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2
ɧɤɯ
F
ɝɞɟ
s
– ɤɚɫɚɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ;
s
– ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ;
ρ
ρ
– ɪɚ-
ɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ.
ɋ ɩɨɦɨɳɶɸ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɪɟɲɚɸɬ ɦɧɨɝɢɟ ɡɚɞɚɱɢ
ɞɢɧɚɦɢɤɢ, ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɫɜɟɫɬɢ ɤ ɞɜɭɦ ɨɫɧɨɜɧɵɦ ɡɚɞɚɱɚɦ. ɂɯ ɧɚɡɵɜɚɸɬ ɩɟɪɜɨɣ (ɩɪɹɦɨɣ) ɢ ɜɬɨɪɨɣ (ɨɛɪɚɬɧɨɣ) ɡɚɞɚɱɟɣ ɞɢɧɚɦɢɤɢ.
ɉɟɪɜɚɹ ɡɚɞɚɱɚ – ɩɨ ɡɚɞɚɧɧɵɦ ɭɪɚɜɧɟɧɢɹɦ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶ-
ɧɨɣ ɬɨɱɤɢ ɨɩɪɟɞɟɥɢɬɶ ɫɢɥɭ, ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɤɨɬɨɪɨɣ ɩɪɨɢɫɯɨɞɢɬ ɞɜɢɠɟɧɢɟɦ ɬɨɱɤɢ.
ɉɨɪɹɞɨɤ ɪɟɲɟɧɢɹ.
1.
Ɉɩɪɟɞɟɥɢɬɶ ɬɪɚɟɤɬɨɪɢɸ ɬɨɱɤɢ, ɟɫɥɢ ɨɧɚ ɧɟ ɡɚɞɚɟɬɫɹ, ɢ ɫɞɟɥɚɬɶ
ɱɟɪɬɟɠ ɡɚɞɚɱɢ.
2.
ɇɚɣɬɢ ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ:
xa
=
x
ɢɥɢ
ya
=
y
za
=
z
ɉɨɞɫɬɚɜɢɬɶ ɷɬɢ ɩɪɨɟɤɰɢɢ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ (4) ɢ
3.
τ
asa=
n
=
2
v
ρ
ɢɥɢ (5) ɢ ɨɩɪɟɞɟɥɢɬɶ ɩɪɨɟɤɰɢɢ ɫɢɥ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ:
=
Fxm
¦
Ɉɩɪɟɞɟɥɢɬɶ ɜɟɥɢɱɢɧɭ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɫɢɥɵ:
4.
2
()
()
ɤɯ
∧
§
¨
©
§
¨
©
ɢɥɢ
=
Fym
¦
ɤ
=
Fzzm
¦
2
()
FFFF
++= ɢɥɢ
¦¦¦
ɤɭ
·
iFcos
,
¸
¹
∧
·
jFcos
,
¸
¹
ɤz
F
¦
ɤɯ
=
=
;
F
F
¦
ɤy
;
F
∧
§·
cos .
F ɤ
¨¸
,
¨¸
©¹
2
;
.
¦
=
=
Fma
¦
ττ
ɤ
=
∧
§
Fcos
τ
,
¨
©
∧
§
nFcos
,
¨
©
ɤz
F
Fma
¦
ɤnn
2
()()
τ
ɤ
F
¦
·
ɤ
τ
=
¸
F
¹
F
¦
·
ɤn
=
¸
¹
F
+=
¦¦
;
ɤn
2
FFF
ȼɬɨɪɚɹ ɡɚɞɚɱɚ – ɩɨ ɡɚɞɚɧɧɵɦ ɫɢɥɚɦ, ɞɟɣɫɬɜɭɸɳɢɦ ɧɚ ɦɚɬɟɪɢ-
ɚɥɶɧɭɸ ɬɨɱɤɭ, ɨɩɪɟɞɟɥɢɬɶ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ.
81
Ɏ
ɉɨɪɹɞɨɤ ɪɟɲɟɧɢɹ.
1.
ȼɵɛɪɚɬɶ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ. ɇɚɱɚɥɨ ɭɞɨɛɧɨ ɫɨɜɦɟɳɚɬɶ ɫ ɧɚ-
ɱɚɥɶɧɵɦ ɩɨɥɨɠɟɧɢɟɦ ɬɨɱɤɢ.
