vladimir pejčev - fizika za studente hemije.pdf
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%60 5%(0 &)%' " $ )( %+$ %) & ) , & ' ) &( ) ( , "0 . !%0 ) %-
6 5%( # ) % : %+$ "($ % *). ) $" )&% ) , ") +$ $ )&% & % %% $" +$.
&, 21 () Zm ) "$ % () m )
0(%+ ) gF , & +$ +$ 6
% "(% 21 ) gF ( 3.3 ).
3.3. + ) #,
(%+ ) ) ) $ "$ * % "* * -). )%$/ . " ) &" ( ) ())% ) " '% ' (' $ %3 )%/ %. 2 0(%+ ) ( 5%( ( 0(%+$) 0) : ( % *). ) &('
) gF $ $ "% &&+ & (" ) % 1m 2m , . -
% &&+ ("% *).0 )%$/ 2r * / .
1 2g 2
m mF
r = , (3.7)
0" $ : 11 2 26,67 10 Nm kg = , '$ $
.$ (")% " 11 2 26,67 10 Nm kg )&% "" ("3 .
%) . (3.7)&3 )
1 2g 2 er
m mF
r =
, (3.8)
& ' '( " ) gF
)&% ) " % $ er ( 3.4 ).
! 3.3
! 3.4
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)% ) )6 "$)%( 0(%+ ) $"0 % "0 % &")%(1 0(%+ &1 %0 %. 2 (%%%( &)(/ 0(%+-0 &1 (" ) ' (' $ ) (
d g
p
F Gm
= , (3.9)
0" $ gF 0(%+ ) $ % ) m "$ &. % ) pm
( pm m ). # ) 6 " $' 0(%+0 &1 "*$ %' .$
$" ' * 0(%+ ) $ %$ %' "$ &. % ) &.0 %.
2 (3.8) (3.9) $' 0(%+0 &1 ".$ )
pg 2 2
p p
1 1 e er r m m m
G F m m r r
= = =
. (3.10)
% G &(+ ) 0(%+ ) gF , " % %% ()
" ) (0 % " $ %' $ ) "*$ . )( (3.9)," $ $' 0(%+0 &1 $" . / , % $ / $"-
+ SI ))% %* % & )" ("% , 2m s .
" &")%(1/ 0(%+0 &1 (" ) . )
31 $ 0(%+ &1 ," $ %0% &(' $"$ %' & $ &(+ $' 0(%+0 &1
G , ) $ & $ /0( ) , " 0) -% $ & $ /0( % %%. " % $ $ . &% (&. 21 &($ &)+$ ), 0(%+ $
) &")%(1 ) &( $ ) +% &% ( 3.5 ), %
) (% G gF ) +% &% )($ %' &1 .
! 3.5
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(3.8),0(%+ ) * 21 () Zm ) % ()
pm ) $
Z pg 2 er
m mF r =
, (3.11)
% $ $' 0(%+0 &1 21 , & (3.10)
Zg 2
p
1 er m
G F m r
= =
. (3.12)
)( "00 5%(0 (3.4), 0(%+ ) 21 &.
% () pm ) )&3%( . / $ ) .' '( ) g ( # )
p gm g F = , (3.13)
& $, )" ) (3.13)
g G= . (3.14)
% %% 0(%+0 . / 21 g ) -
%$/ Zr R h= + " +% 21 ( 3.6 ), 0" $ Z R
&&' 21 , " $ h () $$ ) % () pm ) ")
&(3 21 , # ) &)%, & (3.12) (3.14),
Z2
Z( )
mg
R h =
+. (3.15)
) h % ") &(3 21 (3.15) 1( $ ") &&' 21 , $ )
8Z 6,37 10 m R , (3.16)
% ) (3.15)# &)% )"6 '
Z2Z
mg R
. (3.17)
! 3.6
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2' " ) % %% 0(%+0 . / 21 g # )%% &-
.# )%%, &3% ) m )%% (' , Z R ) % -
/ ())% " 00) 3 (&&' 21 (6 $ (% -
0 &( ). )&% (")% g 6 ) " 29,78 m s (%, 29,81m s 3$ 00)$ 3 , )( " 29,83 m s &-
( , 0( .0 +%0 +$ ) $ 6 .% )$ %.
)( (3.13),0(%+ ) 21 $ "$ % ) m $ $ &(3 21 ( 3.7 ) )
gF mg= . (3.18)
)%( % "$ "* ) &(3 21 , $ $ $"
0(%+$ ) 21 gF $ "$ %
gQ F =
, (3.19)
& ' ) &% ) ( %. %6 5%( ,
+$ ) %# %, $(1 ) ) %& &"0 N , $ %* "$ %
N Q= . (3.20)
! Q N )&%0 ) ) , " $ %% $ "$ % $" , )%0 % & "0 5%( $ ") &(3 21 .
3.4. &
! %/ $(1$ ) &)"+ * %+$ * '% %, * &$" "( )%0 % $ ) *). "" - $ %( &$. $ ) ) , $ ) $(1$ *
! 3.7
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)( &(3 '()% %, ) , " $ . $" " " -" &(3 0)$ %'$ .
!( %/ # " ) $( .
( )% 6 .% % ) ) )(0 %/ / . ! () 0 %/ .6 % )$ .
% % ) m $ '()%$ -%$ &" ( 3.8 ). ) % % " -
$ ) F %0 &(+ &1(0 % %%, ) 6 &% )( " )
")%0 "* 0' (")% grF . ! $ ) )&)%(1 %/ % ") &"0 ( ,
strF , '$ $ ) )&% " ) ) F . % %% strF ) %/ )% )
&)% % %% ) F , " . , " ) % &, )% )($ $(6 (")%
max
str grF F =
. (3.21) $ )&% "* , # ) &)%
maxstr s tr nF F = , (3.22)
0" $ nF % %% &% %% ) $ $ &(-
3 &"0, " $ str , $ () " ()%
% $ ) *). %%, " )%/ "" &(3 ()%& 0')% ). % "$ ) ) F %0 &(+ , " $
n gF F mg= =
. (3.22),$(6 (")%
% %% ) %/ maxstrF () " ('
"" &(3 * % &"0.
" $ % &% ( 3.9 ), " ) $(1
! 3.8
! 3.9
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"' ) %/ , d trF , '$ $ ) )&% " ) . % $ )
6 , " $$ $ % %% "* )"6
maxdtr dtr nF F = , (3.23)
0" $ d tr , $ () " . %/ %.
) "*$ % 3% ) &" "$)%( % ) F )&)%( &( -$) ( %/ ( const.v = ). % )'$ 6 .%
maxd tr d tr nF F F = = . (3.24)
.' $, "% & &(3 , +$% )%%'0 %/ (6 $"
"' : str d tr . &. / ' & ' str 0,8 = dtr 0,4 = ,
' & " str 0,10 = d tr 0,06 = , % & % str 0,04 =
dtr 0,04 = .
! %/ &" ) " (0 '$ . &, )( ()% %/ &(3 21 "($$ ) &))%( ) %/ . *%, %/ # .% 3%%, $ &)% "#(/ %/ )%% . &%. $ "$-)%( )&13/ ) . 2.0 %0 ) " '% 3 %/ /
/$ %/ %1/ , )%( ) (3 &" (/ , ' ) ) %/ )/$ .
3.5. + #
2 0 %, ") ) % m & % V ( ) -
mV
= . (3.25)
0 ' , &) 0)% $ ) $"+ & )&)%+ &)-%0 00 %.
2 0 %, &) 0)% $ %' "% $
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dmdV
= . (3.26)
"+ 0)% SI ))% $)% , 3kg m .
!&+' %# 00 % "* $ )"6 QV
= , (3.27)
0" $ Q %# %. )( (3.25) (3.27), )&+' %# )
".$
mg mg gV V = = = . (3.28)
0)% )&+' %# /$ ) ) %&%.
3.6. - & #
$ &")%(1 )& %$ %' (%), -%/ )(0 % () " %/ $ )% % %0 )& . )
))% ))%$ " n %, ) : 1m , 2m , ..., im , jm , ..., nm . : 1v , 2v , ..., iv ,
jv , ..., nv .
%&)%( $ " )( % &)%0 '0 ))% %0$ * -
). , ) "0 ))% ( &)%0 ))% . i jF $ %3/ )
$ % ) im "$ % ) jm , ) jiF %3/ ) $ %
) jm "$)%($ % ) im . %6 5%( , &% & ) )% % %% &(+ , )&% )
i j jiF F =
. (3.29)
* , $ iF %% )&13/ ) "0 ))% % ) im .
)( "00 5%(0 , )( % &)%0 ))% ) # &)% & $" $"' %/
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)%$ &$( &" $ ) 0 .$ -)% )( "#/ &) . "(
% () 1m 2m ) 0 " . %/ &
%$ &" , " ) * / ).$ )%' &0 ( + , 3 , 3.10 ). &'% %% % $: 0 p = . 4/ &0, % 6 " -
(% %3/ ) )%0 % %% &(+ , )&%0 ) , % 6 ) %, & &)% "$)%( &0, %% )%%
. 1v 2v . "#/ &) , .6
1 1 2 2 0 p m v m v= + = , (3.36)
% $)%
1 1 2 2m v m v= , (3.37)
3% ' " 6 &) % .% $" & % %% &(+ , )&%
)( . ) % %% &) $"
1 1 2 2m v m v= , (3.38)
)"
1 2
2 1
v mv m
= . (3.39)
!' ) &3 ))% % & &1/ . " &%&)%( "))%( )&13/ ) , ))% &3 % & &1/ &")%(1 %(
))%, '$ $ &) 0 p = . &1/ , )( "#/ &) ,
(# (3.36),0" $ 1v . &3 () 1m ), " $ 2v . %
() 2m ). ") . &3 1v . % 2v "* $ (3.39).
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3.7 '
$ " )("( #(% " 3 '/ " &$ " + . + , " ) (3 ) % 6 &" "$)%( ) .
$ % .( &$ s &" "$)%( ) F )%%0 % %%, $ ) &$ & 0 ( 3.11 ), " $ % ) (3
cos A F s F s = = . (3.40)
)( (3.40), " &")%(1 ) (' . " $
cos 2 < . " $ $ 2 = , $ &$ 0s = . " $ -
cos 2 > , ) $ 0 (3 &")%(1 .
! 3.11
$ ) &1( () " )%$/ : ( )F F s=
, &$ ) (3 "# & (1 ( $, " $ "+$ " dA & "+$
&$ ds "* )"6
dA F ds= . (3.41)
& " ".$ ) %0+$ (3.41)& '%( &% s " &-
'%0 $ 1s " $/0 $ 2s
2
1
s
s
A F ds=
. (3.42)
"+ " SI ))% j &, J, $ $, & (3.40), je" : J Nm= .
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3.8. )
$ ' (' $ %3 . $ "* ) (3
" . ) F (3 " A () %( t , )0 $ "
AP
t = . (3.43)
( (# " $ 0/ ") )%%, %$. " $ )0 )% ( -' .
) )0 / ) ( , "3 ) "+$ - .
dAP
dt = . (3.44)
"+ )0 SI ))% $)% , W, $ $, )( (3.43)
$" & : W J s= .
3.9. ' ) . " .
