Resumen Introducción ( ) t ( ) ( )θ ( ) tg - Física re

!"#$%&'#()(*+)$),-&-*
!
*
!
!
!
!
!
!"#$%&'()*"&+'(%,-(.'/&.&,0#'(%,(1&2'(
34-&+$'(
(
)0/<3*87*&190:;5<=>*?<EE/I<0*'7*@ABCD</E!
"#$%&&'()**+,-"$&./'-!""0%1123'&+,-"$&./'-"
#$%&'$(&')&!*+!,-.)/$!"!0!12)3+'.)*$*!,$3$4&'&5!6778!
!
#$%&'$("
!
"#$%! $&'(')*! #%! ('#+! %,! %-! %#$./0*! /%-! 1*2010%,$*! /%! $0&*! *(-03.*! 3*,#0/%&',/*! %-!
&*4'10%,$*! /%-! '0&%! #*(&%! %-! 5&*6%3$0-7! 8'&'! %#*! #%! '5-03'&*,! -*#! 3*,3%5$*#! 9:#03*#! /%!
30,%1;$03'!6!/0,;103'!/%!3.%&5*#!6!9-.0/*#7!!! <%!&%'-04+!.,!%=5%&01%,$*!>.%!3*,#0#$0+!%,!.,'!
#%&0%!/%!/0#5'&*#!/%!5%-*$0$'#!/%!50,?@5*,?A!%,!3'/'!.,*!/%!%--*#!#%!2'&0+!%-!;,?.-*!/%!/0#5'&*7!
B.%?*C!-*#!&%#.-$'/*#!%=5%&01%,$'-%#!*($%,0/*#!#%!','-04'&*,!3*,!.,!#*9$D'&%!!/%!20/%*!6!#%!-*!
3*15'&+!3*,!%-!/%#'&&*--*!$%+&03*C!$',$*!!3*,!3*1*!#0,!&*4'10%,$*7!
!
)(*+,-&../0("
!
!
E,! 5&*6%3$0-! %#! 3.'->.0%&! 3.%&5*! >.%! &%30(%! .,'! 2%-*30/'/! 0,030'-! 6! -.%?*! #0?.%! .,'!
$&'6%3$*&0'!/%$%&10,'/'!5*&!-*#!%9%3$*#!/%!-'!'3%-%&'30+,!?&'20$'30*,'-!6!-'!&%#0#$%,30'!/%-!'0&%7!
E,'! (*-'! ?*-5%'/'! 3*,! %-! 50%C! .,'! 5%-*$'! -',4'/'C! .,! 5'>.%$%! #*-$'/*! /%#/%! .,! '20+,! 6! .,'!
('-'!/0#5'&'/'!5*&!.,!'&1'!#*,!5&*6%3$0-%#7!"-!3'10,*!>.%!#0?.%!.,!5&*6%3$0-!%#!#.!$&'6%3$*&0'7!
<%!','-04'&;!%-!1*2010%,$*!/%!5&*6%3$0-%#!#+-0/*#7!F03G*!1*2010%,$*!%#$'!-010$'/*!'!.,!
5-',*! 2%&$03'-! /%$%&10,'/*! 5*&! -'! /0&%330+,! /%! -'! 2%-*30/'/! 0,030'-7! B'! &'4+,! %#! >.%! -'!
'3%-%&'30+,! /%(0/'! '! -'! ?&'2%/'/! %#! 5.&'1%,$%! 2%&$03'-C! %#$'! ,*! 5.%/%! 1*2%&! .,! 5&*6%3$0-!
-'$%&'-1%,$%7!
!
<0!#%!3*,#0/%&'!>.%!-'!&%#0#$%,30'!/%-!'0&%!%#!/%#5&%30'(-%C!,*#!%,3*,$&'1*#!3*,!.,'#!
%3.'30*,%#!>.%!#*,!#*-.30+,!/%-!5&*(-%1'!/%!$0&*!/%!5&*6%3$0-%#!>.%!#*,!9'10-0'&%#H!
!
