P ro b le m a s
d e
E s tá tic a
F u e rz a s y M o m e n to s
E q u ilib rio d e l p u n to
E q u ilib rio d e l s ó lid o s in ro z a m ie n to
E q u ilib rio d e l s ó lid o c o n ro z a m ie n to
E q u ilib rio d e l s is te m a d e s ó lid o s
E n tra m a d o s y a rm a d u ra s
M e c a n is m o s : p o le a s , c u ñ a s , to rn illo s
M é to d o d e lo s tra b a jo s v irtu a le s
F u e rz a s d is trib u id a s : c a b le s y v ig a s
10 C e n t r o s d e g r a v e d a d
1
2
3
4
5
6
7
8
9
!
"
"
Fuerzas y momentos
W O c _ g N_kPAW d"lc _RU g RTQ cigc _W#UO c k"RTP dKci\gc U g NWiOYS g POTNjWiO
mA=
ih #W ]^W d YRU #c d O ced W_"PO] c U\]^W g j
#UAPON\PAWfkN d Y c O ce N d R l V
N U\] c O`
a£
a)
#a
"! #! "! $! &%
a
a)
α
#a
)( # ) ( # # '
'
3
IJC 9B 8
= GBH
CKDF >
9ABB
MANOPAQ
RTSVUXW#UXQNVY[ZAN\UAW#U\]^WV_a`
0 ++ 2
=
L
+
CED#F
/ ++ 2
>
1
+ ×
0 ++
B ?;A@9
×
/ ++
× ,-.
*+
L
= :
4
>
3
=
=
*+7 1 5
6
;=< 9 8
:
a
a)
¡
α
¢
bced W_"PO] c U\]^WfW_O c _"PY chg WfO c _QNVY[ZANVUW#U\]^W_ g WfQ cigc PU cfg WjO c _kPAW d"lc _m`
~
=
{Aqq }
+
nAqq z xiys v q w }
qVw | r
+ sEt#uv
~
=
pp\q }
+
op\q
n |
d NOTW#Y c _ g Wm_]^]RQ c `
=
=
=
Y RU cd WiOED c ON d g WiO
Y S g POTN O c g R d W QQiRTS\U g
W O c k PAW "d lcGFIH \
PAW A
m
i CB W]W d
c \PAW c ZO RTQ c#d c O
OTNVPW g W O c k"RTP d&c c g 2 PU\] c Z cd&c VPAW O cfd W _&PO ] c U^] W g W c Y c _kPAW d&l c _ _KW c P U c kPAW d"
A
D
9 KK&: _ER
l c W d ]TR Q c O g WJ
WiOYS g POTN g WjO c kPW d"
l cLFNM WV_ g WPOKK6: `
α
t
"!$#&%('$)
]m^`TgX@[0fZ
α
Ta@[
V3
m
i R
Q W ]^W d YRU c#d O cd W_"PO] c U\]^W g WiO _ER _"]^W#Y c g WkPAW d"lc _ QNVUQ#P dd WU\]^W_ VPAW _KW RU g RTQ c W#U
RW#U g
N VPAWS5 MUTWVXY6Z\[&]_^`Tba@YYcZ\[&]edGTgfhYcZjik]ml`TnVfhYcZGo
cg 2 PU\] c _ c O c k"RTP d&c
q
r s
r t
zx3y
zx~y
p
{}|~y
wx3y
r u
r v
"!$#&%('$)
* +,.-0/ `MAW g 1R P32 cc WV_KQ c O c O c _"PY c g WO c _mkPAW "d lc _a`4 R g RTW#U g NW
O=YS g POTN g W
9&:<; YR g RTWU g NWiO#UAPON\PAWfkN d Y c QNVUO ce N d R l NVU\] c OW _N\]RTWUAW>=@?A
O cjd W_"PO ] c U\]^W[_KW N]RTW#UAW6587
s
α
o
Y Y a@YYX
!#"%$'&$ $ )('%+*,.-0/1(324/1 $ ! $ ,5, 0
$ 67 98 & $ - & =<4 &$ *>,.?
< $ @1,(o4AB24)
&C$
i , $67 ;: $ ,)D/%,FE4*>/1 $ ,G, $ /1
H $ $ *>I $ ,<%*JD! $ i $ *LK 6 <%>* /NM1< $ ?)!
/ )(=O-/1 $ * $PQ$ 8I
o RS,T-/1(324/! $ ! $ , &C$ *,T?
< $ @1,
,U/1LV
WYX T
V3a@o =Z\[ X@o Y^]
WEd T
RS` $ ,<%*D! $G$ ,V
@Yo Y −
T
W
Σ
Wba!c
−
_
o aG]
Io Y −
W ^ T
−
V [ o a =Z
WNl`T
a V o X]
V3a [ o ]
d
T
VYYo Y^]
V(a [ o ^]
−
o Y6Z
α
T
a
− _
Yy
4 0OI$ $ )('4*>G $ ,<%*JD! $&$ *>,?
< $ @, $ 2 $ , $ !D & , $ *? 6 <U &P <!Do Y,<3(= & <%*/
i $ *4K 6 <%*>/OM1< $ ?)/!)(=G-/1 $ * $PQ$ 8Io
¡ |(xL
}w(x
|3x3y
}x~y
p
¡ x(x
{x3y
¢ x~y
¡)}xL
(xL
egf3hFiOjGkQlOm
~
=
nLuyx }
+
nLu n |
w
z
{
=
nvun o n t
s
r
=
n1op!q
Yy
4 0'=I$ ,U-0/1(324/1 $ <4?
< $ @IW &$ (= & <%*>/ I
a fhYYPZ $ & /%,#-/!(324/1 $ $ ,W X 5
i WN^+D0* $ ,#M1< $
W X ?)/1)( $ -0/1W<K 6 <%*>/ &$ a@Y iM1< $ ,< & >? $ $ 4
- > &$ (= & %
< *>/g,
] X ]m^, $ 6 4
< 0* VYYY ZGo
I$ $ )('4,<,(= & <%*>/%, i $ *%K 6 <%*/M!< $ ?)/1)(= o
egf3hFiOjGkQlOm
$ 2 $ , $ !D0-0>1 6 K?>- &$ *>,?
< $ @,
r
rs
}x~y
α
r
X
U/1H%* $ (=1, &C$ ,)DKy
-o
I$ *>.* $ i &$ *, $ 4/O2g*>- & ^*! U>K 6 <g*>/ &$ ?g & /I24/1*>, $ ,?
< $ @1,, $ $ $
=
!"$#%&('*)!,+-&.&/10"2#%43!/5%+1#6)"
B 8:9?; => <
=B
A
+
B @ 8:9"; 7
CDE?5%'F/!5%E"4GIH"1)#%&J"2#%-&J"&('*)!,+-&
O N
−O
M
=
KL?L?L
PEQ :!"RI#%-&ST/-&J)!U5%!-&.&/E0"V5%!
f e
=
cY"X"b `Z _
^
f
d
=
aY?X1b `Z
_
^
]
=
X?Y"Z\[ W
uwvx!y{z\|}~I-&/G Q !:)"!
'V)!,+"&wG Q 1-&kw(V#%-&k6)!w'F,GI)?
t!r-)#\6)&/#%sG5"#(t:!r-)#%6)!I',FGI(1#%&4GIH")#%&E?sw1"I()"G Q #%#\
#%5%H$2¡ ¢w#\)#%(#?GIH"1)#%R"2#%-&JG Q !1V-&#%&Jt!r-)#%&J6)!E'F,GIs$ gh$ikjRlEmonRp
q4 Q &/15%Hsr-t'5%E"2#%-&J'V)!:,+-&
¨
£ ¥
α
α
¦©:ª¬« §o­
®
β
¦4¯°o« ©o­
α
£
£¤
®
β
¦§
α
± ¥ ¦³²,« ´ ±
µ
± ¤ ¦³²,« ¨ ±
_
*V5
!"$#&%('*),+.-0/21436527985;:/=<?>:@BAC>/=<ED6@BFHG/BIJLK(MN/=:OG51P365;79852:/=:QR/
1SFUTWVXFZYHQR@([&/
1S\2>/BA]52<]7B/=:
/=:;Q^<_/W1a`b>:dc=:e
>[&5fG/Wg2JihjVk1E>$1l7BmG2>[n51B/
1EQRo=:p/=:d[n@,<_/([n@(3(qnm2:fIsrUgtvu@([&3=>[n@=<B[n@,7B@6e
:wqxQ^>GyG/z[n@
1
Y \2>/SAE52<]7B@=:}3652:9Fft
365279852:/=:;QR/
1{V|[n51vc=:e
>w[n51 T V
α
α
gh$ikjRlEmonRp
q4 Q &/15%Hsr-t'5%E"2#%-&J'V)!:,+-&
©ª£«
§ ¨
α
§
¡ s¢
^£z=
¤
* 2=
β
α
¥(¢
¦ ¤
β
!"~#&%('),Z:[n@
1Gqn@6e;52:@([n/
1G/B>:,8@=<_@([n/[n/=8`8/6G5<_/63!QR@:e
>[n@=<jG/|@=<4q1EQR@
1j
^&@63=Q^@=:Q^<_/
1jA>/=<ED6@21
G/([7q1E7B507BmG;>[n5jt(u@([&3=>[n@=<v[n@b<_/21E>[xQR@:;QR/SFUt
gh$ikjRlEmonRp
q4 Q &/15%Hsr-t'5%E"2#%-&J'V)!:,+-&
»
µ¶
²
±
µ
·
¬
®
´
¯
º
­
°
³
µ2¸
¹
=
+
+
+
+ $<45[n/=7B@21ZG/U1]QRc!Qq&36@;t
!"~#&%(')21B[{3=>5B<_/=8<4/
14/=:;Q@6G5|/=:.[&@0AEq&e
><_@ @6G43>:;QR@Qqn/=:/ G / =@ <4q1EQR@ I7[ 52<qne;/=:65fV,[n51/7Q^<_/=7B5w1
