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Transcript of Γαρμπής Πατάκη reusta
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(. 2121/1993),
, .
(, ) -
, , , , ,
.
.
,
:
: ,
:
:
-:
Copyright . .....( ) , , 2013
, , 2015
8560560/15
ISBN (set) 978-960-16-6460-6
ISBN(vol.1)978-960-16-5047-0
()38,10437A,
TH.:210.36.50.000,210.52.05.600,801.100.2665- FAX:310.36.50.069
KENTPIKH :EMM.MENAKH16,10678 AHNA,TH.:210.38.31.078
YOKATATHMABOPEIAA:KOPYTA(TEPMAONTOY-EPIOXHBKTEO),57009
KAOXPEONIKH,T..1213,TH.:2310.70.63.54,2310.70.67.15-FAX:2310.70.63.55
Web site: http/www.patakis.gr e-mail:[email protected],
, ()
:
: ,
:
:
:
Copyright . ..... () , , 2013
, , 2015
8561 775/15
ISBN (set) 978-960-16-6460-6
ISBN (vol. 2) 978-960-16-5048-7
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3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
8 () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
12/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4
9 () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9.1 ()
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
12/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 ()
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
12/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10 ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
12/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
11 () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
12/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
12 (). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
12/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
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, -
. -
, !
.
, ,
,
.
. -
,
.
-
,
.
, . ,
-
,
, ,
.
. . 6992797955
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A 8
. -
, . ,
.
, .
, , .
, . -
.
.
.
()
V m. H :
=m
V
S.I. 1kg/m3.
V. -
, = m
V= .
H .
.
.
p () -
F
.
p=
F
.
S.I. 1 Pa = 1N/m2.
F ,
, p=F
A.
F = pA,
..
(Atm).
pat
= 1Atm = 105Pa
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10
8
.
, ,
.
-
.
F
.
(-).
: p=gh
: .
h: , .
.
Pascal
. F
-
.
.
F
,
F
A().
Pascal()
.
,
pat
.
Pascal,
.h :
p=p
at
gh
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11
, (-),
().
, -
. .
.
:
.
-
.
.
:
,
,
.
.
, -
.
. .
.
V -
t t.
=V
t
S.I. 1 m3/s.
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8
. t x,
V = Ax (1).
= x
t
(2).
= V
t
(1)
= x
t
(2)
=
.
.
-
.
A1
1,-
A2
-
2
.,
.
m1
1t m
2
2
.
V1,V
2 m
1,m
2.
m = V (1), V = Ax(2), x = t (3).
m1
= V1
(2)
1x
1
(3)
1
1t (4) m
2 = V
2
(2)
2x
2
(3)
2
2t (5).
, , m1 m
2(
).
.
m1
= m2
(4)
(5)
1
1t =
2
2t A11=22 (6) ().
= ,(6) 1=2 =..
, , :
m1
= m2, V
1= V
2, A
1
1 =
2
2,
1 =
2.
.
() ,
,
.
() , -
, .
,
.. m1
1 m
2
2, m
1= m
2
,
( ), ( ).
, ( ),
() ( ).
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13
(2
> 1).
, (A2
< A1),
.
BERNOULLI
-
.
, -
. y1,
1, p
1
,
y2, 2, p2 ,
.
(I).
F1
= p1A
1 F
2 = p
2A
2
F1.
t -
m -
s1,
m, V, s2
.
-
, (I) (II). , W
.= (1).
W.
= W + WB
(2), W ()
() WB
, m
y1 y2.:
WB
= m g y2 y
1( ) = Vg y2 y1( ) (3), . W = W
F1+ W
F2= +F
1s1 F2 s2 = p1A1s1 p2A2s2
V = A1s
1= A
2s
2, W = p
1V p
2 V W = p
1 p
2( )V (4).
(2)(3)
(4)W
.= p
1 p
2( )V Vg y2 y1( ) (5). :
=1
2
m2
2
1
2
m1
2=
1
2
V2
2
1
2
V1
2=
1
2
V 2
2
1
2( ) (6)
(1)(5)
(6)p
1 p
2( )V Vg y2 y1( ) =1
2V
2
2
1
2( ) p1 p2 gy2 +gy1 =1
2
2
2
1
2
1
2
p11
2
1
2 gy1=p2
1
2
2
2 gy2
p
1
22 gy=.
Bernoulli .
Bernoulli, :
(p), 1
2
2
(gy) .
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8
1
22 ,
1
2
2=
1
2 m
V
2=
V.
gy , gy = m
V
gy = U
V
.
Bernoulli .
,
Bernoulli :
p1
1
2
1
2= p
2
1
2
2
2
p
1
22
=
.
, , -
, .
. , -
.
, 1 = 2 = 0, Bernoulli :
p1 gy
1=p
2 gy
2 p gy=. ().
Torricelli()
-
h , .
Bernoulli ()
(), :
pE
+1
2
2+gh = p
K +
1
2
2+ 0 (1) ( y
E = y
1= h, y
K = y
2 = 0)
, , -
, pE
= pK
= pat
(2).
,
= 0 (3).
(1)(4)
(2) p
at + 0 + gh = p
at +
1
2
2+ 0 gh = 1
2
2
= 2gh .
= 2gh Torricelli :
H h
h.
AE
AK
.
AE
=
=
. AK
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15
Bernoulli .
, ,
-
.
(). ,
.
.
, . ,
.
, , -
, .
.
.
..
F . F
, ()
. -
,
.
0 .
:
F
F F
F .
F F=
.
F
.
: S.I. 1 Ns/m2. poise () = 1dyns/cm2 = 101Ns/m2
(1dyn = 10
5N, 1cm2 = 104 m2 ).
.
, .
F=
.-.
, , -
,
.
-
R.
, -
.
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17
.
.1 PASCAL1.
pA
= p.
= gh
2.
pA
= pat
+p.
pA
= pat
+gh
3.
pA = p. + p.
pA
=F
A+gh
4.
pA
= p.
+ p.
pA
=Mg
A+gh
5. m
pA
= p.
+ .
pA
=M + m( )g
A+gh
6.
pA
= 0
p
= 1gh1
p
= 1gd
1
pE
= 1gd
1+
2gh
2
pZ
= 1gd
1+
2gd
2
7.
pA
= pat
p
= pat
+1gh
1
p
= pat
+1gd
1
pE
= pat
+1gd
1+
2gh
2
pZ
= pat
+1gd
1+
2gd
2
.2
.
: p.
= gh h
-
,
.
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8
.3 (
1 >>
2)
h1
h2=
2
1
.
::
p
= pat
+1g h
1+ d( ) p = pat +1gh1 + 1gd (1)
p
= pat
+2gh
2 +
1gd (2). (1). (2) :
pat
+1gh
1+
1gd = p
at +
2gh
2 +
1gd 1gh1 = 2gh2 (3)
h1
h2
=
2
1
(4)
.