K = K 7 + (J 7 ⋅ 41C θ )⋅ D !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%
"
⋅ ; ⋅ D 6 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"&$%
6
J K = J 7 ⋅ 41C(θ ) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!"'$%
J L = J 7 ⋅ C<E(θ ) − ; ⋅ D %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!"($!
L = L 7 + (J 7 ⋅ C<Eθ )⋅ D −
!
)'*#'+,-./%!"#$0!8*#030+,!%,!!!IG*&04*,$'-J!%,!9.,30+,!/%-!$0%15*C!!"&$0%8*#030+,!%,!"!I2%&$03'-J!
%,!9.,30+,!/%-!$0%15*C!!"'$0%K%-*30/'/!%,!!#IG*&04*,$'-JC!!"($0!K%-*30/'/!%,!"!I2%&$03'-J7!F*,/%!JG!
%#!-'!2%-*30/'/!0,030'-C! K % L -'!5*#030+,!%,!-*#!%)%#!!!%!"!&%#5%3$02'1%,$%C!$!%-!$0%15*!6! θ %#!%-!
!1$
;,?.-*!>.%!9*&1'!-'!2%-*30/'/!G*&04*,$'-!3*,!&%#5%3$*!'!-'!2%-*30/'/!$',?%,30'-! 7!
!
<0,! %1('&?*C! 5'&'! 30%&$*#! 3'#*#! %,! -'! 5&;3$03'C! -'! &%#0#$%,30'! /%-! '0&%! ,*! 5.%/%! #%&!
/%#5&%30'/'! 6! %#! ,%3%#'&0*! 0,3-.0&! %,! -'#! %3.'30*,%#! /%! 1*2010%,$*!.,'!&%#5%3$02'!9.%&4'!/%!
&*4'10%,$*7!
8'&'! /%$%&10,'&! -'! 9*&1'! >.%! $%,/&;! %#$'! 9.%&4'! /%! &*4'10%,$*! /%(%1*#! 3'-3.-'&! .,!
5'&;1%$&*!3'&'3$%&:#$03*!/%,*10,'/*!L,M1%&*!/%!N%6,*-/#OC!%-!3.'-!%#$;!/'/*!5*&H!
!
6 ⋅ 0 ⋅ ρ ⋅ J %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!1$!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! &< =
µ
!
",! %#$'! %3.'30+,! ! 0! %#! %-! &'/0*! /%!.,'!%#9%&'!>.%!#%! 1.%2%!3*,!2%-*30/'/!JC!%,!.,!9-.0/*!/%!
!"$
/%,#0/'/!ρ%6!20#3*#0/'/! µ ! 7!
!
!
<0!N%!P!QC!%,$*,3%#!-'!9.%&4'!/%!&*4'10%,$*!%#!/%!-'!9*&1'H!!
./01*-23/451*6*)7*87*&190:;5<=>*?7*@ABCD</E*!+*FGGH!
"!
!"#$%&'#()(*+)$),-&-*
!
*
!
%
%
%%%%
!2$%
+014< = 9 ⋅ π ⋅ µ ⋅ 0 ⋅ J %
%
!2$
)'*#'+3-%!2$!"3.'30+,!/%!<$*R%#! C!/*,/% µ %#!-'!20#3*#0/'/!/%-!1%/0*!6!0!%#!%-!&'/0*!/%!.,'!
%#9%&'!>.%!#%!1.%2%!3*,!2%-*30/'/!J!%,!%-!10#1*7!
!
!
<0!N%!SS!QC!-'!9.%&4'!/%!&*4'10%,$*!%#!-'!#0?.0%,$%H!!
%
"
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! +014< = ⋅ M 9 ⋅ ρ ⋅ ) ⋅ J 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!4$%
6
%
)'*#'+3-% !4$0! T.%&4'! /%! &*4'10%,$*! 5'&'! N%6,*-/#! 1.3G*! 1'6*&! '! QC! /*,/%! )! %#! %-! ;&%'!
$&',#2%&#'-! /%-! 3.%&5*! Iπ* 0FC! 5'&'! .,'! %#9%&'JC! J! %#! -'! 2%-*30/'/! /%-! 10#1*C! ρ*-'! /%,#0/'/! /%-!
