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ΘΟΔΩΡΗΣ ΓΑΡΜΠΗΣ ΦΥΣΙΚΗ Γ ´ Γενικού Λυκείου Ομάδας προσανατολισμού θετικών σπουδών
Transcript of ΘΟΔΩΡΗΣΓΑΡΜΠΗΣblogs.sch.gr/nikanestis/files/2015/11/Γαρμπής...3 ¡ © ù 2 ...
ΘΟΔΩΡΗΣ ΓΑΡΜΠΗΣ
ΦΥΣΙΚΗΓ ´ Γενικού Λυκείου
Ομάδας προσανατολισμού θετικών σπουδών
Το παρόν έργο πνευματικής ιδιοκτησίας προστατεύεται κατά τις διατάξεις της ελληνικής νομοθεσίας (Ν.2121/1993),όπως έχει τροποποιηθεί και ισχύει σήμερα, και τις διεθνείς συμβάσεις περί πνευματικής ιδιοκτησίας. Απαγορεύεταιαπολύτως η άνευ γραπτής αδείας του εκδότη κατά οποιονδήποτε τρόπο ή μέσο (ηλεκτρονικό, μηχανικό ή άλλο) αντι-γραφή, φωτοανατύπωση και εν γένει αναπαραγωγή, εκμίσθωση ή δανεισμός, μετάφραση, διασκευή, αναμετάδοσηστο κοινό σε οποιαδήποτε μορφή και η εν γένει εκμετάλλευση τουσυνόλου ή μέρους του έργου.
Θέση υπογραφής δικαιούχου δικαιωμάτων πνευματικής ιδιοκτησίαςεφόσον η υπογραφή προβλέπεται από τη σύμβαση.
Εκδόσεις Πατάκη – ΕκπαίδευσηΘοδωρής Γαρμπής,Φυσική Γ´ ΓενικούΛυκείου –Ομάδας προσανατολισμού θετικών σπουδώνΔιορθώσεις: Μάγδα ΤικοπούλουΕπιστημονική επιμέλεια: Βέτα Παπαργύρη, Νίκος ΚύροςΥπεύθυνος έκδοσης: Βαγγέλης ΜπακλαβάςΣελιδοποίηση: Κέλλυ ΚαραμανλήΦιλμ-μοντάζ: Μαρία Ποινιού – ΡένεσηCopyright© Σ. Πατάκης Α.Ε.Ε.Δ.Ε. (Εκδόσεις Πατάκη) και Θοδωρής Γαρμπής, Αθήνα, 2013Πρώτη έκδοση από τις Εκδόσεις Πατάκη, Αθήνα, Ιούλιος 2015ΚΕΤ 8560 – ΚΕΠ 560/15ISBN (set) 978-960-16-6460-6ISBN(vol. 1)978-960-16-5047-0
ΠΑΝΑΓΗΤΣΑΛΔΑΡΗ(ΠΡΩΗΝΠΕΙΡΑΙΩΣ)38,10437AΘΗΝΑ,THΛ.:210.36.50.000,210.52.05.600,801.100.2665 -FAX:310.36.50.069KENTPIKH ΔΙΑΘΕΣΗ:EMM.MΠENAKH16,10678AΘHNA,THΛ.:210.38.31.078YΠOKATAΣTHMABOPEIAΕΛΛΑΔA:KOPYTΣAΣ(TEPMAΠONTOY-ΠEPIOXHB´KTEO),57009KAΛOXΩPΙΘEΣΣΑΛONIKH,T.Κ.1213,THΛ.:2310.70.63.54,2310.70.67.15 -FAX:2310.70.63.55Website: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
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
8 ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
– . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1 2 / – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4
9 ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9.1 ( – ) – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
1 2 / – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 ( ) – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
1 2 / – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10 ( – ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
– . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
1 2 / – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
11 ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
1 2 / – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
12 ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
– . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
– . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
– . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
1 2 / – . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
3 3
9
A 8
• . -
, . ,
.
• , .
– , , .– , . -
.
.
.
( )
V m. H :
== m
V
S.I. 1kg/m3.
V. -
, = m
V= .
• H .
– . – .
p ( ) -
F
.
p = F.
S.I. 1 Pa = 1 N/m2.
F ,
, p = F
A.
– F = pA, . .
– (Atm).pat = 1Atm = 105 Pa
10
8
– . , , « »
.
– -
. F .
( - ).
:
p = gh
: .
h: , .
.
Pascal
•
. F -
.
.
F,
F
A( ).
Pascal
( )
.
• , pat .
Pascal, . h :
p = p
at+ gh
11
–
, ( - ),
( ).
• , -
. .
.
• : – – – .
• -
.
.
:
– – – ,
– – , – .
•
.
, -
.
– .
– .– .
V -
t t.
= V
t
S.I. 1m3/s.
12
8
. t x,
V = A x (1).
= x
t (2).
= V
t
(1)
= x
t
(2)
= .
.
-
.
A1
1, -
A2
-
2. ,
.
m1
1
t m2
2
.
V1, V
2 m
1, m
2.
m = V (1), V = A x (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
m2
(
).
.
m1
= m2
(4)
(5) 1 1t =
2 2t A
1 1=
2 2 (6) ( ).
= , (6) 1
=2
= .
.
– , , :m1 = m2 , V1 = V2 , A1 1 = 2 2 , 1 = 2 .
– .– ( ) ,
, .
– ( ) , -, .
– , . . m1
1 m2 2 , m1 = m2
• ,
( – ), ( – ).
• , ( – ),
( ) ( – ).
13
• (2
>1).
, (A2
< A1),
.
BERNOULLI
-
.
, -
. y1,
1, p
1
,
y2,
2, p
2 ,
.
( 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
y1( ) = Vg y
2y
1( ) (3), .
– W = WF1
+ WF2
= +F1
s1
F2
s2
= p1A
1s
1p
2A
2s
2
V = A1
s1
= A2
s2, W = p
1V p
2V W = p
1p
2( ) V (4).
(2) (3)
(4)W
.= p
1p
2( ) V Vg y2
y1( ) (5).
