ΦΥΣΙΚΗ Γ-ΛΥΚΕΙΟΥ ΚΑΤΕΥΘΥΝΣΗΣ 2008 ΟΕΔΒ εναλλακτικό
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ΦΥΣΙΚΗ Γ ΛΥΚΕΙΟΥ ΚΑΤΕΥΘΥΝΣΗΣ2008 ΟΕΔΒ εναλλακτικό ΒΙΒΛΙΟ
Transcript of ΦΥΣΙΚΗ Γ-ΛΥΚΕΙΟΥ ΚΑΤΕΥΘΥΝΣΗΣ 2008 ΟΕΔΒ εναλλακτικό
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2000 . . . , . . () . . .
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, 2008
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:
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, .
, .
, .
,
, .
:
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:
, , , , , :
:
John Bardeen, William Shockley
Walter Brattain, 1947.
.
iv
, . . (),. , . , . , . , . , . . , . . (), . , . , . ,. , . , . . . . . . . , .. . , . . , .. . .. ... , COSMOTE, ... General. Electric
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33
3.13.1
.................................................................................................................................... 3
LC .................................................................................................................. 3
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...... 23
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.
Fourier ................................................................................................................................... 28
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- ............................................................................................................. 43
3.23.2
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v
-
Huygens ...................................................................................................................... 50
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Young ........................................................................................................ 67
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- ............................................................................................................. 88
44
4.14.1
.................................................................................................................................... 95
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vi
-
.......................................................................... 97
............................................................................................................ 97
Bernoulli ................................................................................................................... 99
Torricelli ........................................................................................................ 102
Bernoulli ..................................................................... 104
.......................................................................................................................................... 107
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- ............................................................................................................. 118
4.24.2
.................................................................................................................................... 121
- ................................................................................................... 122
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.......................................................................................................................................... 124
................................................................... 126
- ...................................................... 127
- ( Steiner) ..... 129
Steiner ............................................................................... 131
( ) ... 133
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............................................................................................ 136
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- ........................................................................ 142
vii
-
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: ....................................... 159
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4.34.3
.................................................................................................................................... 171
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, , ..................................... 193
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viii
-
Doppler .................................................................................................................... 201
Doppler .................................................................................. 205
.......................................................................................................................... 207
........................................................................................................................... 208
................................................................................................................................... 210
- ............................................................................................................. 216
4.44.4 EI
.................................................................................................................................... 225
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Michelson - Morley .................................................................................... 229
Michelson - Morley .............................................. 231
.......................................................................... 233
......................................................................................... 234
Lorentz ........................................................................................ 236
....................................................................... 237
: .................................................................................................. 238
- ...................................................................... 242
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ix
-
.......................................................................................................................... 273
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- ............................................................................................................. 279
4.54.5
.................................................................................................................................... 285
................................................................................................ 285
........................................................................................................ 288
- ....................................................... 288
Einstein ..................................................................................... 291
- .................... 292
Compton ............................................................................................................. 295
- Bohr () ................................................................... 298
De Broglie .......................................................................................................... 299
........................................................... 301
Schrdinger .................................................................................................................. 302
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() ......................................................................................................... 314
() ............................................................................................................ 315
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x
-
, , .
~ : ; : . , . . , , , .
. . ,, .
. , , , , , . . . ~ , , , , , . ~ , , ,.. .
, , . , .
~ , . , , (.. , ) , .
. . , .
, , Timeo Hominem unius libri, ( )* . ,
* OHANIAN, . .
xi
-
. , , . , . o . . .
(SI) SI. . , . , . , . resis-tance reactance , , , . mol (o), , . , , . ( ), (molar)(;) , . resis-tore, , , . , . , . , . . . , , u = 5 2 /8 m/s, u = 3,24 m/s. , , , . , .
. 3 .
~ , , , . . . . ~ . 1984 IUPAC ( International Union of Pure andApplied Chemistry, ~ ), Comimission on Thermodynamics ( ), 1 atm (101,325 k Pa) 100 k Pa (1 bar). 22,4 L 22,7 L.
273,15 (0 oC), 273 .
xii
-
101,325 k Pa .
. ., , .. . . . , , .
. =0 cos , 0
.
.
, , 0
q q q/0
.
.
, , , , . , , , , ., , , , , .
.
.
xiii
GB.
-
xiv
-
-
3
-
3.1
~E m (.3.1), , . , . - . T k x, x k, .
. . , r0 (. 3.2). r = r r0 , r r0 , .
C L, . L, C
LC
C, ( ) L . . ~E , , Qm .
t = 0 (. 3.3), , . , . , m . A , , . Qm . t = 0. . .
3
3.1
.
3.2
.
-
q i . C,
q, . i -
R R
3.4.i
RR=
C
qC = 1
4 T
3..3
L, C .
3.4
R .
MA
f (t), t t = t2 t1 , f (t) f = f (t2) f (t1) . f (t) (t1, t2) ,
() t, .
f
t
f
t
f t f t
t t=
2 1
2 1
b g b g
-
m k 3.1, x F = kx.
(3.1)
LC 3.5. C( )
, C = L .
,
L L it
= dd
q
CC =
m
tkx
d
d=
F m m
t= = d
d
5
f (t)
, t .
, t, -
f (t).
t t f .
. :
d cos
dsin
t
t t
+ = +b g b g
d sin
dcos
t
t t
+ = +b g b g
f f t t t
-
(3.2)
(3.1) (3.2) .
(3.3)
H (3.1)
,
, , (3.3)
(3.4)
q: Qm : :
0: , .
(3.4)
(3.5)
(3.6)
t = 0 , Qm 0 = 0 (3.4) (3.5)
(3.7)
q i, , 3.6.
, T
= 2
i I t= m sinq Q t= m cos
I Q I QL C
m m m m= = 1
i I t = +m sin 0b g
iq
tQ t = = +d
dsinm 0b g
q Q t LC
= + =m cos ( )0 1,
k
m=x x t = +0 0cos b g
q x
L m
Ck
i
RS||
T||
UV||
W||
1
Li
t Cq
d
d= 1
q
CL
i
t= d
d
6 T
3..6
q i
-
(3.8)
(3.3) U LC
,
, .
q, i (3.7)
(3.9)
U = UE + UB
(3.9) (3.6)
(3.10)
, (. 3.7).
UC
Q L I= =12
1 1
22 2m m
UC
q L i= +12
1 1
22 2
UC
Q t
U L I t
E
B
==
UV|
W|
1
2
1
1
2
2
2 2
m2
m
cos
sin
U L iB = 12
2
UC
qE = 12
1 2
K m= 12
2U k x= 12
2
T L C= 2
7
3.7
.
UUEU
-
3-1
, , C = 10 F L = 0,10 H. A
= 100 V , : ) , ) ;
) = 100 V
rad / s = 1,0 103 rad / s
.
, (3.7)
) UB = UE , UB + UE = U 2UB = U
(),
(. 3.8).
,
3-2
LC, . i - q.
t t
t t1 1 1
3 410 7 9 10= = = /410
s =
4s = s
-3,
sin
m
m
I
I=
= =
2
2 2
2 4
i I t t= = +m sin sin 1 b g
iI= m 2
22
1
2
1
22 2L i L I= m
i t i t= 1 sin 10 A, s3e j b g,
q t q t= 10 3 cos 10 C, s3e j b g,
I Q I Im m m m A A= = =1 0 10 10 1 03 3, ,
LC
= =
1 1
0 10 10 10 6,
Q C Q Qm m m C C= = = 10 10 10 1 0 106 2 3,
8 T
3.8
-
UE + UB = E
,
Qm Im . 3.9.