(3.8),) $ % ) m &(' &. % ) pm $
pg 2 er
m mF
r =
. (3.45)
+$ " dA $ (3 0(%+ ) & &/ &.0
% ) pm "+$ &$ ds ( 3.12 ) $, )(
(3.41) (3.45),"%
pg 2 er
m mdA F ds ds
r = = . (3.46)
! 3.12 ) (" " $
e | e || | cos cosr r ds ds ds dr = = = , (3.47)
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& $ )%0 "+$ " $"
p2
m mdA dr
r = . (3.48)
"(" $ & " & &/ % %'- 1 %' 2 "# &%/ a $"
2
1
12 p p21 2
1 1(a)r
r
dr A m m m m
r r r
= =
. (3.49)
)( (3.49) $) $ " " 12(a) A 0(%+ ) () "
"$ )%$/ 1r 2r , () " . "# &% & &-
3%/ % $ 1 $ 2. ' " 6 )% " .% (3 " ) % & "# &%/ b, ")
12 12(b) (a) A A= . (3.50)
%$.
12 12(b) (a) 0 A A = . (3.51)
(3.49)# ) 1'% " $
12 21(a) (a) A A= , (3.52)
") " $ " & %/ % & "* &% )&% ) $"
" ) 0%( &" . 2 (3.52) (3.51)".$ ) " $ " A "# %( % L (" %' 1 " %' 2 &% b, % " %' 2 " %' 1
&% )
g p 12 21(b) (a) 0 L L
A F ds m G ds A A= = = + =
. (3.53)
%0 &")%(1 (3.53) ($ ) (% gF ,
") G "# %( $ L .
! 3.12
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). 0(%+ ) # (3.50) (3.53), $ )" $" "0 , )%( ) ). , % $ 0(%+ ) - (%( !
0(%+ , (%( ) .$$ ) - , " $ &. (%( ) .
3.10. ) ) &
' ) "3 (%%%( '% . %/ %$. ())% $ ) . %/ %$ &'( , (" ) "0 -($6 0$: , , , , %". %(- ))% 0$ # " & $"0 . "0 , % " & 0$ )% )%%. ( '/+ #( , $ $ $" " )( .
' 0$ &")%(1 $&)%$ . 0$, "3 ) )&).)% % " (3 " . + $ "( . 0$: ( ) ( - ).
3.10.1. #$ '*
% ) m "$ ) F , / "$)%( 6 ) )&1% &
. % " 1v " 2v . " $ % )'$ (3 ) F # ) &
. %, % & $ %' 0$ %. ' " ) -
%' 0$ % # ""% " ) F " . % & . " 1v " 2v . $ " $, & "+$
2 2 22
11 1 1
2
1
12
2 22 1
2 2
s s ss
ss s s
v
v
dv dv ds A F ds m ds m ds m dv
dt dt dt
m v m vm v dv
= = = =
= =
. (3.54)
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"(" )" " ) " ( )'$ # &")%(% "( ( -' , $ () ) " . % $ &'% &%, .
2
k
2
m v E = , (3.55)
$ ) ( . (3.55))" " %' -
0$ # % 0%( (")% ( k 0 E ).
"+ %' 0$ SI ))% )% $ $"+ " , &.
)( (3.54) (3.55) $ $) " ) " $ ) F (3 % ) m # % & %' 0$ %
12 k2 k1 k A E E E = = . (3.56)
3.10.2. ( &% '*
%+$ 0$ "*0 % () " &-#$ %0 % ") % ())% %) '$ ) &-1 ) .
&. % ) pm &'% %% 0t =
$ &1 0(%+ ) 21 , () h
( Zh R ) " 21 &(3 ( 3.13 ), )
&'% . 1 0v = , % )%0 ) %' 0$ k1 0 E = .
)."0 &" &(10 &% h , %' 0$ % %% " 21 )
2p 2 p 2
k2 p( 2 )2 2m v m
E gh m g h= = = , (3.57)
0" $ . v "* & (1.29) . " ( . 0 %/ . &'% . .
" $ $ (3 0(%+ ) &% * $ 1 2 $" $
12 k k2 k1 p A E E E m g h= = = . (3.58)
! 3.13
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%' 0$ # (3.58)&)"+ $ &
)%$/ * % 21 h .
2 &. % ) pm $ ) 0(%+ &1 21 # )
6 " &%+$ 0$
p p E m g h= , (3.59)
% ) , % )'$ , & (3.58),".$ " $
k k2 k1 p p p p1 p2 p0 A E E E m g h m g h m g E E E = = = = = = . (3.60)
(3.59)# ) ""% )%% ' , &. p 0m g h ( 0h $ () (0
%0 ( " &(3 21 ), ' ) ".$ ( (")% &%+- $ 0$
p p p 0 E E m g h = + . (3.61)
*%, (*/ ( &%+$ 0$ p E , $ ) " &%" p E -
$ )%%, / (3.60) & %' &%+$
0$. (3.59), ") (3.61), &")%(1 ) &)+$ 0(%+
&%+$ 0$ &.0 % ) pm , (# Zh R .
' (")% &%+$ 0$ &.0 % ) pm 0(%+-
&1 21 ".$ ) (3.49) " 0(%+ ) , & ' $ )
Zm ' ) 21 .
12 Z p Z p Z p p1 p2 p1 2 1 2
1 1 1 1 ( ) A m m m m m m E E E r r r r
= = = =
(3.62)
) % $ , " $ 2 Zr R= p2 0 E = . " )
# % " $ 1r r = p1 p E E = , % $ "
Zp Z p Z p Z p
Z Z Z
1 1 r R h E m m m m m m
R r R r R r
= = =
, (3.63)
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0" $ h () $$ ) % ) pm " ( . $ Zh R
&( % )'$ , " $ 2Z Z R r R= , & ) , & (3.17),
&%+$ 0$ )("
Z pp Z p p2 2
Z Z
m mh E m m h m g h
R R = = = , (3.64)
"%' $ ) (3.59).
")%($ &%-+$ 0$ " (3.63)".$ ) ) (3.62) " $ % (
.)')% : 2r . " $
p2 0 E = %, (*/
1r r = p1 p E E = , )"
Z pp( )
m m E r r = . (3.65)
&(3 21 $
Z p Zp Z p Z p Z2
Z Z( )
m m m E R m R m g R
R R = = = , (3.66)
& ' $ 36 g &(3 21 )( (3.16).
' ())% p( ) E r & (3.65)& $ 3.14 .3.10.3. $ ',- $. #$ '*
2. %' k E &%+ j 0$ p E %a ( ) -
%
k p E E E + = . (3.67)
% & ' 0$ )'$ )."0 &" &.0 % ) pm . &'% . ) () h &(3 21 , 3.13 .
! 3.14
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$ 1 ) : k1 0 E = , p1 p E m g h= , % $ & ' 0$
1 k1 p1 p E E E m g h= + = . (3.68)
$ 1
, % $ . $ $ * 1 2
, ) :2
k1 p 1 2 E m v =
,p1 p E m g h = , % $ & ' 0$
2p 1 p
1 k1 p1 p p p2 ( )2 2m v m
E E E m g h g h h m g h m g h
= + = + = + = , (3.69)
- 1v "* $ )( (1.29) . " &($)0 ( -
. 0 %/ , . &'% . .
$ 2 ) : 2k2 p 2 2 E m v= , p2 0 E = , % $ & ' 0$
2p 2 p
2 k2 p2 p22 2m v m
E E E g h m g h= + = = = . (3.70)
)( % $ , $ $ % & , & -' 0$ % '0" )%$ )%%, % ) -
: & ' 0$ )%$ )%% % "$ ) (%( ) # %%' &)%
k p const. E E E = + = (3.71)
% ( ))% %) "$ (%( ) , + ) %-/ , ' 0$ &%( ) %&%, %' , )(%) , %". % )'$ (# "#/ & 0$.
3.11.
" )" "( % &" ( ) 0 & )%/ %/ % % )" /( %+$. )% $ % ( )# &+) $ $ %3 &%%, ) , )'$ )&) %, # )( &"% . , % ) %( &.#($ )"$ )%' -
)%' "+$ , " . &"$ " % (")% . ' -0$ % & "' 0$ "+$ %, "' %&%
-
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( "0 . 0$). , % ) *). "1($ " ) /( %( . &(6( , 3% ' " ' 0$ )%, %% " )%' ) .
())% " )%' ). % $ "( )( %& )" - : .
3.11.1. . % %# '
&)% )%' )" $)% )" "( % & /( ' 0$ )%$ .
" $")%()%, &)%6 +% &)% )%' )" "(
0+ () 1m 2m ), '$ ) (% . & )" 1v 2 0v = , &) ) -" 1v 2v , $ %" # $"$ &($, 3.15 .
! 3.15
"#/ &) "#/ 0$ 0 ) &-)% )"6
1 1 1 1 2 2m v m v m v = + , (3.72)
2 2 21 1 1 1 2 2
2 2 2
m v m v m v = + , (3.73)
0" $ (3.72)&) ) . &3% ) (% . -
. 3(/ 0/0 ))% 0 ) ""% . % &) )" 1v
2v ) & $ . % & )" 1v . &36(/ (3.72) (3.73)
".$ )
1 1 1 2 2( )m v v m v =
, (3.74)
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2 2 21 1 1 2 2( )m v v m v = . (3.75)
) (3.74)&" ) (3.75),)"
1 1 2v v v + = . (3.76)
2 2v (3.76) (3.74).6
1 1 1 2 1 1 1 1 1 1 2 1 2 1
1 2 1 1 2 1
( ) ( )( ) ( )
m v v m v v m v m v m v m v
m m v m m v
= + = +
= +, (3.77)
% $
1 21 1
1 2
m mv vm m
= + . (3.78)
(3.76) (3.78)".$ ) " $
12 1
1 2
2mv v
m m =
+. (3.79)
$ 1 2m m= , " $ )( (3.78) (3.79) 1 0v = , " $ 2 1v v = .
$ 2 1m m (&. 0+ " "),
" ) , (3.78) (3.79), 2m ,
".$ " $ 2 0v = , " $ 1 1v v = . ' " )
0+ & " )( " ".$ " /0 $" . $ $ /0 %, )&%0 ) ( 3.16 ).
3.11.2. . % %# '
&)% )%' )" $
+% )" "( % ) 1m
2m , . 1v 2v , ) % -
)" 6 $" + ,
$"' . v ( 3.17 ). )( "#/
! 3.16
! 3.17
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! 3.18
&) $
1 1 2 2 1 2( )m v m v m m v+ = + , (3.80)
0" $ &) ) . , $ ) (% . ."(" $
1 1 2 2
1 2
m v m vv
m m
+=
+. (3.81)
' , " &)% )%'0 )" "#/ ' 0$.
3.12. ) ) &
% (%$ %' ) ) m , ( & "# r , (3 ( # %/ ( const.v = ), 3.18 . )( %/ "1 - 1.7, % &)"$ ) &% . -
/ na , $ $ ( ) +% #
&%/ '$ $ % %%, & (1.36)
2
nv
ar
= . (3.82)
"0 5%( , %" % "$ ) $ $ ( -0 #0 %/ ( )
cp nF m a= , (3.83))%0 &(+ ) &% . / na , / % %% $
2
cp nm v
F m ar
= = . (3.84)
)( %60 5%(0 , +%&% ) , ) +$,
% ). )&% ) +$, $ ) ( cf
F .
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"$ ( (&) ) $"0 % %%, )&%0 ) , '$ ) &" %' % ( , $ &( &% % )%/
cp cf F F =
. (3.85)
)" %0, & %+$ % ( 0 & & , )6 "$)%( + -%0 ) .
) "0" " ) & &" , %" &)%$ "$)%( +%&% ) , )%( "$)%( +%0 ) . 6 ) "1 %% & +$ ,. )%%0 % %% &(+ %0% $ & %' $$ ) % %% "/ &.
