦɟɯɚɧɢɡɦ (ɨɫɬɚɜɢɦ ɱɢɬɚɬɟɥɸ ɟɝɨ ɤɢɧɟɦɚɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ: ɨɩɪɟɞɟɥɢɬɟ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɫɬɟɪɠɧɟɣ). ɗɬɨɬ ɦɟɯɚɧɢɡɦ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦ. ɗɬɨ, ɜ ɱɚɫɬɧɨɫɬɢ, ɫɥɟɞɭɟɬ ɢɡ ɬɨɝɨ, ɱɬɨ ɩɪɢ ɞɪɭɝɨɣ ɧɚɝɪɭɡɤɟ, ɫ ɛɨɥɟɟ ɜɵɫɨɤɨɣ ɫɬɟɩɟɧɶɸ ɫɢɦɦɟɬɪɢɢ (ɪɢɫ.5.10ɜ) ɢɦɟɟɬɫɹ ɟɞɢɧɫɬɜɟɧɧɨɟ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ (5.2). ȿɝɨ ɪɟɲɟɧɢɟ ɦɨɠɟɬ ɢɦɟɬɶ, ɧɚɩɪɢɦɟɪ, ɜɢɞ

N1=X1, N2=P –X1 – (ɫɥɭɱɚɣ ɞ).