1%/0*!/*,/%!#%!1.%2%!6!M9C!.,!3*%9030%,$%!,.1U&03*!3.6*!2'-*&!/%5%,/%!/%!&<*6!-'!9*&1'!/%-!
!"$
*()%$*7!"-!2'-*&!/%!M9!#%!/%$%&10,'!%=5%&01%,$'-1%,$%7! !!!
!
!!
12*,-,"$34$+/'$(*56""
!
!
"-! %=5%&01%,$*! 3*,#0#$0+! %,! %)%3.$'&! .,'! #%&0%! /%! /0#5'&*#! .$0-04',/*! 3*1*! 5&*6%3$0-!
.,'! 5%-*$0$'! /%! 50,?@5*,?! 6! .,! /0#5*#0$02*! /0#5'&'/*&! 3*,#0#$%,$%! /%! /*#! 1'/%&'#! .,0/'#! 5*&!
.,'!(0#'?&'7!<.)%$'!'!.,'!/%!%#$'#!1'/%&'#!I-'!1+20-J!#%!3*-*3+C!%,!#.!5'&$%!#.5%&0*&C!.,!3'V*!
/%! 8KW! %,! 3.6*! 0,$%&0*&! #%! /0#5.#*! %-! 1%3',0#1*! 5&*5.-#*&! I('#%! 5'&'! %-! 5&*6%3$0-! @! %)%!
1%$;-03*! @! &%#*&$%J! 6! .,0/*! '! #.! 5'&$%! 0,9%&0*&C! *$&'! (0#'?&'! 6! .,'! $&'('! 5'&'! ?&'/.'&! -*#!
/09%&%,$%#!;,?.-*#!/%!/0#5'&*7!
<%! 3*-*3+! .,! #0#$%1'! /%! %)%#! 3**&/%,'/*#! %,! %-! 5.,$*! 0,030'-! /%-! /0#5'&*C! G%3G*! /%!
3'(-%!3','-!(-',3*!6!%#3'-'/*!3'/'!QX!317!3*,!30,$'!'0#-',$%!,%?&'7!
<%!3*1%,4+!%-!%=5%&01%,$*!3*-*3',/*!%-!/0#5*#0$02*!%,!%#$'/*!/%!&%5*#*!!5+6*7#%"#$C!
%,! %-!#0#$%1'!/%!%)%#!3**&/%,'/*#7!B.%?*!#%! 5&*3%/0+!'!3*15&010&!%-!&%#*&$%!!5+6*7#%"&$!6!'!
!8$
&%'-04'&! .,'! #%&0%! /%! /0#5'&*#! >.%! 9.%&*,! 90-1'/*#! 3*,! -'! 3;1'&'! /0?0$'-! A! -*#! ;,?.-*#! '! -*#!
3.'-%#!#%!%9%3$.'&*,!-*#!10#1*#!9.%&*,H!QYZC![XZC!\YZC!]XZ!6!^YZ7!
!4$
W*,!%-!#*9$D'&%! !#%!!1'&3+!%-!*&0?%,!/%!-*#!%)%#!3**&/%,'/*#!6!3+1*!%#$'('!.(03'/*!
3'/'!.,*C!0,/03',/*!$'1(0U,!-'!-%2%!0,3-0,'30+,!/%!QZ7!B.%?*!#%!0,/03+!-'!-*,?0$./!/%!3'/'!%)%C!
/%!$'-!1',%&'!>.%!%-!5&*?&'1'!5./0%&'!&%-'30*,'&!%-!$0%15*!3*,!%-!%#5'30*7!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
5+6*7#% "#% 9% "&0% F0'?&'1'#! %#>.%1;$03*#! /%-! '5'&'$*! /0#5'&'/*&! /%! 5%-*$0$'#! /%! 50,?@5*,?7!!!!!!!!!!!!!!!!!!!!!