; ,
i q . q i, Qm m .
, . , .
, , .
, 3.10. - , , . , ( ),
i
I
q
Q
2
2
2
21
m m
+ =
i
Q
q
Q
Q
Q
2
2 2
2 2
2 2
2 2
2 2m m
m
m
+ =
i q Q2 2 2 2 2+ = m
i Q q2 2 2 2 2= m
i Q q2 2 2 2= me j
i Q q i Q q2 2 2 2 2 2= = m me j
1 2L C
=iL C
Q q2 2 21= me j
1
2
1 1
2
1
2
12 2 2C
q L iC
Q+ = m
9
3.9
3.10
.
-
. () , .
. ,
F = b (3.11)
, , . b ( ) . .
F = F + F
F = kx F = b F = kx b
m = F
(3.12)
, . (3.12)
(3.13)
,
, , .
. , U ,
(3.14)
t = nT, n = 1, 2, 3, ...
1
22k A U=
k
m=
x A e t
b
m
b
m
t= +=
=
U
V
|||
W
|||
cos b g
2
4
22
2
m k x b =
10 T
3.11
.
3.12
.
-
(3.15)
x, t, b (b1 < b2 < b3 < b4 ) (3.13) 3.13.
A (3.13) .)
. -
b. b .
)
b , b, . x(t) [x (t + T ) x (t) ], cos (~t + ) . . , .
)
~ , , . ' , , ( ) ( ).
3-3
, R cm, b, 20 C,
cm,
. k = 36 Nm
b = 6 0 10 4, N sm
b
m
22
24 b2 > b1 3 max < A 2max < A 1 max 3 < 2 < 1 < , 1 , 2 , 3 .
(3.16)
x A t a
A
F
b m k
a
m k
b
= =
+ FHGIKJ
=
U
V
|||||
W
|||||
sin
tan
d
d
m
dd
dd
b g1
22
ma k x b F t= + m dcos
14 T
3.13()
- .
-
().
x (t), t, (3.16),
(3.17)
(3.18)
= 0 cos (d t ) F = Fm cos d t , .
0 d , (3.18) 3.15. (3.18),
k
md =m k
d
d
=
m k
d
d
FHGIKJ =
2
0
F
b m k
0
22
= =+ FHG
IKJ
dm
dd
t a= 0 cos d( )
t a= d dcos b g
x
t= d
d
= bm
22
22
15
3.14
.
3.15
d .
-
(3.19)
(3.16) tan , , tan = 0 = 0, .
, P = F
( 3.16) .
, ( ), ( ), : ) ) .
3-4
m. - 45 . (). 10 m/s2.
A
,
= d
LC . , R , Joule. .
3.18
l = 0 45, ml = 104 712,
m
d rad / s= 4 71,
fd 45
60rad / s= = 2 2
l = gd
2
g=l
P F t= m 2 dcos0
d = ( )
16 T
3.17
3.18
M LC.
3.16
T
-
(3.20)
,
(3.21)
(3.21) (3.12),
(3.13)
, (. 3.19).
.
) ()
) () R
) () .
3-5
L = 5,0 mH C = 2,0 F. R = 1,0 . .
R
L
22
24 2
= , 1 < 2
A A A= 1 2
(3.33)
cos
cos
tansin
cos
x A t
A A A A A
= += + +
= +
U
V||
W||
b g12
22
1 2
2
1 2
2
x A t A t = + +1 2cos cos b gx x x= +1 2
x A t 2 2= +cos b gx A t1 1= cos
24 T
3.24 3.24
.
3.23
-
x = (A1 A2 ) cos t 1 > 2 x = (A2 A1 ) cos (t + ) 1
-
[x2 m, t s]
1
. 1 = 5 m, 2 = 5 m .
( . 3.27)
.
)
. 1 = 2 = f1 f2 . f1 f2 f1 f2.
x = x1 + x2 = A cos 1 t + A cos 2 t = A (cos 1 t + cos 2 t)
(3.34)
(1 > 2 )
. t = 2 A cos mod t .
x (t), t, At cos av t .
av = + 1 2 1 22
mod = 1 22
x A t t= 2 cos cosmod av
x A t t= +2
2 2
1 2 1 2cos cosb g b g
x t= FHGIKJ5 cos 100
3
= 3
A A A A A = + + = + + + FHGIKJ =1
222
1 22 22 5 5 2 5 5
1
25cos m m
= 23
2
3rad
x t2 52= FHGIKJcos 100
3
x t t2 5 5 1006
= FHGIKJ =
FHG
IKJsin 100
6cos
2
26 T
3.27
-
3.28() av , t . 0 2. 3.28
(3.35)
f = f1 f2 (3.36) f1 < f2
f = f2 f1
.
f f
= 1
2 1
f
= 1
f f
= 1
1 2
1 2
2
=T mod =
27
3.28
.
-
, , . (. 3.29).
(. 3.30). () 2, () , : () , () , , () , () . (), () ... , , ' ,
g , l , k . . ' , . k, 1 2.
g k
M2
2= +l
g
1 =l
28 T
3.29
, .
3.30
.
. FOURIER
x1 = A sin 1 t x2 = B sin 2 t, 2 = 2 1.
, x1, x2 , ( )
x A t B t= +sin sin1 2x x x= +1 2
-
29
x1 , .
1 = 0 ,2 = 2 0 , 3 = 3 0 , .... 0(),
. 1822 J.Fourier , , Fourier, : f(t) 0 ,
, f0 ,
( )
, f (t + T0 ) = f (t) t
0 , 1 , 2, ..., 1, 2, 3, ... f(t).
, f (t) = a0 + a1 cos 1t + a2 cos 2 t + ... + b1 cos 1 t + b2 cos 2 t + ...
0, 1, 2, ...., b1, b2, ..... . Fourier.
sin sin cos sin cosa a a+ = + b g
3 03= 2 02= ,
10
02= = ,
f t A A t A t A t b g b g b g b g= + + + + + + +0 1 1 1 2 2 2 3 3 3sin sin sin . . .
00
2= fT
00
1=
T
00
2=
T T
= =11
2
-
) .
, y' y xx'
xy, x , Ay , 1 2, . .
) , 1 = 2 = i) = 0
,
. r (. 3.31)
' .
ii) ,
x A t
y A t y A t
x
y y
== +FHG
IKJ =
UV|W|
cos
cos
2 sin
= 2
r A Ax y02 2= +
r r t= 0 cosr A A tx y= +2 2 cosr x y= +2 2
y x=xy
A
A x
y
= = 1
y A ty= cosx A tx= cos
y A t y= +cos 2b gx A tx= cos 1
30 T
Fourier , . , , , , . Fourier 0 , 1, 2 , ...,b1 , b2 , ... . , , , .
3.31
, .
-
,
(3.37)
(3.37) . ' 2x 2y . ( 3.32)
iii) ,
. 3.33
) .'
, Lissajous. Joules Antoine Lissajous, 1857. 1/2 x=Ay, , 3.34.
: , 1 /2 ,
, .
A
A
y
x
= 2AA
y
x
= 1
= 32
x
A
y
Ax y
2
2
2
21+ =
si n
cos
2
2
tx
A
ty
A
x
y
=
=
2
2
2
2
31
3.32
, /2.
3.33
.
-
, , ... 3.34 ( ) .
.. ,
, "" 2Ax 2Ay , . Lissajous , , .
, 3.34.
Lissajous . , . Lissajous. , . .
1
2
3=
1
2
32 T
3.34
Lissajous .