(pA=p ).
: pA
= pat
+1
gh1
p
= pat
+2
gh2
(3)
pat
+1
gh1
. pA
= p
.
(pE p
Z).
: pE
= pat
pZ
= pat
+2gh, pE pz .
U -
. V=998kg/m
3 .
= 18,4 cm d=1,56 cm. -
;
g=10m/s2 1Atm=10
5Pa.
V = 998kg/m3 , = 18, 4 cm = 0,184 m , d = 1,56cm = 0,0156 m , g = 10m/s2 , 1Atm = 105 Pa ,
= ; , p
A = ;
:
p
= pat
+g + h( ) = pat +g + gh (1)
p= pat + g + d( ) + gh (2)
(1), (2) pat
+g +
gh = p
at +
g + d( ) + gh
g =
g + d( ) = + d( )
=
+ d
= 998
0,184
0,1996kg/m3 =920 kg/m
3
1pA = pat +g pA = 10
5+ 99810 0,184( ) Pa pA=101 836,32 Pa.
2
, , , pA
= p.
p
= pat
+g d + ( ) = 105 + 92010 0,1996( ) Pa = 101.836,32 Pa.
pA=p=101.836,32 Pa.
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.4
, -
U =13.600 kg/m3. -
. h=20cm.
;
g=10m/s2 1Atm=105 Pa.
= 13.600 kg/m2 , h = 20cm = 0,2 m , g = 10m/s2 , 1Atm = 105 Pa , p.
= ;
,
p.
= pA
(1).
, , :
pA =
p (2).p
= p
at +gh = 105 + 13.60010 0,2( ) Pa = 127.200 Pa
p
= 127.200 Pa (3).
(1), (2), (3) p.
=127.200 Pa.
.5
-
h. F
:
F=p
.
F=p
=
gh
F -
h.
, , -
. :F
1= p = gh1
F2
= p = gh
2
F3
= pE = gh
3
h3
> h2
> h1, F
3 > F
2 > F
1.
,
,
.
-
p. F
:
F=pA
.
F=pA=ghA
F
,
. F .
m,
B V -
:
=m
V (1), V = hA (2), B = mg (3)
F = pA = ghA(1)
(2)
m
VgV = mg
(3)
B.
F = B F < B( ) .
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8
r=0,1 m, = 0,4m
=1.000kg/m3.
) -
.) =10
6m
2
.
g=10m/s2, =3,14.
r = 0,1m , = 0,4 m , = 1.000 kg/m3 , g = 10m/s2 , = 3,14
) F = ;
p p = g (1). A = r2 (2).
F = pA(1)
(2) gr2 = 100010 0,4 3,14 0,01N = 125,6 N F=125,6 N.
) = 106
m2
, F = ;
p
= gr (3).
:
F = p
= gr = 1.00010 0,1 106 = 103 F=10-3 N
.6 ( )
F
1
F2
=
A1
A2
d
1
d2
=
A2
A1
WF1 = WF2 = F1d1=F2d2
(F1)-
(d1)
(F2) (d
2).
F2>F1 d21.
, F1
A1.. -
p1
=F
1
A1
(1). Pascal,
A2.,
F2, p
2 =
F2
A2
(2). , Pascal
p1
= p2
(1)
(2)
F1
A1
=F
2
A2
F
1
F2
=A
1
A2
(3).
B = F
2.
d1,
d2,V, .
V = A1d
1= A
2d
2
d1
d2
=A
2
A1
(4).
WF1
= F1d
1
WF2
= F2d
2
(:) WF1W
F2
=F
1
F2
d
1
d2
(3)
(4)
A1
A2
A
2
A1
= 1 WF1 = WF2
.
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21
-
m=1.000 kg.
) F1
m;) ,
d2=0,1m;
) F1, F2. ;
R1
R2=5R
1 g=10m/s2 .
m = 1.000 kg , R 2
= 5R1
, g = 10m/s2
) F1 = ;
F1
A1
= R1
2 (1). . -
p1
= F1A
1
(2).
Pascal,
A2
= R2
2 (3).
, F2, p
2 =
F2
A2
(4). -
, Pascal :
p
1= p
2
(2)
(4) F
1
A1
=F
2
A2
F
1
F2
=A
1
A2
(1)
(3) F
1
F2
= R
1
2
R22
F1
F2
=R
1
2
25R1
2
F1
F2
=1
25 F
1 =
1
25F
2 (5)
(), F2
= B = mg = 10.000 N.
(5) F1
=1
2510.000 N F
1=400 N.
) d2
= 0,1m , d1
= ;
d1,
d2,V, .
V = A1d1 = A2d2 R12d
1= R2
2d2 R
1
2d1
= 25R1
2d2 d
1= 25d
2 d1=2,5m.
) WF1
= ; , WF2
= ;
WF1
= F1d
1= 400 2,5 J = 1.000 J WF1 = 1.000 J
WF2
= F2d
2= 10.000 0,1J = 1.000 J WF2 = 1.000 J
.
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22
8
B.
.7
1=2 A11=22
.
() , -
, .
m1
1 m
2
2, m
1= m
2. V
1= V
2.
1. h . 10 m
2 m/s. 2,5m;
h = . , d
1= 1 0 m ,
1= 2 m/s , d
2 = 2,5m ,
2 = ;
, , . -
:
1
= 2
1
1=
2
2 d
1h
1= d
2h
2 d
1
1= d
2
2
2 =
1
d1
d2
2
= 210
2,5m/s 2=8 m/s
2. . h1=2 m
1=0,8m/s.-
2=4 m/s;
h
1= 2 m ,
1= 0,8 m/s ,
2 = 4 m/s , h
2 = ;
d . , ,
. :
1=
2
1
1=
2
2 dh
1
1= dh
2
2 h
1
1= h
2
2 h
2 = h
1
1
2
h2
= 20,8
4m h2=0,4m
3. 1,44 103
m3/s.-
=12
cm.
) ;
)
1
4. ;
= 1,44103 m3/s , =12
cm =12
102 m
) = ;
, .
= =
2
4 =
4
2 =
41, 44103
122 104
m/s =0,4 m/s.
) 1
= 0,4 m/s , A2
=1
4A
1,
2 = ;
1
1=
2
2
1
1=
2
1
4
1
2 = 4
1
2=1,6 m/s.
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23
4. =102 m3 /s. -
1=10
cm.
) 1 ;
) . ) 0,45m .
g=10 m/s2 .
= 102
m3/s , 1 =
10
cm =10102
m =10
1
m , g = 10m/s2
) 1
= ;
:
A1
=
1
2
4
=
102
4
m2
=10
2
4
m2 A
1 =
10-2
4
m2
= 1
1
1=
1
=102
102
4
m/s 1=4 m/s.
);
(2
> 1).-
, (A2 < A1),.
