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2. ɂɫɤɥɸɱɚɹ ɩɚɪɚɦɟɬɪ
υ
υ
υ
υ
υ
υ
ɥɹɟɦ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɜ ɜɢɞɟ
ɜɢɦ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɜ ɭɪɚɜɧɟ-ɧɢɟ ɤɨɨɪɞɢɧɚɬɵ
t (ɜɪɟɦɹ) ɢɡ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ, ɨɩɪɟɞɟ-
ɭ = f(x). ȼɵɪɚɡɢɦ t ɱɟɪɟɡ ɯ ɢ ɩɨɞɫɬɚ-
ɭ:
x
t
= ; 1
= ; ɭ = ɯ
2
2
x
·
§
4y
−
¸
¨
2
¹
©
2
– 1.
ɉɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɩɚɪɚɛɨɥɵ ɫ ɜɟɪɲɢ-
ɧɨɣ ɜ ɬɨɱɤɟ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ (
0; –1), ɜɟɬɜɢ ɩɚɪɚɛɨɥɵ ɧɚɩɪɚɜɥɟɧɵ ɜɜɟɪɯ.
ȼ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɜɵɱɟɪɱɢɜɚɟɦ ɬɪɚɟɤɬɨɪɢɸ ɬɨɱɤɢ.
1. Ɉɩɪɟɞɟɥɹɟɦ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɩɨɞ-
ɫɬɚɜɥɹɹ ɡɧɚɱɟɧɢɟ
t = 1 c ɜ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ:
ɯ = 2
⋅
1 = 2 (ɫɦ); ɭ = 4 ⋅ 12 – 1 = 3 (ɫɦ); Ɇ(+2; +3) ɫɦ.
ɂɡɨɛɪɚɠɚɟɦ ɬɨɱɤɭ
Ɇ ɧɚ ɬɪɚɟɤɬɨɪɢɢ.
2. ȼɵɱɢɫɥɹɟɦ ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɩɨ ɮɨɪɦɭɥɟ:
2
2
υυυ
M
−= ,
x
y
ɢ υ – ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɫɢ;
ɝɞɟ
x
′
2)t2(x
=
==
ɫɦ/ɫ;
2
υ
y
ȣ =2+8=68 8,25≈ ɫɦ/ɫ.
M
′
−==
22
t8)1t4(y
=
, ɩɪɢ t = 1 c;
= 8 ɫɦ/ɫ;
y
ȼɟɤɬɨɪ
ɲɬɚɛ: ɜ
ȼɟɤɬɨɪ
v ɫɬɪɨɢɬɫɹ ɧɚ ɱɟɪɬɟɠɟ ɩɨ ɟɝɨ ɩɪɨɟɤɰɢɹɦ
M
1 ɫɦ – 4 ɫɦ/ɫ.
ɞɨɥɠɟɧ ɛɵɬɶ ɧɚɩɪɚɜɥɟɧ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɬɪɚɟɤɬɨɪɢɢ ɜ
M
ɢ
. Ɇɚɫ-
x
y
ɞɚɧɧɨɣ ɬɨɱɤɟ, ɱɬɨ ɩɨɞɬɜɟɪɠɞɚɟɬ ɩɪɚɜɢɥɶɧɨɫɬɶ ɪɟɲɟɧɢɹ.
3. ȼɵɱɢɫɥɹɟɦ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɩɨ ɮɨɪɦɭɥɟ:
2
2
aaa
M
+= ,
x
y
ɝɞɟ
ɚ
ɢ ɚɭ – ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɫɢ;
ɯ
61
υ
υ
a=y=ȣ =(8t) =+8(ɦ / ɫ )
yy
M
′
0)2(xa
=
===
xx
′
== .
22
;
2
;
)ɫ/ɦ(88a
ȼɟɤɬɨɪ ɭɫɤɨɪɟɧɢɹ ɫɬɪɨɢɬɫɹ ɧɚ ɱɟɪɬɟɠɟ ɩɨ ɩɪɨɟɤɰɢɹɦ
ɦɚɫɲɬɚɛɟ: ɜ
1 ɫɦ – 4 ɫɦ/ɫ
ȼ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɜɟɤɬɨɪ
2
.
a ɫɨɜɩɚɥ ɫɨ ɫɜɨɟɣ ɩɪɨɟɤɰɢɟɣ ɧɚ ɨɫɶ ɚ
M
ȼɫɟ ɜɟɤɬɨɪɵ ɩɨɤɚɡɚɧɵ ɧɚ ɪɢɫ. 27.