-
33
R ~ LC
q = Qm cos (t + 0)
i = Im sin (t + 0),
R F = k x Fa = b
x = A e t cos (~t + )
,
R ~ F = Fm cosd t,
x = A sin (d t ) = 0 cos (d t ),
0 = d
, , , (, ) d = , .
R LC R
R RLC = V cos d t. q i
i = Im cos (dt) Im = d Qm . K
d = ,
R
LC
= 1
q t Q t
Q
V
R LC
LC
R
b g b g=
=+ FHG
IKJ
=
U
V
||||||
W
||||||
m d
md
dd
dd
sin
tan
1
1
1
22
= RL
22
24
q Q e
R t
L= +m cos2 b g
tand
dam k
b=
AF
b mk
=+ FHG
IKJ
m
dd
22
k
m=
= bm
22
24
b
m=
2
U L i L I t B = = +12
1
22 2 2
0m sin ( )
UC
qC
Q t E m cos ( )= = +12
1 1
2
12 2 20
LC
= 1
drasthriothtesA N A
-
34 T
1. BARTON
8 , 0,25 m 0,75 m. , , . -, . . , 0,50 m , , .
2.
.
1 2. .
3.
.
drasthriothtes
x q m L i b Rk Fm Vm
RLC
,
.
r
0=2
p b = r2
P I R= r2
1
C
-
35
. . , . - . ( , ).
4.
- . -. 45 / . , (
k ), .
(: k
l0 , k0 -
l .
.
.)
5. LISSAJUS
. , 400 Hz 200 Hz. , . , Lissajous. , 1:3, 2:3, 3:4 . .
6.
.. , (.. ). ,
, g
l , k .
g k
M2
2= +l
g
1 =l
1
2450
/
k
m 02. F1 = F2 = F0 cos t ;
30
, . ; .
31
. , b1 . 1. b2 = 2b1
, .() 2 = 1,() 2 = 2 1,() 2 = 1 / 2,
() 2 =
;
32
- , m k, , (b 0), . b. F, , .
33
RLC R R. , () ,() () ; .
34
RLC, t, ,
( ) .
A1
2
-
41
35
RLC, , , ;() (
7/1000 )
()
() Joule, ,
36
:
()
()
()
;
37
: () . . . . . . . () . . . . . . R () . . . . . ..
38
, VR , RLC .
39
34 ( ) .
40
R . , .
41
RLC . () = 100. . q - t. ; . . 7/1000 .
42
1 2. 1 2 . 2 1. LC;
-
42 T
.
43
: () 1)
.
() 2) , - , -
) - 3) ,
) 4) -
44
RLC:() C() L()
R() ;
45
RLC, , . ; RLC, .
46
RLC = V cost. ( ) , ;
() .
() .
() .
() i q
, q = C . :
q = C + q0 cos (t )) .
47
= A cos t = B cos t. ;() 0, () /2, () /3, () ; > 0 > 0
48 = A cos t = B sint. ; (, ).
-
43
() 0, () /2, () 3 /2 ()
49
1 2 , () :
;
50
) ; ;
) ;
51
3.32 . (: ( x , 0) y < 0).
52
, , ;
mod = 1 22
1
LC L = 10 mH C = 1,0 pF. ) , ) .
2
L = 2 ,0 mH C1 = 30 F C2 = 60 F . .
3
q0 = 2,0 Cb, U = 2,0 J. 1,0 mA. L .
4
LC
t = 10-4 s. L = 10 mH.
5
LC . t = 0 Q = 2 C.
,
. C = 2 nF.
6
LC V = 300 V. L = 20 mH C = 2,0 F. . :() () () di /dt.
7
qQ= 3
2
3st = 10 4
-
-
, q . U q U q.
8
LC L = 2,0 mH C = 20 nF t = 0, Q = 2,0 Cb. t .
9
L C. . C1 = 3 nF C2 = 8 nF ,
f1 = 25 kHz -
. . 2 = 10
10
. LC , 1 = 5 10- 6 s 2 = 2 10-6 s . , C1 C2
C1 C2 = 10- 8 F2.
11
, m k, . b.
12
,
.
) . )
.
13
m = 0,10 kg. ,
.
.
14
t = 7,0 s. ;
15
b, 3.10 . k k . d1 d2
. -
16
k 100 N/m m 4,0 kg. , . 0,80 m. b = 0,40 kg/s : ) .) 1/5 . ;
bk m
=
4 2
1
2
2
e j
d
d= 1
2
b = 0 6, kgs
k = 100 Nm
= 12
f250
3= kHz
44 T
-
45
17
(S.I.).
k = 56 N/m, m = 1,0 kg
, :
() ()
()
.
18
, /4. , 20 watt.
19
f1 = 9,0 Hz f2 = 4,0 Hz , . .
20
LC C = 19 F L = 2,5 m. L C 9/10 ;
21
C = 8,0 F V = 320 V L = 5,0 mH R = 0,30 . t = 0 . .
22
RLC, , 25ln2 s,
0 = 5,0 104 rad/s. .
23
200 V i = 2,0 A. R = 5,0 L = 2,0 mH, :) , )
) .
24
. R R = R,
; f0 = 6 105 Hz.
25
x1 = 3 sin 10 t x2 = 4 cos 10 t
;
26
x = 3 sin 10 t y = 3 cos 10 t
, .
33 10 5 Hz
b = 1 kgs
F t= 2 2 8cos
-
46 T
() .
-
3.2
~ , . . , , . ~ , , . .
, . , .
, , ( ).. , , ( ).
, : ) , ( ) .. . ) ( ) .. ) , , .
. , . , .
( ). ( ) ( ) . () , x, y(x, t)
y (x, t) = A0 sin (t kx) , y (x, t) = A0 sin (t kx + ) (3.38)
(t kx + ) , x = 0 t = 0 ( [0 , 2), 0 < 2).
KYMATA 47
3.35
.
-
(3.38) x.
y (x, t) = A0 sin (t + kx) (3.39)
k ( ) () .
(3.40)
.
= 2 ff ( v) ,
.
3.36, . ,
t1 kx1 + = t2 k x2 + (t2 t1) = k (x2 x1)
~
(3.41)
,
. ..
(3.42)
F , ( ) ( ).
F
=
= A B/
= f
k=
x
t
k=
k
= 2
48 T
3.36
t1 t2 (t1 > t2). .
-
, (.. ). y (x, t) Fourier () .
, , , . , , ( ). . .
3-8
m = 2,0 kg L = 10 m. 800 , .
, ,
F .
-
, ,
r .
, , . t kr = . , t r = . , , , . (. 3.38).
, , . , , .
y r tA
rt r,b g b g= sin
st 0 16, st = 2 0 10800
,
tm L
F=t L
=
F L
m= F
=
KYMATA 49
3.38
.
3.37
-
, (. 3.39).
, , (. 3.40), . , .. .
HUYGENS
Huygens, , , . Huygens, .
t1 S1 (.3.41). S1
. t2 S2 (), (t2 t1 ). ~, ~, ~, ... ( ) S2 , t3 S3 , (t3 t2) ... , , , ... ~, ~, ~,..., .
50 T
3.41
Huygens.
Huygens
, . Huygens , , . Huygens Fresnel.
3.39
.
3.40
. S 1,S 2, ... .
-
, , . , , . ( ).
, .
3.42 , 1 2 1 2 . 1 2 :
) 1 > 2 , . (. 3.43).
) 1 < 2 , (. 3.44).
) , 2 , .
, , ' , 3 ( ) , . (. 3.45). . , x x = 0, , ,
y1 = A0 sin (t kx) ( x) y2 = A0~ sin (t + kx) ( x)
x = 0, y1 + y2 = 0,
0 sin t = A0~ sint,
0 sin t = 0~ sin (t + )
.) ,
, . 3.46 , - . ~
KYMATA 51
3.42 .
