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5.2. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ

r, R

Ɂɚɞɚɱɚ 1.

ȼɵɜɟɫɬɢ ɮɨɪɦɭɥɭ ɞɥɹ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɦɭɮɬɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɫɢɦɦɟɬɪɢɢ. Ɇɚɫɫɚ ɦɭɮɬɵ ɬ, ɜɧɭɬɪɟɧɧɢɣ ɢ ɜɧɟɲɧɢɣ ɪɚɞɢɭɫɵ r ɢ R, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.

Ⱦɚɧɨ:

ɬ,

I – ?

Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɹɜɥɹɟɬɫɹ ɚɞɞɢɬɢɜɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ

ɜɪɚɳɚɸɳɟɝɨɫɹ ɬɟɥɚ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɜɫɟɝɨ ɬɟɥɚ

Ɋɟɲɟɧɢɟ:

ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ ɟɝɨ ɱɚɫɬɟɣ.

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, I0 = I + I1, ɝɞɟ I0 – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɫɩɥɨɲɧɨɝɨ ɰɢɥɢɧɞɪɚ ɪɚɞɢɭɫɚ R ɢ ɦɚɫɫɵ ɬ ɫ ɦɨɦɟɧɬɨɦ ɢɧɟɪɰɢɢ I

Ⱦɥɹ ɬɟɥɚ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɮɨɪɦɵ:

Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɦɚɫɫ ɬ

, ɫɨɫɬɨɹɳɟɝɨ ɢɡ ɦɭɮɬɵ ɢ ɰɢɥɢɧɞɪɚ ɪɚɞɢɭɫɚ r ɢ ɦɚɫɫɵ ɬ1

. Ɍɨɝɞɚ I = I0 – I1.

ɢ ɬ1 ɜɵɱɢɫɥɢɦ ɩɥɨɬɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ, ɢɡ ɤɨɬɨɪɨɝɨ

;

ɢɡɝɨɬɨɜɥɟɧɚ ɦɭɮɬɚ:

;()

USSS

 

VhRhr hRr

22 22

Ɉɤɨɧɱɚɬɟɥɶɧɨ:

22 22 2222 22

Rm R rm r mR r R r mR r





22 22 22

()2()2 2() 2

Rr Rr Rr

  

Ɂɚɞɚɱɚ 2.

Ɉɩɪɟɞɟɥɢɬɶ ɦɚɫɫɭ ɨɞɧɨɪɨɞɧɨɝɨ ɫɬɟɪɠɧɹ, ɟɫɥɢ ɩɪɢ ɫɦɟɳɟɧɢɢ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɰɟɧɬɪ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɨɫɢ ɫɬɟɪɠɧɹ, ɪɚɫɫɬɨɹɧɢɟ

'Ɛ = 10 ɫɦ, ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɫɬɪɟɠɧɹ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɧɚ 'I = 100 ɤɝɦ

()

hR r

; h – ɜɵɫɨɬɚ ɦɭɮɬɵ,



mRh Rm



22 22

()

hR r R r



mrh rm



22 22

()

hR r R r



()()()

Ɉɬɜɟɬ:

)(

rRmI

Ⱦɚɧɨ:

Ɛ = 10 ɫɦ

I = 100 ɤɝɦ2

Ɍ – ?

Ɍɟɨɪɟɦɚ ɒɬɟɣɧɟɪɚ ɩɨɡɜɨɥɹɟɬ ɧɚɣɬɢ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɸɛɨɣ ɨɫɢ (I), ɩɚɪɚɥɥɟɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨ- ɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ, ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ (I

) ɢɡɜɟɫɬɟɧ, d – ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɨɫɹɦɢ.

Ɋɟɲɟɧɢɟ:

I = I0 + md2.

ȼ ɧɚɲɟɦ ɫɥɭɱɚɟ:

'I = I – I

'I = m'Ɛ

= md2, d = 'Ɛ.

;

0,1

100 ɤɝ.

Ɉɬɜɟɬ: 100 ɤɝ.

Ɂɚɞɚɱɚ 3. ɇɚ ɛɚɪɚɛɚɧ ɪɚɞɢɭɫɚ R = 0,5 ɦ ɧɚɦɨɬɚɧ ɲɧɭɪ, ɤ ɤɨɧɰɭ ɤɨɬɨɪɨɝɨ

ɩɪɢɜɹɡɚɧ ɝɪɭɡ ɦɚɫɫɨɣ ɬ = 10 ɤɝ. ɇɚɣɬɢ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɛɚɪɚɛɚɧɚ, ɟɫɥɢ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɝɪɭɡ ɨɩɭɫɤɚɟɬɫɹ ɫ ɭɫɤɨɪɟɧɢɟɦ ɚ = 2,04 ɦ/ɫ

Ⱦɚɧɨ:

R = 0,5 ɦ

ɬ = 10 ɤɝ ɚ = 2,04 ɦ/ɫ

I – ?

Ɂɚɞɚɱɚ ɩɨ ɞɢɧɚɦɢɤɟ ɫɢɫɬɟɦɵ ɬɟɥ, ɨɞɧɨ ɢɡ ɤɨɬɨɪɵɯ (ɛɚɪɚɛɚɧ) ɫɨɜɟɪɲɚɟɬ ɜɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ, ɚ ɜɬɨɪɨɟ (ɝɪɭɡ) ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ. ȼɵɩɨɥɧɢɦ ɱɟɪɬɟɠ (ɪɢɫ. 5.7) ɢ ɡɚɩɢɲɟɦ ɫɢɫɬɟɦɭ ɢɡ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ ɨɛɨɢɯ ɬɟɥ:

– ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɛɚɪɚɛɚɧɚ.

Ɇɨɦɟɧɬ ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɛɚɪɚɛɚɧ ɫɢɥɵ F

OY:

ȼɟɥɢɱɢɧɵ

ɢ ɚ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ ɚ = HR;



Ia F R



ma mg F



( ) 10(9,81 2, 04) 0,5

mg aR





ɧɚɬ

Ɋɟɲɟɧɢɟ:

 ®

ma mg F



ɧɚɬ

Ɋɢɫ. 5.7

ɪɚɜɟɧ Ɇ = F

ɧɚɬ

;

 

 ® ¯

ɧɚɬ

ma mg F



ɧɚɬ

ɢɥɢ Ia = m(g – a)  R2.

2, 04

 R, R – ɩɥɟɱɨ ɫɢɥɵ.

ɧɚɬ

, ɟɫɥɢ ɧɢɬɶ ɧɟɪɚɫɬɹɠɢɦɚ.

9, 52 ɤɝ ɦ .

Ɉɬɜɟɬ: 9,52 ɤɝāɦ².

Ɂɚɞɚɱɚ 4. ɉɨɥɧɚɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɞɢɫɤɚ, ɤɚɬɹɳɟɝɨɫɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵ-

Ⱥ –

ɜɚɧɢɹ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɪɚɜɧɚ 24 Ⱦɠ. Ɉɩɪɟɞɟɥɢɬɶ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɞɢɫɤɚ.

Ɋɟɲɟɧɢɟ:

Ɍ Ɍ

Ⱦɚɧɨ:

ɩɨɥɧ

ɩɨɫɬ

– ?

ɜɪ

= 24 Ⱦɠ

– ?

1. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɩɨɥɧɚɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɩɥɨɫɤɨɝɨ

ɞɜɢɠɟɧɢɹ ɞɢɫɤɚ ɪɚɜɧɚ ɫɭɦɦɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɞɢɫɤɚ ɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɟɝɨ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɞɢɫɤɚ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɟɝɨ ɩɥɨɫɤɨɫɬɢ:

= Ɍ

ɩɨɥɧ

ɬ I

ɌɌ

ɩɨɫɬ ɜɪ

+ Ɍɜɪ.

ɩɨɫɬ

2. ɉɪɢ ɞɜɢɠɟɧɢɢ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɥɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɬɨɱɟɤ ɬɨɪɰɚ

ɞɢɫɤɚ ɪɚɜɧɚ:

= ZR; Z = X/R.

3. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɫɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɨɣ ɨɫɢ

ɜɪɚɳɟɧɢɹ ɪɚɜɟɧ:

I = ½mR

4. Ɉɤɨɧɱɚɬɟɥɶɧɨ ɢɡ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɪɚɫɱɟɬɚ ɩɨɥɧɨɣ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ

ɞɢɫɤɚ ɩɨɥɭɱɢɦ ɢɫɤɨɦɵɟ ɮɨɪɦɭɥɵ ɞɥɹ ɪɚɫɱɟɬɚ:

222

ɬɬR

Ɍ T ɌɌ

 

ɩɨɥɧ ɩɨɫɬ ɩɨɫɬ ɩɨɫɬ

22 2 2 2

ɩɨɫɬ

= Ɍ

ɜɪ



= 2/3Ɍ

– Ɍ

ɩɨɥɧ

= 2/324 = 16 Ⱦɠ.

ɩɨɥɧ

= 24 – 16 = 8 Ⱦɠ.

ɩɨɫɬ

Ɉɬɜɟɬ: 8 Ⱦɠ.

Ɂɚɞɚɱɚ 5. Ɇɟɞɧɵɣ ɲɚɪ ɪɚɞɢɭɫɚ R = 10 ɫɦ ɜɪɚɳɚɟɬɫɹ ɫ ɱɚɫɬɨɬɨɣ

= 2 ɫ–1 ɜɨɤɪɭɝ ɨɫɢ ɫɢɦɦɟɬɪɢɢ. Ʉɚɤɭɸ ɪɚɛɨɬɭ ɧɭɠɧɨ ɫɨɜɟɪɲɢɬɶ, ɱɬɨɛɵ ɭɜɟɥɢɱɢɬɶ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɲɚɪɚ ɜɞɜɨɟ?

Ⱦɚɧɨ:

R = 10 ɫɦ

= 2 ɫ–1

= 2

ɌZ

ɜɪ

ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ, ɫɜɹɡɵɜɚɸɳɢɦ ɪɚɛɨɬɭ ɜɧɟɲ-

ɧɟɣ ɫɢɥɵ ɢ ɢɡɦɟɧɟɧɢɟ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ Ⱥ =

ȼ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ȿ

; I – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɲɚɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɢɣ ɱɟɪɟɡ

ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɲɚɪɚ (I = 2/5ɬR

ɦɟɯ

= Ɍɜɪ.

