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elec

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lum

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_λ^c

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#1$8 L P

elec

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elec

∆t = U I ∆t 1T Q = I∆t '#8 62 )!23:' 81826' 4$# 10$M6' 420# 6'# 831$# $6'#<

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³

+<

/0 1M8$'08 410) 4$3')8'*'08 ∆t =

_P^{U Q}_elec

< W<Z< L ∆t =

3×1,5×8,6.10³

10

= 3900 #<

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elec

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,#( +1+ /" %,/ (*'% '%(+6+#(, 444

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#1$8 L P

lum

= 10 S< U$ )!2%&' 56')8310 2##208 420# 62 H.I 4100' &0 !1810D 62 4&35' 4' 710)8$100'*'08 4' 62 62* ' #'32 4$3')8'*'08 6$5' 2& 01*M3' 4K56')8310# V4$# 10$M6'#V 420# 6'# 831$# $6'#< /0 A1$8 )62$3'*'08 %&' 62 V)!23:' &8$6'V 4K&0' $6' '#8 4' 800 *W<!< ,1&3 6'# 831$# $6'#D 10 4$# 1#' 410) 4K&0' 35#'3A' 4' )!23:'# L

Q = 3 × 800 = 2400 *W<!= 8, 6.10

³

+<

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e

=

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-\].U, 2:' 0^_; _O>X

/' 0&1 2' !3435# '&4 #6'1 7$*'7 #3&# 89 :37*' ; Φ =

^P_hν^lum

=

^Plum_hc^λ

< ,'52954 89 2&7=' 2' :35)4$355'*'54

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^Plum_hc^λ

∆t< B5 3?4$'54 235) &5' 2&7=' 269&4353*$' 2' ; ∆t =

_P^{N hc}

lumλ

=

^Q_eP^hc

lumλ

CD') 8'# 473$# $8'# 2' EFF *C<!> 869 8$)94$35 5&*=7$%&' )352&$4 A ∆t =

^(3×_1,6.10^0,8×−³⁶⁰⁰⁾19

×

^6,6.1010×0,6.10^{−34×3.10−68}

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B5 9 )!3$#$ $)$ 89 4'5#$35 U 2' :35)4$355'*'54<

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5&'> #3$4 P

elec

> P

lum

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^hc_λe

/9 D98'&7 5&*=7$%&' 2' 89 4'5#$35 *$5$*98' 5=)'##9$7' '#4 235) 2' U

s

=

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3&7 &5' 835@&'&7 26352' 2' SFF 5*< B5 3?4$'54 9837# U

s

= 2 T <

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)979)4=7$#4$%&'# 7=#&*='# 295# 8' 49?8'9& V<

+3&8'&7 R3*9$5' # ')4798 P5*Q W'5#$35 2' #'&$8 PTQ

T$38'4 XFF(YFF V<Z( G<[

\8'& XYF(YFF V<Y ( V<E

T'74 YVY(YSY V<[ ( V<Y

]9&5' YEY(Y^F V<[

B795@' SFF(SVF V

!"#$ V _ W'5#$35# 2' #'&$8 2' 2$`=7'54'# R./ '5 :35)4$35 2' 8'&7 23*9$5' # ')4798 P#3&7)'# ; !!"#$$%%%&

'()*+,&-(!$(."(,/(-0(12'()23(14,(2!(-1/5-21(4/'& !3 Q

/)0$ (11$)2)'*&$ .

B5 '&4 473&D'7 2' 89 23)&*'5494$35 #&7 8' #&I'4 9&1 927'##'# #&$D954'# ;

( a&'8%&'# 2355='# )35)'75954 8'# &$##95)'#> )9 9)$4= 2'# ?944'7$'# '4 9&4353*$'# ; !!"#$$%%%&'(1*6/(/''(1*783(''(1&+,$(0'8/,87(26!!& !3'

( b5 *34 #&7 89 )35D'7#$35 2' &$##95)'# 8&*$5'&#'# '5 c944> #3&D'54 2355='# '5 8&*'5# 3&7 8'# R./ ; !!"#$$%%%&,8"/)!89'(1&053$08'0$'/7 !$'43(-*!5*%8!!*08'04'8!5,& !3

! "# $%&''()* (& $(++(,- *-+*-' .( /0$-#/(#1- -#1/λ

-de.N, 9@' 5fVX VF[Y

!"#$%&'#( )* +,#-$./*

! "!#$%&#'(

0(#(1!

('2*3% 1#(4,/!

/0' 12$304$5' #' 5'* 6$7 '0 8 *$09 541$0'7 4&:'57 '7 140;' <'5*='9 '7 #' :$;' '0 >? *$09 541$0'7 <'5*= '7 140;' 4&:'57'@

A2 12$304$5' ;=145;'(7('66' #$ 40 4&:5' B 62 <4$# 6' 541$0'7 '7 62 140;' C

-DE.F, 23' 0G?8 ?H>I

! "!#$%&#'(

01*/+$* )* ,!"#$%&'#(

/ 012$3'4 1&5'64 '4 2137' 8'6*9': ;' 4'* # 7' 6'* ;$##<=' '#4 T

e

= 8 min: 13 314' ;' 792$4 *<##$%&' 7&

612$3'4 D

e

:

/ 012$3'4 8'6*9 '4 2137' 1&5'64': ;' 4'* # 7' 5$7<3=' '#4 T

s

= 12 min '4 13 314' ;' 792$4 *<##$%&' 7' ;<

2137' D

s

>

?3 '&4 *179;$#'6 @

/ ;A'<& <6 &3 B&$7' <68<$4 $3)1* 6'##$2;' 7' *<##' 51;&*$%&' µ:

/ ;< 2<$=31$6' <6 &3 <6<;;9;9 $ C7' 7' #')4$13 S '4 7' !<&4'&6 h:

/ ;< 2137' <6 &3 461& 7' #')4$13 σ <& 8137 7' ;< 2<$=31$6'>

D' ;&#: 13 '&4 )13#$796'6 %&' @

/ ;' 792$4 *<##$%&' 7& 612$3'4 ED

e

F '#4 &3' )13#4<34':

/ ;' 792$4 *<##$%&' 7' ;< 2137' ED

s

F 79 '37 7' ;< !<&4'&6 7A'<& z(t) G ;A$3#4<34 t @ ;&# z <&=*'34': ;&#

D

s

'#4 =6<37>

/ H< 5<6$<4$13 7' ;< *<##' 7& #"#4C*' 1&5'64 I'<& 7<3# ;< 2<$=31$6'I '#4 @ dM

dt = µ S dz dt / 7A<&46' <64: &3 2$;<3 7133' $**97$<4'*'34 @

dM

dt = +D

e

− D

s

012$3'4 1&5'64 '4 2137' 8'6*9':

/ ;' 2$;<3 7133'

^dM_dt

= µ S

^dz_dt

= D

e

/ 7134 ;A$349=6<4$13 7133'

D

e

T

e

= µ S h(1) 012$3'4 8'6*9' '4 2137' 1&5'64':

/ ;A'66'&6 '#4 7' )13#$796'6 ;' 792$4 )1**' )13#4<34> +13#$79613# &3' ;$=3' 7' )1&6<34 %&$ <64 7A&3 1$34 7' ;< #&68<)' 7A<;4$4&7' z: 7' 6'##$13 p

atm

E 6'##$13 <4*1# !96$%&'F '4 7' 5$4'##' 39=;$='<2;' '4 6'##164 <6

;< 2137' '3 &3 1$34 7A<;4$4&7' 3&;;': 7' 6'##$13 p

atm

'4 7' 5$4'##' V

s

> HA9)1&;'*'34 '#4 %&<#$( '6*<3'34 E4'* # 7' 5$7<3=' 46C# ;13=F: $3)1* 6'##$2;' )<6 ;' B&$7' ;A'#4 '4 <68<$4 )<6 ;A'<& '#4 '& 5$#%&'&#' J 13 '&4 713) < ;$%&'6 ;' 4!916C*' 7' K'631&;;$ '4 <L6*'6 @

g.z + p

atm

µ = p

atm

µ + V

_s²

2 7A1M V

s

= √

2 g z: 816*&;' );<##$%&' 7' N166$)';;$> H' 792$4 51;&*$%&' #164<34 <6 ;< 2137' '#4 <;16#

D

s

(z) = µ σ V

s

= µ σ p 2 g z / ;' 2$;<3 7133'

^dM_dt

= µ S

^dz_dt

= − D

s

:

/ %&A13 $34C=6' '346' z(0) = h '4 z(T

s

) = 0 @

T

s

= S σ

s 2 h g (2) 012$3'4 1&5'64 '4 2137' 1&5'64':

-0O.P, <=' 3QRS RTUV

/ 0' 1$023 4' *2##' 5M 6 #78)9$: ;

dM

dt = D

e

− D

s

/ .3 98<$*' '9*23'3:= 02 >29$2:$?3 4' 02 *2##' 4& #"#:@*' ?&>'9: '#: 3&0 ;

^∂M_∂t

= µ S

^dz_dt

= 0A B&##$ ; D

e

= D

s

⇔ σ p

2 g z

eq

= S h T

e

(3) C2 98#?0&:$?3 4' (1)= (2) ': (3) 4?33'

r z

eq

h = T

s

2 T

e

B 0$)2:$?3 3&*89$%&' ;

z

eq

h = 0, 56 < 1 +?**' z

eq

'#: 0&# ':$: %&' h= 02 12$<3?$9' 3' 481?94' 2# D

-EF.G, 2<' 3HIJ KJLM

! "#$%& ' ()&*%+#,- _0(#(1!