3.44
(1 < 2).
3.43 (1 > 2 ).
3.45
.
3.46
.
-
, , .
( 1), (1) ( 2), , .~ 1, 1~ 2 , (. 3.47) ,
. :
:
(3.43)
: n21 (2) (1),
(3.44)
, 1, 2 (1)
(2) . Snell. , , ,
, , .
. .
n
2 1
1
2
=
sin
sin
n1
22 1=
1 = 1~
52 T
3.47
.
-
Huygens. , , ~ - Fermat , , ().
,
c , , . (2), (1),
3-9
. ( ) 2.
.
= 2 + 2 ( )
2 + 2 = 180 2y + 180 2x = 360 2(x + y)
x + y = 180o
= 360 2 (180 ) = 2 .
nn
n2 1
2
1
=
nc
=
KYMATA 53
3.48
. (0 = 589 nm)
1,309 1,544 2,417 ( ) 1,52 - 1,80
. 20 CM 1,329 1,333 1,36 1,473 1,501
-
3-10
( ) d. s. n1 n2 .
= , , , sin = sin , = . , , .
s
s = AB sin = sin ( ) s = AB sin ( )
sd
=
cossin cos cos sin
AB =cos cos
d
d
=
sin
sin
sin
sin
=
sin
sin
n
n= 2
1
sin
sin
n
n= 1
2
sin
sin
n
n= 2
1
54 T
3.49
-
1, 2 3 3.50,
. ~ , S1 . ,
, 1 .
t, = t =
BH=2
HE BH= = FHGIKJ t t 1 1 1
t
= 1
s d n
n n =
LNMM
OQPPsin
cos
sin1 1
22
12 2
s d
n
n
n
n
=
L
N
MMMMM
O
Q
PPPPP
sin
cos
sin
1
1
1
2
12
22
2
cos sin sin n
n= = 1 12 1
2
22
2
sin sinn
n= 1
2
s d
= sin
cos sin
cosL
NMOQP
KYMATA 55
3.50
.
-
S1~. , . 1 = 1~ . , 1 = 1~ 1 1~ (. 3.51).
(. 3.52) ,
,
,
S2 ( ).
~
1 = 1 2 = 2 , .
sin
sin
1
2
1
2
=
sin
sin
1
2
1
2
=sinsin
1
2 1
2
= =
sin
2 =sin
1 = ,
HEAZ=2
HEBH
= FHG
IKJ = FHG
IKJ = t
2
12
1 1
2
1
1
2 2
AZ = = t
21
2
t
=1
HE
2
2= =
56 T
3.52
.
3.51
-
(1) (2) (. 3.53),
(1) (2 > 1 ), ( )
c , 2 = 90.
c , . . c 2 = 90
.
(3.45)
3.53 , ( 1) (2). , (2) (1) 90 , c .
. , , , (. 3.54). ' , ' . , ( ) .
3-11
c ( ), - , () n1 1,33 n2 1,00. h = 5,20 m 60 , ;
c
sin c
= 1
2
sin
sin
co
n
902 1=
KYMATA 57
3.53
- .
3.54
( ).
3.55
-
sin c = 0,752 c 48,8 60,
L = 2h tan60o, .
, . , , , (. 3.56).
uygens. . , , , .
. 3.57 , . , , .
, . .. .
, (. 3.58,3.59).
sin cn
n= =2
1
1
1 33,
58 T
3.56
3.57
1882 - 1962. . - - .Sommerfeld. , (1909). 1910 P. Debye - ,. . , . 1912. .
-
. .
( ) : ., '
y1 (x , t) y2 (x , t),
y (x , t) = y1 (x , t) + y2 (x , t)
.
. , , .. , , ().
~ , .
y1 = A0 sin (t kx),y2 = A0 sin (t kx + )
y = y1 + y2 = A0 sin (t kx) + A0 sin (t kx + )
KYMATA 59
3.59
.
3.58
.
-
(3.46)
,
(3.47)
= 2n,
n = 0, 1, 2, ... = 20. .
= (2n + 1)
n = 0, 1, 2, ... = 0. (. 3.60).
3-12
.
y1 = 5 sin (4x 2 t)
(y1 , y2 , x cm, t s)
.
y t x2 5 2 46
= +FHGIKJcos
cos
20=
cos
21=
A A= 22
0 cos
y A
t kx= +FHGIKJ2 2 20 cos sin
sin sin cos sin + = +2
2 2
60 T
3.60
() , (.. = 0) () (.. = n).
-
A
y = y1 + y2
[ y, x cm, t s]
,
x.
.
. , .
(x = 0) , 3.61. ,
y1 = A0 sin (t + kx)
5 3 cm 8,66 cm
y x t= +FHGIKJ5 3 4 2 6sin
y x t= +FHGIKJ10 4 2 6cos
6sin
y x t x t= + +FHGIKJ5 4 2 5 4 2 3sin sin
b g
y x t2 5 4 23
= +FHGIKJsin
y t x t x2 5 2 4 52
2 46
= +FHGIKJ = +
LNM
OQPcos
6sin
KYMATA 61
3.61
-
, x = 0, ,
y2 = 0 sin (t kx) ()
y = y1 + y2 = A0 [sin (t + kx) sin (t kx)]
(3.48)
, . , , . ,
(x) = 2 A0 sin kx (3.49) x
sin kx = 1
(3.50)
n = 0, 1, 2, ...
20 .
x ,
sin kx = 0 k x = n
(3.51)
n = 0, 1, 2, ...
, (. 3.62). .
, x = 0, x = L , ' (. 3.63 - 64). , y = 0 x = 0, y = 0 x = L. (3.48) ,
2 A0 sin kL cos t = 0 sin kL = 0 kL=n
x n
=2
x n
= +2 4
k x n = + 2
y = 2 A0 sin k x cos t
sin sin sin cos = +2
2 2
62 T
3.62
2, 2, 2,... . 1, 1, 1,... .
-
,
(3.52)
n = 1, 2, ...
, (3.52).
()
n = 1, 2, 3, ... (3.53)
L: F: ( ):
,
(3.54)
. fn n - . , , (. 3.65).
, , (. 3.66), :
) , , , .
) f, , .
) x = 0,
, n = 0, 1, 2, 3, ...
) , ,
f=x n =
2
fL
F
1
1
2=
f nL
F
n = 1
2
F
=
f
n
n
=
L
nn = 2
2
L n
n
=
KYMATA 63
3.63 - 64
.
3.65
.
3.66
.
-
.' . , , .
.
3-13
L = 1,00 m ,
(x m, y cm t s)
m = 0,01 kg () :) .) )
.
)
y = 2A0 sinkx cos t
= 2 f
20 = 4,00 cm 0 = 2,00 cm
n = 5
5 .)
F = 625 NF
0 1250
,=
5 1250F
=5 1250
L
F
t t=
n
Lx x
= 5
y An
Lx
n
L
F
t= FHG
IKJFHG
IKJ2 0 sin
cos
n
L
F
= n
L
F
= 2
2
kn
L= k
L
n
= =2 22
y x t= 4 00 5, sin cos 1250 b g b g
64 T
-
) y1 = A0 sin (t + kx) y1 = 2,00 sin (1250 t + 5 x)
y2 = A0 sin (t kx) y2 = 2,00 sin (1250 t 5 x)
, , . .
) . , . 3.67.
n = 1, 2, ... (3.55)
, n = 1, 2, ... (3.56)
L .