Ɋɟɲɟɧɢɟ:

ɦɟɯ

ɍɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɲɚɪɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:

= 2SQ;

= 2

Ɇɚɫɫɚ ɲɚɪɚ ɪɚɜɧɚ: ɬ =

Uɦ

V = 4/3SR

Ɉɤɨɧɱɚɬɟɥɶɧɨ ɞɥɹ ɪɚɫɱɟɬɚ ɢɫɤɨɦɨɣ ɪɚɛɨɬɵ ɩɨɥɭɱɚɟɦ ɜɵɪɚɠɟɧɢɟ:

222 2 22

ImR mR

()



3222 352

34 4 16

RR R

SU SQ S UQ



2525

ɦɦ

35 5 5



2(4 )3 4

ZZ SQ



11 1



16 3,14 0,1 8900 2

35 2



35,3 Ⱦɠ.

Ɉɬɜɟɬ: 35,3 Ⱦɠ.

22600 60



60 20

-1 2

20 c ; 3 ɤɝ ɦ .

Ɉɬɜɟɬ: 3 ɤɝ ɦ².

Ɂɚɞɚɱɚ 6. Ⱦɜɚ ɞɢɫɤɚ ɦɚɫɫɚɦɢ ɬ

= 0,3 ɦ ɜɪɚɳɚɸɬɫɹ ɜɨɤɪɭɝ ɨɛɳɟɣ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɢɯ

ɰɟɧɬɪɵ ɬɹɠɟɫɬɢ, ɫɨɝɥɚɫɧɨ ɭɪɚɜɧɟɧɢɹɦ

= 2 ɤɝ, ɬ2 = 4 ɤɝ ɫ ɪɚɞɢɭɫɚɦɢ r1 = 0,5 ɦ,

= 2t;

= –1,5t. ɉɟɪɜɵɣ ɞɢɫɤ ɩɚɞɚɟɬ

ɜɧɢɡ ɢ ɫɰɟɩɥɹɟɬɫɹ ɫɨ ɜɬɨɪɵɦ ɧɢɠɧɢɦ. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɢɯ ɫɨɜɦɟɫɬɧɨɝɨ ɜɪɚɳɟɧɢɹ.

Ⱦɚɧɨ:

= 2 ɤɝ

= 4 ɤɝ

= 0,5 ɦ

= 0,3 ɦ

– ?

1. ɂɫɩɨɥɶɡɭɟɦ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ:

2. Ⱦɥɹ ɩɟɪɜɨɝɨ ɞɢɫɤɚ: L

Imr

111 1

Ɋɟɲɟɧɢɟ:

const.

= I

2-1

;2c.



;

Ⱥɧɚɥɨɝɢɱɧɨ ɞɥɹ ɜɬɨɪɨɝɨ ɞɢɫɤɚ:

2-1

LI mr



222 22

(1,5)c.

Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɞɥɹ ɞɜɭɯ ɞɢɫɤɨɜ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧ

ɜ ɜɢɞɟ:

+ I

1Z1

2Z2

= IZ.

Ɍɨɝɞɚ ɢɫɤɨɦɚɹ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɫɨɜɦɟɫɬɧɨɝɨ ɜɪɚɳɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ

ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:

11 11

22 22

mr m r mr mr

ZZ Z

 

11 1 2 2 2 11 2 2

22 22

22 2 2

mr m r



11 1 2 2 2

22 2 2

mr mr



11 2 2

§· ¨¸

©¹

2 0,5 2 4 0,3 1,5

20,5 40,3

0, 53 c .

-1

Ɉɬɜɟɬ: 0,53 ɪɚɞ/ɫ.

5.3. Ɂɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ

1. Ɉɩɪɟɞɟɥɢɬɶ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɦɟɞɧɨɝɨ ɲɚɪɚ ɦɚɫɫɨɣ 18 ɤɝ. ɉɥɨɬɧɨɫɬɶ ɦɟ-

ɞɢ 8900 ɤɝ/ɦ

2. Ɉɩɪɟɞɟɥɢɬɶ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɞɧɨɪɨɞɧɨɝɨ ɦɟɞɧɨɝɨ ɫɬɟɪɠɧɹ ɞɥɢɧɨɣ Ɛ = 50 ɫɦ

ɢ ɩɥɨɳɚɞɶɸ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ s

= 5 ɫɦ2 ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɤɨɬɨɪɚɹ ɩɟɪɩɟɧ-

ɞɢɤɭɥɹɪɧɚ ɫɬɟɪɠɧɸ ɢ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɭ, ɨɬɫɬɨɹɳɭɸ ɨɬ ɤɨɧɰɚ ɫɬɟɪɠɧɹ ɧɚ 1/6 ɟɝɨ ɞɥɢɧɵ.

3. Ɇɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ ɦɚɫɫɨɣ 10 ɝ ɫɨɜɟɪɲɚɟɬ ɤɨɥɟɛɚɧɢɹ ɩɨ ɡɚɤɨɧɭ

(t) = S/3sin(St + S/6) ɪɚɞ. ɇɚɣɬɢ ɦɨɦɟɧɬ ɫɢɥɵ ɬɹɠɟɫɬɢ ɱɟɪɟɡ 2 ɫɟɤɭɧɞɵ ɨɬ

ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.

4. ɒɤɢɜ ɪɚɞɢɭɫɨɦ R = 0,2 ɦ ɢ ɦɚɫɫɨɣ ɬ = 10 ɤɝ ɫɨɟɞɢɧɟɧ ɫ ɦɨɬɨɪɨɦ ɩɪɢ

ɩɨɦɨɳɢ ɩɪɢɜɨɞɧɨɝɨ ɪɟɦɧɹ. ɋɢɥɚ ɧɚɬɹɠɟɧɢɹ ɧɟɩɪɨɫɤɚɥɶɡɵɜɚɸɳɟɝɨ ɪɟɦɧɹ F

= 14,7 ɇ. ɋ ɤɚɤɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ ɛɭɞɟɬ ɜɪɚɳɚɬɶɫɹ ɲɤɢɜ ɱɟɪɟɡ t = 10 ɫɟɤɭɧɞ ɩɨɫɥɟ ɜɤɥɸɱɟɧɢɹ ɦɨɬɨɪɚ? Ɍɪɟɧɢɟɦ ɜ ɫɢɫɬɟɦɟ ɩɪɟɧɟɛɪɟɱɶ.