G/[n@
1 AC>/=<ED6@
1vF T VF Y /
1]QRc=:B/:B/([8>:;QR507B/6Gqn5lG/j[n51z[&@6G51t8! 5 1 7B
m G2>[&51vG/j[n@
1vA>/=<_D6@
11452:k T:9%;=<?>@;
ACBEDFHG 9JIK<?>ML ACBNDOFQP 9SRT<VU ACBXWZY\[=]^[7_a`2bdcKe7_gfhei_j[lkjm:fn]^e7c]^[poqW
µ
¿ Â
¿Ã
Á
´
Ä Å
³
8<_/214@=:G59[n@21ZAC>/=<ED6@
1v/=:}365;79852:/=:QR/
1{VL1_>7B@=:G5
=
+
14/P5;Qq&/=:/j[n@b<_/
1E>w[xQR@=:;Q/
+
rts=uKvMwhxCyqz|{C}~ cKemK[_
8eT[ ;=T<RI A[ka]^TbV_bbheef*fe7_j[fhei_j[87]^em:[%KeT[CfmTc]^T[
8_[7cKe8Telkp >K<IK<VUQ eka]^e[Cfmc]^q[8_j[7cKeCelkp ;7<dL<nR ]eCf8`gk[`mK[lk
]_je[7cgfe
blmT_jee84 ¡mc]^e¢W
£ :k'KeCf_j[k[6felkC_j[cKe8elk[lka]^7c¤e8:k[7c¥`[=]_:kW¦Y'[=]^[7_
`2bVcKe7_'[§fH``[7c]¥[¨o_j[lk
K[C=]^¥[§f
_b[c0©«ª0f:k``[7c]^:k¬[po­_[lk
K[87]^[Of:k¬[® [k0¯ <H°±<p² W
µ
¶h¼½¹j¾
¹
°
´
¿
À
³
°
¶¸·º¹®»º¹
[/63!QR52< >:wqxQR@=<4q&5 /:}[&@SGq<_/6363qnm2: VL1/=:;QqnG5 G/P[&@SAC>/=<ED6@P/
1
! @PA>/=<ED6@P/=:}365;79852:/=:QR/
1v/
1
F#"%$=J'&
=
(−
+
+
)
− (*)*+,).-0/
W%e7_®]bVc
RTUU½IKUU:L
f``[7c]^6[¢fe2®mK[7_
8e_j[lk
K[C=]^qT[Cf@_ºb[7c6[ka]^Te8J:_
M = OP ∧ F
< cK[[§fmc]^
mc]^¤7mKe§fm:b[_je T[6feq_j[87]^e#kºK_®][ [ o `e7cKT [Cftmc]^
| >%<?I%<nU«
< [Cf
_j[lk
K[C=]^[Cf:_bh[7c[lk
;=U I
−
K_Qf:k K[8=]^_j[lkQT['fe!Kelk[
femK[7_
8eZ_j[k
K[8=]\T[fh:k¬[® [lk¢¯!# ° # ² %
W $mMkKeCf_j[lkQkc'&
(+ 9
−
``[7c]^ [ fe mK[7_j8e
>0
f_mK=][lkCeCfe7_Q[Cf:K[C=]^_
(*) 9EI:U
[lkZmTc
>KU
,
<"
T_jK_CbcKe'f:k«``[c]^:kHT[
(-, 9U
>KU AED½` m:[peC=]GFKe k HT_j[Zmc
rts=uKvMwhxCyqz2{T{- cfhepjblm_jee84 ¡mc]^e k[Z_j[_j[lk[c]^eimTc\:e7_@[Z``[7c]^
ΜC =
:fe7c: _bd8c]^eCfª]_j'Ke7_@[Z``[7c]^
G 9%;¡IKU AED½` mK[e87]GFKe kH_[ mTc\fe7cKmK[a_a`e UHI Cc
Μ
[Cf _b8c]^eCf®WY'[=]^[_a`2bVcKe7_@l_jC
b8e7`[c]^[[CfT``[7c] _[lk
m:fn]e7c]^[ [e7`Kk:e7_j[lk
O P
O Q
JLKNM
.0/1325476985:
fK``[7c];=<M[lk [Cf[pmc:e7_¬[OmK[_
8elk [ >KU?>
k4bn]mKe8Telk [7c[CfT:fe7c: _bd8c]^eCf:[7cmc:fe7cK'Ke_jeCf[Cfh
eCf _bd8c]eCfªqk[7:e7_je8ek¬mcKe Kb½ka]^e7c:Cbe [Omc`[=]_¢DK[Cf:``[7c]^!; G [lkH[CfMT[mcKe_je [ ®mK[7_
8elkQ[ >KU>
kºbn]mKeCelkt[7c¢[§fM:fhe7cK¢bVcKCfhbdcKe802[c2mc0:fecK:e7_jeCf[§f2e§fM:fhe7cK¢bVcKCfhbdcKe8ª6k[Ke7_je8TelkHmTcKe'Kb½ka]^e7c:Cbe T[ R ` <
]^eCf C`¢k[ `mK[lka]½_je [7cJfe jblm_je'e W @¦e7_je'ae8Cbfhbn]^e7_
fekjm`e [\f:k%``[7c]:kQ[\f:k
:k¬Ke_j[lk < f:k@K[C=]^_j[k
mK[Of:k¬a_
`e7c0k[ e7c\]^`e8T\Cc0k
mMk¬TbV_j[C8Cbc:[lkKe7_jeCfh[Cfelk¬eOfeZ_j[8=]epT[bdc]^[7_4kº[88CBb AcT[fh:k:fe7cKkW
W
¿
TVU
−
¿
RES
e
@ _H:f[7`ek@T[ ka]^=]bh8eW
o k [Ke7_je8Telk
mcKeTb½ka]e7cKCbe gW $m¢``[7c]0[kQmTc
o ª
f¦Ke7_
_j[k
m:fn]^ec]^[[lka]a_a`e8T0K_ZfelkZ®mK[7_
8elk ¨
−
K[8=]^_ K[_aK[7cKKb7m:fe_ eCf¦:fe7c:J[8
bVcb0K_ o ª
o < :
fhe7cKJmK[
_a`e8c fhe _bd8c]^eCf±mcX7cKlm:fh
−
jblm_je-
θ,
.
O
S
W
0U
θ
J9K M
< _je88[Cf::e7_@_j[lkjm:fn]^e7c]^[ < e7:fhbCe7cKfhe
@¦e7_je CeCf7m:fhe7_Hfe Kb½ka]^e7c:Cbe
9
9EI:< >ML `
fVmK[C\[§f``[7c]^[CfTKe_S_[lk
m:fn]e7c]^[[lk
9
;
@¦e7_je8eCfh7m:fe7_@[Cf:7c:lm:f
k[Z]b[7c:[
f[¡ª¢[§f8:k[7c:eCfT]_bcKlm:f
;=U:L<
ADM`
e7fhb8e7c:feOf[¡ª2[Cf k[7cKeCf]_ºb7cKlm:fh
θ,
k[ ]b[7cK[m:[
θ
9.T<?I5I
e Ke7_
_je _bd8c]eCf [ > `ET['fe7_[lka]k`[7]be'eOmcKe'mK[7_j8eK[7_®]®b8eCf e8Cbe'e Ke® =[
r =s uTvMwxCyXzq{ ~ c:
;I AT e7:fhbCe8e¢[7c¤k
m¥[ ] _[7` W Y'[7`:ka]½_je7_OmK[0[lkO[Cm:bnKeCfh[7c]^[0emTcKe2®mK[7_
8e0[ ;¡I AT e8Cbhe2e :e®
e:fhb8eCe[7c0k
m[ ]_j[`
ª eZmcKe_T[ kº[7c]bT _je7_bh\T[ > AT%D½`XW
¿
U
.0/1325476985:
¿
¿
U
¿
;7U
W%e7_®]bVc
RTUU½IKUU:L
Equilibrio del punto
!"#%$'&)(+*-,/.103254625798;:=<>57!2)?%@>)?BA7!CD2)?E8GFDCIH?BAKJL?B>MC:=7N2)OPO):BF<QCD2K?B>MRSH257UTP>3VXWKCD2K?B>3RN:BYZHD7N>M>PC[\HD<I46>^]>57N>
_ H2`2)?a?BA _ H2`CD2cbdZegfhCD2i]2ZjkAGjk2`25<O/H25<I4l7k2g2/<m2 _ H:'?':naD7k:BAGjo:2)?8GFDCHp?BAKCD2c?B>`RSH257UTP>qV!rP2ZjsCD2gtuDegfwv
xZz
xy
|\}o~
α
{
.0/1325476985:
c:TbCbBAcT[p[Cm:bhfhb _b
x y
|\}o~
{
x z
α
= α = +α
α =
=
;T;
%7kAa?B258G>jXC2!j944S:'OP>Iv
!"#%$'&)(+* ,LKMX< 2)? 2Zj _ H258G>1CD21?B> UR :BYZHD7N>2N 2)? p
a ?BA _ H2 CD2G]2ZjkAL jk2G8G>5<I4S:B2/<2;2/< 2 _ H:'?':=a7k:BA-O5H>5<CA kj 2
>5]?':BO)>QHD<> RSH257NTP>PORQSFeDeLf 25< 2)? D] H<I46AUT C)2 ?ijo: jU46258G>+CD2LOP>5a?B2j vE0M24625798;:n<>57M?B>j 4625< j:BA<2ZjK25< ?BAj
OP>5a?B2jEJ;2)?]2ZjoAVcv
X
^\o~
W
Y
[]\o~
Z
x
{
_ H:'?l:=aD7k:BAK25< 2)?]DH<I46A
_ Hp:'?':=aD7o:BAK25< 2)?]DHD<I46A
*;:<
!e _
`
*;:'A
!e _
*-AI
*-A:
= "$#% ! 9
()
"$#% &'
=
+*-,.0/1*-.2,
8
"$#% 7
= "6#% 5
4 =
3
]257N>/<CDA _ H2PC>
*;:<0/
dZb>=
f
?
*@:AB/C*-A:D/
2d E5F
f
?
HG
bbd2=
f
? *-AIJ/ d 2d =
f
)v >57 4S:=< E eSe = eDe5F
_a`bcdfeOgihMjOkml 2on7./p,4q3.srt0uv0xwzy LVW6{| .up}~s72^34q0Ouup}.rt03.
n80^.ua23nO0834t.201 | 7/10
u.rt02X}.2^..2m.O57^ff/v46.2t0,4^unO2t2^3n80830X.}./prq20/Jf0u;}.2uf42.u.2t4^un80^.u(
B
B
B
_ H:'?':=a7k:BAK25< 2)?]DHD<I4SA
_ H:'?l:=aD7k:BAK25< 2)?]DH<I46A
9 G(R
9 :<
]W ¡
AW 9 G(:
9 :G
9 G([
]W ¢ W = = ?