-
.) :
. 3.68.
, n = 1, 2, ... (3.57)
, n = 0, 1, 2, ... (3.58)
L .
f
L1
4=
f n
Ln2 1 2 1
4+ = +b g
f
n
n2 1
2 1+
+=
L
nn2 1
4
2 1+ = +
L
nn2 1
4 2 1+ = +
f
L1
2=
f n
Ln =
2
n
Ln = 1 2
L
nn
2=
KYMATA 65
3.67
.
3.68
.
-
,
f3 = 3 f1 , f5 = 5 f1 , f7 = 7 f1 , ...
. .
3-14
L = 0,75 m. , :
) . ) .
) ,
n = 1, n = 2 n = 3
f1 = 227 Hz, f2 = 453 Hz f3 = 680 Hz
) ,
n = 1, n = 2 n = 3
f1 = 113 Hz, f3 = 340 Hz f5 = 567 Hz
, .
, S1 S2 3.69. , , . , S1 S2, (1 t k r1 + 1) (2 t k r2 + 2 ) . , 1 = 2 f1 = f2 , .
( ) , r1 r2 . , , P, , . S1 S2 ,
f n
Ln2 1 2 1
4+ = +b g
f n
Ln =
2
340 ms
66 T
3.69
-
y1 = A 0 sin (t k r1)y2 = A 0 sin (t kr2)
y = y1 + y2
(3.59)
r2 r1 = n, n = 0, 1, 2, ... (3.60) 2A 0 .
, n = 0, 1, 2, ... (3.61)
() .
, , . r1 r2 = . , . 3.70 () .
YOUNG
, . , 108 s, .
, , , ().
, , . ~, , , .
, , .
Thomas Young, 1801, .
r r n 2 11
2 = +FHG
IKJ
k r rn2 1
2
=b g + 2
cos 0k r r2 1
2
=b g
k r rn2 1
2
=b g cos = 1k r r2 12
b g
y Ak r r
tk r r= +FHG
IKJ2 2 20
2 1 1 2cos sinb g b g
KYMATA 67
3.70
.
-
3.71. 1 S0 . S0 S1
S2 2, . S1 S2 3, , (. 3.72).
E
3.73 3 D S1 S2 D >> , . x
-
r2 r1 = S2 B S2 BS1 S2 B = a sin , ,
r2 r1 = sin () x
-
D = 1,50 m. d = 13,1 cm. (c = 3,00 108 m/s).
f = 5,15 1014 Hz( 5,0934 1014 Hz)
, X, .. . , c.
. , LC ( Tomson) . (3.74) LC, . . , Tomson, , .
, (. 3.75).
, "" , . /2 , . ' , .
/2 ..." ".
.
f = 3 1 5 3 10
2 00 10 13 1 10
8
5 2
,
, ,
fD c
a d= 3c
f
d
D=
3
xd=3
c
f=
Dx=
70 T
3.74
3.75
-
) . , , 3.76. ,
(). , . , . , () . ~ , (), . .
, 3.77, , .
) : , . 3.78 .
, , .
( ) ( ). .
3.79 . , . x, .
(3.65)E E t kx
B B t kx
= =
0
0
sin
sin
b gb g
B
E
KYMATA 71
3.76
.
3.77
3.78
.
-
72 T
. Maxwell .
3-16
1000 W, 6 km () 5,66 10-2 V/m. 4 MHz () = c B, .
= 2 f = 8 106 rad/s (Hz)
= 0 sin (t kx)B = B0 sin (t kx)
, . .
B t x= 1 89 8 10 8 37 10 106 2 10, ,sin (S. I. )e j
E t x= 5 66 8 10 8 37 10 106 2 2, ,sin (S. I. )e j
k = 8 37 10 2, m 1
k
c= =
8 10
3 10
6
8
m -1c
k=
Bc
E0 02
8101 1 89
10
101 89 10= =
, ,T = T
3.79
.
-
, , , , . . ,.. . , , , , . , . (50 Hz) , , , .
. , . .. . ( ), . Fourier . Rngten .
KYMATA 73
, , , c, .
. (. )
: c: 0:
(, , ).r: : r
p0: .
()
I p
c rb g = 0
2 4
30
232 sin
22
-
() , , ' . .
, () , , () (FM - Frequency Modulation ), . .
74 T
, , ,
P = () S :
()
, .
, "" . () p0 p0 = q0l, q0 = e l 10-10 m . , P 10-74 4 .
1014 Hz P 10-18 W
m = Qm . :
p0 = Qml
()
, 0 = 20, l = 30 m
400 W. = 2 5 10 6 rad
s
PI
c= m
2 2 2
0312
l
pI
0 = m l
Pp
c= 0
2 4
0312
-
() , .
3.80 . . , 3 1011 Hz , , , . :
KYMATA 75
(nm) f (1014 Hz)
390 455 7,69 6,59 455 492 6,59 6,10 492 577 6,10 5,20 577 597 5,20 5,03 597 622 5,03 4,82 622 780 4,82 3,84
3.80 ()
(LF)
(VF)
(VLF)
X (LF)
M (F)
Y (F)
(VHF)
- (VHF)
Super (SHF)
E (HF)
Y
()
107-106 m
106-105 m
105-104 m
104-103 m103-102 m
102-101 m
101-1 m1-10-1 m
10-1-10-2 m10-2-10-3 m
0,7-10 m
0,410-6 m0,810-6 m
30 Hz
300 Hz
3 kHz
30 kHz
300 kHz
3 MHz
30 MHz
300 MHz
3 GHz
30 GHz
300 Hz
3000 kHz
30 kHz
300 kHz
3 MHz
30 MHz
300 MHz
3 GHz
30 GHz
3000 GHz
3.80()
-
3.80() .
, :
) : , (. 3.81).
, ' . ' . 30 kHz 3 MHz . ( ) 1500km, , VHF, Very High Frequency ( ).
) : , , ( 30 MHz), (. 3.82).
80 km 500 km ' , . . , .
, (Fade), .
76 T
3.81
3.82
-
) : . , , . VHF, UHF (Ultra HighFrequency, ) . 150 km. , , , . . (. 3.83).
(. 3.84).
(. 3.85) ( ) , ().
3.86 :
KYMATA 77
3.84 3.83
3.85 3.86
-
78 T
TH
3.80() , . . :
1) (ELF - Extremely Low Frequen-cies): 30 Hz 300 Hz. , () 50 Hz, .
2) (VF - Voice Frequencies): 300 Hz 3000 Hz. . 20 Hz 20000 Hz, , VF.
3) X (VLF - Very Low Frequencies): 3 kHz 30 kHz. (15 kHz 20 kHz) . VLF .
4. (LF - Low Frequencies): 30 kHz 300 kHz. . (). , , .
5. (MF - Medium Frequencies): 300 kHz 3000 kHz. ( ) (535 kHz 1605 kHz.), .
6. (HF - High Frequencies): 3 Hz 30 MHz . (), . , , . , , CB.
7. (VHF - Very High Frequencies): - 30 Hz 300 MHz. E . , , , FM( ) (88 Hz 108 MHz) ( 2 13). .
-
KYMATA 79
8. (UHF - Ultra High Frequencies): - 300 Hz 3000 MHz. UHF ( 14 83) ( ). , , , o .
9. (SHF - Super High Frequencies): - 3 GHz 30 GHz . .
10. (EHF - Extremely High Frequen-cies): F 30 GHz 300 GHz. , , . , . EHF .
: 1 GHz ,
R x
y 0 = A sin (t kx + )
= 2 f
R Huygens , ,
.
