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4. Ɍɨɝɞɚ ɢɫɤɨɦɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɬɟɥɚ ɩɪɢɧɢɦɚɟɬ ɜɢɞ:
j
t
x
R
M
G
rt t i H j
() ( ) .
5. ȼɟɤɬɨɪ ɩɟɪɟɦɟɳɟɧɢɹ ɧɚɯɨɞɹɬ ɤɚɤ
rxx yy t H H
()( )( 0)( )
'X
22 2 2
10 10 01
gt
22 2
t
X
01
GG
X
0
'
24 2 4
1
91 5,7ɦ.
44
2
gt
2
GGG
rrr
9,8 1
, ɚ ɟɝɨ ɦɨɞɭɥɶ:
10
2
gt
1
2
Ɉɬɜɟɬ: 5,7 ɦ.
Ɂɚɞɚɱɚ 2. Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢɦɟɟɬ ɜɢɞ:
G
() 3 6rt ti t
GG
2
(ɦ).
ȼɵɱɢɫɥɢɬɶ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ ɢ ɭɝɨɥ ɟɟ ɧɚɤɥɨɧɚ ɤ ɨɫɢ ɯ ɱɟɪɟɡ ɨɞɧɭ ɫɟɤɭɧɞɭ ɨɬ
ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
Ⱦɚɧɨ:
X
D
– ?
1
– ?
1
2
GG
G
() 3 6rt ti tj
= 1 ɫ
1
(ɦ)
1. ɋɤɨɪɨɫɬɶ ɬɟɥɚ ɪɚɜɧɚ:
ɝɞɟ
dx
,
X
x
y
dt
ɉɨ ɭɫɥɨɜɢɸ x = 3t
X
Ɋɟɲɟɧɢɟ:
22
XXX
dy
.
X
dt
2
, y = 6t ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ:
22
(6 ) 6 8,4t
11
,
y
ɦ/ɫ.
ɍɝɨɥ ɧɚɤɥɨɧɚ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɤ ɨɫɢ ɯ ɦɨɠɟɬ ɛɵɬɶ ɧɚɣɞɟɧ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ:
Ɉɬɫɸɞɚ:
D
= arctg1 = 45q.
1
XX
tg ; tg 1.
yy
DD
XX
xx
1
6
1
61
1
Ɉɬɜɟɬ: 45q.
Ɂɚɞɚɱɚ 3. ɍɝɨɥ ɩɨɜɨɪɨɬɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ, ɞɜɢɠɭɳɟɣɫɹ ɩɨ ɤɪɭɝɨɜɨɣ
ɨɪɛɢɬɟ ɪɚɞɢɭɫɚ 2 ɦ, ɨɩɢɫɵɜɚɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ
M
(t) = 4 + 8t + 5t2 (ɪɚɞ). ɇɚɣɬɢ
ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ, ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɢ ɥɢɧɟɣɧɭɸ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɱɟɪɟɡ 1 ɫɟɤɭɧɞɭ ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
Ⱦɚɧɨ:
M
(t) = 4 + 8t + 5t2 (ɪɚɞ)
t
= 1 ɫ
1
= 2 ɦ
Z
– ?
1
H
– ?
X
– ?
1
1. ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ:
Z
= 8 + 10t1 = 8 + 10 = 18 ɪɚɞ/ɫ = 18 ɫ–1.
1
2. ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɭɝɥɨɜɨɝɨ ɭɫɤɨɪɟɧɢɹ:
Z
d
H
dt
Ɋɟɲɟɧɢɟ:
Z
810dt
dt
10 const
(ɪɚɞ/ɫ).
; H = 10 ɪɚɞ/ɫ2 = 10 ɫ–2.
21
3. Ʌɢɧɟɣɧɚɹ ɢ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ:
N
S
W
S
S
W
S
R
X
= ZR;
X
=
Z
R = 182 = 36 ɦ/ɫ.
1
1
Ɉɬɜɟɬ: 36 ɦ/ɫ.
Ɂɚɞɚɱɚ 4. Ʉɚɤɨɜɵ ɧɚɱɚɥɶɧɚɹ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɢ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɞɢɫɤɚ,
ɜɪɚɳɚɸɳɟɝɨɫɹ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ, ɟɫɥɢ ɩɪɢ ɬɨɪɦɨɠɟɧɢɢ ɨɧ ɨɫɬɚɧɨɜɢɥɫɹ
ɡɚ
W
ɫɟɤɭɧɞ, ɫɞɟɥɚɜ N ɨɛɨɪɨɬɨɜ. Ⱦɜɢɠɟɧɢɟ ɫɱɢɬɚɬɶ ɪɚɜɧɨɡɚɦɟɞɥɟɧɧɵɦ.
Ⱦɚɧɨ:
W
Z
– ?
0
H
– ?
Ɇɨɠɧɨ ɩɨɥɨɠɢɬɶ
M(W
) = 2SN.
Ʉɢɧɟɦɚɬɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ
ɜ ɷɬɨɣ ɡɚɞɚɱɟ ɩɪɢɧɢɦɚɸɬ ɜɢɞ:
M
= 0, ɚ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɞɨ ɨɫɬɚɧɨɜɤɢ ɜɵɪɚɡɢɬɶ ɤɚɤ
0
Ɂɚɩɢɲɟɦ ɩɪɢɜɟɞɟɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t =
Z(W
) = 0, ɬɨ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɩɪɢɧɢɦɚɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
SZW
2;
N
°
®
°
0.
ZHW
0
¯
Ɋɟɲɟɧɢɟ:
()
tt
MMZ
°
®
°
ZZH
¯
0
00
()
tt
0
2
HW
2
H
2
t
.
2
W
. Ɍɚɤ ɤɚɤ
ɋɨɜɦɟɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɣ ɫɢɫɬɟɦɵ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɢɬɶ ɢɫɤɨɦɵɟ ɫɨ-
ɨɬɧɨɲɟɧɢɹ ɞɥɹ ɪɚɫɱɟɬɚ ɧɚɱɚɥɶɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɢ ɭɝɥɨɜɨɝɨ ɭɫɤɨɪɟɧɢɹ:
Z
0
N4
,
W
4
.
H
N
2
Ɉɬɜɟɬ:
Z
0
N4
,
W
4
.
H
N
2
Ɂɚɞɚɱɚ 5. ɇɚɣɬɢ ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɧɚ ɨɛɨɞɟ ɞɢɫɤɚ, ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ
Z
ɜɪɚɳɟɧɢɹ ɤɨɬɨɪɨɝɨ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ ɨɩɢɫɵɜɚɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ:
= 23 – 3t
2
(ɫ–1). ɑɟɪɟɡ 1 ɫɟɤɭɧɞɭ ɨɬ ɧɚɱɚɥɚ ɜɪɚɳɟɧɢɹ. Ɋɚɞɢɭɫ ɞɢɫɤɚ 0,1 ɦ.
(t) =
Ⱦɚɧɨ:
Z
(t) = 23 – 3t2 (ɫ–1)
= 1 ɫ
t
1
= 0,1 ɦ
ɚ
– ?
ɩɨɥɧ1
d
2.
