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Ɂɚɞɚɱɚ 2. Ɇɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ ɢɦɟɟɬ ɦɚɫɫɭ ɬ ɢ ɞɥɢɧɭ Ɛ. ȼ ɦɨɦɟɧɬ,
P
X
ɤɨɝɞɚ ɧɢɬɶ ɦɚɹɬɧɢɤɚ ɨɛɪɚɡɭɟɬ ɭɝɨɥ
ɤɨɜɚ ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ ɧɢɬɢ.
Ⱦɚɧɨ:
D, X
ɬ, Ɛ,
– ?
F
ɧɚɬ
1. ȼɵɩɨɥɧɢɦ ɱɟɪɬɟɠ ɤ ɡɚɞɚɱɟ (ɪɢɫ. 3.2) ɢ ɢɫɩɨɥɶɡɭɟɦ ɫɯɟ-
ɦɭ ɪɟɲɟɧɢɹ, ɩɪɢɜɟɞɟɧɧɭɸ ɜ ɡɚɞɚɱɟ 1.
D
ɫ ɜɟɪɬɢɤɚɥɶɸ, ɫɤɨɪɨɫɬɶ ɝɪɭɡɚ ɪɚɜɧɚ X. Ʉɚ-
Ɋɟɲɟɧɢɟ:
2. ɉɨ ɜɬɨɪɨɦɭ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɢɦɟɟɬ
ɜɢɞ:
G
GG
ma F mg
ɧɚɬ
.
Ɋɢɫ. 3.2
3. ȼɵɛɟɪɟɦ ɧɚɩɪɚɜɥɟɧɢɟ ɨɫɢ Ɉɏ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɰɟɧɬɪɨɫɬɪɟɦɢɬɟɥɶɧɨɝɨ
ɭɫɤɨɪɟɧɢɹ ɝɪɭɡɚ. ɇɚɣɞɟɦ ɩɪɨɟɤɰɢɸ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɧɚ ɜɵɛɪɚɧɧɭɸ ɨɫɶ Ɉɏ:
Ɉɏ: ɬɚ = F
4. ɂɫɩɨɥɶɡɭɹ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɰɟɧɬɪɨɫɬɪɟɦɢɬɟɥɶɧɨɝɨ (ɧɨɪɦɚɥɶɧɨɝɨ) ɭɫɤɨɪɟ-
ɧɢɹ, ɢɦɟɟɦ ɚ = ɚ
= ɚɩ =
ɰ
2
X
.
A
5. Ɉɤɨɧɱɚɬɟɥɶɧɨ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɪɚɫɱɟɬɚ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ:
Fmg
ɧɚɬ
– mgcosD.
ɧɚɬ
2
X
D
(cos ).
A
Ɉɬɜɟɬ:
D
Fmg
(cos ).
ɧɚɬ
2
X
A
Ɂɚɞɚɱɚ 3. Ʉɚɤɨɜɚ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɲɚɣɛɵ, ɩɭɳɟɧɧɨɣ ɩɨ ɩɨɜɟɪɯɧɨɫɬɢ ɥɶɞɚ,
ɟɫɥɢ ɨɧɚ ɨɫɬɚɧɨɜɢɥɚɫɶ ɱɟɪɟɡ 40 ɫɟɤɭɧɞ? Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɲɚɣɛɵ ɨ ɥɟɞ
Ⱦɚɧɨ:
= 40 ɫ
t
ɨɫɬ
= 0,05
– ?
0
Ɂɚɞɚɱɚ ɹɜɥɹɟɬɫɹ ɤɨɦɛɢɧɢɪɨɜɚɧɧɨɣ, ɬɚɤ ɤɚɤ ɨɛɴɟɞɢɧɹɟɬ
ɜɨɩɪɨɫɵ ɞɢɧɚɦɢɤɢ ɢ ɤɢɧɟɦɚɬɢɤɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ.
1. ȼɵɩɨɥɧɢɦ ɱɟɪɬɟɠ (ɪɢɫ. 3.3) ɢ, ɢɫɩɨɥɶɡɭɹ ɫɯɟɦɭ ɪɟ-
Ɋɟɲɟɧɢɟ:
P
= 0,05.
ɲɟɧɢɹ ɡɚɞɚɱɢ ɩɨ ɞɢɧɚɦɢɤɟ, ɧɚɣɞɟɦ ɭɫɤɨɪɟɧɢɟ ɲɚɣɛɵ, ɚ ɡɚɬɟɦ ɩɪɢɦɟɧɢɦ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɮɨɪɦɭɥɵ.
GG
GG
ma F mg N
mp
.
31
ma F
t
X
ɪ
X
X
®
Nmg
¯
mp
.
:OXOY
:
ma F
®
¯
mg N
00
mp
ɢɥɢ
Ɋɢɫ. 3.3
2. ɂɫɩɨɥɶɡɭɹ F
3. ɂɡ ɤɢɧɟɦɚɬɢɤɢ
Ⱦɥɹ t = t
ɨɫɬ
= PN, ɢɦɟɟɦ: ɚ = Pg.
mp
X
=
X
– ɚt.
0
: 0 =
X
= ɚt
ɨɫɬ
= Pgt
0
= 0,05 9,8 40 = 19,6 ɦ/ɫ.
ɨɫɬ
X
– ɚt
0
ɨɫɬ
Ɉɬɜɟɬ: 19,6 ɦ/ɫ.
Ɂɚɞɚɱɚ 4. Ɍɟɥɨ ɦɚɫɫɨɣ 5 ɤɝ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ 10 ɦ ɫ ɩɨ-
ɫɬɨɹɧɧɵɦ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ 1 ɫ
ɞɜɢɠɟɧɢɹ.
Ⱦɚɧɨ:
ɬ = 5 ɤɝ
R = 10 ɦ
–2
H
= 1 ɫ
= 40 ɫ
1
– ?
1
1. ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɢɦɩɭɥɶɫɚ
Ⱦɥɹ ɟɝɨ ɦɨɞɭɥɹ ɪ = ɬ
2. ɂɫɩɨɥɶɡɭɹ ɫɜɹɡɶ ɥɢɧɟɣɧɨɣ ɢ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɟɣ ɬɟɥɚ,
X
= ZR.
–2
. ɇɚɣɬɢ ɢɦɩɭɥɶɫ ɬɟɥɚ ɱɟɪɟɡ 10 ɫ ɨɬ ɧɚɱɚɥɚ
Ɋɟɲɟɧɢɟ:
G
G
X
.
mp
.
3. ɇɚɣɞɟɦ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɩɨ ɮɨɪɦɭɥɟ ɤɢɧɟɦɚɬɢɤɢ:
Z
=
Z
+ H t;
0
ɝɞɟ ɭ ɧɚɫ
Z
= 0,
0
Z
1
= H t1.