<0#$%1'!3*,!&%#*&$%!%,!&%5*#*!IQ'J7!<0#$%1'!3*,!&%#*&$%!3*15&010/*!IQ(J7!
!
!
!
B*! 5&01%&*! >.%! #%! &%'-04+! 9.%! 3'-3.-'&! -'! 2%-*30/'/! 0,030'-C! '! 5'&$0&! /%! -'#! _! 5&01%&'#!
3**&/%,'/'#!/%!5*#030+,!6!/%-!0,$%&2'-*!/%!$0%15*!%,$&%!-'#!10#1'#C!/%!-'!9*&1'H!
!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J 7 =
(K" − K7 )6 + (L" − L 7 )6
(D" − D 7 )
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!8$%
!
!
!
./01*-23/451*6*)7*87*&190:;5<=>*?7*@ABCD</E*!+*FGGH!
6!
!"#$%&'#()(*+)$),-&-*
!
*
/*,/%! J 7 !%#!-'!2%-*30/'/!0,030'-C! K 7 ;! L 7 !#*,!-'#!3**&/%,'/'#!'-!$0%15*! D 7 C!6! K" !6! L" !#*,!-'#!
3**&/%,'/'#! '-! $0%15*! D" ! I%#$'#! 3**&/%,'/'#! #*,! %=5%&01%,$'-%#J7! "#$*#! /'$*#! 0,030'-%#! #%!
*($.20%&*,!/%-!#*9$D'&%7!
!
B.%?*C!5*&!$&0?*,*1%$&:'! #%!*($.20%&*,!-'#!2%-*30/'/%#!0,030'-%#!%,!-'#!/0&%330*,%#!!!6!
"C!&%#5%3$02'1%,$%H!
J K 7 = J 7 ⋅ 41C(θ 7 ) !!!:#$!!! J L 7 = J 7 ⋅ C<E(θ 7 ) !!!:&$%
!
/*,/%! θ 7 %#!%-!;,?.-*!0,030'-!'-!>.%!#'-%!/0#5'&'/'!-'!5%-*$0$'7!`!%#$%!;,?.-*!-*!3*,*3%1*#C!6'!
>.%!%-!/0#5*#0$02*!%#$;!?&'/.'/*!'-!10#1*7!
!
!
W*,!-'!2%-*30/'/!0,030'-! J 7 C!#%! /%$%&10,+! %-!,M1%&*!/%!N%6,*-/#C!.$0-04',/*!-'!9+&1.-'!
!1$! 6! #%! *($.2*! .,! N%6,*-/#! /%-! *&/%,! /%! ^XXX7! W*1*! 5*/%1*#! 2%&! %-! N%6,*-/#! %#! 1.3G*!
1'6*&! >.%! -'! .,0/'/! I&<! SSQJC! 5*&! -*! $',$*! #%! 3'-3.-+! 3*,! -'! 9+&1.-'! !4$;! -'! 9.%&4'! /%!
&*4'10%,$*C! +014< 7!
!
`5-03',/*!-'!#%?.,/'!B%6!/%!a%D$*,!%,!-'!/0&%330+,!!!6!"C!$%,%1*#H!
!
+ K 7 = N ⋅ A K 7 = +014<7 ⋅ 41C(θ 7 ) !!!<#$!!!!!!!!!!!!! + L 7 = N ⋅ A L 7 = +014<7 ⋅ C<E(θ 7 ) − N ⋅ ; !!!<&$%
!
/*,/%! + K 7 C! A K 7 6! + L 7 C! A L 7 C!#*,!-'!9.%&4'!6!-'!'3%-%&'30+,!'-!$0%15*! D 7 !%,!-'#!/0&%330+,!!!6!"C!
&%#5%3$02'1%,$%C!N!%#!-'!1'#'!/%!-'!5%-*$0$'!6!;!%#!-'!'3%-%&'30+,!?&'20$'$*&0'7!
!
F%#5%)',/*!-'#!'3%-%&'30*,%#H!
!
⋅ 41C(θ 7 )
⋅ C<E(θ 7 ) − N ⋅ ;
+
+
A K 7 = 014<7
!!!!!=#$!!!!!!!!!!! A L 7 = 014<7
!!!!=&$%
N
N
!