Ɉɬɜɟɬ: ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɩɚɪɚɛɨɥɟ, ɭɪɚɜɧɟɧɢɟ ɤɨɬɨɪɨɣ:
2
ɭ = 2ɯ
– 1;
= 8,25 cɦ/c; aM = 8 cɦ/c2.
M
2. ɄɈɇɌɊɈɅɖɇɕȿ ȼɈɉɊɈɋɕ
1.
ɑɬɨ ɧɚɡɵɜɚɟɬɫɹ ɦɟɯɚɧɢɱɟɫɤɢɦ ɞɜɢɠɟɧɢɟɦ?
2.
ɑɬɨ ɬɚɤɨɟ ɬɪɚɟɤɬɨɪɢɹ?
3.
Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɪɚɟɤɬɨɪɢɹ ɬɨɱɤɢ ɩɪɢ ɤɨɨɪɞɢɧɚɬɧɨɦ ɫɩɨɫɨɛɟ
ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ?
4.
ɑɬɨ ɬɚɤɨɟ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ? ȿɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ.
5.
Ʉɚɤ ɜɵɱɢɫɥɹɟɬɫɹ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ ɩɪɢ ɟɫɬɟɫɬɜɟɧɧɨɦ ɢ ɤɨɨɪɞɢ-
ɧɚɬɧɨɦ ɫɩɨɫɨɛɚɯ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ?
6.
Ʉɚɤ ɧɚɩɪɚɜɥɟɧ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɬɪɚɟɤɬɨɪɢɢ?
7.
ɑɬɨ ɬɚɤɨɟ ɭɫɤɨɪɟɧɢɟ? ȿɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ.
8.
Ʉɚɤ ɨɩɪɟɞɟɥɢɬɶ ɭɫɤɨɪɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɩɪɢ ɟɫɬɟɫɬɜɟɧɧɨɦ
ɢ ɤɨɨɪɞɢɧɚɬɧɨɦ ɫɩɨɫɨɛɚɯ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ?
9.
Ʉɚɤ ɧɚɩɪɚɜɥɟɧɵ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɬɪɚɟɤɬɨɪɢɢ ɧɨɪɦɚɥɶɧɨɟ, ɤɚ-
ɫɚɬɟɥɶɧɨɟ ɢ ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɹ?
10.
Ɏɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɤɚɫɚɬɟɥɶɧɨɝɨ ɢ ɧɨɪɦɚɥɶɧɨɝɨ ɭɫɤɨɪɟɧɢɣ
ɬɨɱɤɢ.
ɚ
ɢ ɚɭ ɜ
ɯ
.
ɭ
62
ɉɊȺɄɌɂɑȿɋɄɈȿ ɁȺɇəɌɂȿ ʋ 5
υ
Ɍɟɦɚ: ɉɨɫɬɭɩɚɬɟɥɶɧɨɟ ɢ ɜɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ.
ɐɟɥɶ: Ɉɫɜɨɢɬɶ ɨɩɪɟɞɟɥɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɩɪɢ
ɩɨɫɬɭɩɚɬɟɥɶɧɨɦ ɢ ɜɪɚɳɚɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɹɯ ɬɟɥɚ.
ȼɪɟɦɹ ɩɪɨɜɟɞɟɧɢɹ: 2 ɱɚɫɚ.
1. ɈɋɇɈȼɇɕȿ ɉɈɅɈɀȿɇɂə ɌȿɈɊɂɂ
ɉɪɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ ɥɸ-
ϕ
ω
ε
a
M
ɰ
a
M
r
M
z
ɜɪ
ɚ
Ɇ
Ɋɢɫ. 28.
ɛɚɹ ɩɪɹɦɚɹ, ɩɪɨɜɟɞɟɧɧɚɹ ɜ ɬɟɥɟ, ɞɜɢɠɟɬɫɹ
ɩɚɪɚɥɥɟɥɶɧɨ ɫɚɦɨɣ ɫɟɛɟ. ȼɫɟ ɬɨɱɤɢ ɬɟɥɚ ɜ
ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɢɦɟɸɬ ɝɟɨɦɟɬɪɢɱɟɫɤɢ
ɪɚɜɧɵɟ ɫɤɨɪɨɫɬɢ ɢ ɭɫɤɨɪɟɧɢɹ, ɬɪɚɟɤɬɨɪɢɢ
ɜɫɟɯ ɬɨɱɟɤ ɬɟɥɚ ɨɞɢɧɚɤɨɜɵ.
