Propiedades elasticas de los materiales3

!"#$%&'('&)*&+,)-%.()*'&*+#)*/(-&"%(+&)*
*
0*
!"#$%&'('&)*&+,)-%.()*'&*+#)*/(-&"%(+&)*
*
L*
*
!"#$%&'()*)(+"(,-+.'/*+")(",%+"0*.($'*,(+"
!1;"<(9%$0*/'64"
*
N(*/!#'%012+,*(3'(/!#'%012+,*($*+)1%+14* "*)&*'&6%2&*.#/#*&+*.#.%&2-&*&2-"&*+(*
!12"34.$%)5//'64"
12*&+*"&)-#*'&+*.($3-4+#*'&*/&.,2%.(*)&*5(*&)-4'%('#*.#/#*+()*64&"7()*(.-8(2*
)#9"&* #9:&-#)* %2'&6#"/(9+&);* <9:&-#)* =4&* $#"* +(* (..%>2* '&* +()* 64&"7()* 2#*
C("%(.%>2*'&*+#2A%-4'*$"#'4.%'(*$#"*+(*-&2)%>2*&2*42(*'%"&..%>2*E*)4*+#2A%-4'*(2-&)*
'&*)&"*)#/&-%'#*(*+(*-&2)%>2G**
.(/9%(9(2* 2%* '&* -(/(?#* '&* 6#"/(;* @%2* &/9("A#B* +#)* #9:&-#)* .#2)-"4%'#)* .#2*
/(-&"%(+&)* "&(+&)* )%&/$"&* $4&'&2* '&6#"/(")&* &* %2.+4)#* "#/$&")&* .4(2'#* )&* +&)*
($+%.(2*64&"7();*
12*&)-&*($("-('#B*&)-4'%("&/#)*.4&"$#)*=4&*$#"* +(* (..%>2* '&* +()* 64&"7()* 2#*
)>+#* )&* /4&C&2B* )%2#* =4&* -(/9%D2* )&* '&6#"/(2* E* =4&B* 42(* C&7* '&)($("&.%'(* +(*
64&"7(B* "&.4$&"(2* -#-(+* #* $(".%(+/&2-&* )4* 6#"/(* E* -(/(?#* %2%.%(+;* F42=4&* &+*
$"#.&)#* '&* '&6#"/(.%>2* '&* +#)* '%C&")#)* /(-&"%(+&)* &)-,* "&+(.%#2('#* .#2* )4*
&)-"4.-4"(* /%."#).>$%.(B* &2* &)-&* ($("-('#* &)-4'%("&/#)* +(* '&6#"/(.%>2* '&* +#)*
.4&"$#)*'&)'&*42*$42-#*'&*C%)-(*/(."#).>$%.#;*
!=
!l
*
l0
*HLI*
'#2'&*56*&)*&+*%2."&/&2-#*'&*+#2A%-4'B*E*6O*&)*+(*+#2A%-4'*%2%.%(+;*
N(* '&6#"/(.%>2* &)* 42* $(",/&-"#* ('%/&2)%#2(+* H2#* -%&2&* 42%'('&)I* E* &)*
6"&.4&2-&*'("+#*&2*P;*
*
!1=">$*//'64"?"/%0&$(+'641"
1+* &)64&"7#* '&* -"(..%>2* )&* '(* .4(2'#* )&* )#/&-&* 42* .4&"$#* (* '#)* 64&"7()*
%A4(+&)*E*'&*)&2-%'#*.#2-"("%#*=4&*-%&2'&2*(*&)-%"("*>*(+("A("*&+*.4&"$#*H6%A4"(*0(I;*
Q&* 6#"/(* (2,+#A(B* &+* &)64&"7#* '&* .#/$"&)%>2* )&* '(* .4(2'#* +()* 64&"7()* &)-,2*
"
&:&".%'()* 5(.%(* &+* .4&"$#* '&* 6#"/(* =4&* -%&2'&2* (* .#/$"%/%"+#* H6%A4"(* 09I;* 12*