3.13. ) &
%/ 5%( " "1 3.2. . $ )%-% " (# ) % ))% . !" 6 % .% % ' &+) ( ) % ))% .
3.13.1. '- '% #$ '"' $ $'+ .'! %$ , '! ' /'
) %$ %' 6 % ))% $ ) 6 &( -
$) , . / 0a . $ &% ))% S ( # -
' 3 , &% ))% S (0
) "( 0 ( B) ) m ,
& ' $ 0 )." , 0 B( )%' &0 " (0 ( 3.19 ). %&)%(1 $ " $ &"-0 (0 0% (%/ * 0 &"0 # ) %).
2 &)%' $ ) &-
% ))% S , )." 0 $ ( 0 0v = ) ) 6 )%% . ( 0 const.v = ) ())% "
! 3.19
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&'% )( %/ (0 , &3% 0 "$ ) ) 0(%+$ ) +$ &"0, $ ) *). (%#($ .
3% ) ))% S 6 . , 0 6 )%$% ") (0
&'6 " . & &" &(+ )&% " %/ (0 .)%' $ ) ))% S ( (0 ) .$3/( / 0 &
&" &)%$/ ) inF $ $$ )&3%( . / 0a ") (0 . (
%( ) ( )
in 0F m a= , (3.86)
$ " ) (" +$ ))% S " . / (# "0 5% -( .
)" "$)%( +$ ) inF 0 B, &0 6 ) )% % )( "
)%' ) &0 elF )&)%( (%# ) inF , % $: in elF F =
.
)&)%(1/ (%# , 0 B 6 ))% S (% , " 6 ) ))% S
%% )%% . / 0a (0 .
/ +$ ) inF (3 ) "% , & / )%'
) &0 elF .
+$ ) $(1$ ) ( $ ) 6 . - ()& ). &) , . -
/ ( $ ) %-/ , % &% "$ -+$ ) )&% ) - . )%(1/ , . / $ )&% ) " ) %/ , & &% + -
$ ) "$ &" .
! 3.20
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'% $ # "$)%( +$ ) " )% )() ." & (%0 %/ ) &(3 21 . % %% ) -
%, &" 0(%+ ) gF mg= , "$ +$ ) in 0F m a=
) # , 0" $ 0a . / )()0 ." ) (3 . (" %# )% ) 0 0( )mg ma m g a+ = + , %, $ &. 0 3a g= , &("
%# 6 .% 4mg , 3% % &%6$ "() ))% )% .
!%0 )% )() ." #6 $ & %/ ,
" . (% . / 0a . (6 ( )"( . %&)% (
)"( () " ' ")/ $ $ %* %# , .0 (0 & & 3/ 0"0 3 .
% &("0 &(6/ %# $(1 ) & (#/ % " ) 6 (3 ( 3.20 ). (" %# % )'$ )
a 0 0( )Q mg ma m g a= + = + . (3.87)
) % 6 # ( 3.20b ), &(" )// %# $ "
a 0 0( )Q mg ma m g a= = . (3.88)
2 )'$ " $ . / % 0a g= (% )." &"), &(" %#
% $" $ , %$. " " &$( .
3.13.2. '- '% #$ '"' $ ' '
) %$ %' (%) 6 % ))% $ % 0-
. . ) m ( $ & $ 0 &)1( " $ ") ( %$6 &' &% z ) , " x )
y ) # ( &' ) / %$.
") &% +$ ))% S ('$ ) z ) &"" )
z ) &%0 ))% S ), % ) m (3 ( # %/
( 3.21 ). $ "$)%( & % # $ ) cpF cf F . (
) $ +%&% ) $ % "$ "# &&' +% %-
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+$, &($6 % " ) 6 & 0 &&' r ; / % %% $,& (3.84) (2.12) $"
22
cpm v
F m r r
= = . (3.89)
! 3.21
%6 5%( , & "$ +%0 ) cf F , -
$ )% % %% &(+ cpF , " $$ $ ) )&% (" +%). . &% ) ) %( ) /%()0 %&, &3% ) $(1$ &%
(+$ ) ))% , & ' cpF "$ %, cf F &. ) &
$ "# % &, &)%' 6 ) % %% &($) ( ) %/ * % &' # %), &(+ %0% &%/ %-% "/ , %$. % (3 "$ ( ) .
") &% "% ))% S ( 3.21b ) %, & &-)%' $. )%' )%%$ " % "$ "* ) $ %# " 0 "1 " +%, 3% ) )&1( % / &. 5 % %% # ) % "% , ( )
2cfiF m r = . (3.90)
"* " "/ &, % 6 ) %% . & ($ $.
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%+$ )&) %, ')%+ & "# )%' ) .
&3% )'$ )( ')%+ % ) im "$ +%0 +$
) , '$ % %%, & (3.90), ) 2cfi( )i i iF m r = , 3% ' " ,
&)% ')%+ $ $" ) im , () " ir . 3% +%-0 +$ ) ')%+ % "$ '% % %% (&3%
)( ')%+ '% ir , " $ )% )( ')%+ ), % ) )%($
' & / $ 0 0 , & ' .% & (
(")% 0 . % . 2.0 %0 $ " %$6 ))% "$)%( .)/ +%0 +$ ) " %0 '$ .
%0 +$ ) )% ) )"%/ - . ")& ))% )" $ $, )( ')%+ 6 "(% ) 0(%+$ mg # , ) ) ')%+ , % " "
/(0 ( ) " )" . *%, %#/ 0(%+- ) ) $ ')%+ " $ ( 0,27m< ), &3% $ " /
$ # ( ).
!3%/ )" +%0, ')%+ 6, ) 0(%+$ mg # ,
"(% +%0 +$ ) 2m r , $ $ %* ) )
')%+ , )%$6 " &6 3% "1 " +% %+$. -"6 " -
&% . / ( 2na r = ) 0 " ." 0 &% (6 " 0(%+-
0 . / g (&. " 2 60000r g > ), ))% %#/
0 $ &(6 , % $ 06 %#% " ')%+ ")& ))% , 3% $ " (0 '$ .$ $ . -0$.
%0 +$ ) $(1$ ) &)"+ %+$ 21 )($ ) , & ' $ 0 . %+$ 21 $"
5
ZZ
2 2 rad7,29 1024 60 60 sT
= = = . (3.91)
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% ) m $ ) &(-3 21 , "$ 0(%+ ) 21
gF , %* +%0 +$
) cfiF ( 3.22 ), % " $ %# %
Q $ $" (%) . )
gF cfiF .
3% $ 2cfiF m r = , $) $ "
)($ $/ (")% &( ( cfi 0F = ),
$(6 (% ( max 2cfi Z ZF m R = ). $
max 2 23cfi Z Z Z Z
g3,5 10F m R R
F mg g = = = , (3.92)
(" " $ ($ % 1( (/ $ " 0,35% ).
" +%0 +$ ) ))% $ % )%%
0 . , $(1 ) KorF . ( ) "$
% ) m ) 6 %( . v ") ))% $ %- )%% 0 . ; ") $
Kor 2 ( )F m v = . (3.93)
) 0+ ) m 6 & #$ &'
(. %/ ) "# &&' r )%% . - r v ( 3.23 ). ( )'$ r v -
&$ &( 0, % $
Kor 2 r F m v = . (3.94)
2 &)%' $ ) &' , 0+
) 6 & ($)$ &%/ , 3% . - $3/( "(/ )( ) . . ) 0+ % " ) 6 "#
! 3.22
! 3.23
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! 4.1
&&' &% ))% S , . ) &)%(% ( ) . 3& (0 ".1/ "# &&' &' .
0'/ "# &&' , &)%' $ ) ( &'
( &% +$ ))% S ) (" &%/ 0+ ,&3% / "$ ( ) . ") &' &)%' ( S ) ("
)% ($) &%/ &)%' ))% S , *% ( -1)% &%/ .$3/( +/ &' )&" %.
4.
4.1. .
% % ) m # ) &")%(% ' ))% ))%(1 " % )
1m , 2m , ..., im , ..., nm , '$ )
(% $ 1r , 2r , ..., ir , ..., nr
( 4.1 ). & ) ( -(0 ))% $, % )'$
1
n
ii
m m=
= . (4.1)
( % "3 ) 0%$) %' '$ $ ( -% $ "*
CM1
1 ni i
i
r m r m =
= , (4.2)
")
CM1
n
i ii
m r m r =
= . (4.3)
+/ (4.3)& ( ".$ )
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CM
1
ni
ii
dr dr m m
dt dt ==
, (4.4)
% $)%
CM1
n
i ii
m v m v=
= , (4.5)
0" $ CMv . +% ) , " $ iv . % ) im .
+/ (4.5)& ( )"
CM
1
ni
ii
dv dvm m
dt dt ==
, (4.6)
%
CM1
n
i ii
m a m a=
= , (4.7)
0" $ CMa . / +% ) , " $ ia . / % ) im .
)( "00 5%(0 $
1 1 1
n n ni
i i i ii i i
dvm m a F F
dt = = = = = = , (4.8)
0" $ F %% )&13/ ) $ "$ %, " $ ) %3/ ) $" (&0(1 3.6).
% )'$ # ) &)%
CM1
n
ii
m a F F =
= = . (4.9)
/ ' " ) +% ) % 6 %$ %' '$ $ ) $" &$ ) ))% , &" "$)%( %% )&13/ ) $ "$ ))% (%). $ %/ % %)%, (4.9) &%&)% &)$ /0( %/ , % ) %/ % )(" %/ %-
$ %' . ) % 6 ($) , " ) /0( %/ #
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%)% %/ +% ) %+ %/ +% ) &-% %' .
)'$ " $ % ( ( 0F = ), (4.8))" " $
CM CMCM
( ) 0dv d m vm a mdt dt
= = =
, (4.10)
" )"
CM const.m v = , (4.11)
3% ' " ) +% ) (0 % 6 ( &($) .
% ) % && ) ) %#3% % ) % 0 0(%+ &-1 . % % $ &" %' %% ( -%)0 . 0(%+ ) () %#) )(
% ) im g ( 4.2 )
g1 1
n n
i ii i
m g g m mg F = =
= = = . (4.12)
#3% )%' 0 % (&%,(1 , &&&") ) +%. )'$ % &(0 . ,%#3% ) "*$ )&% , (3/ % $/ "( %' , & ' %#3% # &) (% &(' %' (3/ (% ).
4.2. ) & ) #
$$")%($ . %+0 %/ %0 % $) %+$ -&% ) . ( 6 .% ") "* ' (' &" "1 %/ .
4.2.1. '&
% +$ I %$ %' ") &% ) OO (
4.3 ) "3 ) )"6 '
! 4.2
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2 I m r = , (4.13)
0" $ m ) %$ %' , " $ r % %% (% $ %$ %' ") &-
% ) OO .% +$ % ) m ") &%-
) OO ( 4.4 ) "3 ) ) % +$ % ) %
2
1
n
i ii
I m r =
= , (4.14)
& ' $ ir % %% (% $ % ) im ")
&% ) OO . )'$ " $ )&" ) % (&" ), # ) &% % ' , % ) -% +$ % "3
2 2
1 0lim
mn
i ini
I m r r dm
=
= = , (4.15)
0" $ r )%$/ % ) dm " ) OO . )( (3.26) 0)%, -
% ) # ) &)% dV , 0" $ dV
% &, % % +$ (4.15)
)"
2
0
V
I r dV = , (4.16)
0" ) %0+$ (3 & '%($ & %. 2 0 % 0)% $ )%%, % # &)% )&" &)0 %0
20
V
I r dV = . (4.17)
! 4.3
! 4.4
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) & (4.17)# & % " % +$ &. -
00 (1 ) m , &&' R () h ) 2 2mR , " 0
&% ) m &&' R ) 22 5mR .