• :
= 1
2m
2
2 1
2m
1
2 = 1
2V
2
2 1
2V
1
2 = 1
2V
2
2
1
2( ) (6)
(1) (5)
(6)p
1p
2( ) V Vg y2
y1( ) = 1
2V
2
2
1
2( ) p1
p2
gy2
+ gy1
= 1
22
2 1
21
2
p1
+ 1
21
2 + gy1
= p2
+ 1
22
2 + gy2
p + 1
2
2 + gy = .
Bernoulli . Bernoulli, :
(p), 1
2
2
( gy) .
14
8
• 1
2
2 , 12
2 = 12
mV
2 =V
.
• gy , gy = mV
gy = UV
.
• Bernoulli .• ,
Bernoulli :
p1
+ 1
21
2 = p2
+ 1
22
2
p + 1
2
2 = .
, , - , . . , -
.• , 1 = 2 = 0, Bernoulli :
p1
+ gy1
= p2
+ gy2
p + gy = . ( ).
Torricelli ( )
-
h , .
Bernoulli ( )
( ), :
pE
+ 1
2
2 + gh = pK
+ 1
2
2 + 0 (1) (yE
= y1
= h, yK
= y2
= 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
<< AE, << .
.
• Bernoulli -
. ,
. -
2
1,
3. -
Bernoulli, 2
1,
3 ,
, , ,
. ,
.
15
• Bernoulli .
, ,
-
.
( ).
,
.
.
, . ,
.
, , -
, .
.
.
. .
F . F
, ( )
. -
,
.
0 .
• :
– F
– F
– F
– F .
F F = .
F
.
: – S.I. 1 N s/m2.
– poise ( ) = 1dyn s/cm2 = 10 1 N s/m2
(1dyn = 10 5 N, 1cm2 = 10 4 m2 ).
– .
, .
• F = . -
.
, , -
,
.
• -
R.
, -
.
16
8
.
m kg
V m3
kg/m3
F N
m2
p Pa = N/m2
pat
= 1Atm 1 tm = 105 Pa
p. Pa = N/m2
m/s
m3/s
1
2
2 J/m3
gy
J/m3
N s/m2
m
.
= m
V= m
V (1)
p = F
A= F
A (2)
pat
= 1Atm = 105
Pa (3)
p.
= gh (4)
= V
t= A (5)
A1 1
=2 2 1
=2 (6)
Bernoulli p1
+ 1
21
2 + gy1
= p2
+ 1
22
2 + gy2 (7)
Torricelli = 2gh (8)
F = (9)
.
• = m
V. ,
= m
V= ., .
• p = F, F
. F ,
, p = F
A.
• p.
h
. p.
.
• A1 1
=2 2 1
=2
.
• p1
+ 1
21
2 + gy1
= p2
+ 1
22
2 + gy2
Bernoulli .
– , , y1
= y2
= 0,
Bernoulli p1
+ 1
21
2 = p2
+ 1
22
2.
– , 1
=2
= 0, Bernoulli p1
+ gy1
= p2
+ gy2
( ).
•
R A = R2 =2
4.
V = Ah = R2h.
17
.
.1 – PASCAL
1.
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 =1gh
1
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
-
,
.
18
8
.3 ( 1 > > 2)
• h
1
h2
= 2
1
.
: :
p = pat
+1g h
1+ d( ) p = p
at+
1gh
1+
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
+1gh
1p = p
at+
2gh
2
(3)
pat
+1gh
1. pA = p .
• (pE
pZ).
: pE
= pat
pZ
= pat
+2gh, pE pz .
U -
. V
= 998 kg/m3
.
= 18,4 cm d = 1,56 cm. -
;
g = 10 m/s2
1Atm = 105
Pa.
V= 998 kg/m3 , = 18, 4 cm = 0,184 m , d = 1,56 cm = 0,0156 m , g = 10 m/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 = pat
+ g + d( ) + gh
g = g + d( ) = + d( )
=+ d
= 9980,184
0,1996kg/m3 = 920 kg/m
3
• 1 p
A= p
at+ g p
A= 105 + 998 10 0,184( ) Pa p
A= 101 836,32 Pa.
2 , , , p
A= p .
p = pat
+ g d +( ) = 105 + 920 10 0,1996( ) Pa = 101.836,32 Pa.
pA
= p = 101.836,32 Pa.
19
.4
, -
U = 13.600 kg/m3. -
. h = 20 cm.
;
g = 10 m/s2
1Atm = 105
Pa.
= 13.600 kg/m2 , h = 20 cm = 0,2 m , g = 10 m/s2 , 1Atm = 105 Pa , p.
= ;
,
p.
= pA
(1).
, , :
pA
= p (2).
p = pat
+ gh = 105 + 13.600 10 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.
, , -
. :
F1
= p = gh1
F2
= p = gh2
F3
= pE
= gh3
h3
> h2
> h1, F3 > F2 > F1.
,
,
.
-
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 ( ) .
20
8
r = 0,1 m, = 0,4 m
= 1.000 kg/m3.
) -
.
) = 106
m2
.
g = 10 m/s2, = 3,14.
r = 0,1m , = 0,4 m , = 1.000 kg/m3 , g = 10 m/s2 , = 3,14
) F = ;
p p = g (1). A = r2 (2).
F = pA(1)
(2)g r2 = 1000 10 0,4 3,14 0,01 N = 125,6 N F = 125,6 N.
) = 10 6 m2 , F = ;
p = gr (3).
:
F = p = gr = 1.000 10 0,1 10 6 = 10 3
F = 10-3
N
.6 ( )
F
1
F2
= A1
A2
d
1
d2
= A2
A1
WF1
= WF2
= F1d
1= F
2d
2
(F
1) -
(d1)
(F2) (d
2).
F2
> F1
d2
< d1 2
>1.
• , F1
A1. . -
p1
= F1
A1
(1). Pascal,
A2. ,
F2, p
2= F
2
A2
(2). , Pascal
p1
= p2
(1)
(2)
F1
A1
= F2
A2
F1
F2
= A1
A2
(3).
B = F
2.
• d1,
d2, V , .
V = A1d
1= A
2d
2
d1
d2
= A2
A1
(4).
•
WF1
= F1d
1
WF2
= F2d
2
(:) WF1
WF2
= F1
F2
d1
d2
(3)
(4)
A1
A2
A2
A1
= 1 WF1= WF2
.