5. Ʉ ɨɛɨɞɭ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɫɤɚ ɪɚɞɢɭɫɚ R = 0,2 ɦ ɩɪɢɥɨɠɟɧɚ ɤɚɫɚɬɟɥɶɧɚɹ ɫɢɥɚ F = 98,1 ɇ. ɉɪɢ ɜɪɚɳɟɧɢɢ ɧɚ ɞɢɫɤ ɞɟɣɫɬɜɭɟɬ ɦɨɦɟɧɬ ɫɢɥɵ ɬɪɟɧɢɹ Ɇ = 4,9 ɇ

ɦ, ɚ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɞɢɫɤɚ ɪɚɜɧɨ

= 100 ɫ–2. ɇɚɣɬɢ ɦɚɫɫɭ ɞɢɫɤɚ.

ɬɪ

6. ɒɚɪ ɞɢɚɦɟɬɪɨɦ D = 6 ɫɦ ɢ ɦɚɫɫɨɣ ɬ = 0,25 ɤɝ ɤɚɬɢɬɫɹ ɛɟɡ ɫɤɨɥɶɠɟɧɢɹ ɩɨ

ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ, ɜɪɚɳɚɹɫɶ ɫ ɱɚɫɬɨɬɨɣ

= 4 ɫ–1. ɇɚɣɬɢ ɤɢɧɟɬɢɱɟɫɤɭɸ

ɷɧɟɪɝɢɸ ɲɚɪɚ.

7. Ʉɚɤɢɦ ɦɨɦɟɧɬɨɦ ɢɦɩɭɥɶɫɚ ɨɛɥɚɞɚɥ ɞɢɫɤ, ɜɪɚɳɚɸɳɢɣɫɹ ɜɨɤɪɭɝ ɨɫɢ

ɫɢɦɦɟɬɪɢɢ, ɟɫɥɢ ɨɧ ɨɫɬɚɧɨɜɢɥɫɹ ɱɟɪɟɡ ɬɨɪɦɨɠɟɧɢɢ ɪɚɜɧɵɦ

= 3,5 ɫ–2.

= 15 ɫɟɤɭɧɞ? ɉɪɢɧɹɬɶ ɭɫɤɨɪɟɧɢɟ ɩɪɢ

8. Ɉɩɪɟɞɟɥɢɬɶ ɪɚɛɨɬɭ ɫɢɥɵ ɬɪɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɜɪɚɳɚɸɳɢɣɫɹ ɜɨɤɪɭɝ ɫɜɨɟɣ ɨɫɢ ɫɢɦɦɟɬɪɢɢ ɦɚɯɨɜɢɤ, ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɤɨɬɨɪɨɝɨ 30 ɤɝ

ɦ

/ɫ ɡɚ ɜɪɟɦɹ

ɞɨ ɩɨɥɧɨɣ ɨɫɬɚɧɨɜɤɢ ɦɚɯɨɜɢɤɚ.

9. ɇɚɣɬɢ ɥɢɧɟɣɧɵɟ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɨɜ ɲɚɪɚ, ɞɢɫɤɚ ɢ ɨɛɪɭɱɚ, ɫɤɚɬɢɜɲɢɯɫɹ ɛɟɡ ɫɤɨɥɶɠɟɧɢɹ ɫ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɜɵɫɨɬɨɣ h = 0,5 ɦ.

10. ɑɟɥɨɜɟɤ ɦɚɫɫɨɣ ɬ

= 60 ɤɝ ɧɚɯɨɞɢɬɫɹ ɧɚ ɧɟɩɨɞɜɢɠɧɨɣ ɩɥɚɬɮɨɪɦɟ ɦɚɫ-

ɫɨɣ ɬ = 100 ɤɝ ɫ ɪɚɞɢɭɫɨɦ R = 10 ɦ. ɋ ɤɚɤɨɣ ɱɚɫɬɨɬɨɣ ɛɭɞɟɬ ɜɪɚɳɚɬɶɫɹ ɩɥɚɬɮɨɪɦɚ, ɟɫɥɢ ɱɟɥɨɜɟɤ ɩɨɣɞɟɬ ɫɨ ɫɤɨɪɨɫɬɶɸ

= 4 ɤɦ/ɱ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɥɚɬɮɨɪɦɵ

ɜɨɤɪɭɝ ɨɫɢ ɜɪɚɳɟɧɢɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ r = 5 ɦɟɬɪɨɜ.

5.4. Ɇɟɯɚɧɢɤɚ ɬɜɟɪɞɨɝɨ ɬɟɥɚ.

Ɍɟɫɬɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ɡɧɚɧɢɣ

ɍɪɨɜɟɧɶ I

1. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɤɚɤɨɝɨ ɬɟɥɚ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ I = mr2:

ɚ) ɬɨɧɤɢɣ ɨɛɪɭɱ; ɛ) ɲɚɪ; ɜ) ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ; ɝ) ɩɨɥɨɜɢɧɚ ɞɢɫɤɚ; ɞ) ɰɢɥɢɧɞɪ?