3'-3.-'1*#!'G*&'!-'#!2%-*30/'/%#! J K" !6! J L" !'-!$0%15*! D" H!
!!
J K" = J K 7 + A K 7 ⋅ 9D !!!!>#$!!!!!!!!!!!! J L" = J L 7 + A L 7 ⋅ 9D !!!!>&$%
!
!
/*,/%! 9D = D" − D 7 7!
!
B'!2%-*30/'/! J" !%#$;!/'/'!5*&H!
J" = J K" 6 + J L" 6 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"?$%
6!%-!;,?.-*! θ " !-*!%#$;!5*&H!
!
!
 J L" 
 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!""$%
!!! θ " = A DAE

 J K" 
!
`G*&'C!#%!3'-3.-',!-'#!3**&/%,'/'#!'-!$0%15*! D" !/%!-'!9*&1'H!
K" = K7 + J K 7 ⋅ 9D !!!!!!!!!!!!! L" = L 7 + J L 7 ⋅ 9D !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"1$%
!
6! -.%?*C! 3*,! $*/*#! %#$*#! /'$*#C! #%! 2.%-2%! '! &%5%$0&! %-! 5&*3%/010%,$*! 5'&'! %-! $0%15*! D 6 ! 6! '#:!
#.3%#02'1%,$%7!!
!
#$%&6*5-,%"7"-/%.&%/0("
!
!4$
<%! ','-04+! .,*! /%! -*#! 20/%*#! I;,?.-*! /%! /0#5'&*! ]XbJ! 3*,! %-! #*9$D'&%! ! >.%! $'1(0U,!
5*#0(0-0$+!-'!%/030+,!/%!U#$%C!5%&10$0%,/*!','-04'&!#*-*!%-!1*2010%,$*!/%#%'/*!6!/%#3'&$'&!-*!>.%!
-*!>.%!,*!%&'!M$0-7!F03G*!#*9$D'&%!,*#!0,/03+!$'1(0U,!>.%!#%!$%,:',!QY!01;?%,%#!5*&!#%?.,/*!
/%!90-1'30+,7!W'/'!.,'!/%!%--'#!&%5&%#%,$'('!.,!0,#$',$%!/%!$0%15*!3*,!#.!&%#5%3$02'!5*#030+,7!
`!5'&$0&!/%!/03G*#!$0%15*#!6!5*#030*,%#!#%!&%'-04+!.,'!$'(-'!6!?&;903*!3*,!"=3%-!!5+6*7#%1$7!!
!
./01*-23/451*6*)7*87*&190:;5<=>*?7*@ABCD</E*!+*FGGH!
:!
!"#$%&'#()(*+)$),-&-*
!
*
f!I1J
Q
XCY
X
X
XC_
XC\
XC]
e!I1J
XCd
Q
QC_
5+6*7#%1H!c&;903*!/%!5*#030+,!%=5%&01%,$'-
f!I1J
!
!
E$0-04',/*! /%#/%! -'! 9+&1.-'! !8$% G'#$'! -'! !"1$% ! #%! &%'-04+! %-! /%#'&&*--*! $%+&03*C!
/%$%&10,',/*!3*,!%--'#!-'#!5*#030*,%#!>.%!/%(%&:'!'-3',4'&!-'!5%-*$0$'!6!?&'903',/*!-'#!10#1'#!
!5+6*7#%2$7!
_
QCY
Q
XCY
X
X
XC_
XC\
XC]
XCd
Q
QC_
e!I1J
5+6*7#%2H!c&;903*!/%!5*#030+,!$%+&03'
!
!
!
<%!3*15'&'&*,!'1('#!3.&2'#C!*($%,0%,/*!3*1*!&%#.-$'/*!.,'!/09%&%,30'!,*$'(-%!
!5+6*7#%4$7!