2.
ɂɡɨɛɪɚɡɢɬɶ ɩɨɥɨɠɟɧɢɟ ɞɜɢɠɭɳɟɣɫɹ ɬɨɱɤɢ ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɬɟ-
ɤɭɳɟɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
3.
Ɂɚɩɢɫɚɬɶ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ.
4.
ɋɨɫɬɚɜɢɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ (1)
ɢɥɢ (2).
5.
ɉɪɨɢɧɬɟɝɪɢɪɨɜɚɬɶ ɩɨɥɭɱɟɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ.
6.
ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɨɩɪɟɞɟɥɢɬɶ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢ-
ɪɨɜɚɧɢɹ.
7.
ɍɫɬɚɧɨɜɢɬɶ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɢ ɨɩɪɟɞɟɥɢɬɶ ɢɫɤɨɦɵɟ ɜɟɥɢ-
ɱɢɧɵ.
1.2. ɉɪɢɧɰɢɩ Ⱦɚɥɚɦɛɟɪɚ. Ɇɟɬɨɞ ɤɢɧɟɬɨɫɬɚɬɢɤɢ
1.2.1. ɋɢɥɚ ɢɧɟɪɰɢɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ
ɋɢɥɚ
Ɏ , ɪɚɜɧɚ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ
M
ɚ
ɚ)
Ɏ
n
Ɏ
a
τ
Ɏ
τ
τ
ɫɢɥ, ɫ ɤɨɬɨɪɨɣ ɞɜɢɠɭɳɚɹɫɹ ɬɨɱɤɚ ɞɟɣɫɬɜɭɟɬ
ɧɚ ɬɟɥɚ, ɫɨɨɛɳɚɸɳɢɟ ɟɣ ɭɫɤɨɪɟɧɢɟ, ɧɚɡɵɜɚɟɬɫɹ ɫɢɥɨɣ ɢɧɟɪɰɢɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ:
amɎ −= .
ɋɢɥɚ ɢɧɟɪɰɢɢ ɢɦɟɟɬ ɧɚɩɪɚɜɥɟɧɢɟ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɟ ɧɚɩɪɚɜɥɟɧɢɸ ɭɫɤɨɪɟɧɢɹ
ɋɢɥɭ ɢɧɟɪɰɢɢ ɭɫɥɨɜɧɨ ɩɪɢɤɥɚɞɵɜɚɸɬ ɤ
ɞɜɢɠɭɳɟɣɫɹ ɬɨɱɤɟ (ɪɢɫ. 33,
ȼ ɫɥɭɱɚɟ ɤɪɢɜɨɥɢɧɟɣɧɨɝɨ ɞɜɢɠɟɧɢɹ
ɫɢɥɚ ɢɧɟɪɰɢɢ ɢɦɟɟɬ ɞɜɟ ɫɨɫɬɚɜɥɹɸɳɢɟ
(ɪɢɫ. 33,
ɛ):
ɚ).
ɎɎɎ +=
,
n
τ
M
22
ɎɎɎ +=
,
n
dv
==
,
mmaɎ
dt
2
v
.
mmaɎ ==
ρ
a
n
ɛ)
Ɋɢɫ. 33.
τ
a
ττ
nn
a .
82
Ɏ
Ɏ
Ɏ
ɰ
a
Ɇ
ɰ
Ɏ
ɜɪ
Ɉ
ɜɪ
a
R
ȼ ɫɥɭɱɚɟ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫɢɥɚ
ɢɧɟɪɰɢɢ ɪɚɫɤɥɚɞɵɜɚɟɬɫɹ ɧɚ ɜɪɚɳɚɬɟɥɶɧɭɸ
ɫɢɥɭ ɢɧɟɪɰɢɢ
ɢɧɟɪɰɢɢ
Ɏ ɢ ɰɟɧɬɪɨɛɟɠɧɭɸ ɫɢɥɭ
ɜɪ
Ɏ (ɪɢɫ. 34), ɬ. ɟ.:
ɰ
ɎɎɎ += ,
ɰɜɪ
Ɋɢɫ. 34.
ɎɎɎ += ,
ɜɪ
==
ɜɪɜɪ
ω
== .