) h = 0,45 m , A2 = ;
11 = 22 2 = 1
1
2
(1).
1,
2-
h = 0,45m.
.... , , :
.... (, ): K
.( )
.() = W
B
1
2m
2
2
1
2m
1
2= +mgh
2
2
1
2= 2gh
2
2=
1
2+ 2gh
2
= 1
2+ 2gh
(2)
2
= 42 + 210 0,45 m/s 2
= 5 m/ s
(1) A2
=102
4
4
5m2 =
102
5m2 = 2103 m2 A2=210
-3m
2.
A2:
= 1
1=
2
2
2 =
2
2
=102
5m2 A
2=210-3 m2
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24
8
.8 () BERNOULLI
p1
1
2
1
2 gy
1=p
2
1
2
2
2 gy
2
,
.
T y1,
1, p
1,
. y2,
2, p
2, -
.
1.
, , h=5 m.
2Atm
10 m/s, 1,82 Atm.
) ;
) , ;
103 kg/m3 , g 10m/s2 , 1Atm 105 Pa.
h = 5 m , p
1= 2Atm = 2 105 Pa ,
1= 10 m/s , p
2=1,82Atm =1,82 105 Pa , =103 kg/m3 , g =10 m/s2 , 1Atm =105 Pa
) 2
= ;
ernoulli , , p1 +1
21
2+ gy1 = p2 +
1
22
2+ gy2 (1).
, y1
= 0 y2
= h.
(1) p1
+1
2
1
2+ 0 = p
2 +
1
2
2
2+gh 2105 +
1
2103 102 = 1,82105 +
1
2103
2
2+ 10
3 10 5
200.000 + 50.000 = 182.000 + 5002
2+ 50.000 500
2
2= 18.000
2
2= 36 m
2/s
2 2=6 m/s .
)
A1
A2
= ;
, , :
1
= 2
1
1=
2
2
1
2
=
2
1
1
2
=6
10
1
2
=
3
5
2.
. ,
5Atm 3Atm .
h , ;
103 kg/m2 , g 10m/s2 , 1Atm 105 Pa.
p1=
5Atm =
5 105
Pa , p2 =
3Atm =
3 105
Pa , =
10
3
kg/m
3
, g =
10m/s
2
, 1Atm =
10
5
Pa , h =
; Bernoulli , , p
1+
1
2
1
2+ gy
1= p
2 +
1
2
2
2+ gy
2 (1).
, y1
= 0 y2
= h.
, A1
= A2
= A (2).
, , 1
= 2
1
1=
2
2
(2)
1
= 2
(3).
(1)(3)
p1
+1
2
1
2+ 0 = p
2 +
1
2
1
2+gh p
1= p
2 + gh (4)
gh = p1 p
2 h =
p1 p
2
g h =
5 105 3 105
103 10m =
2105
104m = 20 m h=20 m.
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25
(4) Bernoulli p1
= p2
+gh.
p2
< p1,h .
p = p1 p
2 = gh,h.
3. .
1=18cm
2 () 2=6 cm
2.
1=4 m/s.
) 2
;
) ;
103 kg/m3 , g=10m/s2 , 1Atm=105 Pa , -
.
A
1= 18cm2 , A
2 = 6 cm2 , 1 = 4m/s , = 10
3 kg/m3 , g = 10m/s2 , 1Atm = 105 Pa
) 2 = ;
, , :
1
= 2
1
1=
2
2
2 =
1
1
2
2= 4
18
6m/s 2=12 m/s
)p1 = ;
Bernoulli , , p1 +1
21
2+ gy1 = p2 +
1
22
2+ gy2 (1).
, , y1 = y2 = 0.
, p
= pat
= 1Atm = 105 Pa.
(1) p1
+1
2
1
2= p
2 +
1
2
2
2 p1
= p2
+1
2
2
2
1
2
1
2 p1
= 105 +1
2103 122
1
2103 42
Pa =
= 100.000 + 72.000 8.000( ) Pa = 164.000 Pa = 1,64105 Pa p1=1,64105
Pa.
4. 20m/s.
) p ;
) F A=100m2;
) ;
1,2 kg/m3, 1Atm=10
5Pa.
= 20m/s ,
= 0 , = 1,2 kg/m3 , 1Atm = 105 Pa
) p = ;
. ,
= 0.
Bernoulli , , pB
+1
2
2+ gy
1= p
+
1
2
2+ gy
2 (1).
, , y1
= y2
= 0.
(1) pB
+1
2
2+ 0 = p
+ 0 + 0 p
p
=
1
2
2 p =1
2
2 p =1
21,2 202 Pa p=240 Pa
) A = 100 m2 , F = ;
p F -
F = Ap F = 100 240 N F=24.000 N.) p
B = ;
, ,
p
= p
at = 1Atm = 105 Pa. p = p
p
p
= p
p p
B = 100.000 240( ) Pa pB=99.760 Pa.
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26
8
.9 VENTOURI
1=
2gh
A1A
2
2
1
()
Ventouri .
A1, A
2 h , .
, , .
pB
= p1
= pat
+gh1 p = p2 = pat +gh2 , :
p1p
2 = p
at +gh1pat gh2 = g h1 h2( ) = gh p1p2 = gh (1)
, , 1 = 2 11 = 22
2 =
1
2 1 (2).
Bernoulli , , p1
+1
2
1
2+gy
1= p
2 +
1
2
2
2+gy
2 (3).
, , y1
= y2
= 0.
(3) p1
+1
2
1
2= p
2 +
1
2
2
2 (2) p1
+1
2
1
2= p
2 +
1
2
1
2
22
1
2
p1p
2 =
1
21
2
2
2
1
2 1
2
1
2 (1)
gh =1
2
1
2 12
2
2 1
gh =
1
2 12 1
2
22 1
12
=2gh
A12
A2
2 1
1 =
2gh
A1
A2
2
1.
A1
= 3A2
, h = 0,1 m , g = 10m/s2 , 1
= ;
1 =2gh
A1
A2
2
1
=210 0,1
32 1m/s =
2
8m/s = 0,5 m/s
1=0,5 m/s.
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27
.10 TORRICELLI
.
= 2gh
(, . 14 Torricelli)
.
Smax=h, d=h
2,
= gh , = 2gh , = 45
.
.
, -
, ,
.
B.
Torricelli,
= 2g h d( ) (1). d.
x
, y
. :
x = , x = t (2), y = gt (3), y =
1
2gt2 (4)
x = S y = d.
(2)x = S
S = t t =
S
(5)
(4)y = d
d =1
2gt2
(5)
d =1
2g
S2
2
(1)
d =gS2
4g h d( ) d =
S2
4 h d( ) S2 = 4hd 4d2 4d
24hd +S2 = 0
(6)
d, -
, 0 4h( )2
4 4S2 0 16 h
216S
2 0 16S
216 h
2S h, :
Smax
= h (7).