2. ɇɚɣɬɢ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɞɢɫɤɚ ɩɨɫɥɟ ɬɨɝɨ, ɤɚɤ ɜ ɧɟɝɨ ɩɨɩɚɥɚ ɢ ɡɚɫɬɪɹɥɚ ɧɚ ɪɚɫɫɬɨɹɧɢɢ r ɨɬ ɰɟɧɬɪɚ ɩɭɥɹ ɦɚɫɫɨɣ ɬ. Ɇɚɫɫɚ ɞɢɫɤɚ Ɇ, ɪɚɞɢɭɫ ɞɢɫɤɚ R.

3. ɇɚ ɪɢɫ. 5.8 ɢɡɨɛɪɚɠɟɧɨ ɬɟɥɨ, ɢɦɟɸɳɟɟ ɨɫɶ ɜɪɚɳɟɧɢɹ Ɉ.

Ɋɢɫ. 5.8

ɇɚ ɧɟɝɨ ɞɟɣɫɬɜɭɸɬ ɞɜɟ ɪɚɜɧɵɟ ɩɨ ɦɨɞɭɥɸ ɫɢɥɵ (F

= F2), ɬɨɱɤɢ ɩɪɢɥɨɠɟ-

ɧɢɹ ɤɨɬɨɪɵɯ ɧɚɯɨɞɹɬɫɹ ɧɚ ɨɞɢɧɚɤɨɜɵɯ ɪɚɫɫɬɨɹɧɢɹɯ ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ. ɋɪɚɜɧɢɬɶ ɦɨɞɭɥɢ ɦɨɦɟɧɬɨɜ ɷɬɢɯ ɫɢɥ (Ɇ

ɚ) Ɇ

= Ɇ2;

> Ɇ2;

ɛ) Ɇ

ɜ) Ɇ

< Ɇ2;

ɝ) Ɇ

= 0;

= 0.

ɞ) Ɇ

ɢ Ɇ2):

4. ɒɚɪ ɦɚɫɫɨɣ 1 ɤɝ ɢ ɪɚɞɢɭɫɚ 0,1 ɦ ɜɪɚɳɚɟɬɫɹ ɜɨɤɪɭɝ ɨɫɢ ɫɢɦɦɟɬɪɢɢ ɩɨ ɡɚɤɨɧɭ

(t) = 0,3 + t2 + 0,1t3. ɇɚɣɬɢ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɲɚɪɚ ɱɟɪɟɡ 10 ɫɟɤɭɧɞ ɩɨɫɥɟ

ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.

5. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɫ ɧɨɪɦɚɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ ɚ

~ t4. ɉɪɢ ɷɬɨɦ ɦɨɦɟɧɬ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɬɨɱɤɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ

ɜɪɚɳɟɧɢɹ, ɜɵɪɚɠɚɟɬɫɹ ɫɬɟɩɟɧɧɨɣ ɮɭɧɤɰɢɟɣ ɜɪɟɦɟɧɢ Ɇ ~ t

. ɇɚɣɬɢ ɡɧɚɱɟɧɢɟ ɩ.

6. Ɉɞɧɨɪɨɞɧɵɣ ɫɬɟɪɠɟɧɶ ɞɥɢɧɨɣ Ɛ ɢ ɦɚɫɫɨɣ ɬ ɦɨɠɟɬ ɜɪɚɳɚɬɶɫɹ ɜɨɤɪɭɝ ɡɚ-

ɤɪɟɩɥɟɧɧɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɤɨɧɟɰ. Ʉɚɤɭɸ ɞɥɢɧɭ ɞɨɥɠɟɧ ɢɦɟɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ ɦɚɫɫɨɣ ɬ, ɱɬɨɛɵ ɩɪɢ ɪɚɜɧɵɯ ɨɬɤɥɨɧɟɧɢɹɯ ɨɬ ɜɟɪɬɢɤɚɥɢ ɨɛɚ ɦɚɹɬɧɢɤɚ ɢɦɟɥɢ ɨɞɢɧɚɤɨɜɵɟ ɭɝɥɨɜɵɟ ɭɫɤɨɪɟɧɢɹ?

7. Ⱦɜɚ ɞɢɫɤɚ, ɦɚɫɫɵ ɤɨɬɨɪɵɯ ɪɚɜɧɵ, ɚ ɪɚɞɢɭɫɵ

ɬɚɤɨɜɵ, ɱɬɨ R1 = 2R2, ɪɚɫɤɪɭɱɢɜɚɸɬ ɢɡ ɫɨɫɬɨɹɧɢɹ ɩɨɤɨɹ ɞɨ ɨɞɢɧɚɤɨɜɵɯ ɭɝɥɨɜɵɯ ɫɤɨɪɨɫɬɟɣ. ɇɚɣɬɢ ɨɬɧɨɲɟɧɢɟ ɩɪɨɢɡɜɟɞɟɧɧɵɯ ɪɚɛɨɬ Ⱥ

1/Ⱥ2

8. ɒɚɪ ɢ ɫɩɥɨɲɧɨɣ ɰɢɥɢɧɞɪ ɢɡ ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɦɚɬɟɪɢɚɥɚ ɨɞɢɧɚɤɨɜɨɣ

ɦɚɫɫɵ ɤɚɬɹɬɫɹ ɛɟɡ ɫɤɨɥɶɠɟɧɢɹ ɫ ɨɞɢɧɚɤɨɜɨɣ ɫɤɨɪɨɫɬɶɸ. ȼɨ ɫɤɨɥɶɤɨ ɪɚɡ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɰɢɥɢɧɞɪɚ ɛɨɥɶɲɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɲɚɪɚ.

