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ȼ ɮɨɪɦɭɥɚɯ (4.9), (4.10) ɡɚɤɥɸɱɟɧɨ ɫɭɳɟɫɬɜɨ ɨɞɧɨɝɨ ɢɡ ɨɫɧɨɜɧɵɯ ɡɚɤɨɧɨɜ
F
G
G
G
G
G
GGG
ɦɟɯɚɧɢɤɢ –
ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ, ɤɨɬɨɪɵɣ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ
ɨɛɪɚɡɨɦ: ɩɨɥɧɚɹ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɵ ɬɟɥ, ɜ ɤɨɬɨɪɨɣ ɞɟɣ-
ɫɬɜɭɸɬ ɬɨɥɶɤɨ ɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɥɵ, ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ.
ȿɫɥɢ ɜ ɫɢɫɬɟɦɟ ɜɦɟɫɬɟ ɫ ɤɨɧɫɟɪɜɚɬɢɜɧɵɦɢ ɫɢɥɚɦɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɞɟɣɫɬɜɭɸɬ ɧɟɤɨɧɫɟɪɜɚɬɢɜɧɵɟ ɫɢɥɵ (ɧɚɩɪɢɦɟɪ, ɫɢɥɚ ɬɪɟɧɢɹ, ɫɢɥɚ ɬɹɝɢ), ɬɨ, ɪɚɫɫɦɚɬɪɢɜɚɹ ɷɬɢ ɫɢɥɵ ɤɚɤ ɜɧɟɲɧɢɟ, ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ:
n
EE A
21
¦
i
,
ɜɧɟɲ
1
ɝɞɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ – ɚɥɝɟɛɪɚɢɱɟɫɤɚɹ ɫɭɦɦɚ ɪɚɛɨɬ ɤɚɠɞɨɣ ɜɧɟɲɧɟɣ ɫɢɥɵ, ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɬɟɥɚɦ ɫɢɫɬɟɦɵ.
Ʉɚɠɞɨɣ ɬɨɱɤɟ ɩɨɬɟɧɰɢɚɥɶɧɨɝɨ ɩɨɥɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ, ɫ ɨɞɧɨɣ ɫɬɨɪɨɧɵ, ɧɟɤɨɬɨɪɨɟ ɡɧɚɱɟɧɢɟ ɜɟɤɬɨɪɚ ɤɨɧɫɟɪɜɚɬɢɜɧɨɣ ɫɢɥɵ
ɫɬɨɪɨɧɵ, ɧɟɤɨɬɨɪɨɟ ɡɧɚɱɟɧɢɟ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɬɟɥɚ
, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɬɟɥɨ, ɫ ɞɪɭɝɨɣ
U. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
ɦɟɠɞɭ ɤɨɧɫɟɪɜɚɬɢɜɧɨɣ ɫɢɥɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɟɣ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɩɨɥɟ
ɞɨɥɠɧɚ ɫɭɳɟɫɬɜɨɜɚɬɶ ɨɩɪɟɞɟɥɟɧɧɚɹ ɫɜɹɡɶ. ɍɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ ɤɨɧɫɟɪɜɚɬɢɜɧɚɹ
ɫɢɥɚ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɩɨɬɟɧɰɢɚɥɶɧɨɝɨ ɩɨɥɹ ɪɚɜɧɚ
ɷɧɟɪɝɢɢ
, ɜɡɹɬɨɦɭ ɫ ɨɛɪɚɬɧɵɦ ɡɧɚɤɨɦ:
G
. (4.11)
gradUF
ɝɪɚɞɢɟɧɬɭ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
Ƚɪɚɞɢɟɧɬ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ – ɷɬɨ ɜɟɤɬɨɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɧɚɩɪɚɜɥɟɧɢɟ
ɤɨɬɨɪɨɣ ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɧɚɢɛɨɥɟɟ ɛɵɫɬɪɨɝɨ ɜɨɡɪɚɫɬɚɧɢɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ, ɚ ɦɨɞɭɥɶ ɱɢɫɥɟɧɧɨ ɪɚɜɟɧ ɫɤɨɪɨɫɬɢ ɜɨɡɪɚɫɬɚɧɢɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɢ ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ ɜɞɨɥɶ ɷɬɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ (ɩɪɨɢɡɜɨɞɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɧɚɢɛɨɥɟɟ ɛɵɫɬɪɨɝɨ ɜɨɡɪɚɫɬɚɧɢɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɢ).
ȼ ɥɸɛɨɣ ɬɨɱɤɟ ɩɨɬɟɧɰɢɚɥɶɧɨɝɨ ɩɨɥɹ ɧɚɩɪɚɜɥɟɧɢɟ ɤɨɧɫɟɪɜɚɬɢɜɧɨɣ ɫɢɥɵ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɧɚɩɪɚɜɥɟɧɢɸ ɝɪɚɞɢɟɧɬɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɢ.
ȼ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɞɥɹ ɡɚɩɢɫɢ ɝɪɚɞɢɟɧɬɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɢ ɩɪɢɦɟɧɹɸɬ ɜɟɤɬɨɪɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ, ɨɛɨɡɧɚɱɚɟɦɵɣ
ɫɢɦɜɨɥɨɦ
(ɧɚɛɥɚ) ɢ ɧɨɫɹɳɢɣ ɧɚɡɜɚɧɢɟ ɨɩɟɪɚɬɨɪɚ ɧɚɛɥɚ ɢɥɢ ɨɩɟɪɚɬɨɪɚ Ƚɚ-
ɦɢɥɶɬɨɧɚ:
U
w
U
x
w
U
w
i
y
w
U
w
j
k
.
z
w
ȼ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɜɵɪɚɠɟɧɢɟ (4.11) ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ:
;UF
G
§
w
U
¨
ɝɞɟ
k
F
U
U
U
w
x
w
w
w
,,
– ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɩɨ ɧɚɩɪɚɜɥɟ-
z
y
w
w
i
¨
w
x
©
ɧɢɹɦ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ
ɧɚɬɧɵɯ ɨɫɟɣ
x, y, z, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
w
U
w
y
x, y, z, ɚ
GG
j
G
·
w
U
¸
,
¸
w
z
¹
– ɨɪɬɵ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢ-
kji
,,
41
ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɩɨ
ɧɚɩɪɚɜɥɟɧɢɹɦ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ ɫ ɨɛɪɚɬɧɵɦ ɡɧɚɤɨɦ ɨɩɪɟɞɟɥɹɸɬ
ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɫɢɥɵ ɧɚ ɤɨɨɪɞɢɧɚɬɧɵɟ ɨɫɢ:
w
F
U
x
F
w
x
w
U
y
;;
w
y
w
U
F
z
.
w
z
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɧɚɹ ɜɢɞ ɮɭɧɤɰɢɢ
ɫɢɥɭ, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɱɚɫɬɢɰɭ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɩɨɬɟɧɰɢɚɥɶɧɨɝɨ ɩɨɥɹ.
