Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Теоретическая механика. Практикум. Учебное пособие
.pdf
ɊȺɁȾȿɅ II
«ɄɂɇȿɆȺɌɂɄȺ»
ȼ ɪɚɡɞɟɥɟ «Ʉɢɧɟɦɚɬɢɤɚ» ɩɪɢɧɹɬɵ ɫɥɟɞɭɸɳɢɟ ɨɫɧɨɜɧɵɟ ɨɛɨɡɧɚɱɟɧɢɹ: x, y, z – ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ; v – ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ; ɚ – ɭɫɤɨɪɟɧɢɟ ɬɨɱ-
ɤɢ; ɚ
– ɤɚɫɚɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ; ɚn – ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱ-
τ
ɤɢ;
ρ
– ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ; ϕ – ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɬɟɥɚ; ω – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɬɟɥɚ;
ɫɤɨɪɨɫɬɟɣ.
ε
– ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɬɟɥɚ; ɦɰɫ – ɦɝɧɨɜɟɧɧɵɣ ɰɟɧɬɪ
ɉɊȺɄɌɂɑȿɋɄɈȿ ɁȺɇəɌɂȿ ʋ 4
Ɍɟɦɚ: Ʉɢɧɟɦɚɬɢɤɚ ɬɨɱɤɢ.
ɐɟɥɶ: ɂɡɭɱɢɬɶ ɫɩɨɫɨɛɵ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ, ɧɚɭɱɢɬɶɫɹ ɨɩɪɟ-
ɞɟɥɹɬɶ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɬɨɱɤɢ ɩɪɢ ɤɨɨɪɞɢɧɚɬɧɨɦ ɫɩɨɫɨɛɟ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ, ɩɨɥɨɠɟɧɢɟ ɬɨɱɤɢ ɧɚ ɬɪɚɟɤɬɨɪɢɢ ɜ ɡɚɞɚɧɧɵɣ ɦɨɦɟɧɬ
ɜɪɟɦɟɧɢ, ɚ ɬɚɤɠɟ ɨɫɧɨɜɧɵɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɞɜɢɠɟɧɢɹ
ɬɨɱɤɢ ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɫɩɨɫɨɛɚɯ ɡɚɞɚɧɢɹ ɟɟ ɞɜɢɠɟɧɢɹ.
ȼɪɟɦɹ ɩɪɨɜɟɞɟɧɢɹ: 2 ɱɚɫɚ.
1. ɈɋɇɈȼɇɕȿ ɌȿɈɊȿɌɂɑȿɋɄɂȿ ɉɈɅɈɀȿɇɂə
Ʉɢɧɟɦɚɬɢɤɚ – ɷɬɨ ɪɚɡɞɟɥ ɦɟɯɚɧɢɤɢ, ɜ ɤɨɬɨɪɨɦ ɢɡɭɱɚɟɬɫɹ ɞɜɢɠɟ-
ɧɢɟ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɟɥ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɫ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ,
ɛɟɡ ɭɱɟɬɚ ɫɢɥ, ɨɩɪɟɞɟɥɹɸɳɢɯ ɞɜɢɠɟɧɢɟ.
Ɉɫɧɨɜɧɵɦɢ ɦɚɬɟɪɢɚɥɶɧɵɦɢ ɨɛɴɟɤɬɚɦɢ ɤɢɧɟɦɚɬɢɤɢ, ɬɚɤ ɠɟ ɤɚɤ ɢ
ɜɫɟɣ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ, ɹɜɥɹɸɬɫɹ: ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ, ɫɢɫɬɟ-
ɦɚ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ, ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ. ɉɟɪɜɵɦ ɦɚɬɟɪɢɚɥɶɧɵɦ ɨɛɴɟɤɬɨɦ ɹɜɥɹɟɬɫɹ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ,
ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɜ ɞɚɧɧɨɦ ɭɱɟɛɧɨɦ ɩɨɫɨɛɢɢ.