! "#$%&!$'(!)'%* "+( #$+&),%*( -!, &%*./0!'(%*

/01 23# 4'# '5621$6# 7' 7$8'2# 126$)4'# 9$:$ ;7$1 !"#$%&'( )*$!+,%&'( -./#-.",'

234$&*,(4(1* $'4'"#( "'/+$* 5 )#%-$*& $'4(& )4(" %(* /#$!1%$* *"& %(* 6-#7&* 8

!$*1&,#(6 8 %(* )'/*("'#(

.< )!$*$'= 41 )><?&@1$#>< 7;#$@<' &<' 7;4>)1(

4$#16$>< 70;4')62><# 71<# 4'# *>4;)&4'#= %&' 40>< 2'(

2;#'<6' 12 &<' )>*A$<1$#>< 8$26&'44' 7' #62&)(

6&2'# 1&5 ;4')62><# 4>)14$#;# 1 '4;'# *;#>*32'#=

4>2#%&0>< 1 12 '5'* 4' &<' 146'2<1<)' 4$1$#><

#$* 4' ( 4$1$#>< 7>&A4' ( 4$1$#>< #$* 4'= )>**' 71<# &< 7$3<' )><?&@&;B

C' 7$3<' )><?&@&; 4' 4&# #$* 4' '#6 4' A&61(D=E(

7$3<' F'< 7'##>&# G #'# H>2*&4'# *;#>*32'#I= &<

!"72>)12A&2' 7' H>2*&4' C

4

H

6

@1J'&5 $<)>4>2' '6

$<K1**1A4'B +0'#6 &< $* >261<6 2;1)6$H %&$ '#6

&6$4$#; 71<# 41 #"<6!3#' 7' <>*A2'&5 >4"*32'#B C1 *>4;)&4' 7' β()12>63<' C

40

H

56

>##37' ><J' 7>&A4'# 4$1$#><# )><?&@&;'#B .44' '&6 1A#>2A'2 &<' 4&*$32' A4'&($<7$@> '6 7><<' #1 )>&4'&2 >21<@' L 41 )12>66'B C1 )>4>216$>< 7'# 4&*'# 7& K1*1<6 2>#' '#6 7&' L 401))&*&416$>< 7' )12>63<' )><6'<& 71<#

#>< 14$*'<616$>< FL 72>$6'= !>6> 7' M12>< C>@1<IB

('9*4% 1#(:,/!

N814&'2 40>272' 7' @21<7'&2 7' 41 4><@&'&2 70><7' 701A#>2 6$>< 7& A&61(D=E(7$3<' 71<# 4' 2>)!' &4621(8$>4'6B

-OP.Q, 1@' <RSE TUDV

!"#$%&'#( )* +,#-$./*

! "#$%& ' ()&*%+#,- 01*/+$* )* ,!"#$%&'#(

/' 0&12(345(6$78' 9:#'81' &8' 2;1'9828)' ;$2$#<8 6<&0;' ( ;$2$#<8 #$* ;' ( ;$2$#<8 6<&0;'4 2=') 6'# :;')19<8#

π 6:;<)2;$#20;'# 2& 8$='2& 6'# 6'&> 6<&0;'# ;$2$#<8#?

@8 '&1 *<6:;$#'9 )'1 '# 2)' A &8' 6$*'8#$<8 )<**' &8 &$1# %&281$%&' 9')128B&;2$9' 6' ;<8B&'&9 L 6' 9<C<86'&9 $8D8$'?

E; #F2B$1 6' )2;)&;'9 ;F:8'9B$' E

c

6'# :;')19<8# <&9 2=<$9 &8 <969' 6' B9286'&9 6' ;F:8'9B$' h ν =

^{h c}_λ

6'#

!<1<8#?

!"#$%!" #&'()*" +

! "#$!%&'( )*+#&,#(-#". ∆k

x

∆x ≈ 2 π )'((#/ 01&,21# p = ~ k #% E =

₂^p_m²

∆p

x

∆x ≈ h ⇒ 2 m E L

²

≈ h

²

)'(3 h c

λ ≈ h

²

2 m L

²

⇒ λ ≈ 2 m c L

²

h 45 6#( 0"#(!(% 7 8'&, $! )&,%!(3# #(%"# )#19 !%':#, ; L ≈ 3 × 10

⁻¹⁰

m< ;

λ ≈ 2 × 9, 1 × 10

⁻³¹

× 3 × 10

⁸

× "

3 × 10

⁻¹⁰

2

6, 62 × 10

⁻³⁴

≈ 0, 1 µm λ #,% -&#( )!(, $*1$%"! =&'$#% 0"'3># 6< 400 nm<?

,"-.$%#" #&'()*" +

! 8'(3%&'( )*'()# #,% 1(# ,10#"0',&%&'( )*@AAB ;

ψ ˜ (x, t) = ˜ ψ

0

e

^{−i(ω t±k x)}

'( 0#1% !1,,& $! "CC3"&"# ,'1, 8'":# )# ,&(1, '1 3',&(1, ;

ψ ˜ (x, t) = ˜ ψ

0

sin (k x + ϕ) e

^{−i ω t}

!=#3 $#, 3'()&%&'(, !19 $&:&%#, ψ ˜ (x = 0, t) = ˜ ψ (x = `, t) = 0? @( 0#1% 0!" #9#:0$# 3>'&,&" ϕ = 0? D*!1%"# 0!"%

k ` = n π !=#3 n 1( #(%&#"?

E#$! "#=&#(% F

2π

λ ` = n π ⇒ ` = n λ 2

!=#3 n 1( #(%&#"? G$ H ! #( #I#% )#, (J1), F $! 8'(3%&'( )*'()# #( x = 0 #% x = `/ 3'::# 0'1" $! 3'")# )#

B#$)#?

K( "C&(L#3%!(% )!(, $*C21!%&'( )# M3>"N)&(.#"/ '( %"'1=#

k

²

= 2 m ω

~ = 2 m E

~

²

= n π

`

2

⇒ E = E

n

= 1 2 m

n π ~

`

2

@( 0#1% 0!" #9#:0$# 0"#()"# $! %"!(,&%&'( #(%"# $#, )#19 0"#:&#", (&=#!19 n = 1 → n = 2/ ,'&%

h c

λ ≈ (E

2

− E

1

) = 3 2 m

π ~

`

2

⇒ λ ≈ 4 3

2 m c `

²

h = 4

3

2 × 9, 1 × 10

⁻³¹

× 3 × 10

⁸

× "

3 × 10

⁻¹⁰

2

6, 62 × 10

⁻³⁴

≈ 0, 1 µm λ #,% -&#( )!(, $*1$%"! =&'$#% 0"'3># 6< 400 nm<?

/!)$0$%#" #&'()*" +

@( 3!$31$# $#, C(#".&#, )#, (&=#!19/ 3'::# )!(, $! ,#3'()# :C%>')# ; E = E

n

= 1

2 m n π ~

`

2

@( 3':0%# $#, C$#3%"'(, π )C$'3!$&,C, ; &$ H ! )#19 )'1-$#% )'(3 O C$#3%"'(,? A1&, '( "#:0$&% $#, (&=#!19/ )#19 C$#3%"'(, 0'1=!(% 3'#9&,%#" ,1" $# :P:# (&=#!19 6Q 0"'L#3%&'(, )# ,0&(<? 41,,&/ $#, (&=#!19 R #% Q ,'(% '3310C,?

G$ ,*!.&% )'(3 )# $! %"!(,&%&'( #(%"# $#, (&=#!19 n = 2 #% n = 3/ ,'&%

h c

λ ≈ (E

3

− E

2

) = 3

²

− 2

²

2 m

π ~

`

2

⇒ λ ≈ 4 5

2 m c `

²

h = 4

5

2 × 9, 1 × 10

⁻³¹

× 3 × 10

⁸

× "

3 × 10

⁻¹⁰

2

6, 62 × 10

⁻³⁴

≈ 0, 1 µm λ #,% -&#( )!(, $*1$%"! =&'$#% 0"'3># 6< 400 nm<?

5S ; #( "C!$&%C λ = 217 nm?

TUGKMA 0!.# (VOO QWRX

! "!#$%#!&'() *+, !"'+#, _0(#(1!