(6) S = Smax
= h,4d 2 4 hd +h 2 =0 2d h( )2
=0 2d h =0 2d =h d =h
2 (8).
(1)(8)
= 2g hh
2
= 2g
h
2
= gh (9)
(5)(7)
(9)t =
h
gh t =
h
g (10). (3)
(10)
y
= gh
g y = gh (11)
= 2
+ y
2 (9)
(11)gh + gh = 2gh
= 2gh .
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28
8
1
m ,
d =h
2
(8). .... , , :
.... (, ): = W K.()
.()
= WB
1
2m
2 1
2m
2= +mgd
(8)
2
2= gh =
2+ gh
(9)
= 2gh
2Bernoulli , , :
p
+1
2
2+ gy1 = p +
1
2
2+ gy2 (12)
,
y1
= d =h
2(8) y2 = 0., ,
, p = p = pat (13).
(12)(13)
pat
+1
2
2+g
h
2
= pat
+1
2
2
+ 0
2=
2+ gh
(9)
2= 2gh
= 2gh .
(9), (11) = y = gh , = 45.
. Torricelli,
= 2g h h1( ) (1).
h1.-
,
, :
x
= S
=
t
(2) y
= h1
=1
2gt
2 t
=2h
1
g (3)
(2)
(1)
(3) S = 2g h h1( ) 2h
1
g S = 2 h1 h h1( ) (4)
(), S
= 2 h2
h h2( ) (5).
, :
S
= SA
(4)
(5)2 h
1h h
1( ) = 2 h2 h h2( )
h1
h h1( ) = h2 h h2( ) h1h h1
2= h
2h h
2
2
h1h h
2h = h
1
2 h2
2 h h1 h
2( ) = h1 h2( ) h1 + h2( ) h = h
1+ h
2 (6). h = h
1+ d (7).
(6), (7) h2
= d.
E AB = ZH, S1
= S5.
E = , S2
= S4.
E = , S3
= Smax
.
....
Bernoulli, -
. -
,
Bernoulli
-
.
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29
1. .
, h=1,25 m, -
=2 cm
2. 5 L ;
g=10 m/s2 .
h = 1,25 m , A = 2 cm2 = 2104 m2 , V = 5 L = 5 103 m3 , g = 10m/s2 , t = ; Torricelli, = 2gh (. -
. 14).
= 2101,25 m/s = 5 m/s.
= (1). =V
t (2).
(1), (2) A =V
t t =
V
A t =
5 103
2104 5s t=5 s.
2. . -
, h=1,8 m , -
10cm2,
4 10 3 m3/s.
)
.
) h
.
g=10 m/s2.
h = 1,8 m , A = 10cm2 = 10104 m2 = 103 m2 , . = 4103 m3/s , g = 10m/s2
) ;
Torricelli, :
= 2gh = 2101,8 m/s = 6m/s
= 6 m/s.:
= = 610
3 m3/s. .
= 4103 m3/s.
> .
,
, ()
.
) h = ;
, -
, .
.
=
.=
=
.
= 410
3
103 m/s = 4 m/s.
= 2g h
2= 2g h h =
2
2g=
42
210m = 0,8 m h =0,8m.
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30
8
.11
h. .
T W. P . :
W.
=
1
2m
2 mgh
P.
=
1
2
2 gh
A
.
1
Bernoulli , , :
pB
+ 12
2+ gy1 = p +
12
2+ gy2 (1)
,
= 0,,
= .
, y1
= 0 y2
= h.
(1) pB
+ 0 + 0 = p
+1
22 +gh p
B p
=
1
22 +gh p =
1
2
2+ gh (2).
m:
W.
= pV(2) 1
22 +gh
V =1
2
m
V2 +
m
Vgh
V =1
2m2 + mgh W
.=
1
2m
2+ mgh
F F = Ap(2)
F =1
2
2+ gh
A (3).
P. = F(3)
P. = 122 + gh A
.
2
.... , m, :
.... (, ): = W K.()
.()
= W .
+ WB
1
2m2 0 = W
. mgh W
.=
1
2m2 + mgh (4)
:
P.
= W
.
t
(4)
P.
=
m
2
2+ mgh
t P
.=
2
2+gh
m
t (5)
= Vt
= A (6).
m
t=
V( )t
= V
t
(6)
m
t= (7).
(5)(7)
P.
=
2
2+ gh
P
.=
1
2
2+ gh
A.
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31
1. -
=1.000kg/m
3 =900kg/m
3 .
d1=20cm, d
2=30 cm.
) -;
) ;
) -
=1 cm2
-.
) -
A=400 cm2 .
)
264 N.;
1Atm=105
Pa, g=10m/s2.
= 1.000 kg/m3
,
= 900 kg/m3
, d1
= 20cm = 0,2 m , d2
= 30cm = 0,3 m , 1Atm = 105
Pa , g = 10m/s2
)p
= ;
.1
p
= pat
+gd
1= 105 + 90010 0,2( ) Pa = 101.800 Pa
p=101.800 Pa
)p
= ;
.1
p
= p
at +
gd
1+
gd
2 = 105 + 90010 0,2 + 1.00010 0,3( ) Pa = =
104.800 Pa p
=104.800 Pa
) = 1cm2
= 104 m2 , F1
= ;
.5
,
p
= 101.800 Pa p
= pat
= 1Atm = 100.000 Pa,. -
p1 = p pE = 1.800 Pa. F1:
F
1= p1 = 1.80010
4 = 0,18 F
1=0,18 N
) A = 400 cm2 = 4102 m2 , F2
= ;
.5
, p
= 104.800 Pa
pZ
= pat
= 1Atm = 100.000 Pa,.
p2
= p p
= 4.800 Pa. F
2:
F
2 = p
2A = 4.800 4102N = 192 N F2=192 N
) F.
= 264 , V = ;
d, p,
F -
V = Ad (1).
F.
= 264. -
F = F. = 264.
F = p p =F
A p =
264
4102Pa p = 6.600 Pa.
p = p p
p
= p +p
Z p
= 6.600 + 100.000( )Pa
p
= 106.600 Pa.
p
= pat
+g d + d
1( ) + gd2 106.600 = 100.000 + 90010 d + 0,2( ) + 1.00010 0,3
106.600 = 104.800 + 9.000d 9.000d = 1.800 d = 0,2 m.
(1) V = 4102 0,2 m3 V=810-3 m3 .
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32
8
2.
A1=1 cm
2 A2=10cm
2.
,
k=1.000 N/m. -
-.
) -
= 0,1m;
) .
) (1) .
g=10m/s2.
A1
= 1cm2 , A2
= 10cm2 , k = 1.000 N /m , g = 10m/s2
.6
) = 0,1m , m = ;
= 0,1m,.
F1 , F1 = B = mg
(1).
p1 =F
1
A1
(2).