Ɋ1 = ɬH t
R = 5 1 10 10 = 500 ɤɝɦ/ɫ.
1
Ɉɬɜɟɬ: 500 ɤɝɦ/ɫ.
Ɂɚɞɚɱɚ 5. Ɍɟɥɟɠɤɚ ɫ ɩɟɫɤɨɦ ɦɚɫɫɨɣ 10 ɤɝ ɤɚɬɢɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ
ɇɚɜɫɬɪɟɱɭ ɥɟɬɢɬ ɲɚɪ ɦɚɫɫɨɣ 2 ɤɝ ɫ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ
ɤɨɣ ɫɤɨɪɨɫɬɶɸ ɛɭɞɟɬ ɞɜɢɝɚɬɶɫɹ ɬɟɥɟɠɤɚ ɫ ɡɚɫɬɪɹɜɲɟɦ ɜ ɩɟɫɤɟ ɲɚɪɨɦ?
Ⱦɚɧɨ:
= 10 ɤɝ
ɬ
Ɍ
X
= 1 ɦ/ɫ
1
ɬ
= 2 ɤɝ
ɲ
= 7 ɦ/ɫ
2
Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɞɥɹ ɫɢɫɬɟɦɵ
ɬɟɥɟɠɤɚ – ɲɚɪ ɞɨ ɢ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɬɟɥɟɠɤɢ ɢ ɲɚɪɚ:
mm ɬɬ
Ɉɫɶ ɯ ɜɵɛɟɪɟɦ ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ
Ɋɟɲɟɧɢɟ:
GG G
XX X
12
T ɲɌɲ
()
X
= 7 ɦ/ɫ. ɋ ɤɚ-
2
.
G
X
.
1
– ?
32
X
= 1 ɦ/ɫ.
1
Ɉɏ: ɬ
ɋ
X
ɬɬ
XX
Ɍɲ
X
ɯ
ɬɬ
– ɬ
ɌX1
12
Ɍɲ
= (ɬɌ + ɬɲ)
ɲX2
10 1 2 7
10 2
X
0, 3
.
ɯ
ɦ/ɫ.
Ɉɬɪɢɰɚɬɟɥɶɧɚɹ ɜɟɥɢɱɢɧɚ ɩɪɨɟɤɰɢɢ ɢɫɤɨɦɨɣ ɫɤɨɪɨɫɬɢ ɝɨɜɨɪɢ ɨ ɬɨɦ, ɱɬɨ ɨɧɚ
ɧɚɩɪɚɜɥɟɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɨɫɢ (ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɩɟɪɜɨɧɚɱɚɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ
ɲɚɪɚ). ȼɟɥɢɱɢɧɚ (ɦɨɞɭɥɶ) ɫɤɨɪɨɫɬɢ
X
= 0,3 ɦ/ɫ.
Ɉɬɜɟɬ:
X
= 0,3 ɦ/ɫ.
Ɂɚɞɚɱɚ 6. Ɍɟɥɨ ɫɤɨɥɶɡɢɬ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ, ɨɛɪɚɡɭɸɳɟɣ ɫ ɝɨɪɢɡɨɧ-
ɬɨɦ ɭɝɨɥ Į = 45q (ɪɢɫ. 3.4). ɉɪɨɣɞɟɧɧɵɣ ɬɟɥɨɦ ɩɭɬɶ ɞɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ s = ɋt
ɝɞɟ ɋ = 1,73 ɦ/ɫ
2
. ɇɚɣɬɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɬɟɥɚ ɨ ɩɥɨɫɤɨɫɬɶ.
2
,
Ɋɢɫ. 3.4
Ⱦɚɧɨ:
D
= 45q
2
s = ɋt
ɦ
= 1,73 ɦ/ɫ2
P
– ?
1. Ɍɚɤ ɤɚɤ ɩɪɢ ɨɞɧɨɧɚɩɪɚɜɥɟɧɧɨɦ ɞɜɢɠɟɧɢɢ ɩɭɬɶ s = ɯ – ɯ0,
ɬɨ ɩɪɢɧɹɜ ɯ
= 0, ɢɦɟɟɦ ɯ = s = Ct2.
0
Ɍɨɝɞɚ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ
ɚ ɭɫɤɨɪɟɧɢɟ ɚ =
= 2ɋ = const.
Ɋɟɲɟɧɢɟ:
X
2. ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɜɟɥɢɱɢɧɚ ɭɫɤɨɪɟɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɧɚɣɞɟɧɚ ɩɭɬɟɦ ɪɟ-
ɲɟɧɢɹ ɞɢɧɚɦɢɱɟɫɤɨɣ ɡɚɞɚɱɢ ɩɨ ɭɤɚɡɚɧɧɨɣ ɪɚɧɟɟ ɫɯɟɦɟ:
GG
ma mg N F
GG
mp
.
ɍɞɨɛɧɨ ɜɵɛɪɚɬɶ ɨɫɶ ɯ ɜɞɨɥɶ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ, ɚ ɨɫɶ y ɩɟɪɩɟɧɞɢɤɭɥɹɪ-
ɧɨ ɤ ɧɟɣ. Ɍɨɝɞɚ:
ma mg F
:
OX
®
:
Ɍɚɤ ɤɚɤ F
OY
= PN, ɢɦɟɟɦ
mp
0cos 0.
¯
ma mg N
®
Nmg
¯
Ⱥ = g(sin
Ɉɤɨɧɱɚɬɟɥɶɧɨ g(sin
sin 2 9,8 0,7 2 1,73
gC
P
D
cos 9,8 0, 7
g
D
D
sin 0 ;
D
mg N
D
sin ;
DP
cos .
D
D
– P cosD).
mp
– P cosD) = 2C.
0, 5.
= ɯ = 2ɋt,
Ɉɬɜɟɬ: 0,5.
33
3.3. Ɂɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ
1. Ⱥɜɬɨɦɨɛɢɥɶ, ɢɦɟɸɳɢɣ ɫɤɨɪɨɫɬɶ 10 ɦ/ɫ, ɧɚɱɢɧɚɟɬ ɞɜɢɝɚɬɶɫɹ ɜɜɟɪɯ ɩɨ ɞɨ-
ɪɨɝɟ ɫ ɭɝɥɨɦ ɧɚɤɥɨɧɚ ɤ ɝɨɪɢɡɨɧɬɭ, ɫɨɫɬɚɜɥɹɸɳɢɦ 10q. Ɉɩɪɟɞɟɥɢɬɶ ɩɭɬɶ, ɩɪɨɣɞɟɧɧɵɣ ɞɨ ɨɫɬɚɧɨɜɤɢ, ɢ ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ, ɟɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɩɪɢ ɞɜɢɠɟɧɢɢ 0,5.