ɍɪɚɜɧɟɧɢɟ ɪɚɜɧɨɦɟɪɧɨɝɨ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ:
V
M
s = s0 +
⋅
t.
Ɂɞɟɫɶ s – ɞɭɝɨɜɚɹ ɤɨɨɪɞɢɧɚɬɚ ɰɟɧɬɪɚ
ɬɹɠɟɫɬɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ.
ɍɪɚɜɧɟɧɢɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ
ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɨɞɧɨɣ ɬɨɱɤɢ – ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɢ ɦɨɝɭɬ ɛɵɬɶ ɡɚɞɚɧɵ ɬɪɟɦɹ
ɫɩɨɫɨɛɚɦɢ:
)t(zzcc= – ɜɟɤɬɨɪɧɵɣ;
=
°
®
°
z
¯
c
)t(fx
1c
−=
)t(fy
2c
=
)t(f
3
ɵɣ;ɤɨɨɪɞɢɧɚɬɧ
sɫ = sɫ(t) – ɟɫɬɟɫɬɜɟɧɧɵɣ.
ɋɤɨɪɨɫɬɶ ɢ ɭɫɤɨɪɟɧɢɟ ɥɸɛɨɣ ɬɨɱɤɢ ɬɟɥɚ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ ɬɚɤ ɠɟ, ɤɚɤ ɢ ɜ ɤɢɧɟɦɚɬɢɤɟ ɬɨɱɤɢ.
ɉɪɢ ɜɪɚɳɚɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɩɪɹɦɚɹ, ɩɪɨɜɟɞɟɧɧɚɹ
ɱɟɪɟɡ ɞɜɟ ɬɨɱɤɢ, ɨɫɬɚɟɬɫɹ ɧɟɩɨɞɜɢɠɧɨɣ ɢ ɧɚɡɵɜɚɟɬɫɹ ɨɫɶɸ ɜɪɚɳɟɧɢɹ,
63
ɜɫɟ ɨɫɬɚɥɶɧɵɟ ɬɨɱɤɢ ɬɟɥɚ ɨɩɢɫɵɜɚɸɬ ɨɤɪɭɠɧɨɫɬɢ ɜ ɩɥɨɫɤɨɫɬɹɯ, ɩɟɪ-
υ
υ
ɩɟɧɞɢɤɭɥɹɪɧɵɯ ɨɫɢ, ɫ ɰɟɧɬɪɚɦɢ, ɥɟɠɚɳɢɦɢ ɧɚ ɨɫɢ (ɪɢɫ. 28).
ɍɪɚɜɧɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ:
ϕ
= f(t),
ɝɞɟ ϕ – ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɜɨɤɪɭɝ ɨɫɢ.
ɍɪɚɜɧɟɧɢɟ ɪɚɜɧɨɦɟɪɧɨɝɨ ɜɪɚɳɟɧɢɹ:
ϕ
=
ϕ
+ ω ⋅ t.
0
ɍɪɚɜɧɟɧɢɟ ɪɚɜɧɨɩɟɪɟɦɟɧɧɨɝɨ ɜɪɚɳɟɧɢɹ:
ωϕϕ
00
2
⋅
ε
t
.
+⋅+=
t
2
ɍɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ:
ϕ
d
ω
= .
dt
ɍɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ:
2
ϕω
d
d
ε
dt
== .
2
dt
Ʌɢɧɟɣɧɚɹ (ɜɪɚɳɚɬɟɥɶɧɚɹ) ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ Ɇ:
= ω ⋅ r,
ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ
ɧɭ
ω
(ɪɢɫ. 28).
ɧɚɩɪɚɜɥɟɧ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɨɤɪɭɠɧɨɫɬɢ ɜ ɫɬɨɪɨ-
M
M
ɍɫɤɨɪɟɧɢɟ ɬɨɱɤɢ Ɇ:
ɜɪ
ɰ
M
aaa += ,
MM
ɜɪ
ɝɞɟ
a – ɜɪɚɳɚɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ (ɧɚɩɪɚɜɥɟɧɨ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ
M
ɨɤɪɭɠɧɨɫɬɢ ɜ ɫɬɨɪɨɧɭ
ε
). Ɇɨɞɭɥɶ
ɜɪ
ar
ε
=⋅;
M
ɧɨɟ ɭɫɤɨɪɟɧɢɟ (ɧɚɩɪɚɜɥɟɧɨ ɤ ɰɟɧɬɪɭ ɨɤɪɭɠɧɨɫɬɢ). Ɇɨɞɭɥɶ
ɰ
a – ɰɟɧɬɪɨɫɬɪɟɦɢɬɟɥɶ-
M
ɰ
2
ra
⋅=
ω
M
.