Pascal, -
.,
F2 , p2 =F
2
A2
(3). ,
Pascal p1
= p2
(2)
(3)
F1
A1
=F
2
A2
F
1
F2
=A
1
A2
F
1
F2
=1
10 F
2 = 10F
1 (4).
, F.
= k., F2 = F . = k (5).
(4)(1)
(5)k = 10mg m =
k10g
m =1.000 0,1
1010kg m=1 kg
) d = ;
d2
= = 0,1m.-
d1, d
2,
V, .
V = A1d
1= A
2d
2 d
1= d
2
A2
A1
d1
= 0,1 10
1m d
1= 1m.
d = d1
+ d2 d=1,1m.
) WF1
= ;
, F.
.
WF1
= WF.
= U.
=1
2k2 =
1
21.000 0,12 J = 5 J W
F1= 5 J.
WF1
, WF2
WF1
= F1d
1 W
F2= F
2d
2, -
.
.
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33
3.
H=2 m. h -
10cm2,
=2 m/s.
) h .) -
;
) 2tm, ,
.
1.000kg/m3 , g=10m/s2 , 1Atm=105 Pa, 204 =14,3, 240 =15,5.
H = 2 m , A = 10cm2 = 103 m2 ,
= 2m/s , = 1.000 kg/m3 , g = 10m/s2 , 1Atm = 105 Pa ,
204 = 14,3 , 240 = 15,5
) = ; , h = ;
.10
=
=2
10
-3m
3/s.
Bernoulli (
) (), :
p
Z +
1
2
Z
2+ gy
1= p
+
1
2
2+ gy
2 (1)
, ,
, pZ = p = pat (2).
,
= 0 (3).
, y1
= h y2
= 0.
(1)(2)
(3)p
at + 0 + gh = p
at +
1
2
2+ 0 h =
2
2g h =
22
210m h=0,2m.
)S = ;
.10
d = H h = 1,8m.-, , x = S =
t (4)
y = d =1
2gt2 t =
2d
g t = 0,6 s.
(4) S = 2 0,6 m S=1,2m
) pZ
= 2Atm = 2105 Pa ,
= ; , = ; ,
= ;
.10
Bernoulli , :
p
Z +
1
2
Z
2+ gy
1= p
+
1
2
2+ gy
2 (5)
() : pZ
= 2Atm = 2105 Pa, p
= 1tm = 105 Pa,
Z
= 0, y1
= h, y2
= 0
(5) pZ
+ 0 + gh = p
+1
2
2+ 0
200.000 + 1.00010 0,2 = 100.000 +1
21.000
2
= 204 m/s =14,3m/s.
() :
S =
t S = 14,3 0,6 m/s S = 8,58 m
= S S = 8,58 1,2( ) m = 7,38 m =7,38m.
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34
8
Bernoulli , , :
p
+
1
2
2+ gy1 = pE +
1
2E
2+ gy2 (6)
, y1 = d = 1,8 m y2 = 0.
, , , : p
= p
E = p
at (7)
(6)(7)
pat
+1
2
2+gd = p
at +
1
2
2+ 0
1
2
2+gd =
1
2
E
2
2=
2+ 2gd
=
2+ 2gd
= 204 + 2101,8 m/s
= 240 m/s =15, 5 m/s.
.... ,
Bernoulli .
4. -
0,6 cm
10 m .
0,2cm,-
9 m/s.
) .
) ;
) .
) :
i) F1 F2-
ii) W
V=103
m3.
)
.
=1.000kg/m3, 1Atm=10
5Pa, g=10m/s
2, =3,14 -
.
r1
= 0,6 cm = 6103 m , r2
= 0,2 cm = 2103 m , h = 1 0 m , 2
= 9m/s , = 1.000 kg/m3
, 1Atm = 105
Pa ,
g = 10m/s2 , = 3,14
) 1
= ;
.7
A1 = r12 (1), A2 = r22 (2). , , :
1=
2
1
1=
2
2
(1)
(2)r
1
21
= r2
22
1=
2
r2
r1
2
1
= 91
3
2
m/s 1=1 m/s
) .
= ;
.
= 1
1
(1)
r1
21
= 3,14 6103( )2
1m3 /s =
= 113,04106 m3/s .=113,0410-6
m3/s.
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35
) p1
= ;
.8
Bernoulli , , :
p1
+1
21
2+ gy
1 = p
2 +
1
22
2+ gy
2 (3)
, p
= p2
= pat
= 1Atm = 105 Pa.
, y1
= 0 y2
= h (4).
(3)
(4)
p1
+1
2
1
2+ 0 = p
2 +
1
2
2
2+gh p
1= p
2 +
1
2
2
2 1
2
1
2+gh
p1
= 100.000 +1
21.000 92
1
21.00012 + 1.0001010
Pa
p1=240.000 Pa=2,4 10
5Pa=2,4Atm.
) i) F1 = ; , F2 = ; Bernoulli, . 13
,
F1=
p1A1 o F2
= p2A
2.
F1
= p1A
1= p
1r
1
2= 2,4105 3,14 6103( )
2
N = 27,1296 N
F1=27,1296 N
F2
= p2A
2 = p
2r
2
2= 105 3,14 2103( )
2
N = 1,256 N
F2=1,256 N
ii)V = 103 m3 , W = ;
Bernoulli, . 13
t m -
s1,m, V,
s2
. W ()
() :
W = WF1 + WF2 = +F1s1 F2s2 = p1A1s1p2A2s2 W = p1A1s1 p2A2 s2 (5) V = A
1s
1= A
2 s
2 (6).
(5)
(6)
W = p1V p
2V W = p
1p
2( )V
W = 2,4105 105( )103 J W=1,4102 J=140 J.
)
V
B
= ; , U
t
B
= ;
1
2
1
2=
1
21.00012 J/m 3 = 500 J/m3.
K
V
B
=
1
2
1
2= 500 J/m
3.
gy1 = 0, y1 = 0.
U
V
B
= gy1=0.
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36
8
5. 0,2m3/s 1 Atm.
H=20m.
) ;
) ;
) h=
15m; =10
3kg/m
3, g=10m/s
2, 1 Atm=10
5Pa.
= 0,2 m3/s , H = 20 m , p
1= 1Atm ,
= 103 kg/m3 , g = 10m/s2 , 1Atm = 105 Pa
) 1
= ;
.7
1, 3 = 0 -
H = 20m. .... ,
, :
.... (, ): K.()
.()
= WB
1
2m
3
2 1
2m
1
2=mgH
0
1
2m
1
2=mgH
1= 2gH
1= 210 20 m /s 1=20 m/s
) 1
= ;
.7
= 1
1
1=
1
A1
=0,2
20m2 A1=10
-2m
2.