X
ɚ
W
dt
1. ɉɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɬɚɧɝɟɧɰɢɚɥɶ-
ɧɨɟ ɭɫɤɨɪɟɧɢɟ ɚ
ɮɨɪɦɭɥɟ
a ɚɚ
ɢɥɢ ɢɫɩɨɥɶɡɭɹ X =
H
.
ɩɨɥɧ ɩ
Z
R,
aR Rt
W
Ɋɟɲɟɧɢɟ:
ɢ ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɚɩ ɫɨɝɥɚɫɧɨ
22
W
Z
d
dt
(6)
.
22
222
R
3.
XZ
aR
n
R
2
.
Z
R
4. Ɉɤɨɧɱɚɬɟɥɶɧɨ:
a ɚɚ Rt tR
ɩɨɥɧ ɩ
22 2 2 242
W
111 1 1
36 20 0,1 20 40R |
(6) (23 3 )
42
ɦ/ɫ.
Ɉɬɜɟɬ: 40 ɦ/ɫ.
2.3. Ɂɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ
1. Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɡɚɞɚɧ ɭɪɚɜɧɟɧɢɟɦ
ɫɤɨɥɶɤɨ ɜɪɟɦɟɧɢ ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ ɩɪɨɟɤɰɢɹ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɧɚ ɨɫɶ
ɫɢɬ ɜ 3 ɪɚɡɚ ɩɪɨɟɤɰɢɸ ɫɤɨɪɨɫɬɢ ɧɚ ɨɫɶ
2. ɉɭɬɶ, ɩɪɨɣɞɟɧɧɵɣ ɬɟɥɨɦ, ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ:
s = 6 – 3t + 2t
2
+ t3 (ɦ). Ɉɩɪɟɞɟɥɢɬɶ ɫɪɟɞɧɸɸ ɫɤɨɪɨɫɬɶ ɟɝɨ ɞɜɢɠɟɧɢɹ ɜ ɢɧɬɟɪɜɚɥɟ
ɯ.
G
GG
3
9rtitj
y ɩɪɟɜɵ-
. ɑɟɪɟɡ
ɜɪɟɦɟɧɢ ɨɬ 1 ɫ ɞɨ 4 ɫ.
3. ɇɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ, ɞɜɢɠɭɳɟɣɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ 4 ɦɟɬ-
ɪɚ, ɡɚɞɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ
ɚ
= 1 + 6t + 9t2. Ɉɩɪɟɞɟɥɢɬɶ ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ
ɩ
ɬɨɱɤɢ ɢ ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ ɱɟɪɟɡ 1 ɫ ɨɬ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ.
4. Ɍɟɥɨ ɛɪɨɲɟɧɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ 15 ɦ/ɫ. Ɉɩɪɟɞɟɥɢɬɶ ɪɚɞɢɭɫ
ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ ɬɟɥɚ ɱɟɪɟɡ 2 ɫ ɩɨɫɥɟ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ. ɋɨɩɪɨɬɢɜɥɟɧɢɟɦ
ɜɨɡɞɭɯɚ ɩɪɟɧɟɛɪɟɱɶ.
5. Ɍɟɥɨ ɛɪɨɲɟɧɨ ɩɨɞ ɭɝɥɨɦ ɤ ɝɨɪɢɡɨɧɬɭ, ɨɤɚɡɚɥɨɫɶ, ɱɬɨ ɦɚɤɫɢɦɚɥɶɧɚɹ ɜɵ-
ɫɨɬɚ ɩɨɞɴɟɦɚ ɜ ɱɟɬɵɪɟ ɪɚɡɚ ɦɟɧɶɲɟ ɞɚɥɶɧɨɫɬɢ
ɩɨɥɟɬɚ. ɉɪɟɧɟɛɪɟɝɚɹ ɫɨɩɪɨɬɢɜ-
ɥɟɧɢɟɦ ɜɨɡɞɭɯɚ, ɨɩɪɟɞɟɥɢɬɶ ɭɝɨɥ ɤ ɝɨɪɢɡɨɧɬɭ, ɩɨɞ ɤɨɬɨɪɵɦ ɛɵɥɨ ɛɪɨɲɟɧɨ ɬɟɥɨ.
6. Ɉɩɪɟɞɟɥɢɬɶ ɧɚɱɚɥɶɧɭɸ ɫɤɨɪɨɫɬɶ, ɫ ɤɨɬɨɪɨɣ ɧɟɨɛɯɨɞɢɦɨ ɛɪɨɫɢɬɶ ɬɟɥɨ
ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ, ɱɬɨɛɵ ɨɧɨ ɜɟɪɧɭɥɨɫɶ ɨɛɪɚɬɧɨ ɱɟɪɟɡ 10 ɫ.
7. ɂɡ ɨɞɧɨɣ ɬɨɱɤɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɛɪɨɫɢɥɢ ɞɜɚ ɬɟɥɚ ɫɨ ɫɤɨɪɨɫɬɶɸ 10 ɦ/ɫ, ɨɞ-
ɧɨ ɩɨɞ ɭɝɥɨɦ 30°, ɚ ɜɬɨɪɨɟ ɩɨɞ ɭɝɥɨɦ 60
q ɤ ɝɨɪɢɡɨɧɬɭ. ɇɚɣɬɢ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ
ɧɢɦɢ ɱɟɪɟɡ 2 ɫ ɩɨɫɥɟ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
8. ɋɤɨɥɶɤɨ ɨɛɨɪɨɬɨɜ ɫɞɟɥɚɥɢ ɤɨɥɟɫɚ ɚɜɬɨɦɨɛɢɥɹ ɩɨɫɥɟ ɧɚɱɚɥɚ ɬɨɪɦɨɠɟɧɢɹ
ɞɨ ɩɨɥɧɨɣ ɨɫɬɚɧɨɜɤɢ, ɟɫɥɢ ɨɧ ɞɜɢɝɚɥɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ 60 ɤɦ/ɱ ɢ ɨɫɬɚɧɨɜɢɥɫɹ
ɡɚ 3 ɫɟɤɭɧɞɵ. Ⱦɢɚɦɟɬɪ ɤɨɥɟɫ 0,7 ɦɟɬɪɚ. Ⱦɜɢɠɟɧɢɟ ɫɱɢɬɚɬɶ ɪɚɜɧɨɡɚɦɟɞɥɟɧɧɵɦ.
9. ɇɚɣɬɢ ɪɚɞɢɭɫ ɤɨɥɟɫɚ, ɟɫɥɢ ɥɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɬɨɱɤɢ ɧɚ ɟɝɨ
ɨɛɨɞɟ ɜ 2,5 ɪɚɡɚ ɛɨɥɶɲɟ, ɱɟɦ ɥɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ, ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɧɚ 5 ɫɦ
ɛɥɢɠɟ ɤ ɨɫɢ.
10. Ʉɨɥɟɫɨ, ɜɪɚɳɚɹɫɶ ɪɚɜɧɨɭɫɤɨɪɟɧɧɨ, ɞɨɫɬɢɝɥɨ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ 20 ɫ
–1
ɱɟɪɟɡ 10 ɨɛɨɪɨɬɨɜ ɨɬ ɧɚɱɚɥɚ ɜɪɚɳɟɧɢɹ. ɇɚɣɬɢ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɤɨɥɟɫɚ.
11. Ɍɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ 20 ɫɦ ɫ ɩɨɫɬɨɹɧɧɵɦ ɬɚɧɝɟɧɰɢɚɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ 5 ɫɦ/ɫ
2
. ɑɟɪɟɡ ɤɚɤɨɟ ɜɪɟɦɹ ɩɨɫɥɟ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ ɧɨɪ-
ɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɛɭɞɟɬ ɪɚɜɧɨ ɬɚɧɝɟɧɰɢɚɥɶɧɨɦɭ?
12. Ɍɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɬɚɤ, ɱɬɨ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɫɨ ɜɪɟɦɟɧɟɦ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ
M
(t) = 5 + t + t2 + t3 (ɪɚɞ). ɇɚɣɬɢ ɪɚɞɢɭɫ ɤɨɥɟɫɚ, ɟɫɥɢ ɤ ɤɨɧɰɭ
ɜɬɨɪɨɣ ɫɟɤɭɧɞɵ ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ ɧɚ ɨɛɨɞɟ ɤɨɥɟɫɚ ɞɨɫɬɢɝɥɨ ɜɟɥɢɱɢ-
2
10
ɧɵ 3,46
ɦ/ɫ2.
23
2.4. Ʉɢɧɟɦɚɬɢɤɚ. Ɍɟɫɬɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ ɞɜɢɠɭɳɟɣɫɹ ɩɨ ɩɥɨɫɤɨɫɬɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢɦɟɟɬ ɜɢɞ:
G
() ( 1) 2rt t i tj
GG
(ɦ). ɇɚɣɬɢ ɦɨɞɭɥɶ ɩɟɪɟɦɟɳɟɧɢɹ ɬɨɱɤɢ ɡɚ ɩɟɪɜɵɟ ɭ ɫɟɤɭɧɞ
ɍɪɨɜɟɧɶ I
ɞɜɢɠɟɧɢɹ.
2. Ɉɩɪɟɞɟɥɢɬɶ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɜ ɤɨɧɰɟ ɩɟɪɜɨɣ ɫɟɤɭɧɞɵ ɞɜɢɠɟɧɢɹ,
ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɤɨɬɨɪɨɝɨ ɢɡɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɩɨ ɡɚɤɨɧɭ
3. Ɂɚɜɢɫɢɦɨɫɬɶ ɤɨɨɪɞɢɧɚɬɵ ɬɟɥɚ ɨɬ ɜɪɟɦɟɧɢ ɢɦɟɟɬ ɜɢɞ ɯ(t) = 5 + 3t – t
GG
G
32
() 3rt ti t j
.
3
(ɦ).
ɇɚ ɤɚɤɨɦ ɪɚɫɫɬɨɹɧɢɢ ɨɬ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ ɨɧɨ ɨɫɬɚɧɨɜɢɬɫɹ?
4. Ɍɟɥɨ ɫɜɨɛɨɞɧɨ ɩɚɞɚɟɬ ɫ ɜɵɫɨɬɵ 40 ɦɟɬɪɨɜ? Ɂɚ ɤɚɤɨɟ ɜɪɟɦɹ ɜɵɫɨɬɚ ɬɟɥɚ
ɧɚɞ ɩɨɜɟɪɯɧɨɫɬɶɸ Ɂɟɦɥɢ ɭɦɟɧɶɲɢɬɫɹ ɜ 2 ɪɚɡɚ (
5. Ɍɟɥɨ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ 2 ɦ. ɉɭɬɶ, ɩɪɨɣɞɟɧɧɵɣ ɬɟɥɨɦ,
ɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɩɨ ɡɚɤɨɧɭ
ɜɪɟɦɟɧɢ
t = 1 ɫ ɟɝɨ ɧɨɪɦɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ.
s(t) = 0,4t
g = 10 ɦ/ɫ
2
+ 0,2t (ɦ). Ɉɩɪɟɞɟɥɢɬɶ ɞɥɹ ɦɨɦɟɧɬɚ
6. Ɍɟɥɨ ɛɪɨɲɟɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ 10 ɦ/ɫ ɩɨɞ ɭɝɥɨɦ 30
2
)?
q ɤ ɝɨɪɢɡɨɧɬɭ. ɇɚɣɬɢ
ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɟɥɚ ɱɟɪɟɡ 1 ɫ ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
M
7. Ⱦɜɢɠɟɧɢɟ ɬɟɥɚ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ ɡɚɞɚɧɨ ɭɪɚɜɧɟɧɢɟɦ
+ 10
2
t – t
(ɪɚɞ). ɇɚɣɬɢ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɬɟɥɚ.
8. ȼɪɚɳɟɧɢɟ ɤɨɥɟɫɚ ɪɚɞɢɭɫɨɦ 0,1 ɦ ɨɩɢɫɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ:
(t) = 1 +
M
(t) = 1 + t2.
ɇɚɣɬɢ ɥɢɧɟɣɧɭɸ ɫɤɨɪɨɫɬɶ ɬɨɱɟɤ ɧɚ ɨɛɨɞɟ ɱɟɪɟɡ ɞɜɚ ɩɨɥɧɵɯ ɨɛɨɪɨɬɚ.
9. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɤɨɥɟɫɚ, ɟɫɥɢ ɡɚ 5 ɫ ɟɝɨ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ
ɭɜɟɥɢɱɢɥɚɫɶ ɧɚ 20 ɫ
10. Ⱦɢɫɤ ɜɪɚɳɚɟɬɫɹ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ ɩɨ ɡɚɤɨɧɭ
–1
.
M
(t) = 0,5t2 (ɪɚɞ).
Ɉɩɪɟɞɟɥɢɬɶ ɤ ɤɨɧɰɭ ɜɬɨɪɨɣ ɫɟɤɭɧɞɵ ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ ɬɨɱɤɢ, ɧɚɯɨɞɹɳɟɣɫɹ ɧɚ
ɪɚɫɫɬɨɹɧɢɢ 0,8 ɦ ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ.
ɍɪɨɜɟɧɶ II
1. Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ ɞɜɢɠɭɳɟɣɫɹ ɩɨ ɩɥɨɫɤɨɫɬɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢɦɟɟɬ ɜɢɞ
G
() ( 1) 2rt t i t j
ɦɨɝɨ ɫɨɨɬɧɨɲɟɧɢɟɦ
GG
3
(ɦ). ɇɚɣɬɢ ɧɚɱɚɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ.
2. ɇɚɣɬɢ ɦɨɞɭɥɶ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɱɟɪɟɡ 1 ɫ ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ, ɨɩɢɫɵɜɚɟ-
G
3. ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɢɦɟɟɬ ɜɢɞ ɯ(t) = 5 + 6t + t
GG
32
() 3rt ti t j
.
2
(ɦ). Ɂɚ ɤɚɤɨɟ ɜɪɟɦɹ
ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ ɭɜɟɥɢɱɢɬɫɹ ɜ 3 ɪɚɡɚ?
4. ɋ ɛɚɲɧɢ ɜɵɫɨɬɨɣ 40 ɦɟɬɪɨɜ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ ɛɪɨɲɟɧɨ ɬɟɥɨ ɫɨ ɫɤɨɪɨɫɬɶɸ 20 ɦ/ɫ. Ɉɩɪɟɞɟɥɢɬɶ ɜɪɟɦɹ ɩɚɞɟɧɢɹ ɬɟɥɚ ɧɚ Ɂɟɦɥɸ (
5. Ɍɟɥɨ ɛɪɨɲɟɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ 10 ɦ/ɫ ɩɨɞ ɭɝɥɨɦ 30
g = 10 ɦ/ɫ
q ɤ ɝɨɪɢɡɨɧɬɭ. ɇɚɣɬɢ ɟɝɨ
2
).
ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɜ ɜɟɪɯɧɟɣ ɬɨɱɤɟ ɬɪɚɟɤɬɨɪɢɢ.
6. Ɂɚɜɢɫɢɦɨɫɬɶ ɩɭɬɢ, ɩɪɨɣɞɟɧɧɨɝɨ ɬɟɥɨɦ, ɞɜɢɠɭɳɢɦɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɚ 3 ɦ, ɡɚɞɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ
s(t) = 0,4t
2
+ 0,1t. Ɉɩɪɟɞɟɥɢɬɶ ɩɨɥɧɨɟ ɭɫɤɨɪɟɧɢɟ
ɬɟɥɚ ɱɟɪɟɡ 1 ɫ ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ.
24
H
7. Ɍɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɫ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ
Z
ɥɨɜɚɹ ɫɤɨɪɨɫɬɶ
ɱɟɧɢɟ
ɤ.
= 0. Ɇɨɞɭɥɶ ɧɨɪɦɚɥɶɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɚ
~ t. ɉɪɢ t = 0 ɭɝ-
ɤ
~ t
. ɇɚɣɬɢ ɡɧɚ-
ɩ
8. ɇɚɣɬɢ ɱɢɫɥɨ ɩɨɥɧɵɯ ɨɛɨɪɨɬɨɜ, ɫɞɟɥɚɧɧɵɯ ɡɚ 5 ɫ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ
H
ɫ ɦɨɦɟɧɬɚ ɧɚɱɚɥɚ ɜɪɚɳɟɧɢɹ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ
9. Ⱦɢɫɤ ɧɚɱɢɧɚɟɬ ɜɪɚɳɚɬɶɫɹ ɢɡ ɫɨɫɬɨɹɧɢɹ ɩɨɤɨɹ. ɋ ɩɨɫɬɨɹɧɧɵɦ ɭɝɥɨɜɵɦ
ɭɫɤɨɪɟɧɢɟɦ 5 ɫ
ɜɚɹ ɫɤɨɪɨɫɬɶ ɞɨɫɬɢɝɧɟɬ ɜɟɥɢɱɢɧɵ 60 ɫ
–2
. ɇɚ ɤɚɤɨɣ ɭɝɨɥ ɨɧ ɩɨɜɟɪɧɟɬɫɹ ɤ ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɭɝɥɨ-
–1
?
10. Ⱦɢɫɤ ɜɪɚɳɚɟɬɫɹ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ ɩɨ ɡɚɤɨɧɭ
= 2 ɫ–2.
M
(t) = 0,5t2 ɪɚɞ.
Ɉɩɪɟɞɟɥɢɬɶ, ɧɚ ɤɚɤɨɣ ɭɝɨɥ ɨɧ ɩɨɜɟɪɧɟɬɫɹ ɤ ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɧɨɪɦɚɥɶɧɨɟ
ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɹ ɬɨɱɤɢ ɧɚ ɟɝɨ ɨɛɨɞɟ ɫɪɚɜɧɹɸɬɫɹ.
ɍɪɨɜɟɧɶ III
1. ȼɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɢɡɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɩɨ ɡɚɤɨɧɭ
G
X
()2tti
G
1
2
ɇɚɣɬɢ ɩɭɬɶ, ɩɪɨɣɞɟɧɧɵɣ ɟɸ ɡɚ ɩɟɪɜɵɟ 10 ɫɟɤɭɧɞ.
2. ɉɪɢ ɞɜɢɠɟɧɢɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɟɟ ɤɨɨɪɞɢɧɚɬɵ ɢɡɦɟɧɹɸɬɫɹ ɩɨ ɡɚɤɨɧɭ
ɯ(t) = A t, y(t) = At(1 + Bt), ɝɞɟ Ⱥ ɢ ȼ – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɤɨɧɫɬɚɧɬɵ. ɇɚɫɤɨɥɶɤɨ ɢɡɦɟ-
ɧɢɬɫɹ ɞɥɢɧɚ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɷɬɨɣ ɬɨɱɤɢ ɡɚ ɩɟɪɜɭɸ ɫɟɤɭɧɞɭ ɞɜɢɠɟɧɢɹ?
3. Ʉɨɨɪɞɢɧɚɬɚ ɞɜɢɠɭɳɟɣɫɹ ɬɨɱɤɢ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ
ɯ(t) = 16t – 4t
2
+ 8 (ɦ).
Ʉɚɤɨɜɨ ɟɟ ɦɚɤɫɢɦɚɥɶɧɨɟ ɭɞɚɥɟɧɢɟ ɨɬ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ?
4. Ɍɟɥɨ ɩɚɞɚɟɬ ɫ ɜɵɫɨɬɵ 100 ɦ ɫ ɧɭɥɟɜɨɣ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ. Ɂɚ ɤɚɤɨɟ
ɜɪɟɦɹ ɬɟɥɨ ɩɪɨɣɞɟɬ ɩɨɫɥɟɞɧɢɟ 10 ɦɟɬɪɨɜ ɫɜɨɟɝɨ ɩɭɬɢ (
g = 10 ɦ/ɫ
2
)?
5. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɧɚɱɢɧɚɟɬ ɞɜɢɝɚɬɶɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ 12,5 ɦ
ɚ
ɫ ɩɨɫɬɨɹɧɧɵɦ ɬɚɧɝɟɧɰɢɚɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ
ɦɟɧɢ, ɤɨɝɞɚ ɜɟɤɬɨɪ ɩɨɥɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɨɛɪɚɡɭɟɬ ɫ ɜɟɤɬɨɪɨɦ ɫɤɨɪɨɫɬɢ ɭɝɨɥ 45
= 0,5 ɦ/ɫ2. Ɉɩɪɟɞɟɥɢɬɶ ɦɨɦɟɧɬ ɜɪɟ-
W
q.
6. Ɍɟɥɨ ɛɪɨɫɢɥɢ ɩɨɞ ɭɝɥɨɦ ɤ ɝɨɪɢɡɨɧɬɭ. ȼ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɞɨɫɬɢɝɥɚ ɜɟɥɢɱɢɧɵ 30 ɦ/ɫ, ɚ ɪɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɬɪɚɟɤɬɨɪɢɢ ɫɨɫɬɚɜɢɥ 100 ɦ. Ɉɩɪɟɞɟɥɢɬɶ ɷɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
7. Ɍɟɥɨ ɜɪɚɳɚɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ ɫ ɭɝɥɨɜɵɦ ɭɫɤɨɪɟɧɢ-
H
= 2t2c–2. ɉɪɢ t = 0 ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ Z = 0. Ɉɩɪɟɞɟɥɢɬɶ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɭɝ-
ɟɦ
ɥɨɜɨɣ ɫɤɨɪɨɫɬɢ.