Ʉɥɸɱɟɜɵɟ ɩɨɧɹɬɢɹ ɢ ɬɟɪɦɢɧɵ
Ƚɪɚɞɢɟɧɬ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ
Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ
Ɇɨɳɧɨɫɬɶ
Ɋɚɛɨɬɚ
Ɋɚɛɨɬɚ ɪɟɡɭɥɶɬɢɪɭɸɳɟɣ ɫɢɥɵ
Ɋɚɛɨɬɚ ɭɩɪɭɝɨɣ ɫɢɥɵ
ɗɧɟɪɝɢɹ
ɗɧɟɪɝɢɹ ɤɢɧɟɬɢɱɟɫɤɚɹ
(, ,),Uxyz
ɦɨɠɟɦ ɧɚɣɬɢ ɤɨɧɫɟɪɜɚɬɢɜɧɭɸ
ɗɧɟɪɝɢɹ ɤɢɧɟɬɢɱɟɫɤɚɹ
ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ
ɗɧɟɪɝɢɹ ɩɨɬɟɧɰɢɚɥɶɧɚɹ
ɗɧɟɪɝɢɹ ɩɨɬɟɧɰɢɚɥɶɧɚɹ
ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ
ɗɧɟɪɝɢɹ ɩɨɬɟɧɰɢɚɥɶɧɚɹ
ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ ɩɪɭɠɢɧɵ
42
Ɏɨɪɦɭɥɵ ɢ ɟɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ ɜ ɫɢɫɬɟɦɟ ɋɂ
Ⱥ
F
A
X
F
N
X
X
g
G
G
GGG
ɨɫɧɨɜɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ
ɇɚɡɜɚɧɢɟ Ɏɨɪɦɭɥɚ
1. Ɋɚɛɨɬɚ ɫɢɥɵ
2. Ɋɚɛɨɬɚ
ɪɟɡɭɥɶɬɢɪɭɸɳɟɣ ɫɢɥɵ
3. Ɋɚɛɨɬɚ
ɭɩɪɭɝɨɣ ɫɢɥɵ
4. Ɇɨɳɧɨɫɬɶ
5. Ʉɢɧɟɬɢɱɟɫɤɚɹ
ɷɧɟɪɝɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ
ɬɨɱɤɢ
6. ɉɨɬɟɧɰɢɚɥɶɧɚɹ
ɷɧɟɪɝɢɹ ɬɟɥɚ ɜ ɩɨɥɟ
ɫɢɥɵ ɬɹɠɟɫɬɢ
7. ɉɨɬɟɧɰɢɚɥɶɧɚɹ
ɷɧɟɪɝɢɹ
ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ
ɩɪɭɠɢɧɵ
8. ɉɨɥɧɚɹ
ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ
ɬɟɥɚ
9. Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ
ɩɨɥɧɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ
ɷɧɟɪɝɢɢ ɬɟɥɚ
10. ɋɜɹɡɶ
ɤɨɧɫɟɪɜɚɬɢɜɧɨɣ ɫɢɥɵ
ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɢ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ
ɩɨɥɟ – ɜ ɞɟɤɚɪɬɨɜɨɣ
ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
FdS
s
³
s
GG
F
¦
p
n
n
¦
1
i
kxA
2
G
m
X
Ɍ
2
mU
kxU
2
UɌE
E const
;
ɜɧɟɲ
0
gradUF
Ⱥ
§
¨
k
F
¨
©
w
w
w
U
U
i
w
y
x
ȿɞɢɧɢɰɵ
ɢɡɦɟɪɟɧɢɹ
ɜ ɫɢɫɬɟɦɟ ɋɂ
n
[A] = [Ⱦɠ]
[F] = [ɇ];
A
i
2
[A] = [Ⱦɠ]
[k] = [ɇ/ɦ];
[x] = [ɦ];
[A] = [Ⱦɠ]
[F] = [ɇ];
H
G
] = [ɦ/ɫ];
[
[N] = [ȼɬ]
2
[m] = [ɤɝ];
G
] = [ɦ/ɫ];
[
[Ɍ] = [Ⱦɠ]
[m] = [ɤɝ];
h
[g] = [ɦ/ɫ²];
[U] = [Ⱦɠ]
2
[k] = [ɇ/ɦ];
[x] = [ɦ];
[U] = [Ⱦɠ]
[m] = [ɤɝ];
[g] = [ɦ/ɫ²];
[U] = [Ⱦɠ]
[A] = [Ⱦɠ];
[ȿ] = [Ⱦɠ]
[F] = [ɇ];
·
w
U
¸
j
¸
w
z
¹
[U] = [Ⱦɠ]
43
4.2. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
t
A
Ɂɚɞɚɱɚ 1. ɉɪɢ ɞɜɢɠɟɧɢɢ ɬɟɥɚ ɦɚɫɫɨɣ 10 ɤɝ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɟɣ ɫɢɥɵ,
ɫɨɫɬɚɜɥɹɸɳɟɣ ɭɝɨɥ 60
ɟɬɫɹ ɩɨ ɡɚɤɨɧɭ
ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
ɯ(t) = 15 + t
Ⱦɚɧɨ:
ɬ = 10 ɤɝ
D
= 60q
= 3 ɫ
1
Ⱥ
– ?
1
q ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɩɟɪɟɦɟɳɟɧɢɹ, ɤɨɨɪɞɢɧɚɬɚ ɬɟɥɚ ɢɡɦɟɧɹ-
3
(ɦ). ɇɚɣɬɢ ɪɚɛɨɬɭ ɷɬɨɣ ɫɢɥɵ ɡɚ ɜɪɟɦɹ 3 ɫɟɤɭɧɞɵ ɨɬ
Ɋɟɲɟɧɢɟ:
1. ȼɟɥɢɱɢɧɚ ɫɢɥɵ ɢɡɦɟɧɹɟɬɫɹ, ɩɨɷɬɨɦɭ ɩɨɥɧɚɹ ɪɚɛɨɬɚ ɦɨɠɟɬ
ɛɵɬɶ ɧɚɣɞɟɧɚ ɤɚɤ ɫɭɦɦɚ ɷɥɟɦɟɧɬɚɪɧɵɯ ɪɚɛɨɬ
ɝɞɟ
dr – ɦɨɞɭɥɶ ɩɟɪɟɦɟɳɟɧɢɹ ɬɟɥɚ.
t
1
. ȼ ɧɚɲɟɦ ɫɥɭɱɚɟ dr = dx = 3t
dA
³
0
dA = Fdr cos
2
dt.
2. Ɂɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɞɟɣɫɬɜɭɸɳɟɣ ɫɢɥɵ ɧɚɣɞɟɦ, ɢɫɩɨɥɶɡɭɹ ɜɬɨɪɨɣ ɡɚɤɨɧ
ɇɶɸɬɨɧɚ. ȼ ɩɪɨɟɤɰɢɢ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɞɜɢɠɟɧɢɹ (ɨɫɶ
ddx
X
3.