2. ɑɟɪɟɡ ɧɟɩɨɞɜɢɠɧɵɣ ɛɥɨɤ ɩɟɪɟɤɢɧɭɬɚ ɬɨɧɤɚɹ ɧɟɪɚɫɬɹɠɢɦɚɹ ɧɢɬɶ, ɧɚ ɤɨɧ-
ɰɚɯ ɤɨɬɨɪɨɣ ɜɢɫɹɬ ɝɪɭɡɵ ɦɚɫɫɚɦɢ ɬ
= 200 ɝ ɢ ɬ2 = 300 ɝ. Ʉɚɤɨɣ ɩɭɬɶ ɩɪɨɣɞɟɬ
1
ɤɚɠɞɵɣ ɢɡ ɝɪɭɡɨɜ ɡɚ ɩɟɪɜɭɸ ɫɟɤɭɧɞɭ? Ɇɚɫɫɨɣ ɛɥɨɤɚ ɢ ɬɪɟɧɢɟɦ ɜ ɧɟɦ ɩɪɟɧɟɛɪɟɱɶ.
3. ɇɚ ɤɚɤɨɣ ɜɵɫɨɬɟ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ Ɂɟɦɥɢ ɭɫɤɨɪɟɧɢɟ ɫɜɨɛɨɞɧɨɝɨ ɩɚɞɟɧɢɹ
ɞɨɫɬɢɝɚɟɬ ɜɟɥɢɱɢɧɵ 1 ɦ/ɫ
2
.
4. Ɍɟɥɨ ɦɚɫɫɨɣ ɬ ɞɜɢɠɟɬɫɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɫ ɩɨɦɨɳɶɸ ɩɪɭɠɢɧɵ ɠɟɫɬɤɨ-
ɫɬɶɸ k. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɪɚɜɟɧ
P
. Ɉɩɪɟɞɟɥɢɬɶ ɭɫɤɨɪɟɧɢɟ ɬɟɥɚ, ɟɫɥɢ ɩɪɭɠɢ-
ɧɚ ɪɚɫɬɹɧɭɬɚ ɧɚ ɜɟɥɢɱɢɧɭ 'Ɛ.
5. Ʉɚɤɨɜɚ ɫɤɨɪɨɫɬɶ ɩɭɥɢ ɩɪɢ ɜɵɥɟɬɟ ɢɡ ɞɭɯɨɜɨɝɨ ɪɭɠɶɹ, ɟɫɥɢ ɟɟ ɦɚɫɫɚ ɬ = 2,5 ɝ,
ɞɥɢɧɚ ɫɬɜɨɥɚ Ɛ = 0,7 ɦ, ɤɚɥɢɛɪ D = 5 ɦɦ, ɚ ɫɪɟɞɧɟɟ ɞɚɜɥɟɧɢɟ ɜɨɡɞɭɯɚ ɜɨ ɜɪɟɦɹ
ɜɵɫɬɪɟɥɚ ɜ ɫɬɜɨɥɟ ɪ = 9,8 ɉɚ?
6. Ɍɟɥɨ ɦɚɫɫɨɣ ɬ = 0,5 ɤɝ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ. ɉɪɢ ɷɬɨɦ
ɬɟɥɚ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ ɯ(t) = 6 + 9t
2
– 7t3 (ɦ). ɇɚɣɬɢ ɫɢɥɭ, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ
ɤɨɨɪɞɢɧɚɬɚ
ɬɟɥɚ ɜ ɤɨɧɰɟ ɜɬɨɪɨɣ ɫɟɤɭɧɞɵ ɞɜɢɠɟɧɢɹ.
7. ɇɚɣɬɢ ɨɬɧɨɲɟɧɢɟ ɫɢɥ ɞɚɜɥɟɧɢɹ ɧɚ ɫɟɪɟɞɢɧɭ ɜɵɩɭɤɥɨɝɨ ɢ ɜɨɝɧɭɬɨɝɨ ɦɨ-
ɫɬɨɜ ɫ ɨɞɢɧɚɤɨɜɵɦ ɪɚɞɢɭɫɨɦ ɤɪɢɜɢɡɧɵ, ɪɚɜɧɵɦ 40 ɦ, ɩɪɢ ɞɜɢɠɟɧɢɢ ɚɜɬɨɦɨɛɢɥɹ ɫɨ ɫɤɨɪɨɫɬɶɸ 36 ɤɦ/ɱ.
8. ɒɚɪɢɤ ɦɚɫɫɨɣ ɬ ɥɟɬɢɬ ɫɨ ɫɤɨɪɨɫɬɶɸ
G
X
ɩɨɞ ɭɝɥɨɦ 30q ɤ ɩɨɜɟɪɯɧɨɫɬɢ
0
ɫɬɟɧɵ. ɇɚɣɬɢ ɢɦɩɭɥɶɫ, ɩɟɪɟɞɚɧɧɵɣ ɫɬɟɧɤɟ, ɫɱɢɬɚɹ ɭɞɚɪ ɲɚɪɢɤɚ ɨ ɫɬɟɧɤɭ ɚɛɫɨɥɸɬɧɨ ɭɩɪɭɝɢɦ.
9. Ɉɩɪɟɞɟɥɢɬɶ ɩɟɪɢɨɞ ɨɛɪɚɳɟɧɢɹ ɤɨɧɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ, ɟɫɥɢ ɟɝɨ ɞɥɢɧɚ 45 ɫɦ,
ɚ ɭɝɨɥ, ɨɛɪɚɡɭɟɦɵɣ ɧɢɬɶɸ, ɫ ɜɟɪɬɢɤɚɥɶɸ 60q.
10. Ɍɟɥɨ ɦɚɫɫɨɣ 300 ɝ ɫɜɨɛɨɞɧɨ ɩɚɞɚɟɬ ɫ ɜɵɫɨɬɵ 10 ɦ. Ʉɨɝɞɚ ɜɵɫɨɬɚ
ɭɦɟɧɶɲɚɟɬɫɹ ɜɞɜɨɟ, ɜ ɬɟɥɨ ɩɨɩɚɞɚɟɬ
ɢ ɡɚɫɬɪɟɜɚɟɬ ɜ ɧɟɦ ɩɭɥɹ ɦɚɫɫɨɣ 10 ɝ, ɥɟɬɟɜɲɚɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ 400 ɦ/ɫ. ɇɚɣɬɢ ɭɝɨɥ, ɤɨɬɨɪɵɣ ɫɨɫɬɚɜɢɬ ɫ
ɝɨɪɢɡɨɧɬɨɦ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɩɨɫɥɟ ɭɞɚɪɚ ɩɭɥɢ.
34
3.4. Ⱦɢɧɚɦɢɤɚ. Ɍɟɫɬɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ɡɧɚɧɢɣ
ɍɪɨɜɟɧɶ I
1. Ʉɚɤɢɦ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ ɜɵɪɚɠɚɟɬɫɹ ɬɪɟɬɢɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ?