*;:'A Q -* A:
?
" &(*
! %
" &()
= ! $
" &('
# = !
+-,./10234657.830
9;:<>=@?BAC?BAED
F
9C:GH=I9JG(:K=MLNOLPQD
F
9JG(RTSU?VXWVYD
FZ9JG([\=MLP]^VHD
E
¡ /v4^.rt0uJ3.aup}~}fn80X
_a`bcdfeOgihM
j ml 2on7./p,4q3.srt0uv0xwzy LVW6{| .up}~s72^34q0Ouup}.rt03.
n80^.ua23nO0834t.201 | 7/10
u.rt02X}.2^..2m.O57^ff/v46.2t0,4^unO2t2^3n80830X.}./prq20/Jf0u;}.2uf42.u.2t4^un80^.u(
B
B
B
?(A
H0/ }2
¢ WWP LWW^V
_a`bcdfeOgihij#" J2 .O .uv57.rt03.60 | 7/v0083$ 72X}0#72o^457^.3. ]WD 3.,.uv4q.up}~
72^34o0} /1.u n80^.u
34^u3.-.Off4^un842X}.2^34^u.26726,^f024&%4/v('842X}0O .}./prq20/0u}.2u4X2.u.2t4^un80 .u(
>
; 7
97
87
<
:7
:7
?
=
9 .2u2 .2 .O^n80^f. ?
y
9 .2u2 .2 .O^n80^f. L
y
9 .2u2 .2 .O^n80^f.¢
y
J2z.O .uv57.rt0 3. 0 | 7/10 083*$ 72X}0 4 uM} /1.u n7./p,^4^uM72^34^uM,4/in80^.u .u }~2x.2
^457^.u-,K/. ,^.uv02 n 083072^4q.^457^.10 ,.uv0 NW-D
O.}./ rq20/.O3204/3.
] W-D 8
_`bcd eOgMh j )
.O57^ff/v4+ ^
4 u
54
α
6
y
V } |
α !
?(LW uv.2
α
y
NW
y
? ¢ A
r
?8V
¡ /v4^.rt0uJ3.aup}~}fn80X
3 4^un80^.u(
_`bcd eOgMh j J2i. .uv57.rt063.m06 | 7/106083*$B72X}0# 72^457^. 3. ]WW D 3.
,^.uv4q,.23. 3. .}./prq20/^0 .O 2X}./ 204q3. 20Of4/1.ua3.60 7./'80 ,0/v057^.0rm^4^u nO0 ^.ua.up}2t}.2uv4^u F .O-20O4/
3.-f0u;}.2uf42.u.2t4^un80^.uC,0/v0 y VXWWD 0}4O} |
y A& ¢
α
;
:
13254
α
<
=
h 70234 0}.2u2t.2t.O nO0^.%4X/v '842X}0O uv.8027 0#.2t.O,72X}4 0YnO42n7/ /1.26} /1.u 7./'80u ,^0/10-57.-.up}
.2 .857^f /v4tu 7ou 7rt0 %^03.u./Jn8./14
_
y W
u.234 sf0 7./'80-57.-. $./1n8.-.O^n80^.72^34 0,72X}4 ,>.2 .O,72X}4 0 .O 20O4/J3 e
423fnO2 | /1~81n80-3.-.857^ f/vf4
+-,
* #
$%(' " 1325476
= $&%(' ! +α
.
α
y
/0
)
¢ L] L
D
89254
70234Hf0M}.2uX2K.2K.nO0^. , 0
uv.80 27 0 .2
.O,72X}4
0
n842n7/p/1.2
} /1.u 7./'80u
,0/10 57^. .u } .2
.857^f /v4tu 7ou 7rt0 %^03.u./Jn8./14
_J: y
u.234
W
0-7./'80-57.-. $(./1n8.-.O^n80^f.72^3460O,72}4 . .2m.,72X}4/0 .O 20Of4/J3 e
?]
! "$#&%')(+*-,+&#-."/./-0'132$45
{
sytMu
vwPx
= s?tMu
z
r
.
:Op
InmoZ
'TR
α
:qb
67+.0'1/82 :9 " 9 #-42&/ 9 / 9 <;=</ 959?> 2$A@B(C31"D"/82&E,1/+F-GH2$&#-"JILKG"/ 9 M 9 ,N-#-/O2&.#
PQ > SR
≤
I
≤
"&#&%P
'TR
UV)675W/2AX2 Y
I
Z
[R > 2& 9 / 9 '
/ 9 /E2 :9 " 9 -# 4:2&/ 9\95 ]" 9 < 9 "/W#-/ _^O]/2`H1P
#1a+/)#13 , 1/+F- 9?> 2
1 /-( H +.0'1/./ 9 </./ )/-0'1$2345 B9 1D+ / 9 12C<P</.."/ 9 /_#-/ #
<
! "$#&%')(+*-,+&#-."/./-0'132$45
|
α
:~}
: b
IcbdZ
-ef gAR
h
IjikZl: GR
6~ 42&/@B 9 "/A^ 9 <*M$#-P
3546 798;:=<?>A@CBADEF5GHCIKJMLCEFNL HPO0QRHCSUTVIWFYXPILCIWFZEF\[^]`_bac dPegfih [9jk_ml c0nCegfoIWFpQrqKOsHPOutVXPEFkvwIxXCtVSKO0QrI
HPOyGxSKzuTVI{h|F+I\SKLuE=hSKO}FZE~zPJIHPOCSFpHPLuIKJptVGUtVI\GUt;T;OCX0JZtVGxSbT;t
FZSQrS=TAGxE~vwEyFZICI\IOTVSptVWH JS\S=XHPOPQrS0
DIQrIKJpvAtOCSKJTVSYQrIKO9F+tV~OXPI=TuGxSzuTVI=PT;SWFOCE~JMvwS=T;IWFIKOwTVEuFSKLCE=hEFhAI=TuqKOCWHTVEXPI
Ix~Hut;TRtzPJZtVE9
cCe
−α
α
[j
[]
/)/2H1 ^n0'1$2$ 45
/)/2H1P ^j0'1:$2$4?
fg
α
fg
H2$&#"
[^]
α
c e
−α
α
Z
)
-
$
% #
(+*, α = &' ( α =
"
α = α = α
2&82&/ "/2 9 / B9 /A&//
!
. H/+"
[ j
0'1/-"
> QP0/
h
1 Z
Q: e
R
h
2.i Z
e> R
h
2AbdZ
ff Q > R
e
={SKJ Q tO e ePc
lCePe
3546 798;:=<?>g@+*,^OATVS ptVWH JS SxXHPO0QrS I=TCzTVEP~HCI X I ~ePe
fXPI LCIWF+EF+IYvwSO0Q tVIKOuI IOI=~Hut;T;t zPJZtVEIKOwTVS LCEuFtVG=tV~O
tOCX tVG=SxXPSzCS E TVSgSxGxG=t;~O X IgTVSg HCIJ/.xS10 SKLT;tVGxSxX S IKO I=T I2PQRJZIKvwEsTRtzPJIgXPITVSGHCIKJXPSg~HCILCSWF+S LCE0JNTVS LCE TVIxS
3
DIQ IKJMvAtOCSJIUT0CSUTVE~J^XPI TVS
HCIKJ4.xS0
JLK
M
I
T
O/QSR
OPK
N
N
O4QSR
f
g
$!"# h Y R
&
= "!$# %
= "!$# \-]_^`Fabcaed
3546 798;:=<?> @P@5,^O I=TYIWFZ~HuIKvwS XPI TVS tVWHPJS SxXH O0QrS=kI=TG=SKzuTVI-687IWFMQrq HPOutVXPEmLCE~JFpH I2PQRJIvwE97S HPO
vHCI=T;TVIsGKHWhSsGxE~O9FMQrSKO0Q IAXPIwJZtVPt;XPI.AI~F;:s_<~ewf=Rv ?>utNFZIwSLuT;tVGxSAIKO IUT^I2PQ
JIKvwE@7 XPI=TGxSKzuTVIHPOCSA HCIKJ/.xS
CIKJ Q tVG=S=T9XPI~FZGxIKOCX IKO0QrB
I A ( D
_ CWe f}I=T`F+trFMQrIKvwSIWFMQrqIKOAIx~Ht;T;tzPJ+tVEGKHCSOCXPEI=T9qKOCWHuTVE _ dPeFEN=DgIQrIKJMvAt OCSKJkTVS
θ
TVE~OCPt Q
HCXOCSQRHPJZS=T 'HG XPIUTPvHuI=T;TVI
I
[+K
UWV
X$Y R
M
T
NFZ
')(
_|lC d dbv
ac
[9JZE~zuTVIKvwS~F^X I ,5FMQrqQ t;GxS0
Equilibrio del sólido sin rozamiento
3546 798;:=<?>\@ [ SKJS TVEF F tVWHutVIKOPQrIWF X
t FpLuEuF+t Q tCEuF+ GxSUTVGKHuTVSJ?T;S Q IKO F+tV0O X I=TgGxSKzuTVImh TVS JZIxSxGxG=t;~ObIO TVS
SKJ Q tVGHuTVSxG=t;~O,kT LC~I FZEXPI TVS Cz SKJMJSFZI
GxE0O +F tVXPIJS
XPIWFpLPJZIxG=tVSKzTVI0
T
J Y R
>
X$Y R
h Y R
J Y R
lue
={SKJ Q tO e ePc
lCePe
l^a
[9JZE~zuTVIKvwS~F^X I ,5FMQrqQ t;GxS0
3Y46 7 8;:U<?>s@ [ SKJZSTVEuF X t
FLCEuF+t Q t CEuF XPI=T L JE~zuTVIvwSSKO0QrIKJ+tVE~JZGxS=TVGKHTVSKJ VT SQrIO F+tV~O?XPIUT G=SKzuTVI h TVSJIxSxGxGUtV~O
IKOwTVS
SKJMQ tVGKHuT;SxG=tV~OsF+tTVS~F zuKS JMJSWFNFZE0OuE~vwEPKOCI=SWF h FpH LCIWFZE IWFXPIAalCe f
l l
={SKJ Q tO e ePc
lCePe