;3;6;6.
atatFmt
X
2
ɯ) ɨɧ ɢɦɟɟɬ ɜɢɞ ɬɚ = F.
dt dt
4. Ɉɤɨɧɱɚɬɟɥɶɧɨ:
tt
11
A mt t dt m t dt
³³
00
18 18 18 10 81
mt m t
cos cos 1823 Ⱦɠ.
44 42
DD
23
63 cos 18cos
t
1
44
0
DD
1
Ɉɬɜɟɬ: 1823 Ⱦɠ.
Ɂɚɞɚɱɚ 2. ɇɚɣɬɢ ɪɚɛɨɬɭ, ɤɨɬɨɪɭɸ ɧɭɠɧɨ ɫɨɜɟɪɲɢɬɶ, ɱɬɨɛɵ ɫɠɚɬɶ ɭɩɪɭɝɭɸ
ɩɪɭɠɢɧɭ ɫ ɠɟɫɬɤɨɫɬɶɸ
k = 2,94 ɤɇ/ɦ ɧɚ ɜɟɥɢɱɢɧɭ 'Ɛ = 20 ɫɦ.
Ⱦɚɧɨ:
k = 2,94 ɤɇ/ɦ
'Ɛ = 20 ɫɦ
Ⱥ – ?
ȼ ɡɚɞɚɱɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɛɨɬɚ ɧɟɤɨɧɫɟɪɜɚɬɢɜɧɨɣ ɜɧɟɲ-
ɧɟɣ ɫɢɥɵ.
Ɋɟɲɟɧɢɟ:
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɪɚɛɨɬɚ ɦɨɠɟɬ ɛɵɬɶ ɧɚɣɞɟɧɚ ɤɚɤ ɢɡɦɟɧɟɧɢɟ ɦɟɯɚɧɢɱɟɫɤɨɣ
ɷɧɟɪɝɢɢ ɬɟɥɚ (ɩɪɭɠɢɧɵ).
1.
Ⱥ = 'ȿ
ȿ
2.
ɦɟɯ
ɧɚɹ ɷɧɟɪɝɢɹ ɨɩɢɫɵɜɚɟɬɫɹ ɮɨɪɦɭɥɨɣ
= ȿ
– ȿ
ɦɟɯ
ɦɟɯ2
ɦɟɯ1
.
= Ɍ + U, ɝɞɟ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɩɪɭɠɢɧɵ Ɍ = 0, ɚ ɟɟ ɩɨɬɟɧɰɢɚɥɶ-
2
'
kx
, ɝɞɟ 'ɯ – ɞɟɮɨɪɦɚɰɢɹ ɩɪɭɠɢɧɵ.
U
2
22
''
kx kx
3.
A
21
, ɝɞɟ
22
A
232
k
'
A
22
'
ɯ
= 0, ɚ
'
1
2,94 10 0, 2
44
ɯ
= 'Ɛ.
2
0,059 ɤȾɠ.
Ɉɬɜɟɬ: 0,059 ɤȾɠ.
D
,
Ɂɚɞɚɱɚ 3. Ɍɟɥɨ, ɛɪɨɲɟɧɧɨɟ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ, ɭɩɚɥɨ ɨɛɪɚɬɧɨ ɱɟɪɟɡ 4 ɫɟ-
t
E
ɤɭɧɞɵ ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ. Ɉɩɪɟɞɟɥɢɬɶ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɬɟɥɚ ɜ ɦɨɦɟɧɬ ɩɚɞɟɧɢɹ ɧɚ ɡɟɦɥɸ ɢ ɟɝɨ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɜ ɜɟɪɯɧɟɣ ɬɨɱɤɟ. Ɇɚɫɫɚ ɬɟɥɚ 200 ɝ.
ɋɨɩɪɨɬɢɜɥɟɧɢɟɦ ɜɨɡɞɭɯɚ ɩɪɟɧɟɛɪɟɱɶ.
Ⱦɚɧɨ:
ɬ = 200 ɝ
= 4 ɫ
ɞɜ
U
– ?
max
T
– ?
ɩɚɞ
Ɍɚɤ ɤɚɤ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɜɨɡɞɭɯɚ ɧɟ ɭɱɢɬɵɜɚɟɬɫɹ, ɬɨ ɜɵɩɨɥɧɹɟɬɫɹ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɜ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɟ ɬɟɥɨ – Ɂɟɦɥɹ (
ɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɬɟɥɚ ɫ ɩɨɜɟɪɯɧɨɫɬɶɸ Ɂɟɦɥɢ.
Ɋɟɲɟɧɢɟ:
ȿ
= const). ɋɜɹɠɟɦ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɩɨɬɟɧ-
ɦɟɯ
ɉɨɥɧɚɹ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɪɚɜɧɚ ɫɭɦɦɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɣ ɬɟɥɚ:
ɦɟɯ
UT
.
ȼ ɜɟɪɯɧɟɣ ɬɨɱɤɟ ɬɪɚɟɤɬɨɪɢɢ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɪɚɜɧɚ ɧɭɥɸ,
ɚ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɢɦɟɟɬ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ. ȼ ɦɨɦɟɧɬ ɩɚɞɟɧɢɹ
ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɪɚɜɧɚ ɧɭɥɸ, ɚ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɢɦɟɟɬ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɚɤɫɢɦɚɥɶɧɚɹ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɪɚɜɧɚ
ɦɚɤɫɢɦɚɥɶɧɨɣ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɬɟɥɚ.
ȼ ɦɨɦɟɧɬ ɛɪɨɫɚɧɢɹ ɩɨɥɧɚɹ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɪɚɜɧɚ ɧɚɱɚɥɶɧɨɣ ɤɢɧɟɬɢɱɟɫɤɨɣ
ɷɧɟɪɝɢɢ (ȿ
ɤɨɬɨɪɚɹ ɪɚɜɧɚ ɦɚɤɫɢɦɚɥɶɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ
= Ɍ0), ɚ ɜ ɦɨɦɟɧɬ ɩɚɞɟɧɢɹ – ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɩɚɞɟɧɢɹ Ɍ
ɦɟɯ
U
(Ɍ
= U
ɦɚɯ
ɩɚɞ
ɦɚɯ
).
ɩɚɞ
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɜ ɦɨɦɟɧɬ ɩɚɞɟɧɢɹ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
22
ɬɬ
XX
ɩɚɞ
T ɌɌ
ɩɚɞ ɩɚɞ
; ɧɨ .
22
ɂɫɩɨɥɶɡɭɹ ɤɢɧɟɦɚɬɢɱɟɫɤɨɟ ɫɨɨɬɧɨɲɟɧɢɟ
ɜɪɟɦɟɧɢ ɞɜɢɠɟɧɢɹ ɬɟɥɚ
t = t
, ɢɦɟɟɦ
ɞɜ
X
–
0
=
X
0
X
y(tɞɜ
– gtɞɜ;
) = –
X
0
X
X
0
(t) =
y
= –
0y
gt
ɞɜ
2
0
X
– gt ɢ ɡɚɩɢɫɚɜ ɟɝɨ ɞɥɹ
0y
X
.