! "!#$%#!&'() *+ ,-!"'+#

/01 23# 4,5.67. +8996:. /. ;<,. =>?? ( ,@5;<. / !"#$%!&'( ")* '( "!+( ,"-(!.-$%-$)*"/

23',* )'4%"*, )% 13,-#(* +#%, )%,1', $531'*,6

A'# 2B)CDC# D' E21$E'*'FE# E!'2*B)!$*$%&'# D' #&2G1)' 12 2C(

1)E$BF !CEC2BH3F' H1I(#BJ$D' )BF#$#E'FE K 'L B#'2M K &F' E'* C21E&2'

$FGC2$'&2' K J1 E'* C21E&2' D' G&#$BF D' J01)$'2 %&' )BF#E$E&' J1 $3)'

*CE1JJ$%&' K &F 'FN$2BFF'*'FE H1I'&L )BFE'F1FE J0'# 3)' )!$*$%&' K

$FE2BD&$2'O +' E21$E'*'FEM J12H'*'FE &#$EC 'F *$J$'& $FD&#E2$'JM '#E &F 2B)'##&# J'FE 'E E!'2*$%&'*'FE 1)E$NC %&$ '2*'E D0$FE2BD&$2' &F CJC(

*'FE 'F #&2G1)' D0&F' $3)' *CE1JJ$%&' )B**' J'# 1)$'2# #1F# 1##'2 12 J1 !1#' J$%&$D' P B C21E$BF %&$ #'21$E E23# )B&E'&#' )B* E' E'F&

D'# E'* C21E&2'# D' G&#$BF CJ'NC'# D'# 1)$'2#O <J '2*'E D01*CJ$B2'2 FBE1QJ'*'FE J'# 2B 2$CEC# *C)1F$%&'# D'# $3)'# *1$# FC)'##$E' D'#

E'* # D' E21$E'*'FE 2'J1E$N'*'FE JBFH# %&0$J )BFN$'FE D0B E$*$#'2O R E$E2' D0'L'* J'M #$ J0BF #B&!1$E' )BFF1SE2' J' E'* # D' E21$E'(

*'FE FC)'##1$2' K &F' D$T&#$BF D& )12QBF' D1F# J01&#ECF$E' K U>>V+

D0&F 1)$'2 $F$E$1J'*'FE K &F B&2)'FE1H' *1##$%&' 'F )12QBF' >MWX D' E'JJ' G1YBF %&' )' B&2)'FE1H' 1EE'$HF' 1 23# E21$E'*'FE ?X 'F *1##' D' )12QBF' K ?** D' J1 #&2G1)'M &F E'* # D' D$T&#$BF D0'FN$2BF Z[

!'&2'# '#E FC)'##1$2' \ +'E 'L'* J' *BFE2' %&' J'# )BFD$E$BF# D' D$T&(

#$BF 'F !1#' #BJ$D' )BFD&$#'FE K D'# E'* # D' E21$E'*'FE E23# JBFH#M D$])$J'*'FE )B* 1E$QJ'# 1N') J1 2BD&)E$BF $FD&#E2$'JJ'O

+'# D&2C'# '&N'FE ^E2' )' 'FD1FE )BF#$DC21QJ'*'FE 2CD&$E'# #$

J0BF 1&H*'FE' J1 E'* C21E&2' D' E21$E'*'FEO @$F#$ #$ BF CJ3N' D' W>V+

J1 E'* C21E&2' D' E21$E'*'FE 12 21 B2E 1& )1J)&J 2C)CD'FEM J' E'* # D' D$T&#$BF #' 2CD&$E K ?[ !'&2'#O .F 'T'EM J' )B'])$'FE D' D$T&#$BF DC 'FD 'L BF'FE$'JJ'*'FE D' J1 E'* C21E&2' T #'JBF &F' JB$ D' E" ' @22!CF$&# _

D = D

0

exp − Q

RT

1N') R = 8, 314 J · K

⁻¹

· mol

⁻¹

M D

0

J' G1)E'&2 2C('L BF'FE$'J %&$ )B22'# BFD 1& )B'])$'FE D' D$T&#$BF B&2

&F' E'* C21E&2' $F`F$' 'E Q J0CF'2H$' D01)E$N1E$BFO /1F# J' )1# D' J1 D$T&#$BF D& )12QBF' D1F# J01&#ECF$E'M D

0

= 1, 5 × 10

⁻⁶

cm

²

· s

⁻¹

'E Q = 313 kJ · mol

⁻¹

O

('7*3% '('&'3&'#(

a$ J1 D&2C' D' E21$E'*'FE '#E D0'FN$2BF Z[ !'&2'# K U>>V+M )1J)&J'2 J0CJCN1E$BF D' E'* C21E&2' FC)'##1$2' B&2 G1$2' EB*Q'2 J1 D&2C' D& E21$E'*'FE K ?[ !'&2'#M 'E J1 )B* 12'2 K )'JJ' DBFFC' 12 J' E'LE'O

-5<.a, 1H' FVb= =>?W

!"#$%&'#( )* +,#-$./*

! "!#$%#!&'() *+, !"'+#, 01*/+$* )* ,!"#$%&'#(

/0 )102'34$4 D

0

'0 &0$45# 67 8 D

0

= 1, 5 × 10

⁻¹⁰

m

²

· s

⁻¹

9

:;5%&<4$10 =' =$>&#$10 ?<$4 < <3<@43' &0 4'* # )<3<)453$#4$%&' τ 1&3 &0 #"#4A*' =' 4<$BB' )<3<)453$#4$%&' L 4'B# %&'

_τ¹

∼

^D

L²

#1$4 τ ∼

^L2

D

9 :' 3< 134 ='# 4'* # =' =$>&#$10 C T

0

= 900 D C '4 T

0

+ ∆T 2<&4 8 τ (T

0

)

τ (T

0

+ ∆T) = D (T

0

+ ∆T ) D (T

0

) = 74

14 = 5, 3 /3 B' 3< 134 ='# ='&E )1'F)$'04# =100' 8

^D(T_D(T^0+∆T₀₎ ⁾

= exp

Q R

∆T T0(T0+∆T)

≈ exp

Q R

∆T T0²

#$ T

0

∆T G )1**' B' B<$##' '04'0=3' B' 4'E4'9 7B #;<H$4 =10) =' 35#1&=3'

^Q_R^∆T_T²

0

= ln 5, 3 ⇔

∆T = ln 5, 3 RT

₀²

Q = ln 5, 3 8, 314 × (273 + 900)

²

113 × 10

³

= 61 K I#1$4 JKD+G <& B$'& =' LMD+G )' %&$ 3'#4' 31)!'N9

-O7.6, <H' 0DJP QMKL

! "#$%&'()*"

0(#(1!

! "#$ %! &'()*%'

2(* &,34*1&#',* +3,53'&*

/' 0 12)'*34' 5667 +4$#8$9:; <;:9=1; *94%&' &: );& >49:) 1' ?? * >9)' @ A94#'$=='B /' 39==;: 9##' 9&(1'##&# 1& *&4 C#$8&2 @ 7DEF * 1& 8$4'&4G ;&4 9=='4 #' 4'84;&H'4 19:# =9 =&)94:' C#;&# =9 3944' 849:#H'4#9=' 1& 3&8D @ 5DII * 1' !9&8'&4GB

('6*3% 1#(7,/!

JH9=&'4 =9 H$8'##' 1& 39==;: =;4# 1& 8$4 1' <;:9=1;B

-<K.L, 9M' :NO0 56EF

x ¨ = 0

¨

y = − g ⇒

x ˙ = v

x0

˙

y = − g t + v

y0

⇒

x = v

x0

t + x

A

y = −

¹2

g t

²

+ v

y0

t + y

A

y − y

A

= − 1 2 g

x − x

A

v

x0

2

+ v

y0

x − x

A

v

x0

B C



 

 

h = −

¹2

g

l vx0

2

+ v

y0

l vx0

h

⁰

= −

¹2

g

l⁰ vx0

²

+ v

y⁰

l⁰ vx0

⇒

( h v

²_x0

= −

^{g l}2²

+ v

y0

v

x0

l h

⁰

v

_x²₀

= −

^{g l}2⁰²

+ v

y0

v

x0

l

⁰







v

_x²₀

=

^g

(

^−l2l0+l02l

)

2 (h l⁰−h⁰l)

v

y0

=

^h_l

v

x0

+

₂^{g l}_vx0

( v

x0

= q

_10×(−9,152×33+33²×9,15)

2×(2×33−2×9,15)

= 27, 5 m · s

⁻¹

v

y0

=

_9,15²

× 27, 5 +

^10×9,15_2×27,5

= 7, 7 m · s

⁻¹

v = q

v

²_x0

+ v

_y²0

≈ v

x0

= 28, 6 m · s

⁻¹

v ≈ 100 km/h

!"#$%&'()$*+

0(#(1!

! "!# $%&'"()*&+,-!