!17"8+95($:%"%".(4+'64"
(/9#)*.()#)B*&+*&)64&"7#*&)*+(*64&"7(*'%C%'%'(*$#"*&+*,"&(*-"(2)C&")(+;*
1+*!"#$!%&'('()!*"+,**!*)&*'&6%2&*.#/#*+(*64&"7(*$#"*42%'('*'&*,"&(G**
*
*
!=
F
*
A
@%*2#*(.-8(2*#-"()*64&"7()B*+(*64&"7(*2&-(*)#9"&*&+*.4&"$#*&)*24+(*$#"*+#*=4&*&+*
H0I*
'#2'&* -( &)* +(* 64&"7(* ($+%.('(* E* .* &)* &+* ,"&(* '&+* .4&"$#B* =4&* '&$&2'&* '&+* -%$#* '&*
-&2)%>2*'&*=4&*)&*-"(-&;*
#9:&-#*.#2-%24(",*&2*"&$#)#B*$&"#*&2*.(/9%#*&+*.4&"$#*)%*=4&*$4&'&*'&6#"/(")&;*
*
*
12* &+* )%)-&/(* %2-&"2(.%#2(+* +(* -&2)%>2* )&* /%'&* &2* JK/L* =4&* )&* '&2#/%2(*
!().(+*H!(I;*
@&A82* .#/#* &)-D2* #"%&2-('()* +()* 64&"7()* =4&* $"#'4.&2* +(* -&2)%>2* .#2*
"&)$&.-#* (* +()* )4$&"6%.%&)* '&+* .4&"$#* )#9"&* &+* =4&* )&* ($+%.(2B* )&* '%)-%2A4&2*
'%6&"&2-&)* -%$#)* '&* &)64&"7#);* F)3B* -&2&/#)* &)64&"7#)* '&* -"(..%>2B* '&* .#/$"&)%>2B*
'&*6+&M%>2B*'&*.%7(++(*E*'&*-#")%>2;*
F* .#2-%24(.%>2* '&)."%9%"&/#)* 9"&C&/&2-&* .('(* 42#* '&* &)-#)* -%$#)* '&*
&)64&"7#)B*$&"#*(2-&)*'&6%2%"&/#)*&+*.#2.&$-#*'&*'&6#"/(.%>2;*
*
R%A4"(*0;*1)=4&/(*'&+*&)64&"7#*'&*-"(..%>2*H(I*E*'&+*&)64&"7#*'&*.#/$"&)%>2*H9I;*
!"#$%&'('&)*&+,)-%.()*'&*+#)*/(-&"%(+&)*
*
S*
!"#$%&'('&)*&+,)-%.()*'&*+#)*/(-&"%(+&)*
*
V*
12* +(* '&6#"/(.%>2* =4&* )&* $"#'4.&* &2* 42* )>+%'#B* '&9%'(* (* +(* (..%>2* '&+*
'&2#/%2(* ,!-(.*/'&01+ 2"#$%&'/;* F+* %"* (4/&2-(2'#* +(* -&2)%>2* )&* (+.(27(* 42*
&)64&"7#*)&*'%)-%2A4&2*S*"&A3/&2&)*#*7#2()*642'(/&2-(+&)*=4&*)&*$4&'&2*#9)&"C("*
C(+#"* H$42-#* TI* $("(* &+* .4(+* &+* .4&"$#* )&* "#/$&B* &)* +(* '&2#/%2('(* %!1$&01+ (+
&2*+(*6%A4"(*L;*
!$-3!.4(+ ,!+ .(%3./* H#* '&* "4$-4"(I* E* +(* .#""&)$#2'%&2-&* '&6#"/(.%>2* )&* ++(/(*
*
,!-(.*/'&01+,!+.(%3./;**
!("(* +#)* '%6&"&2-&)* -%$#)* '&* /(-&"%(+&)* +(* (/$+%-4'* '&* +()* '%6&"&2-&)* 7#2()*