ɰɰ
22
ɰ
,
RmmaɎ
ε
2
RmmaɎ
1.2.2. ɉɪɢɧɰɢɩ Ⱦɚɥɚɦɛɟɪɚ
1. Ⱦɥɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ.
ȼ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɞɜɢɠɟɧɢɹ ɧɟɫɜɨɛɨɞɧɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ
ɬɨɱɤɢ ɝɟɨɦɟɬɪɢɱɟɫɤɚɹ ɫɭɦɦɚ ɚɤɬɢɜɧɵɯ ɫɢɥ, ɪɟɚɤɰɢɣ ɫɜɹɡɟɣ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɨɱɤɭ, ɢ ɫɢɥɵ ɢɧɟɪɰɢɢ ɪɚɜɧɚ ɧɭɥɸ, ɬ. ɟ.
0ɎNR =++ .
ȼ ɩɪɨɟɤɰɢɹɯ ɧɚ ɞɟɤɚɪɬɨɜɵ ɨɫɢ ɤɨɨɪɞɢɧɚɬ:
0ɎNR
=++ ,
xxx
,
0ɎNR
=++
yyy
0ɎNR
=++ .
zzz
ȼ ɩɪɨɟɤɰɢɹɯ ɧɚ ɟɫɬɟɫɬɜɟɧɧɵɟ ɨɫɢ:
0ɎNR =++
,
τττ
0ɎNR
=++ ,
nnn
0NR
=+ .
BB
2. Ⱦɥɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ:
ȼ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɞɥɹ ɜɫɹɤɨɣ ɧɟɫɜɨɛɨɞɧɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ
ɫɢɫɬɟɦɵ ɝɟɨɦɟɬɪɢɱɟɫɤɚɹ ɫɭɦɦɚ ɝɥɚɜɧɵɯ ɜɟɤɬɨɪɨɜ ɚɤɬɢɜɧɵɯ ɫɢɥ, ɪɟɚɤ-
83
ɰɢɹ ɫɜɹɡɟɣ ɢ ɫɢɥ ɢɧɟɪɰɢɢ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɢɫɬɟɦɭ
ɫɢɥ, ɷɤɜɢɜɚɥɟɧɬɧɭɸ ɧɭɥɸ, ɢ ɤ ɧɟɣ ɦɨɠɧɨ ɩɪɢɦɟɧɹɬɶ ɜɫɟ ɭɪɚɜɧɟɧɢɹ
ɫɬɚɬɢɤɢ.
ɉɨɪɹɞɨɤ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɫ ɩɪɢɦɟɧɟɧɢɟɦ
ɩɪɢɧɰɢɩɚ Ⱦɚɥɚɦɛɟɪɚ
1.
ɋɞɟɥɚɬɶ ɪɢɫɭɧɨɤ.
2.
ɂɡɨɛɪɚɡɢɬɶ ɜɫɟ ɡɚɞɚɜɚɟɦɵɟ ɫɢɥɵ.
3.
ɉɪɢɦɟɧɢɬɶ ɩɪɢɧɰɢɩ ɨɫɜɨɛɨɠɞɚɟɦɨɫɬɢ ɨɬ ɫɜɹɡɟɣ, ɧɚɥɨɠɟɧɧɵɯ
ɧɚ ɫɢɫɬɟɦɭ, ɢɡɨɛɪɚɡɢɬɶ ɪɟɚɤɰɢɢ ɫɜɹɡɟɣ.
4.
Ⱦɨɛɚɜɢɬɶ ɤ ɡɚɞɚɜɚɟɦɵɦ ɫɢɥɚɦ ɢ ɪɟɚɤɰɢɹɦ ɫɜɹɡɟɣ ɫɢɥɵ ɢɧɟɪɰɢɢ
ɬɨɱɟɤ ɫɢɫɬɟɦɵ.
5.
ȼɵɛɪɚɬɶ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ.
6.
ɋɨɫɬɚɜɢɬɶ ɭɪɚɜɧɟɧɢɟ ɩɨ ɩɪɢɧɰɢɩɭ Ⱦɚɥɚɦɛɟɪɚ.
7.