2. Ʉɚɤɨɜɨ ɧɚɩɪɚɜɥɟɧɢɟ ɭɫɤɨɪɟɧɢɹ ɬɟɥɚ?
3. Ɋɚɡɦɟɪɧɨɫɬɶ ɫɢɥɵ ɜ ɫɢɫɬɟɦɟ ɋɂ.
4. ɇɚɣɞɢɬɟ ɫɢɥɭ ɞɚɜɥɟɧɢɹ ɝɪɭɡɚ ɦɚɫɫɨɣ ɬ = 50 ɤɝ ɧɚ ɩɨɥ ɥɢɮɬɚ, ɩɨɞɧɢɦɚ-
ɸɳɟɝɨɫɹ ɜɜɟɪɯ ɫ ɭɫɤɨɪɟɧɢɟɦ 1,2 ɦ/ɫ
2
.
5. ɇɚɣɞɢɬɟ ɫɢɥɭ ɧɚɬɹɠɟɧɢɹ ɠɟɫɬɤɨɣ ɧɢɬɢ, ɟɫɥɢ ɝɪɭɡɵ ɧɚ ɪɢɫɭɧɤɟ ɞɜɢɠɭɬɫɹ
ɩɪɚɤɬɢɱɟɫɤɢ ɛɟɡ ɬɪɟɧɢɹ ɫ ɭɫɤɨɪɟɧɢɟɦ ɚ.
6. ɑɟɦɭ ɪɚɜɟɧ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɤɨɥɟɫ ɚɜɬɨɦɨɛɢɥɹ ɨ ɞɨɪɨɝɭ, ɟɫɥɢ ɩɪɢ
ɫɤɨɪɨɫɬɢ 10 ɦ/ɫ ɬɨɪɦɨɡɧɨɣ ɩɭɬɶ ɪɚɜɟɧ 8 ɦɟɬɪɨɜ?
7. Ʉɨɨɪɞɢɧɚɬɚ ɬɟɥɚ ɦɚɫɫɨɣ 1 ɤɝ ɩɪɢ ɞɜɢɠɟɧɢɢ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ ɯ(t) =
= (5 + t)
2
ɦ. ɇɚɣɞɢɬɟ ɢɦɩɭɥɶɫ ɬɟɥɚ ɤ ɤɨɧɰɭ ɬɪɟɬɶɟɣ ɫɟɤɭɧɞɵ ɞɜɢɠɟɧɢɹ.
8. ɇɚɣɞɢɬɟ ɢɡɦɟɧɟɧɢɟ ɢɦɩɭɥɶɫɚ ɬɟɥɚ ɦɚɫɫɨɣ 5 ɤɝ, ɪɚɜɧɨɦɟɪɧɨ ɞɜɢɠɭɳɟɝɨ-
ɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɫɨ ɫɤɨɪɨɫɬɶɸ 2 ɦ/ɫ, ɡɚ 1/4 ɩɟɪɢɨɞɚ.
9. Ɍɟɥɨ ɞɜɢɠɟɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ 2 ɦ/ɫ ɢ ɭɞɚɪɹɟɬɫɹ ɨ ɧɟɩɨɞɜɢɠɧɨɟ ɬɟɥɨ ɬɚɤɨɣ
ɠɟ ɦɚɫɫɵ. ɋɱɢɬɚɹ ɭɞɚɪ ɰɟɧɬɪɚɥɶɧɵɦ ɢ ɧɟɭɩɪɭɝɢɦ, ɨɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ ɬɟɥ ɩɨɫɥɟ ɭɞɚɪɚ.
10. Ⱥɜɬɨɦɚɬ ɜɵɩɭɫɤɚɟɬ 600 ɩɭɥɶ ɡɚ ɫɟɤɭɧɞɭ. Ɇɚɫɫɚ ɩɭɥɢ 4 ɝ, ɟɟ ɫɤɨ-
ɪɨɫɬɶ 500 ɦ/ɫ. ɇɚɣɬɢ ɫɪɟɞɧɸɸ ɫɢɥɭ ɨɬɞɚɱɢ ɩɪɢ ɫɬɪɟɥɶɛɟ.
ɍɪɨɜɟɧɶ II
1. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɬɪɚɟɤɬɨɪɢɢ Ⱥȼ ɩɨɞ
ɞɟɣɫɬɜɢɟɦ ɨɞɧɨɣ ɫɢɥɵ (ɪɢɫ. 3.5). Ʉɚɤɢɟ ɧɚɩɪɚɜɥɟɧɢɹ ɫɢɥɵ ɧɟɜɨɡɦɨɠɧɵ?
Ɋɢɫ. 3.5
2. Ʉɚɤɨɜɚ ɜ ɫɢɫɬɟɦɟ ɋɂ ɟɞɢɧɢɰɚ ɫɤɨɪɨɫɬɢ ɢɡɦɟɧɟɧɢɹ ɢɦɩɭɥɶɫɚ
3. Ɍɟɥɨ ɦɚɫɫɨɣ 0,5 ɤɝ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ. ɉɭɬɶ, ɩɪɨɣɞɟɧɧɵɣ ɬɟɥɨɦ,
ɨɩɢɫɵɜɚɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ s(t) = 20 – 8t + 5t
2
– t3 (ɦ). ɇɚɣɬɢ, ɤɚɤɚɹ ɫɢɥɚ ɞɟɣ-
§·
¨¸
©¹
ɫɬɜɭɟɬ ɧɚ ɬɟɥɨ ɱɟɪɟɡ 1 ɫ ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
4. Ʉ ɧɢɬɢ ɩɨɞɜɟɲɟɧ ɝɪɭɡ ɦɚɫɫɨɣ 1 ɤɝ. ɇɚɣɬɢ ɫɢɥɭ ɧɚɬɹɠɟɧɢɹ ɧɢɬɢ, ɟɫɥɢ
ɧɢɬɶ ɫ ɝɪɭɡɨɦ ɩɨɞɧɢɦɚɬɶ ɜɜɟɪɯ ɫ ɭɫɤɨɪɟɧɢɟɦ 5 ɦ/ɫ
2
(g = 10 ɦ/ɫ2).
35
dp
dt
?
5. Ʉɚɦɟɧɶ, ɩɪɢɜɹɡɚɧɧɵɣ ɤ ɜɟɪɟɜɤɟ, ɪɚɜɧɨɦɟɪɧɨ ɜɪɚɳɚɟɬɫɹ ɜ ɜɟɪɬɢɤɚɥɶɧɨɣ
p
ɩɥɨɫɤɨɫɬɢ. ɇɚɣɬɢ ɦɚɫɫɭ ɤɚɦɧɹ, ɟɫɥɢ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɪɚɡɧɨɫɬɶ ɫɢɥ ɧɚɬɹɠɟɧɢɹ ɜɟɪɟɜɤɢ ɜ ɧɢɠɧɟɣ ɢ ɜɟɪɯɧɟɣ ɬɨɱɤɚɯ ɬɪɚɟɤɬɨɪɢɢ ɪɚɜɧɚ 10 ɇ (g = 10 ɦ/ɫ
2
).