Ɉɫɧɨɜɧɚɹ ɡɚɞɚɱɚ ɤɢɧɟɦɚɬɢɤɢ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨɛɵ ɡɧɚɹ ɡɚɤɨɧ
ɞɜɢɠɟɧɢɹ ɞɚɧɧɨɝɨ ɬɟɥɚ (ɬɨɱɤɢ), ɨɩɪɟɞɟɥɢɬɶ ɜɫɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɜɟɥɢɱɢɧɵ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟ ɟɝɨ ɞɜɢɠɟɧɢɟ. ɑɬɨɛɵ ɤɢɧɟɦɚɬɢɱɟɫɤɢ ɡɚɞɚɬɶ
ɞɜɢɠɟɧɢɟ ɬɨɱɤɢ, ɧɚɞɨ ɡɚɞɚɬɶ ɟɟ ɩɨɥɨɠɟɧɢɟ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɜɵɛɪɚɧɧɨɣ
ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɜ ɡɚɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ. ɋɭɳɟɫɬɜɭɸɬ ɬɪɢ ɫɩɨɫɨɛɚ
ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ: ɜɟɤɬɨɪɧɵɣ, ɤɨɨɪɞɢɧɚɬɧɵɣ ɢ ɟɫɬɟɫɬɜɟɧɧɵɣ.
Ʌɢɧɢɹ ɩɪɟɞɫɬɚɜɥɹɸɳɚɹ
ɬɟɥɶɧɵɯ ɩɨɥɨɠɟɧɢɣ ɞɜɢɠɭɳɟɣɫɹ ɬɨɱɤɢ ɜ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ
ɧɚɡɵɜɚɟɬɫɹ ɬɪɚɟɤɬɨɪɢɟɣ.
ɫɨɛɨɣ ɝɟɨɦɟɬɪɢɱɟɫɤɨɟ ɦɟɫɬɨ ɩɨɫɥɟɞɨɜɚ-
ɤɢɧɟɦɚɬɢɤɚ ɤɨɬɨɪɨɣ ɢ
51
ɉɨ ɜɢɞɭ ɬɪɚɟɤɬɨɪɢɣ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɦɨɠɧɨ ɪɚɡɞɟɥɢɬɶ ɧɚ ɩɪɹɦɨ-
r
ɥɢɧɟɣɧɨɟ ɢ ɤɪɢɜɨɥɢɧɟɣɧɨɟ.
ɋɤɨɪɨɫɬɶ – ɷɬɨ ɜɟɤɬɨɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɛɵɫɬɪɨɬɭ ɢ
ɧɚɩɪɚɜɥɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ.
ɍɫɤɨɪɟɧɢɟɦ ɬɨɱɤɢ ɧɚɡɵɜɚɟɬɫɹ ɜɟɤɬɨɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɢɡɦɟɧɟɧɢɟ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɦɨɞɭɥɹ ɢ ɧɚɩɪɚɜɥɟɧɢɹ ɜɟɤɬɨɪɚ
ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ.
1.1. ɋɩɨɫɨɛɵ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ
1.1.1. ȼɟɤɬɨɪɧɵɣ ɫɩɨɫɨɛ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ
ɉɨɥɨɠɟɧɢɟ ɬɨɱɤɢ Ɇ ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɨɦ
ɧɵɦ ɢɡ ɧɟɤɨɬɨɪɨɝɨ ɰɟɧɬɪɚ «Ɉ» (ɪɢɫ. 22). Ⱦɜɢɠɟɧɢɟ ɬɨɱɤɢ Ɇ ɡɚɞɚɟɬɫɹ
ɭɪɚɜɧɟɧɢɟɦ:
)t(rr = .
M
r
O
r
1
M
1
Ɋɢɫ. 22.
ɋɤɨɪɨɫɬɶ ɬɨɱɤɢ:
υ
M
.
dt
rd
=
ɍɫɤɨɪɟɧɢɟ ɬɨɱɤɢ:
2
a ==
M
d
υ
dt
rd
.
2
dt
Ɍɪɚɟɤɬɨɪɢɟɣ ɬɨɱɤɢ ɹɜɥɹɟɬɫɹ ɝɨɞɨɝɪɚɮ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ.
ɋɪɚɜɧɟɧɢɟ ɫɩɨɫɨɛɨɜ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɢ ɮɨɪɦɭɥ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ
ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɭɞɨɛɧɨ ɩɪɢɜɟɫɬɢ ɫ ɩɨɦɨɳɶɸ ɬɚɛɥ. 8.