/01 23#

!"#$%"$&'()&$*$)+(%$,(*$)&-.(/,"(01

2(* 1#%13* )* 4*$ +#%, !5'&*, $6 ,!7*8'#( )%* 9 $:6',;

405)!6721 !$' &8$9$#' 9'# &9821#6:#; +' #6:8 <'# 6:<'# 1)6&#8$%&'#

!1&8'# =25%&':)'# %&$ #6:8 $:1&<$>9'# 12 90!6**' ?9'# &9821#6:# 6:8

&:' =25%&':)' <' 20 kHz @ 1 GHzA;

B:' 128$' <'# 6:<'# #6:62'# ?'8 <6:) 9'# &9821#6:#A 2'>6:<$##':8 962#%&' 90$* 5<1:)' 1)6&#8$%&' )!1:7'C )0'#8(@(<$2' 1&D $:8'2=1)'#

':82' 9'# <$E52':8# *$9$'&D; +0'#8 6&2 )'88' 21$#6: %&' #6&# 90'1&C 6: ':8':< 823# '& )' %&$ #' 1##' @ 91 #&2=1)' )12 &:' 721:<' 128$'

<'# 6:<'# #6:62'# #' 25F5)!$##':8 #&2 90'1& '8 823# '& #6:8 821:#*$#'#

#6&# 90'1& )12 9'# $* 5<1:)'# <' 90'1& '8 <' 901$2 #6:8 823# <$E52':8'#;

*1##' G69&*$%&' H$8'##' <& #6: I* 5<1:)' 1)6&#8$%&'

?kg · m

⁻³

A ?m · s

⁻¹

A ?Pa · s · m

⁻¹

A

1$2 JCKLM NMN MJO

'1& JLLL JMOL 1, 48 × 10

⁶

4' 2$:)$ ' <' 905)!6721 !$' 2'G$':8 @ >191"'2 @ 901$<' <0&9821#6:#

&:' P6:' @ 58&<$'2 ? 12 'D'* 9' 90&852&# <0&:' =&8&2' *1*1:A '8 <'

*'#&2'2 '8 <01:19"#'2 9'# 6:<'# 1"1:8 585 25F5)!$'# #&2 9'# <$E52':8'#

$:8'2=1)'# ?905)!6 <'# &9821#6:#A <' *1:$32' @ 2')6:#8$8&'2 &:' $*17'

<'# <$E52':8# *$9$'&D 821G'2#5#;

Q: 1 9$%&' &: 7'9 5)!6721 !$%&' #&2 91 P6:' @ 1:19"#'2; +' 7'9 6##3<' &:' $* 5<1:)' 1)6&#8$%&' 26)!' <' )'99' <' 91 '1&; 405)!6(

721 !' *1:$ &9' 91 #6:<' <' *1:$32' @ )' %&0$9 :0" 1$8 1# <01$2 ':82'

91 '1& '8 91 #6:<' 2')6&G'28' <' 7'9 )12 901$2 1"1:8 &:' $* 5<1:)' 823# =1$>9' =1&##'21$8 9'# *'#&2'#; +0'#8 5719'*':8 6&2 )'88' 21$#6: %&' 905)!6721 !$' :' '&8 1# R82' &8$9$#5' )6**' *58!6<' <0$*17'2$' *5<$)19' 6&2 9'# 6&*6:# 6& 91 821)!5' %&$ )6:8$'::':8 <' 901$2;

('5*6% 1#(<,/!

.: 5G19&1:8 9' 21 628 <' <'&D $:8':#$85# #6:62'# 122$G1:8 #&2 9' )62 # ?': 25#':)' '8 ': 901>#':)' <01$2 #&2 9' 821S'8 <' 906:<'AC $99&#82'2 %&1:8$818$G'*':8 9'# 25)1&8$6:# 2$#'# 12 905)!6721 !';

-TI.U, 17' :VOW KLJX

!"#$%&'#( )* +,#-$./*

!"#$%&'()$*+

01*/+$* )* ,!"#$%&'#(

/0 '&1 *2345$#'6 5' )26 # )2**' 3' 57'8&9

:'# 3$;46'01'# 203'# %&$ #201 '0 <'& 380# 5' 62=5>*' #201 #)!4*81$#4'# )$(3'##2&# ?

@ A5 #78B$1 3' )85)&5'6 5' )2'C)$'01 3' 1680#*$##$20 T '0 40'6B$' 37&0' 203' 8)2&#1$%&' D 57$01'6E8)' '8& (8$69

@ A5 E8&1 '0#&$1' #2**'6 5'# 40'6B$'# 3' 12&1'# 5'# 203'# 866$F801 #&6 5' )26 #G #$ 20 #& 2#' 57$0)2!46'0)' 3' )'55'#()$ H 8# 37$01'6E46'0)'I9

p = Z

c

.v

z

2&6 &0' 203' 580' #' 62 8B'801 F'6# 5'# z )62$##801#

'1

p = − Z

c

.v

z

2&6 &0' 203' 580' #' 62 8B'801 F'6# 5'# z 34)62$##801#

:8 6'##$20 '#1 =$'0 '01'03& )201$0&'G )2**' 20 '&1 #7'0 8##&6'6 '0 $*8B$0801 58 64#'0)' 37&0 $#120 3'

*8##' 0&55' '016' 5'# 3'&J B8K9 .1 5' 34=$1 F25&*$%&' '#1 )201$0& 8&##$9 L20)G D 57$01'6E8)' z = 0 3' #')1$20 SG P(z = 0

⁻

, t) = P (z = 0

⁺

, t) ∀ t

'1

S.v

z

(z = 0

⁻

, t) = S.v

z

(z = 0

⁺

, t) ∀ t L20) 5'# )2'C)$'01# 3' 64M'J$20 '0 6'##$20 r

p

'1 '0 F$1'##' r

v

F46$N'01 ?

r

p

= − Z

1

− Z

2

Z

1

+ Z

2

= − r

v

/0 '0 343&$1 %&' 5' )2'C)$'01 3' 64M'J$20 '0 40'6B$' '#1 R = (Z

1

− Z

2

)

²

(Z

1

+ Z

2

)

²

'1 320) %&' 5' )2'C)$'01 3' 1680#*$##$20 '0 40'6B$' '#1

T = 4 Z

1

Z

2

(Z

1

+ Z

2

)

²

= 1, 12 × 10

⁻³

:7$01'0#$14 %&$ 8##' 8F') 578$6 '#1 4B85' D 58 #2**' 3'# $01'0#$14# I

n

3' )!8%&' 203' 1680#*$#' I

a

= I

1

+ I

2

+ ... =

n=∞

X

n=1

I

n

2O 5' 6'*$'6 1'6*' 3' 58 #46$' '#1 ? I

1

= T T I

0

= T

²

I

s

'1 58 6'581$20 3' 64)&66'0)' '#1 ? I

n

= I

n−1

R R = I

1

. "

R

²

n

= I

1

R

²ⁿ

:' )85)&5 3' )'11' #46$' B42*416$%&' 3200' ?

I

a

= T

²

I

s n=∞

X

n=1

R

²ⁿ

= T

²

I

s

1 − R

²

-PA.Q, 8B' 0RST UTVW

I

s

1 − (1 − T )

²

/ 3$)27$56 6&*:0$%&' 9

I

a

I

s

=

"

1, 12 × 10

⁻³

2

1 − (1 − 1, 12 × 10

⁻³

)

²

= 5, 60 × 10

⁻⁴

;< 9 #$ 3=56 >5&?3$'> @' #5**'0 75&7'# 3'# $67'6#$7:#4 56 705&8' #$* 3'*'67 9 I

a

I

s

= T

²

/ 3$)27$56 6&*:0$%&' 9

I

a

I

s

= "

1, 12 × 10

⁻³

2

= 1, 3 × 10

⁻⁶

%&$ 2?5&7$7 A 32 *B*' )56)3&#$56C

Ia

Is

1C +'32 D&#7$E' 32 6:)'##$7: @& F'3 :)!5F02 !$%&' 5&0 :8$7'0 32 0:#'6)' @=2$0 #&0 3' 702D'7C

-GH.I, 2F' 6JKL MNLO

!"#$%&'#( )* +,#-$./*

!"#$#% &!# !'((# ))) *'!* "#* '**+#,,#*

0(#(1!

/0 )100234 41&# 56'7 89$'0)' $* 9'##$100204' )10#$#4204 : 9'4$9'9 &0' 02 ' ;6&0' 42<5' #20# =2$9' 41*<'9 52 >2$##'55' 1#8' ;' ;'##&#? @' =2A10 5&# *1;'#4'B 10 '&4 #10C'9 : '05'>'9 &0' #'9>$'44' #20# =2$9' 41*<'9 562##$'44' 1#8' ;'##&# D: C2&)!'E 1& '0)19' &0' ='&$55' ;' 2 $'9 #20# =2$9' 41*<'9 5' 4858 !10' 1942<5' ;' 52 FC&9' ;' ;91$4'?

('2*3% '('&'3&'#(

G$ 10 4$9' H#&I#2**'04 ;1&)'*'04H #&9 52 ='&$55' ;' 2 $'9B 5' 4858 !10' 1942<5' 9'#4'92 #15$;2$9' ;' 52 ='&$55' '4 #'92 '0492308 29 5&$? J&'55' =19)' *$0$*25' =2&4($5 '7'9)'9 #&9 52 ='&$55' 1&9 %&' 5' 4858 !10' C5$##' 29 92 194 : 52 ='&$55' K

('2*3% 1#(4,/!