%'&)(+*-,/.103254#687:9#;=<#;?>@>A;CBEDFCDGIH?9#J1;LK+J=MENNLK+JCOEJPADRQTSUD9#GVXW/YEK5Z[PAJL;?O#D1Q\;3PAD<>AJ=KD\P]P^Y+O#J?>`_^VUa1VbJP
S/VPA;PcWd;1S-aDFCDPAJFeY#JP^W/>`;fJ?9Sb;f_^VbGY+>`;f;KhgiY9IWj;k1l]J?WjJ?>@F=V9E;?>nm;opJ1S#q#;1SbDI>rKJ:Sb;:_dY#J?>^s;ftuO#;?>A;FC;?9IWjJv9#J?>rSb;
<E;?>@>`;J?9wJxY\VUSUV<>yVbDzJ?9CSb;nO#DEPyVba{Vb|9CV9#K+VUa;K;]<#o}Sb;P}>`J;1aa1VbD9EJPpJ?9CSbDEPp;?O#D1Q\D\P~k
t
+N
N
>CE~zPJI TVS zCSKJpJS?SxGQCSKO GKHuSQRJE HCIKJ/.xSWF ,kT LCIWF+E 3NT;SWF OCE~JMvwSUTVIWFIKO TVEFSKLCE=hEuF ~ h{TVS HCIKJ4.xS
SKLuT;tVG=SxXPS
IKOI=TuI2PQRJZIKvwE
DgtVSxWJZSKvwS
XPI=T9FZ T;t;XPET;tzPJI
t
NI
NI
%
E~OCX t;G=tV~OXPI
Ix~Ht;T;tzPJ+tVE
fe@
h{L
f
i {
N
%
I db
h1
y
)
f
~ nX@
~ i U~
t
E~vwSKOuXPEvwE~vwIKO0QrEF JZIWFpLCIxGKQrEXPI
LCIJSKOCXPE~HuIxXPS
_
'
−
'
G=EuF eFE _
e
!"#$
l [P@W)WdVUa;Ik
>AD<ESbJ?FC;PcK+J
)(
NNL
%[&)(+* ,U.103254 87:9#;=<#;?>^>`;=B#DFCDGIH?9EJ;KJ
SUVPA;P
Wd;1S aDFCD5PAJ
FeYEJP@W/>`;eJv95Sb;e_^VbGY>A;e;Khg Y+9IWj;Ik
+*
Y9zFeY#J1SUSbJnY9\VbKDz;:P^YwJ W/>`JvFCD: KJaDI9-P@Wj;?9WjJ
J
KJ=O#JPADRQ
SbD9#GVXWY#K5Z[PAJL;?O#D1Q\;3PAD<+>`J=K+DEP]P`YO#J?>`_`Vba1VbJP
FC;?9IWdVUJ?9#JeJ?93JxY\VUSUV<>yVbD=<#; g~DRSb;z;a1a1Vb|93xY#JeSbJzJ g)J?>`aJ
-,/F3kl]J)WjJv>@F=V9#;?>cJ1SE;1SU;?>`GI;?F=VbJv9IWjDzKJ1S+FeY#J1S/SbJIk
INN
N
N
%
DI<>`J3Sb;L<#;v>@>`;R;a)W #;?9ua?Y#;)W>`D
;?OESUVba1;K;J?9wJ1SEJ W/>AJ?FCD
l
_dY#J?>^s;Pwm SpO#JPAD5%
cSb;P9ED>@FC;1SbJPfJv9uSUDEP:;?O#D1Q\DEPI zQ
Sb;_dY#J?>`s;
k
Vb;G>A;?FC;KJ1S PA|+SUVUKDeSUV<>`J
t
N
N
%
D9#K+VUa1Vb|9wKJJxY\VUSUV<>yVbD
^
y
/
i {L
%
.1Le^
y1LR
fC
i {
t
0132
I db
h1
y
)
X@
~ i1 b)
pDFC;?9EKD]FCDFCJ?9IWjD\P
O#Jv>`;?9#KDzxYEJK;
>AJP^O#Ja?WjDzKJn
!
{
@
h
−
∆
ZE
Z i 11
'
&
=
Z
i 11="N
% $ !#"
M
, ,
1k$;?> WdV9
N dN M#NNE
%[&)(+* ,U.1032C4
PAJ
;vO#D1Q\;
J?9
PyD<>`J
:
7:9E;<#;?>^>`;B#DIFCDGIH?9#J;]KJ
P`YJ +W/>`J?FCD
Y9#;
KJO#JPAD]Q3SUD9#GVXW/YEKCZJP@Wj;:;v> WdVba?Y\Sb;K;]J?93P^YJ W/>AJ?FCD
P^YO#Jv>`_^Vba1VUJ
SUVjPA;
Wd;1SwaDIFCDPAJFeY#JP@W/>A;
J?9
Sb;
_^VUGY>`;
Q
;1Khg Y9Wj;Ik
l]J)WjJ?>^F=V9#;?>pSb;>`J;aa{Vb|9wJ?9CSb;;?> WdVUa?YESb;a{Vb|9k
1
11
DI<>`JSb;n<#;?>@>A;;1a)W #;?9
l
W/>AJPp_dY#J?>^s;P[m S+O#JPAD]%
Sb;n9#DI>@FC;1SEJ?9wJ1SE;?O#D1Q\D
Vb;G>A;?FC;zKJ1ScPA|+SUVUKDRSUV<>`Jkr;zaD9#K#Vba1Vb|99#J1aJPA;?>yVb;]O#;?>`;zxY#J]Y9
QLSb;n>`J;a1a1Vb|9wJ?9
PA|#SUVbKD3PADFCJ)W VbKDL;W/>AJP
PAJaD> WdJ?9wJ?9zY9zF=VP^FCD]OY9IWdD DCPAJ;v9
J?9wJxYEVUS/V<>AVbDzJPpxYEJSU;P W/>`JPp_dY#Jv>^s;P
k
_ Y#J?>^s;P[JP@WdJ
OE;?>`;1SbJ{Sb;Ppo
1{
1
%
D9#K+VUa1Vb|9wKJJxY\VUSUV<>yVbD
(
"$#&% 0 - '
!
= "$#&% *,+-/.
) = +α
α
%
α
I db
h1
y
)
~ nX@
~ i U~
1
pDFC;?9EKD]FCDFCJ?9IWjD\P
O#Jv>`;?9#KDzxYEJK;
>AJP^O#Ja?WjDzKJ
M})
@
h
{ Z
α
2
rf
Mc) M
−
Z
i {L
N
M-
[P@W)WdVUa;Ik
>AD<ESbJ?FC;PcK+J
)+*-,/.1032547698: ;<=?>=@A@B=DC!EFGEIHKJ<L5=DMLNE )PORQSE<!HITVUXWY9Z[LN\U]=?L<^L`_W T3Q3Ta>I@bTSE9L<9WI<=?c5=-d!TSYI=`Y
NLF
TSLNbeAJ@TSc5=fQ3TgN=hYILi@=`Y/TSEfjkU]=lQmc`EFGEnNLiFhWLNAUX@B=hL<oQS=pe\T3HWI@B=p=`YrqsWI<KU]=KtIuL-U]L@\F
Ta<=@+L5Q1d=5QSEK@vY/L5Qxw<HW!QSE
YILyL`_W!T3QXTa>I@TSE α NbT!Z!z7{5jyt
ª
©
α
¨
DI<>`JSb;n<#;?>@>A;;1a)W #;?9
l
W/>AJPp_dY#J?>^s;P[m
S+O#JPAD]%
Sb;n9#D>@FC;{S#Jv9eJ{S#;vO#D1Q\D
Vb;G>A;?FC;zKJ1ScPA|+SUVUKDRSUV<>`Jkr;zaD9#K#Vba1Vb|99#J1aJPA;?>yVb;]O#;?>`;zxY#J]Y9
J?9wJxYEVUS/V<>AVbDzJPpxYEJSU;P W/>`JPp_dY#Jv>^s;P
f'QLSb;n9#D>^FC;1SEJ?9
PA|#SUVbKD3PADFCJ)W VbKDL;W/>AJP
PAJaD> WdJ?9wJ?9zY9zF=VP^FCD]OY9IWdD DCPAJ;v9
k
_ Y#J?>^s;P[JP@WdJ
OE;?>`;1SbJ{Sb;Ppo
¥
¦
£
}i~
¤
α
α
¨
D9#K+VUa1Vb|9wKJJxY\VUSUV<>yVbD
α
-
)
|
α
=
-K
α
=
α
KaSr55'b'sa-
'VA' s¡`¢ 3'
}i~
α
pDFC;?9EKD]FCDFCJ?9IWjD\P
)
α
α
§
α
©
§
O#Jv>`;?9#KDzxYEJK;
>AJP^O#Ja?WjDzKJ
a1DEPcM
α
α
! M
α −
aDEP
α −
α −
=
α
α
"N
"$#%'& #'(
ME
5t =@ UTa< {
)+*-,/.10325476G83 ;<!= >=@A@B=CEFGEIHKJ<!L`=Y/LiQSEK<HITVUXWY Z O MLNbEf) LNAUw WI< TSYI=ME@WI<E YILpN\W1N L54IUg@BLFGE!N =WI<
M=N=`Y/E@ _KWL M/WL`YILDYILNbQXT6!`=@ NbTa< @BE.!`=F
TSL<KU]E ME@ WI<=DHW+S= dL@ UTSc5=5Qt- =k>=@A@=?NL =ME5O =?NE>I@L WI<=
N\WIML@e\TSc5TSL clT3Q7+a<YK@TSc`= Q3TgNb= Y/Lo@=`Y/TSE jyt T Q3= Q3E<HITVUXW!Y Y/L Q3=o>!=@A@B= LNG{5j8&vYILU]L@AF
Ta<!=@fL5Q w<HW!QSE α YIL
L`_W!T3QXTa>I@TSE1t
l
α
m
n
α
o
k
EK>I@BLQS=>=@A@B==`c-U =<hUX@BLN eWL@"!`=N$#
%QMLNEi)&/QS= <!E@AFG=5Q L<GLlQ =ME5O E vO QS= @BL`=`c5c5T('<GL<*) YTa@TSHITSY/=
ML@AML<Y/TScW QS=@AFGL<IU]Ly=yQ3= HW+S=Kt
uiTS=`H@=FG=pYIL5Q N,'/Q3T3YIE[Q3Ta>[email protected] p
= c`E<YTSc5T('< <L5c`LN=@bTS=M=@B=p_WLWI< N,'Q3TSYIE7NEFGL-U TSYIE =UX@LNve WL@"!`=N+LNAUL
L<fL`_W!T3QXTa>I@TSEpLN _W!LyQ3=NmUX@BLN eWL@"!`=N NyL c`E@ UL<fL<pWI<pF
TgN\FGEMIWI<KUE0/mEGNL`=<iM!=@B=5QSLlQS=N21
i
g
h
¦ E<Y/T3c5T('<fYILyL`_W T3Q3Ta>I@bTSE
α
d*e
f
b c
j
d e
α
f
< ;
5> ?@
α
=
=
ACBED α
=
<
9:
F.GIHJ(KMLONQPRKQPRS
HTKUWV
NUXGISCY
K5H
USCY$J[Z\KH]G^YEUX_a`Y
G^J7NH
b0c
!