, ɩɪɨɜɟɞɟɧ-
52
Ɍɚɛɥɢɰɚ 8
r
x
x
υ
υ
υ
υ
υ
ɋɩɨɫɨɛ
ɡɚɞɚɧɢɹ
ȼɟɤɬɨɪɧɵɣ
)t(rr =
Ʉɨɨɪɞɢɧɚɬɧɵɣ
=
°
®
°
¯
)t(fx
1
=
)t(fy
2
=
)t(fz
3
Ɍɪɚɟɤɬɨɪɢɹ ɋɤɨɪɨɫɬɶ ɍɫɤɨɪɟɧɢɟ
Ƚɨɞɨɝɪɚɮ
ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ
Ɉɩɪɟɞɟɥɹɟɬɫɹ
ɢɫɤɥɸɱɟɧɢɟɦ t
ɢɡ ɭɪɚɜɧɟɧɢɣ
ɞɜɢɠɟɧɢɹ
ɉɚɪɚɦɟɬɪɵ ɬɨɱɤɢ
rd
υ
=
dt
222
υ υυυ
υ
++=
x
=
,
x
υ
,
yz
yυ=
,
y
zz=
aaaa++=
ax=
x
a =
222
,
z
τ
ȿɫɬɟɫɬɜɟɧɧɵɣ
s = s(t)
Ɂɚɞɚɧɚ
ɩɨ ɭɫɥɨɜɢɸ
ɡɚɞɚɱɢ (ɜɦɟɫɬɟ
ɫ ɪɚɞɢɭɫɨɦ
ɤɪɢɜɢɡɧɵ
ρ
)
ds
υ
=
dt
a =
τ
a =
n
τ
1.1.2. Ʉɨɨɪɞɢɧɚɬɧɵɣ ɫɩɨɫɨɛ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ
ɍɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ Ɇ ɡɚɞɚɸɬɫɹ:
=
°
®
°
¯
)t(fx
1
=
=
)t(fy
2
),t(fz
3
ɝɞɟ x, y, z – ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ, ɡɚɜɢɫɹɳɢɟ ɨɬ ɜɪɟɦɟɧɢ (ɪɢɫ. 23).
ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɭɬɟɦ ɢɫɤɥɸɱɟɧɢɹ ɩɚɪɚɦɟɬɪɚ t ɢɡ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ. ɋɤɨɪɨɫɬɶ ɬɨɱɤɢ:
2
M
x
,
υυυυ
++=
z
y
2
2
,
,
ɝɞɟ
x
ɧɚɬ x
=
x
– ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɫɢ ɤɨɨɪɞɢ-
y
z
,
y
=
υ
y
, z
=
z
.
2
rd
2
dt
yz
=
y
za
=
aaa +=
n
2
aaa +=
2
sd
2
dt
2
v
ρ
,
ya
,
,
2
,
n
,
53
υ
Ɇ
Ɇ
x
z
y
M
z
M
x
M
M
y
Ɋɢɫ. 23.
ȼɟɤɬɨɪ
ɜɫɟɝɞɚ ɧɚɩɪɚɜɥɟɧ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɬɪɚɟɤɬɨɪɢɢ ɜ ɞɚɧ-
M
ɧɨɣ ɬɨɱɤɟ.
ɍɫɤɨɪɟɧɢɟ ɬɨɱɤɢ:
2
2
ɝɞɟ xa
x
= ,
M
, za
ya
=
y
z
= .
2
x
++= ,
y
aaaa
z
1.1.3. ȿɫɬɟɫɬɜɟɧɧɵɣ ɫɩɨɫɨɛ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ
ɉɪɢ ɟɫɬɟɫɬɜɟɧɧɨɦ ɫɩɨɫɨɛɟ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɬɪɚɟɤɬɨɪɢɟɣ
ɦɨɠɟɬ ɛɵɬɶ ɤɚɤ ɩɪɹɦɚɹ (ɪɢɫ. 24), ɬɚɤ ɢ ɤɪɢɜɚɹ (ɪɢɫ. 25) ɥɢɧɢɢ.