4.2.2. % . $' . .'* %.
% # " % ) O $ $ ( +%# (
4.5 ). ! F ( ( +%#) "$ % %' A $ $ "* (% $ r ") ) %+$. $ -
F ") ) O $)% (%
M , $ $ $"
M r F =
. (4.18)
$ ( $ . $ ( -
% r F , # &(+ ) %+$ ( (% 0 . . /
), " $ /0( ) "* &( ")0 (%/ ( 4.5 M $ - $%) &)%' ).
% %% % ) $" $
sin M rF F d = = , (4.19)
&3% ) . : sind r = .
% "$ n ) ( ( $ ) &(+ ) %+$),
& % ) $" $ (%) . &$"' % )
1 1
n n
i i ii i
M M r F = =
= = , (4.20)
0" $ ir (% $ &" %' ) iF .
! 4.5
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( & ) $" % -%%, )&%0 ) , $ "$ "# )% &( , &")%(1$ .% )&0 ) $" $, )(
(4.20), . % $" 1 M
"0 ) 2 M $ . $ )&0 (
4.6 ). # .% "* ") & (1 ) ( $$
# ) . . ) # )%$ ( ( 4.6 % . . ( +-
%#), .6 % )% ) , . % ) $ )% &(+ ) , % $ "(1 ).% % %%
1 2 1 2 1 2( ) M M M F d F d F d d F d = + = + = + = . (4.21)
)%$/ 1d 2d $(1$ ) ' %%, 3% ' " $ %
)&0 ) $" )( ) ( ) $ . $ )&0, /( % %% "* $ & (" % %% F ) $" " ) -0 )%$/ d * /( $ "(/ .
4.2.3. ! # $ ' & * $'- $' * % $
. $'
% % & (10 . % ) O ( +-
%# ( 4.7 ) &" "$)%( %- $6 )&13/ ) F . &3% ) -'$ , % 0 "(% )&13/ ) $ # ( &(+ ) %+$. *%, ) &% %( ) $ # ( &(+ ) %-+$ ) %+$ % ) . &% ) $ ) & ) )
! 4.6
! 4.7
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("0($6 % ) ) %+$) %# " ) %+$ &.% %( ) "* ) "0($6 % ) +$ $ ) $(1$ %' '(36/ .
) % ) 4.7 ))%$ " % ) 1m , 2m , ..., im , ...,nm '$ ) (% $ 1r , 2r , ..., ir , ..., nr ") ) %+$. )&-
13/ ) iF (# ( +%#) "$ im , ) # #%
&%: 8iF &(+ %0% niF &(+ (% $ ( &(+ -
%0%), ")
8 ni i iF F F = +
. (4.22) )( ( +$ "+0 (4.20)# ) &)%
8 n 8 n 8( )i i i i i i i i i i i i M r F r F F r F r F r F = = + = + = , (4.23)
&3% $ n 0i ir F = (0 * ir niF $ ). &% ) 8iF
"$ &(+ %0%, "$6 %$ ) im 0 . / .
"0 5%( (2.16): 8i ia r = , % &% ) .6
8 8 ( )i i i i iF m a m r = =
, (4.24)
% $, & (4.23) (4.24)
8 ( )i i i i i i M r F r m r = = . (4.25)
$ ir
8i ir F , % %% % ) i % ')%+ ) 2
i i i i i i M r m r m r = = . (4.26)
3% 8iF # ( +%# ( 4.7 ), " , )( (4.25), i M &(+
) %+$ ) ) &)%' , %$. &(+ ) % ) )% )
&(+ ) (%
2i i i M m r = . (4.27)
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!/ (4.27)& )( % ) ".$ ) & % )( )&13/ )
2 2
1 1 1
n n n
i i i i ii i i
M M m r m r I = = =
= = = =
. (4.28)
/ (4.28) &")%(1 )( $"' " %+0 %/ %0 % &% ) 0 $ "0 5%(
( F m a= ) %)% %/ ; % ) M 0 $ ) F , %
+$ I 0 $ ) m , " $ 0 . / 0 . / a .
4.2.4. #$ '* , ' * $ ' & * $'- $' * %
$ . $'
%' 0$ %0 % ) m $
% &% ) 0 . ( 4.8 ) $" $ ) %' 0$
)( % ) im $ ) 6 -
'% %0+$ . iv & # -
&%/ (% $ ir
2
k k1 1
2 22 2 2
1 1
212 2 2
n ni i
ii i
n n
i i i ii i
m v E E
I m r m r
= =
= =
= =
= = =
, (4.29)
& ' $ 36 (2.12) - . : i iv r = .
!" 6 .% "* " )&13/ ) & %+$ %0 % )%-
) OO ( 4.9 ). )( % ) % im "$ )&13 -
/ ) iF $ # ( $ &(+ ) %+$. 2 ()
%( t % ) im &6 6 &%
i ids r d = , (4.30)
! 4.8
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% $ " )&13/ ) iF &% ids ,
& "+$ (3.40)
i i i si i si i idA F ds F ds F r d = = = , (4.31)
0" $ siF &$+$ ) iF &(+ &%
ids . $ si i iF r M = % %% %
) ") ) %+$ '$ ) &-(+ && ) ) %+$, " $
i i idA M d = . (4.32)
!/ . (4.32) )( % ) % ".$ ) " )( )&13/ ) $ "$ % () %( t
1 1 1
n n n
i i i ii i i
dA dA M d d M M d = = =
= = = = , (4.33)
0" $ M % %% %% % ) '$ &(+ # "# ) %+$. '/ ( ) , )( % ) & %( 0%(
())% " %0 " %# " % % & %( 0%( ) - ( & %( ) %+$ ) $ )&% %/ 1 )%).
& " )( )&13/ ) ' 0 &$ ".$ ) %-01/ (4.33)
2
1
A M d
= . (4.34)
!0 % $ % &% ) $, & "+$ )0 (3.43)
dA d P M M
dt dt
= = = , (4.35)
&3% $, & (4.33),dA M d = .
! 4.9
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4.2.5. .% . $'
% &) %$ %' ") &% ) %+$ OO ( 4.10 ) "3 )
L r p= , (4.36)
0" $ r (% $ %$ %' ") ) %+$, " $ p
&) %$ %' ) m , ( . v ) ) %+$.
(+ (% L $ ( $$ # r p , ") &(+ ) %+$
) ")0 (%/ , % &&" 0& )$ (%, .3 (% 0 . .
)'$ %+$ %0 % &%
) OO ( 4.11 ), )( % ) im ,
'$ $ $ ") ) %+$ "*
(% $ ir '$ $ &) i i i p m v= ,
)($ % &) ") &% ) OO
( )i i i i i i i i i L r p r m v r m r = = =
, (4.37)
0" $ iv / )( (2.15).
$ i iv r ir
, " $ % %% % -
&) $"
2i i i i i i L r m r m r = = , (4.38)
% ) (% i L # &")%(%
2i i i L m r = . (4.39)
% &) % + L ") &% ) OO ".$ )
)/ % &) i L )( % ) ") &% )
! 4.10
! 4.11
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56
2 2
1 1 1
n n n
i i i i ii i i
L L m r m r I = = =
= = = = . (4.40)
(4.40) % &) & %+$ 0 $ ) (3.1)
&) % & %)+$ ( p m v=
). (4.40),)( $"' " %+0 %/
&% ) (4.28)0)
( )d d I dL M I I
dt dt dt
= = = =
, (4.41)
0 $ ) (3.4)& %)% %/ .
(4.41), $ 0 M = , )" " $
const. L I = = , (4.42)
3% $ &) % & %+$ % &-% ) : $ & % )( )&13/ ) $ "$ % $ % &% ) $" , %" $ % &) % ")
&% ) )%%. (0 . . &. '( $ )%$ % &)%1
$ % (% ) . '( )($ "# )( %0(,
/0( % +$ 1 I .6 (6 " ) ) %0( 3 "
% +$ 2 I " ) &1.1 %. "#/
% &) (4.42).6
1 1 2 2 I I =
, (4.43)
)%0 6 &( )'$ 0 . &)%1 1 .% / 0 "0 ) -
'$ , " ) 2 .
!' ) "3( ) '+ " $ % 0 . . # )($ 0 . &(6% 0 )&3%/ % &. -#(/ $" 0 "0$ (' )/$ % +$ )(0 %). # "
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)($ %+$ )& , 0 " 3 $" 0 .+ )% (' &(6( % +$ , )/$ 0 . )(0 %).
5.
5.1. ) & )
% % "$ )&13/ ) , 1F , 2F , ..., iF , ..., nF , '$ ) &"
%' ") & (1 ) %+$ "* (% $ 1r , 2r , ...,
ir , ..., nr , )( %(0 (/ %0 % ) )"6 :
( ) %% )( )&13/ ) $ "$ % $" $
10
n
ii
F F =
= = 0 0v = , (5.1)
0" $ 0v &'% . %;
() %% % )( )&13/ ) . $ ) $" $
10
n
ii
M M =
= = 0 0 = , (5.2)
0" $ 0 &'% 0 . %.
&) $ "(1 " )( (5.2)." )&/ $ % &% ) $ # )%$ ( (&. "% ) % $)0 "%0
))% ).)( (5.1) " (1( % " (3 %)% %/ , " )( (5.2)
"&3% " (3 %+ %/ .
3.2. ()
) ( % % $ # " % ) &(' " -* %' , )# " %)3 &/ ) ' )%& )(0 &-/ . $ ) &%, &0 ) # %% %)%. '
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" 6 &0 .% (%# $ )&/ )( (5.2)" $ 0 M = , ") $ 06 %+$ &% ) .
&. $ &0 . '()%0 &($)0 3%& ( 5.1 ), $
"$ "( ) 1F 2F $ # )%$ ( ( ( +%# 5.1 ), - ) &(+ &0. ) %+$ $ &% ( $$ ) "$ .
3% $ ) %+$ &%, &0 ) # %% %)%, 3% ' " 6 &0 .% (%# $ )&/ )(
(5.2)
1 2 1 1 2 2 0 M M M r F r F = + = + =
. (5.3)
. ) # ( $ &(+ ) %+$, % /( % ) $ &(+ ) %+$, % (5.3)# " ) &3 ) . ,
1 2 1 1 2 2 0 M M M d F d F = = = , (5.4)")
1 1 2 2d F d F = , (5.5)
" $
2 1
1 2
F d k
F d
= = , (5.6)
& ' ) k ( .
0 & 5.1 , " $ ) "$ ) '% )% ) %+$, ( ) , # , &3% )'$ ,
.% ( 1 2d d ) ( 1 2d d = ,
&. " % $, ("% ).
! 5.1
! 5.2
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. ) "$ ) )% )% ) %+$, &0 ) ( ( 5.2 ). ( &+& +3 &"%+ " '( . 0 ' )% &"%+, $ ) .6 0. %. 36 "%+ "$ ) -
1F " . ) &"0 %% '$ $ %# &")%(1 ) 2F .
6.
2 " &)% %0 '()%0 %, '()% % "3 ) (/$ )($ " $ . ) &" "$)%( )&13/ ) . '()%0 %, & &)% "$)%( )&13/ ) % ) (6 )($ &(.% " $ . . )&13/ ) &3 $" "* (")% ( ), " " %, $ ) % (3 # (%% )($ &(.% " $ . .