J&'55' =19)' *$0$*25' =2&4($5 '7'9)'9 #&9 52 ='&$55' ;' 2 $'9 1&9 %&' 5' 4858 !10' 1942<5' C5$##' #&9 )'44'

;'90$L9' K

('2*3% *5+*,&

J&'55' =19)' '7'9)'9 #&9 52 ='&$55' ;' 2 $'9 1&9 52 9'4$9'9 #20# =2$9' 41*<'9 5' 4858 !10' ;& <&9'2& K

!"#$%&! ' +1**' )104'74&25$#24$10B 10 '&4 2&##$ #$* 5'*'04 =2$9' 56'7 89$'0)' ;' 52 ='&$55' '4 ;& 4858 !10' 1942<5' ;'>204 5'# 84&;$204# '4 ;'*20;'9 '0#&$4' 52 )10;$4$10 08)'##2$9' : 52 98&##$4' ;' 56'7 89$'0)'

-MN.G, 2C' 0OPQ RSTQ

!"#$#% &!# !'((# ))) *'!* "#* '**+#,,#*

01*/+$* )* ,!"#$%&'#(

/0 12343$55' 630# 5' 27872'01$'5 R 5$7 9 53 13:5' ;1'22'#12'<= #& >#7 ?35$57'0@

/0 )>**'0)' 32 #)!7*31$#'2 53 #$1&31$>0 #&2 53 A?&2' B@ /0 3 '55' /= >2$?$0' 6' 5C3D' !>2$E>0135 6'#

x= 53 >#$1$>0 $0$1$35' 6& )'012' 6C$0'21$' 6& 1757 !>0' >213:5'@ F'# 6$#130)'# )323)172$#1$%&'# R= r '1 D #>01 2' 27#'017'# #&2 53 4&' 6' 6'##&# 6' 53 A?&2' B;:<@

!"#$% B G ;3< H&' 6' )I17 6' 53 #$1&31$>0 G ;:< H&' 6' 6'##&#@ ,>&2 0' 3# 35>&26$2 53 A?&2'= 5'# 3&12'# 8>2)'#

0C>01 3# 717 2' 27#'017'#@

!"#$%&!' &()*%' + /0 3 '55' m ! "!##$ %$ ! &$'( $) M *$ $ %' +, ,-./0$ $+ g = 10 "1#

⁻²

2!**, ,3!+(/0

%$ ! -$#!0+$'31 40 0/+$ − → F = F − u →

x

5F > 06 ! &/3*$ $7$3*,$ -!3 2/-,3!+$'3 #'3 ! &$'( $1 40 #'--/#$ 8'$ *$++$

&/3*$ $#+ 0' $ -/'3 t < 0 -'(# */0#+!0+$ 53,-/0#$ 9 '0 ,*.$ /061 40 0/+$3! µ $ */$:*($0+ %$ &3/++$"$0+ %' +, ,-./0$ #'3 ! &$'( $ %$ -!-($31 40 0,; (;$3! ! %(<,3$0*$ $0+3$ */$:*($0+ #+!+(8'$ $+ %=0!"(8'$1

>/0*$30!0+ $# ?! $'3# 0'",3(8'$#) /0 -3$0% m = 5 ; @ -/'3 0$ -!# %,-!##$3 ! ("(+$ %2!<3!0*.(##$"$0+

%2'0$ $++3$ %$ AB ; %$ ! -/#+$) ( 0$ &!'+ -!# %,-!##$3 C &$'( $# DC1 1 E/'3 $ +, ,-./0$) /0 -$'+ -3$0%3$

M = 100 ;1

^!

F0G0) /0 -$'+ */0#(%,3$3 8'$ µ $#+ */"-3(# $0+3$ 0, 2 $+ 0, 41 E/'3 /H+$0(3 *$+ /3%3$ %$ ;3!0%$'3) /0 -$'+

-/#$3 #/0 +, ,-./0$ #'3 '0 -!-($3) -/#, '(I"J"$ #'3 '0 *!.($3 3(;(%$ ./3(K/0+! 1 40 (0* (0$ 2$0#$"H $ %2'0

!0; $ α1 L( $ +, ,-./0$ ; (##$ -/'3 '0 !0; $ α

c

) /0 ! µ = tan α

c

1

^"

F7-,3("$0+! $"$0+) /0 /H+($0+ %$# !0; $#

α

c

$0+3$ MB

^◦

$+ AB

^◦

) #/(+ 0, 18 < α

c

< 0, 361

40 */0#(%,3$3! '0 "/'?$"$0+ 9 '0 #$' %$;3, %$ (H$3+, $ /0; %$ 2!7$ 5476 */0&/0%' !?$* ! %3/(+$ %2!*+(/0

%$ ! &/3*$ − → F ) */""$ $ "/0+3$ ! G;'3$ N5H61

^#

O!0# ! #'(+$) ( $#+ ("-/##(H $ %$ #,-!3$3 $# */"-,+$0*$# %2!0! =#$ $+ %$ 3,! (#!+(/0 @ $ $# #/0+ .!'+$"$0+

("H3(8',$#1

40 $<$*+'$ $0#'(+$ '0 H( !0 %$# &/3*$# #'3 $# #/'#I#=#+P"$# Q+, ,-./0$R $+ Q&$'( $R @

!"#$ %&' ()*+&' ',* "& -."./0)$& 1 #/0 -/(%# @ − → P = − M g − u →

z

S ! 3,!*+(/0 %$ ! &$'( $ #'3 $ +, ,-./0$)

*/"-/#,$ %$ #! */"-/#!0+$ 0/3"! $ $+ +!0;$0+($ $ @ − → R = N − → u

z

+ T − u →

x

1 T!0+ 8'$ $ +, ,-./0$ 0$ %,*/ $ -!#) /0 ! N > 0 111 3$"!38'$ !##$K +3(?(! $ (*(111 E!3 */0+3$) 9 *$ #+!%$) ( &!'+ 0/+$3 8'$ T $#+ !"#$%&'()

$+ &*+#,)%-&*#) ! -3(/3(1

!"#$ %&' ()*+&' ',* "# (&,!""& 1 #/0 -/(%# @ − → P = − mg − u →

z

S ! 3,!*+(/0 %' +, ,-./0$ #'3 ! &$'( $) */"-/#,$ %$

#! */"-/#!0+$ 0/3"! $ $+ +!0;$0+($ $ @ − → R = − N − u →

z

− T − u →

x

5/0 ! '+( (#, $ -3(0*(-$ %$# !*+(/0# 3,*(-3/8'$#

!' -!##!;$6 S U/3*$ %$ 2/-,3!+$'3 @ − → F = F − u →

x

S 0$ -!# /'H ($3 H($0 #V3 ! 3,!*+(/0 %$ ! +!H $ @ /0 */0#(%P3$

8'2( 02= ! -!# %$ &3/++$"$0+ %$ ; (##$"$0+) #/(+ − R →

t

= N

t

− u →

z

1

^$%

40 !-- (8'$ $0#'(+$ $ +.,/3P"$ %$ ! 8'!0+(+, %$ "/'?$"$0+ 5/' %' "/'?$"$0+ %' *$0+3$ %2(0$3+($6 3$#-$*+(?$"$0+ !' +, ,-./0$ $+ 9 ! &$'( $ %$ -!-($31 F0 -3/W$*+(/0 #'3 $# !7$# 5476 $+ 54K6) /0 /H+($0+

! "#$% &' ()*)+'&,-'.,/* 0$- $1',&2 ,& *3$-. 4'- -5+ 67$ %$..$ +)8)+$*%$ -/,. $*%/+$ 03'%.7'&,.)!

9! :;2 4'+ %/*.+$2 %<'67$ ).70,'*. %/**'=. -'*- 0/7.$ &' 1'--$ 0$ -/* 4/+.'>&$ ;±1( ?

@! A$..$ $B4)+,$*%$ $-. 7* $B$+%,%$ %&'--,67$ -7+ &$ 8+/..$1$*. 0$ (&,--$1$*.

C! D* *3).70,$+' 4'- ,%, &$- )#$*.7$&&$- +/.'.,/*- 0$- -/&,0$- 0'*- &$ 4&'* </+,E/*.'& F %3$-. 7* ./7. '7.+$ 4+/>&G1$!

HI! J744/-$+ 673,& *3K ' 4'- 0$ 8+/..$1$*. $-. 8/+.$1$*. 0,-%7.'>&$2 -7+./7. 673/* #$++' $* L* 0$ +)-/&7.,/* 67$ &$ 8+/..$1$*.