.(/9%(2;* F)3B* &2* */%!.&/"!$+ -.#5&"!$* +(* "#-4"(* )&* $"#'4.&* '&2-"#* '&* +(* 7#2(*
&+,)-%.(* #* .#2* /4E* $#.(* '&6#"/(.%>2* $+,)-%.(;* 12* */%!.&/"!$+ ,6'%&"!$* +(* 7#2(* '&*
'&6#"/(.%>2* $+,)-%.(* &)* /4E* A"(2'&B* #9-&2%D2'#)&* A"(2'&)* '&6#"/(.%#2&)* (2-&)*
'&*=4&*)&*$"#'47.(*+(*"#-4"(;*H12*&)-#)*/(-&"%(+&)*-(/9%D2*&)*-3$%.#*=4&*&2*+(*7#2(*
$"&C%(*(*+(*"#-4"(*+(*-&2)%>2*'%)/%24E(*.#2*+(*'&6#"/(.%>2*&2*C&7*'&*(4/&2-(";I*
12*+(*7#2(*'&*.#/$#"-(/%&2-#*&+,)-%.#*+%2&(+*)&*.4/$+&*+(*+&E*'&*W##X&* 0E*)&*
-%&2&G*
" = E #*
*
HSI*
'#2'&*7*&)*&+*'&2#/%2('#**0,3"(+,!+7(315;L*
T#/#* +(* '&6#"/(.%>2* "* &)* ('%/&2)%#2(+B* +()* 42%'('&)* '&+* />'4+#* '&* Y#42A*
!
)#2*%A4(+&)*(*+()*'&*+(*-&2)%>2;*F)3B*&2*&+*)%)-&/(*%2-&"2(.%#2(+*&+*/>'4+#*'&*Y#42A*
*
R%A4"(*L;*T#/$#"-(/%&2-#*-3$%.#*'&*+(*'&6#"/(.%>2*6"&2-&*(+*&)64&"7#*&2*42*/(-&"%(+*2#"/(+;*
*
)&*/%'&*&2*JK/L*>*!().(+*H!(I;*
12*+(*)%A4%&2-&*-(9+(*)&*/4&)-"(2*42#)*C(+#"&)*-3$%.#)*'&*C(+#"&)*'&*/>'4+#*
'&* Y#42A* $("(* (+A42()* )4)-(2.%()* E* )4* -&2)%>2* '&* "4$-4"(;* 12* +(* /(E#"3(* '&* +#)*
•@%4*"(,-+.'/*",'4(*,A*5()-(*&+*$42-#*FB*+(*-&2)%>2*&)*+%2&(+*.#2*+(*'&6#"/(.%>2;*1+*
-D"/%2#*!"#$%&'(*)%A2%6%.(*=4&*.4(2'#*'&)$("&.&*+(*-&2)%>2*&+*.4&"$#*C4&+C&*(*)4*
6#"/(*#"%A%2(+*E*2#*=4&'(*'&6#"/('#;*1)-&*.#/$#"-(/%&2-#*&+,)-%.#*)&*/(2-%&2&*
%2.+4)#*$("(*-&2)%#2&)*)4$&"%#"&)*(+*$42-#*F;*
*
)>+%'#)B* 42(* A"(2* -&2)%>2* $"#'4.&* /4E* $#.(* '&6#"/(.%>2* $#"* &)#* +#)* C(+#"&)* '&+*
/>'4+#*'&+*Y#42A*)#2*-(2*(+-#);*J#"/(+/&2-&*/4.5#*/(E#"&)*=4&*+#)*C(+#"&)*'&*
+(*-&2)%>2*'&*"#-4"(;*HJ>-&)&*&+*.(/9%#*'&*42%'('&)*&2*+(*-(9+(I;*
N#)*C(+#"&)*'&+*/>'4+#*'&*Y#42A*)#2*$",.-%.(/&2-&*%A4(+&)*$("(*+(*-"(..%>2*E*
$("(* +(* .#/$"&)%>2B* $&"#* 5(E* /(-&"%(+&)* .#/$4&)-#)* $("(* +#)* =4&* 5(E* 2#-(9+&)*