6. ɉɨɟɡɞ ɦɚɫɫɨɣ 360 ɬɨɧɧ ɩɪɢ ɬɨɪɦɨɠɟɧɢɢ ɞɜɢɝɚɟɬɫɹ ɬɚɤ, ɱɬɨ ɟɝɨ ɫɤɨɪɨɫɬɶ
ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɜɪɟɦɟɧɢ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɝɪɚɮɢɤɨɦ (ɪɢɫ. 3.6). ɇɚɣɬɢ
ɫɢɥɭ ɬɨɪɦɨɠɟɧɢɹ.
Ɋɢɫ. 3.6
7. Ʉɨɨɪɞɢɧɚɬɵ ɞɜɭɯ ɬɟɥ, ɦɚɫɫɵ ɤɨɬɨɪɵɯ ɬ
ɢɡɦɟɧɹɸɬɫɹ ɩɨ ɡɚɤɨɧɭ ɯ
(t) = 10t2 – 20, x2(t) = 40 – 5t2. ɇɚɣɞɢɬɟ ɢɯ ɢɦɩɭɥɶɫ ɩɨɫɥɟ
1
= 3 ɤɝ, ɬ2 = 2 ɤɝ, ɩɪɢ ɞɜɢɠɟɧɢɢ
1
ɧɟɭɩɪɭɝɨɝɨ ɭɞɚɪɚ.
8. ɒɚɪɢɤ ɦɚɫɫɨɣ 0,1 ɤɝ, ɞɜɢɝɚɹɫɶ ɩɨ ɧɨɪɦɚɥɢ ɤ ɫɬɟɧɟ ɫɨ ɫɤɨɪɨɫɬɶɸ 6 ɦ/ɫ
ɚɛɫɨɥɸɬɧɨ ɭɩɪɭɝɨ ɭɞɚɪɢɥɫɹ ɨɛ ɧɟɟ ɢ ɨɬɫɤɨɱɢɥ. ɇɚɣɬɢ ɢɡɦɟɧɟɧɢɟ ɢɦɩɭɥɶɫɚ ɲɚɪɢɤɚ.
9. ɂɦɩɭɥɶɫ ɞɜɢɠɭɳɟɝɨɫɹ ɬɟɥɚ ɢɦɟɟɬ ɜɢɞ
G
(5 8)
G
2
ti
. ɇɚɣɬɢ ɞɟɣɫɬɜɭɸ-
ɳɭɸ ɧɚ ɧɟɝɨ ɫɢɥɭ ɱɟɪɟɡ 6 ɫɟɤɭɧɞ ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
10. ɋ ɤɚɤɨɣ ɫɤɨɪɨɫɬɶɸ ɩɨɫɥɟ ɝɨɪɢɡɨɧɬɚɥɶɧɨɝɨ ɜɵɫɬɪɟɥɚ ɢɡ ɜɢɧɬɨɜɤɢ ɫɬɚɥ
ɞɜɢɝɚɬɶɫɹ ɫɬɪɟɥɨɤ, ɫɬɨɹɳɢɣ ɧɚ ɝɥɚɞɤɨɦ ɥɶɞɭ? Ɇɚɫɫɚ ɫɬɪɟɥɤɚ ɫ ɜɢɧɬɨɜɤɨɣ 70 ɤɝ,
ɦɚɫɫɚ ɩɭɥɢ 10 ɝ ɢ ɟɟ ɫɤɨɪɨɫɬɶ ɩɪɢ ɜɵɥɟɬɟ 700 ɦ/ɫ.
ɍɪɨɜɟɧɶ III
1. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ Ɇ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɱɚɫɨɜɨɣ
ɫɬɪɟɥɤɢ ɬɚɤ, ɱɬɨ ɬɚɧɝɟɧɰɢɚɥɶɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɫɤɨɪɨɫɬɢ ɢɡɦɟɧɹɟɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɝɪɚɮɢɤɨɦ (ɪɢɫ. 3.7).
Ɋɢɫ. 3.7
36
Ɉɩɪɟɞɟɥɢɬɟ ɧɚɩɪɚɜɥɟɧɢɟ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɬɨɱɤɭ Ɇ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟ-
ɧɢ t = t
(ɪɢɫ. 3.8).
1
Ɋɢɫ. 3.8
2. Ɍɟɥɨ, ɛɪɨɲɟɧɧɨɟ ɜɜɟɪɯ, ɜɨɡɜɪɚɳɚɟɬɫɹ ɜ ɢɫɯɨɞɧɭɸ ɬɨɱɤɭ. ɋɪɚɜɧɢɬɶ ɜɪɟɦɹ ɩɨɞɴɟɦɚ (t
) ɢ ɜɪɟɦɹ ɩɚɞɟɧɢɹ (t
ɩɨɞ
), ɭɱɢɬɵɜɚɹ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɜɨɡɞɭɯɚ.
ɩɚɞ
3. ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ F ɬɟɥɨ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ ɬɚɤ, ɱɬɨ ɡɚɜɢɫɢɦɨɫɬɶ ɩɪɨɣɞɟɧɧɨɝɨ ɬɟɥɨɦ ɩɭɬɢ ɨɬ ɜɪɟɦɟɧɢ ɢɦɟɟɬ ɜɢɞ s = A – Bt + Ct
(Ⱥ; ȼ; ɋ – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɤɨɧɫɬɚɧɬɵ). ɇɚɣɬɢ ɦɚɫɫɭ ɬɟɥɚ.
4. ɉɨɟɡɞ ɦɚɫɫɨɣ ɬ = 500 ɬɨɧɧ ɩɨɫɥɟ ɩɪɟɤɪɚɳɟɧɢɹ ɬɹɝɢ ɬɟɩɥɨɜɨɡɚ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɬɪɟɧɢɹ F
= 100 ɤɇ ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɱɟɪɟɡ t = 1 ɦɢɧ. ɋ ɤɚɤɨɣ ɫɤɨ-
ɬɪ
ɪɨɫɬɶ ɲɟɥ ɩɨɟɡɞ?
5. Ƚɢɪɶɤɚ, ɩɪɢɜɹɡɚɧɧɚɹ ɤ ɧɢɬɢ ɞɥɢɧɨɣ 60 ɫɦ, ɨɩɢɫɵɜɚɟɬ ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ
ɩɥɨɫɤɨɫɬɢ ɨɤɪɭɠɧɨɫɬɶ ɫ ɪɚɞɢɭɫɨɦ 30 ɫɦ. ɇɚɣɬɢ ɩɟɪɢɨɞ ɨɛɪɚɳɟɧɢɹ ɝɢɪɶɤɢ.