9. ɉɨ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɫ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɜɵɫɨɬɵ ɫɩɭɫɤɚɟɬɫɹ ɨɛɪɭɱ: ɚ) ɫɤɨɥɶɡɹ ɛɟɡ ɤɚɱɟɧɢɹ; ɛ) ɤɚɬɹɫɶ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ.

ɨɬɧɨɲɟɧɢɟ ɫɤɨɪɨɫɬɟɣ ɰɟɧɬɪɚ ɨɛɪɭɱɚ ɜ ɨɞɧɨɦ ɢ ɬɨɦ ɠɟ ɦɟɫɬɟ ɞɥɹ

ɇɚɣɬɢ

ɞɜɭɯ ɫɥɭɱɚɟɜ (

10. ȼ ɤɨɧɟɰ ɫɬɟɪɠɧɹ ɦɚɫɫɨɣ Ɇ = 1 ɤɝ ɢ ɞɥɢɧɨɣ Ɛ = 1 ɦ, ɜɢɫɹɳɟɝɨ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɤɨɧɟɰ, ɩɨɩɚɞɚɟɬ ɩɭɥɹ ɢ ɡɚɫɬɪɟɜɚɟɬ ɜ ɧɟɦ. ɇɚɣɬɢ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɫɬɟɪɠɧɹ ɩɨɫɥɟ ɭɞɚɪɚ ɩɭɥɢ, ɟɫɥɢ ɦɚɫɫɚ ɩɭɥɢ ɬ = 10 ɝ, ɚ ɫɤɨɪɨɫɬɶ

= 500 ɦ/ɫ.

ɍɪɨɜɟɧɶ II

1. Ʉɚɤɨɜɨ ɧɚɡɜɚɧɢɟ ɮɢɡɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ, ɨɩɪɟɞɟɥɹɟɦɨɣ ɜɵɪɚɠɟɧɢɟɦ Ɇ/H ,

– ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ; Ɇ – ɦɨɦɟɧɬ ɞɟɣɫɬɜɭɸɳɟɣ ɫɢɥɵ?

ɝɞɟ

2. ɇɚ ɪɢɫ. 5.9 ɢɡɨɛɪɚɠɟɧɵ ɬɟɥɚ, ɫɨɫɬɚɜɥɟɧɧɵɟ ɢɡ ɨɞɢɧɚɤɨɜɵɯ ɨɞɧɨɪɨɞɧɵɯ ɬɪɟɭɝɨɥɶɧɵɯ ɩɥɚɫɬɢɧ. ɍɤɚɡɚɬɶ ɮɢɝɭɪɵ ɫ ɦɢɧɢɦɚɥɶɧɵɦ ɢ ɦɚɤɫɢɦɚɥɶɧɵɦ ɦɨɦɟɧɬɚɦɢ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɈɈ'.

Ɋɢɫ. 5.9

3. Ɍɟɥɨ ɜɪɚɳɚɟɬɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɦɨɦɟɧɬɚ ɫɢɥ, ɝɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɤɨɬɨɪɨɝɨ ɨɬ ɜɪɟɦɟɧɢ ɩɪɟɞɫɬɚɜɥɟɧ ɧɚ ɪɢɫ. 5.10. ɉɪɢ t = 0

> 0. Ʉɚɤ ɢɡɦɟɧɹɟɬɫɹ ɭɝɥɨ-

ɜɚɹ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɜ ɢɧɬɟɪɜɚɥɚɯ ɜɪɟɦɟɧɢ ɚ) ɢ ɛ)?

Ɋɢɫ. 5.10

4. ɇɚɣɞɢɬɟ ɨɬɧɨɲɟɧɢɟ ɭɝɥɨɜɵɯ ɫɤɨɪɨɫɬɟɣ (

), ɩɪɢɨɛɪɟɬɚɟɦɵɯ ɩɨɤɨɹ-

ɳɢɦɫɹ ɬɟɥɨɦ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɪɚɳɚɸɳɢɯ ɦɨɦɟɧɬɨɜ, ɝɪɚɮɢɤɢ ɤɨɬɨɪɵɯ ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ. 5.11.

Ɋɢɫ. 5.11

5. Ⱦɜɚ ɞɢɫɤɚ ɨɞɢɧɚɤɨɜɨɣ ɬɨɥɳɢɧɵ ɫ ɪɚɜɧɵɦɢ ɦɚɫɫɚɦɢ ɢɡɝɨɬɨɜɥɟɧɵ ɨɞɢɧ ɢɡ ɠɟɥɟɡɚ (

= 7800 ɤɝ/ɦ3), ɚ ɞɪɭɝɨɣ ɢɡ ɞɟɪɟɜɚ (

= 2000 ɤɝ/ɦ3). Ɉɧɢ ɜɪɚɳɚɸɬ-

ɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɪɚɜɧɵɯ ɩɨ ɦɨɞɭɥɸ ɫɢɥ, ɤɚɫɚɬɟɥɶɧɵɯ ɤ ɢɯ ɨɛɨɞɚɦ. ɇɚɣɬɢ ɨɬɧɨɲɟɧɢɟ ɭɝɥɨɜɵɯ ɭɫɤɨɪɟɧɢɣ ɞɢɫɤɨɜ

6. ȼɟɥɢɱɢɧɚ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ ɢɡɦɟ-

ɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ

~Lt . Ʉɚɤ ɢɡɦɟɧɹɟɬɫɹ ɩɪɢ ɷɬɨɦ ɦɨɦɟɧɬ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ

ɧɚ ɬɟɥɨ?