$( )%' "+$ '()% % 0 ) .$)% /( %3 -/ )%%. % )%$ 3% '()%0 % &( ) )&* ( (%# )%/ ) )" "$)%( $ ) %' &" . ) % " -$ "* )&13/ ) , " " &- * )%$/ , %' -
* ) %# " (% % &%" (%# )%-/ . )%' ) ) ) )&) )%+$ %' * - ) . )%' * ) ) )%/ " (% % &%" (%# )%/ , " $ % &3 )($ .
)%' "+$ '()% % 0 ) &"% "( )( ()%: ( ) .
! 6.1
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6.1. & (%& )
0 3& )%%0 &&'0 &) , &(3 S , "$
) nF "# / ) , ( )&* & &(3 (
6.1 ). ' (' ") ") % %% ) nF
&(3 &&'0 &) S , ( )
nn
F S
= . (6.1)
)" "$)%( ) nF , "# 3& l 6 ) &(6% &
( 6.1 ), )/% & ( 6.1b ). &)% & "# - ' ) l , " 6 %( & "# ( )
.%
ll l
= , (6.2)
& ' $ 0l > & )% / , " $ 0l < & ).$/ .
%( )%' "+$ 3& &&+ $ &-, 3% &")%(1 )
nl = , (6.3)
0" $ +$% )%')% $ () ) " ). %$ " 0 $ 3& '/ . (6.3)(# &" )( " $ &* 0+ -
"+$ .+&' (")% +$% )%')% ( ) * -
1 y E
= , (6.4)
% ) % )'$ ,( # &)%
n y l E = . (6.5)
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& 0( " SI ))% $ )% $"+ ,
, 2N m .
+ ())% 0 & n " %( "+$ l
%$ ( ) ( 6.2 ). $ n np < , () -
)% $ # $ ,( (6.5); $ n np = , ")%0%
$ ; $ np n ne < < , ())% $ ;
$ n ne = ")%0% $ ; $ n ne > " " -
, % " "/ %$ , nk . )& -
" ( &")%(1 "$0 & & ).$/ .
! 6.2
) (6.5) n (6.1), l (6.2),".$ )
n y
l l
E S F k
l
= = , (6.6)
%$. &)% & "# l &&+-
$ $ ) nF , 0" $ k )%% &-
&+)% .
6.2. &
$ 0 % . &(00 &&&" $" )($ )% '()% ( -
! 6.3
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&"0, " )&% )% "$ %0+$ ) 8F , '$ $
"$)%( ( )&* & &(3 S ( 6.3 ).
&)"+ "$)%( ) 8F " " &$( , $ ) )%$
& &/ )$( % $"0 ") "0 . ") % %%
%0+$ ) 8F &(3 S $ % ) "$ ( ) -
88
F S
= , (6.7)
&" '$ ) "$)%( % "3 , %( "+$ 6 .%, ) 6.3 , $"
tg xl
= . (6.8)
)&% $ %(* " &)%$ )"6 ())% * %0+-
$0 & 8 %( "+$ % " "%0 %$, (
)
8 s E = , (6.9)
0" $ s E " )+/ %$. 50( $"+ SI ))% )% $
%0+$ &, 2N m .
!&+$ )'$ )%' "+$
)+/ $)% ( ). &. $ 0 3& . (1 ( #+ ) "# -
l &&' R $" )( $ '(36 , " / "0 $ "$
)&0 ) F F , $ $ %0+$ &(+ ") &&' &) (1 , &(-
3 S ( 6.4 ).! 6.4
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% M 0/ ) "(" " &$( %0+$0 & 8 $ $
+% &(3 $" , )% & &$. # ) & % " $
+$ % ) M , 3% ) ( )
4s
2 1l M M
R E c
= = , (6.10)
0" $ c % )%%, " $ s E " )+/ %$. $ ) ')%
)% )&%$ +, )%+$ &$" ' )%%.
7.
" $ " $ ) .( &'(/ (%# ( ) %/ ( ) %')% 0)( ( ).
2 " '()% %, $ )&) &)% $ "* . - &, %' % $ "* &, "* . ,(6 $ . )" ) . % ) % - , $ ) &)%(1 &(+ %% )( )&13/ ) $ %')% "$ . %')% "$ ) 0(%+ ) , " 6 &(3 %')% .% &(+ 0(%+ ) ( )" 6 .% %, " 6 " (6 &(3 , 3% ) , .% 0"0 . - ). (( &)%(1/ %')% &)"+ $ %0 3% %')% $ -
)%' %& %0+$ & - " )+/ $ $" ( s 0 E = ).
%, &(3 %')% %0+$ "$ ( ) , -% %')% &6 ) $" ") "0 , 3% ' " $ %')% &%1( - %'!
)(% % $ "* . % "* & (6 $ + &)% $ ) )%( )&0/ .
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7.1. (
. ) &) "$)%( ) $ %')% "$ &(3 '()% % ) $ ) %%, . ( 31 &(3 %
%')%, (" $ ' (' $ ) ( F
pS
= , (7.1)
0" $ F ) ( )&* & &(3 S . ) $ ( )&* & &(3 $ "$ , &%) ) "3 "+$ .
dF p
dS = . (7.2)
%) $ ) (' ; SI ))% $"+ &%) $ :2Pa N m= .
7.2. (/& # . ( .
) %')% )" & (10 . - , $ "( &% & '% &&'-
&) (&(3 1S 2S ) '% &(-
+ ) ( 7.1 ). (" ) )"6 &%&)%(:
1. %')% $ )%31( ;2. %')% "$ 0(%+ )
21 ;3. %3/0 %/ %')%.
&( &, &(3 1S , "$ )&13/ ) 1F , 6 ) &%
"* "# 1s , & ' & %')% (3% &%) 1 1 1 p F S = .
" $ & % ) .(1 )
1 1 1 1 1 1 1 1 A F s p S s p V = = =
, (7.3)0" $ 1V & & &%)% %')%.
! 7.1
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)" &/ &(0 &, &6 ) & &(3 2S "# 2s
&" "$)%( ) 2F $ %')% "$ "0 &, &%) 2 2 2 p F S = .
" $ .(1 "0 ) $
2 2 2 2 2 2 2 2 A F s p S s p V = = = . (7.4)
"#/ 0$, $6 (" " $ %/ , )"
1 2 A A = , (7.5)
% $)%
1 1 2 2 p V p V = . (7.6)
)( &%&)%( " $ %')% % )" " $ 1 2V V = , % )
' ".$ " $
1 2 p p= . (7.7)
)( &%" #0 # ) -)% , & ) &%) &-/ %( %')% $ $ &) -/ )( &(+ . )( # ) 1'% " 6 &%) "(% . $ %3/ &(3 )" ) %')%
, %* $ 31 &(3 % ) %')%, & ' $ &(+ % &(3 & (1 .
)( &%0 +3 , 3%) &"-)%(1 7.2 . $ &%) %')% )%% & (7.7),)"
1 2
1 2
F F S S
= , (7.8)
%
! 7.2
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1 1
2 2
F S F S
= , (7.9)
3% ' " $ ") ) $" ") &(3 $ "$ . )"+
$ " ) %( ) 1F $ &%) & &(3 1S # " -.% ( ) 2F "0 & ( &(3 2S .
" %')% ())% %( & & " &%) "% $ ) %
V V p E V
= , (7.10)
0" $ V E " )%31()% . 2 ) &)"+ $ 0%( & &-
. " $ " )%31()% " %')% " (' / 0 " '()% %, $ & " (6 0 " 0)( . !%0 ) %')% )%$ &%' )%31( .
7.3. ( # %
)%$ %')% )%%- 0)% $ ) )%/
(/ ( (%# ) )" &- 7.3 .
$ %')% (%# ," $ )( &) -
% )% %* (%# . $" %( % %')%
. % &' ".1 dh ) )%+ &(3 S , ) ". h , '% ") &(3 %')%. , % ) &%) (%#($ ) & )( ( $" ) & % %%, )&%
& ) ). (% &(+ &" ) &%) 1F 2F "$ 0(%+-
! 7.3
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) gF '0 %, % &)%$ )"6 $")% % %% &-
% ) ($ )( $ )% &(+ )
1 g 2F F F + = . (7.11)
3% $ 1F p S = , 2 ( )F p dp S = + , gF mg Vg S dh g = = = , (7.11)&)%$
( ) p S S dh g p dp S + = + . (7.12)
"(" $ $) " $
dp gdh = . (7.13)
%0+$ 0/ ".$ )
atm 0
p h
p
dp g dh = , (7.14)
0" $ atm p %)) &%) &(3 %')% ( 0h = ). 1 )"
atm p p g h = + . (7.15)
%) "% (7.15) ( ) - , &3% () " . - )" ) %')% (6 ) " ". . &. )'$ ($ ) &(3 ) ) &/ %'36 )% 0)%, ( %')% .6 )% )( )" -
( () " . )" ( 7.4 ). ( &$( ( ) .
) + U +( ( 7.5 ) "( %' -
)% '% 0)% ( 1 2 > ) $ ) 3$ (&. #(
(" ), )." &(3 ( %')% + -
6 ) '% () ( + 1h 2h ) ")
( a b . " $ &%) ($ +( $"
! 7.4
! 7.5
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1 atm 1 1 p p g h = + , (7.16)
" $ &%) ")$
2 atm 2 2 p p g h = + . (7.17)
$")% &%) )"
atm 1 1 atm 2 2 p g h p g h + = + , (7.18)
")
1 2
2 1
hh
= . (7.19)
"(" ) # '% 0)% $" %')%, $ & % 0)%
"0 %')%, &)% / () )%.( %')% 1h 2h .
)%& "*(/ 0)% &*/ $" %')% ) "0 )&%- )'$ , " ) %')% 3$ , % $ .
7.4. ( )
3% $ (6 ' , 0)( $ "* . % "* &- , (6 $ . & )" ) , 3% ' " $ )." &(3 . ( $ &)"+ %'0 %/ 0) $ ) % )." , &3% ) *). )%$/ $ ) * - ) . %) 0) &)"+ $ " "( )" , () " /( 0)% %&%.
2 " %')% $ ) &%' )%31( , 0)( ) % - . $ %&% 0) )%%, " $, )( )(0 +- $ -
0 0 const. ( , const.) pV p V m T = = = (7.20)
($ 0' $ & 7.6 .
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! 7.6
)% 0) % )'$ )
00 0 0
m m p pV p V p = = = , (7.21)
0" $ 0 0m V = &'% 0)% 0) .
)$ 6 .% & " -$ -%( (# (% ) 0)( " $ $ &%&)%(1 " ) )" )%' , " $ 0) 1( ) ") /( *). )%$/ * - ) ) 1( ). ! $ ). " 0)( ) &%)+ () %&%, *% 0) -
( (&. 2H , 2O , 2N , He, ( " ) $ ) &3$ " 0)( ).
%&% %)) &%).
%) 0)( ) . +& " $'36 36 -0 %(0 % .$3/ $ .
7.5. #
2 #(% 21 $(#$ " )( 0)( $, ( , ( " . "3 %' 21 ( ) # &")%(1 0) )3 $,
( , )"# ( &) &+%): 78% 2N , 21% 2O , 0,94%Ar,
0,02% 2CO ," )% 0)( (" & % / ' . %-
) 21 ( $ ) 0(%+$. ) &)% " %& 200 km , # ) 0. &"% % )$: , .