0
-
+
0
Ɋɢɫ. 24.
Ⱦɜɢɠɟɧɢɟ ɬɨɱɤɢ ɡɚɞɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ:
s = s(t),
ɝɞɟ s – ɞɭɝɨɜɚɹ ɤɨɨɪɞɢɧɚɬɚ ɬɨɱɤɢ.
M
+
0
)
t
(
s
=
s
ɉɪɢ ɷɬɨɦ ɞɨɥɠɧɵ ɛɵɬɶ ɡɚɞɚɧɵ:
ɬɪɚɟɤɬɨɪɢɹ ɬɨɱɤɢ, ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɢ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɞɭɝɨɜɨɣ ɤɨɨɪɞɢɧɚɬɵ (ɪɢɫ. 25).
ɋɤɨɪɨɫɬɶ ɬɨɱɤɢ: ɦɨɞɭɥɶ ɜɟɤɬɨɪɚ
Ɋɢɫ. 25.
ɫɤɨɪɨɫɬɢ ɪɚɜɟɧ ɚɛɫɨɥɸɬɧɨɦɭ ɡɧɚɱɟɧɢɸ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɞɭɝɨɜɨɣ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɩɨ
ɜɪɟɦɟɧɢ:
54
ds
s
t
s
t
~
υυ
,
==
dt
ds
=
υ
M
,
dt
ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɧɚɩɪɚɜɥɟɧ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɬɪɚɟɤɬɨɪɢɢ ɜ ɫɬɨɪɨɧɭ
ɞɜɢɠɟɧɢɹ.
ɍɫɤɨɪɟɧɢɟ ɬɨɱɤɢ:
aaa
+=
,
ɝɞɟ
a – ɤɚɫɚɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɟ ɢɡɦɟɧɟɧɢɟ ɫɤɨɪɨ-
τ
M
n
τ
ɫɬɢ ɩɨ ɜɟɥɢɱɢɧɟ, ɧɚɩɪɚɜɥɟɧɨ ɜɞɨɥɶ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɬɪɚɟɤɬɨɪɢɢ.
a – ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɟ ɢɡɦɟɧɟɧɢɟ ɫɤɨɪɨɫɬɢ ɩɨ
n
2
d
sd
υ
a
τ
2
dt
,
==
dt
ɧɚɩɪɚɜɥɟɧɢɸ, ɧɚɩɪɚɜɥɟɧɨ ɩɨ ɝɥɚɜɧɨɣ ɧɨɪɦɚɥɢ ɤ ɰɟɧɬɪɭ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ:
2
υ
a = ,
n
ρ
ɝɞɟ ρ – ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ.
Ɍɨɝɞɚ ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ:
2
() ()
τ
2
aaa +=
.
n
Ɂɚɞɚɧɢɟ 1. Ɉɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ ɢ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɜ ɞɚɧɧɵɣ ɦɨ-
ɦɟɧɬ ɜɪɟɦɟɧɢ, ɢɫɯɨɞɧɵɟ ɞɚɧɧɵɟ ɩɪɢɜɟɞɟɧɵ ɜ ɬɚɛɥ. 9.