M
8. Ⱦɜɢɠɟɧɢɟ ɬɟɥɚ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ ɡɚɞɚɧɨ ɭɪɚɜɧɟɧɢɟɦ
S
(6t – 3t2) (ɪɚɞ). ɋɤɨɥɶɤɨ ɨɛɨɪɨɬɨɜ ɫɞɟɥɚɟɬ ɬɟɥɨ ɞɨ ɦɨɦɟɧɬɚ ɢɡɦɟɧɟɧɢɹ
= 2
(t) =
ɧɚɩɪɚɜɥɟɧɢɹ ɜɪɚɳɟɧɢɹ?
9. Ⱦɢɫɤ ɪɚɞɢɭɫɨɦ
ɨɛɨɞɟ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ
ɬɨɪ ɩɨɥɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɨɛɪɚɡɭɟɬ ɫ ɪɚɞɢɭɫɨɦ ɤɨɥɟɫɚ ɭɝɨɥ 60
ɦ ɜɪɚɳɚɟɬɫɹ ɬɚɤ, ɱɬɨ ɥɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɬɨɱɟɤ ɧɚ
3
X
(t) = 2t2. Ɉɩɪɟɞɟɥɢɬɶ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɜɟɤ-
q.
10. Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɨɦ 0,1 ɦ ɬɚɤ, ɱɬɨ
M
ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɨɩɢɫɵɜɚɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ
(t) = 0,3t + 0,1t3 (ɪɚɞ). Ɉɩɪɟɞɟɥɢɬɶ
ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɥɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ ɞɨɫɬɢɝɧɟɬ
ɜɟɥɢɱɢɧɵ 0,4 ɦ/ɫ.
.
25
3. ȾɂɇȺɆɂɄȺ ɆȺɌȿɊɂȺɅɖɇɈɃ ɌɈɑɄɂ
3.1. Ʉɪɚɬɤɢɟ ɫɜɟɞɟɧɢɹ ɢɡ ɬɟɨɪɢɢ
Ⱦɢɧɚɦɢɤɚ ɢɡɭɱɚɟɬ ɞɜɢɠɟɧɢɟ ɬɟɥ ɜ ɫɜɹɡɢ ɫ ɬɟɦɢ ɩɪɢɱɢɧɚɦɢ (ɜɡɚɢɦɨɞɟɣɫɬɜɢɹɦɢ ɦɟɠɞɭ ɬɟɥɚɦɢ), ɤɨɬɨɪɵɟ ɨɛɭɫɥɨɜɥɢɜɚɸɬ ɬɨɬ ɢɥɢ ɢɧɨɣ ɯɚɪɚɤɬɟɪ ɞɜɢɠɟɧɢɹ. Ʉɥɚɫɫɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ ɹɜɥɹɟɬɫɹ ɦɟɯɚɧɢɤɨɣ ɬɟɥ ɛɨɥɶɲɢɯ (ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ
ɦɚɫɫɨɣ ɚɬɨɦɨɜ) ɦɚɫɫ, ɞɜɢɠɭɳɢɯɫɹ ɫ ɦɚɥɵɦɢ (ɩɨ ɫɪɚɜɧɟɧɢɸ ɫɨ ɫɤɨɪɨɫɬɶɸ ɫɜɟɬɚ)
ɫɤɨɪɨɫɬɹɦɢ.
ȼɨɡɞɟɣɫɬɜɢɟ ɧɚ ɞɚɧɧɨɟ ɬɟɥɨ ɫɨ ɫɬɨɪɨɧɵ ɞɪɭɝɢɯ ɬɟɥ ɜɵɡɵɜɚɟɬ
ɢɡɦɟɧɟɧɢɟ
ɟɝɨ ɫɤɨɪɨɫɬɢ, ɬ. ɟ. ɫɨɨɛɳɚɟɬ ɞɚɧɧɨɦɭ ɬɟɥɭ ɭɫɤɨɪɟɧɢɟ. Ɇɟɪɨɣ ɞɟɣɫɬɜɢɹ ɨɞɧɨɝɨ
ɬɟɥɚ ɧɚ ɞɪɭɝɨɟ ɹɜɥɹɟɬɫɹ ɜɟɤɬɨɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɤɨɬɨɪɨɣ ɩɪɢɫɜɨɟɧɨ ɧɚɡɜɚɧɢɟ ɫɢɥɚ.
ɉɪɢ ɢɡɭɱɟɧɢɢ ɞɜɢɠɟɧɢɹ ɤɚɤɨɝɨ-ɥɢɛɨ ɬɟɥɚ ɞɟɣɫɬɜɢɟ ɧɚ ɧɟɝɨ ɞɪɭɝɢɯ ɬɟɥ ɢɡɨɛɪɚɠɚɟɬɫɹ ɩɪɢɥɨɠɟɧɧɵɦɢ ɤ ɷɬɨɦɭ ɬɟɥɭ ɜɟɤɬɨɪɚɦɢ-ɫɢɥɚɦɢ. ȿɫɥɢ ɫɢɥɭ ɢɥɢ ɫɢɫɬɟɦɭ
ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɟɥɭ, ɦɨɠɧɨ
ɡɚɦɟɧɢɬɶ ɞɪɭɝɨɣ ɫɢɥɨɣ ɢɥɢ ɫɢɫɬɟɦɨɣ ɫɢɥ, ɧɟ
ɢɡɦɟɧɹɹ ɩɪɢ ɷɬɨɦ ɫɨɫɬɨɹɧɢɹ ɞɜɢɠɟɧɢɹ ɬɟɥɚ, ɬɨ ɬɚɤɢɟ ɫɢɥɵ ɢɥɢ ɫɢɫɬɟɦɵ ɫɢɥ
ɧɚɡɵɜɚɸɬɫɹ ɷɤɜɢɜɚɥɟɧɬɧɵɦɢ. ȼ ɱɚɫɬɧɨɫɬɢ, ɤɨɝɞɚ ɫɢɫɬɟɦɚ ɫɢɥ ɡɚɦɟɧɹɟɬɫɹ ɨɞɧɨɣ
ɫɢɥɨɣ, ɬɨ ɷɬɚ ɫɢɥɚ ɧɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ.
Ɉɩɵɬ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɪɚɜɧɵɟ ɫɢɥɵ ɫɨɨɛɳɚɸɬ ɪɚɡɧɵɦ ɬɟɥɚɦ ɪɚɡɧɵɟ ɩɨ ɜɟ-
ɥɢɱɢɧɟ ɭɫɤɨɪɟɧɢɹ. ȼɫɹɤɨɟ ɬɟɥɨ ɩɪɨɬɢɜɢɬɫɹ ɩɨɩɵɬɤɚɦ
ɢɡɦɟɧɢɬɶ ɟɝɨ ɫɨɫɬɨɹɧɢɟ
ɞɜɢɠɟɧɢɹ. ɗɬɨ ɫɜɨɣɫɬɜɨ ɧɚɡɵɜɚɟɬɫɹ ɢɧɟɪɬɧɨɫɬɶɸ. ȼ ɤɚɱɟɫɬɜɟ ɤɨɥɢɱɟɫɬɜɟɧɧɨɣ
ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɢɧɟɪɬɧɨɫɬɢ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜɟɥɢɱɢɧɚ, ɧɚɡɵɜɚɟɦɚɹ ɦɚɫɫɨɣ ɬɟɥɚ.