R ~ , . 1 = 1~, ( = )
1, 2 1, 2
sin
sin
n
1
221
1
2
= =
F
=
k
= 2
drasthriothtesA N A
-
80 T
.
,
c . , , . c
R , .
y = 2 sinkx cos t
n = 0, 1, 2, ... ( 20) ( ) . :
, n = 1, 2, 3, ...
L
, n = 1, 2, ...
:
, n = 0, 1, 2, ...
.
R
() :
r2 r1 = n n = 0, 1, 2, ..., ( )
, n = 0, 1, 2, ...
( )
R Young, x , D ,
R , . , . Maxwell. () :
= 0 sin(t kx)B = B0 sin (t kx)
, . , , , .
Dx=
r r n 2 11
2 = +FHG
IKJ
f n
Ln2 1 2 1
4+ = +b g
f n
Ln =
2
f nL
F
n = 1
2
x n
=2
x n
= +2 4
sin
c = 1
2
nc
=
-
KYMATA 81
, , . ,
, , .
, . , .
. - , .
. , . .
, .. 80 Hz. 500mV. ,
. . 1000 . , .
.
. , . d .
f
F
= 1
drasthriothtes
-
( ).
82 T
F . . , , .
, . 1 cm 1 cm, .
. , . .
, . .
.
, , , .. . , ,
. . /8, , . d . d : ,.. 100, , .
. . , . , . - 523,3 Hz (;). - , . , L0, 0,6 . L = L0 + 0,6 D ( ) . I
= 2 f (L0 + 0,6 D) .
, .
L
f= =
2 2
-
1
;
2
; () y (x, t) = 5 cos (3x 5 t)() y (x, t) = 2 sin2 (2 t x)() y (x, t) = 3 sin (10 t 0,1x) + 2 cos (10 t = 0,10 x)() y (x, t) = 6e- sin (2 t x)
() y (x, t) = 8-3 x sin (5x 3 t)3
.() y (x, t) = 6 cos (5 t 3x)() y (x, t) = 7 sin (6x 10 t)() y (x, t) = 12 sin (2x 5 t)() y (x, t) = 6 cos (3x + 2 t) + 7 sin (3x 2 t)
4
i) 6sin (3 t 2x) () 2,00 m/sii) 7 cos (4x + 5t) () 1,25 m/siii) 6 cos ( 5x + 6 t) () 1,50 m/siv) 2sin ( 3x 6 t) () 1,20 m/s
5
; ;
6
. .()
.
() .
() .
() ,
.
7
, , 1 2 r1 r2 , :
8
, ;() () () () ()
9
. :() 1
2.()
.()
.()
.
10
d2 d1
I
I
r
r
1
2
22
12
=
KYMATA 83
-
, d2 = 2 d1. ;() 1 = 22 ,
()
() 1 = 42
()
11
. .
12
:() () () () .
13
Huygens.
14
: () . . . . . . , () . . . . . . . () . . . . . . () . . . . . . .
15
: () . . . . . . , () . . . . . . , () . . . . . . .
16
.
17
Huygens .
18
.
19
.
() (), ;
20
;
1 21
2=
1 22=
84 T
-
21
; .
22
;
23
( ).
24
, .
25
.
( ), . .()
.()
.()
, , .
26
( ), . (
).
27
t = 0 .
y x ;
28
. ( )
2
KYMATA 85
-
29
.
, sin,
() , () , () , () , ()
.
30
.
.
31
: , (). . . . . . . () . . . . . . () . . . . . .
32
. .()
.()
.()
.
() .
33
. 1 cm/s. () 2 s () 2,5 s () 3 s () 3,5 s () 4,5 s 2 cm 4 cm.
34
, L, F. . .()
() .
() , .
1
2L
F
3
2
4
3
16
27
8
9
3
4
2
3
1
2
86 T
-
() .
35
, , , .
36
. . ;
37
; ( , ).
38
, ;
39
, ;
40
% % , :
Hooke.
41
:
1) ()
2) ()
3) ()
42
: , () . . . . . . . () . . . . . .
43
Young
. ;
44
Young ; ;
45
Young () ()
. () 61014Hz () :
L4
L2
1
2L
F
f
f1
2
100
100= ++
b gb g
KYMATA 87
-
1 :
:() () y() ( ).
2
16 z 20 000 z, ,
. 1450 m/s 340 m/s .
3
:() () () ()
( ).
y x t x y t= 10 100sin cm sb g b g,
y t x x y t= 10 3 0 1sin cm s, ,b g b g
i) 7,5 1014 Hzii) 5,5 1014 Hziii) 5 1014 Hz
46
:() 1) () 2)
() 3)
47
;
48
: , () . . . . . . () . . . . . . () . . . . . . . () . . . . . . .
49
.
50
;
51
, ;
() , .
() .
() Huygens ( ).
52
( ) : () 10 %, () 50 %, () 0 %;
53
;
54
:()
, .()
.() . .
88 T
-
-
KYMATA 89
4
, , 10 m 4,0 s. 10 20 s .
5
()
() x = 5,0 cm, t = 1,0 s.
6
, , 16,0 cm . 0,210 kg.m-1 F = 30,0 N : ()
() y (x, t)
.
7
0,15mm F = 200 . . 7,8 103 kg.m-3.
8
. 1,44 m . g = 9,80 m/s2
9
30 m 15 m. 1,0 mm.
. 8,9 103 kg.m-3 ,7,8 103 kg.m-3 600 N.
10
589nm. ) () 1,50; ) 60 , , . 1,00.
11
. 60, . 1,52 1,00, .
12
40 1,58. . n = 1,00
13
560 nm . 50,0 30,0. (n = 1,00)
14
1,660 1,620. 50 ( ) 60, , .
15
30 .
y x t x y t= FHGIKJ10 2sin
4 cm sb g b g,
-
4/3 () ( )
16
450 nm 400 nm. - . 4/3() 1 ().
17
o
. . . 1,00.
18
. =10 m/s. .() t = 4,5 s() t = 5,0 s() t = 5,5 s() t = 7,0 s
19
() x = 1,75 cm, t = 1,50 s() x = 8,75 cm, t = 2,25 s.
20
y1 y2
y1 + y2. .
21
y1 + y2 x = 1,65 cm.( )
22
y1 + y2
; x = 2,5 cm .
y t x
y t xx y t
1
2
3 2 3
3 2 3
= = + +FHG
IKJ
sin
sin
3
cm, sb g
b g,
y x t
y t x
y x
t s
1
2
1 5 5 2
1 5 2 5
= =
FHG
IKJ
,
,
,sin
sin
cm
b gb g
y x t
y x t
y x
t s
1
2
2 5
2 5
= FHGIKJ
= FHGIKJFHG
IKJ
,
,
,sin
10
0,001
3
sin
10
0,001
cm
y x t
y x t
y x
t s
1
2
3 3 2
4
= = +
FHG
IKJ
sin
cos
cm
b gb g
,
90 T
-
KYMATA 91
23
.
.( ).
24
0,5 m 8100 f1 . , 10000 f2 = 100 Hz. f1 .
25
40,0 cm 1,00 mm f 360 Hz , . , = 7,86 103 kg .m- 3.
26
F f. .
27
0,600 m 60,0 . 0,0100 kg.m-1 . 16 Hz 20000 Hz.
28
1,00 m 0,800 mm. F = 800 N, :() () ()
, 1 = 4,00 cm 2 = 3,00 cm .
8,89 103 kg m-3.29
100 Hz .
4,00 %. .
30
444 Hz .
. . 340 m/s-1 .
31
350 Hz. 340 m.s-1. 3 .
32
0,600 m 698Hz. .