) h = 15 m , A2
= ; , p2
= ;
.8
1,
2-
h = 15m. .... , :
.... (, ): K
.()
.() = W
B
1
2m
2
2 1
2m
1
2=mgh
2
2 1
2=2gh
2 =
1
2 2gh (1) 2
= 202 21015 m /s 2
= 10m/s
= A2
2 A
2 =
2
A2
=0,2
10
m2 A2=0,02 m
2.
Bernoulli , , p1
+1
2
1
2+
gy
1= p
2 +
1
2
2
2+
gy
2
(2).
, y1 = 0 y2 = h.
1
(2) p1
+1
2
1
2+ 0 = p
2 +
1
2
2
2+
gh 100.000 +
1
21.000 202 = p
2 +
1
21.000102 + 1.0001015
p
2 = 100.000 Pa p2=1 Atm
2
(2) p1
+1
21
2+ 0 = p
2 +
1
22
2+gh
(1)
p
1+
1
21
2= p
2 +
1
2 1
2 2gh( ) + gh
p
1+ 1
2
12
= p2
+ 12
12 gh + gh p1 = p2 p2=1 Atm
, -
, p
= p
= 1Atm.
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6. =0,1 m2 ,
m=5,4 kg-
.
20m/s.
) p ; ) F ;
) 2
, -
;
.
=1,2 kg/m3, g=10m/s
2.
= 0,1m2 , m = 5,4 kg , 1 = 20 m/s , . = 1,2 kg/m
3 , g = 10m/s2.
) p = ;
.8
-
.
,
= 0 p
= patm
.
Bernoulli , , :
p
+
1
2
.2
+ .
gy1
= p
+1
2.
2+ .gy2
(1)
, , y1 = y2 = 0.
(1) p
+1
2.
1
2+ 0 = p
+ 0 + 0 p
p
=
1
2.
1
2 p =1
2.
1
2
p =
1
21,2 202 Pa p=240 Pa.
) F = ;
.5
pF
F = Ap F = 0,1 240 N F=24 N.
) 2
= ;
p =1
2.22
(2).
, F = 0 F = w A p = mg(2)
A1
2.
2
2= mg
2 =
2mg
A.
2
=2 5,410
0,1 1,2m/s
2=30 m/s.
7. 1kg/m3.
120 m/s 100m/s.
) p ;
) =50 m2 .
g=10 m/s2 .
. = 1kg/m3 ,
= 120 m/s ,
= 100 m/s , g = 10m/s2
)p = ;
.8
,
.-
.
Bernoulli , , :
pB
+1
2.
2+ .gy1 = p +
1
2.
2+ .gy2
(1)
, , y1
= y2
= 0.
(1) pB
+1
2.
2= p
+
1
2.
2 p
= pB
+1
2
.
2 1
2
.
2
(2).
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8
Bernoulli , , :
p
= pB
+1
2
.
2 1
2
.
2(3)
p = p p(2)
(3) p = p
B +
1
2.
2 1
2.
2 pB
1
2.
2+
1
2.
2
p =
1
2.
2
2( ) p = 12
1 1202 1002( ) Pa p=2.200 Pa.
) = 50 m2 , m = ;
.5
pF
F = Ap F = 2.200 50 N F = 110.000 N. , :
F = 0 F = w F = mg m =F
g m=11.000 kg
, ()
.
.
8. U -
. 24m/s .
;
=800kg/m
3,
.=1,2 kg/m
3, = =
g=10m/s2, 1 Atm=10
5Pa.
1
= 24 m/s ,
= 800 kg/m3 , .
= 1,2 kg/m3 , g = 10m/s2 , 1Atm = 105 Pa , h = ;
.8
, . ,
(2
= 0)
(p2
= patm
).
Bernoulli , , :
p1
+1
2
.1
2+
.gy
1= p
2 +
1
2
.2
2+
.gy
2
(1)
, , :
y1
= y2
= 0 (, ).
(1) p1
+1
2.
1
2+ 0 = p
atm + 0 + 0 p
1+
1
2
.
1
2= p
atm
(2).
, , :
p
= pE p
1 +
gh = p
atm (3)
(2), (3) p1
+1
2.
1
2= p
1+
gh
1
2.
1
2=
gh h =
.
1
2
2g
h = 1,2 242
2 80010m h = 0,0432 m h=4,32 cm.
h p, -
(3) p = patm
p1
= gh h =
p
g
.
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39
9. . -
. -
1=20 m/s.
) ;
) .
) h -
.
=10
3kg/m
3,
.
=1,2 kg/m3, g=10m/s
2, 1 Atm=10
5Pa.
1
= 20 m/s ,
= 103 kg/m3 , .
= 1,2 kg/m3 , g = 10m/s2 , 1Atm = 105 Pa
)p1 = ; .8
, .
,
(2
= 0)
(p2
= patm
).
Bernoulli , , :
p1 +1
2
.1
2+
.gy
1 = p
2 +
1
2
.2
2+
.gy
2
(1)
,
, y1
= y2
= 0.
(1) p1
+1
2.
1
2+ 0 = p
at + 0 + 0 p
1= p
atm
1
2.
1
2
p1
= 100.0001
21,2 202
Pa p1=99.760 Pa.
);
()
p1 = 99.760 Pa. p
atm = 100.000 Pa.
p = patm
p1
= 240 Pa
.
(-
.)
) h = ;
.8
, ,
= 0,
= 0pB
= p1
= 99.760 Pa,
pE
= patm
= 105 Pa.
Bernoulli , p1 +
1
2
2+
gy
1= p
atm +
1
2
2+
gy
2
(2)
, y1=
h y2 =
0. (2) p
1+ 0 +
gh = p
at + 0 + 0
gh = p
atm p
1 h =
patm
p1
g
h =240
103 10m h = 0,024 m h=2,4cm.
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40
8
10. -
=0,01m2 h1=0,55m h
2=0,45 m.
)
;
) .
) h1 h2 . -
h1
.
=103
kg/m3, g=10m/s
2, 1 Atm=10
5Pa.
= 0,01m
2 , h1
= 0,55 m , h2
= 0,45m ,
= 103 kg/m3 , g = 10m/s2 , 1Atm = 105 Pa
) = ; , = ; .7, .8
Bernoulli () (), :
pA
+1
2
2+
gy
1= p
+
1
2
2+
gy
2
(1)
, , , pA
= p
= pat
(2).
,
= 0(3).
, y1
= h2
y2
= 0.
(1)(2)
(3)p
at + 0 +
gh
2 = p
at +
1
2
2+ 0
gh
2 =
1
2
2
= 2gh
2
= 210 0, 4 5 m/s
=3 m/s.
= = 0,01 3 m3 /s =0,03m3/s.
) = ; , p = ; .7, .8
, , :
1 = 2 A = = = 2gh2 (4) =3 m/s
Bernoulli () , :
p
A +
1
2
2+ gy1 = p +
1
2
2+ gy2
(5)
,
= 0(6).
, y1
= 0 y2
= h1.