Ɍɚɛɥɢɰɚ 9
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ
ʋ
1
2
s = f(t), ɦ
2
= 3
– 2t + 5 2 12,5
= 4t –
ȼɪɟɦɹ
t, c
2
+ 1 1 4
Ɋɚɞɢɭɫ
ɤɪɢɜɢɡɧɵ
ɦ
,
ɉɪɢɦɟɱɚɧɢɟ
ρ
3
t
cos2s ⋅=
2
π
2
55
ɉɪɨɞɨɥɠɟɧɢɟ ɬɚɛɥɢɰɵ 9
/
s
t
t
s
t
s
t
s
t
s
t
s
/
s
t
t
s
s
t
/
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ
ʋ
4
5
s
s
s = f(t), ɦ
1
t
3
1
t
4
1
2
2
3
6 s = –2cost
3
7
=
8
2
– 2
+ t 2 12,5
2
= 2
– 4t 3 8
9
10
11
12
13
= 4t – 2
2
= 3
– 2t + 4 2 20
14 s = 4sin2t
1
1
15
16
17
3
t
s −+=
2
3
= 6t – 2
= –3cos⋅ 0,5t 0,5π
18 s = –2sin2t
19
20
=t+2t3
= 4
1
4
2
–
2
ȼɪɟɦɹ
t, c
23
2t
+⋅+⋅=
34
+⋅−⋅=
t2cos2s ⋅−=
tsin3s =
tcos32s =
2
t4t
2
0,5 8
3
–
2 9
2 12
t6t
π
2
π
/8
π
/6
π
/3
0,5 8
π
/8
2 4
π
8
1 5
8 12
Ɋɚɞɢɭɫ
ɤɪɢɜɢɡɧɵ
ɦ
0,5
4,5
0,55
21 s = t3 – 2t2 + 3t 1 5
22
23 s = 2sin3t
24
2
=2
+ 4t – 5 2 36
π
π
t2cos2s =
/8
9
4,5
,
ɉɪɢɦɟɱɚɧɢɟ
ρ
4
2
Ⱦɥɹ ɜɫɟɯ ɜɚɪɢɚɧɬɨɜ
ɫɱɢɬɚɬɶ,
ɱɬɨ ɡɚɞɚɧɨ:
1) ɬɪɚɟɤɬɨɪɢɹ
ɬɨɱɤɢ:
8
+
0
-
2) ɧɚɱɚɥɨ «0»
ɢ ɧɚɩɪɚɜɥɟɧɢɟ
ɨɬɫɱɟɬɚ ɞɭɝɨɜɨɣ
2
ɤɨɨɪɞɢɧɚɬɵ
1
56
Ɉɤɨɧɱɚɧɢɟ ɬɚɛɥɢɰɵ 9
/
/
s
t
t
s
t
s
t
ɯ
f
ɭ
f
t
ɯ
y
t
/
/
6
/
x
t
y
x
t
y
t
/
/
x
y
t
/
/
/
/
x
t
y
t
1
2
25
s ++−=
26 s = –4sint
27 s = 3cos2t
28
29
30
3
t
3
3
= 2
– 4
= 5t –
2
= 6
– 8t + 1 1 4
t2
t
2
+ t 2 9
2
+ 4 0,5 8
2 4
π
π
3
12
1
1
Ɂɚɞɚɧɢɟ 2. Ɉɩɪɟɞɟɥɢɬɶ ɬɪɚɟɤɬɨɪɢɸ, ɚ ɬɚɤɠɟ ɫɤɨɪɨɫɬɶ ɢ ɭɫɤɨɪɟɧɢɟ
ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɢɫɯɨɞɧɵɟ ɞɚɧɧɵɟ ɩɪɢɜɟɞɟɧɵ ɜ ɬɚɛɥ. 10.
Ɍɚɛɥɢɰɚ 10
ʋ
=
(t), ɦ
1
=
(t), ɦ
2
, ɫ
1 2 3 4
1
= 2t
2 ɯ = 2sint y = 4sint
3 ɯ = sin2t y = cos 2t
4 ɯ = 2cos t y = 3sint
5
6
= 3
= 8
2
– 2
2
+3
7 x = 4 cos2t y = 4 sin2t
8 x = 3sin t y = 5cos t
9
= 3t
10 x = 3cos4t y = 6cos4t + 2
11 x = 3sin t y = 3cos t
12 x = 2sin2t
13 x = 3t2 + 5
14
15
= 2
2
+ 1
t2cos5x =
2
= 4
+ 1 0,5
π
4
π
π
3
= 2t 1,5
2
= 2
0,25
π
8
π
4
2
= 6
– 4
y = 4cos2t
= 4
t3y =
2
– 2 0,5
t2sin5x =
1
3
π
12
π
4
π
12
1
/3
π
/6
16
t3sin3x =
t3cos2y =
π
/9
57
Ɉɤɨɧɱɚɧɢɟ ɬɚɛɥɢɰɵ 10
x
t2 y
t
/
/
x
y
t
/
x
t
y
t
/
6
x
t
y
t
/
x
t
t
y
t
/
/
1 2 3 4
17
=
18 x = 4cos 3t – 2 y = 5cos 3t
2
– 3
tcos7x =
t3cos2x =
19 x = 6sin2t
20
21
22
= 3t + 2
= 4
23
24 x = 3cos2t y = sin2t
25
=
3
+ 2
26 x = cos2t y = 3cos2t
27
t3sin3x =
4
=
– 3 1
π
9
y = 6cos2t
= 9
= 2
= 2
tsin5y =
2
2
+ 1 0,5
t3sin2y =
6
– 4 1
t3cos3y =
π
π
π
1
/12
π
π
π
8
/6
3
8
/9
28
29
=
cos5x =
4
– 2
t
2
2
siny =
= 2
t
2
2
30 x = 2sint – 2 y = 3sint + 3
π
/2
1
2
π
4
ɉɪɢɦɟɪ ʋ 1
Ɉɩɪɟɞɟɥɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɬɨɱɤɢ
ɩɪɢ ɟɫɬɟɫɬɜɟɧɧɨɦ ɫɩɨɫɨɛɟ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ.