•@%4*" (,-+.'/*" 4%" ,'4(*,A* 5()-(* &+* $42-#* UB* +(* '&6#"/(.%>2* '&+* /(-&"%(+* )%A4&*
)%&2'#* &+,)-%.(* H(+* "&-%"("* +(* -&2)%>2* '&)$("&.&* +(* '&6#"/(.%>2* E* &+* .4&"$#*
"&.4$&"(*)4*6#"/(*#"%A%2(+IB*)%2*&/9("A#B*&+*.#/$#"-(/%&2-#*&2-"&*F*E*U*E(*2#*
&)*+%2&(+;*N(*-&2)%>2*'&+*$42-#*U*)&*.#2#.&*.#2*&+*2#/9"&*'&*")*&%!+!"#$%&'(B*&)*+(*
/,M%/(*-&2)%>2*=4&*2#*$"#'4.&*'&6#"/(.%>2*$&"/(2&2-&*(+*"&-%"("*+(*-&2)%>2;*
*
•@%4*"&,-+.'/*A*&2-"&*+#)*$42-#)*U*E*TB*.4(2'#*+(*-&2)%>2*($+%.('(*&)*/(E#"*=4&*
&+* +3/%-&* &+,)-%.#;* F+* "&-%"("* +(* -&2)%>2* &+* .4&"$#* =4&'(* .#2* 42(* '&6#"/(.%>2*
"&/(2&2-&;*1)-&*-%$#*'&*'&6#"/(.%>2B*=4&*2#*'&)($("&.&*(+*"&-%"("*+(*-&2)%>2B*)&*
'%6&"&2.%()B*.#/#*$#"*&:&/$+#*$("(*&+*54&)#*HC&"*C(+#"&)*&2*+(*-(9+(I;*
Z&)$&.-#* (* +(* -&2)%>2* '&* "4$-4"(B* %2'%.("* =4&* 2#"/(+/&2-&* &)* /(E#"* &2*
.#/$"&)%>2*=4&*&2*-"(..%>2;*
*
********************************************************
0* Q&*+(*&.4(.%>2*HSI*)&*$4&'&*#9-&2&"*+(*+&E*'&*W##X&*)%2*/,)*=4&*)4)-%-4%"*+(*-&2)%>2*E*+(*
'&6#"/(.%>2* $#"* +#)* C(+#"&)* '&-&"/%2('#)* '&* )4* '&6%2%.%>2B* &.4(.%#2&)* H0I* E* HLI*
"&)$&.-%C(/&2-&;*
L*>B%0*+"C%54D"E7FF=G7H;IJA*R3)%.#B*/D'%.#B*6%)%>+#A#*E*&A%$->+#A#*9"%-,2%.#;*1)-4'%#*+(*
6%)%#+#A3(* '&+* #:#;* Q&).49"%>* &+* ()-%A/(-%)/#* E* &+* /&.(2%)/#* '&* (.#/#'(.%>2* '&+* #:#;*
1)-4'%>* 6&2>/&2#)* .($%+("&)* E* 6&2>/&2#)* '&* %2-&"6&"&2.%()* '&* #2'()B* ()3* .#/#* +(*
&+()-%.%'('*'&*+#)*/(-&"%(+&);*T#+(9#">*&2*&+*'&).%6"(/%&2-#*'&*+(*$%&'"(*Z#)&--(;*
!"#$%&'('&)*&+,)-%.()*'&*+#)*/(-&"%(+&)*
*
[*
!"#$%&'('&)*&+,)-%.()*'&*+#)*/(-&"%(+&)*
*
b*
\(9+(*0G*](+#"&)*'&*/>'4+#*'&*Y#42A*E*-&2)%>2*'&*"4$-4"(*$("(*(+A42#)*/(-&"%(+&);*
K*.($'*,"
K6)5,%")("C%54D"
H&2*^JK/LI*
F+4/%2%#*
F.&"#*
W%&""#*6#":('#*
T#9"&*
!+#/#*
N(->2*
W4&)#*H-"(..%>2I*