21
-
m = 1.000 kg.
) F1
m;
) ,
d2
= 0,1 m;
) F1, F
2. ;
R1
R2
= 5R1
g = 10 m/s2.
m = 1.000 kg , R2
= 5R1
, g = 10 m/s2
) F1
= ;
F1
A1 = R12 (1). . -
p1 = F1
A1
(2).
Pascal,
A2 = R22 (3).
, F2, p2 = F2
A2
(4). -
, Pascal :
p
1= p
2
(2)
(4)
F1
A1
= F2
A2
F1
F2
= A1
A2
(1)
(3)
F1
F2
= R1
2
R2
2
F1
F2
= R1
2
25R1
2
F1
F2
= 1
25F1 = 1
25F2 (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 = A1d
1= A
2d
2R
1
2d1
= R2
2d2
R1
2d1
= 25R1
2d2
d1
= 25d2
d1
= 2,5 m.
) WF1
= ; , WF2
= ;
• W
F1= F
1d
1= 400 2,5 J = 1.000 J W
F1= 1.000 J
• W
F2= F
2d
2= 10.000 0,1 J = 1.000 J W
F2= 1.000 J
.
22
8
B.
.7
1=
2 A
1 1=
2 2
• .
• ( ) , -
, .
• m1
1
m2
2, m1 = m2 . V1 = V2 .
1. h . 10 m
2 m/s. 2,5 m;
h = . , d1
= 10 m ,1
= 2 m/s , d2
= 2,5 m ,2
= ;
, , . -
:
1=
2 1 1=
2 2d
1h
1= d
2h
2d
1 1= d
2 2 2=
1
d1
d2
2= 2
10
2,5m/s
2= 8 m/s
2. . h1
= 2 m
1
= 0,8 m/s. -
2= 4 m/s;
h1
= 2 m ,1
= 0,8 m/s ,2
= 4 m/s , h2
= ;
d . , ,
. :
1=
2 1 1=
2 2dh
1 1= dh
2 2h
1 1= h
2 2h
2= h
11
2
h2
= 20,8
4m h
2= 0,4 m
3. 1,44 103
m3/s. -
= 12cm.
) ;
)
1
4. ;
= 1,44 10 3 m3/s , = 12cm = 12
10 2 m
) = ;
, .
= =2
4= 4
2= 4 1, 44 10 3
122 10 4m/s = 0,4 m/s.
) 1
= 0,4 m/s , A2
= 1
4A
1,
2= ;
1 1
=2 2 1 1
=2
1
41 2
= 41 2
= 1,6 m/s.
23
4. = 102
m3/s. -
1
= 10cm.
) 1
;
) .
) 0,45 m .
g = 10 m/s2.
= 10 2 m3/s ,1
= 10cm = 10 10 2
m = 10 1
m , g = 10 m/s2
) 1
= ;
:
A
1= 1
2
4=
10 2
4m2 = 10 2
4m2 A1 = 10-2
4m2
=1 1 1
=1
= 10 2
10 2
4
m/s1
= 4 m/s.
) ;
(2
>1). -
, (A2
< A1), .
) h = 0,45 m , A2
= ;
1 1
=2 2 2
=1
1
2
(1).
1,
2 -
h = 0,45 m.
. . . . , , :
. . . . ( , ): K
.( ) .( )= W
B
1
2m
2
2 1
2m
1
2 = +mgh2
2
1
2 = 2gh2
2 =1
2 + 2gh
2=
1
2 + 2gh(2)
2= 42 + 2 10 0,45 m/s 2 = 5 m/s
• (1) A2
= 10 2
4
4
5m2 = 10 2
5m2 = 2 10 3 m2
A2
= 2 10-3
m2.
A2
:
=1 1
=2 2 2
=2
2= 10 2
5m2
A2
= 2 10-3
m2
24
8
.8 ( ) BERNOULLI
p1
+ 1
21
2 + gy1
= p2
+ 1
22
2 + gy2
,
.
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/s
2, 1Atm = 10
5Pa.
h = 5 m , p1
= 2Atm = 2 105 Pa ,1
= 10 m/s , p2
= 1,82Atm = 1,82 105 Pa , = 103 kg/m3 , g = 10 m/s2 , 1Atm = 105 Pa
) 2
= ;
ernoulli , , p1 + 12 1
2 + gy1 = p2 + 12 2
2 + gy2 (1).
, y1
= 0 y2
= h.
(1) p1
+ 1
21
2 + 0 = p2
+ 1
22
2 + gh 2 105 + 1
2103 102 = 1,82 105 + 1
2103
2
2 + 103 10 5
200.000 + 50.000 = 182.000 + 500
2
2 + 50.000 5002
2 = 18.0002
2 = 36 m2/s2
2= 6 m/s .
) A
1
A2
= ;
, , :
1
=2 1 1
=2 2
1
2
= 2
1
1
2
= 6
101
2
= 3
5
2.
. ,
5Atm 3Atm .
h , ;
103
kg/m2, g = 10 m/s
2, 1Atm = 10
5Pa.
p
1= 5Atm = 5 105 Pa , p
2= 3Atm = 3 105 Pa , = 103 kg/m3 , g = 10 m/s2 , 1Atm = 105 Pa , h = ;
Bernoulli , , p1 + 12 1
2 + gy1 = p2 + 12 2
2 + gy2 (1).
, y1
= 0 y2
= h.
, A1 = A2 = A (2).
, , 1
=2 1 1
=2 2
(2)
1 = 2 (3).
(1) (3)
p1
+ 1
21
2 + 0 = p2
+ 1
21
2 + gh p1 = p2 + gh (4)
gh = p1
p2
h = p1
p2
gh = 5 105 3 105
103 10m = 2 105
104m = 20 m h = 20 m.
25
(4) Bernoulli p1
= p2
+ gh.
• p2
< p1, h .
• p = p1
p2
= gh, h.
3. .
1= 18 cm
2 ( )
2= 6 cm
2.
1
= 4 m/s.
) 2
;
) ;
103
kg/m3, g = 10 m/s
2, 1Atm = 10
5Pa , -
.
A1
= 18 cm2 , A2
= 6 cm2 ,1
= 4 m/s , = 103 kg/m3 , g = 10 m/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 + 12 1
2 + gy1 = p2 + 12 2
2 + gy2 (1).