ȼ ɨɫɧɨɜɟ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ ɥɟɠɚɬ ɬɪɢ ɡɚɤɨɧɚ ɞɢɧɚɦɢɤɢ, ɫɮɨɪɦɭɥɢ-
ɪɨɜɚɧɧɵɟ ɂ. ɇɶɸɬɨɧɨɦ ɜ 1687 ɝ.
ɉɟɪɜɵɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: ɜɫɹɤɨɟ ɬɟ-
ɥɨ ɧɚɯɨɞɢɬɫɹ ɜ ɫɨɫɬɨɹɧɢɢ ɩɨɤɨɹ ɢɥɢ ɪɚɜɧɨɦɟɪɧɨɝɨ ɩɪɹɦɨɥɢɧɟɣɧɨɝɨ ɞɜɢɠɟɧɢɹ,
ɩɨɤɚ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɫɨ ɫɬɨɪɨɧɵ ɞɪɭɝɢɯ ɬɟɥ ɧɟ ɡɚɫɬɚɜɢɬ ɟɝɨ ɢɡɦɟɧɢɬɶ ɷɬɨ
ɫɨɫɬɨɹɧɢɟ. Ɏɨɪɦɭɥɢɪɨɜɤɟ ɷɬɨɝɨ ɡɚɤɨɧɚ ɦɨɠɧɨ ɩɪɢɞɚɬɶ ɫɥɟɞɭɸɳɢɣ ɜɢɞ: ɫɤɨɪɨɫɬɶ ɥɸɛɨɝɨ ɬɟɥɚ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ (ɜ ɱɚɫɬɧɨɫɬɢ, ɪɚɜɧɨɣ ɧɭɥɸ), ɟɫɥɢ
ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɜɫɟɯ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɟɥɭ, ɪɚɜɧɚ ɧɭɥɸ:
n
F10G. (3.1)
¦
i
i
ɋɢɫɬɟɦɚ ɨɬɫɱɟɬɚ, ɜ ɤɨɬɨɪɨɣ ɜɵɩɨɥɧɹɟɬɫɹ ɩɟɪɜɵɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ, ɧɚɡɵɜɚɟɬ-
ɫɹ ɢɧɟɪɰɢɚɥɶɧɨɣ, ɚ ɩɟɪɜɵɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɜ ɫɜɹɡɢ ɫ ɷɬɢɦ ɧɚɡɵɜɚɸɬ ɡɚɤɨɧɨɦ
ɢɧɟɪɰɢɢ.
Ɉɩɵɬɧɵɦ ɩɭɬɟɦ ɭɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ, ɰɟɧɬɪ ɤɨɬɨɪɨɣ ɫɨɜɦɟɳɟɧ ɫ ɋɨɥɧɰɟɦ, ɚ ɨɫɢ, ɧɚɩɪɚɜɥɟɧɧɵɟ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɨɛɪɚɡɨɦ ɜɵɛɪɚɧɧɵɟ
ɡɜɟɡɞɵ, ɹɜɥɹɟɬɫɹ ɢɧɟɪɰɢɚɥɶɧɨɣ. ɗɬɚ
ɫɢɫɬɟɦɚ ɧɚɡɵɜɚɟɬɫɹ ɝɟɥɢɨɰɟɧɬɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɨɣ ɨɬɫɱɟɬɚ. Ʌɸɛɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ, ɞɜɢɠɭɳɚɹɫɹ ɪɚɜɧɨɦɟɪɧɨ ɢ ɩɪɹɦɨɥɢɧɟɣɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɟɥɢɨɰɟɧɬɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ, ɛɭɞɟɬ ɢɧɟɪɰɢɚɥɶɧɨɣ. ɋɢɫɬɟɦɭ
ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ Ɂɟɦɥɟɣ, ɦɨɠɧɨ ɩɪɢɛɥɢɠɟɧɧɨ ɫɱɢɬɚɬɶ ɢɧɟɪɰɢɚɥɶɧɨɣ
ɜɫɥɟɞɫɬɜɢɟ ɦɚɥɨɣ ɜɟɥɢɱɢɧɵ ɭɫɤɨɪɟɧɢɹ, ɫ ɤɨɬɨɪɨɣ ɷɬɚ ɫɢɫɬɟɦɚ ɞɜɢɠɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɟɥɢɨɰɟɧɬɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
26
ȼɟɤɬɨɪɧɭɸ ɜɟɥɢɱɢɧɭ, ɪɚɜɧɭɸ ɩɪɨɢɡɜɟɞɟɧɢɸ ɦɚɫɫɵ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ
X
F
p
F
t
p
t
ɟɟ ɫɤɨɪɨɫɬɢ, ɧɚɡɵɜɚɸɬ
ɢɦɩɭɥɶɫɨɦ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ:
G
G
. (3.2)
mp
ȼɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
ɂɡɦɟɧɟɧɢɟ ɢɦɩɭɥɶɫɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜɨ ɜɪɟɦɟɧɢ ɪɚɜɧɨ ɪɟɡɭɥɶɬɢɪɭɸ-
ɳɟɣ (ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ)
n
Ɂɞɟɫɶ
¦
ɪ
1
i
G
ɜɫɟɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɨɱɤɭ:
p
G
G
pd
F
. (3.3)
p
dt
GG
– ɪɟɡɭɥɶɬɢɪɭɸɳɚɹ ɜɫɟɯ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɦɚɬɟɪɢɚɥɶ-
FF
i
ɧɨɣ ɬɨɱɤɟ.
ɂɡ ɷɬɨɣ ɮɨɪɦɭɥɢɪɨɜɤɢ ɜɬɨɪɨɝɨ ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ ɫɥɟɞɭɟɬ ɜɵɪɚɠɟɧɢɟ ɞɥɹ
ɧɚɯɨɠɞɟɧɢɹ ɩɪɢɪɚɳɟɧɢɹ ɢɦɩɭɥɶɫɚ ɡɚ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ, ɩɪɨɬɟɤɲɢɣ ɨɬ ɦɨɦɟɧɬɚ t
ɞɨ ɦɨɦɟɧɬɚ t2:
1
ȼ ɱɚɫɬɧɨɫɬɢ, ɟɫɥɢ
G
=const
ɪ
t
G
GG
21
2
pFdt
³
p
t
1
.
, ɬɨ ɩɪɢɪɚɳɟɧɢɟ ɢɦɩɭɥɶɫɚ ɡɚ ɩɪɨɦɟɠɭɬɨɤ
ɜɪɟɦɟɧɢ IJ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
GG
ppF
21
G
W
p
.
ȼɵɪɚɠɟɧɢɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɷɬɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɢɦɩɭɥɶɫɨɦ ɫɢɥɵ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɚɧɚɥɢɡ ɜɵɪɚɠɟɧɢɹ (3.3) ɩɪɢɜɨɞɢɬ ɤ ɜɚɠɧɨɦɭ ɜɵɜɨɞɭ:
ɟɫɥɢ ɢɡɜɟɫɬɧɨ, ɤɚɤ ɢɡɦɟɧɹɟɬɫɹ ɢɦɩɭɥɶɫ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɫɨ ɜɪɟɦɟ-
G
ɧɟɦ )(
, ɬɨ ɦɨɠɧɨ ɧɚɣɬɢ ɫɢɥɭ, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɬɟɥɨ, ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟ-
ɦɟɧɢ.