Ɉɩɪɟɞɟɥɢɬɶ ɧɟɢɡɜɟɫɬɧɵɟ ɜɟɥɢɱɢɧɵ.
2. ɄɈɇɌɊɈɅɖɇɕȿ ȼɈɉɊɈɋɕ
1.
ɑɬɨ ɬɚɤɨɟ ɞɢɧɚɦɢɤɚ?
2.
ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɚɤɫɢɨɦɵ ɞɢɧɚɦɢɤɢ.
3.
ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɨɫɧɨɜɧɨɣ ɡɚɤɨɧ ɞɢɧɚɦɢɤɢ.
4.
Ɏɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɦɚɫɫɵ ɬɟɥɚ.
5.
ɑɬɨ ɬɚɤɨɟ ɢɧɟɪɬɧɨɫɬɶ?
6.
ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɩɟɪɜɭɸ ɡɚɞɚɱɭ ɞɢɧɚɦɢɤɢ.
7.
ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɜɬɨɪɭɸ ɡɚɞɚɱɭ ɞɢɧɚɦɢɤɢ.
8.
Ɂɚɩɢɲɢɬɟ ɨɫɧɨɜɧɨɣ ɡɚɤɨɧ ɞɢɧɚɦɢɤɢ ɞɥɹ ɫɜɨɛɨɞɧɨɣ ɦɚɬɟɪɢɚɥɶ-
ɧɨɣ ɬɨɱɤɢ; ɞɥɹ ɧɟɫɜɨɛɨɞɧɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ.
9.
ɑɬɨ ɬɚɤɨɟ ɫɢɥɚ ɢɧɟɪɰɢɢ?
10.
Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɢɥɚ ɢɧɟɪɰɢɢ ɜ ɫɥɭɱɚɟ ɤɪɢɜɨɥɢɧɟɣɧɨɝɨ ɞɜɢ-
ɠɟɧɢɹ?
11.
ɋɮɨɪɦɭɥɢɪɨɜɚɬɶ ɩɪɢɧɰɢɩ Ⱦɚɥɚɦɛɟɪɚ.
12.
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɦɟɬɨɞ ɤɢɧɟɬɨɫɬɚɬɢɤɢ?
84
Ɂɚɞɚɧɢɟ 1. ɉɟɪɜɚɹ ɡɚɞɚɱɚ ɞɢɧɚɦɢɤɢ
ɭ
ɝ
t
ɍɫɥɨɜɢɟ ɡɚɞɚɱɢ
Ⱦɜɢɠɟɧɢɟ ɬɨɱɤɢ ɩɨ ɩɪɹɦɨɣ ɡɚɞɚɧɨ ɭɪɚɜɧɟɧɢɟɦ:
ɯ = ɚ ⋅ t3 + ɜ ⋅ t + ɫ,
ɝɞɟ ɚ, ɜ, ɫ (ɜ ɦ) – ɩɨɫɬɨɹɧɧɵɟ.
ɇɚɣɬɢ ɫɢɥɭ
Ɇɚɫɫɚ ɬɨɱɤɢ
ʋ ɜɚɪ. ɚ ɜ ɫ m, ɤ
1 3 1 0 5 2
2 5 3 3 15 0
3 7 9 0 10 5
4 0 8 4 13 4
5 8 12 5 12 0
6 10 2 0 3 9
7 0 1 10 7 1
8 11 0 12 9 2
9 14 2 17 11 0
10 20 3 3 10 0
Ⱦɜɢɠɟɧɢɟ ɬɨɱɤɢ m = 1 ɤɝ ɡɚɞɚɧɨ ɭɪɚɜɧɟɧɢɹɦɢ:
F, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɬɨɱɤɭ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t.
«m» ɤɝ. ɑɟɦ
ɪɚɜɧɚ ɫɤɨɪɨɫɬɶ ɜ ɷɬɨɬ ɦɨɦɟɧɬ?
t, c
ɍɫɥɨɜɢɟ ɡɚɞɚɱɢ
ɯ = ɚ ⋅ t2 + ɜ ⋅ t + ɫ,
ɭ = ɚ
⋅ t2 + ɜ2 ⋅ t + ɫ2,
2
ɝɞɟ ɯ ɢ ɭ – ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ (ɜ ɦ).