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&) ) &)% 0" " 10km. %&) ) $(6 "
(" & $ ")&( %) . &% &" ) () 10km
) %& 60 C . &)"+ 0(/ % )'( + $(1 )
(% , %* % )%$/ ( " ( ). !( #(% 21 "($ ) %&).
!%%) ) &)% " 10 40 km . /$ )%$/ , " $ '
(" & %. &% &" " 80 C .
) ) &)% " 40km. ( $ ". .0 $ ( 0)(
)% )" %1.')%0 '/ ) !+ .
)% ( " ) 31,29kg m = , # ) ""% /
) . & % & ) . ( " %.
%)) &%) &")%(1 &%) $ (3 ( " 21 %) )" )($ %#.( 0 $ ' 1645.0" , )"6
' : #( ) & )% &(% "# 1m. 2% ) %( &(% %( &)%, +( )
& )&)% )" ) #( , % ) &)% ( 7.7 ). ) #( )6 760mm " &(-
3 #( )" , & ' $ )&% (" ( . " #( +( )%$ & &)% (( , ) &-
.# , ."6 " 0 )%( )&/( & #( , '$ $ &%) 1( ),)" " $ ( #( +( '$ $ ")%%' &%) (%# ) %-)) &%) $"
atm 101.325 Pa p g h = = . (7.22)
2())% %))0 &%) ())% " () " ( $
7970atm 0atme
h
p p
= , (7.23)
! 7.7
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0" $ 0atm p %)) &%) ( ( 0h = ), " $ atm p &%)
() h ( %) " ( .
%)) &%) "* )% () " .
(6/ (#)% ( " (%$. &(6/ ' (" & ( " ), "( -" " &" %))0 &%), $ $ 0)% (" & / " 0)% ( -" . )( ())% &%) ( " " (#)% , % 0 " &"(" & ( .
%)) &%) ) . $'36 ) &%.1( - , $ $ )3% '$( +( .
$ %)) &%) 101,325 kPa ," $ &.# ) ( "30
)%. &(3 21 m , &" &%&)%( " $ 29,81m sg = " () " -
")% km, $"
g 4
2
10 N 10 kgm9,81s
F m
g
5
= = . (7.24)
('$ % $ )%/ " "# ( ) &3% $ %3/)% 6$ &%) $" %)) , % ) *). & $.
7.6. #
) " )%% 0)% f )%/ (/ ( 7.8 ).
" " $, )( /0( % & %* $. $" -
31 &(3 .(% % &)%0 " , '$ $ & V . )( % &(3 S &)%0 % " "$ ) iF $
$ S , % %%
i iF p S = , (7.25)
0" $ i p &%) " $ () ) " ". $$ ) S )&"
)." &(3 . &)% % " $, %% )( &(3) ) &%)
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1
n
p ii
F F =
= , (7.26)
.% $" & &(+ % %%, )&% & ) %# &)-
%0 % " f pF Q= , (7.27)
" &(+ "$)%( & % /0( %#3%.
)( (7.27))" " $
f f f pF Q m g V g = = = , (7.28)
0" $ f m ) &)%0 % " .
) )" &)% % " '()% % )%0 . %# ( 7.8b ). ! &%) $ " "$ % )%6 -
&/ , % 6 " /( %% pF "(% %, ) -
(3 . % ) ( ) , % %% "%0 (7.28),%$. , : % &%&1 " "$ (% (3 ) &%) $" %# " )%)%0 % %. %$6 ) $ "$ %, %$. % )
t f t f ( ) pQ Q F Vg Vg Vg = = = . (7.29)
! 7.8
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2' " &(" %# % () " 0)% % " ,& ' ) $ % )'$ :
1. t f > , % % % " : pQ F > ;
2. t f = , % ." % " : pQ F = ;
3. t f < , % &( : pQ F < .
) +3 &+& ) &%), 3% $ " -%1$ &) .
7.7. (/ #
)%$ &$( &" 0" ) &(3 %')% &3 . &, 0 )%+/ %')% . &, %/ )% & &(3 (" , &(1/ " )&+ , %". (( )' &$( %% ) "$)%( ) % , %$. "$)%( * &(-' ) . ( ) %0 ) "%, (6 )%$/ " -%( &&' /( "$)%( $ 1( . % )%-
$/ ( ) , R , " ) ) %0 &-&' ( .
) "( 1M 2M
)%$/ 1d 2d " &(3 %'-
)%, & ' $ 1d R , 2d R<
7.9 . )" )%'0 )&"
))" 1M , -
%$6 ) /(0 "$)%( $" $ . )&" ))"
2M $ )%' , &3% $ 0)% 0) , $ ) /0($
) 0 "$)%(, % / " 0)% %')%. -
%% ((0 )&" , 2M "(6 %% 2F ))" -
, $%) %3/)% %')%.
! 7.9
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)( &%"0, # ) 1'% " ) &(3) )$ -%% * ) ) %3/)% %')%, %$. &(3 -) )$ "$ +& %')% . %')% "$ )&13/ ) , (%# $ .6 $ ) %% * - ) &(3) )$ &(3 , )." %')% "* ) % $ . . ) " )( 0%$) % $/ &(3 & "%$ &, & "* ) %')% 0 . $ $ ) , &)"+ )// / &(3 . )" %"+$ &(3 %')% " ) )/$ , # ) )%% " $ )%/ &, ) "$ )( &(+ , & &(3 %')%. ($ & ( ) % , "0($6 ) ) ) &(3)0 &- .
! &(3)0 & 0 ) (%%%( ""% & &(00 " #+ ) $" &% )%, &%&10 )&+ ( 7.10 ). ! &(3)0 & %#6 " )/ &(-3 . " )&+ , % 6 "(% &%
)% %% F , $ ) # (%#%
"* )&13/ ) F #
F F =
, (7.30)
, "0'$
F F = . (7.31)
)&% $ %(* " $ ) &(3)0 & &&+ "# "" 0+ , % " % )'$ )
2F l = , (7.32)
0" $ 2l "# "" 0+ , " $ % $
() " &" %')%. "(" )" " $
! 7.10
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2F
l = , (7.33)
&")%(1 ) & $"+ "# "" 0+ .
/ &% )% s # , ) F (3 " "
2 2 A F s F s l s S = = = = , (7.34)
0" $ S &(6/ )." &(3 . ) $" )%, " $ 2 S
& &(6/ )." &(3 . . (7.34).6
2
A
S
=
, (7.35)
% &")%(1 " &%. " ) )." &(3 &(6 $"+ .
)( (7.33) (7.35), $"+ ) N m 2J m SI ))%
$"+ .
)%% &(3)0 & &" ) &(6/ %&%.
7.8. ( ) )
" %')% "" '()% % $(1$ ) "( ()% ) : . ) ) ) &('/ * )% ()%, " ) " ) ) &('/ * '% ()%.
) ) / " " , # ) " %')% () '()% %, + & "" (" )% ( 7.11 ).
! 7.11
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%% F $ "$ &(3) )$ %')% "% $
(%) . ) $ koF ) " $ adF . ( .% -
0' &(3 %')%, ' . &)%$ &% ) (
)." &(3 $ . "(" " %/ %')%. - % -* %0% &(3 %')% %0% &(3 '()%0 %, -
7.11 / $ " 2 . $ 0 = , (3/ $ &%&, &.
7.11b & (" %$ &(3 )%.
) ) (6 " " , %" %')% () '()% %,
" , + , #( )% ( 7.12 ). $ 0 (3/ (6 " 2 .
$ = , " ) " &%& (3/ .
! 7.12
" )% %')% () $", () "0 '()% %. &, (" () )% () & , #( () ')% &(3 0(#* ( -) )%.
7.9. "#
&)% $ &$( " ) %')% ) - +( (&' 1mm / )
&3 & )&$ )"( , (6 - $ " ( %')% () +( ), )&" ( %')% () +( ) ( %' -
)% )" ( 7.13 ). 2(1 )." ! 7.13
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&(3 %')% & ( ) ( . %')% () & , () %')% () & .
%')% &%& () " & ( 0 = ), 3% $ )'$ ) ("
)%$ & ( 7.14 ). 3% ) & %')%, " - ) &(' %')% ", 3% "(" " )%(/ (0
)) . 50( 6 &(3 ) &(3)0 & %#% " )/ , % 6 "(% & /0( . (3 , 3% 6 "()% " &" / ( %')% & , )( " ) %% ) &(3)0 & $"' & %-
%% ) %# )%. %')% " ( %')% )"
F Q = . (7.36)
! 7.14
) &(3)0 & "$ & "# "" 0+ ((" $ % . 0 2r & ), )"
2F r
= . (7.37)
# )%. %')% & " ( %')% )" )
2Q mg Vg r h g = = = , (7.38)
0" $ r &&' & , 0)% %')%, " $ V & )%.
%')% & .
(7.36).6
22r r h g = , (7.39)
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" $
2h
r g
= . (7.34)
& &$( " %0 ) '$ &) )%( ) 1( %$ " )%( .1 .
7.10. )
7.10.1. ! . ! . # $ .
. ( ) " $ " $ )
.( &'(/ %/ &" $ ) &" ( &)-% )%31( &)% () " . "+ " &"% -%/ $ %/ " ( ") "0 %), (6 %/ $"0 " , $ ) )% &" .)' ( - , ") .
$( %", )%/ %/ %')% # ) &)% ) "" (% . +$ ( )( %' &)%
( , )v v r t = . (7.41)
( &)% ( ) . % )'$ ')%+ " ) 6 "#
'$ ) %0% )($ %' && ) &(+ . v ( -
7.15 ). )% )%$ $ (6 $ % 0" $ (6 . " . &3% )'$ , )( %' &)%- %. & (%, &" . , $3 &%) 0) -%
( , ) p p r t = , ( , )r t =
. (7.42)
$ )%$/ " , . " )%$ )%% % ( . % )'$ , )&-" )%$ $ .6 )%, ") % " ) 6 $" " -
! 7.15
! 7.16
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0 & )% )%$ $ , % ) )%$ $ &&$ ) % " . !%+ )%$/ $(1 ) & / . .
(3 $ . $ )%$ $ &(' )( %' %(
%, ( ) ( 7.16 ). "+$ )%$ $,# ) 1'% " " # &6 &(3 )%$ +( , %$. " $ ) 6 % )%$ +( 3 ) ) " ( / .
) )%31( " 6 )%$ +(
& 7.17 . ) 1v 2v )"/ . -
)%$/ " &&' &) &(-
3 1S 2S . 2& " 1V , $ &* &(3 1S ( t , $" $ &
(1)%0 % )( 1S () 1v t
1 1 1V S v t = . (7.43)
)% () %( , &(3 2S &%6 6 & "
2V , $ $ "* 0
2 2 2V S v t = . (7.44)
$ " )%31( , .6
1 2V V = , (7.45)
" , & (7.43) (7.44),)"
1 1 2 2S v t S v t = , (7.46)
")
1 1 2 2S v S v= . (7.47)
( (# $ "( &) )%$ +( , &3% )'$ ) )%$ +( # &)% )"6
const.S v = (7.48)
! 7.17
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(7.47) (7.48)& % ) .
' ('
V Q S v
t
= =
, (7.49)
( ) , & ' $ V & %')% $ % ( t &-
% &&' &) &(3 S . "+ &% SI ))% $ -
, 3m s .
7.10.2. '%! #
" ()%31( () -) " , 0)% , )%$ )%+
"# )%$ +( $ $ 0% ") % ( 7.18 .) " * &&' &) 1
2 (&(3 1S 2S ) ) m , )"/ . -
" ) 1v 2v , " ) () " -
) % ( 1h 2h .