W4&)#*H.#/$"&)%>2I*
^"(2%-#*
]%'"%#*
W#"/%A>2*
_('&"(*
!#+%&)-%"&2#*
T(4.5#*
`O*
LOO*
0aO*
0LO*
0b*
aO*
0`*
0c*
[O*
`O*
LS*
c*
S*
O;OO`*
>(4+'64")("
$5&.5$*"(4"
.$*//'64"
H&2*_JK/LI*
LOO*
[LO*
SaO*
SOO*
0L*
S`O*
0LO*
*
*
VO*
L*
VO*
[O*
*
>(4+'64")("
$5&.5$*"(4"
/%0&$(+'64"
H&2*_JK/LI*
*
*
*
*
*
*
*
LLO*
LOO*
*
0`*
0OO*
0OO*
*
*
*
R%A4"(*S;*Q#)*.#26%A4"(.%#2&)*-3$%.()*'&*42*&)64&"7#*'&*6+&M%>2G*H(I*.#2*42*&M-"&/#*6%:#*E*64&"7(*
($+%.('(*&2*&+*#-"#B*H9I*.#2*($#E#*&2*+#)*'#)*&M-"&/#)*E*64&"7(*($+%.('(*&2*+(*$("-&*.&2-"(+*
*
*
F+* &)-%"("* #* .#/$"%/%"* 42* #9:&-#* &2* 42(* '%"&..%>2* )&* $"#'4.&* -(/9%D2* 42(*
*
*
R%A4"(*V;*@&$("(.%>2*'&*7#2()*'&*-"(..%>2*E*.#/$"&)%>2*&2*42*&)64&"7#*'&*6+&M%>2*-3$%.#*
'&6#"/(.%>2*&2*+(*'%"&..%>2*-"(2)C&")(+*H$&"$&2'%.4+("I*(*+(*-&2)%>2*($+%.('(;*F)3B*
$#"*&:&/$+#B*(+*-"(..%#2("*42*/(-&"%(+B*('&/,)*'&+*(+("A(/%&2-#*'&*)4*+#2A%-4'B*)&*
$"#'4.&* 42(* '%)/%24.%>2* '&* )4* )&..%>2* -"(2)C&")(+;* N(* '&6#"/(.%>2* -"(2)C&")(+*
H !t I* $"#'4.%'(* E* +(* '&6#"/(.%>2* +#2A%-4'%2(+* H !l I* &)-,2* "&+(.%#2('()* /&'%(2-&* +(*
/%)/#* $&)#;* Q&9%'#* (* &)-(* '&$&2'&2.%(* .#2* +(* 6#"/(* '&* .4&"$#* +#)* &)64&"7#)* '&*
)%A4%&2-&*&M$"&)%>2G*
6+&M%>2*)#2*/,)*.#/$+&:#)*'&*&)-4'%(";*
!t = !" ·!l *
*
12*+(*6%A4"(*S*C&/#)*'#)*&:&/$+#)*-3$%.#)*'&*6+&M%>2;*12*&+*$"%/&"#*&+*.4&"$#*
HVI*
'#2'&* ! * &)* &+* '&2#/%2('#* 0,/$6'( /!( 8'+""'*9( !#"* +#* -(2-#B* $("(* .("(.-&"%7("*
&)-,*6%:#*$#"*42#*'&*)4)*&M-"&/#)*E*)&*($+%.(*+(*64&"7(*&2*&+*#-"#;*12*&+*)&A42'#B*&+*
.#/$+&-(/&2-&*+(*$"#$%&'('&)*&+,)-%.()*'&*42*/(-&"%(+*2#*9()-(*.#2*'&-&"/%2("*)4*
.4&"$#*&)-,*6%:#*$#"*)4)*'#)*&M-"&/#)*E*+(*64&"7(*&)*($+%.(*&2*+(*$("-&*.&2-"(+;*
J#)#-"#)* 2#* $"&-&2'&/#)* &)-4'%("* &2* '&-(++&* +(* 6+&M%>2B* )%/$+&/&2-&B* $("(*
/>'4+#*'&*Y#42AB*)%2#*=4&*-(/9%D2*&)*%/$#"-(2-&*)4*/>'4+#*'&*!#%))#29*