$%&'(&!)*+-,./'$i
0
<(=
?>-
@+A'BC&.
HJIGK
L ,.MB&.'/NA'
STU α (STU
e
j i
− 1
687
5
2+3.4 α
9;:
EGD F
α =
R
α +
Q
)=O
P
V
α WYXZ\[ ]\X_^
fijklmonqpsr.t
j o
A(ou,+vwGxsNA.y,'A.'v,.z
{+|A|A'B{'{
q>B
@+A'B}~
M&G
&
w)x('oo
(+( ::': bq
?wGx
c,B+
!&8NA.?,A.'z,.Ju
?B
u' " ::': by b
&G'B%.,.+c'
.B'aiu).,.+c'
.B=
:': b)-A8
&.@'x|A.C
?@!N!A.,.&.'
'ow'|*
i " :':': b .Bc,.=
.-Bo
!M
`)acbed/f?g\h
d
"#
o Jx& :: " :':
fijkGlmnqprC&MJo '=,.
@A.
@!C'
A'&q,.J?o y=cA.|B(& (>-
@+A'C_A'&!'
j
A
M
&G
!&yo =
o& M&y
zJJ
A
oo
& h )
&G&8w)xzNA.z
,'A'
,)
oM
?+= :':': b ! j A.o+%o,.+Cw)x/'ooNA.C,'A.o'?
@!&i
1
"
C
`)acbed/f?g\h
d
"
!
Equilibrio del sólido con rozamiento
!"$#&%(')+*$'-,.$.-/0*'"1('32+465
!#$798;:(*<=2'-1(4>?2!$@'-*$>2A2@:$1'@B@7DC7D'32E8FG5
C)!"?2'H:'328I@J'?5DJBK7D#3:$@J)*MLM:$&8N&O&PQ"2K:$1('@BK7DCG7D'H7RC5
7S)*$J'32T5
7I2UE45D!U7SV)!&8FG5K:#&(2&O!WA'8F'X7SZY
)[\'5]5D!*$'H5DJBN:'^KV)J*$'@&V)X7D'^&8F-'_`1@"&8N''=5DK@J'')a!:(7
5;7R$7D'5Db1(2U7DCG7Dc!7R*7DC)*$"d
[e'5(C)$')BK7
C7D'&8N'f*'=&V)X7D'$8FXgR(7R"1f'G5'a!:(7
5
7S$7DhO
$.!
.!
i>jHkmlJnEoqpJr
!s
t!@'5Dub)C8IvC^:8;@XBN:'@V)32EY$wx5>1'!2Hmy(5
32Q&"5D'32m'-5
(2\1(4>(2 rZz y rZ{ 4+5DuB|:'KV)b}
$* 'T@&V)X7D'^&8FJ'T_AO
WA7D)#3"E*$'52c5
7
*$H5
7R$@'
¯
}
.!
.!
r {
r z
~ !*7
C7Dc!*$'E')a!:>7
5
7R$U7D
r {
zu {EEG)
.!
{f ¢¡3£G) ¤
r z
¥&¦ N§D¨Z©xªxª¡3£q U«3©q ¦ ¡3
q¡T§9¬q ¦ ! ¢­)®^ ¦ §
©q
}
!"(*$"!"'&8F>2e'32K1')C8FJ*$'=
1'^@*$Ja!:(')*$ { 3$yR/
E{<
−
<(C(2.!+.
B?/
:
O |8N7R .N.I,.$.(
(
s @ '5bC$')BK7DCG7D'&8F' *$' @&V)X7
'&8FXgR(7R"yf5
K@ '32K8 ' '328F*$*$' "3](7RX7D'^&8F 7S$X7R'$8F' 4`5
C)!@'!2K1!*7D'&8F'EBN:('KV)E*$'T@&V)X7
'&8FJ'32Q5DT>7S"y$5R:'#&2U'fC^:$H1(5D'Ea!:'`¤b
µ
wx5$]5D! *$'x '32^!$yRf/ y&*$'E*$!*'fa&:')*$E'5$]5D&
µ
?.$yh
$"PxZK@Z&"$#&%')\_*'5DBK7D#3:$@)*L¢:$&8F *$'u"32` "!#`4+5
!#$798;:(* <h?,Xy>2'
"&8N7D'^'H'?')a!:>7
5
7R$U7D+14>*$H'?'5£!*$'J*$':$2!1!8F' .>O H *'2@:+'#$8;@'"u_ 4+"')*7D&8F':$
C)(5D'X:$>7D*$ ?2K: ' $8;@'^"$ZOhW'5x' $8I@'"_ 1'*$'X:$ C:('1 *$'X"32"%x !#O>WA'8F'X7SHY )[
*7R$:¢Lb'5\*7
)#3@"+*$'G5=2c5
7D* 5
7R$@'+*'5Dd [EC)5DC:>5Db5D"8F'h27Dc! *$'5xC)>5D' d C)[Z5D-BN:'KV)-*$'
@&V)X7D'^&8F-''5 14> d>*[m2U7 '5 14>2'AC)!h27D*$'@J5
7I2Uy>*$')*!:(C7RT2U7£#' >7I28F'^]5D!@'!2m*$&
' '
4 % 1@
a!:'Z5DTu2U'="^&8F'#&E'')a&:(7
5
7R7DJ'"5DT1(2U7
C7Dc!"7R*7DC))*$$O
2
(4)+*
3
()+*
1
576
i>jHkmlJnEoqpJr
W 7D#3@"-*$'52c5;7D*$ 5
7R$'&O t!@'+5DX@-)C8;v BN:'@V)32"Ye5
(2Z1'32(2*$'+5DXJ46'5(5Da!:'y 5
A
8F'h2U7
c!*$'G5C)>5D'T45
Z@'32@:(598F$8F' ''5h1$:$&8N"*$'14>y(a!:''32m5DH2K:$"*$'5Du!"G54+*$'A5DbB|:'KV)u*'
@&V)X7D'^&8FhO
()+*
0
,
_
()+*
-/.
%
}
~ !*7
C7Dc!*$'E')a!:>7
5
7R$U7D
!(5D'"!2
*'=w288N7
C)&O
mFnFoqpfr3sutvAwhx$y'zGx"z|{~}1Vrvw
V
opZ
rvAwGxyzGx"z{}r3st_vAwA
D r5Csu_AsArs
tsr
th3s
r t&Vt =
r
vw
{
vw
!"(*$"!"'&8F>2e'32K1')C8FJ*$' ~
= + "!$#&%'
.
+
=
)
) -
=
** )
,
(
/"%103254627%89'!:<;=>0?%""@%@A4CBDE';=F@A!&'4C@4CB&<"#&%G47;=46H"I4=!<0J2K4C
R
P
= Q =
O M M L
N
/94SG!UTGV8WI!DXYZ4C"2[!'\]0I^@%GA;C0$0I^;=0_Y`0Iba
8caEde;C0>@!&YE!&GD2[0>"f;C!G00I!90J8g"b;=
2[104=B&<`;C'46@A@A4CB&<"_;ChHh!`0I@@;CG\";i%?D!j!j"%'kTGHS#&%94=;=46H"4C!1l
$,
l K2546
/ H3I>T!DY`!"?NMOP "$;CQ%4C?%":SRV%"D2[UT '>Y`0 a 8 ;C!&?"42 %WV7\0I Y`D254=:
#&%94=;=46H"4C! !A89'bW;SHG!&"X, "`%"g0!&!&K25A\2[A; #&%-Y2;Z9[V\] 8WY`'4C"2[<%' @HG;C`%"G4C"! 0%
(5"2 Y`!^Pl A;h("
5 27IY`!_O G" %' @%G3! " Y`&0I aEd>9`#ga`l
2[3YZ4ia' A;b1;C!& YKciG46Y`!g'A;
@!"G% 4C@A4CD2["h!DXYZ4=D2[!
#&%_;ChH3 0IhY`D2[?D<#&%G4=;746H"I4C!$`;Ch!904C@A4CBD<4i'4C@A"Dl
µ
e
<>=@?ACBEDGFIHKJL
klknm ioi
f
gUhji
d
prq
C4 A?Y` "A;_0IB';=4C'! ;=4iH"Dl /!DH" ;C HG3 @2"!G$#&%5%X&0(' A; 0I! " ;C H3Z8 "A; H9;C!"#&%\;=
2[104=B&Z"A;@HG;Ch8 ;C0%G;2["2[)+*_Z;G"%"D25!<"_!89!-, \"#&%_&0;Cj0%"Y`_">;=F!&3Y`A;D8b">;=.%5%X_'
!DXYZ4CD2[!1l /0
%K%X"h!DXYZ4CD2[! 254= 0%21A;C!& Y43(5146Y`!6%879
*
µ :
P
w hxi
y
z
{|
,
~Ux
}
<
O
< d
;J!&4C@A4CB&<"#D%G4=;=46H'I4C!
~x
m n oqpfr3sut.stJx$y'zGx"z { }1Vrvut
~x
µ
z
D r5Csu_AsArs
tsr
th3s
r t&Vt =
r
x
m n xqpZ
rvstqo$yzx'z { }rstvut
{|
!
%&(')$*'!+,-/.)01$32
B
< B
; B
< 5
456
C EHFG < 5
C
D = =?> E-F)G
D + 4 A 78:9
4 @
I .),(')$1JK)
O
R S PQ LNM
=
T KUVWK-XY')Z('[.V,3(K)0
\!']3^0')
H
\'[$3JKWV$[?_
')
bZc
d-f d-e `&a
=
gih,V')0V('!3$j
kl-K$'$3K)H
\'[$3JKWV$
µ
mon-prq(s
º»¼½¾¿0ÀÁÂ)Ã[Ä Å
0V,)©$·!Æ«ª) º ¡¢,)
ZÇZ)0¡Y[,ª«)¢V)
0)$[)·!