6. Ⱦɜɢɠɟɧɢɟ ɬɟɥɚ ɦɚɫɫɨɣ 0,5 ɤɝ ɡɚɞɚɧɨ ɭɪɚɜɧɟɧɢɹɦɢ ɯ(t) = 10t
3
(ɦ); y(t) = 5t (ɦ).
ɇɚɣɬɢ ɫɢɥɭ, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɬɟɥɨ ɜ ɤɨɧɰɟ ɜɬɨɪɨɣ ɫɟɤɭɧɞɵ ɞɜɢɠɟɧɢɹ.
7. Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ ɞɜɢɠɭɳɟɣɫɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ 2 ɤɝ ɢɦɟɟɬ ɜɢɞ
G
() 4 3rt ti tj
ɩɨ ɨɤɪɭɠɧɨɫɬɢ, ɪɚɜɧɨ 10 ɫ
GG
2
8. ɍɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ 0,2 ɤɝ, ɞɜɢɠɭɳɟɣɫɹ
. ɇɚɣɬɢ ɢɦɩɭɥɶɫ ɬɟɥɚ ɱɟɪɟɡ 3 ɫɟɤɭɧɞɵ ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
–2
. ɇɚɣɬɢ ɢɡɦɟɧɟɧɢɟ ɢɦɩɭɥɶɫɚ ɬɨɱɤɢ ɩɨɫɥɟ ɩɨɜɨɪɨɬɚ
ɧɚ 180q.
9. ȼ ɹɳɢɤ ɫ ɩɟɫɤɨɦ ɦɚɫɫɨɣ 9 ɤɝ, ɞɜɢɠɭɳɢɣɫɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɫɨ ɫɤɨ-
ɪɨɫɬɶɸ 6 ɦ/ɫ, ɩɚɞɚɟɬ ɝɢɪɹ ɦɚɫɫɨɣ 1 ɤɝ, ɨɬɩɭɳɟɧɧɚɹ ɫ ɜɵɫɨɬɵ 10 ɦɟɬɪɨɜ. Ɉɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ ɹɳɢɤɚ ɫ ɝɢɪɟɣ.
10. ɋɬɚɥɶɧɨɣ ɲɚɪɢɤ ɦɚɫɫɨɣ 20 ɝ, ɩɚɞɚɹ ɫ ɜɵɫɨɬɵ 1 ɦ ɧɚ ɫɬɚɥɶɧɭɸ ɩɥɢɬɭ,
ɨɬɫɤɚɤɢɜɚɟɬ ɨɬ ɧɟɟ
ɭɞɚɪɟ (g = 10 ɦ/ɫ
ɧɚ ɜɵɫɨɬɭ 0,81 ɦ. ɇɚɣɬɢ ɢɦɩɭɥɶɫ, ɩɨɥɭɱɟɧɧɵɣ ɩɥɢɬɨɣ ɩɪɢ
2
).
2
37
4. ɊȺȻɈɌȺ ɂ ɗɇȿɊȽɂə
F
G
G
G
G
Ⱥ
s
p
A
4.1. Ʉɪɚɬɤɢɟ ɫɜɟɞɟɧɢɹ ɢɡ ɬɟɨɪɢɢ
ɉɭɫɬɶ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ
ɜɟɪɲɚɟɬ ɷɥɟɦɟɧɬɚɪɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ
ɮɢɡɢɱɟɫɤɨɟ ɬɟɥɨ (ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ) ɫɨ-
Sd
. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɫɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ
ɬɟɥɨ, ɧɟ ɹɜɥɹɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɨɣ. ɋɤɚɥɹɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɨɩɪɟɞɟɥɹɟɦɚɹ
ɫɤɚɥɹɪɧɵɦ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɬɟɥɨ ɧɚ ɟɝɨ ɷɥɟɦɟɧɬɚɪɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɷɬɨɣ ɫɢɥɵ, ɧɚɡɵɜɚɟɬɫɹ ɷɥɟɦɟɧɬɚɪɧɚɹ ɪɚɛɨɬɚ:
.
),( SdFdA
Ɋɚɛɨɬɚ ɫɢɥɵ, ɤɨɬɨɪɚɹ ɢɡɦɟɧɹɟɬɫɹ ɧɚ ɩɭɬɢ S, ɦɨɠɟɬ ɛɵɬɶ ɨɩɪɟɞɟɥɟɧɚ ɩɨ
ɮɨɪɦɭɥɟ:
, (4.1)
FdS
S
ɝɞɟ
– ɩɪɨɟɤɰɢɹ ɫɢɥɵ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɩɟɪɟɦɟɳɟɧɢɹ ɬɟɥɚ.
F
³
S
Ƚɪɚɮɢɱɟɫɤɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɮɨɪɦɭɥɵ (4.1) ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ. 4.1, ɧɚ ɤɨɬɨ-
ɪɨɦ ɩɪɟɞɫɬɚɜɥɟɧ ɝɪɚɮɢɤ F
ɤɚɤ ɮɭɧɤɰɢɢ ɩɨɥɨɠɟɧɢɹ ɬɨɱɤɢ ɧɚ ɬɪɚɟɤɬɨɪɢɢ (ɝɨɪɢɡɨɧ-
s
ɬɚɥɶɧɭɸ ɨɫɶ ɦɨɠɧɨ ɧɚɡɜɚɬɶ ɨɫɶɸ, ɞɥɢɧɚ ɨɬɪɟɡɤɚ ɷɬɨɣ ɨɫɢ ɦɟɠɞɭ ɬɨɱɤɚɦɢ 1 ɢ 2
ɪɚɜɧɚ ɩɨɥɧɨɣ ɞɥɢɧɟ ɩɭɬɢ). ɂɡ ɪɢɫɭɧɤɚ ɜɢɞɧɨ, ɱɬɨ ɷɥɟɦɟɧɬɚɪɧɚɹ ɪɚɛɨɬɚ ɱɢɫɥɟɧɧɨ
ɪɚɜɧɚ ɩɥɨɳɚɞɢ ɡɚɲɬɪɢɯɨɜɚɧɧɨɣ ɩɨɥɨɫɤɢ. Ɋɚɛɨɬɚ ɧɚ ɩɭɬɢ ɨɬ ɬɨɱɤɢ 1 ɞɨ ɬɨɱɤɢ 2
ɱɢɫɥɟɧɧɨ ɪɚɜɧɚ ɩɥɨɳɚɞɢ ɮɢɝɭɪɵ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɤɪɢɜɨɣ F
ɩɪɹɦɵɦɢ 1 ɢ 2 ɢ ɨɫɶɸ S.