.)&)4</*$M.'>&$ $-. 8/*0'1$*.'& 4/7+ &' +)7--,.$ 0$ &3$B4)+,$*%$! N/7.$8/,-2 &$7+ 4+,-$ $* %/14.$ *$ %<'*($ 4'- +'0,%'&$1$*. &' 1).</0$ 0$ +)-/&7.,/* $. *$ 8',. O673'&/7+0,+O ,*7.,&$1$*. &$- $B4+$--,/*- 0$ &' 4+$1,G+$ 4<'-$!

XYZFLE -!;$ 0[\] ABM^

/'# ')0$1'*'20 3

M x ¨

t

= T (1) 0 = N − M g (2)

m x ¨

f

= F − T (3) 0 = N

t

− N − mg (4)

45 /'650$72 89: )72;&$0 N = mg 3 65 /<5)0$72 ;' 65 )7* 7#520' 27/*56' ;' 6=5)0$72 ;' 65 >'&$66' #&/ 6' 0<6< !72' )7* '2#' 6' 7$;# ;& 0<6< !72'?

45 /'650$72 8@:A '2 &0$6$#520 89:A )72;&$0 B N

t

= (M + m)g? C20&$0$1'*'20A 65 )7* 7#520' 27/*56' ;' 6=5)0$72

;' 65 05D6' )7* '2#' 6' 7$;# ;' 6='2#'*D6' E0<6< !72'A >'&$66'F !"#$%$!" #& '($))&*&"% #+ %,(,-.!"& )+/ (0 1&+$((& 2

G52# &2 /'*$'/ 0'* #A $6 >5&0 2<)'##5$/'*'20 %&' 6' 0<6< !72' H6$##' #&/ 65 >'&$66'A #$272A )' ;'/2$'/ '#0 '20/5I2< '0 72 )7&/0 B 65 )505#0/7 !'? +'00' )72;$0$72 /<6$*$25$/' )7//'# 72; B 65 %&'#0$72 7#<' 5& 2$1'5&

J$2$0$50$72J '0 J)72K/*<J?

!"#$%&'( ) L 5/0$/ ;' t = 0A 72 #& 7#' %&=$6 2=" 5 5# H6$##'*'20 ;& 0<6< !72' #&/ 65 >'&$66'? M2 5 ;72) B 07&0 $2#0520 x

t

(t) = x

f

(t) = x(t)A 0520 %&' | T | < µN?

.2 #7**520 8N: '0 8O: 7&/ <6$*$2'/ T A 72 7D0$'20 (M + m)¨ x = F? M2 /')7225I0 5& 5##5H' 65 )7* 7#520'

#'672 − u →

x

;& 0!<7/P*' ;& *7&1'*'20 ;& )'20/' ;=$2'/0$' 7&/ 6' #"#0P*' E0<6< !72'A >'&$66'F? L1') 8N:A 72 7D0$'20 6='Q /'##$72 3 T = M

M + m F > 0

45 )72;$0$72 ;' 272 H6$##'*'20 8+R-: #=<)/$0 ;72) 3 M

M + m F < µM g?

,7&/ %&' 6' 0<6< !72' H6$##' #&/ 65 >'&$66'A $6 >5&0 F > µ(M + m)g 8+-: ?

L1') 6'# 156'&/# 2&*</$%&'# )!7$#$'#A 72 5 &2' >7/)' *$2$*56' )7* /$#' '20/' SA9 R '0 SA@ R #'672 65 156'&/

;' µA #7$0 &2' >7/)' ;' 6=7/;/' ;' H/52;'&/ ;& 7$;# ;' 6='2#'*D6' E0<6< !72'A >'&$66'FA 72;</< ;& )7'T)$'20 µ?

U$ )'00' )72;$0$72 '#0 >5)$6'*'20 /'* 6$' 7&/ 6' 0<6< !72' #&/ 65 >'&$66'A '66' ;'1$'20 6&# ;<6$)50' ;52# 6' )5# ;' 27*D/'&Q )7&1'/0# '0 5##$'00'# #&/ &2' 25 '???

!"#$%$!" )+/ (& /&%/0$% #& (0 1&+$((& 2

VW*' #$ 6' 0<6< !72' H6$##' #&/ 65 >'&$66'A $6 15 #' ;< 65)'/ 1/5$#'*D65D6'*'20A )7**' 65 >'&$66'A ;52# 6' #'2#

;' − → F ? C6 >5&0 ;72) 5&##$ %&' 6' )7205)0 >'&$66'X0<6< !72' )'##' 51520 %&' 6' 0<6< !72' 2=500'$H2' 6' D7/; ;' 65 05D6'?

!"#$%&'( ) L 5/0$/ ;' t = 0A 72 #& 7#' %&=$6 " 5 H6$##'*'20 ;& 0<6< !72' #&/ 65 >'&$66'? M2 5 ;72)

| T | = µN = µM g B 07&0 $2#0520? M2 >5$0 '2 7&0/' 6=!" 70!P#' /5$#7225D6' %&' 0 < x ˙

t

< x ˙

f

A )='#0(B(;$/' %&' 65 1$0'##' ;& 0<6< !72' 5/ /5 7/0 B 65 >'&$66' '#0 #'672 −− u →

x

A )' %&$ )72;&$0 B <)/$/' T = µM g > 0? +'# !" 70!P#'#

#'/720 1565D6'# 0520 %&' x ˙

t

− x ˙

f

< 0?

.2 /' /'2520 8N: '0 8O: 51') )'# !" 70!P#'# ;' H6$##'*'20A 72 7D0$'20 3







¨

x

t

= µg

¨

x

f

= F − µM g m

⇒







˙

x

t

= µgt

˙

x

f

= F − µM g

m t ⇒



 

 

x

t

(t) = 1 2 µgt

²

x

f

(t) = F − µM g

2m t

²

U$ 65 )72;$0$72 ;' H6$##'*'20 #&/ F '#0 1</$K<'A 6=!" 70!P#' #&/ 6' #'2# ;& H6$##'*'20 '#0 07&Y7&/# 1</$K<' &$#%&' x ˙

t

− x ˙

f

= µ(m + M )g − F

m t < 0?

M2 $20/7;&$0 65 ;$Z</'2)' δ(t) = x

f

(t) − x

t

(t) = 1

2m [F − µ(M + m)g]t

²

? M2 /'0/7&1' <1$;'**'20 '2)7/'

%&' δ(t) > 0 0520 %&' 65 /'650$72 8+-: '#0 1</$K<'?

45 >'&$66' #'/5 /'0$/<' ;' #7&# 6' 0<6< !72' B 5/0$/ ;' 6=$2#0520 t

0

7[ δ(t

0

) = R + r? M2 7D0$'20 567/#

t

0

=

s 2m(R + r) F − µ(M + m)g ?

,'2;520 )'00' !5#' ;='20/5I2'*'20A $6 2' >5&0 5# %&' 6' )'20/' ;=$2'/0$' ;& 0<6< !72' %&$00' 65 05D6'? C6 >5&0

;72) %&' x

t

(t

0

) < D #7$0 1

2 µgt

²₀

< D? .2 &0$6$#520 6='Q /'##$72 ;' t

0

0/7&1<' /<)<;'**'20A 72 7D0$'20 %&' 65

>7/)' F ;7$0 W0/' 0'66' %&' 3 F > µmg(R + r) + µ(M + m)gD

,7&/ #$* 6$K'/ )'00' 'Q /'##$72 6$00</56' #52# )!52H'/ /5;$)56'*'20 65 H<2</56$0< ;& /7D6P*'A 72 '&0 <0&;$'/ D 6' )5# 7[ D = RA )' %&$ )72;&$0 B 3 F > µmg

1 + r R

+ µ(M + m)g = µ h

M + m 2 + r

R i g?

VW*' #$ 6='Q /'##$72 ;' )'00' >7/)' 6$*$0' '#0 6&# )7* 6'Q' %&' )'66' 7D0'2&' 5/ 65 /'650$72 8+-: ;' 6=5 /7)!' /<6$*$25$/'A $6 2=" 5 '2 /50$%&' %&' '& ;' )!52H'*'20 )5/ m M A )' %&$ 500<2&' )72#$;</5D6'*'20

-\C.U, 5H' 2]^_ 9SN`

!"#$%$!" &'( )*+((,% #' %-)-./!"0 +1+"% )0 2!(# #0 )+ %+2)0 3

DE*' #$ /9 6'&$//' 9 ;2; 4'2$4;' 3' #5&# /' 2;/; !5:' 989:2 %&0$/ :0922'$F:' /' G543 3' /9 29G/'< )' 3'4:$'4 9&49

&:' 8$2'##' x ˙

t

(t

0

) = µg

s 2m(R + r)

F − µ(M + m)g :5: :&//' = /09G#)$##' x

t

(t

0

) = µmg(R + r)

F − µ(M + m)g 7 H0$/ :0" 9 9# 3' 64522'*':2 3' /9 29G/' #&4 /' 2;/; !5:'< )' 3'4:$'4 89 922'$:34' /' G543 3' /9 29G/' = 8$2'##' )5:#29:2'7 I/ 69&2 35:) 4':34' ': )5* 2' /' )5'J)$':2 3' 64522'*':2 29G/'K2;/; !5:'< :52; µ