0
.
Ɉɤɨɧɱɚɬɟɥɶɧɨ ɞɥɹ ɢɫɤɨɦɵɯ ɡɧɚɱɟɧɢɣ ɦɚɤɫɢɦɚɥɶɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɩɚɞɟɧɢɹ ɢɦɟɟɦ:
22
UT
ɬɚɯ ɩɚɞ
mg t
0, 2 9,8 4
ɞɜ
88
22
38,4 Ⱦɠ.
Ɉɬɜɟɬ: 38,4 Ⱦɠ.
Ɂɚɞɚɱɚ 4. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɢ ɩɭɥɢ ɢɫɩɨɥɶɡɭɸɬ ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ
ɛɚɥɥɢɫɬɢɱɟɫɤɢɣ ɦɟɬɨɞ. ɇɚ ɬɨɧɤɨɣ ɠɟɫɬɤɨɣ ɧɢɬɢ ɜɢɫɢɬ ɧɟɭɩɪɭɝɨɟ ɬɟɥɨ. ɉɭɥɹ ɥɟɬɢɬ ɝɨɪɢɡɨɧɬɚɥɶɧɨ. ɉɨɫɥɟ ɟɟ ɩɨɩɚɞɚɧɢɹ ɬɟɥɚ ɨɬɤɥɨɧɹɸɬɫɹ ɧɚ ɜɵɫɨɬɭ 40 ɫɦ.
Ɇɚɫɫɚ ɬɟɥɚ 5 ɤɝ. Ɇɚɫɫɚ ɩɭɥɢ 20 ɝ. Ɉɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ ɩɭɥɢ.
Ⱦɚɧɨ:
ɬ
= 5 ɤɝ
Ɍ
ɬ
= 20 ɝ
ɩ
h = 40 ɫɦ
X
– ?
ɩ
1. ɍɞɚɪ ɧɟɭɩɪɭɝɢɣ, ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ ɢ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɩɨɫɥɟ ɭɞɚɪɚ ɩɭɥɢ
ɧɨ ɧɚɣɬɢ, ɢɫɩɨɥɶɡɭɹ ɬɨɥɶɤɨ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ:
GG
mm mm
XX
nn T T n
Ɋɟɲɟɧɢɟ:
0( )
0
X
ɦɨɠ-
0
.
,
45
G
X
0
mm
X
, ɢɦɟɟɦ:
n
m
X
nn
.
Tn
ȼ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ
Ɉɏ: ɬ
ɯ, ɧɚɩɪɚɜɥɟɧɧɭɸ ɜɞɨɥɶ ɜɟɤɬɨɪɚ
+ 0 = (ɬɌ + ɬ
ɩXɩ
)
X
ɩ
0
X
2. ɉɨɫɥɟ ɭɞɚɪɚ ɬɟɥɨ ɫ ɩɭɥɟɣ ɩɨɞɧɢɦɚɟɬɫɹ ɜɜɟɪɯ ɩɨ ɤɪɭɝɨɜɨɣ ɬɪɚɟɤɬɨɪɢɢ.
ɋɨɩɪɨɬɢɜɥɟɧɢɟ ɜɨɡɞɭɯɚ ɧɟ ɭɱɢɬɵɜɚɟɬɫɹ, ɩɨɷɬɨɦɭ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ. ȼɵɛɪɚɜ ɜ ɤɚɱɟɫɬɜɟ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɵ ɬɟɥ
Ɂɟɦɥɸ ɢ ɬɟɥɨ, ɜ ɫɢɥɭ ɪɚɫɫɭɠɞɟɧɢɣ, ɩɪɢɜɟɞɟɧɧɵɯ ɜ ɡɚɞɚɱɟ ʋ 3, ɢɦɟɟɦ ɞɥɹ ɬɟɥɚ
ɫ ɩɭɥɟɣ
ȿ
= const.
ɦɟɯ
ȼ ɧɢɠɧɟɣ ɬɨɱɤɟ ɬɪɚɟɤɬɨɪɢɢ
ȼ ɜɟɪɯɧɟɣ ɬɨɱɤɟ ɬɪɚɟɤɬɨɪɢɢ
()
ɬ m
Ɍɨɝɞɚ:
Tn
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ:
X
n
2
X
0
()
mmgh
2( )
mm
m
n
Tn
22
m
X
nn
Tn
2
()2
mm gh
Tn
()
ɬɬ
ȿɌ
ɦɟɯ
ȿ
= U = (mT + m
ɦɟɯ
Ɍɩ
2
.
;
gh
2
(4 0, 02) 2 9, 8 0,4
0, 02
X
)gh.
n
2
0
.
562,8 ɦ/ɫ.
Ɉɬɜɟɬ: 562,8 ɦ/ɫ.
Ɂɚɞɚɱɚ 5. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɩɪɢ ɭɩɪɭɝɨɦ ɧɟɰɟɧɬɪɚɥɶɧɨɦ ɭɞɚɪɟ ɞɜɭɯ ɨɞɢɧɚɤɨ-
ɜɵɯ ɲɚɪɨɜ, ɨɞɢɧ ɢɡ ɤɨɬɨɪɵɯ ɩɨɤɨɢɬɫɹ, ɨɧɢ ɪɚɡɥɟɬɚɸɬɫɹ ɩɨɞ ɭɝɥɨɦ 90
q ɞɪɭɝ ɤ
ɞɪɭɝɭ.
Ⱦɚɧɨ:
ɬ
= ɬ2 = ɬ
1
X
=
X
10
= 0
20
Ⱦɨɤɚɡɚɬɶ:
D
= 90q
1. ɉɪɢ ɭɩɪɭɝɨɦ ɭɞɚɪɟ ɜɵɩɨɥɧɹɸɬɫɹ ɨɞɧɨɜɪɟɦɟɧɧɨ ɡɚ-
ɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɢ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ
ɷɧɟɪɝɢɢ. ɉɨ ɡɚɤɨɧɭ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ:
Ɋɟɲɟɧɢɟ:
GGG
0mm ɬ m
XXX
.
12
ɉɨ ɡɚɤɨɧɭ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ:
222
XXX
ɬɬɬ
222
GG G
XXX
ɉɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ
Ʉɚɤ ɜɢɞɧɨ, ɜɟɤɬɨɪ
X
®
222
XXX
¯
G
ɞɨɥɠɟɧ ɪɚɜɧɹɬɶɫɹ ɜɟɤɬɨɪɧɨɣ ɫɭɦɦɟ ɜɟɤɬɨɪɨɜ
ɇɨ ɦɨɞɭɥɶ ɫɭɦɦɚɪɧɨɝɨ ɜɟɤɬɨɪɚ ɛɭɞɟɬ ɨɩɪɟɞɟɥɹɬɶɫɹ ɜɵɪɚɠɟɧɢɟɦ
ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ ɭɝɨɥ ɦɟɠɞɭ ɫɤɥɚɞɵɜɚɟɦɵɦɢ ɜɟɤɬɨɪɚɦɢ
ɪɚɜɟɧ 90
q (ɱɬɨ ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɞɨɤɚɡɚɬɶ).
0
12
12
12
.
.
G
X
X
ɢ
1
22
XXX
1
ɢ
12
G
X
ɛɭɞɟɬ
2
G
X
G
2
.
46
F
4.3. Ɂɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ
1. Ʌɶɞɢɧɚ, ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɤɨɬɨɪɨɣ ɪɚɜɧɚ S = 1 ɦ2, ɚ ɬɨɥ-
ɳɢɧɚ
h = 0,4 ɦ, ɩɥɚɜɚɟɬ ɜ ɜɨɞɟ. Ʉɚɤɭɸ ɪɚɛɨɬɭ ɧɚɞɨ ɫɨɜɟɪɲɢɬɶ, ɱɬɨɛɵ ɩɨɥɧɨ-
ɫɬɶɸ ɩɨɝɪɭɡɢɬɶ ɥɶɞɢɧɭ ɜ ɜɨɞɭ.
2. Ƚɪɭɡ ɦɚɫɫɨɣ
D
= 90q ɢ ɨɬɩɭɫɤɚɸɬ. ɇɚɣɬɢ ɫɢɥɭ ɧɚɬɹɠɟɧɢɹ ɫɬɟɪɠɧɹ ɜ ɦɨɦɟɧɬ ɩɪɨɯɨɠɞɟɧɢɹ
ɬ, ɩɨɞɜɟɲɟɧɧɵɣ ɧɚ ɧɟɜɟɫɨɦɨɦ ɫɬɟɪɠɧɟ, ɨɬɤɥɨɧɹɸɬ ɧɚ ɭɝɨɥ
ɝɪɭɡɨɦ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ.
3. Ɍɟɥɨ ɦɚɫɫɨɣ
ɫɨ ɫɤɨɪɨɫɬɶɸ
ɬ = 100 ɝ, ɛɪɨɲɟɧɧɨɟ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɫ ɜɵɫɨɬɵ h = 20 ɦ
X
= 10 ɦ/ɫ, ɭɩɚɥɨ ɧɚ Ɂɟɦɥɸ ɫɨ ɫɤɨɪɨɫɬɶɸ
1
X
= 20 ɦ/ɫ. ɇɚɣɬɢ ɪɚɛɨ-
2
ɬɭ ɩɨ ɩɪɟɨɞɨɥɟɧɢɸ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ.
4. ɋ ɥɟɞɹɧɨɣ ɝɨɪɵ ɜɵɫɨɬɨɣ
ɤɨɬɨɪɵɟ ɨɫɬɚɧɚɜɥɢɜɚɸɬɫɹ, ɩɪɨɣɞɹ ɩɨ ɥɶɞɭ ɝɨɪɢɡɨɧɬɚɥɶɧɵɣ ɩɭɬɶ
h = 1 ɦ ɢ ɨɫɧɨɜɚɧɢɟɦ b = 5 ɦ ɫɴɟɡɠɚɸɬ ɫɚɧɢ,
Ɛ = 95 ɦ. ɇɚɣɬɢ
ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɚɧɟɣ ɨ ɥɟɞ.
5. Ɍɟɥɨ ɛɪɨɲɟɧɨ ɩɨɞ ɭɝɥɨɦ ɤ ɝɨɪɢɡɨɧɬɭ ɫɨ ɫɤɨɪɨɫɬɶɸ
ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ, ɨɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɧɚ ɜɵɫɨɬɟ
X
. ɂɫɩɨɥɶɡɭɹ ɡɚɤɨɧ
0
h ɧɚɞ
ɝɨɪɢɡɨɧɬɨɦ.
6. ɋɤɨɪɨɫɬɶ ɬɟɥɚ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɧɟɤɨɬɨɪɨɣ ɫɢɥɵ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ
= 30 + 6
3
t
(ɦ/ɫ). Ɉɩɪɟɞɟɥɢɬɶ ɪɚɛɨɬɭ ɷɬɨɣ ɫɢɥɵ ɡɚ ɩɟɪɜɵɟ 10 ɫɟɤɭɧɞ ɞɜɢɠɟɧɢɹ.
X
(t) =
7. Ʉɚɤɚɹ ɪɚɛɨɬɚ ɩɪɨɢɡɜɟɞɟɧɚ ɞɥɹ ɫɠɚɬɢɹ ɛɭɮɟɪɧɨɣ ɩɪɭɠɢɧɵ ɧɚ 5 ɫɦ, ɟɫɥɢ
ɞɥɹ ɟɟ ɫɠɚɬɢɹ ɧɚ 1 ɫɦ ɬɪɟɛɭɟɬɫɹ ɫɢɥɚ 30 ɤɇ?
8. ɇɚɣɬɢ ɪɚɛɨɬɭ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɤɚɧɚɬɚ, ɩɨɞɧɢɦɚɸɳɟɝɨ ɝɪɭɡ ɦɚɫɫɨɣ 50 ɤɝ
ɧɚ ɜɵɫɨɬɭ 10 ɦɟɬɪɨɜ ɡɚ 7 ɫɟɤɭɧɞ?
9. ɇɚ ɬɟɥɨ ɦɚɫɫɨɣ 10 ɤɝ ɞɟɣɫɬɜɭɟɬ ɩɨɫɬɨɹɧɧɚɹ ɫɢɥɚ
5,0 ɇ. Ɉɩɪɟɞɟɥɢɬɶ ɤɢ-
ɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɬɟɥɚ ɱɟɪɟɡ 2 ɫɟɤɭɧɞɵ ɩɨɫɥɟ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ. ɋɨɩɪɨɬɢɜɥɟɧɢɟɦ ɩɪɟɧɟɛɪɟɱɶ.
10. Ɍɟɥɨ ɩɨɫɥɟ ɚɛɫɨɥɸɬɧɨ ɧɟɭɩɪɭɝɨɝɨ ɭɞɚɪɚ ɨ ɧɟɩɨɞɜɢɠɧɨɟ ɬɟɥɨ ɫɬɚɥɨ
ɞɜɢɝɚɬɶɫɹ ɜ 4 ɪɚɡɚ ɦɟɞɥɟɧɧɟɟ. Ɉɩɪɟɞɟɥɢɬɶ ɞɨɥɸ ɷɧɟɪɝɢɢ, ɩɟɪɟɲɟɞɲɟɣ ɜɨ ɜɧɭɬɪɟɧɧɸɸ ɷɧɟɪɝɢɸ ɬɟɥɚ.