(5)(2)
(6)p
at + 0 + 0 = p
+
1
2
2+
gh
1
(4)
p
= p
at
1
2 2gh
2
gh
1
p
= at g h1 + h2( )(6) p = 100.000 1.000 10 0,55 + 0,45( ) Pa p = 90.000 Pa p 9 10
4 Pa.
) h1(max) = ;
p 0 (6) pat g h1 + h2( ) 0 g h1 + h2( ) pat
h1
+ h2
pat
g
h1
pat
g
h2 h
1(max)=
pat
g
h2 h
1(max) =
100.000
1.00010 0,45
m h1(max)=9,55 m.
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41
8
1
8.1.1.: )
)
)
) .
;
8.1.2.: )
) ,
)
) -.
;
8.1.3.: )
)
)
)
.
;
8.1.4.: )
)
)
) .
;
8.1.5.p : )
)
)
) .
;
8.1.6.:
)
)
)
) p = pat
+gh -
.
;
8.1.7. p.
h
:
) )
) )
;
8.1.8.: ) -
)
)
) .
;
8.1.9.-
: ) -
) -
)
)
.
;
8.1.10.: ) -
) -
) -
-
) -
.
;
-
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42
8
8.1.11.: )
) -
)
) -
, .
;
8.1.12. -:
)
) -
)
) . ;
8.1.13.: ) 1 = 2 ) 11 = 22
) 1
1=
2
2 ) A
1
1=
2
2
;
8.1.14.: )
)
) -
) ,
-
.
;
8.1.15. , m
1 -
A1t m
2
A2
:
) m1
= m2
) m1
> m2
) m1
< m2 )m
1 m
2
;
8.1.16., :
) , -
) ,
) , ,
) , ,
.
;
8.1.17.: ) , -
) ,
) ,
) , -
.
;
8.1.18.Bernoulli : )
)
)
)
.
;
8.1.19.H Bernoulli :
) p + 2
+gy = .
) p +1
2 + gy
2= .
) p +1
2
2+gy = .
) p + +1
2gy
2= .
;
8.1.20. ,
:
)
)
)
) .
;
8.1.21.Bernoulli, -:
) , -
) ,
) ,
)
.
;
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43
8.1.22. , :
) , -
) , -
) Bernoulli, -
) Bernoulli, -
.
;
8.1.23.Bernoulli,
-:
) ,
) ,
) ,
) ,
.
;
8.1.24.
Bernoulli. :
) -
-
) -
-
)
) -
. ;
8.1.25. - -
.
,
.
) .
)
, pat
.
) -
, pat .
;
8.1.26.-
.
) -
pat
.
) -
pat .) .
) .
;
8.1.27. -
-
.
-
,
:
)
) ()
) ().
;
8.1.28. .
F , -
, -
. :
)
)
,
)
,
)
-
0 .
;
-
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44
8
8.1.29. . -
F
, ,
. F :
)
)
)
)
.
;
8.1.30.-
. F ,
,
. F :
) F = A ) F = A
) F = A
) F =
;
8.1.31. F = A
:
)
)
)
) .
;
8.1.32.R : )
) -
-
-
) -
-
, -
) -
-
, .
;
8.1.33.: )
)
) -
)
.
;
8.2.1.
. , , ,
h :
) p
= p
= p
= p
) p
> p
> p
> p
) p
< p
< p
< p
-
.
8.2.2.
-
. .
:
) F
> F
> F
> F
) F < F < F < F ) F
= F
= F
= F
-
.
8.2.3.-U -
,
1
2
-
. :
2
-
7/24/2019 reusta
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45
)h
1
h2
=
1
2
)h
1
h2
=
2
1
)h
1
h2
=
1
2
2
2 )
h1
h2
=
2
2
1
2
-
.
8.2.4., -, U,
1
= 1.000 kg/m3.
2
= 800 kg/m3,
h = 5cm.-
:
) 1cm
) 2 cm
) 4cm ) 5 c m-
.
8.2.5. h = 3 m
R =10
cm.
1.030 kg/m3, g = 10m/s2.
-
:
) 200
) 273 ) 300
) 309
-
.
8.2.6.
2
= 101.
F1
= 5 N,
F2
:
) 0,5 ) 5 ) 10 ) 50
-
.
8.2.7. -
-
F1
= 4 N
F2
= 12N.
d1, d
2 = 5cm.
d
:
) 5cm ) 20 cm ) 1 0 cm )15cm
-
.
8.2.8. -
-
-
d1
d2.
WF1
, WF2
:
)W
F1
WF2
=A
1
A2 )
WF1
WF2
=A
2
A1
)W
F1
WF2
= 1 )W
F1
WF2
< 1
-
.
8.2.9.h . 8 m
1,5 m/s.
2 m
:
) 1,5 m/s ) 3 m/s ) 4,5m/s ) 6 m / s
-.
8.2.10.. h
1= 2 m
1
= 2 m/s.
, 0,5 m,
:
) 0,5m/s ) 1 m/s ) 4 m /s ) 8 m / s
-
.
8.2.11.
2 m/s.
1/3. -
:
)2
3m/s ) 4 m /s ) 6 m / s ) 8 m / s
-
.
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46
8
8.2.12.-
1= 4 m/s.-
-
1
= 9 cm2.
2 = 6 m/s, -
:
) 6 c m2 )12cm
2 ) 3 cm2 ) 27 c m
2
-
.
8.2.13.
1= 2 cm
20 , 2
= 0,2 cm-
.
1
= 0,5 m/s,
:
) 1 m/s ) 2 m /s ) 2,5m/s ) 5m/s -
.
8.2.14.-
4:1. 1,
1
2,
2
1
2, :
) 1
= 42
1
= 42
) 2
= 41
2
= 41
) 1
= 2
1= 4
2
) 1
= 2
2 = 4
1
-.
8.2.15. 10 m3/s 8,64107 m3.-:
) 100
) 120
) 150
) 200
-
.
8.2.16.--
4 m/s. g = 10m/s2 ,
:
) 0,4 m ) 0,8 m ) 1 m ) 1,2 m
-
.
8.2.17.- (
m3/s) -
-
.
:
) 6 m3/s
) 6 m3/s
) 8 m3/s
) 8 m3/s
-
.
8.2.18.-.
: )
) ( -
)
) ().
-
.
8.2.19. .
-
-
: ) ) ) .
-
.
8.2.20. :
)
) .
-
.
8.2.21.-h
1.
2h, 2. -
, :
) 1
> 2
)1
= 2 )
1<
2
-
.
-
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47
8.2.22. -
-
.
-
-
,
:
) ) ) .
-
.
8.2.23.
.
1, , ,
2.
.:
)
=
=
=
)
>
=
=
)
>
>
>
)
>
>
>
.:
) p
= p
= p
= p
) p
> p
> p
> p
) p
> p
> p
> p
) p
> p
> p
> p
-
.
8.2.24.