0

7

L 4M# &: :5&8'9& G$/9: 3'# 654)'# #&4 /' 2;/; !5:'< /9 *;2!53' /9 /&# 'J)9)' )5:#$#2' = &2$/$#'4 /' 2!;54M*' 3' /0;:'4F$' )$:;2$%&'7 L89:2 %&' /' 2;/; !5:' :0922'$F:' /' G543 3' /9 29G/'< $/ 69&2 %&' /' 24989$/ 4;#$#29:2 3' /9 654)' 3' 64522'*':2 #&4 /9 3$#29:)' R − x

t

(t

0

) 3$##$ ' /0;:'4F$' )$:;2$%&' 1

2 M x ˙

t2

(t

0

)9)%&$#' 94 /' 2;/; !5:' 39:# /' 4;6;4':2$'/ /$; = /9 29G/'< #5$2 N 1

2 M x ˙

t2

(t

0

) < µ

0

M g[R − x

t

(t

0

)]7 O&'/%&'# /$F:'# 3' )9/)&/# )5:3&$#':2 =

&:' :5&8'//' 'P 4'##$5: 3' /9 89/'&4 *$:$*9/' 3' /9 654)' = 9 /$%&'4 N F > µg

M + m

1 + µ

0

+ µ µ

0

+ 1 + µ µ

0

r R

Q: 4'245&8' /5F$%&'*':2 %&' 5&4 &: µ

0

24M# 69$G/'< $/ 69&2 &:' 654)' 24M# F49:3' 5&4 ;8$2'4 3' 69$4' )!&2'4 /' 2;/; !5:'7 >&*;4$%&'*':2< ': 4':9:2 µ

0

= µ = 0, 2< 5: 5G2$':2 &:' 654)' F

lim

= 0, 285 >< 3& *E*' 5434' 3' F49:3'&4 %&' /'# )5:3$2$5:# 4;);3':2'#7

!"#$%&! ' ()*+*, +* -!&. #&//) &.)0)/!$ 0! .12+$3"! 4& "+&5!"!*. 4& 6!*.$! 47)*!$.)! %&) 6+*4&). 8 x ¨

t

=

− µ

0

g .#*. %&! x ˙

t

(t) > 09 !/.! 8 )*.2:$!$ !. 8 52$);!$ %&! 0# 5).!//! /7#**&0! #5#*. %&! 0! .202-1+*! *7#..!):*!

0! <+$4 4! 0# .#<0!9

Q: )5:#292' )/9$4'*':2 %&0$/ 69&2 985$4 &: )5'J)$':2 3' 64522'*':2 µ /' /&# 69$G/' 5##$G/' 5&4 :' :;)'##$2'4

%&0&:' 69$G/' 654)' F 7 +')$ 3$2< 39:# /' )9# 5R $/ :0" 9 %&0&:' 9##$'22'< /9 654)' = 'P'4)'4 '#2 3' 25&2' 69S5: 24M#

49$#5::9G/'7 L& 9##9F'< 39:# /' )9# 9#"* 252$%&' µ = 0< 5: 4'245&8' ;8$3'**':2 %&0$/ " 9 25&T5&4# F/$##'*':2 ':24' /' 2;/; !5:' '2 /9 6'&$//' '2 %&0&:' 654)' F *$:$*' #&J2 = 4;9/$#'4 /0'P ;4$':)'7 U09$//'&4#< /9 4;#5/&2$5:

)5:3&$49$2 = x

t

(t) = 0 N /' 2;/; !5:' :' G5&F' 9# V

,94 )5:24'< &:' 65$# /9 6'&$//' 4'2$4;' 3' #5&# /' 2;/; !5:'< $/ 69&2 49/':2$4 'J)9)'*':2 /9 8$2'##' 9)%&$#' 94 /' 2;/; !5:' 39:# R 7 I/ 69&2 35:) 3'# 64522'*':2# W$* 5429:2#W ':24' /' 2;/; !5:' '2 /9 29G/'7 H9:# )'# 3'4:$'4#<

*E*' 98') &:' 8$2'##' x ˙

t

(t

0

) *$:$*'< /' 2;/; !5:' F/$##'49$2 $:;8$29G/'*':2 T&#%&09& G543 3' /9 29G/'777

!"# $%%&# '%"( %!)* +

Q: '&2 ':)54' 9J:'4 /9 *53;/$#92$5: ': )5:#$3;49:2 %&' /9 654)' 'P'4);' #&4 /9 6'&$//' :0'#2 9# &: ;)!'/5:

*9$# 9&F*':2' /$:;9$4'*':27 X: 'P'* /' 30&:' 2'//' ;2&3' '&2 E24' 245&8;' 9& 3;G&2 3& #&T'2 3' ,!"#$%&' II 3'# D$:'# D, @??A7

.:Y:< &:' ;2&3' 5&##;' 3' )' F':4' 30'P ;4$':)'# 9 ;2; 4;#':2;' 39:# /0;*$##$5: =>* *7!/. -#/ 4!/ 6+<#?!/=

#&4 Z[L>+. \ /' AAK?\K@?A@7

! "#$%& '#()**$+,% -$ %+.$& /+) ( )0%1),2 -$&&$ ,2%3*+&)30 '$ 1,34*56$

-[I.H, 9F' :]^_ @?A\

!"#$%&'#( )* +,#-$./*

! "#$%& ' &!# _0(#(1!

('2*3% 1#(4,/!

!" #$"%!" & !'$

./012$0# 3' 45210$)4' 3& 6&44'0$7 3' 4587$97 3'# ,!"#$)$'7# 7:;<= >94? @A *21# BCCB ? AB=(A<=

!" #$%&' ( '"$) *"+ ,-.- /01!234

5(* 6%"!* 7%' "* +,#+%$"*888 9 $:*3%

DE24$#'1 &7' F&#E' 3' #02G$4$0E )97>'72G4' 09&0 2& 497H 3' #97 >94 '&0 #' F2$1' #$* 4'*'70? I'&/ G9&0'$44'# '7 42#0$%&' 9&1 G9$##97 H2J'&#' KH'71' ,' #$L +9)2???M 3' 1, 5 LL 9& *N*' 2 LL )97#0$0&'70 4' *20E1$2& 3' G2#' O #' 19)&1'1? P5&7' 3'# 3'&/ '#0 3E)9& E' 9&1 F9&17$1 459H$>' '0 42 Q& ' #&1 42%&'44' #'1970 R/E# 019$# 9& %&201' 2$4'197#

9&1 F91*'1 42 %&'&'?

S& 0'1*' 35&7 E>'70&'4 )9* 0' O 1'G9&1# KT2 F2$0 4&# #E1$'&/ UM 42 4$GE120$97 3& )127 35211N0 '7012V7' 42 3E)9* 1'##$97 3& G9&)!97 '0 42 1'##$97 O 45$70E1$'&1 2)%&$#' 21 H97W2H' F2$0 4' 1'#0' X 3EH2H'*'70 3' 42 0&"Y1' 21 42 F91)' 1'##270'L EQ')0$97 G1&024' 3' 45'2& '0 *9&>'*'70 3' 42 F&#E' 21 1E2)0$97?

5"+"67&+8%789$'% "**+:;8<"78='% >' ?" #$%&' @ A <"%%' ( =8>' "='6 :B8=' '7 9$'$' 100 g C A >8"<D7+' >$ 6:+*% @ 8, 5 cm C

A >8"<D7+' >' ?" 7$ED+' @ 2 cm C A =:?$<' 7:7"? >$ +&%'+=:8+ @ 1, 5 L C A F"$7'$+ @ 40 cm C

A *+'%%8:G %$**:+7"H?' @ I$%9$J( 40 bar-

Z7 " *'0 0, 5 L 35'2& '0 97 910' 42 1'##$97 3' 452$1 $70E1$'&1 O [ G21#?

\E1$R'1 %&270$020$>'*'70 %&' 4'# )973$0$97# #970 1E&7$'# 9&1 %&' 42 F&#E' 3E)944'?

-D].^, 2H' 7:B_C =_BA

!" #$"%!" & !'$

./012$0# 3' 45210$)4' 3& 6&44'0$7 3' 4587$97 3'# ,!"#$)$'7# 7:;<= >94? @A *21# BCCB ? AB=(A<=

!" #$%&' ( '"$) *"+ ,-.- /01!234

* # +%,-# .%! ,# '(/'%0,#111 2 03#$%

DE24$#'1 &7' F&#E' 3' #02G$4$0E )97>'72G4' 09&0 2& 497H 3' #97 >94 '&0 #' F2$1' #$* 4'*'70? I'&/ G9&0'$44'# '7 42#0$%&' 9&1 G9$##97 H2J'&#' KH'71' ,' #$L +9)2???M )97#0$0&'70 4' *20E1$2& 3' G2#' N #' 19)&1'1? O5&7' 3'# 3'&/ '#0 3E)9& E' 9&1 F9&17$1 459H$>' '0 42 P& '

#&1 42%&'44' #'1970 Q/E# 019$# 9& %&201' 2$4'197# 9&1 F91*'1 42 %&'&'?