(*
/#)-"("* 42* $#.#* &+* A"('#* '&* '%6%.4+-('* '&+* &)-4'%#* '&* +(* 6+&M%>2B* C&/#)* &2* +(*
!1L"M,(N'641"
6%A4"(*V* .#/#* &2* 42* .4&"$#* )#/&-%'#* (* 42* &)64&"7#* '&* 6+&M%>2* 42()* $("-&)* &)-,2*
T4(2'#*)#9"&*42*.4&"$#*(.-8(2*-&2)%#2&)*+(-&"(+&)B*D)-()*$4&'&2*5(.&"*=4&*
)#/&-%'()*(*-"(..%>2*E*#-"()*&)-,2*)#/&-%'()*(*.#/$"&)%>2;*
!("(* (.(9("B* .#/&2-("* =4&* (+* %A4(+* =4&* .#2* +#)* &)64&"7#)* '&* -"(..%>2* E*
&+* .4&"$#* )&* '#9+&* #* )&* $(2'&&;* F* '%.5()* -&2)%#2&)* +(-&"(+&)* )&* +&)* '&2#/%2(*
&)64&"7#)*'&*6+&M%>2;*
.#/$"&)%>2* .4(2'#* 42* .4&"$#* &)* )#/&-%'#* (* 42(* 6+&M%>2* )&* -%&2&* 42(* 7#2(* '&*
N#)* &)64&"7#)* '&* 6+&M%>2* )#2* 64&"-&/&2-&* '&$&2'%&2-&)* '&* +(* 6#"/(* '&* +#)*
.#/$#"-(/%&2-#*&+,)-%.#B*&2*&+*=4&*+(*'&6#"/(.%>2*$"#'4.%'(*)&*"&.4$&"(*(+*.&)("*
.4&"$#)*H(*'%6&"&2.%(*'&*+#)*&)64&"7#)*'&*-"(..%>2*#*.#/$"&)%>2*=4&*)>+#*'&$&2'&2*
+(* -&2)%>2B* E* #-"(* '&* .#/$#"-(/%&2-#* $+,)-%.#B* &2* &+* =4&* &+* .4&"$#* )&* =4&'(*
'&* +(* )&..%>2* -"(2)C&")(+* '&+* #9:&-#B* 2#* '&* )4* 6#"/(* &2* A&2&"(+I;* !#"* &:&/$+#B* 42*
'&6#"/('#*(+*.&)("*+(*-&2)%>2;*
-49#* 54&.#* "&)%)-&* /&:#"* (* .%&"-#)* &)64&"7#)* '&* 6+&M%>2* =4&* 42* -49#* /(.%7#* '&+*
*
!1O"P':*,,*1"
!"#$%&'('&)*&+,)-%.()*'&*+#)*/(-&"%(+&)*
*
`*
!"#$%&'('&)*&+,)-%.()*'&*+#)*/(-&"%(+&)*
@&* $"#'4.&* .4(2'#* )#9"&* &+* .4&"$#* (.-8(* (+A42(* 64&"7(* -(2A&2-&* (* +(*
*
c*
*
.4(+=4%&"(*'&*+()*)4$&"6%.%&)*'&+*.4&"$#;**
12*+(*6%A4"(*[*)&*"&$"&)&2-(*42(*'%)$#)%.%>2*-3$%.(*'&*42*&)64&"7#*'&*.%7(++(;*
N(*64&"7(*($+%.('(*-*&)*$("(+&+(*(*+(*)4$&"6%.%&*'&+*.4&"$#*'&*,"&(*.B*$#"*+#*-(2-#*+(*
-&2)%>2*#*&)64&"7#*'&*.%7(++(**H!I*&)G*
*
"=
F
*
A
H[I*
*
!1!">%$+'641"
d2*&)64&"7#*'&*-#")%>2*)&*$"#'4.&*.4(2'#*(*42*.4&"$#*)&*+&*($+%.(*42*$("*'&*
.(
!