, ɜɟɪɬɢɤɚɥɶɧɵɦɢ
s
Ɋɢɫ. 4.1
ȿɫɥɢ ɧɚ ɬɟɥɨ ɨɞɧɨɜɪɟɦɟɧɧɨ ɞɟɣɫɬɜɭɟɬ ɧɟɫɤɨɥɶɤɨ ɫɢɥ, ɪɟɡɭɥɶɬɢɪɭɸɳɚɹ ɤɨ-
ɬɨɪɵɯ ɪɚɜɧɚ
GG
FF ¦
, ɬɨ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɪɚɛɨɬɵ ɪɟɡɭɥɶɬɢɪɭɸɳɟɣ ɫɢɥɵ ɛɭɞɟɬ
n
n
ɢɦɟɬɶ ɜɢɞ:
n
A
ɝɞɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɧɚɯɨɞɢɬɫɹ ɚɥɝɟɛɪɚɢɱɟɫɤɚɹ ɫɭɦɦɚ ɪɚɛɨɬ ɤɚɠɞɨɣ ɢɡ ɫɢɥ
GG G
,,...,
FF F
12
, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɨ.
n
, (4.2)
¦
i
1
i
38
Ɋɚɛɨɬɚ ɭɩɪɭɝɨɣ ɫɢɥɵ, ɩɨɞɱɢɧɹɸɳɟɣɫɹ ɡɚɤɨɧɭ Ƚɭɤɚ, ɩɪɢ ɪɚɫɬɹɠɟɧɢɢ ɢ
F
G
X
X
F
N
X
A
ɫɠɚɬɢɢ ɩɪɭɠɢɧɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
2
kxA
, (4.3)
2
ɝɞɟ k – ɤɨɷɮɮɢɰɢɟɧɬ ɠɟɫɬɤɨɫɬɢ ɩɪɭɠɢɧɵ,
x – ɜɟɥɢɱɢɧɚ ɞɟɮɨɪɦɚɰɢɢ ɩɪɭɠɢɧɵ.
ɇɚ ɩɪɚɤɬɢɤɟ ɢɦɟɟɬ ɡɧɚɱɟɧɢɟ ɧɟ ɬɨɥɶɤɨ ɜɟɥɢɱɢɧɚ ɫɨɜɟɪɲɟɧɧɨɣ ɪɚɛɨɬɵ, ɧɨ ɢ
ɜɪɟɦɹ, ɜ ɬɟɱɟɧɢɟ ɤɨɬɨɪɨɝɨ ɨɧɚ ɫɨɜɟɪɲɚɟɬɫɹ. ɉɨɷɬɨɦɭ ɞɥɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɦɟɯɚɧɢɡɦɨɜ, ɩɪɟɞɧɚɡɧɚɱɟɧɧɵɯ ɞɥɹ ɫɨɜɟɪɲɟɧɢɹ ɪɚɛɨɬɵ, ɜɜɨɞɢɬɫɹ ɜɟɥɢɱɢɧɚ, ɩɨɤɚɡɵɜɚɸɳɚɹ, ɤɚɤɭɸ ɪɚɛɨɬɭ ɞɚɧɧɵɣ ɦɟɯɚɧɢɡɦ ɫɨɜɟɪɲɚɟɬ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ. ɗɬɚ ɜɟɥɢɱɢɧɚ ɧɚɡɵɜɚɟɬɫɹ ɦɨɳɧɨɫɬɶɸ.
Ɇɝɧɨɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɦɨɳɧɨɫɬɢ ɜɜɨɞɢɬɫɹ, ɤɨɝɞɚ ɡɚ ɪɚɜɧɵɟ ɩɪɨɦɟɠɭɬɤɢ
ɜɪɟɦɟɧɢ ɫɨɜɟɪɲɚɟɬɫɹ ɧɟɨɞɢɧɚɤɨɜɚɹ ɪɚɛɨɬɚ, ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɫɤɚɥɹɪɧɨɟ ɩɪɨ-
ɢɡɜɟɞɟɧɢɟ ɜɟɤɬɨɪɚ ɫɢɥɵ
ɧɚ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ
G
, ɫ ɤɨɬɨɪɨɣ ɞɜɢɠɟɬɫɹ ɬɨɱɤɚ
ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ:
G
G
. (4.4)
Ɏɢɡɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɬɟɥɚ ɢɥɢ ɫɢɫɬɟɦɵ ɬɟɥ
ɫɨɜɟɪɲɚɬɶ ɪɚɛɨɬɭ, ɧɚɡɵɜɚɟɬɫɹ ɷɧɟɪɝɢɟɣ. ɗɧɟɪɝɢɹ ɬɟɥɚ ɦɨɠɟɬ ɛɵɬɶ ɨɛɭɫɥɨɜɥɟɧɚ,
ɜɨ-ɩɟɪɜɵɯ, ɞɜɢɠɟɧɢɟɦ ɬɟɥɚ ɫ ɧɟɤɨɬɨɪɨɣ ɫɤɨɪɨɫɬɶɸ ɢ, ɜɨ-ɜɬɨɪɵɯ, ɧɚɯɨɠɞɟɧɢɟɦ
ɬɟɥɚ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɩɨɥɟ ɫɢɥ. ɗɧɟɪɝɢɹ ɩɟɪɜɨɝɨ ɜɢɞɚ ɧɚɡɵɜɚɟɬɫɹ ɤɢɧɟɬɢɱɟɫɤɨɣ
ɷɧɟɪɝɢɟɣ, ɚ ɜɬɨɪɨɝɨ ɜɢɞɚ – ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɟɣ. ɉɨɥɟ ɫɢɥ ɧɚɡɵɜɚɟɬɫɹ ɩɨɬɟɧɰɢɚɥɶɧɵɦ, ɟɫɥɢ
ɪɚɛɨɬɚ ɫɢɥ ɩɨɥɹ ɧɚɞ ɬɟɥɨɦ, ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɩɭɬɢ, ɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɶɤɨ ɧɚɱɚɥɶɧɵɦ ɢ ɤɨɧɟɱɧɵɦ ɩɨɥɨɠɟɧɢɟɦ ɬɟɥɚ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ. ȼ ɷɬɨɦ
ɫɥɭɱɚɟ ɫɚɦɢ ɫɢɥɵ ɧɚɡɵɜɚɸɬɫɹ ɤɨɧɫɟɪɜɚɬɢɜɧɵɦɢ.