. , 3Atm
2Atm .
103 kg/m3, g = 10m/s2 1Atm = 105 Pa. -
h , :
) 5 m ) 1 0 m ) 1 5 m ) 2 0 m
-
.
8.2.25. -
10 m/s. ,
1,2 kg/m3,
, 1
tm =
10
5
Pa. A. p
:
) 20 Pa ) 30 Pa ) 40 Pa ) 60 Pa
. F
= 50 m2 :
) 2.000 ) 3.000
) 4.000 ) 5.000
-
.
8.2.26. -
.
3>
1>
2.
.-
(1), (2), (3) :
) 1
= 2
= 3
) 2
> 1
> 3
) 2
< 1
< 3
.(1), (2), (3) :
) p1
= p2
= p3
) p2
> p1
> p3
) p2
< p1
< p3
., ,
: ) h1 = h2 = h3 ) h
2 > h
1> h
3
) h2
< h1
< h3
-
.
8.2.27.
. ,
.
, :
) h1
= h2
) h1
> h2
) h1
< h2
-
.
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48
8
8.2.28. A1
1.-
F1
, ,
1.
2,
F2,
. :
) F2
= 2F1
) F2
= 4F1
) F2
= 8F1 ) F
2 = 16F
1
-
.
A
8.3.1. 100 m -. , 1 m 0,6 m,-
;
1.020 kg/m3, -
1Atm = 105 Pa , g = 10m/s2.
8.3.2. =20
cm
= 1.000 kg/m2
= 920 kg/m3.
d1
= 10cm, d2
= 20cm.
) -;
) ;
) ; (
.)
g = 10m/s2.
8.3.3.-
= 1.000 kg/m3
= 800 kg/m3 .
d1
= 30cm, d2.
103.400 Pa.
) -; ) d2 ;
) F1-
= 2 cm2
-.
)
F2
= 34 N.
1Atm = 105 Pa , g = 10m/s2.
8.3.4. 4 cm2. d
1= 0,1m. -
. -
2, d
, ; 1.000 kg/m3, g = 10m/s2.
8.3.5.U -.
= 1.000 kg/m3
= 900 kg/m3.
- d = 0,1m.
) .
) ;
) ;
1Atm = 105
Pa , g = 10m/s2.
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49
8.3.6. h = 10 m R =20
cm.
) ;
)
.
1.030 kg/m3, g = 10m/s2.
8.3.7. -
U,
= 13.600 kg/m3.
.
p.
= 154.400 Pa,h.
1Atm = 105 Pa , g = 10m/s2.
8.3.8. m = 2.000 kg.
) F1
-
m;
)
d2
= 0,01m;
) F1, F2.;
R1 R
2 = 8R
1, g = 10m/s2.
8.3.9., A
1 A
2 = 5A
1. -
,
k .
.
m = 2 kg,-
=
0,1m.) k ;
) .
) (1) .
g = 10m/s2.
8.3.10.1 =10
cm.
1
= 4 m/s.
) ;
) , 2
=5
cm.-
;
) , ;
8.3.11. 1,44103 m3/s.-
1= 0,4 m/s.
) 1;
) .
) 0,192 m .
g = 10m/s2.
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50
8
8.3.12.. 3Atm-
8 m/s, 2,84Atm.
, A1 = 10cm2
A2 =
20cm
2
.) ;
) ;
) h , ;
)
, , F1()
F2()
.
)
.
= 1.000 kg/m3, 1Atm = 10
5Pa, g = 10m/s
2.
.
8.3.13.. 1,82Atm-
6 m/s, 10 m/s.
5 cm2.
, h = 5 m.
) ;
) .
)
, , W
V = 103
m3.
= 1.000 kg/m3, 1Atm = 105 Pa, g = 10m/s2.
8.3.14.- = 8103 m3/s . A = 103 m2.
) , ;
) , 4tm 3tm ,
h , ;
)
, ,W -
m = 1kg.
= 1.000 kg/m3, 1Atm = 105 Pa, g = 10m/s2.
8.3.15. = 48104 m3/s. -
A1=
12cm
2
() A2. 2,2Atm.
) 1
;
) 2
;
) 2
.
= 1.000 kg/m3, 1Atm = 10
5Pa.
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51
8.3.16., 30m/s.) p ;
) F A = 100 m2;
) ;
, 1,2 kg/m
3
, 1Atm =
10
5
Pa, g =
10m/s
2
.
8.3.17. = 2,4103 m3/s. A1, A2 , 1 = 22.
1,26Atm, 1,2Atm.
) 1,
2, .
) 1,
2, .
= 1.000 kg/m3, 1Atm = 105 Pa.
8.3.18.Ventouri . A
1, A
2
A1
= 2A2
h = 0,15 m, h
, . ) , ;
) 1 ;
) ;
= 1.000 kg/m3, g = 10m/s2.
8.3.19. . h = 0,8 m -
3cm2.
) ;
) 4,8 L;
g = 10m/s2.
8.3.20. . , h = 0,8 m , -
A = 20cm2,
6103 m3/s.) .
) h ,
.
) A1
= 1,5m2 ,
().
g = 10m/s2.
8.3.21. . -, h = 1,8m ,
10cm2 ,
8103 m3/s. ) .
) ;
= 3,2 m, g = 10m/s2.
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8
8.3.22. H = 1,25 m. , h = 0,45 m
,
20cm2 , .
) .
)
;
) ;
)
;
g = 10m/s2.
8.3.23. h = 0,9 m. , d
, , .
) d
Smax ;) .
)
. ;
g = 10m/s2.
8.3.24. H = 5 m. - 1Atm 0,04 m2.
) ;
) ;
) h = 1,8m;
= 103 kg/m3, g = 10m/s2 , 1Atm = 105 Pa.
8.3.25. = 0,2 m2 -. -
-
10 m/s.
) p;
) F ;
) ,
;
.
= 1,2 kg/m3, g = 10m/s2.
8.3.26. 0,8 kg/m3. 100 m/s 80 m/s.
) p;
) = 60 m2 ,.
g = 10m/s2.
8.3.27.U . 18m/s.
;
= 900 kg/m3, . = 1,2 kg/m3,
g = 10m/s2 , 1Atm = 105 Pa.
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8.3.28.. -.
1
= 25 m/s.
) ;
) . ) h -
.
= 103 kg/m3,
.
= 1,2 kg/m3, g = 10m/s2 , 1Atm = 105 Pa.
8.3.29.- = 0,02 m2. -
= 4 m/s.
8104 Pa. ) h
2;
) ;
) h1; ) h1 h2.
h1
.
= 103 kg/m3, g = 10m/s2 , 1Atm = 105 Pa.
8.3.30. A = 40cm2 - h = 10m. = 4 m/s.
) ;
) 500 kg.
)
, 500 kg;
= 1.000 kg/m3, g = 10m/s
2.
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