Ɉɩɪɟɞɟɥɢɬɶ ɫɢɥɭ, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɬɨɱɤɭ, ɢ ɫɤɨɪɨɫɬɶ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
= 1 ɫ.
ʋ ɜɚɪ. ɚ, ɦ ɚ
, ɦ
2
ɜ, ɦ ɜ2, ɦ ɫ, ɦ ɫ2, ɦ
11 0 3 5 3 5 8
12 2 0 3 5 8 9
13 3 4 0 2 6 6
14 4 3 2 0 2 7
15 5 2 4 6 0 4
85
ʋ ɜɚɪ.
ɑɟɦɭ
ɝ
ɚ, ɦ ɚ2, ɦ
ɜ, ɦ ɜ2, ɦ ɫ, ɦ ɫ2, ɦ
16 6 3 6 4 8 0
17 5 4 2 0 9 5
18 4 0 3 3 7 6
19 3 5 4 5 0 7
20 2 3 0 2 8 5
ɍɫɥɨɜɢɟ ɡɚɞɚɱɢ
Ⱦɜɢɠɟɧɢɟ ɬɨɱɤɢ ɦɚɫɫɨɣ «m» ɡɚɞɚɧɨ ɭɪɚɜɧɟɧɢɹɦɢ:
ɯ = ɚ ⋅ tn, y = ɜ ⋅ tɤ.
Ɉɩɪɟɞɟɥɢɬɶ ɫɢɥɭ, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɬɨɱɤɭ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ «
t».
ɪɚɜɧɚ ɫɤɨɪɨɫɬɶ ɜ ɷɬɨɬ ɦɨɦɟɧɬ?
ʋ ɜɚɪ. m, ɤ
ɚ, ɦ ɜ, ɦ n ɤ t, c
21 3 0,1 0,05 1 3 4
22 2 0,05 0,2 2 1 5
23 1 1,0 0,05 3 1 3
24 4 0,05 0,1 1 3 2
25 5 0,01 0,2 2 2 2
26 2 0,02 0,1 3 1 4
27 3 0,1 0,03 1 3 5
28 1 0,3 0,05 2 3 3
29 5 0,02 0,4 3 1 1
30 4 0,2 0,3 1 3 6
ɉɪɢɦɟɪ ʋ 1
Ⱦɜɢɠɟɧɢɟ ɬɨɱɤɢ ɦɚɫɫɨɣ
Ɉɩɪɟɞɟɥɢɬɶ ɫɢɥɭ, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɬɨɱɤɭ ɢ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɬɨɱ-
ɤɢ ɜ ɦɨɦɟɧɬ
t = 1 ɫ.
m = 1 ɤɝ ɡɚɞɚɧɨ ɭɪɚɜɧɟɧɢɹɦɢ:
4
– t3 + 5 (ɦ),
ɯ = 2t
3
ɭ = t
+ t2 – 7 (ɦ).
86
Ɋȿɒȿɇɂȿ
Ⱦɥɹ ɫɨɫɬɚɜɥɟɧɢɹ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ
ɤɨɨɪɞɢɧɚɬ ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦ ɞɜɚ ɪɚɡɚ ɢɫɯɨɞɧɵɟ ɭɪɚɜɧɟɧɢɹ:
23
−=
,t3t8x
2
,t2t3y
+=
2
−=
,t6t24x
.2t6y
+=
ɍɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ:
,Fma
=
xx
Fma
=
,yy
x
y
2
,t6t24xa
−==
.2t6ya
+==
ȼ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t = 1 ɫ:
ɚɯ = 24 – 6 = 18,
ɚɭ = 64 + 2 = 8.
Ɍɨɝɞɚ, ɩɪɢ m = 1 ɤɝ:
Fx = 18 H, Fɭ = 8 H.
Ɇɨɞɭɥɶ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɣ ɧɚ ɬɨɱɤɭ ɜ ɦɨɦɟɧɬ t = 1 ɫ, ɪɚɜɟɧ:
2
y
x
222
=+=+= ,
74,19818FFF
F = 19,74 H.