11. ɇɚɫɨɫ ɦɨɳɧɨɫɬɶɸ
ɦɚɫɫɭ ɠɢɞɤɨɫɬɢ, ɩɨɞɧɹɬɨɣ ɡɚ ɜɪɟɦɹ
12. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɦɚɫɫɨɣ
ɥɵ. ɉɪɢ ɷɬɨɦ ɩɪɨɣɞɟɧɧɵɣ ɩɭɬɶ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ
Ɉɩɪɟɞɟɥɢɬɶ ɦɨɳɧɨɫɬɶ, ɡɚɬɪɚɱɢɜɚɟɦɭɸ ɧɚ ɞɜɢɠɟɧɢɟ ɬɨɱɤɢ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
13. Ɍɟɥɨ ɦɚɫɫɨɣ
G
GG
2
23
ti t j
ɜɪɟɦɟɧɢ
(ɇ). Ɉɩɪɟɞɟɥɢɬɶ ɦɨɳɧɨɫɬɶ, ɪɚɡɜɢɜɚɟɦɭɸ ɷɬɨɣ ɫɢɥɨɣ ɤ ɦɨɦɟɧɬɭ
t = 10 ɫ.
N ɩɨɞɧɢɦɚɟɬ ɠɢɞɤɨɫɬɶ ɧɚ ɜɵɫɨɬɭ h. Ɉɩɪɟɞɟɥɢɬɶ
t, ɟɫɥɢ ɄɉȾ ɧɚɫɨɫɚ
ɬ ɞɜɢɠɟɬɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɧɟɤɨɬɨɪɨɣ ɫɢ-
K
.
s = A + Bt + Ct
2
– Dt3.
W
.
ɬ = 3 ɤɝ ɧɚɱɢɧɚɟɬ ɞɜɢɝɚɬɶɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ
14. ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɜ ɰɟɧɬɪɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ ɩɪɟɞɫɬɚɜɥɟɧɚ
21
ɜɵɪɚɠɟɧɢɟɦ
()Ur
(ɦɤȾɠ). Ɉɩɪɟɞɟɥɢɬɶ, ɩɪɢ ɤɚɤɢɯ ɡɧɚɱɟɧɢɹɯ r ɦɚɤɫɢ-
2
rr
ɦɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɞɨɫɬɢɝɚɸɬ: 1) ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ; 2) ɫɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɬɟɥɨ.
47
4.4. Ɋɚɛɨɬɚ ɢ ɷɧɟɪɝɢɹ. Ɍɟɫɬɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ɡɧɚɧɢɣ
ɍɪɨɜɟɧɶ I
1. ɑɟɦɭ ɪɚɜɧɚ ɪɚɛɨɬɚ ɫɢɥɵ ɬɹɠɟɫɬɢ ɩɪɢ ɩɨɞɴɟɦɟ ɬɟɥɚ ɦɚɫɫɨɣ 2 ɤɝ ɧɚ ɜɵɫɨ-
ɬɭ 25 ɦ?
2. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ, ɤɨɬɨ-
ɪɚɹ ɡɚɜɢɫɢɬ ɨɬ ɩɪɨɣɞɟɧɧɨɝɨ ɩɭɬɢ
ɜɞɨɥɶ ɩɟɪɟɦɟɳɟɧɢɹ ɬɟɥɚ. ɇɚɣɬɢ ɪɚɛɨɬɭ ɫɢɥɵ ɧɚ ɩɭɬɢ
s ɩɨ ɡɚɤɨɧɭ
Fs
(D = const) ɢ ɧɚɩɪɚɜɥɟɧɚ
D
s
.
0
3. Ʉɚɤɭɸ ɪɚɛɨɬɭ ɧɭɠɧɨ ɫɨɜɟɪɲɢɬɶ, ɱɬɨɛɵ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɦɚɫɫɨɣ 2 ɤɝ ɭɜɟɥɢ-
ɱɢɥɚɫɶ ɜ 2 ɪɚɡɚ? ɇɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɪɚɜɧɚ 3 ɦ/ɫ.
4. Ɍɟɥɟɠɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɞɨɪɨɝɟ ɫɨ ɫɤɨɪɨɫɬɶɸ 10 ɦ/ɫ ɢ
ɜɴɟɡɠɚɟɬ ɧɚ ɩɨɞɴɟɦ. ɇɚ ɤɚɤɨɣ ɜɵɫɨɬɟ ɧɚɞ ɭɪɨɜɧɟɦ ɞɨɪɨɝɢ ɨɫɬɚɧɨɜɢɬɫɹ ɬɟɥɟɠɤɚ?
ɋɨɩɪɨɬɢɜɥɟɧɢɟɦ ɩɪɟɧɟɛɪɟɱɶ.
5. Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɜ ɦɨɦɟɧɬ ɛɪɨɫɚɧɢɹ ɪɚɜɧɚ
200 Ⱦɠ. Ɉɩɪɟɞɟ-
ɥɢɬɶ, ɞɨ ɤɚɤɨɣ ɜɵɫɨɬɵ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ Ɂɟɦɥɢ ɦɨɠɟɬ ɩɨɞɧɹɬɶɫɹ ɬɟɥɨ, ɟɫɥɢ ɟɝɨ
ɦɚɫɫɚ ɪɚɜɧɚ 500 ɝ.
6. Ɉɩɪɟɞɟɥɢɬɶ ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɟɥɚ ɦɚɫɫɨɣ 1 ɤɝ ɪɚɜɧɨɦɟɪɧɨ ɞɜɢɠɭ-
ɳɟɝɨɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ 0,5 ɦ ɫ ɷɧɟɪɝɢɟɣ 50 Ⱦɠ.
7. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɦɚɫɫɨɣ
ɬ ɞɜɢɠɟɬɫɹ ɜ ɩɥɨɫɤɨɫɬɢ ɯy ɩɨ ɡɚɤɨɧɭ
ɯ = cos2t; y = sin2t. Ɇɝɧɨɜɟɧɧɚɹ ɦɨɳɧɨɫɬɶ ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɧɟɟ ɫɢɥɵ ɜ ɧɚɱɚɥɶ-
ɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɜɧɚ 1 ȼɬ. ɇɚɣɬɢ ɡɧɚɱɟɧɢɟ ɫɢɥɵ ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
8. Ʉɚɦɟɧɶ ɦɚɫɫɨɣ
ɬ = 2 ɤɝ ɫɜɨɛɨɞɧɨ ɩɚɞɚɟɬ ɫ ɜɵɫɨɬɵ ɜ ɬɟɱɟɧɢɟ ɜɪɟɦɟɧɢ
W
= 2 ɫ.
ɇɚɣɬɢ ɤɢɧɟɬɢɱɟɫɤɭɸ ɢ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɢ ɤɚɦɧɹ ɜ ɫɪɟɞɧɟɣ ɬɨɱɤɟ ɩɭɬɢ.
9. ȼɟɪɬɢɤɚɥɶɧɨ ɫɬɨɹɳɢɣ ɲɟɫɬ ɩɚɞɚɟɬ, ɩɨɜɨɪɚɱɢɜɚɹɫɶ ɜɨɤɪɭɝ ɧɢɠɧɟɝɨ ɤɨɧɰɚ. ɇɚɣɬɢ ɫɤɨɪɨɫɬɶ ɜɟɪɯɧɟɝɨ ɤɨɧɰɚ ɲɟɫɬɚ ɜ ɦɨɦɟɧɬ ɩɚɞɟɧɢɹ ɧɚ Ɂɟɦɥɸ. Ɇɚɫɫɚ
ɲɟɫɬɚ
ɬ, ɚ ɞɥɢɧɚ Ɛ.
10. ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɞɜɭɯ ɨɞɢɧɚɤɨɜɵɯ ɦɚɬɟɪɢɚɥɶ-
2
m
ɧɵɯ ɬɨɱɟɤ ɢɦɟɟɬ ɜɢɞ:
ɦɟɠɞɭ ɬɨɱɤɚɦɢ
r ɫɢɥɚ ɢɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ?
UG
. Ʉɚɤ ɢɡɦɟɧɹɟɬɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɪɚɫɫɬɨɹɧɢɹ
r
ɍɪɨɜɟɧɶ II
1. ɑɟɦɭ ɪɚɜɧɚ ɪɚɛɨɬɚ ɫɢɥɵ ɬɹɠɟɫɬɢ, ɟɫɥɢ ɬɟɥɨ ɦɚɫɫɨɣ 1 ɤɝ ɨɩɢɫɵɜɚɟɬ ɜ ɜɟɪɬɢɤɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɨɥɧɭɸ ɨɤɪɭɠɧɨɫɬɶ ɪɚɞɢɭɫɚ 1 ɦɟɬɪ.
2. ɋɤɨɪɨɫɬɶ ɬɟɥɚ ɦɚɫɫɨɣ 1 ɤɝ ɩɪɢ ɩɪɹɦɨɥɢɧɟɣɧɨɦ ɞɜɢɠɟɧɢɢ ɩɨɞ ɞɟɣɫɬɜɢɟɦ
ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɫɢɥɵ ɢɡɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɩɨ ɡɚɤɨɧɭ
X
(t) = 16 – 2t3 (ɦ/ɫ).
Ʉɚɤɭɸ ɪɚɛɨɬɭ ɫɨɜɟɪɲɚɟɬ ɷɬɚ ɫɢɥɚ ɞɨ ɩɨɥɧɨɣ ɨɫɬɚɧɨɜɤɢ ɬɟɥɚ?
3. Ʉɚɤɭɸ ɪɚɛɨɬɭ ɧɭɠɧɨ ɫɨɜɟɪɲɢɬɶ, ɱɬɨɛɵ ɭɜɟɥɢɱɢɬɶ ɱɚɫɬɨɬɭ ɜɪɚɳɟɧɢɹ ɬɟɥɚ ɦɚɫɫɨɣ 0,1 ɤɝ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ 50 ɫɦ, ɥɟɠɚɳɟɣ ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ
ɩɥɨɫɤɨɫɬɢ ɫ
4. ɇɚ ɧɢɬɢ ɞɥɢɧɨɣ
Q
= 3 ɫ–1 ɞɨ
1
Q
= 5 ɫ–1?
2
Ɛ = 0,8 ɦ ɩɨɞɜɟɲɟɧ ɲɚɪ. Ʉɚɤɭɸ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɫɤɨ-
ɪɨɫɬɶ ɧɚɞɨ ɫɨɨɛɳɢɬɶ ɲɚɪɭ, ɱɬɨɛɵ ɨɧ ɨɬɤɥɨɧɢɥɫɹ ɞɨ ɜɵɫɨɬɵ ɬɨɱɤɢ ɩɨɞɜɟɫɚ?
5. Ɍɟɥɨ ɦɚɫɫɨɣ 0,1 ɤɝ ɛɪɨɫɚɸɬ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ. Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ
ɜ ɦɨɦɟɧɬ ɛɪɨɫɚɧɢɹ 20 Ⱦɠ. ȼɨ ɫɤɨɥɶɤɨ ɪɚɡ ɨɧɚ ɭɦɟɧɶɲɢɬɫɹ ɱɟɪɟɡ 1 ɫɟɤɭɧɞɭ?
48
6. ȼ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɟɥɚ ɦɚɫɫɨɣ 0,5 ɤɝ,
ɞɜɢɠɭɳɟɝɨɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ 0,5 ɦ, ɪɚɜɧɨ 2 ɦ/ɫ
2
, ɚ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪ-
ɝɢɹ 0,25 Ⱦɠ. ɇɚɣɬɢ ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɟɥɚ ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
7. ɇɚ ɬɟɥɨ ɦɚɫɫɨɣ 10 ɤɝ ɞɟɣɫɬɜɭɟɬ ɩɨɫɬɨɹɧɧɚɹ ɫɢɥɚ 5 ɇ. Ɉɩɪɟɞɟɥɢɬɶ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɬɟɥɚ ɱɟɪɟɡ 2 ɫ ɩɨɫɥɟ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
8. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ. ɉɪɢ ɷɬɨɦ ɩɪɨɣɞɟɧɧɵɣ ɟɸ
ɩɭɬɶ
ɇɚɣɬɢ ɡɧɚɱɟɧɢɟ
3
s ~ t
. Ɇɨɳɧɨɫɬɶ ɪɟɡɭɥɶɬɢɪɭɸɳɟɣ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɬɨɱɤɭ, N ~ tn.
ɩ.
9. Ʉɚɦɟɧɶ ɦɚɫɫɨɣ 0,1 ɤɝ ɩɪɢɜɹɡɚɧ ɤ ɜɟɪɟɜɤɟ ɞɥɢɧɨɣ 0,5 ɦ ɢ ɪɚɜɧɨɦɟɪɧɨ
ɜɪɚɳɚɟɬɫɹ ɜ ɜɟɪɬɢɤɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ. ɋɢɥɚ ɧɚɬɹɠɟɧɢɹ ɜɟɪɟɜɤɢ ɜ ɧɢɠɧɟɣ ɬɨɱɤɟ
ɨɤɪɭɠɧɨɫɬɢ 50 ɇ. ɇɚ ɤɚɤɭɸ ɜɵɫɨɬɭ ɩɨɞɧɢɦɟɬɫɹ ɤɚɦɟɧɶ, ɟɫɥɢ ɜɟɪɟɜɤɚ ɨɛɨɪɜɟɬɫɹ
ɜ ɬɨɬ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɤɚɦɧɹ ɧɚɩɪɚɜɥɟɧ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ?
10. ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɷɥɟɤɬɪɨɫɬɚɬɢɱɟɫɤɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɞɜɭɯ ɦɚ-
ɥɵɯ ɨɞɢɧɚɤɨɜɨ ɡɚɪɹɠɟɧɧɵɯ
ɨɬ ɞɪɭɝɚ, ɢɦɟɟɬ ɜɢɞ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɪɚɫɫɬɨɹɧɢɹ ɜ 3 ɪɚɡɚ?