QCd
QC]
f!I1J
QC\
QC_
"=5%&01%,$'-
Q
g%+&03'!0/%'-
XCd
XC]
XC\
XC_
X
X
XC_
XC\
XC]
XCd
Q
QC_
e!I1J
5+6*7#%4H!W*15'&'30+,!/%!-'#!3.&2'#!%=5%&01%,$'-!3*,!-'!$%+&03'!
!
!
!
<%! 0,$%,$+! >.%! '1('#! 3.&2'#! 3*0,30/0%&',C! 5'&'! %#$*! #%! 9.%!2'&0',/*!?&'/.'-1%,$%!-'!
2%-*30/'/!0,030'-!6!%-!;,?.-*!/%!/0#5'&*!3*,!.,'!'5-03'30+,!/%-!"=3%-7!N%#.-$+!>.%!-'!2%-*30/'/!
0,030'-! /%(%&:'! #%&! /%-! Y]h! /%! -'! %=5%&01%,$'-! 6! >.%C! 3*,! .,! ;,?.-*! 0,030'-! /%! YQZC! -'#! 3.&2'#!
3*0,30/:',! !5+6*7#% 8$7! i$&'! 9*&1'! /%! 0?.'-'&! -'#! 3.&2'#! 9.%! 1*/0903',/*! %-! 3*%9030%,$%! /%!
&*4'10%,$*C!/*,/%!%-!>.%!1;#!').#$+!%#!X7\7!
!
./01*-23/451*6*)7*87*&190:;5<=>*?7*@ABCD</E*!+*FGGH!
<!
!"#$%&'#()(*+)$),-&-*
!
*
XCj
XCd
XC^
f!I1J
XC]
XCY
XC\
XC[
XC_
XCQ
X
X
XC_
XC\
XC] e!I1J XCd
Q
QC_
5+6*7#%8H!W*15'&'30+,!/%!-'#!3.&2'#!%=5%&01%,$'-!3*,!-'!$%+&03'C!3*,!
3'1(0*#!%,!;,?.-*!6!2%-*30/'/!0,030'-!
!
!
<0!'-!3*1%,4'&!,.%#$&*!',;-0#0#!/%!-*#!/'$*#!,*!G.(0U#%1*#!&%'-04'/*!.,!%#$./0*!5&%20*!
'3%&3'!/%-!,M1%&*!/%!N%6,*-/#!6!/%-!1*/%-*!#*(&%!%-!3.'-!$&'(')'&:'1*#C!G'(&:'1*#!*($%,0/*!
.,'! /09%&%,30'! 1.3G*! 1;#! #0?,0903'$02'! %,$&%! -*#! &%#.-$'/*#! $%+&03*#! 6! -'! 5&;3$03'7! "#*! -*!
/%1.%#$&'! -'!5+6*7#%:C!%,!-'!3.'-!#%!?&'903+!-'#!5*#030*,%#!*($%,0/'#!3*,!%-!10#1*!/%#'&&*--*!
$%+&03*!',$%&0*&C!5%&*!/%#5&%30',/*!-'!&%#0#$%,30'!/%-!'0&%!6!5*&!-*!$',$*!G'30%,/*! +014< = 7 7!
!
_
QCd
QC]
f!I1J
QC\
QC_
Q
XCd
XC]
XC\
XC_
X
X
XC_
XC\
XC] e!I1J XCd
Q
QC_
5+6*7#%:H!W*15'&'30+,!/%!-'#!3.&2'#!%=5%&01%,$'-!3*,!-'!$%+&03'!
/%#5&%30',/*!&*4'10%,$*
!
!
8,(.6&%/0(!
!
`! 5'&$0&! /%! -*#! &%#.-$'/*#! *($%,0/*#! %,! -'! 5&;3$03'C! 3*,$&'#$'/*#! 3*,! %-! /%#'&&*--*!
$%+&03*C!#%!G'!--%?'/*!'!-'!3*,3-.#0+,!/%!>.%!-'!&%#0#$%,30'!/%-!'0&%!).%?'!.,!&*-!9.,/'1%,$'-!%,!
%#$%!%=5%&01%,$*7!"-!&*4'10%,$*!'9%3$'!/%!1',%&'!,*$'(-%!-*#!&%#.-$'/*#7!