R%A4"(*b;**Q&-(++&*(4/&2-('#*'&+*.#"-&*/&'%(2-&*-%:&"();*
64&"7()*.#/#*&+*=4&*)&*/4&)-"(*&2*+(*6%A4"(*`*=4&*-%&2'&*(*5(.&"+&*A%"("*&2-#"2#*(*
)4*&:&;*Q&9%'#*(+*&=4%+%9"%#*/&.,2%.#B*&+*#-"#*&M-"&/#*'&+*.4&"$#*5(*'&*&)-("*6%:#*E*
&2* D+* ($("&.&2* 42* $("* '&* 64&"7()* #$4&)-()* =4&* %/$%'&2* =4&* &+* .4&"$#* A%"&;* F)3B*
*
*********************************
*
.#/#*"&)4+-('#*'&+*&)64&"7#*'&*-#")%>2*&+*.4&"$#*)&*-#")%#2(*&2-#"2#*(*)4*&:&;*
R%A4"(*[;*1)=4&/(*-3$%.#*'&*42*&)64&"7#*&2*.%7(++(;*
1)-4'%("* +(* -#")%>2* 42* '&-(++&* &)* '%63.%+* $#"=4&* +(* '&6#"/(.%>2* &)* '%6&"&2-&*
$("(*+()*'%6&"&2-&)*$("-&)*'&+*.4&"$#;*@%/$+&/&2-&*.#/&2-("*=4&*(+*%A4(+*=4&*.#2*
*
Q&9%'#* (+* &=4%+%9"%#* /&.,2%.#* &2* +(* 9()&* '&+* .4&"$#* ($("&.&* #-"(* 64&"7(*
'&6#"/(.%>2*$"#'4.%'(*)&*"&.4$&"(*(+*.&)("*+(*-&2)%>2B*E*#-"(*'&*.#/$#"-(/%&2-#*
#$4&)-(*=4&*%/$%'&*=4&*&+*.4&"$#*)&*'&)$+(.&;*
!#"* #-"#* +('#B* +(* '&6#"/(.%>2* &2* .%7(++(* C%&2&* '('#* $#"* &+* ,2A4+#* #B* E* $("(*
$+,)-%.#B*&2*+(*=4&*&+*.4&"$#*)&*=4&'(*'&6#"/('#*(+*.&)("*+(*-#")%>2;*
*
'&6#"/(.%#2&)*$&=4&?()*)&*-%&2&G*
*
+#)* #-"#)* &)64&"7#)* )&* -%&2&* 42(* 7#2(* '&* .#/$#"-(/%&2-#* &+,)-%.#B* &2* +(* =4&* +(*
" # tan " =
$x
*
h
HbI*
F+*%A4(+*=4&*$("(*+#)*#-"#)*&)64&"7#)B*&2*&+*.()#*'&*+(*.%7(++(*-(/9%D2*)&*-%&2&*
42* .#/$#"-(/%&2-#* &+,)-%.#* $("(* -&2)%#2&)* $&=4&?()* E* 42* .#/$#"-(/%&2-#*
!
$+,)-%.#*$("(*-&2)%#2&)*A"(2'&);*!("(*&+*.#/$#"-(/%&2-#*&+,)-%.#*+%2&(+*)&*-%&2&G*
" = G# *
*
H`I*
'#2'&*:*&)*&+**0,3"(+,!+'&4/""/+(+'(.%/1%!;**
1+* />'4+#* '&* .%7(++(* &)-,* "&+(.%#2('#* .#2* &+* />'4+#* '&* Y#42A* /&'%(2-&* +(*
!
)%A4%&2-&*&M$"&)%>2G*
*
G=
E
*
2(1+ " )
HcI*
'#2'&*"*&)*&+*.#2#.%'#*.#/#*0,/$6'(/!(8'+""'*;*
d2* &:&/$+#* '&* &)64&"7#* &2* .%7(++(* &)* &+* .#"-&* /&'%(2-&* -%:&"()B* -(+* E* .#/#*
!
$4&'&*C&")&*(/$+%('#*&2*+(*6%A4"(*b;**
*
**
R%A4"(*`;**1)=4&/(*'&*42*&)64&"7#*'&*-#")%>2;*