vladimir pejčev - fizika za studente hemije.pdf

7/26/2019 Vladimir Pejev - Fizika za studente hemije.pdf

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6 5%( # ) % : %+$ "($ % *). ) $" )&% ) , ") +$ $ )&% & % %% $" +$.

&, 21 () Zm ) "$ % () m )

0(%+ ) gF , & +$ +$ 6

% "(% 21 ) gF ( 3.3 ).

3.3. + ) #,

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) 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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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

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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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42

'% $ # "$)%( +$ ) " )% )() ." & (%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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43

+$, &($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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44

%+$ )&) %, ')%+ & "# )%' ) .

&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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45

% ) 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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46

! 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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48

%)% %/ +% ) %+ %/ +% ) &-% %' .

)'$ " $ % ( ( 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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49

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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50

) & (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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51

( & ) $" % -%%, )&%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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52

("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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53

!/ (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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54

% $ " )&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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55

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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57

)($ %+$ )& , 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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58

" 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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59

. ) "$ ) )% )% ) %+$, &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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60

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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61

& 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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62

&"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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71

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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72

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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73

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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74

)( &%"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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75

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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76

%% 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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77

&(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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78

" $

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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79

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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81

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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82

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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83

)

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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84

(/ &(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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94

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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