ȿɫɥɢ ɦɚɫɫɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɩɨɫɬɨɹɧɧɚ ɜɨ ɜɪɟɦɟɧɢ ɢɥɢ ɧɟɢɡɦɟɧɧɚ
cons
m
, ɬɨ (3.3) ɩɪɢɧɢɦɚɟɬ ɩɪɨɫɬɨɣ ɢɡɜɟɫɬɧɵɣ ɜɢɞ:
n
G
Fma
¦
i
1
i
G
(3.4)
ɢ ɢɦɟɟɬ ɦɟɫɬɨ ɫɥɟɞɭɸɳɚɹ ɮɨɪɦɭɥɢɪɨɜɤɚ ɜɬɨɪɨɝɨ ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ.
ɍɫɤɨɪɟɧɢɟ ɜɫɹɤɨɝɨ ɬɟɥɚ ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɧɟɝɨ ɫɢɥɟ
ɢ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɦɚɫɫɟ ɬɟɥɚ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜɫɹɤɨɟ ɬɟɥɨ ɞɜɢɠɟɬɫɹ ɫ ɭɫɤɨɪɟɧɢɟɦ, ɟɫɥɢ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɜɫɟɯ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɷɬɨɦɭ
ɬɟɥɭ, ɧɟ ɪɚɜɧɚ ɧɭɥɸ.
ɍɪɚɜɧɟɧɢɟ (3.4) ɧɚɡɵɜɚɸɬ ɭɪɚɜɧɟɧɢɟɦ ɞɜɢɠɟɧɢɹ ɬɟɥɚ. ɉɪɢ ɭɫɤɨɪɟɧɢɢ,
ɪɚɜɧɨɦ ɧɭɥɸ, ɚG = 0 (ɫɨɫɬɨɹɧɢɟ ɩɨɤɨɹ ɢɥɢ ɩɪɹɦɨɥɢɧɟɣɧɨɟ ɪɚɜɧɨɦɟɪɧɨɟ ɞɜɢɠɟɧɢɟ) ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɩɟɪɟɯɨɞɢɬ ɜ ɭɪɚɜɧɟɧɢɟ (3.1) ɞɥɹ ɩɟɪɜɨɝɨ ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ.
27
Ɍɪɟɬɢɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: ɜɫɹɤɨɟ ɞɟɣ-
f
f
f
p
F
ɫɬɜɢɟ ɧɨɫɢɬ ɯɚɪɚɤɬɟɪ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ; ɫɢɥɵ, ɫ ɤɨɬɨɪɵɦɢ ɞɟɣɫɬɜɭɸɬ ɞɪɭɝ ɧɚ
ɞɪɭɝɚ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɟ ɬɟɥɚ, ɜɫɟɝɞɚ ɪɚɜɧɵ ɩɨ ɜɟɥɢɱɢɧɟ ɢ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵ
ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ:
ɂɡ ɡɚɤɨɧɚ ɫɥɟɞɭɟɬ, ɱɬɨ
ɫɢɥɵ ɜɫɟɝɞɚ ɜɨɡɧɢɤɚɸɬ ɩɨɩɚɪɧɨ: ɩɪɢ ɜɡɚɢɦɨɞɟɣ-
ɫɬɜɢɢ ɬɟɥ 1 ɢ 2 ɜɫɹɤɨɣ ɫɢɥɟ, ɩɪɢɥɨɠɟɧɧɨɣ ɬɟɥɨɦ 1 ɤ ɬɟɥɭ 2 (
ɫɬɚɜɢɬɶ ɪɚɜɧɭɸ ɟɣ ɩɨ ɜɟɥɢɱɢɧɟ ɢ ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɫɢɥɭ (
GG
12 21
. (3.5)
f
G
), ɦɨɠɧɨ ɫɨɩɨ-
12
G
),
21
ɩɪɢɥɨɠɟɧɧɭɸ ɬɟɥɨɦ 2 ɤ ɬɟɥɭ 1.
ȿɫɥɢ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɢɫɬɟɦɭ ɬɟɥ, ɬɨ ɜɫɟ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɬɟɥɚ ɫɢɫɬɟɦɵ, ɦɨɠɧɨ ɪɚɡɞɟɥɢɬɶ ɧɚ ɜɧɭɬɪɟɧɧɢɟ ɢ ɜɧɟɲɧɢɟ. ȼɧɭɬɪɟɧɧɢɟ ɫɢɥɵ – ɷɬɨ ɫɢɥɵ,
ɫ ɤɨɬɨɪɵɦɢ ɬɟɥɚ ɫɢɫɬɟɦɵ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɞɪɭɝ ɫ ɞɪɭɝɨɦ. ȼɧɟɲɧɢɟ ɫɢɥɵ – ɷɬɨ
ɫɢɥɵ, ɫ ɤɨɬɨɪɵɦɢ ɬɟɥɚ ɫɢɫɬɟɦɵ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɫ ɬɟɥɚɦɢ, ɧɟ ɜɯɨɞɹɳɢɦɢ ɜ
ɞɚɧɧɭɸ ɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ
ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ, ɫɢɫɬɟɦɚ ɬɟɥ ɧɚɡɵɜɚɟɬɫɹ
ɫɢɫɬɟɦɭ. ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɢɯ ɫɢɥ
ɡɚɦɤɧɭɬɨɣ.
ɂɦɩɭɥɶɫɨɦ ɫɢɫɬɟɦɵ ɧɚɡɵɜɚɟɬɫɹ ɜɟɤɬɨɪɧɚɹ ɫɭɦɦɚ ɢɦɩɭɥɶɫɨɜ ɬɟɥ ɫɢɫɬɟɦɵ:
GG G G G
pp p p
12
...
n
.
¦
ni
i
1
ɂɦɩɭɥɶɫ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɵɦ. ɗɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɨɞɟɪɠɚɧɢɟ
ɢɦɩɭɥɶɫɚ:
G
ɜɧɟɲ
p
G
0.
pconst
ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ
Ɏɨɪɦɭɥɢɪɨɜɤɟ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɦɨɠɧɨ ɩɪɢɞɚɬɶ ɫɥɟɞɭɸɳɢɣ
ɜɢɞ: ɜ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɟ ɜɟɤɬɨɪɧɚɹ ɫɭɦɦɚ ɢɦɩɭɥɶɫɨɜ ɬɟɥ ɞɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
ɪɚɜɧɚ ɜɟɤɬɨɪɧɨɣ ɫɭɦɦɟ ɢɦɩɭɥɶɫɨɜ ɬɟɥ ɩɨɫɥɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ:
GG G G G G
pp p pp p
... ' ' ...
12 1 2
nn
.
Ɉɬɦɟɬɢɦ, ɱɬɨ ɡɚɤɨɧ ɜɵɩɨɥɧɹɟɬɫɹ ɢ ɞɥɹ ɫɭɦɦ ɩɪɨɟɤɰɢɣ ɢɦɩɭɥɶɫɨɜ ɬɟɥ ɞɨ ɢ
ɩɨɫɥɟ ɢɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɫɢ ɤɨɨɪɞɢɧɚɬ.