, , y1
= y2
= 0.
, p = pat
= 1Atm = 105 Pa.
(1) p1
+ 1
21
2 = p2
+ 1
22
2 p1
= p2
+ 1
22
2 1
21
2 p1
= 105 + 1
2103 122 1
2103 42 Pa =
= 100.000 + 72.000 8.000( ) Pa = 164.000 Pa = 1,64 105 Pa p
1= 1,64 10
5Pa.
4. 20 m/s.
) p ;
) F A = 100 m2;
) ;
1,2 kg/m3, 1Atm = 10
5Pa.
= 20 m/s , = 0 , = 1,2 kg/m3 , 1Atm = 105 Pa
) p = ;
. , = 0.
Bernoulli , , pB + 12
2 + gy1 = p + 12
2 + gy2 (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 = A p F = 100 240 N F = 24.000 N.
) pB
= ;
, ,
p = p
at= 1Atm = 105 Pa. p = p p p = p p p
B= 100.000 240( ) Pa p
B= 99.760 Pa.
26
8
.9 VENTOURI
1= 2gh
A1
A2
2
1
( )
Ventouri .
A1, A
2 h , .
, , .
pB
= p1
= pat
+ gh1
p = p2
= pat
+ gh2, :
p1
p2
= pat
+ gh1
pat
gh2
= g h1
h2( ) = gh p
1p
2= gh (1)
, , 1
=2 1 1
=2 2 2
= 1
2
1 (2).
Bernoulli , , p1
+ 1
21
2 + gy1
= p2
+ 1
22
2 + gy2
(3).
, , y1
= y2
= 0.
(3) p1
+ 1
21
2 = p2
+ 1
22
2 (2)
p1
+ 1
21
2 = p2
+ 1
21
2
2
2 1
2
p1
p2
= 1
21
2
2
2 1
2 1
21
2 (1)
gh = 1
21
2 1
2
2
21
gh = 1
21
2 1
2
2
21
1
2 = 2gh
A1
2
A2
21
1 = 2gh
A1
A2
2
1
.
A
1= 3A
2, h = 0,1m , g = 10 m/s2 ,
1= ;
1= 2gh
A1
A2
2
1
= 2 10 0,1
32 1m/s = 2
8m/s = 0,5 m/s
1= 0,5 m/s.
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 = 12
gt2 (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 4d2 4hd + S2 = 0
(6)
• d, -
, 0 4h( )2
4 4S2 0 16h2 16S2 0 16S2 16h2 S h, :
Smax = h (7).
• (6) S = Smax
= h, 4d2 4hd + h2 = 0 2d h( )2
= 0 2d h = 0 2d = h d = h2
(8).
• (1) (8)
= 2g hh
2= 2g
h
2= gh (9)
• (5) (7)
(9)t = h
ght = h
g (10). (3)
(10)
y= g
h
gy = gh (11)
= 2 +y
2 (9)
(11)gh + gh = 2gh = 2gh .
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
2 Bernoulli , , :
p + 12
2 + gy1 = p + 12
2 + gy2 (12)
,
y1 = d = h2
(8) y2 = 0. , ,
, p = p = pat (13).
(12) (13)
pat
+ 1
2
2 + gh
2= p
at+ 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
2gt2 t = 2h1
g (3)
(2) (1)
(3) S = 2g h h
1( ) 2h1
gS = 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( ) = h
2h h
2( ) h1h h
1
2 = h2h h
2
2
h1h h
2h = h
1
2 h2
2 h h1
h2( ) = h
1h
2( ) h1
+ h2( )
h = h1 + h2 (6). h = h1 + d (7).
(6), (7) h2 = d.
• E AB = ZH, S1
= S5.
• E = , S2
= S4.
• E = , S3
= Smax
.
. . . .
Bernoulli, -
. -
,
Bernoulli
-
.
29
1. .
, h = 1,25 m , -
= 2 cm2. 5 L;
g = 10 m/s2.
h = 1,25 m , A = 2 cm2 = 2 10 4 m2 , V = 5 L = 5 10 3 m3 , g = 10 m/s2 , t = ;
Torricelli, = 2gh ( . -
. 14).
= 2 10 1,25 m/s = 5 m/s.
= (1). = Vt
(2).
(1), (2) A = V
tt = V
At = 5 10 3
2 10 4 5s t = 5 s.
2. . -
, h = 1,8 m , -
10 cm2,
4 103
m3/s.
)
.
) h
.
g = 10 m/s2.
h = 1,8 m , A = 10 cm2 = 10 10 4 m2 = 10 3 m2 ,.
= 4 10 3 m3/s , g = 10 m/s2
) ;
Torricelli, :
= 2gh = 2 10 1,8 m/s = 6 m/s = 6 m/s. :
= = 6 10 3 m3/s.
.= 4 10 3 m3/s. >
.,
, ( )
.
) h = ;
, -
, .
.
=.
= = . = 4 10 3
10 3m/s = 4 m/s.
= 2gh 2 = 2gh h =2
2g= 42
2 10m = 0,8 m h = 0,8 m.
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
2
2 + gh pB
p = 1
2
2 + gh p = 12
2 + gh (2).
• m :
W.
= pV(2) 1
2
2 + gh V = 1
2
m
V
2 + m
Vgh V = 1
2m 2 + mgh W . = 1
2m 2 + mgh
• F F = A p(2)
F = 12
2 + gh A (3).
P.
= F(3)
P . = 12
2 + gh A .
2 • . . . . , m , :
. . . . ( , ): = W K.( ) .( )
= W.
+ WB
1
2m 2 0 = W
.mgh W . = 1
2m 2 + mgh (4)
• :
P.
=W
.
t
(4)
P
.=
m 2
2+ mgh
tP . =
2
2+ gh m
t (5)
= Vt
= A (6).
m
t=
V( )t
= V
t
(6) mt
= (7).
(5) (7)
P.
=2
2+ gh P . = 1
22 + gh A .
31
1. -
= 1.000 kg/m3 = 900 kg/m
3.
d1
= 20 cm, d2
= 30 cm.
) - ;
) ;
) -
= 1 cm2
- .