64
ɉɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ:
ɭ
r
r
x
t
x
t
x
t
t
x
t
x
t
x
t
t
t2 7
x
t
x
t
x
t
x
t
x
t
x
t
t
x
t
x
t
x
t
x
t
x
t
x
/6t
t
x
t
x
t
x
t
x
t
ɜɪ
()
M
2
2
ɰ
()
raaa
MM
42
ωε
+⋅=+= .
Ɂɚɞɚɧɢɟ 3. ɇɚɣɬɢ ɫɤɨɪɨɫɬɶ ɢ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ Ɇ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ
ɜɪɟɦɟɧɢ, ɢɫɯɨɞɧɵɟ ɞɚɧɧɵɟ ɜɡɹɬɶ ɢɡ ɬɚɛɥ. 11.
Ɍɚɛɥɢɰɚ 11
ʋ
ɫɵ (ɫɦ)
R
2
R
2
3
t, c
3
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ 1
x = x(t) (x – ɦ, t – c)
Ɋɚɞɢ
1 2 3 4 5 6 7
1 60 45 36 – 1
2 80 – 60 45 0,5
3 100 60 75 30 1
4 58 45 60 – 1
5 80 – 45 30 2
6 100 60 30 – 1
45 35 105 – 0,5
8 35 10 10 – 1
9 40 30 15 – ½
10 15 – 40 35 2
11 40 25 20 – 1
12 20 15 10 – 1
13 30 20 40 – 0,5
14 15 10 15 – 1
15 15 10 15 –
16 20 15 15 – 0,25
17 15 10 20 – 2
18 20 15 10 – 1
19 15 10 20 – 1
20 25 15 10 – 0,5
21 20 10 30 15 0,25
22 40 20 35 – 0,5
= 4
= 6
= 2
= 3
= 0,3
= ¼
= 3
= 3
= 6
=
=
= 5
= 2
= 3
=
=
= 6
= 8
= 6
2
+ 2t – 1
2
– 4t – 2
3
– 3
2
+ 2t + 4
2
+ 2t + 4
4
–
2
– 2t + 1
2
– 2t + 2
2
+ t – 1
2
– t + 2
= 2t + 3
2
– 2t + 2
2
– 4t + 2
3
– 3t + 1
2
– 2t + 1
2
= 8
– 2t
2
– t + 2
1
3
– ½
= 4t + 2
2
– 2t + 5
2
+ 6t – 2
2
– 4t – 2
2
+ 2t
3
+ ½
2
2
+ 2t
2
2
65
Ɉɤɨɧɱɚɧɢɟ ɬɚɛɥɢɰɵ 11
x
/6t
t
t
x
t
x
t
t
x
t
x
t
x
t
t
x
t
x
t
x
x
x
1 2 3 4 5 6 7
1
3
= ¼
= 4
= 4
= 4
= 2
= 4
= 7 + 9
2
– ½
2
3
– 2
2
– 3t + 2
3
– 4
2
23 40 30 30 15 1
24 30 15 40 20 3
25 50 20 60 – ½
26 32 15 32 16 0,5
27 40 18 40 18 2
28 40 20 40 15 2
29 25 20 50 25 2
30 30 15 20 –
=
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 4
+ ½t
2
+ 4t
2
– 2t
2
– 4t
+ 8t
2
+ 2t
2
1
2
1
x
3
2
3
M
1
2
2
M
M
3
3
4
3
M
1
2
1
66
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 4
x
x
x
x
5
M
2
7
2
x
1
3
1
x
M
6
M
3
2
8
2
M
3
1
3
1
9
2
M
1
10
3
2
M
1
3
67
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 4
x
x
x
x
11
3
M
1
13
2
M
1
15
2
12
2
M
3
14
3
M
2
16
2
1
3
1
2
M
1
x
M
x
3
1
3
68
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 4
x
x
17
2
1
M
3
19
1
x
M
3
18
2
20
M
1
x
3
3
M
2
1
x
2
21
M
1
2
22
1
3
2
M
3
69
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 4
x
x
2
x
23
2
24
2
M
1
M
x
3
1
26
25
2
M
M
3
1
3
2
1
x
3
27
3
M
1
x
28
2
1
M
3
70
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