Ⱦɚɧɨ: ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɡɚɞɚɧɧɨɣ ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɬɪɚɟɤɬɨɪɢɢ ɫɨ-
ɝɥɚɫɧɨ ɡɚɤɨɧɭ:
ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ
Ɉɩɪɟɞɟɥɢɬɶ: ɫɤɨɪɨɫɬɶ ɢ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɧɚ ɬɪɚɟɤɬɨɪɢɢ ɜ ɞɚɧɧɵɣ
ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
2
S = 4t
– 2t + 3 ɦ,
ρ
= 6 ɦ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t = 1 c.
58
Ɋȿɒȿɇɂȿ
υ
a
υ
M
ɦ
+5
+
=
0
-
s
a
M
n
τ
a
M
τ
n
Ɋɢɫ. 26.
Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɫɥɟɞɭɸɳɟɦ ɩɨɪɹɞɤɟ:
1. Ɉɩɪɟɞɟɥɹɟɦ ɩɨɥɨɠɟɧɢɟ ɬɨɱɤɢ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ:
S = 4 ⋅ 12 – 2 ⋅ 1 + 3 = 5 (ɦ),
ɢɡɨɛɪɚɠɚɟɦ ɬɨɱɤɭ Ɇ ɫ ɞɭɝɨɜɨɣ ɤɨɨɪɞɢɧɚɬɨɣ S = 5(ɦ) ɧɚ ɬɪɚɟɤɬɨɪɢɢ
(ɪɢɫ. 26).
2. ɑɟɪɟɡ ɬɨɱɤɭ Ɇ ɩɪɨɜɨɞɢɦ ɟɫɬɟɫɬɜɟɧɧɵɟ ɨɫɢ:
τ
(ɤɚɫɚɬɟɥɶɧɭɸ) ɢ n
(ɝɥɚɜɧɭɸ ɧɨɪɦɚɥɶ). ɉɪɢ ɩɟɪɟɦɟɳɟɧɢɢ ɬɨɱɤɢ ɩɨ ɬɪɚɟɤɬɨɪɢɢ ɷɬɢ ɨɫɢ
ɞɜɢɠɭɬɫɹ ɜɦɟɫɬɟ ɫ ɧɟɣ.
3. Ɉɩɪɟɞɟɥɹɟɦ ɚɥɝɟɛɪɚɢɱɟɫɤɭɸ ɜɟɥɢɱɢɧɭ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɤɚɤ ɩɟɪ-
ɜɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ ɞɭɝɨɜɨɣ ɤɨɨɪɞɢɧɚɬɵ S ɩɨ ɜɪɟɦɟɧɢ ɢ ɢɡɨɛɪɚɠɚɟɦ
ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ
ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɨɫɢ τ:
M
υ
2
+−==
′
2t8)3t2t4(S
−=
,
ɩɪɢ t = 1 ɫ, υ = 8 ⋅ 1 – 2 = 6 (ɦ/ɫ).