7. Ɉɞɧɨɪɨɞɧɵɣ ɫɬɟɪɠɟɧɶ ɪɚɫɤɪɭɱɢɜɚɸɬ ɢɡ ɫɨɫɬɨɹɧɢɹ ɩɨɤɨɹ ɞɨ ɨɩɪɟɞɟɥɟɧ-

ɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ. ɇɚɣɬɢ ɨɬɧɨɲɟɧɢɟ ɩɪɨɢɡɜɟɞɟɧɧɵɯ ɩɪɢ ɷɬɨɦ ɪɚɛɨɬ (Ⱥ

ɚ/Ⱥɛ

ɟɫɥɢ ɨɫɶ ɜɪɚɳɟɧɢɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɫɬɟɪɠɧɸ ɢ ɩɪɨɯɨɞɢɬ ɚ) ɱɟɪɟɡ ɟɝɨ ɤɨɧɟɰ; ɛ) ɱɟɪɟɡ ɟɝɨ ɫɟɪɟɞɢɧɭ.

8. Ɇɚɯɨɜɢɤ ɜ ɜɢɞɟ ɫɩɥɨɲɧɨɝɨ ɞɢɫɤɚ, ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɤɨɬɨɪɨɝɨ I = 150 ɤɝ ɤɨɧɭ

ɦ

, ɜɪɚɳɚɟɬɫɹ ɬɚɤ, ɱɬɨ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɢɡɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɩɨ ɡɚ-

(t) = 8 + 15t2 – t4 (ɪɚɞ). ɇɚɣɬɢ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɦɚɯɨɜɢɤɚ ɱɟɪɟɡ 2 ɫ

ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.

9. Ɍɨɧɤɢɣ ɩɪɹɦɨɣ ɫɬɟɪɠɟɧɶ ɞɥɢɧɨɣ Ɛ = 1 ɦ ɩɪɢɤɪɟɩɥɟɧ ɤ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɤɨɧɟɰ. ɋɬɟɪɠɟɧɶ ɨɬɤɥɨɧɢɥɢ ɧɚ ɭɝɨɥ

= 60q ɨɬ ɩɨ-

ɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɢ ɨɬɩɭɫɬɢɥɢ. ɋ ɤɚɤɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ ɨɧ ɩɪɨɣɞɟɬ ɩɨɥɨɠɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ?

10. ɉɥɚɬɮɨɪɦɚ, ɢɦɟɸɳɚɹ ɮɨɪɦɭ ɞɢɫɤɚ, ɦɨɠɟɬ ɜɪɚɳɚɬɶɫɹ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ. ɇɚ ɤɪɚɸ ɩɥɚɬɮɨɪɦɵ ɫɬɨɢɬ ɱɟɥɨɜɟɤ ɦɚɫɫɨɣ ɬ ɭɝɨɥ

ɩɨɜɟɪɧɟɬɫɹ ɩɥɚɬɮɨɪɦɚ, ɟɫɥɢ ɱɟɥɨɜɟɤ ɨɛɨɣɞɟɬ ɩɥɚɬɮɨɪɦɭ ɩɨ ɤɪɚɸ ɢ ɜɟɪ-

ɧɟɬɫɹ ɜ ɢɫɯɨɞɧɭɸ ɬɨɱɤɭ. Ɇɚɫɫɚ ɩɥɚɬɮɨɪɦɵ ɬ

= 240 ɤɝ. ɑɟɥɨɜɟɤɚ ɩɪɢɧɹɬɶ ɡɚ

= 60 ɤɝ. ɇɚ ɤɚɤɨɣ

ɦɚɬɟɪɢɚɥɶɧɭɸ ɬɨɱɤɭ.

ɍɪɨɜɟɧɶ III

1. ȼ ɤɚɤɢɯ ɟɞɢɧɢɰɚɯ ɢɡɦɟɪɹɟɬɫɹ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɬɟɥɚ?

2. ɂɡ ɫɩɥɨɲɧɨɝɨ ɨɞɧɨɪɨɞɧɨɝɨ ɰɢɥɢɧɞɪɚ ɫɞɟɥɚɥɢ ɩɨɥɵɣ, ɭɞɚɥɢɜ ɩɨɥɨɜɢɧɭ

ɦɚɫɫɵ. ȼɨ ɫɤɨɥɶɤɨ ɪɚɡ ɢɡɦɟɧɢɬɫɹ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɰɢɥɢɧɞɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɝɨ ɨɫɢ?

3. Ⱦɢɫɤ ɪɚɞɢɭɫɨɦ R = 1 ɦ ɢ ɦɚɫɫɨɣ ɬ = 100 ɤɝ ɜɪɚɳɚɟɬɫɹ ɜɨɤɪɭɝ ɨɫɢ ɫɢɦ-

ɦɟɬɪɢɢ ɩɨ ɡɚɤɨɧɭ

(t) = A + Bt(1 + t2), ɝɞɟ Ⱥ, ȼ – ɤɨɧɫɬɚɧɬɵ. Ɉɩɪɟɞɟɥɢɬɶ ɦɨɦɟɧɬ

ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɞɢɫɤ ɫɢɥ ɤ ɤɨɧɰɭ ɬɪɟɬɟɣ ɫɟɤɭɧɞɵ ɞɜɢɠɟɧɢɹ.

4. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɫ ɧɨɪɦɚɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ, ɜɟɥɢɱɢɧɚ ɤɨɬɨɪɨɝɨ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ ɚ

~ t4. ɉɪɢ

ɷɬɨɦ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɨɤɪɭɠɧɨɫɬɢ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ L ~ t ɇɚɣɬɢ ɡɧɚɱɟɧɢɟ k.

5. ɋ ɛɥɨɤɚ ɪɚɞɢɭɫɚ R ɪɚɡɦɚɬɵɜɚɟɬɫɹ ɧɢɬɶ ɫ ɝɪɭɡɨɦ ɦɚɫɫɨɣ ɬ (ɪɢɫ. 5.12). ɍɤɚɡɚɬɶ ɜɟɪɧɵɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɝɪɭɡɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ:

ɚ) ɬ ɛ) ɬ

;

ɜ) J

ɝ)

6. Ɍɟɥɨ ɜɪɚɳɚɟɬɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɦɨɦɟɧɬɚ ɫɢɥ, ɡɚɜɢɫɢɦɨɫɬɶ ɤɨɬɨɪɨɝɨ ɨɬ ɜɪɟɦɟɧɢ ɩɪɟɞɫɬɚɜɥɟɧɚ ɧɚ ɪɢɫ. 5.13.

Ɋɢɫ. 5.12 Ɋɢɫ. 5.13

ɉɪɢ t = 0 ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ Z = 0. ɍɤɚɠɢɬɟ ɬɨɱɤɭ ɧɚ ɝɪɚɮɢɤɟ, ɫɨɨɬɜɟɬɫɬɜɭ-

ɸɳɭɸ ɦɚɤɫɢɦɚɥɶɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ.

7. ɇɚ ɫɬɟɪɠɧɟ ɞɥɢɧɨɣ Ɛ, ɦɚɫɫɚ ɤɨɬɨɪɨɝɨ ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɚ, ɡɚɤɪɟɩɥɟɧɵ ɞɜɚ ɦɚɥɟɧɶɤɢɯ ɲɚɪɢɤɚ ɦɚɫɫɚɦɢ ɬ ɢ 3ɬ (ɪɢɫ. 5.14). ɋɬɟɪɠɟɧɶ ɪɚɫɤɪɭɱɢɜɚɸɬ ɢɡ ɫɨɫɬɨɹɧɢɹ ɩɨɤɨɹ ɞɨ ɨɩɪɟɞɟɥɟɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ. ɉɪɢ ɤɚɤɨɦ ɩɨɥɨɠɟɧɢɢ ɨɫɢ ɜɪɚɳɟɧɢɹ (I, II, III) ɫɨɜɟɪɲɚɟɬɫɹ ɧɚɢɦɟɧɶɲɚɹ ɢ ɧɚɢɛɨɥɶɲɚɹ ɪɚɛɨɬɚ?

Ɋɢɫ. 5.14

8. Ɉɛɪɭɱ ɢ ɞɢɫɤ ɨɞɢɧɚɤɨɜɨɣ ɦɚɫɫɵ ɤɚɬɹɬɫɹ ɛɟɡ ɫɤɨɥɶɠɟɧɢɹ ɫ ɨɞɢɧɚɤɨɜɨɣ ɫɤɨɪɨɫɬɶɸ. Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɨɛɪɭɱɚ Ɍ

= 39,2 Ⱦɠ. ɇɚɣɬɢ ɤɢɧɟɬɢɱɟɫɤɭɸ

ɷɧɟɪɝɢɸ ɞɢɫɤɚ.

9. ɋ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɧɚɱɢɧɚɸɬ ɫɤɚɬɵɜɚɬɶɫɹ ɛɟɡ ɫɤɨɥɶ-

ɠɟɧɢɹ ɞɜɚ ɰɢɥɢɧɞɪɚ:

ɚ) ɫɩɥɨɲɧɨɣ; ɛ) ɬɨɧɤɨɫɬɟɧɧɵɣ ɩɨɥɵɣ. ɇɚɣɬɢ ɨɬɧɨɲɟɧɢɟ ɥɢɧɟɣɧɵɯ ɫɤɨɪɨɫɬɟɣ ɢɯ ɰɟɧɬɪɨɜ ɬɹɠɟɫɬɢ ɜ ɞɚɧɧɵɣ ɦɨ-

ɦɟɧɬ ɜɪɟɦɟɧɢ (

10. ȼɪɚɳɚɸɬɫɹ ɬɪɢ ɨɞɢɧɚɤɨɜɵɯ ɤɨɥɟɫɚ ɜɨɤɪɭɝ ɨɛɳɟɣ ɨɫɢ. ɂɯ ɭɝɥɨɜɵɟ ɫɤɨɪɨɫɬɢ ɨɞɢɧɚɤɨɜɵ, ɧɨ ɧɚɩɪɚɜɥɟɧɢɟ ɜɪɚɳɟɧɢɹ ɨɞɧɨɝɨ ɢɡ ɤɨɥɟɫ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɞɜɭɦ ɞɪɭɝɢɦ. Ʉɚɤ ɢɡɦɟɧɢɬɫɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ, ɟɫɥɢ ɤɨɥɟɫɚ ɫɰɟɩɢɬɶ?

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