) -
A = 400 cm2.
)
264 N. ;
1Atm = 105
Pa, g = 10 m/s2.
= 1.000 kg/m3 , = 900 kg/m3 , d1
= 20 cm = 0,2 m , d2
= 30 cm = 0,3 m , 1Atm = 105 Pa , g = 10 m/s2
) p = ;
.1
p = p
at+ gd
1= 105 + 900 10 0,2( ) Pa = 101.800 Pa
p = 101.800 Pa
) p = ;
.1
p = p
at+ gd
1+ gd
2= 105 + 900 10 0,2 + 1.000 10 0,3( ) Pa = =
= 104.800 Pa p = 104.800 Pa
) = 1cm2 = 10 4 m2 , F1
= ;
.5
,
p = 101.800 Pa p = pat
= 1Atm = 100.000 Pa, . -
p1
= p pE
= 1.800 Pa. F1
:
F
1= p
1= 1.800 10 4 = 0,18 F
1= 0,18 N
) A = 400 cm2 = 4 10 2 m2 , F2
= ;
.5
, p = 104.800 Pa
pZ
= pat
= 1Atm = 100.000 Pa, .
p2
= p p = 4.800 Pa. F2
:
F
2= p
2A = 4.800 4 10 2 N = 192 N F
2= 192 N
) F.
= 264 , V = ;
d, p ,
F -
V = Ad (1).
F.
= 264 . -
F = F.
= 264 .
F = p p = F
Ap = 264
4 10 2Pa p = 6.600 Pa.
p = p p p = p + pZ
p = 6.600 + 100.000( ) Pa
p = 106.600 Pa.
p = pat
+ g d + d1( ) + gd
2106.600 = 100.000 + 900 10 d + 0,2( ) + 1.000 10 0,3
106.600 = 104.800 + 9.000d 9.000d = 1.800 d = 0,2 m.
(1) V = 4 10 2 0,2 m3V = 8 10
-3m
3.
32
8
2.
A1
= 1 cm2
A2
= 10 cm2.
,
k = 1.000 N/m . -
-
.
) -
= 0,1 m;
) .
) (1) .
g = 10 m/s2.
A1
= 1cm2 , A2
= 10 cm2 , k = 1.000 N/m , g = 10 m/s2
.6
) = 0,1m , m = ;
= 0,1m, .
F1
, F1 = B = mg (1).
p1 = F1
A1
(2).
Pascal, -
. ,
F2, p2 = F2
A2
(3). ,
Pascal p1
= p2
(2)
(3)
F1
A1
= F2
A2
F1
F2
= A1
A2
F1
F2
= 1
10F2 = 10F1 (4).
, F.
= k .
, F2 = F . = k (5).
(4) (1)
(5)k = 10mg m = k
10gm = 1.000 0,1
10 10kg m = 1 kg
) d = ;
d2
= = 0,1m. -
d1, d
2,
V , .
V = A1d
1= A
2d
2d
1= d
2
A2
A1
d1
= 0,110
1m d
1= 1m.
d = d1
+ d2
d = 1,1 m.
) WF1
= ;
, F ..
WF1
= WF .
= U.
= 1
2k 2 = 1
21.000 0,12 J = 5 J W
F1= 5 J.
WF1
, WF2
WF1
= F1d
1W
F2= F
2d
2, -
.
.
33
3.
H = 2 m. h -
10 cm2,
= 2 m/s.
) h .
) -
;
) 2 tm, ,
.
1.000 kg/m3, g = 10 m/s
2, 1Atm = 10
5Pa, 204 = 14,3, 240 = 15,5.
H = 2 m , A = 10 cm2 = 10 3 m2, = 2 m/s , = 1.000 kg/m3 , g = 10 m/s2 , 1Atm = 105 Pa ,
204 = 14,3 , 240 = 15,5
) = ; , h = ;
.10
• = = 2 10-3
m3/s.
• Bernoulli (
) ( ), :
pZ + 1
2 Z2 + gy1 = p + 1
22 + gy2 (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
2gh = 22
2 10m h = 0,2 m.
) S = ;
.10
d = H h = 1,8 m. -
, , x = S = t (4)
y = d = 1
2gt2 t = 2d
gt = 0,6 s.
(4) S = 2 0,6 m S = 1,2 m
) pZ
= 2Atm = 2 105 Pa , = ; , = ; , = ;
.10
• Bernoulli , :
pZ + 1
2 Z2 + gy1 = p + 1
22 + gy2 (5)
( ) :
pZ
= 2Atm = 2 105 Pa, p = 1 tm = 105 Pa,
Z= 0, y
1= h, y
2= 0
(5) pZ
+ 0 + gh = p + 1
2
2 + 0
200.000 + 1.000 10 0,2 = 100.000 + 1
21.000 2 = 204 m/s = 14,3 m/s.
• ( ) :
S = t S = 14,3 0,6 m/s S = 8,58 m
= S S = 8,58 1,2( ) m = 7,38 m = 7,38 m.
34
8
• Bernoulli , , :
p + 1
22 + gy1 = pE + 1
2 E2 + gy2 (6)
, y1 = d = 1,8 m y2 = 0. , , , :
p = pE
= pat
(7)
(6) (7)
pat
+ 1
2
2 + gd = pat
+ 1
2
2 + 0
1
2
2 + gd = 1
2E
2 2 = 2 + 2gd = 2 + 2gd
= 204 + 2 10 1,8 m/s = 240 m/s = 15, 5 m/s.
. . . . ,
Bernoulli .
4. -
0,6 cm
10 m .
0,2 cm, -
9 m/s.
) .
) ;
) .
) :
i) F1
F2
-
ii) W
V = 103
m3.
)
.
= 1.000 kg/m3, 1Atm = 10
5Pa, g = 10 m/s
2, = 3,14 -
.
r1
= 0,6 cm = 6 10 3 m , r2
= 0,2 cm = 2 10 3 m , h = 10 m ,2
= 9 m/s , = 1.000 kg/m3 , 1Atm = 105 Pa ,
g = 10 m/s2 , = 3,14
) 1
= ;
.7
A1 = r12 (1), A2 = r2
2 (2).