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ m, ɤɨɬɨɪɚɹ ɞɜɢɠɟɬɫɹ
ɫɨ ɫɤɨɪɨɫɬɶɸ
G
, ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
Ɍ
2
m
X
. (4.5)
2
Ɋɚɛɨɬɚ, ɫɨɜɟɪɲɚɟɦɚɹ ɧɚɞ ɬɟɥɨɦ, ɪɚɜɧɚ ɩɪɢɪɚɳɟɧɢɸ ɟɝɨ ɤɢɧɟɬɢɱɟɫɤɨɣ
ɷɧɟɪɝɢɢ:
ɝɞɟ
A ɌɌ
,ɌɌ
– ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɜ ɤɨɧɟɱɧɨɦ ɢ ɧɚɱɚɥɶɧɨɦ ɫɨɫɬɨɹɧɢɹɯ,
21
,
21
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
Ɋɚɛɨɬɚ, ɫɨɜɟɪɲɟɧɧɚɹ ɧɚɞ ɬɟɥɨɦ ɤɨɧɫɟɪɜɚɬɢɜɧɵɦɢ ɫɢɥɚɦɢ ɩɨɬɟɧɰɢɚɥɶɧɨɝɨ
ɩɨɥɹ ɧɚ ɥɸɛɨɦ ɩɭɬɢ, ɧɚɱɢɧɚɸɳɟɦɫɹ ɜ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɟ 1 ɢ ɡɚɤɚɧɱɢɜɚɸɳɟɦɫɹ ɜ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɟ 2, ɨɤɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨɣ ɭɛɵɥɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ
ɬɟɥɚ:
ɝɞɟ
UU
,UU
– ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɜ ɧɚɱɚɥɶɧɨɦ ɢ ɤɨɧɟɱɧɨɦ ɫɨɫɬɨɹɧɢ-
12
,
12
ɹɯ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
39
Ʉɨɧɤɪɟɬɧɵɣ ɜɢɞ ɮɨɪɦɭɥɵ ɞɥɹ ɪɚɫɱɟɬɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɬɟɥɚ ɡɚɜɢ-
gmU
U
UɌE
E
t
E
ɫɢɬ ɨɬ ɩɪɢɪɨɞɵ ɫɢɥɨɜɨɝɨ ɩɨɥɹ. ȼ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ ɜɛɥɢɡɢ ɡɟɦɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɦɚɫɫɵ m ɢɦɟɟɬ ɜɢɞ:
h
, (4.6)
h – ɜɵɫɨɬɚ, ɨɬɫɱɢɬɚɧɧɚɹ ɨɬ ɧɭɥɟɜɨɝɨ ɭɪɨɜɧɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ( 0
ɝɞɟ
).
ɇɚɱɚɥɨ ɨɬɫɱɟɬɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɦɨɠɧɨ ɜɵɛɢɪɚɬɶ ɩɪɨɢɡɜɨɥɶɧɨ, ɩɨɷɬɨɦɭ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɨɠɢɬɟɥɶɧɨɣ, ɪɚɜɧɨɣ ɧɭɥɸ,
ɚ ɬɚɤɠɟ ɢɦɟɬɶ ɨɬɪɢɰɚɬɟɥɶɧɵɟ ɡɧɚɱɟɧɢɹ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɧɟ
ɦɨɠɟɬ ɛɵɬɶ ɨɬɪɢɰɚɬɟɥɶɧɨɣ.
ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɦɚɫɫɨɣ ݉
ɩɨɥɟ, ɫɨɡɞɚɧɧɨɦ ɬɨɱɟɱɧɨɣ ɦɚɫɫɨɣ
మ
ʜήˏ
ɝɞɟ
ܩൌ6,67 ή 10
ିଵଵ
– ɝɪɚɜɢɬɚɰɢɨɧɧɚɹ ɩɨɫɬɨɹɧɧɚɹ,
మ
ˍˆ
ܯ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɝɥɚɫɧɨ ɜɵɪɚɠɟɧɢɸ:
ܷሺݎሻ ൌ െG
୫
୰
,
ɜ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ
ܯ – ɦɚɫɫɚ ɬɟɥɚ, ɫɨɡɞɚɸɳɚɹ ɝɪɚɜɢɬɚɰɢɨɧɧɨɟ ɩɨɥɟ,
݉ – ɦɚɫɫɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ,
r – ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɝɪɚɜɢɬɚɰɢɨɧɧɨɝɨ ɰɟɧɬɪɚ ɞɨ ɦɚɫɫɵ
݉.
ȼɢɞɢɦ, ɱɬɨ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɜɵɛɢɪɚɸɬ ɧɚ ɛɟɫɤɨɧɟɱɧɨɫɬɢ, ɩɨɫɤɨɥɶɤɭ
ܷሺݎ՜λሻൌ 0.
ɉɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɟɣ ɦɨɠɟɬ ɨɛɥɚɞɚɬɶ ɨɬɞɟɥɶɧɨ ɜɡɹɬɨɟ ɭɩɪɭɝɨ ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɟ ɬɟɥɨ (ɧɚɩɪɢɦɟɪ, ɫɠɚɬɚɹ ɢɥɢ ɪɚɫɬɹɧɭɬɚɹ ɩɪɭɠɢɧɚ). ȼ ɷɬɨɦ ɫɥɭɱɚɟ
ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɡɚɜɢɫɢɬ ɨɬ ɜɡɚɢɦɧɨɝɨ ɪɚɫɩɨɥɨɠɟɧɢɹ ɨɬɞɟɥɶɧɵɯ ɱɚɫɬɟɣ
ɬɟɥɚ (ɧɚɩɪɢɦɟɪ, ɨɬ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɜɢɬɤɚɦɢ ɩɪɭɠɢɧɵ).
ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ ɩɪɭɠɢɧɵ ɦɨɠɟɬ ɛɵɬɶ ɪɚɫɫɱɢɬɚɧɚ ɩɨ ɮɨɪɦɭɥɟ:
2
kxU
, (4.7)
2
ɝɞɟ k – ɤɨɷɮɮɢɰɢɟɧɬ ɠɟɫɬɤɨɫɬɢ ɩɪɭɠɢɧɵ;
x – ɜɟɥɢɱɢɧɚ ɞɟɮɨɪɦɚɰɢɢ ɩɪɭɠɢɧɵ.
ɋɭɦɦɚ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ T ɬɟɥɚ ɢ ɟɝɨ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ U ɨɛɪɚ-
ɩɨɥɧɭɸ ɦɟɯɚɧɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɬɟɥɚ:
ɡɭɟɬ
. (4.8)
ɉɪɢɪɚɳɟɧɢɟ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɬɟɥ, ɦɟɠɞɭ ɤɨɬɨɪɵɦɢ ɞɟɣɫɬɜɭɸɬ
ɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɥɵ, ɨɤɚɡɵɜɚɟɬɫɹ ɪɚɜɧɵɦ ɪɚɛɨɬɟ ɜɧɟɲɧɢɯ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ
ɤ ɬɟɥɚɦ ɫɢɫɬɟɦɵ:
EA
21 ɜɧɟɲ
. (4.9)
ȿɫɥɢ ɫɢɫɬɟɦɚ ɡɚɦɤɧɭɬɚ (ɜɧɟɲɧɢɟ ɫɢɥɵ ɨɬɫɭɬɫɬɜɭɸɬ), ɬɨ ɩɪɢɪɚɳɟɧɢɟ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɵɦ ɧɭɥɸ, ɨɬɤɭɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ
.cons
(4.10)
40
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