§)ª!©jÈ$È$É+
!Ê
3)!!0¬)~)j)
!W,j(
)
0
)N$^©-®V©«
(![V$©«)!^_Ë|[
j0
0
¸
$ª
j)(/j§)ª
jË|,
0j0Y
^)!!0)$Y
$30
0),1Ê/1)k/$)?
̬ ¶ ©«)0$®)0
$0
$,
0
$3$,!©«
(! ¬
µ
Ð
Ñ
Ò
Ï
Í
qÎ
tYu]vxw1y3z|{1}
)~ ,0^
0V,
W)(/-N0)-?$0
0),x/)!!0 $0
¢¡£
,-/V!^¤¥j(0)$$!]$j)0¡Y¦o§)j-¨*/$©j$ª
)!©0¡3^_)/3$,!(©«
!¬(­N3),
$,!©«
(!k
ª,*®)H
¯©°±YV©²³-´
¥
µ µ
¶
0·-,©3$0¸W
$]V$¹¡«
)
0
¸[$3§YV$?
U¬
Ñ
Í
Ó
Ó
ÙÚÜÛ
qÎ
Ô Õ
Ô Õ
Ø Õ
ÖW×
Í
µ
qÎ
mÞÝß?à^s!ámZn-p
Íâ
Ñ
"$#
¬_V
¡-/°1)0¡Y«
º»¼½¾¿0ÀÁ«Â
$§!)*)©$·!Æ¢$]
1/
·-$,1ÜËÜ$!ª_
)*$¢)-?Ê
µ
$)),Å®)_
0¬ &
$/
0
!$[,!©«
(!(!W,«©])-*/$)/
0
3-
´$ ¬ +
$§!)¢
)
)(!W,i§YV$?
Þ)_Ë|
0
0
¸i$
)(/ 0Z
© ¹©])-/Wµ
,
/
·-$0:ËÜ$$!¬
¶ /©«V)0YV!_®)0
1$®)0,-&$^)^§!)30
$§ª?©(!)·!3[§!V
U¬
É
tYu]vxw1y3z|{1}
))
0
-N$^©-®V©«![V©«V)!0
^
0¸«
^)(,]/©^
0
)©0$[°·-
0§
θ
´ ´$o-&$
HV ´|È$É?¬
θ
µ
θ
~),«H+$§)Ê
W )(/-[ 0)-? )(/[ ¡
[,-/V![¤ 0()0¡Y §)«-j
/ $©«
[)©0 ¡ $[),[$1,$©«
!U¬
))
0
-^$1©®V©«
!V$©«V!0
[_)/[
,!©«
(!k
ª,*®)H
¯©°±YV©²³-´
¬
µ µ
¶
0·-,©3$0¸W
$]V$¹¡«
)
0
¸[$©-®V©«!]V$©«)!)0
0
3)_Ë
}
¤!
-
¤
³
#
#
)+*
,#
º
.
Î
â
qÎ
º
nÎ
))
0
¸[$©-®©«
!]V©«V)!
0
3?V)
}/'
)
)1*
qÎ
Î
¤
'
) *
n Î
0'
â
kY
$]
j
Ì¡«$0)13©])Y
³
º
nÎ
°)·-Y
&§)
´ %&
"$# µ ≤
"
≤
"('
´ÅÈ$È
µ
©
°
!¬
º»¼½¾¿0ÀÁ Â Ä )Ê)©$·!Æ)0Ê- º ¡
·$ W)?Ê()0¡Y«)Z/±$ ,©[¦ $,«$
/)0
)?!HU·!Y
/
0!*$j,$©«
! ]¡ /ʱW,© -°j$$ª$Ê
(
0§)ª)-
)+$)^
00Y
)HU
j_Ë|(,
$)3)(/^§)©µ!)j
)^([?
0¸V)
0$3Ê? W
0
¸
$©-®V©«
$]V$©«V)(!!¬ ¶ ©«)0$®)0
$ [)
0
¸[$ ¡ ¬
µ
α β
α
¦
β
tYu]vxw1y3z|{1}
~)$,[Z),]0
W)
()W,Å_)/-*
)/^Y
^(*¬
- º k©0 }
1/1$*,!©«!¢³Y´
µ µ
*¡
¶
·-,(©^0¸
1WV$,
Ñ
}
¤
³
-
α−β
º
(
α
β
)
0
¸[$3§YV$?
" !
à
Ñ
α
}
º
$#&%'
"
-
³
©$*©©!Y¯-/)
1$
3)(,)$1§$
´"
α
0
µ =
α −
´
α
β
´
µ
"
α − β β
¬_V
¡«)·$ W ©!)3[§WV$Z()0¡Y$
º»¼½¾¿0ÀÁZÂ*Ä )^)/,^)©$·!Æ()j$^)´
)3/±W,©k¦ !$,Z$)Z,«®)
0 /)·! /µ ±W,© -°*$Y
$¢*$
1/ Ë|¢]
*),
0N$$$
-¡Y)·$ W - & §)Å/©Å
¹
Ê)(, $ °·-
$ ɬ ¶ /©«V)|¯0
°)·-
ZU?
¸!]0
x¡Ê
3),^!©«
$U¬
α
s0sHÎ
¦
α
tYu]vxw1y3z|{1}
°0
Y
1$0°)·-
α
¬
Ù|Ù|Û
= α
ÌÛ
− α
ß?à
%
α
m
Í
*
Î
¦
­N1)/,[/©[
Z,Ê$°()·-
Å$]É¡
*(U?
¸0¯
]©]
!
]V,0
0
¸Z),Z
°)·-
1$ Hɬ
¶
·-,(©]$0&¸W
$V,ª¡
)
0
¸$]0§$
U¬~)$,]
*)(,1
WÊ_)/-:U0 )- º
U?¸ $H
Y
¡
1,0
0
¸¤ 0)0¡Y*¦ §)[-j
Ê/$©$
[_)/[1,!©«!-, )V
·$
)
0
3/V)0k©-N
)!©0¬
&
©
°
!¬
Ù|Ù|Û
º
º
,/!03#"$&24%(50'),1"+687*&0"59%,.1- %,/0"1
¤
ϕ
¤
=
=
ϕ − ϕ Ù|Ù|Û
©)$*©©(!+,-,)
1$¦3!_
)j
?
¸[$0
(
C
<
A
:
=@D1?
=1>@?
;
´
µ
B
$~ U/ W-¡Y()$1¡)(,)$ª
*0U®)0
! $H °·-
´ ÌÉx¬ )$
H
$H°)·-Y
¢*!)kH
ϕ
®)0
$j
,0
0
¸Yǹ´ ¬(­N3)(/3$,!©«
(!1- /1$,0¡Y0
0
¸Z®)
0
µ
µ
E q u ilib r io d e l s is te m a d e s ó lid o s
B,+.-/1032465879;:;=<>?=@BADCE>FHG'IG=JGKLC BNMPOBCEQ3@ISRGKDUT'
BV@OB'[email protected] :
MYKLG'JCMZK[CE<?G
@HIKLV\CWO3KLC!IG]?IBCZ^_MHJGKLCW`baUOV\CEFHCbKLG=F!@GKD3?'GHOB'ADBQ@L]Idce@UfghOIUKLV\CHKLG'JCM;KLCf<?G=@
K?@JG?A?BVBG%iCELkj\CEUOlOmUKL;MnKD@oV\Gf3?CP@BICYWOB!>BV@Om\VBpEYFHGIBfGq@oV\<OmGfr8GUG>Fb=OB8UGsK[BpE
IGOV\<OBG#MPOB=?G\\VVBpEWGHOB'
BV@OBVBpEt
;
^
u
:
!#"%$'&)(%*
sLCE<OBGFHEKlIG,g K>UT.mV\f
B
+.-/s0m2465 W:;YVmOmIE?CHIG JGK[C MP?\ImC8^ K[G;GV@Gf3L;G G E@mO3<LBCbG Om JCKdBVBpEYImV\\I8G BO ADBQ@?\Iece@ff GV\CEsK[BIGL E@G UC
I K OBK DK @JGLADBVBGEK KLCf mO UKLK)%r8GUG>Fb OB LG\\V\VmpE G BO BV@OB\VmpE;MbOB,UG1K[BpEWIGOV<OBGf
u
2+67
;
^
<
5
8:9;7
:
!#"%$'&)(%*
B
!
#"
%
+.-/s0m2465 Zg OsG.X?GFHC
IG @ <>? C!iCEFHCQ G\8IG8OBCfQ 3@I oMPJGKLC
K[GZJCM%KLCE<?G
@8J?GI G
BV\O O3KL MoGYGOsJ@fUC u a KLCE<?G8@ KLGFbBV3OmIELCWF!< oOmUKLC IG8?\IBC!^ na bF M6JGKLC
sghO,K[GFbBVmO3IE?C_GEK>UTYV\COmCV\\IC K[CE<?GP@NKD@JGLADBVBGPiCELkj\CEUO >@QfCKLa OhV\CEFHCnKLG G GnOB ADBQ@?
\Idce@fUa GKDUIC GO V\CE c @fUC G_G E@3Om<LmC1r8G.UG>Fb r%<@ c.'GO Im\Q?FHPIGO#KLpOmmIC O3<?GbIGbOB
<>?]MYIG.G>Fb OBZJCK[BVBpE IGOsVGf3?CbIG;QL G\I\IbIG8Om <D? < OBEK ?G\V\VBCEGK GYOBCKhJCMCK
M
u V OB8A@GDj\8IG LCfj\FbBGUC E@G;\V. %KLCf<?GZGOsVmOmIE?C I hK[1GOJGKLCWIGO KLGFbmVmOmIE?CWGK Ha
@TO
GK GO OBCE F kFHCqIG
J? E@G'GOsK[KDUGFH'GK>UG'GWG f@mOm<LBC
$
) +'
2
µ
%
3
& '
&) *'
.) /'
/4
(
-,
/01
:
u
^
:
=
aB'F
!#"%$'&)(%*
,
,
:
%
5 CE<?GHOB%<>?WV. BV\FHGfUGd+.2ZA@GDj\K[a OBK=CE>FHOmGK]G
M u
M GO JGEKLC
IGbOB%<D?fh
V\CEImVBpE
12\2e5+m5PIGqG E@3Om<LmCbGK E@G%OBKh3?GEK#A@GDj\K]KLGqV\CE
UG GP@PJ@fC1 K CE>FHOBGK G G
M
u
KLG'V\CE
U!G
aO@G\QfC8JCf IBViC8J@UC%GOJGKLC%IG OB=<D?f
=
:
rqB\QLFH'IGOsKLpOmmIC!Om<?G
B
!