, , :
1=
2 1 1=
2 2
(1)
(2)r1
2
1= r
2
2
2 1=
2
r2
r1
2
1= 9
1
3
2
m/s1
= 1 m/s
) .
= ;
.
=1 1
(1)
r1
2
1= 3,14 6 10 3( )2
1m3/s =
= 113,04 10 6 m3/s
.= 113,04 10
-6m
3/s.
35
) p1
= ;
.8
Bernoulli , , :
p1 + 1
2 12 + gy1 = p2 + 1
2 22 + gy2 (3)
, p = p2 = pat = 1Atm = 105 Pa. , y1 = 0 y2 = h (4).
(3)
(4)
p1
+ 1
21
2 + 0 = p2
+ 1
22
2 + gh p1
= p2
+ 1
22
2 1
21
2 + gh
p1
= 100.000 + 1
21.000 92 1
21.000 12 + 1.000 10 10 Pa
p
1= 240.000 Pa = 2,4 10
5Pa = 2,4Atm.
) i) F1
= ; , F2
= ;
Bernoulli, . 13
,
F1
= p1A
1o
F2
= p2A
2 .
•
F1
= p1A
1= p
1r1
2 = 2,4 105 3,14 6 10 3( )2
N = 27,1296 N
F1
= 27,1296 N
•
F2
= p2A
2= p
2r2
2 = 105 3,14 2 10 3( )2
N = 1,256 N
F2
= 1,256 N
ii) V = 10 3 m3 , W = ;
Bernoulli, . 13
t m -
s1, m, V,
s2
. W ( )
( ) :
W = W
F1+ W
F2= +F
1s
1F
2s
2= p
1A
1s
1p
2A
2s
2W = p1 A1 s1 p2 A2 s2 (5)
V = A1 s1 = A2 s2 (6).
(5)
(6)
W = p1
V p2
V W = p1
p2( ) V
W = 2,4 105 105( ) 10 3 J W = 1,4 10
2J = 140 J.
)V
B
= ; ,U
tB
= ;
•
1
21
2 = 1
21.000 12 J/m3 = 500 J/m3.
K
VB
= 1
21
2 = 500 J/m3.
• gy1
= 0, y1
= 0.
U
VB
= gy1
= 0.
36
8
5. 0,2 m3/s 1 Atm.
H = 20 m.
) ;
) ;
)
h = 15 m;
= 103
kg/m3, g = 10 m/s
2, 1 Atm = 10
5Pa.
= 0,2 m3/s , H = 20 m , p1
= 1Atm , = 103 kg/m3 , g = 10 m/s2 , 1Atm = 105 Pa
) 1
= ;
.7
1,
3= 0 -
H = 20 m. . . . . ,
, :
. . . . ( , ): K.( ) .( )
= WB
1
2m
3
2 1
2m
1
2 = mgH
0
1
2m
1
2 = mgH1
= 2gH1
= 2 10 20 m/s1
= 20 m/s
) 1
= ;
.7
=1 1 1
=1
A1
= 0,2
20m2
A1
= 10-2
m2.
) h = 15 m , A2
= ; , p2
= ;
.8
• 1,
2 -
h = 15 m. . . . . , :
. . . . ( , ): K
.( ) .( )= W
B
1
2m
2
2 1
2m
1
2 = mgh2
2
1
2 = 2gh
2 = 12 2gh (1)
2= 202 2 10 15 m/s
2= 10 m/s
= A2 2
A2
=2
A2
= 0,2
10m2
A2
= 0,02 m2.
• Bernoulli , , p1 + 12 1
2 + gy1 = p2 + 12 2
2 + gy2 (2).
, y1 = 0 y2 = h.
1
(2) p1
+ 1
21
2 + 0 = p2
+ 1
22
2 + gh 100.000 + 1
21.000 202 = p
2+ 1
21.000 102 + 1.000 10 15
p
2= 100.000 Pa p
2= 1 Atm
2
(2) p1
+ 1
21
2 + 0 = p2
+ 1
22
2 + gh
(1)
p
1+ 1
21
2 = p2
+ 1
21
2 2gh( ) + gh
p
1+ 1
21
2 = p2
+ 1
21
2 gh + gh p1
= p2
p2
= 1 Atm
, -
, p = p = 1Atm.
37
6. = 0,1 m2,
m = 5,4 kg -
.
20 m/s.
) p ;
) F ;
) 2
, -
;
.
= 1,2 kg/m3, g = 10 m/s
2.
= 0,1m2 , m = 5,4 kg ,1
= 20 m/s ,.
= 1,2 kg/m3 , g = 10 m/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
p F
F = A p F = 0,1 240 N F = 24 N.
) 2
= ;
p = 12 . 2
2
(2).
, F = 0 F = w A p = mg(2)
A1
2. 2
2 = mg2
= 2mg
A.
2= 2 5,4 10
0,1 1,2m/s
2= 30 m/s.
7. 1 kg/m3.
120 m/s 100 m/s.
) p ;
) = 50 m2
.
g = 10 m/s2.
.= 1kg/m3 , = 120 m/s , = 100 m/s , g = 10 m/s2
) p = ;
.8
,
. -
.
• Bernoulli , , :
pB + 12 .
2 + .gy1 = p + 12 .
2 + .gy2 (1)
, , y1 = y2 = 0.
(1) pB
+ 1
2.
2 = p + 1
2.
2 p = pB + 12 .
2 12 .
2
(2).
38
8
• Bernoulli , , :
p = pB + 12 .
2 12 .
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 = 1
21 1202 1002( ) Pa p = 2.200 Pa.
) = 50 m2 , m = ;
.5
p F
F = A p F = 2.200 50 N F = 110.000 N. , :
F = 0 F = w F = mg m = F
gm = 11.000 kg
, ( )
.
.
8. U -
. 24 m/s .
;
= 800 kg/m3,
.= 1,2 kg/m
3, = =
= g = 10 m/s2, 1 Atm = 10
5Pa.
1= 24 m/s , = 800 kg/m3 ,
.= 1,2 kg/m3 , g = 10 m/s2 , 1Atm = 105 Pa , h = ;
.8
, . ,
(2
= 0)
(p2
= patm
).
Bernoulli , , :
p1 + 12 . 1
2 + .gy1 = p2 + 12 . 2
2 + .gy2 (1)
, , :
y1 = y2 = 0 ( , ).