Urr
ɬɟɥ (q1 = q2 = q), ɧɚɯɨɞɹɳɢɯɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ r ɞɪɭɝ
2
kq
()
(k = const). ȼɨ ɫɤɨɥɶɤɨ ɪɚɡ ɢɡɦɟɧɢɬɫɹ ɫɢɥɚ ɢɯ
ɍɪɨɜɟɧɶ III
1. Ɍɟɥɨ ɦɚɫɫɨɣ ɬ = 5 ɤɝ ɦɟɞɥɟɧɧɨ ɜɬɚɫɤɢɜɚɸɬ ɜ ɝɨɪɭ ɫ ɭɝɥɨɦ ɧɚɤɥɨɧɚ ɤ ɝɨɪɢɡɨɧɬɭ 60
q. ɑɟɦɭ ɪɚɜɧɚ ɪɚɛɨɬɚ ɫɢɥɵ ɬɪɟɧɢɹ ɧɚ ɩɭɬɢ ɜ 1 ɦ, ɟɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬ
ɬɪɟɧɢɹ 0,2.
2. Ʉɨɨɪɞɢɧɚɬɚ ɩɪɹɦɨɥɢɧɟɣɧɨ ɞɜɢɠɭɳɟɝɨɫɹ ɬɟɥɚ ɦɚɫɫɨɣ 2 ɤɝ ɢɡɦɟɧɹɟɬɫɹ ɫɨ
ɜɪɟɦɟɧɟɦ ɩɨ ɡɚɤɨɧɭ
ɧɚ ɜɞɨɥɶ ɨɫɢ
ɯ. ɇɚɣɬɢ ɪɚɛɨɬɭ ɷɬɨɣ ɫɢɥɵ ɞɨ ɨɫɬɚɧɨɜɤɢ ɬɟɥɚ.
ɯ(t) = 10 + 4t – 5t
3. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɫɨ ɫɤɨɪɨɫɬɶɸ
ɬɚ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɬɨɱɤɭ, ɡɚ ɜɪɟɦɹ
4. ɉɪɭɠɢɧɚ, ɠɟɫɬɤɨɫɬɶ ɤɨɬɨɪɨɣ ɪɚɜɧɚ
ɞɚɟɬ ɧɚ ɡɟɦɥɸ ɫ ɜɵɫɨɬɵ
h = 5 ɦ. ɇɚɫɤɨɥɶɤɨ ɫɨɠɦɟɬɫɹ ɩɪɭɠɢɧɚ ɩɪɢ ɭɞɚɪɟ ɨ ɡɟɦ-
3
(ɦ). Ⱦɟɣɫɬɜɭɸɳɚɹ ɧɚ ɬɟɥɨ ɫɢɥɚ ɧɚɩɪɚɜɥɟ-
2
X
~ t
. Ɋɚɛɨ-
tA ~ t
n
. ɇɚɣɬɢ ɩ.
k = 100 ɤɇ/ɦ, ɚ ɦɚɫɫɚ ɬ = 400 ɝ, ɩɚ-
ɥɸ, ɟɫɥɢ ɟɟ ɨɫɶ ɨɫɬɚɧɟɬɫɹ ɜɟɪɬɢɤɚɥɶɧɨɣ?
5. Ɍɟɥɨ ɦɚɫɫɨɣ
ɪɚɞɢɭɫɨɦ
R = 10 ɫɦ. ɇɚɣɬɢ ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ (ɚ
ɜɬɨɪɨɝɨ ɨɛɨɪɨɬɚ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ ɞɨɫɬɢɝɥɚ
6. Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ, ɞɜɢɠɭɳɟɣɫɹ ɩɨ ɨɫɢ
(ɯ> 0). Ʉɚɤ ɡɚɜɢɫɢɬ ɨɬ ɯ ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɬɨɱɤɭ ɫɢɥɚ?
~Tx
7. ɋ ɜɟɪɲɢɧɵ ɢɞɟɚɥɶɧɨ ɝɥɚɞɤɨɣ ɫɮɟɪɵ ɪɚɞɢɭɫɚ
ɬ = 10 ɝ ɧɚɱɚɥɨ ɞɜɢɝɚɬɶɫɹ ɪɚɜɧɨɭɫɤɨɪɟɧɧɨ ɩɨ ɨɤɪɭɠɧɨɫɬɢ
) ɬɟɥɚ, ɟɫɥɢ ɤ ɤɨɧɰɭ
W
Ɍ = 3,14 ɦȾɠ.
Ɉɏ
R ɫɨɫɤɚɥɶɡɵɜɚɟɬ ɧɟɛɨɥɶɲɨɟ
ɬɟɥɨ. Ɉɩɪɟɞɟɥɢɬɶ, ɧɚ ɤɚɤɨɣ ɜɵɫɨɬɟ, ɨɬɫɱɢɬɵɜɚɹ ɨɬ ɜɟɪɲɢɧɵ ɫɮɟɪɵ, ɬɟɥɨ ɨɬ
ɫɮɟɪɵ ɨɬɨɪɜɟɬɫɹ?
8. Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ, ɞɜɢɠɭɳɟɝɨɫɹ ɜ ɜɹɡɤɨɣ ɫɪɟɞɟ, ɭɛɵɜɚɟɬ ɫɨ
ɜɪɟɦɟɧɟɦ ɩɨ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɦɭ ɡɚɤɨɧɭ
-Dt
T ~ e
(D = const). Ʉɚɤ ɡɚɜɢɫɢɬ ɨɬ ɫɤɨ-
ɪɨɫɬɢ ɬɨɱɤɢ ɦɨɳɧɨɫɬɶ ɫɢɥɵ ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ?
9. Ʉɚɤɭɸ ɦɢɧɢɦɚɥɶɧɭɸ ɪɚɛɨɬɭ ɧɟɨɛɯɨɞɢɦɨ ɫɨɜɟɪɲɢɬɶ, ɱɬɨɛɵ ɥɟɠɚɳɢɣ ɧɚ
ɫɬɨɥɟ ɝɪɭɡ ɦɚɫɫɨɣ
ɤ ɧɟɦɭ ɪɟɡɢɧɨɜɨɝɨ ɠɝɭɬɚ ɠɟɫɬɤɨɫɬɶɸ
ɬ = 1 ɤɝ ɩɨɞɧɹɬɶ ɧɚ ɜɵɫɨɬɭ h = 1 ɦ ɩɪɢ ɩɨɦɨɳɢ ɩɪɢɜɹɡɚɧɧɨɝɨ
k = 10 ɇ/ɦ.
49
10. ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɟɥɚ, ɫɨɜɟɪɲɚɸɳɟɝɨ ɤɨɥɟɛɚɧɢɹ ɜɞɨɥɶ ɨɫɢ
ɨɩɢɫɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ
ɧɵ. ɇɚɣɬɢ ɤɨɨɪɞɢɧɚɬɭ
U(x) = Ax
ɯ
ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɬɟɥɚ.
0
2
+ B(h – x), ɝɞɟ Ⱥ; ȼ; h – ɩɨɫɬɨɹɧɧɵɟ ɜɟɥɢɱɢ-
ɯ,
50
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