ɇɚɩɪɚɜɥɹɸɳɢɟ ɤɨɫɢɧɭɫɵ ɫɢɥɵ:
F
18
cos
x
F
7,19
cos,91,0
ɉɪɢ t = 1 ɫ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɧɚ ɨɫɢ ɪɚɜɧɵ:
x
ɭ
23
2
87
F
18
y
βα
F
=−=−−=
7,19
,538t3t8xV
.41,0
======
.523t2t3ɭV
=+=+−=
Ɇɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɪɚɜɟɧ:
ɝ Ɋ
2
y
x
222
=+=+= ,
1,755VVV
V = 7,1 ɦ/ɫ.
Ɂɚɞɚɧɢɟ 2. ɉɟɪɜɚɹ ɡɚɞɚɱɚ ɞɢɧɚɦɢɤɢ
ɍɫɥɨɜɢɟ ɡɚɞɚɱɢ
Ⱥɜɬɨɦɨɛɢɥɶ ɦɚɫɫɨɣ «m» ɞɜɢɠɟɬɫɹ ɩɨ ɜɵɩɭɤɥɨɦɭ ɦɨɫɬɭ ɫ ɪɚɞɢɭɫɨɦ ɡɚɤɪɭɝɥɟɧɢɹ
Ɉɩɪɟɞɟɥɢɬɶ ɫɢɥɭ ɞɚɜɥɟɧɢɹ ɚɜɬɨɦɨɛɢɥɹ ɧɚ ɦɨɫɬ ɜ ɧɚɢɜɵɫɲɟɣ ɬɨɱɤɟ.
ɉɪɢ ɤɚɤɨɣ ɫɤɨɪɨɫɬɢ ɚɜɬɨɦɨɛɢɥɶ ɨɬɨɪɜɟɬɫɹ ɨɬ ɦɨɫɬɚ?
ʋ ɜɚɪ. m, ɤ
1 1000 90 72
2 800 100 108
3 900 200 144
4 600 50 36
5 1200 120 90
6 1300 50 54
7 1100 80 72
8 850 100 108
9 950 90 90
10 1050 85 90
11 800 50 36
12 600 80 54
13 1100 100 72
14 1050 90 90
15 700 120 108
Ⱥɜɬɨɦɨɛɢɥɶ ɦɚɫɫɨɣ «m» ɞɜɢɠɟɬɫɹ ɩɨ ɜɨɝɧɭɬɨɦɭ ɦɨɫɬɭ ɫ ɪɚɞɢɭɫɨɦ ɡɚɤɪɭɝɥɟɧɢɹ
Ɉɩɪɟɞɟɥɢɬɶ ɫɢɥɭ ɞɚɜɥɟɧɢɹ ɚɜɬɨɦɨɛɢɥɹ ɧɚ ɦɨɫɬ ɜ ɧɢɡɲɟɣ ɬɨɱɤɟ.
ɉɪɢ ɤɚɤɨɣ ɫɤɨɪɨɫɬɢ ɫɢɥɚ ɞɚɜɥɟɧɢɹ ɞɨɫɬɢɝɧɟɬ ɞɜɨɣɧɨɝɨ ɜɟɫɚ ɚɜɬɨɦɨɛɢɥɹ?
«R» ɫɨ ɫɤɨɪɨɫɬɶɸ «v».
, ɦ v, ɤɦ/ɱ
ɍɫɥɨɜɢɟ ɡɚɞɚɱɢ
«R» ɫɨ ɫɤɨɪɨɫɬɶɸ «v».
88
2
ʋ ɜɚɪ.
Ɋ
V
m, ɤɝ
, ɦ v, ɤɦ/ɱ
16 900 200 144
17 100 90 72
18 1200 120 90
19 1100 80 72
20 950 90 90
21 800 50 36
22 800 100 108
23 1100 100 72
24 700 120 108
25 600 50 36
26 1300 50 54
27 850 100 108
28 1050 95 90
29 600 80 54
30 1050 90 90ɠ
ɉɪɢɦɟɪ ʋ 2
1
m
Ɋɢɫ. 35.
1
mq
a
N
n
1
n
Ɋɢɫ. 36.