*
,
u
a ;F
*
φ
=
aB'F
:
φ
"
u GK r8G8OB8ADBQ@L%KLG8IG\IE@VG E@GqOB;IK>UVB
a F aO@G\QfCHGOtTQ@OmCHGK
GK u
a 8
Q oa 1'F6 ghOV\Gf3?C%IG'Q? GI\IHKLG'GV@Gf3?' aE F
<
"
"
(
!
<
a [ ;IK>UVB u
IGOG.X?GFHC
IG OB=<>?
:
/
*
1
CEImVBpEWIG'G E@mOm<[BC%IG OB=<>? " 8UQ "
"!
*
!
B
φ
*
$&%% #
'() ϕ =
"
+
*
`
!
`
-
φ
=
*
*
*.
.
!
φ
sLCE<OBGFHEKlIG,g K>UT.mV\f
r8G OB'V\CEImVBpEWIG'G E@mOm<[BCKLG,mGG
=
) ϕ = % #
2
? GO OBCfhF kFHC IGO V\CGADBVBGfUG%IG8?Cfj\FbmGfUCatOBHK[ 3@VBpE6GK,IG8FHC FbmGfUCYFbkGfUG %
A@G?j\
IG ?Cj\FbBGfC%BGG%KD@ OBCE FHT.XFHC E@G8GK#BQ@Os
bJCE OBZCE>FHO ?8GO OmCEhIICWIG
aGO OBCE IG
OB=CE>FHOHGK VCK 6
r8G
"
%
KLG,BGG
"
"
$
aB!
"!$#&%(')+*,.-/0(1&2314/65798+5;:=<>:@?A8CBED79F(5"1G8H?A:IFJ<68(DKF=L>8;0MFIFJL6FONP8Q?A8L;7A:=R;/6FS?TR;<6:J79UT1G0(1&:=V6871XWM8/ZY[F\2>Y[F(2
0M8]8^?A:Q]SR6:@?_Y37AF:`/^2GFaUT1Gb@R;79FaFM5cNdR"/ZY[FZegfh/i?TRj14/ZY[:`71G87V6FPDj5;8>?k:@?U_:`79F@?l5;:C79FM561G8nmkDo<6:?A8Cpq0MFM5;FR"/6FZe
r:Y[:`7T]+14/6F`7h:(2"<6:@?8k5":(2>0(1&2&14/>5798<6F`79FtsR>:u:?_Y[:v/68Iw;R6:(2GsZR6:Ze(xy8;5;F@?y2GF@?z?TR"<6:`79UT1&0(1G:@?z?A:t0M8/$?1G5;:`7AF`/2&1{?AF@?e
m
m
!#"%$'&)(%*
(ejF`7OY 14/
"!$#&%(')
*"!$#68L"79:CR;/
L>2G8;sR>:@?5;:<6:@?8>?+*+,.-
<K2GF`/68
0/
1X/60(2&14/>FM5;8
sR>:U_87_]FR"/&%`/6b@R>2G8
D1*32-546/
0MF`L>2G:;e9 8>?t0M8":MUT1G0(1&:`/ZY[:@?=5;:S7A8ZWMF`]+1G:`/;Y[8a:`/;Y379:+2&8>?vL>2&8;sR6:@?D
79:@?T<6:(0Y 1 w6F`]:J/ZY[:Ze r:Y[:`7T]+14/6F`7+2GF
Y :`/ ?1?>Z/
5":'"(t0M8/o2GFV68Z7A1XWM8/ZY F(2 ?A:
79:@?T<6:(0Y 1 w6F`]:J/ZY[:Ze7]SL68>?I:@?_Y8%J/
:(2<>2GF`/68
5;:(2t0MF`LK2G:D2GF@?+U R6:J7TWMF@?+5;:
14/60(2314/6FM5;8
?A8/
?1 Y)6F`/
5;8>?
R"/>1G5;8>?=]:M561GF`/ZY[:+R;/
µ
, -:; D
798;WMF`]+1G:`/ZY 8^sZR6:nFM0Y@)6FJ/
2 -:;=<;
µ
?8L;79:
2G8>?
L>2G8;sR>:@?Pe
* 2
* ,
ACBCD
$7A8L>2G:`]F?h5": f?_Y8%Y 1&0MFZe
4 / D 2G8/6b;1 Y{R65
:@?_Yk
% FJ7OY 1G0`RK2GFl:`/ ?TR: ;Y379:J]8
D U_87_]F
%`/6b@R>2&8S5;: "( 0M8/+2GFuV>87A1XWM8/;Y[F(2 e#;R: ;Y37A:`]8
?A:=F`<68\DKF?A8L;79:2&F?TR;<6:`7AUT1G0(1G:231{?AF=5;:vR;/lL>2G8"sR6:t5;:v<6:@?A8
< / z
; ?1 Y3R>FM5;8o?A8L"79:R;/6F ?TR"<6:`79UT1&0(1G:14/60\2&14/6FM5"F7_R6bZ8>?AFQsZR6:QU_87_]FR /&%`/6b@R>2G8 5": <C(t0M8/o2GFV68Z7A1XWM8/ZY F(2 e
r:Y[:`7T]+14/6F`7 2GF 79:MF(0M0(1?
> /q:`/ 2GFF`7_Y 1G0`R>2&FM0(1?>/ Dq:\20M8":MUT1G0(1&:`/ZY[: 5;: 798ZWMF`]+1G:J/ZY[8 ] 4/>14]8 <6F`79F sR6: V6F DKF
"!$#&%(')Q*"*+-I/6FkL6F`7T79FkR;/>1GUT87_]:S5;:k<6:?A8
R;/
:MsR>1&2314L;7A1G8$e
D
ACB D
(ejF`7OY 14/
"!$#&%(')
*C-/6F:@?0MF(2G:`7AF5;:.<6:@?A8p
7_R6bZ8>?A8$e f 20M8;:MUT1&0(1G:`/ZY :
<6:`7c?A8Z/6F
5;:n<>:@?A8
5;:
?A:.FJ<68(DKF.:`/aR;/6Fl<6FJ79:M5w6:`7_Y 1G0MF(2 2&1{?AFID
798ZWMFJ]+1G:`/ZY[8i:?
BF?A0(1G:`/>5;:n<687C2GF
µ
-
; 46e 9yF
5"1{?TY[F`/60(1&F
:@?A0(F(2G:`79F;e r:`Y[:`7_]+14/>F`7.V6F@?TY[F
:`/ZY379:
sR6:
?8L;79:lR"/
?TR6:(2&8CV687A1GWM8/ZY[F(2
<>:(2G5;F>8>?+:@?+5;:
j0J]e-I/6F
<6:(2G5;F68^<;R6:M5":o?TR;L>147C?c14/
sR6:
2GF
:@?A0MF(2&:`79FI?A:t0MF(1GbZF;e
$7A8L>2G:`]F?h5": f?_Y8%Y 1&0MFZe
*C-/
"!$#&%(')
?A:`]+1&0(1&2&14/>5798a5":S<6:@?8
7_R6bZ8>?AF;e9 FSL6F`7T79FSV68Z]8;b`/6:MF+5;:S<>:@?A8
*
-
-:4 /
/
79F(5"1G8+H ;?:F`<68\DKFC?A8L"79:SR;/>F?TR;<6:J79UT1G0(1&:lV687A1GWM8/ZY[F(2
D
D
`H ; :@?TY8%.F`7OYO1G0`R>2GF(5;F.:`/
2G8/6b;14Y3R65
F`<68(DKF?A8L"79:S2GF?TR;<6:`7AUT1G0(1G:S2&1[?AFl5;:(2h?A:`]+1G0\1&2&14/65Z798C:`/CR;/C<;R;/ZY[8
]F`/ZY 1G:J/6:Q:`/
:MsRK1&2&14L;71G8
sR6:k5"1{?TY[FuH
52GFSU R6:J7TWMFS5;:k798ZWMFJ]+1G:`/ZY[8
"
D
?A:
L#t5"14L"R NPF`7:(2
α "
0! :(2>w6F(2&87 5;:k2GF=/>87_]F(2 :`/$oDQ5;:2&F=79:MF(0M0(1?>/C:`/C2&FIF`7OYO1G0`R>2GF(0(1?>/
4/K14]8C5;:(2
?8L;79:l:(2 0(1&2314/65798SD
:(2$w6F(2G87z]
:MsR>1&2314L;7A1G8$e
0M8;:MU91G0(1G:`/;Y[:l5;:798ZWMF`]+1G:J/ZY[8
α
: ;Y37A:`]8
5;:(2g: ;Y{79:`]8 kef 2?1{?_Y[:J]FS?A:
:`/o2GF<68K?1G0(1?>Z/o14/65"1G0(FM5;FZe r:Y[:`7T]+14/6F`7F!=:\2 w6F(2G87I5;:\2.%`/6b@R>2G8
5"1GFMb@79FJ]F5;:\2 ? >62&1G5;8C2&14L;7A:5;:l2GFIL6F`7_7AF
?9R
%
- ; µ
<6F`79Fl:(2
"
H
H
)
&('
)
α
−
'+*-,
α .
&0/21*
α
34576
f 2G:w6FJ/65;8lF(2>0`R6FM5Z79FM5;8;(D0M8Z]8
@A B 345 6
H
α
α−
3C45
α −
A
.
α +
= ?
<2=> ; α
α 8:9+.
B
α
α
-
; (
!