R& 0'1*' 35&7 E>'70&'4 )9* 0' N 1'G9&1# KS2 F2$0 4&# #E1$'&/ TM 42 4$GE120$97 3& )127 35211U0 '7012V7' 42 3E)9* 1'##$97 3& G9&)!97 '0 42 1'##$97 N 45$70E1$'&1 2)%&$#' 21 H97W2H' F2$0 4' 1'#0' X 3EH2H'*'70 3' 42 0&"Y1' 21 42 F91)' 1'##270'L EP')0$97 G1&024' 3' 45'2& '0 *9&>'*'70 3' 42 F&#E' 21 1E2)0$97?

Z7 910' 42 1'##$97 3' 452$1 $70E1$'&1 N [ G21#?

\E1$Q'1 %&270$020$>'*'70 %&' 4'# )973$0$97# #970 1E&7$'# 9&1 %&' 42 F&#E' 3E)944'?

-D].^, 2H' 7:BB_ =_BA

!"#$%&'#( )* +,#-$./*

! "#$%& ' &!# 01*/+$* )* ,!"#$%&'#(

/0 )!1$#$2 )1**' #"#23*' 1&4'52 6' 57)$ $'028 ,1&5 %&9$6 #1$2 :;'< 10 #' 6=)' >10) >=0# &0 57?75'02$'6 @)'6&$

>' 6= ?&#7'A %&$ 09'#2 =# B=6$67'0 *=$# '0 *1&4'*'02 5')2$6$B0' =))76757< >9=))7675=2$10 =5 5= 152 =& #16

@B=6$67'0A C ~ Γ = Γ~u

z

@1D ~u

z

'#2 4'5# 6' !=&2A8

E' 416&*' >9'=& '#2 0127 V < µ = 1, 0 × 10

³

kg · m

⁻³

'#2 6= *=##' 416&*$%&' >' 69'=&< S

i

=

^πd

2 i

4

'#2 6= #')2$10

$0275$'&5' 1D d

i

= 8, 5 cm '#2 6' >$=*325' >& )15 #< 2=0>$# %&' 6= #')2$10 ';275$'&5' '#2 S

e

=

^πd

2 e

4

1D d

e

= 2 cm '#2 6' >$=*325' >' 6= 2&"35'8

E' >7F$2 *=##$%&' '#2 D

m

= RR

S

µ. − → v . −−→ d

²

S >91D D

m

= µv

e

S

e

= µv

i

S

i

@GA =4') 6'# 4$2'##'# >' 69'=& H 697I')2$10

@v

e

A '2 $0275$'&5' @v

i

v

e

)=5 S

i

S

e

A8

E' #"#23*' ?'5*7 @$0>$)7

f

A %&$ )1J0)$>' H 69$0#2=02 t =4') 6' #"#23*' 1&4'52 @$0>$)7

o

A = )1**' %&=02$27 >'

*1&4'*'02 C

P ~

f

(t) = P ~

o

(t)

K &0 $0#2=02 &6275$'&5< 6= %&=02$27 >' *1&4'*'02 >& #"#23*' ?'5*7 '#2 >'4'0&' C P ~

f

(t + dt) = P ~

o

(t + dt) + δm

s

~v

s

/0 251&4' >10) C

D ~ P

f

Dt = d ~ P

o

dt + δm

s

dt ~v

s

/5

• 1&5 6' #"#23*' 1&4'52< 6' 6$%&$>' '#2 =& 5' 1# >=0# 6' 57?75'02$'6 @#$ 10 )10#$>35' 697)1&6'*'02 )1**' %&=#$

#2=2$100=$5'A< >10)

^{d ~}_dt^Po

= ~ 0 L

•

^δmdt^s

~v

s

= − D

m

v

e

~u

z

< 1D D

m

'#2 6' >7F$2 *=##$%&' '2 v

e

6= 015*' >' 6= 4$2'##' >97I')2$10 >' 69'=& >=0# 6' 57?75'02$'6 >' 6= F1&2'$66' L

•

^{D ~}Dt^Pf

= Σ F ~

ext

= − µV g~u

z

− µ V Γ~u

z

+ P

e

S

e

~u

z

− P

i

S

i

~u

z

8 M1$2< #&$4=02 ~u

z

C

µV Γ = − µV g + P

e

S

e

− P

i

S

i

+ D

m

v

e

E= ?15)' >' 1&##7' '#2 >10) F ~ = D

m

v

e

~u

z

@NA >10) F ~ = µ S

e

v

²_e

~u

z

8 O'#2' H >72'5*$0'5 v

e

8 P0 F$6=0 >970'5B$' )$072$%&' >100' C

DE

c

Dt = dE

c

dt + δm

s

dt v

_s²

2 )=5 $6 09" = =# >' *=##' '025=02'8 /5

• 1&5 6' #"#23*' 1&4'52< 6' 6$%&$>' '#2 =& 5' 1# >=0# 6' 57?75'02$'6< >10)

^dE_dt^c

= 0 L

•

^δmdt^s v²_s

2

= D

mve 2

2

L

•

^DE_Dt^c

= ΣP u

ext

+ ΣP u

int

=4') ΣP u

int

= 0 #$ 10 =>*'2 %&' 697)1&6'*'02 '#2 =5?=$2 '2 $0)1* 5'##$F6'8 K&##$< D

mve

2

2

= ΣP u

ext

= P

i

S

i

v

i

− P

e

S

e

v

e

8 E' >7F$2 >100' C D

mve 2

2

= ΣP u

ext

= (P

i

− P

e

)

^D_µ^m

#1$2 v

e2

=

^2(Pi_µ^−Pe)

@QA %&910 =&5=$2 & 251&4'5 =4') 6= 5'6=2$10 >' R'501&66$8 K$0#$< '0 )1*F$0=02 @GA< @NA '2 @QA< 10 251&4' C

F ~ = µS

e

v

²_e

~u

z

= 2S

e

(P

i

− P

e

) ~u

z

#1$2 F = 2

^πd

2 e

4

(P

i

− P

e

) =

^π×0,022₂^×6×105

= 3, 8 × 10

²

N %&$ '#2 F$'0 #& 75$'&5' =& 1$># >' 6= F1&2'$66' C m g ≈ 1 × 10 = 10 N8 K$0#$< 6= ?&#7' >7)166'8

SR C 6' ?=$2 %&' 6'# 25=4=&; >& 1$># '2 >' 6= ?15)' >9$0'52$' >9'025=T0'*'02 #102 07B6$B'=F6'# >'4=02 6' 25=4=$6

>'# ?15)'# >' 5'##$10 '&2 )10#2$2&'5 &0' 4=6$>=2$10 >'# !" 12!3#'# =& 0$4'=& '; '528

-OU.M, =B' 0VGGG NWGX

! "#$% &#'(!$' )!*&+',-!.%#(/

0(#(1!

!"#$%&'&()% *+ ,'+&-.'!/$+! "/$0&!)*1%'2(3+$

!!" #$ %&'() *%$ + , -.# ' !"( %$ /'")0-'(%$"(

23%&4+3,$*%,"

/0 !1&2( 134'&3 5& !1&2 134'&3 '#2 &0 2310#6&)(

2'&3 74')2351)5&#2$%&' 6'#2$07 8 356&$3' 6'# #50#

8 132$3 69&0 #$:014 74')23$%&'; <4 '#2 '0 )'41 49$0(

='3#' 6& *$)35 !50';

<4 '>$#2' 6' 05*?3'&> 2" '# 6' !1&2( 134'&3# @

• 74')2356"01*$%&'

• 74')235#212$%&'A

• $7B574')23$%&'

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'0#$50;

('7*3% 1#(8,/!

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134'&3 *$# 8 =523' 6$# 5#$2$50 K50 &2$4$#'31 6'# 37#$#210)'#A )506'0#12'&3# '2 ?5?$0'#L;

M0 37 50631A '0 '> 4$)$2102 4' 3525)54' 165 27 '2 '0 G5&30$##102 &0 :31 !$%&' #&3 4'%&'4 N:&3'3502 4'#

5$02# '> 73$*'021&> K#$ 5##$?4' 1=') &0' 3' 37#'0212$50 6'# $0)'32$2&6'#L '2 &0' )5&3?' 6' *5674$#12$50; M0 6500'31 4'# =14'&3# 6'# 131*J23'# 6' )'22' *5674$#12$50;

!"#$%&' ( )%*+,*%"%,- .

• &0 !1&2 134'&3 74')2356"01*$%&' 8 72&6$'3

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• &0' ?5$2' 6' 37#$#210)'# 8 67)16'# K6' 1 Ω 8 11 kΩL

• &0 5#)$445#)5 ' 0&*73$%&'

• 6$='3# N4# 74')23$%&'# '2 )500'>$50#

• &0 536$012'&3 1=') 21?4'&3 '2 )132' 691)%&$#$2$50

-P<.Q, 1:' 0RSTS UVSW