" ) 6 ' ) &" "$)%( 0(%+ ) )
&%) $ "$ &(3 1S 2S . 2 ( t &)% " "
&6 ) "# )%$ +( $ 1 2 $ 1 2 . $ "
)%31( , )( $"' %%% (7.47) &(3 1S 2S # )
&)%
1 1 2 2S v S v= . (7.50)
( ") )% (7.50) ) t ( const. = ), ".$ )
1 1 2 2S v t S v t = , (7.51)
%
1 2m m = , (7.52)
! 7.18
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0" $ 1 2m m m = = ) " $ % ( t &% &&'
&) &(3 1S , ") 2S . * $ 1 2 0$ " )%$
)%, % ) # )%% " $ ) m * $ 1 1 &3 &)-
% * $ 2 2 . % )'$ 6 & & %' &%+- $ 0$ ) m .% $"
2 22 2 1 1
1 12 2
E m v m g h m v m g h
= + + . .(7.53)
" $ (3 ) &%) % ( t )
1 1 2 2 1 1 1 2 2 2 1 2 A F s F s S p s S p s p V p V = = = , (7.54)
0" $ 1s &/ &(3 1S $ 1 1 , 2s &/ &(3 2S
$ 2 2 ; 1 1 1F S p= $ ) $ "$ &(3 1S (3 & %(
" ( &(+ %), " $ 2 2 2F S p= ) $ "$ &(3 2S (3 0-
%( " ()&% %).
)( "#/ ' 0$, & & 0$ -% " $" $ " )&13/ )
E A = , (7.55)
" , & (7.53) (7.54),)"
2 22 2 1 1 1 2
1 1 ( )2 2
m v m g h m v m g h p p V + = . (7.56)
1/ ( ) V , &% (*/ ) 0)% m V = ,".$ )
2 21 1 1 2 2 2
1 12 2
p g h v p g h v + + = + + , (7.57)
% $)%
21const.2 p g h v + + = , (7.58)
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3% $ & % + , $ #( "#/ 0$ &
)%+ %/ "0 " . p (7.58) ( ) -
, ' g h , ' 2 2v .
7.10.3. (' '%! #
)
7.19 & $ )" &/ %'36 0)% , $ )%' %( ('$ &&' &-
) &(3 2S ) $ ) ". h ) -
&" )." &(3 %')% ('$ &&' &)
&(3 1S ). $ 1 2S S , " , & $"' -
%%% (7.47),)" " $ 1 2v v , % ) #
% " $ 1 0v . -$( $"' , $6 (" " . &-
(3 1S 2S "$ )% %)) &%) atm p , ".$ )
2atm atm 2
12 p g h p v
+ = +, (7.59)
" )" " $
2 2v gh= . (7.60)
(7.60)&")%(1 , $ # " $ . )%-
+/ %')% &.30 )" )% " %')% )." &" ) () h .
. )%6 " . 2v $ )%%, (6 &" ) ( , $ () h &" ) ( .
! 7.19
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)
7.20 & $ %$( +( ,3% $ )%( $" % +( ) &&'-
&)+ '% &(3 1S 2S , $ ) )%$ () ") . - $ % ( . !%0 ) & - - $( $"' ".$
2 21 1 2 2
1 12 2
p v p v + = + , (7.61)
%
2 21 2 2 1( )2
p p p v v = = . (7.62)
$, & $"' %%% (7.47)
12 1
2
S v v
S = , (7.63)
(7.62))"
2 2 22 2 2 2 21 1 11 1 1 1 22 2 2
2 2 21 ( )
2 2 2S S v
p v v v S S S S S
= = =
, (7.64)
" $ . 1v $"
1 2 2 21 2
2( )
pv S S S
= , (7.65)
% 6 &% " , & (7.49),.%
1 1 1 2 2 21 2
2( )
pQ S v S S
S S
= =
. (7.66)
! 7.20
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(/ &(3 1S 2S &&' &) %$( +( , 0) -
% %')%, / &%) p &&' &)+ -
%, # ) , 36/ (7.66),""% &% " .
)
7.21 , 06( / &0 &%) ( )%%'0
"'0 ). %( %( +( )%( ) ( 2 0v = ). - -
$( $"' # " ) & "* &) +( &(3 1S
$ &%' " , $ $ " " %( %( +( (0" $ . "
' ) 1v , &%) ) 1 p ), &) &(3 2S $ )
( %( %( +( (0" &%) -
) 2 p , " $ . 2 0v = ). ) -
% ( ) %( +( ( 1 2 0h h= = ),
" , & -$($ $"' , )"
21 1 2
1
2 p v p + = . (7.67)
% & $ * &-
%) 2 p %))0 &%) atm p
2 atm p p p = , (7.68)
(7.67) (7.68))" " $ & &%) " +( ( )%%'
"' )2
1 1 2 atm12
p v p p p + = = + . (7.69)
! 7.21
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) ( )
&& ) (" ( 7.22 ) ))%$ ) " +( B &1(0 &&-
'0 &) '$ ) # " , &(3 1S "(" (" '$ $ . %-
( ( )" 0 0 %( +(
. ) ) 1v ' . ("0
&) &(3 1S +( B, ) 0v .
("0 &&' &) &(3
0S ) +( B, & $"' -
%%% (7.47))"
1 1 0 0S v S v= , (7.70)
" $
01 0 0
1
S v v v
S = > , (7.71)
&3% $ 0 1S S > .
) )" &&' &) &(3 1S 0S & -$( $"-
' (7.58), (/ ())0 ' &3% $ ()) *
1S 0S , .6
22 011 02 2
vv p p + = + , (7.72)
" $
2 21 0 0 1( )2
p p v v = . (7.73)
$, )( (7.71) 1 0v v> , 0/0 )"
1 0 p p< , (7.74)
! 7.22
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3% ' " 6, )'$ " $ 0 p %)) &%), &%) 1 p .%
/ " %))0 .
( &+& " % , " $ ( " &(' ())( )
%')% )" )&3( $ &)%.
7.11. )
%')% 0)( &&"$ 0& . ) )% ) "3 %3/ %/ " , &)$ ) ) %/ , $ ) $(1 & %( %/ $"0 )$ " ") "0 . , ( (#. .$ 4, &'( ) () %/ * )$( %')% $ ) * & &' " $ $" $, " ) "0 6 . v .! () 0 %/ & )$(% ( ) %/ "* $ " -
vF S
x
=
, (7.75)
0" $ S "" &(3 "( )$ %')%, v x $ 0"$% . , %$. -
. "( ))" )$ $ ) *). )%$/ x , "
&")%(1 +$% () )% %0 " , $ () " ()%
" /0( %&%. 50( $"+ SI ))% $)%
, 2Ns m .
) () , )%31( " 6
)$(% ( ) +"' +( -0 &&' R ( 7.23 ). ) "#
+( ' ) l , 1 p 2 p ." &%)+ -
$( +( , & 5%( (7.75),# ) & % " $ . " )%( &. -
' +$ )%$/ r " ) +(
2 21 2 ( )4
p pv R r
l
= . (7.76)
! 7.23
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- " $ $(6 "# ) +( (" $ 0r = )
21 2
4 p p
v Rl
= , (7.77)
" $ $" "# "( +( (" $ r R= ).36/ (7.76),# ) & % " $ &% +(
7.23 "*
41 2
8 p pdV R
Qdt l
= = , (7.78)
$ # " $ &% +( &&+ 0"$% &%) 1 2( ) p p l .
$( # " ) & , + , & . 0 $+$.
&% Q () " 4 R , " $ 0 0 ) (#$ " &%) &)%
&+ . & &%+/ %')% 0.
$"' %%% (7.48),&% )%$ +( $ )%%, % ) , & (7.78),# &)%
41 2 const.
8 p p R
Ql
= = (7.79)
$ 4 8 R )%% ' , )" " $
1 2 const. p p pl l
= = , (7.80)
3% ' " )%%' &%) " () 0 " ( &" "# )%$ +( )%%0 &) ( const. p l = ). )'$ "0 " ,
)( -$( $"' (7.57) +( 7.23 , .6
2 21 2
1 22 2v v
p p + = + , (7.81)
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$ $ 1 2h h= . $, & $"' %%%
const.S v = , / . '/+ " )%$ +(
( )% &) ( const.S = ), )" " $ const.v =
( 1 2v v= (7.81)),% (7.81)&& .
1 2 p p= , (7.82)
")
1 2 0 p p p = = . (7.83)
)( $(0 # ) ""% +$% () )% 36/ / - , & 0 7.24 . )( (7.78) $
40
0 08 pV R
t l
= , (7.84)
4
8V R pt l
= , (7.85)
0" $ 0t ( &%. " %')% & %0 +$% () )% 0 &%-
+( "# l &&' R , " $ t ( &%. " %')% &-
%0 +$% () )% &% )% %( +( . 1/ -
(7.84)) (7.85)".$ )
0 0 00 0 0 0 0 0
p t g h t t p t g h t t
= = =
, (7.86)
0" ) 0 0)% & % & % %')%.
$( (# )1'( ')% %')%, % $ &1( )) -
& $, ")& $ 3( '% )&)%+. ')% $ (# - $( ($ ) , " ) $ (# -
! 7.24
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($ , &&% 0 .3 %')% (( , ,(6 )( ). " )%+$, 7.25 & $ ())% &-% (" (/%() %')%) ( (/%() %')%) ())% " 0"$% &%). ( &$( # ) .$)% .+ )) -& $. , " "0' ( . 3%&), & )+/ ) -(1( /( $%+$ ) %, )/$6 % () ) .
8.
+) $ ) &(1$ )% ' &) "*0 () %-( ($ ) , , &, )+(/ %, %/ &% !+ , " )'0 36 , ")/ , %".
"')% (' # ) %%' &)% () () +$ )"60 .
( ) ( ) f t T f t + = , (8.1)
0" $ T &" &% +$ .
"' %/ $ ) (3 "# )% &-
%/ , ' "( )&% ) (%#0 $ , ( ) .
$ % ) m .3 )%' &0 8.1 . #$ % $ .6 ' )
0 x = ( ). " ) % (" ( -%#0 $ )% / &0 )%$/ x A=
, % ) &)% 6 &'% " ) . 6 & (%# $ &" "$)%( )%' ) &0, % %%
! 7.25
! 8.1
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elF k x= , (8.2)
0" $ k +$% )%')% &0, $ () " " $ &0 )%' ). " $ $ &0 '/ . %) . )%' )
0)
elF k x= , (8.3)
0" $ x (% )%$/ % " (%#0 $ . 2 &%' " %0
3% $ ) )%' ) )&% " ) (% )%$/ x .
%% " % ")%0 (%# $ 0 x = , &0 $ " -
) , % ) % %% )%' ) , & (8.2), . / %, & "0 5%( , $" . *%, % )%(1 " ) 6 &(+ )%-' . v ( ) )&% " ) x ) ) ' )%' %' 0$, ).$$6 &0, % % ($6 /$ )%' ) $ %# " )&' "1 ).$/ &0. " ) & %' 0$ &%( &-%+$, % ) %% )%(1 ( 0v = ) $ x A= . %0 &-#$ % &'/ " ) . 6 &" "$)%( )%' ) ).$ &0,&% (%# $ ( 0 x = ). (%#0 $ % )%(1
%/ &(+ )%' . v ()" ) $" ) x ) ) &% ' )%' %' 0$, )%#6 &0, % % ($6 /$ )%' ) $ %# " )&' "1 )% / &0. " ) & -%' 0$ &%( &%+$, % ) %% )%(1 ( 0v = )
$ x A= . %0 $ % &( &'/ " ) . 6 ( -%# $ &" "$)%( )%' ) (' &0, &% ( -%# $ , % ) '%( &+) &"' &(1 .