Ʉɥɸɱɟɜɵɟ ɩɨɧɹɬɢɹ ɢ ɬɟɪɦɢɧɵ
ȼɡɚɢɦɨɞɟɣɫɬɜɢɟ
Ɂɚɤɨɧ ɇɶɸɬɨɧɚ ɜɬɨɪɨɣ
Ɂɚɤɨɧ ɇɶɸɬɨɧɚ ɩɟɪɜɵɣ
Ɂɚɤɨɧ ɇɶɸɬɨɧɚ ɬɪɟɬɢɣ
ɂɦɩɭɥɶɫ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ
ɂɦɩɭɥɶɫ ɫɢɥɵ
ɂɦɩɭɥɶɫ ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ
ɬɨɱɟɤ
ɂɧɟɪɬɧɨɫɬɶ
ɋɢɥɚ
ɋɢɥɵ ɜɧɟɲɧɢɟ
ɋɢɥɵ ɜɧɭɬɪɟɧɧɢɟ
ɋɢɥɚ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ
ɋɢɥɵ ɷɤɜɢɜɚɥɟɧɬɧɵɟ
ɋɢɫɬɟɦɚ ɨɬɫɱɟɬɚ
ɝɟɥɢɨɰɟɧɬɪɢɱɟɫɤɚɹ
ɋɢɫɬɟɦɚ ɨɬɫɱɟɬɚ ɡɚɦɤɧɭɬɚɹ
ɋɢɫɬɟɦɚ ɨɬɫɱɟɬɚ ɢɧɟɪɰɢɚɥɶɧɚɹ
Ɇɚɫɫɚ
28
Ɏɨɪɦɭɥɵ ɢ ɟɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ ɜ ɫɢɫɬɟɦɟ ɋɂ
F
X
G
X
G
f
p
F
p
ɨɫɧɨɜɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ
ȿɞɢɧɢɰɚ
ɢɡɦɟɪɟɧɢɹ
ɜ ɫɢɫɬɟɦɟ ɋɂ
ɇɚɡɜɚɧɢɟ
Ɏɨɪɦɭɥɚ
(ɢɥɢ ɨɛɨɡɧɚɱɟɧɢɟ)
Ɇɚɫɫɚ m Ʉɢɥɨɝɪɚɦɦ (ɤɝ)
ɋɢɥɚ
ɂɦɩɭɥɶɫ
ɦɚɬɟɪɢɚɥɶɧɨɣ
ɬɨɱɤɢ
ɉɟɪɜɵɣ ɡɚɤɨɧ
ɇɶɸɬɨɧɚ
ȼɬɨɪɨɣ ɡɚɤɨɧ
ɇɶɸɬɨɧɚ
Ɍɪɟɬɢɣ ɡɚɤɨɧ
ɇɶɸɬɨɧɚ
ɉɪɢɪɚɳɟɧɢɟ
ɢɦɩɭɥɶɫɚ
ɂɦɩɭɥɶɫ ɫɢɥɵ
ɂɦɩɭɥɶɫ
ɫɢɫɬɟɦɵ
ɦɚɬɟɪɢɚɥɶɧɵɯ
ɬɨɱɟɤ
Ɂɚɤɨɧ
ɫɨɯɪɚɧɟɧɢɹ
ɢɦɩɭɥɶɫɚ
n
F10
¦
i
i
GG
GG
ppF
21 ɪ
GG G G G
pp p p p
G
Fpconst
GG G G G G
12 1 2
... ' ' ...
pppp p
G
G
G
mp
G
(
i
211ɪ
const
n
G
F
¦
i
1
G
G
F
pd
p
dt
GG
pFdt
f
12 21
t
2
³
G
t
G
12
ɜɧɟɲ
p
G
(
W
...
G
0
ɬ n
,
= 0)
ɚ
G
am
const
¦
i
)
n
1
ɪ
ni
ɇɶɸɬɨɧ (ɇ)
[m] = [ɤɝ];
[ȣ] = [ɦ/ɫ];
[p] = [ɇāɫ]
[F] = [H]
[m] = [ɤɝ];
[a] = [ɦ/ɫ²];
[F] = [H]
[F] = [H]
[p] = [ɇāɫ];
[F] = [H]
[p] = [ɇāɫ];
[F] = [H]
[p] = [ɇāɫ]
[p] = [ɇāɫ]
29
3.2. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
P
g
Ɂɚɞɚɱɚ 1. ɉɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɞɜɢɠɟɬɫɹ ɝɪɭɡ ɦɚɫɫɨɣ 10 ɤɝ ɩɨɞ
ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ 50 ɇ, ɧɚɩɪɚɜɥɟɧɧɨɣ ɩɨɞ ɭɝɥɨɦ 60
q ɤ ɝɨɪɢɡɨɧɬɭ. Ɉɩɪɟɞɟɥɢɬɶ
ɭɫɤɨɪɟɧɢɟ ɝɪɭɡɚ, ɟɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɩɪɢ ɞɜɢɠɟɧɢɢ ɪɚɜɟɧ 0,1.
Ⱦɚɧɨ:
ɬ = 10 ɤɝ
F = 50 ɇ
D
= 60q
= 0,1
ɚ – ?
Ȼɭɞɟɦ ɩɪɢɞɟɪɠɢɜɚɬɶɫɹ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɩɨ ɞɢɧɚɦɢ-
ɤɟ ɭɞɨɛɧɨɣ ɨɛɳɟɩɪɢɧɹɬɨɣ ɫɯɟɦɵ.
1. ȼɵɩɨɥɧɢɦ ɱɟɪɬɟɠ (ɪɢɫ. 3.1).
2. ɂɡɨɛɪɚɡɢɦ ɜɫɟ ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɬɟɥɨ ɫɢɥɵ.
3. Ɂɚɩɢɲɟɦ ɜɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɜ ɜɟɤɬɨɪɧɨɣ ɮɨɪɦɟ:
GG
ma F mg N F
Ɋɟɲɟɧɢɟ:
GGG
.
mp
Ɋɢɫ. 3.1
4. ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɧɚɣɞɟɦ ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɧɵɯ ɜɟɥɢɱɢɧ ɢ
ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɭɸ ɫɢɫɬɟɦɭ ɫɤɚɥɹɪɧɵɯ ɭɪɚɜɧɟɧɢɣ:
:
ma F F
®
:OXOY
0sin 0
¯
cos 0 0
FmgN
D
D
mp
.
5. ɋ ɭɱɟɬɨɦ ɫɜɹɡɢ ɫɢɥɵ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɫ ɫɢɥɨɣ ɪɟɚɤɰɢɢ ɨɩɨɪɵ (F
ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ:
ma F N
®
0sin
¯
cos
Fm
DP
D
N
.
ɋɨɜɦɟɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɣ ɫɢɫɬɟɦɵ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɢɬɶ ɮɨɪɦɭɥɭ ɞɥɹ
ɪɚɫɱɟɬɚ ɢɫɤɨɦɨɝɨ ɭɫɤɨɪɟɧɢɹ:
F
(cos sin ) 50(0, 5 0,1 0,85)
DP D
ag
m
P
10
0,1 9,8 1,95
ɦ/ɫ2.
Ɉɬɜɟɬ: 1,95 ɦ/ɫ
30
=PN),
mp
2
.
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