<%! G'! 20#$*! >.%! #0! ,*! #%! -'! 3*,#0/%&'! %,! -'#! %3.'30*,%#! /%! 1*2010%,$*! %,! .,!
%=5%&01%,$*!3*1*!%-!,.%#$&*C!%,!%-!>.%!%#!,%3%#'&0*!0,3-.0&-'C!#%!--%?',!'!&%#.-$'/*#!>.%!/0#$',!
#0?,0903'$02'1%,$%!/%!-*!>.%!#%!*(#%&2'!%,!-'!5&;3$03'7!
!
!
./01*-23/451*6*)7*87*&190:;5<=>*?7*@ABCD</E*!+*FGGH!
8!
!"#$%&'#()(*+)$),-&-*
!
*
",!-'!%=5%&0%,30'!%=0#$%!.,'!9.%&4'!>.%!%#!5&*5*&30*,'-!'-!3.'/&'/*!/%!-'!2%-*30/'/!/%-!
5&*6%3$0-! 6! >.%! #0%15&%! #%! *5*,%! '-! 1*2010%,$*! /%! U#$%7! "#$'! 9.%&4'! %#! 3*,$&'&0'! '-!
1*2010%,$*C! *! #%'! %#! *5.%#$'! '! -'! 2%-*30/'/7! "#*! G'3%! >.%! -'! 2%-*30/'/! G*&04*,$'-! I%,! =J! ,*!
#%'!3*,#$',$%!#0,*!2'6'!/%3&%30%,/*!'!-*!-'&?*!/%-!$0%15*7!
<0!/%#5&%30'1*#!-'!&%#0#$%,30'!/%-!'0&%!6!?&'903',/*!-'#!5*#030*,%#!"!2#7!!C!*(#%&2'1*#!
>.%!-'!$&'6%3$*&0'!$%+&03'!,*!#%!5'&%3%!'!-*#!/'$*#!*($%,0/*#7!!!
K01*#! >.%! #0! 3*,#0/%&'1*#! -'! 9.%&4'! /%! &*3%C! 3*&&%#5*,/0%,$%! '! .,! ,M1%&*! /%!
N%6,*-/#! 1.3G*! 1'6*&! >.%! Q! %,! %-! /%#'&&*--*! $%+&03*C! -'! 3.&2'! '/>.0%&%! .,'! 9*1'! >.%! %#!
3*15'$0(-%!3*,!-*#!&%#.-$'/*#!*($%,0/*#7!!!!
!
#$9$+$(./5%%
!
Q7! <7!c0-C!"7!N*/&:?.%4C!LT:#03'!&%@W&%'$02'OC!8&%,$03%!k'--C!l.%,*#!`0&%#C!_XXQ7!
_7! T7!<%'&#C!m7!n%1',#R6C!k7!f*.,?C!LT:#03'!E,02%&#0$'&0'OC!"/7!`//0#*,!o%#-%6@
B*,?1',C!m%=03*!FTC!Qjjd7!
[7! c7o7!8'&R%&C!L8&*)%3$0-%!1*$0*,!D0$G!'0&!&%#0#$',3%!>.'/&'$03!0,!$G%!#5%%/OC!`17!p7!
8G6#7C!K*-7!\YC!a*7!^C!]X]!Ip.-6!Qj^^J7!
\7! K0/%*8*0,$!8G6#03#!T.,/'1%,$'-#C!m7!B.%$4%-#3GD'(C!87!B'D#C!m7!c0-%C!o*&R#G*5!
8G6#03#!8&*?&'1!'$!F03R0,#*,!W*--%?%!0,!W'&-0#-%C!B%,*=!<*9$D*&R#!q,3C!
G$$5HrrDDD7259.,/'1%,$'-#73*1r!
Y7! s*/'R!"'#6#G'&%!We^[[XC![7Q!1%?'50=%-#C!QY!95#7!
./01*-23/451*6*)7*87*&190:;5<=>*?7*@ABCD</E*!+*FGGH!
9!