(1) p1
+ 1
2. 1
2 + 0 = patm
+ 0 + 0 p1 + 12 . 1
2 = patm (2).
, , :
p = pE
p1 + gh = patm (3)
(2), (3) p1
+ 1
2. 1
2 = p1
+ gh1
2. 1
2 = gh h = . 1
2
2 g
h = 1,2 242
2 800 10m h = 0,0432 m h = 4,32 cm.
h p, -
(3) p = patm
p1
= gh h = p
g.
39
9. . -
. -
1
= 20 m/s.
) ;
) .
) h -
.
= 103
kg/m3,
.
= 1,2 kg/m3, g = 10 m/s
2, 1 Atm = 10
5Pa.
1= 20 m/s , = 103 kg/m3 ,
.= 1,2 kg/m3 , g = 10 m/s2 , 1Atm = 105 Pa
) p1
= ;
.8
, .
,
(2
= 0)
(p2
= patm
).
Bernoulli , , :
p1 + 12 . 1
2 + .gy1 = p2 + 12 . 2
2 + .gy2 (1)
,
, y1 = y2 = 0.
(1) p1
+ 1
2. 1
2 + 0 = pat
+ 0 + 0 p1
= patm
1
2. 1
2
p1
= 100.0001
21,2 202 Pa p
1= 99.760 Pa.
) ;
( )
p1
= 99.760 Pa.
patm
= 100.000 Pa.
p = patm
p1
= 240 Pa
.
( -
.)
) h = ;
.8
, , = 0, = 0 pB
= p1
= 99.760 Pa,
pE
= patm
= 105 Pa.
Bernoulli , p1 + 1
22 + gy1 = patm + 1
22 + gy2
(2)
, y1 = h y2 = 0.
(2) p1
+ 0 + gh = pat
+ 0 + 0 gh = patm
p1
h =p
atmp
1
g
h = 240
103 10m h = 0,024 m h = 2,4 cm.
40
8
10. -
= 0,01 m2
h1
= 0,55 m h2
= 0,45 m.
)
;
)
.
) h1
h2. -
h1
.
= 103
kg/m3, g = 10 m/s
2, 1 Atm = 10
5Pa.
= 0,01m2 , h1
= 0,55 m , h2
= 0,45 m , = 103 kg/m3 , g = 10 m/s2 , 1Atm = 105 Pa
) = ; , = ;
.7, .8
• Bernoulli ( ) ( ), :
pA + 1
22 + gy1 = p + 1
22 + gy2
(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 gh2
= 1
2
2
= 2gh
2= 2 10 0, 45 m/s = 3 m/s.
• = = 0,01 3 m3/s = 0,03 m3/s.
) = ; , p = ;
.7, .8
• , , :
1=
2A = = = 2gh2 (4) = 3 m/s
• Bernoulli ( ) , :
pA + 1
22 + gy1 = p + 1
22 + gy2
(5)
, = 0 (6).
, y1 = 0 y2 = h1.
(5) (2)
(6)p
at+ 0 + 0 = p + 1
2
2 + gh1
(4)
p = p
at
1
22gh
2gh
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
4Pa.
) h1(max)
= ;
p 0 (6)
pat
g h1
+ h2( ) 0 g h
1+ h
2( ) pat
h1
+ h2
pat
gh
1
pat
gh
2h
1(max)=
pat
gh
2h
1(max)= 100.000
1.000 100,45 m h
1(max)= 9,55 m.
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. :
) -
) -
) -
-
) -
.
;
42
8
8.1.11. :
)
) -
)
) -
, .
;
8.1.12. -
:
)
) -
)
) .
;
8.1.13. :
) 1
=2
) 1 1
=2 2
) 1 1
=2 2
) A1 1
=2 2
;
8.1.14. :
)
)
) -
) ,
-
.
;
8.1.15. ,
m1
-
A1
t m2
A2
:
) m1
= m2
) m1
> m2
) m1
< m2 ) m
1m
2
;
8.1.16. ,
:
) , -
) ,
) , ,
) , ,
.
;
8.1.17. :
) , -
) ,
) ,
) , -
.
;
8.1.18. Bernoulli :
)
)
)
)
.
;
8.1.19. H Bernoulli
:
) p + 2 + gy = .
) p + 1
2+ gy2 = .
) p + 1
2
2 + gy = .
) p + + 1
2gy2 = .
;
8.1.20.
,
:
)
)
)
) .
;
8.1.21. Bernoulli, -
:
) , -
) ,
) ,
)
.
;
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 .
;
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
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 = 5 cm. -
:
) 1cm
) 2 cm
) 4 cm
) 5 cm
-
.
8.2.5. h = 3 m
R = 10cm.
1.030 kg/m3, g = 10 m/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
= 12 N.
d1, d
2= 5 cm.
d
:
) 5 cm ) 20 cm ) 10 cm ) 15 cm
-
.
8.2.8. -
-
-
d1
d2.
WF1
, WF2
:
) W
F1
WF2
= A1
A2
)W
F1
WF2
= A2
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 ) 4,5 m/s ) 6 m/s
-
.
8.2.10. .
h1
= 2 m
1
= 2 m/s.
, 0,5 m,
:
) 0,5 m/s ) 1m/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
-
.
46
8
8.2.12. -
1= 4 m/s. -
-
1
= 9 cm2.
2
= 6 m/s, -
:
) 6 cm2 ) 12 cm2 ) 3cm2 ) 27 cm2
-
.
8.2.13.
1= 2 cm
20 , 2
= 0,2 cm -
.
1= 0,5 m/s,
:
) 1m/s ) 2 m/s ) 2,5 m/s ) 5 m/s
-
.
8.2.14. -
4:1. 1,
1 2,
2
1 2
, :
) 1
= 42 1
= 42
) 2
= 41 2
= 41
) 1
=2 1
= 42
) 1
=2 2
= 41
-
.
8.2.15. 10 m3/s
8,64 107 m3. -
:
) 100
) 120
) 150
) 200
-
.
8.2.16. -
-
4 m/s. g = 10 m/s2,
:
) 0,4 m ) 0,8 m ) 1m ) 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
-
.