33
L = 0,50 m , 675 Hz 1010 Hz . .
y x t x y t= 12 0 6 150sin ( , ) cos ( ) c m, s,b g
-
34
. . 80 cm 80 cm 2,10 m, , 100 Hz 2000 Hz. 340 m/s.
35
500 m. ; 340 m/s.
36
3,00 m. A 10,0 m . 0,100 m. , 342 m/s-1.
37
256 Hz.
. = 340 m/s-1.
38
Young 590 nm. 0,30 mm x = 0,30 cm. .
39
( ) 0,50 mm, , 650 nm. D = 1,5 m. : . 2 cm ;
40
Young - . 0,3000 mm 3,000 m. 3,584 cm 6 ,
92 T
-
4
-
4.1
() , . , . Pascal, , " , , (), ( ).
, , .
(4.1)
(4.2)
p y, p0, y0 (. 4.2) g, .
, , .
, .
. . , () , . , , . , .
. , , . , ( ) (. 4.3).
1 atm V0 10- 9 mm3. ,
p p g y y= + 0 0 b g
d
d
p
yg=
95
4.2
.
4.3
.
4.1
.
-
V m ' .
. . ~, , .
: , , . (. 4.4). , . , ,' .
: , , , . , , , . , ,, , , , . , . .
, , . . .. , ..
: , , (). , , , (. 4.5).
. ~ (. 4.6) . ~
=
lim
V V
m
V0
96 MHXANIKH
4.4
(1)
4.5
. 4.6
, , .
-
, , , .
. ' . ( ) , , (. 4.7). , , ., , .
. . , . . , . , , , , , .
: , .
) ) ) ,
( ) , ,
. , Bernoulli.
: V , , , t, t.
(4.3)
S.I. m3/s L3 T -1. , ,
. (. 4.8) t , t. ,
(4.4)
( ) 4.9. 1, 1 1. Q A 2 , 2 2 . t 1 ,
= Vt
A t
t= =
V
t=
97
4.7
.
4.8
t t
-
t 2
V2 = 2 t = A2 2 t
:) ,
, . , 1, 2.
) . , . , 1, 2,
m1 = m2
1 1 2 t = 2 2 2 t
(4.5)
P Q
(4.6)
.
, , 1 = 2 ,
(4.7)
(4.7) (), , . .
A A 1 1 2 2=
= .
1 1 1 2 2 2 =
m V A t2 2 2 2 2 1= =
m V t1 1 1 1 1 1= =
V t A t1 1 1= =
98 MHXANIKH
4.9
.
-
, , , .
4-1
2,0 m2 . 3,0 m3 s-1, . 12 m.s -1. .
=
A (4.7)
= 0,25 m2
, 0,25 m2.
BERNOULLI
, (. 4.10), , 1, Q, 2.
t (. 4.10) t + t (. 4.10), t . l1 m1 = A1 1 t l2 m2 = A2 2 t.
m1 = m2 = 1 1 t = m (4.8)
H m F1 F2 (. 4.10). , , m
m Q
,
E U K m g y m Q Q Q = + = +2 2212
E U K m g y m p p p = + = +1 1212
= A 2 0 1 512
, ,m 2
= AA
A =
= = 3 02 0
1 51 1,
,,m s m s
=
99
-
(4.9)
F1 = p1 A1 ( ) F2 = p2 A2 ( Q) ,
(4.10)
(4.11)
(4.12)
(4.10) (4.8), (4.9), (4.11) (4.12)
A tg y y A t p p A t1 1 2 1 1 1 22
12
1 2 1 11
2 + = b g e j b g
m g y y m p A t p A t2 1 22
12
1 1 1 2 1 11
2 + = b g e j
W F l p A t p A tF2 2 2 2 2 2 2 1 1= = =
W F l p A tF1 1 1 1 1 1= =
E W WF F= +1 2
Q PE m g y y m = = + 2 1 22 1212
b g e j
100 MHXANIKH
4.10
4.10 4.10 t t + t.
-
(4.13)
Q , ,
(4.14)
: , . , .
4.14 Bernoulli, :
,
". .
.
.
.
' , ' Bernoulli, F1 F2 () , . .
y1 = y2 (4.13)
(4.15)
, ( - ), 1 = 2 = 0
(4.16)
.
4-2
0,60 cm, 10 m. 0,15 cm, , 8,0m.s-1 : ) . ) . .
g y p g y p1 1 2 2+ = +
1
2
1
212
1 22
2 p p+ = +
pp V
V=
g ym
Vg y
U
V= =
1
2
1
22
2
m
V
V= =
1
22 g y p+ +
1
22
1 g y p+ + = .
1
2
1
212
1 1 22
2 2 g y p g y p+ + + +=
101
-
) (1) (2)
A1 1 = A2 2
2 = 0,50 m.s-1
) Bernoulli (1) (2)
(1) , . p1 = 1 atm = 10
5 .m-2
( p2, ;)
TORRICELLI
4.12 h . , . Bernoulli Q . y1 = h, y2 = 0, 1 = 0 2 = 4.13
p1 p2 ,
g h= 2 g h = 12
2
g h p p+ = +1 2 212
p252 32 10= , Pap2 5 3 3 2 210 10 10 10 1
210 8 0 5= + + LNM
OQP,e j Pa
p p g h 2 1 12
221
2= + + e jp p g h 2 22 1 121
2
1
2+ = + +
218= (0,15)
(0,60)ms
2
2
A
2
1 1
2
=
102 MHXANIKH
4.11
4.12
To , .
-
, h. Torricell.
: , 4.13.
, 1 2 1 > 2,
A1 1 = A2 2 1 > 2
1 < 2 Bernoulli
1 < 2 p1 > p2
, () , (). , , , .
4-3
. h1 h2, , . h1 h2.
Torricelli, 1, (1),
1
2
1
212
1 22
2 p p+ = +
103
4.13
.
4.14
-
. , , ( ).
,
x1 = 1t1
, x1 = x2 ( h1) h1 = (H h2) h2 h1 + h2 = H
BERNOULLI
Bernoulli . , , . .
) : 4.15 . . , . , (1) , . (1) , , .
x H h h2 2 24= b g
x H h h1 1 14= b g
th
g1
12=h g t1 1212
=
g H h1 12= b g
104 MHXANIKH
4.15
.
4.16
. .
-
) : Venturi , . , . , . , . h (.4.16).
. Bernoulli ,
,
, p p .
(1) (2) p1 = p2
h .
) Pilot: , . , () (. 4.17). (), , () (). ' () , (). ~, . () () . Bernoulli
(4.17)g h
=
FHGIKJ
LNMM
OQPP
2
12
b g
1
2
1
22
22 g h
+ =
FHGIKJb g
1
2
1
22 2 p p + =
p p g h = b gp g H p g h g H h+ = + + b g
p p g h g H h2 = + + b gp p g 1 = +
1
2
1
22 2 p p+ = +
=
A =
105
-
p1 = p2. p1 = p p 2 = p + g h, ~
. ,
(4.18)
h , ' , .
) - : 4.18. . .
, 4.19.
g h= 2
1
22 p p g h+ = +
1
202 p p+ = +
106 MHXANIKH
4.17
Pitot. .
4.18
. .
4.19
.
-
, , . , .
) : Bernoulli . 4.20, , . , , . ( ).
, d . , 0 (. 4.21). , , 0 . 4.21. F ,
0. d .
(4.19)
. 4.19
(4.20) d
=
0
F
A
d
=
0
F A
d= 0
107
4.20
.
4.21
( ) 0 ( ).