Ɍɚɤ ɤɚɤ v
ɧɚɩɪɚɜɥɟɧɢɟɦ ɨɫɢ
> 0, ɬɨ ɧɚɩɪɚɜɥɟɧɢɟ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɫɨɜɩɚɞɚɟɬ ɫ
M
τ
.
4. Ɉɩɪɟɞɟɥɹɟɦ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ Ɇ ɤɚɤ ɝɟɨɦɟɬɪɢɱɟɫɤɭɸ ɫɭɦɦɭ
ɞɜɭɯ ɭɫɤɨɪɟɧɢɣ:
+= ,
nM
aaa
τ
ɝɞɟ ɚn – ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɟ ɢɡɦɟɧɟɧɢɟ ɫɤɨɪɨɫɬɢ
ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ:
22
ȣ 6
a= = =6(ɦ / ɫ )
n
ȡ 6
2
.
59
y
+3
+2
+1
-2 -1 +1 +2
-1
Ɋɢɫ. 27.
M
Ɉɬɜɟɬ: v
ɧɵ ɧɚ ɪɢɫ. 26).
M
ɇɚ ɱɟɪɬɟɠɟ
V
V
a
M
M
y
M
V
x
ɩɨ ɝɥɚɜɧɨɣ ɧɨɪɦɚɥɢ ɤ ɰɟɧɬɪɭ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ.
a – ɤɚɫɚɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ,
τ
ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɟ ɢɡɦɟɧɟɧɢɟ ɫɤɨɪɨɫɬɢ ɩɨ ɜɟɥɢɱɢɧɟ:
υ
τ
x
ɇɚ ɱɟɪɬɟɠɟ
ɧɚɩɪɚɜɥɟɧɢɟɦ ɨɫɢ
ɱɟɧɢɟ 8 > 0.
ɉɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ
ɥɟɧɨ ɩɨ ɞɢɚɝɨɧɚɥɢ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ,
ɩɨɫɬɪɨɟɧɧɨɝɨ ɧɚ ɜɟɤɬɨɪɚɯ
ɤɚɤ ɧɚ ɫɬɨɪɨɧɚɯ, ɩɨɷɬɨɦɭ:
n
τ
==+=+=
= 6 ɦ/ɫ, ɚɆ = 10 ɦ/ɫ2. (ȼɟɤɬɨɪɵ ɜɫɟɯ ɭɫɤɨɪɟɧɢɣ ɩɨɤɚɡɚ-
a ɧɚɩɪɚɜɥɹɟɬɫɹ
n
′
=
−===
a ɫɨɜɩɚɞɚɟɬ ɫ
τ
τ
, ɬ. ɤ. ɟɝɨ ɡɧɚ-
a ɧɚɩɪɚɜ-
22222
.
)ɫ/ɦ(1010086aaa
ɉɪɢɦɟɪ ʋ 2
Ɉɩɪɟɞɟɥɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ
ɬɨɱɤɢ ɩɪɢ ɤɨɨɪɞɢɧɚɬɧɨɦ ɫɩɨɫɨɛɟ ɡɚɞɚɧɢɹ ɞɜɢɠɟɧɢɹ.
Ⱦɚɧɨ: ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɜ ɩɥɨɫɤɨɫɬɢ ɫɨɝɥɚɫɧɨ ɭɪɚɜɧɟɧɢɹɦ:
®
¯
Ɉɩɪɟɞɟɥɢɬɶ: ɬɪɚɟɤɬɨɪɢɸ, ɫɤɨɪɨɫɬɶ ɢ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɜ ɦɨɦɟɧɬ
ɜɪɟɦɟɧɢ
t = 1 c.
Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɫɥɟɞɭɸɳɟɦ ɩɨɪɹɞɤɟ:
1. ɂɡɨɛɪɚɠɚɟɦ ɞɟɤɚɪɬɨɜɭ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɧɚ ɩɥɨɫɤɨɫɬɢ ɫ ɢɡɨɛ-
ɪɚɠɟɧɢɟɦ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ (ɪɢɫ. 27).
=
)ɫɦ(t2x
2
−=
Ɋȿɒȿɇɂȿ
).ɫɦ(1t4ɭ
2
M
a ɢ na ,
τ
)ɫ/ɦ(8)2t8(sa
.
60
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]