HJI
G
D E
H I
β
G
S:1 5T7U / UWVX50T ,Y,2ZX[ UW\ UP]^_U 1
DFE
KML2NPORQ
α
4
(ejF`7OY 14/
0!Qr:=2GFt0M8Z/65"1G0(1=>/.5;:t:MsR>1&2314L;7A1G8?A: Y 1&:`/6:
=
=
β
−β
%\')"!$%v,.%#%$ )&# (' % #M)' ! '*#+&,(' ) .-0/ b@R6F(2GF`/>5;8lFu0(:`798S2G8>?]8]:`/;Y[8>?79:@?T<>:M0Y[8l5;:
−
?A@)BDCFE9G)H&I"JLK)BH(E
3 ; < =
MONQPSRLTVU
2 >1 ;
+
W
:
57698
α =
β XZY)[&\[L]
?A: Y 1G:`/>:
4
W
^`_QP
[&a
U
bdcfehg EiCFEDjkEAH&B%l*m7n)l g H&Iol gp CFBAH&B%l*m7BDj grqg l g G)HsCFI
¥§¦D¨L©%ª«¬ª­D¨A©®¯®°D±ª²ª£³´yª¦
α
¡£¢.¤
e BtlrE q ILG)H gq%g nLGuH&BABJLK g l gp C g Ikm7B0v g BDG)BAJLKBAlExwyK)BDCzEAH&B{CFIszE9j g BDGsv|I"Bm~}QMONOm7BDGu
[&\%]
(TxU
DE CFEoB%l
)ElrILCjG g jkIH&B%l q I&B%w grq%g B9Gsv|BoH&BhCFI&zEDj g BDGsvI lrEm g vK)E qg nLGfBm0H&BjkI
g j g BDGsv|I g G&j g G)BDGsv|BtlrE"wK)BDCFzE
\
H&B{CFIszEDj g B9Gsv|I q K&jo@lB}
Mu
µ
e BtlrE q ILG)H gq%g nLGuH&BABJLK g l g p C g Ikm7B0v g BDG)BB%l&
)E%lrILC`H&BtlrE{G)IsCjkE%l
\
MT&TxU
µ
T
\
C7I p lrBDjkELm`H(Bmvv gq Es
dfeCg>hjilk)monqp>rtsuwv'xwyWz+{}|#uwv~y9xwy#{}|q#w'{
wy{}|#>zy9
uvW/u|#|Wuwy)|~>y9j)~>yxy{}|v).|#|
{}yq
wy#{zvtyWu/vt/
z}vv.~$H
uv1yCyWzQ:uwvWz'wv.6lv
wyWz¡.v~yz|¡.vWQ/yWu |yWuQy)¢{
wy)/|W
wvWuw~>|/vxwyz${}|#uwv
{}ytyuwW
wyWuzvxwvWzv.~>vtvT£7¤q¥~y)¦yC§zy|¨£©{$©y)¦vWxw|1ª«|QyWuo¬­{}yt)|#uS{4/~>yWzvTl®{}|«¯Pw°{4jy)©.|y./)/yWuy~y
z|¡.vWQ/yu|yWu¨±ª²¬³y{ µ £«µ~>yCyWzQ:uwvWz5/v~>®{vuw)/v¶W§«7v'·'vlv¸#
wy'x
wy)~yt{
:z6/v'xwyWz${4|#uwvt{47u
´
¸#
wyy)vWl#u~y{4ll.y
ÉÄËÊfÆ
Â
¬
Ç+ÄÈÆ
¨
Ã#Ä Å9Æ
EDCFE"lrE @)Im grqg nLG g G)H rg q E%H&ElrEtv|E p lEtG)IBmvBDG q IsG)H grq%g ILG)BmH&BtjkI
g j g BDGsv|I g G&j g GBDGsv|Bs`G B%l*E9@)I%dIkB9G
\
EH&BDjm H&BtlrE{G)ILCjkEl)E q v )EtlrEAwK)B9CzEAH&B{CFIszEDj g B9Gsv|IofH g C g i g H(EE q%g EtlrEAH&BDC7B q )s
E
!#"$%'&)(+*),+-.*/&10*/32."4(5&)(6*798:.2)*:
½
;<0.*/= >0?#09&1(@A1($"BCC,
Â
À
>£¿¤
¾
D
Á
¨
½
¬
º ¼
α
º »
º ¼
d
d
º »
¹
e B%l(@)I(lisILG)I"H&BAwK)BDCFzEm m7BAH&BHLK q B g G&jkBH g EDv|EDjkBDGsvBxBls
)ElrILC`H&BtlrEAwK)BDCFzExH(B0C7IszEDj g BDG&v|I"BDGuB%l*mK)B%lrI
N
K
=
J I EFHG
L ! M
α
E G)IsCjkE%l*B9GPO w ILCjkE q ILG rl EA
)BDCyv grq E%lK&GQ9G)iKlrI
t
JLK)BOBLm B%lj g mFjkIuJLK)BOwILCjkE rl EAv|E p lrE q L
I GB%l m K)B%lrISR)H&B lrIdm
α
H&Ev|Im m7BuH&BHsK q B m K
)E%lrILC
e BklrE mFK&jkEuH&B"jkIsjkBDGsv|Im{CFBm@)B q v|I H&BT g iK)E%l`E q BDCFI m7u
B I p v g BDG)BklE
\ α
[&\aL]
B q K)E qg nLG
acb ['`
=
Z _ [5^
+
] ['\ Y UWV#X
α
) vWz 7u> ®wµ
ty /v /
z}v {}y /yWuwy /|{wv)/|#zy#{ xwyzvWuw~| ¸#
wy.~>v xwvWz}v /v uw|#zv)yWucy1
= =
vWxw|)ª«|± ´ «µC£/µ o9ylv y.W
v.)/#u C {4yf /yWuwy
wyWz¡.v~y9z|¡.vWQ/yWu | % "$#" !
=
ty)¦xw|>' &/|#uw|²~yq
wyWz¡.v{{4y| /yWuyT:uy.~>/vWvWyWu yTlvu|#zv)¢yWu)(
/0,1,32
, +
+,- * 87«
6 5 =
− . 6 4 =
y)«.|y./)/yWuy ~y'z|¡.vWQlyWu|qyWuy)j{
y)/|Ty{
>£3«µ£y) wv)/|#z¶C§S7|T~y/v
wyWz¡.v~y9z|¡.vWQ/yWu |y#{ £7
ªP/v9xwyWz${}|uwv9x
wy.~y{}y)
7z5{
lyWuw~|txw|#z</vµv´ /v{47u¸#
wyy{v~y{4ll.y
h 9<; uo/vx|{4/)l#u²'&7Q¿y)£«/vvWlvy{ ¶yWu {4¿
wv))/#u~y|:Q/yWu|P7uQ7uyWuy)£#ªolv{
yWz¡.v{°~yz|¡.vWQ/yWu |
:
/yWuwyu {
=wv1/|#z¶C§«7|w£ µ xw|z/vuw|#zv) ?>6vP
wyWz¡.vQ~yz|¡.vQ/yWu| yWuA@ /yWuyP/vP~>7z}y..)/u ~y/vvWlv
~>7z}/l~v9wv.)/vvWzz}7wv
ÁBDCFE?GHCIJCLK:MON>·0PNQBRK:STNUBDV3GWMXK
MYNZC[ZC\V\NZC
]^SNQ_ E`S3a
SLK
M1b1cMGedC#·
α
½
À
¶ ¼
¿
½
Â
Ã
¾ ¼
º
» ¼
Á
¶ ·
» ¼
θ
¶ ¼ ¸
¾ ·
» ·
¶ · ¸
v)/vWuw~>|v.yz|q/v{
v~y9|#yWu |{ zy{xwy)C|T~y( £w{}y /yWuy
n
» ·
gml
p
=
v j
o
rFs:t
u
gmh
k
+
q
α
o
n
=j
p
rFs:t
gml
α
− j
=
¡
¢
£
+
= W
0
0 W
·
k
µ ¹
gih
ty/v/yHªQ~y)j{}yuw|TvWxl/)v.~vv)z}/¶uw
/|T~y
wyz¡.v{<|#zv)~|txw|#z</|{wy.C |#zy{ dxw ½zyJ{i| £
W
wyWu v'¸
wy QO = :
= ~ 8D } 3
F
¤
¼
º
θ
f
µ¹
{ ªTyu/yWuw~>|yu
=
8¥§¦U¨D¦'©? ª
\«m¬
­Z«3®e¥¯¥$¦U¨° ª
¡
=
¥
=
¡
=
+
=
\
±0²³1´Tµ
Entramados y armaduras
®$«:­Z µ 3¥ \ ¥§¦0¦U¨O
º!"$#&%('*)
Á
ª Z­ =3¥§¦'®§+ ¦'®e ­Z-,e¨/.®e ² \0¦Q ® µ ¨ ª
®10 3 2 Z­ ¥ ®eO33¨Z«3ª
¥ ª ­° ® ¦ ¨/ :­Z$F¨/43ª
© ­Z
Á ¿BFÁ Ã B*C B Ã
i
ª
«3® µ F­?ªzF­?65
«F©?«87:9
;2 ­Z¥<,U
®>=O¥ 3
Y30¦@?
ª ªz­Z¥A
® ®e
¥
7 7
© ¥¨¦'®e6:EDW ª¦'®e$3F¨/43ª
«TFO« µ 5®e¥¨43ªG
±
*NM
.
L
O
±
J
µ
*NM
.
H
I
K
´Lµ
´Tµ
´Lµ
´Tµ
® ¦U¨Dª
³
º!"$#&%
+*;
' K
0¦Q ® µ ¨Dª
® ­Z ,U: >® =O 1
3 ¦ ?
ª
¨Dª
\¨/F ª
«z¥¨¦'®e
EDWª¦'®$33F¨/41ª« F
O« µ \
5 ®e¥¨/41ªG
±
OO\Aª
­°¥ ®§®e¥ ­Z ® µ O1®e F D ª¦Q ²
µ
*
H
I
µ
±
*
*
J
µ
K
µ
»
±
®$«:­Z µ 3¥ \ ¥§¦0¦U¨O
º!"$#&%
K
0 ¦Q ® µ ¨Dª
®i­Z ,U: >® =Oz1
z3 ¦ ?
zª OO\ ª
\z­Z¥: ®§®e¥Yz­Z® µ O3®$z­Z , ¨/.®e
O D \ª1¦Q^© ¥ 1
¨ ¦ ®e6
ED0 ª¦'®e33¨/43ª« F
« µ 5®e¥ ¨/43ªG
® ¦U¨Dª
³
+*;
H
O
´ µ
I
*
µ
³
µ
K
J
µ
»
³
®$«:­Z µ 3¥ \ ¥§¦0¦U¨O
´
® ¦U¨Dª
³
+*;
M e c ani smos : p ol e as, c uñ as, torni l l os y di sc os
M é todo de l os trab aj os v i rtual e s
F ue rz as di stri b ui das: c ab l e s y v i g as
C e ntros de g rav e dad
0
Puede agregar este documento a su colección de estudio (s)
Iniciar sesión Disponible sólo para usuarios autorizadosPuede agregar este documento a su lista guardada
Iniciar sesión Disponible sólo para usuarios autorizados(Para quejas, use otra forma )