ɒɚɪɢɤ ɦɚɫɫɨɣ
ɜɪɚɳɚɟɬɫɹ ɜ ɜɟɪɬɢɤɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɫɨ ɫɤɨɪɨɫɬɶɸ
V = 12 ɦ/ɫ (ɪɢɫ. 35). Ⱦɥɢɧɚ ɧɢɬɢ Ɛ = 1 ɦ. Ɉɩɪɟɞɟɥɢɬɶ
ɧɚɬɹɠɟɧɢɟ ɧɢɬɢ ɜ ɬɨɱɤɚɯ
ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ, ɟɫɥɢ ɪɚɡɪɵɜɧɨɟ ɭɫɢɥɢɟ ɞɥɹ ɧɢɬɢ
ɪɚɜɧɨ
20 ɇ?
ȼ ɬɨɱɤɟ 1 ɧɚ ɲɚɪɢɤ ɞɟɣɫɬɜɭɟɬ ɫɢɥɚ ɬɹɠɟɫɬɢ ɢ
ɪɟɚɤɰɢɹ ɧɢɬɢ (ɪɢɫ. 36).
ɒɚɪɢɤ ɢɦɟɟɬ ɭɫɤɨɪɟɧɢɟ
ɋɨɫɬɚɜɥɹɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɜ ɩɪɨɟɤɰɢɢ ɧɚ
ɧɨɪɦɚɥɶ:
anm = mq + N
m = 0,1 ɤɝ ɧɚ ɧɢɬɢ ɪɚɜɧɨɦɟɪɧɨ
1 ɢ 2. Ʉɚɤɨɜɚ ɞɨɩɭɫɤɚɟɦɚɹ
Ɋȿɒȿɇɂȿ
2
v
= .
a
n
A
,
1
89
ɨɬɤɭɞɚ:
N = a m mq = m(a q) = m q = 0,1 10 = 13,4–
1n n
––––
22
§·§ ·
V12
¨¸¨ ¸
¨¸¨ ¸
l1
©¹© ¹
=
1
.H4,13N
,
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɞɥɹ ɬɨɱɤɢ 2 (ɪɢɫ. 37):
a
m = N
n
– mq,
1
ɨɬɤɭɞɚ:
§
¨
n2
m mq ma N
¨
©
·
V
¸
+−=+=
¸
A
¹
=
2
.H4,15N
22
§
12
¨
1,0q
¨
1
©
·
¸
4,1510
=
+−=
¸
¹
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɪɚɡɪɵɜɧɨɣ ɫɤɨɪɨɫɬɢ ɢɫɩɨɥɶɡɭɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢ-
ɠɟɧɢɹ ɞɥɹ ɲɚɪɢɤɚ ɜ ɬɨɱɤɟ 2:
n
a
n
N
2
2
V
A
,mqNm
−=−
ɨɬɤɭɞɚ:
mq
=
V
ɪɚɡ
−
m
ɪɚɡ
)mqN(
=
=
.ɫ/ɦ9,13V
⋅−
)101,020(1
1,0
A
ρ
2
Ɋɢɫ. 37.
Ɂɚɞɚɧɢɟ 3. ȼɬɨɪɚɹ ɡɚɞɚɱɚ ɞɢɧɚɦɢɤɢ
ɍɫɥɨɜɢɟ ɡɚɞɚɱɢ
Ɍɟɥɨ ɞɜɢɠɟɬɫɹ ɩɨ ɧɚɤɥɨɧɧɨɣ ɲɟɪɨɯɨɜɚɬɨɣ ɩɥɨɫɤɨɫɬɢ, ɤɨɬɨɪɚɹ ɨɛɪɚɡɭɟɬ ɫ ɝɨɪɢɡɨɧɬɨɦ ɭɝɨɥ
ɥɢɬɶ ɫɤɨɪɨɫɬɶ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
ɜɪɟɦɟɧɢ
t ɢ ɩɭɬɶ, ɩɪɨɣɞɟɧɧɵɣ ɬɟɥɨɦ ɡɚ ɜɪɟɦɹ t, ɟɫɥɢ ɬɟɥɨ ɧɚɱɚɥɨ ɞɜɢ-
ɠɟɧɢɹ ɢɡ ɫɨɫɬɨɹɧɢɹ ɩɨɤɨɹ.
α
. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ f. Ɉɩɪɟɞɟ-
t. Ɉɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ ɜ ɦɨɦɟɧɬ
90
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