(( %/ % "))%( )&13/ ) ( ) - ))% %- &0. *%, )( )&13/ ) %/
0 ) %, % ' 0$ )+%0 %/ & %&-% 0$, % 0 ( )+% %/ &)%$.
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)+%0 %/ 0 .% ) $ ) )%' &" . &, %%' ( ' ) % )+$ &" "$)%( %( &% 0(%+ ) , & ' $ , )+ -
+$, ) &/ x " (%#0 $ (%%' -' % % ) ).
8.1. * &
$) )+(/ $$")%($ $ . )+%0
%/ , $ ) "($ & &($)$ &%/ , &" "$)%( )%' ) elF
$ $ ) )%$/ " (%#0 $ x , & (8.3).
) ) %/ 0 %, )( "00 5%(0 % ) m 8.1 # ) &)%, $6 (" (7.3)
m a k x= , (8.4)
0" $ a . / %. ) (% a x , (3.4)# " ) &3 )"6 .
m x k x= , (8.5)
")
k x x
m= . (8.6)
(*/ )
2k m
= (8.7)
(8.6)&)%$
2 0 x x + = . (8.8)
(8.3)&")%(1 "+$ $"' "00 " ) )%%- +$%, '$ $ 3/
cos( ) x A t = + , (8.9)
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0" $ x ( $ )%$/ % " (%#0 $), A $ ($-(6 )%$/ % " (%#0 &-
#$ ),t
+
$ )+(/ ,
)+(/ ( 0t = ). (8.9)&")%(1 $"' 0 $ -)0 )+(/ , 0' &")%(1 8.2 .
$ )) ( )) , )($" ) &"' +$ ) &" 2 ,# ) &)%
2cos( ) cos( 2 ) cost t t
+ = + + = + +
. (8.10)
(8.1))" " $ &" )+(/ T "*
2T
= , (8.11)
0" $ # (+$ , $ ) # ""% (8.7).
-$ )++$ $"+ ( ( ) ( )
1T
= . (8.12)
(8.11) (8.12)".$ )
2 2T = = . (8.13)
8.2. ) ) ) &
( 6 .% "* " 12 A )%' ) &0 )+%0 ))% )
8.1 . )( "+0 " ) (3.42) (8.3),# ) &)%
! 8.2
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2 2
1 1
2 212 el 2 1( )2
s x
s x
k A F ds k x dx x x= = = . (8.14)
2' " " )%' ) &0 () ) " &'%0 $/0 $ 1 x
2 x , 3% ' " $ )%' ) &0 (%( . % )'$ , 0 -
(3.60),.6
12 p p1 p2 A E E E = = , (8.15)
0" $, & (8.14)
2
p( ) 2k x
E x =
, (8.16)
)%' &%+$ 0$. ( 0$ $ $(6 x A= )
2
p( ) 2k A
E A = , (8.17)
$" $ 0 x =
p(0) 0 E = . (8.18)2 . % ) , )( (8.9),".$
sin( )dxv x A t dt
= = = + . (8.19)
% )'$ ) %' 0$ % $ ) )%$/ x " ( -%#0 $ , $6 (" (8.7),".$
2 22 2 2 2
k
2 2 2 2 2
( ) sin ( ) 1 cos ( )2 2 2
cos ( ) ( )2 2
mv m k A E x A t t
k k A A t A x
= = + = +
= + =
, (8.20)
$ $ $(6 0 x =
2
k (0) 2k A
E =
, (8.21)" $ $" x A=
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k ( ) 0 E A = . (8.22)
& 0$ % $ ) )%$/ x " (%#0 $ $" $, & (8.16) (8.20)
2 2 2 2 22 2
p k( ) ( ) ( ) ( )2 2 2 2 2 2k x k k x k A k x k A
E x E x E x A x= + = + = + = , (8.23)
)%% $ )( ( %/ .
8.3 &- $ &%-+$, %' & 0$ ( -) " )%$/ % " (%#0 &-
#$ x .
8.3. ()/ &
"' 0 $)0 )+(/ (8.9) (" $ &" &%&-)%( " ) ) %/ # %. $ &%" %(0 )+(/ )%% ( , ) ( % ( ) $) )+(/ . *%, " )(0 0 )+%0 ))% ) %/ -
# ) .6 , 3% ( "(" " &"/ &%" )++$ . ( )+(/ ( ) % ( - ) $) )+(/ .
%( $")%( %%' &% # " ) % )+% ))% & 8.4 , $, &" )%' ) &&+ )%$/ " ( -%#0 $ , "$ ) () 0 %/ . 2 . %/ )+% , ) %/ () )" &- ) $ . , ) $$ $ )&-% " ) .
! 8.3
! 8.4
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trF rv= , (8.24)
0" $ r +$% %& $ () " " $ . %, " () %)% )" .
"0 5%( , &(+ x ) )+(/ ))% # " ) &3
m a k x rv= , (8.25)
, &3% ) )( % (% , $"' (8.25)# " ) &3 )( ) .
m x k x r x= , (8.26)% $)%
0r k x x xm m
+ + = . (8.27)
(*/ ) : 2 r m = 2 k m = , (8.27)&)%$
2
2 0 x x x + + =
, (8.28)3% $ )%( 0 "+$ $"' "00 " , '$ 3/
4r km< )
1e cos( )t x A t = + , (8.29)
0" ) A &%" &'% , $ ) "*$ &'% )( -
%/ , " $ 1 "* 2 2
1 = . (8.30)
(8.29)& $ 0' 8.5 , 0 = .
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! 8.5
$ 1 < , (8.11))" " $ 1T T > .
0 ( 0r ), &03 )++$ &.#($ ) &03 .
% )'$ (8.29)&& .
1cos( ) x A t = + . (8.31)
2 4r km %/ $ , % (" (%#0 $
(6 ) /0 , )" ( ) %/ , &"' .
8.4. ( &
" $) %/ )+%0 ))% ' )++$ $ ) ( "$)%( "* )&13/ &"' ) . " %( )+% ))% & $ 8.6 , '$ )&13/ &"' ) # " )
&")%( )) +$
0 pcosF F t = , (8.32)
0" $ 0F &%", " $ p # (+$ &"
) . "' %/ %(0 )+%0 ))% 6 .%
m a F k x r v= . (8.33)
) )( (% , (8.33)# " ) &3 ) .
! 8.6
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0 pcosm x k x r x F t + + = , (8.34)
%
0
pcosF r k
x x x t m m m + + =
. (8.35)
(*/ ) : 2 r m = 2 k m = , 0/ && .
2 0p2 cos
F x x x t
m + + = , (8.36)
3% $ )%( 0 "+$ $"' "00 " . 5 )%+- 3/ $
pcos( ) x A t = + , (8.37)
0" $
02 2 2 2 2
p p( ) 4F
Am
= +
(8.38)
p2 2
p
2tg
=
. (8.39)
2())% &%" )+( -
/ A " p & $ -
8.7 . &%" )
(")% " $ p r = , %$. " $
(+$ &" ) $"
2 2r 2 = , (8.40)
% &$( ( ) .! 8.7
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$( +$ " 00 $ '$ , ."6 " $ &" % $ . & )%+$ )%( , "%+ , )%+, &%+, %)$ -$ +.
9.
9.1.
&%" &0(1 . $ % )+(/ $"0 %, ("6 ' &)%$/ "0 % /0 . ( (0 &0(1 6- . &)%$/ "0 % (')%+ ) ) $ $ % (')%+ ) $ )+ -
$ ( . )%'$ )" (% # .% '()% %, %')%, 0) ), 0" ) ')%+ )" *). &( )%' ( , ' "+$ ()+(/ ) $" " )" &)6 ) )% "( )" .(( ' &3/ "+$ ()+(/ ) )%' )" -( ) , ) &$( . . & % " ) ) -" $ ) %) &)% 6 + &(+ &)%/ %),(6 ')%+ )" )+$ )( )(0 (%#0 $ . , -0$ " '0 %) 6 ) )" &3/ )+%0 -%/ ) ')%+ ')%+ % )" , ) %/ ')%+ % ) -" . %$ ' , 0$ %)0 %/ # " ) &) ( ( )%$/ .
' %) )6 )(" ) . &, %/ %) & &(3 (" , &)%/ (' %), %".
" ' %) &)%$ "0 ()% %) " $ ) $(#$ . 2 " ' %), '$ &)%/ $ &%. ' ))% ()" $ ) ))%$ " ')%+ ), %-0% %) 0 " ) &)% &)% . ')%+ (( ). %-0% %) &)"+ ) )+(/ .6 -% ( .
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' %) 0 ) &"% , ())% " &(+ )+(/ ')%+ )%' )" ") &(+
&)%/ %).
" %) %) ')%+ )%' )" )+$ &(+ &)%/ %). (" 6 .% % " )%' )" $ ' % *). &( &0 )%( ' &0 9.1 , & 0 ) $")%( ' # )(%% " &-) )+(/ " %) 0 %).
) % &( (" (%#0 $ (3 ( # ), %-
&'/ " "$ "( ) : &( ) $(1 & (3/ , %# " (% % &(.% $, " ) "0 $(1 & * % & "0 % (%#0 $. %$ ' "0 % &'/ " ) 6 &(+ 0-" . 0 , )+(/ "00 % "(" " )+(/ %60 %,%". %$ ' , )+(/ &(0 % "(" )+(/ $" "0 % $ ) &(+ &(+ /(0 )+(/ , 3% ' " $ &(+ &)%/ %) &(+ )+(/ &(0 ( &% )( )%) %. . & % " )+(/ "00 % ) )+( -/ &(0, ") " )( )"6 % ) )+(/ ") &%-" (# ) " ) ). ( ' " &)%/ %) . .
)( % %) # )(%% (3 $ % " & 9.1 %)%' () %-( , $ ) 3%) & 9.1b .
&'% %% ( 0t = ) ." $ " ) )( % (%# &-#$ , & ' $ % 1 ". . / ) (3 , &'/ " )+$
$) )(0 (%#0 $ . %% 4t T = , % 1 ")%0
$ ) "1/ " (%#0 $ (&%"), " $ % 2 ". - . / ) (3 .
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%% 2t T = %-
1 )( %/ -# & (%# -
$, % 2 $ ") -%0 &%", " $ %- 3 ". . / ) - (3 . %%
3 4t T = % 1 ")%0 $
&%" # , % 2
)( %/ # & (%# $, % 3 ")%0 $ &%" (3 , " $ % 4 ". . / -) (3 . %-
% t T = % 1 & (%# $,% 2 ")%0 $ &-%" # , % 3 )( %/ # & (%# $, %- 4 ")%0 $ &%" (3 , " $ % 5 ". . / (3 , & 6 (%#0 $. , % 5 &'/ )($ %/ (3 )%( " % 1. $ $" % 3% % 5 &'/ )($ &(, % 1 )($ "0 )++$ . 0 ' , % 1 5 )+$ )%$ . &3% ) -'$ , "( ')%+ )" $ ) )( %% 6 )% . , ) -% ) &"$" ) "1 " (%#0 $ , # ) " ) .)%$/ * "( $.# ')%+ $ ) ( ) . ' " $ )%$/ * % 1 5 $" %)$ "# '0 %) $ ) &)% " % 1 % 5. $ %) ")& " % 1 " % 5
( $"0 &" )+(/ T , " $ %) "# $" & (" &" )+(/ . %)
! 9.1
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