-
-
. , , S.I. Pa.s ML-1 T - 1 . poise (P) CGS , 1 poise =10 -1 Pa.s.
(. 4.22). . , . F1 , F2 ., , . ' . . , , . ' , .
, : . , . , , , , , . , , , ,
(. 4.23). , . , . ,
' , .
4.24
. 20 C .1 8 10 5, Pa s
F
A=
108 MHXANIKH
4.22
4.23
m 1 (). m 2 ().
4.24
poise, 0 C.
-
, ' , .
, , . . , , ...
, , (. 4.25).
,
,
,
,
(4.21)
(4.22)
C1 C2 .
C1 C2 . C1 C2 , R .
. ,
Mk + L- 3 k - + + 1 - - 1 = 1 L1 T - 2
,
(4.23)
(4.21)
( Stokes) (4.24)F R = 6
C R1 6=
C R1
k 0
1
1
===
k 1
3k 1 1
1 2
+ = + + =
=
UV|W|
(ML ) (ML T ) (L) (LT ) MLT3 k 1 1 1 2 =
k
R F=C F1 =
C R1 k
F C = 2 2
F C = 1
109
4.25
.
-
= 1, = 0, = 2
H
(4.25)
C2
(4.26)
C , , () . 4.26 C .
: , , .: .: ,
. , . , , Stokes, .
4-4
R = 10 cm m = 2,0 kg . . C2
2.
, . . . ,
F = mg
= 43 m s-1
1m s= 2
0 1
2 0 9 8
3 14 1 3,
, ,
, ,
R
m g
= 2
42 R m g2 =
C C 22
2=
C R22=
4
C R22
2
R F=[ ] [ ]C F2 2 =C R2
110 MHXANIKH
4.26
.
4.27
-
111
4-5
R = 0,50 mm. 1 = 6,0 m
.s-1. , ,) , ) 2 = 4,0 m s
-1,) . . ( = 1,8 10-5 Pa.s)
) F = C1 1 F = 6R1
F = 1,0 10-6 N) 2 , , (1 - 2). ,
F = C1 (1 - 2) F = 6 R (1 - 2)
F = 0,3 10-6 N) , () .
F = 0
= 1 = 6,0 m.s-1
' , . , () . , , F . , 4.31. F 4.31 . F ( ) F . : , 4.32. ., (1 > 2). , Bernoulli, . , ' . .
F N= 6 3 14 1 8 10 0 5 10 2 05 3, , , ,
F ( ) N= 6 3 14 1 8 10 0 5 10 6 05 3, , , , 4.28 - 30
4.31
4.32
.
-
112 MHXANIKH
R :
) , , ( ) .
1 1 = 2 2) Bernoulli,
, ,
, y1, y2 p1, p2 o . .
R Bernoulli , , Pitot ...
R , d . 0 , .
= F/A d .
R ,
R,
.
R , .
C R22=
4
C R1 6=
F C = 2 2
F C = 1
d
=
0
1
2
1
212
1 1 22
2 2 g y p g y p+ + + +=
drasthriothtesA N A
BERNOULLI1. ,
5 cm. . .
2. 5 cm 5 cm .
drasthriothtes
-
113
, 1 mm 2 mm . . .
3. : 250mL . . , , , . , , , , . , , . . 5 mm.
4. . , ,
. , . , , . , , ( ). , . , , , . , . . , . , .
( ;).
-
114 MHXANIKH
1
(). () ().
i) .
ii) .
2
;
3
. , () . . . . . () . . . . ., . () . . . . . () . . . . . .
4
. - () ().
() 1 2
5.
, .. (),
30 cm . , . 30 cm .
, Stokes .
6.
. , , . . , , , , 7 m - 80 kgf (kp).
-
115
1 2 ( ).
() 1 2 .
() 1 2.
5
;
6
1 2 3:1. 1, 1 2, 2 1 2.
.() 1 = 2 1 = 32() 1 = 32 2 = 31() 1 = 2 2 = 31() 2 = 31 1 = 2
7
Bernoulli () . . . . . , () . . . . . () . . . . ., () . . . . . () . . . . ..
8
. .
() Bernouli .
() , .
() m ( ) m , Bernoulli .
() .
() .
9
() ,
, . ;
() , , ;
-
116 MHXANIKH
10
.
11
.
; H .
12
;
13
() ().
14
h1, h2, h3 .
.
15
, , ,
() 2 , () , () 4 () / 2;
16
( , , ).
17
140 2,0 mm2 . 3,010-3m3s-1, ;
2
-
117
18
() A = .() 1/ 2 2 + g y + p = .,() 1/ 2 2 + p = .(1) N Bernouli .(2) Bernoulli .(3) .(4) Bernoulli, .
19
() . . , ;
() , , , ;
20
, ()
() () ;
21
, , () . . . . . () . . . . . () . . . . . (). . . . . .
22
F = C2
2.
() () () ()
23
0.
;
24
, , W .
4
3
7
9
16
3
4
=F F916
-
118 MHXANIKH
= 2 W () 2 W, () W, () 4 W, () 8 W C2
2.
25
D1 D2 0 - 0 ,
( ).
26
R 2R . .() C1 ,
(i) , (ii) 2 , (iii) / 4, (iv) 4 ;
() , C22,
(i) , (ii) 2, (iii) /2, (iv) 4 ;
27
, , , .(1) n (1) (2) C1 (2) kg.m
- 1 . s- 1
(3) C2 (3) kg.s
- 1
(4) C (4) kg.m
- 1
28
, () . . . . . (). . . . . . () . . . . . () . . . . ..
-
1
8000 m3s-1 44 106 m3. .
2
6,0 m s-1. A ; g = 10 m.s-2.
3
. 1 /2 = 5,0 h = 15 cm, 1. H
g = 10 m.s- 2 .
4
. 0,20 m2 0,050 m2
. 5,0 ms-1 2,0 105 N m-2 :() () . 1,0 103 kg m-3.
5
1,0 mm2 75 mm2. ,
-
119
, 3,5 m, 1,0 m. 10 , ; 1,0 103 kgm3 g = 9,8 m.s- 2 .
6
. , , . ;
7
0,010 m2. 2,0 10-4 m3s-1, 1,0 cm2. . . g = 10 ms-2.
8
30 cm 15 cm. 4,0 104 Pa 3,0 104 Pa, . 1,0 103 kg m - 3 .
9
H 1,75 105 Pa.
= 6 , , . (
, ). 103 kg m-3.
10
Pitot . 26,5 cm. km h-1. 0,800 103 kg m- 3 1,30 kgm-3. g = 9,80 m s- 2.
11
. h = 100 m 200 m3 s-1, . g = 10 ms-2 103 kg m- 3.
12
5,0 103 N m- 2 . m3 , 1,0 m.
13
20 m2 ( ). , 40 m.s-1, 50 m.s-1. . 1,3 kg m-3.
-
14
1,5 10-3 m. 1,0 103 kgm-3 g = 9,8 ms- 2. 1,3 kgm-3. = 1,8 10- 5 Pas
15
A .() ,
;() 20 %
;
16
5,0 cm 0,5 kg . . 1,3 kg m- 3 g = 9,8 m.s-2. c1 .
17
R = 40 m
.s-1. R = 2R , . :() F = C1
() F = C2 2
.
18
2,5 m. r = 2,0 m C . 80,0 kg. 1,3 kg m-3 10 m.s-2.
19
. c1 ,
20
. m R. . . g.
m g
C = + FHG
IKJ
2
1
2
120 MHXANIKH
-
4.2
. (), . , , . , , . .
(. . 4.33) , (
) . , , , . . ,