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 = 10 m/s2 1Atm = 105 Pa. -
h , :
) 5 m ) 10 m ) 15 m ) 20 m
-
.
8.2.25. -
10 m/s. ,
1,2 kg/m3,
, 1 tm = 105 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
) h2
> h1
> h3
) h2
< h1
< h3
-
.
8.2.27.
. ,
.
, :
) h1
= h2
) h1
> h2
) h1
< h2
-
.
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 -
. , 1m 0,6 m, -
;
1.020 kg/m3, -
1Atm = 105 Pa, g = 10 m/s2.
8.3.2. = 20cm
= 1.000 kg/m2 = 920 kg/m3.
d1
= 10 cm, d2
= 20 cm.
) - ;
) ;
) ; (
.)
g = 10 m/s2.
8.3.3. -
= 1.000 kg/m3 = 800 kg/m3 .
d1
= 30 cm, d2.
103.400 Pa.
) - ;
) d2
;
) F1
-
= 2 cm2
- .
)
F2
= 34 N.
1Atm = 105 Pa, g = 10 m/s2.
8.3.4. 4 cm2.
d1
= 0,1m. -
. -
2 , d
, ;
1.000 kg/m3, g = 10 m/s2.
8.3.5. U -
. = 1.000 kg/m3 = 900 kg/m3.
- d = 0,1m.
) .
) ;
) ;
1Atm = 105 Pa, g = 10 m/s2.
49
8.3.6. h = 10 m R = 20cm.
) ;
)
.
1.030 kg/m3, g = 10 m/s2.
8.3.7. -
U,
= 13.600 kg/m3.
.
p.
= 154.400 Pa, h.
1Atm = 105 Pa, g = 10 m/s2.
8.3.8.
m = 2.000 kg.
) F1
-
m;
)
d2
= 0,01m;
) F1, F
2. ;
R1
R2
= 8R1, g = 10 m/s2.
8.3.9. ,
A1
A2
= 5A1. -
,
k .
.
m = 2 kg, -
= 0,1m.
) k ;
) .
) (1) .
g = 10 m/s2.
8.3.10. 1
= 10cm.
1
= 4 m/s.
) ;
) , 2
= 5cm. -
;
) , ;
8.3.11. 1,44 10 3 m3/s. -
1
= 0,4 m/s.
) 1
;
) .
) 0,192 m .
g = 10 m/s2.
50
8
8.3.12.
. 3Atm -
8 m/s, 2,84Atm.
, A1
= 10 cm2
A2
= 20 cm2 .
) ;
) ;
) h , ;
)
, , F1
( )
F2
( )
.
)
.
= 1.000 kg/m3, 1Atm = 105 Pa, g = 10 m/s2.
.
8.3.13.
. 1,82Atm -
6 m/s, 10 m/s.
5 cm2.
, h = 5 m.
) ;
) .
)
, , W
V = 10 3 m3.
= 1.000 kg/m3, 1Atm = 105 Pa, g = 10 m/s2.
8.3.14. -
= 8 10 3 m3/s .
A = 10 3 m2.
) , ;
) , 4 tm 3 tm ,
h , ;
)
, , W -
m = 1kg.
= 1.000 kg/m3, 1Atm = 105 Pa, g = 10 m/s2.
8.3.15. = 48 10 4 m3/s. -
A1
= 12 cm2 (
) A2. 2,2Atm.
) 1
;
) 2
;
) 2
.
= 1.000 kg/m3, 1Atm = 105 Pa.
51
8.3.16. , 30 m/s.
) p ;
) F A = 100 m2;
) ;
, 1,2 kg/m3, 1Atm = 105 Pa, g = 10 m/s2
.
8.3.17. = 2,4 10 3 m3/s.
A1, A
2 ,
1= 2
2.
1,26Atm, 1,2Atm.
) 1,
2 , .
) 1,
2 , .
= 1.000 kg/m3, 1Atm = 105 Pa.
8.3.18. Ventouri
. A1, A
2
A1
= 2A2
h = 0,15 m, h
, .
) , ;
) 1
;
) ;
= 1.000 kg/m3, g = 10 m/s2.
8.3.19. .
h = 0,8 m -
3cm2.
) ;
) 4,8 L;
g = 10 m/s2.
8.3.20. . ,
h = 0,8 m , -
A = 20 cm2,
6 10 3 m3/s.
) .
) h ,
.
) A1
= 1,5 m2,
( ).
g = 10 m/s2.
8.3.21. . -
, h = 1,8 m ,
10 cm2,
8 10 3 m3/s.
) .
) ;
= 3,2 m, g = 10 m/s2.
52
8
8.3.22.
H = 1,25 m. , h = 0,45 m
,
20 cm2, .
)
.
)
;
) ;
) ;
g = 10 m/s2.
8.3.23.
h = 0,9 m. , d
, , .
) d
Smax
;
) .
)
. ;
g = 10 m/s2.
8.3.24. H = 5 m. -
1Atm 0,04 m2.
) ;
) ;
) h = 1,8 m;
= 103 kg/m3, g = 10 m/s2, 1 Atm = 105 Pa.
8.3.25. = 0,2 m2 -
. -
-
10 m/s.
) p ;
) F ;
) ,
;
.
= 1,2 kg/m3, g = 10 m/s2.
8.3.26. 0,8 kg/m3.
100 m/s 80 m/s.
) p ;
) = 60 m2, .
g = 10 m/s2.
8.3.27. U .
18 m/s .
;
= 900 kg/m3, .
= 1,2 kg/m3,
g = 10 m/s2, 1 Atm = 105 Pa.
53
8.3.28. . -
.
1
= 25 m/s.
) ;
) .
) h -
.
= 103 kg/m3,
.
= 1,2 kg/m3, g = 10 m/s2, 1 Atm = 105 Pa.
8.3.29. -
= 0,02 m2. -
= 4 m/s.
8 104 Pa.
) h2;
) ;
) h1;
) h1
h2.
h1
.
= 103 kg/m3, g = 10 m/s2, 1 Atm = 105 Pa.
8.3.30. A = 40 cm2 -
h = 10 m. = 4 m/s.
) ;
) 500 kg .
)
, 500 kg ;
= 1.000 kg/m3, g = 10 m/s2.
