Curso: Física (DI)

Anuncio
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 _RŽU g RTQ cigc _W#UO c k"RTP dKci‘\gc U g N’WiOYS g POTNj“ŒWiO
€mƒ‚A„=…‡†iˆŠ‰Œ‹h #W ]^W d YŒRU #c d O ced W_"PO] c U\]^W g j
”#UAPO‡N–•\PAWfk—N d Y c O ce˜ N d R l V
N U\] c O`
ƒ›a›£ž
ša›)œ
Ÿ#›a›ž
"! #! "! $! &% ™
a›ƒ›ž
ša›)œ
α
Ÿ#›a›ž
)( # ) ( # # '
'
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›ƒ›ž
ša›)œ
¡
α
¢
Ÿƒ›ƒ›ž
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_—]^”ƒ]R‡Q c `
=
=
=
Y RU cd WiOED c O‡N d g WiO 
Y S g POTN “ O c g R d W QQiRTS\U g Š
W O c k PAW "d lcGFIH \
• PAW A
€mƒ‚„ …‡†iˆ ‰CBŠ Wƒ]W d Œ
˜ c “ •\PAW c ZO RTQ c#d c O
OTN•VPW g W O c k"RTP 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
WiOYS g POTN g WjO c kPW d"
l cLFNM WV_ g WPOKK6: `
‹€Œ
Ž
α
‹ t
"!$#&%('$)
]m^`TgX€@[0fZ
‚
α
Tƒa„@[…V3†
€mƒ‚„ …‡†iˆ R
‰ Q  ƒW ]^W d YŒRU c#d O cd W_"PO] c U\]^W g WiO _ER _"]^W#Y c g WŒkPAW d"lc _ QNVUQ#P d—d WU\]^W_ •VPAW _KW RU g RTQ c W#U
R‡W#U g ’
N •VPAWS5 MUTWVXY6Z\[&]_^`Tba@YYcZ\[&]edGTgfhYcZjik]ml`TnVfhYcZGo
cg 2 PU\] c _ c O c k"RTP d&c
q
r s
r t
zx3y
zx~y
p
{}|~y
wx3y
r u
r v
"!$#&%('$)
* +,.-0/‰ `MAW g 1R P32 c–c WV_KQ c O c O c _"PY c g W’O c _mkPAW "d lc _a`4 R g RTW#U g NŒW
O=YS g POTN g W
9&:<; YŒR g RTWU g N’WiO”#UAPO‡N–•\PAWfk—N d Y c QNVUO ce˜ N d R l NVU\] c OW _N\]RTWUAW>=@?A
‡ ‰
O cjd W_"PO ] c U\]^W[_KW N]RTW#UAW6587
‡ s
‡ Š
‡ ˆ
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 †
− _
ƒY„y…4† ‡0ˆ‰‹ŠOŒI$ $ )('4*>G $ ,<%*JD! $Ž&$ *>,?
< $ @,‚ $ 2 $ , $ !D & , $ *Ž? 6 <UŽ &P <!Do Œ Y,<3(=€ & <%*/
i $ *4K 6 <%*>/OM1< $ ?)/!)(=G-/1 $ * $PQ$ 8Io
¡ |(xL
Ÿ}w(x 
ž
|3x3y
Ÿ}x~y
p
¡ x(x 
{x3y
¢ x~y
¡)Ÿ}xL
œ(xL
egf3hFiOjGkQlOm
~
=
nLuyx }
+
nLu n |
w
z
{
=
nvun o n t
s
r
=
n1op!q
ƒY„y…4† ‡0ˆ‰'‘=ŒI$ ,U-0/1(324/1 $ ’<4?
< $ @IW &$ (=€ & <%*>/ I
a fhYYPZ $ & /%,#-/!(324/1 $ $ ,W X 5
i WN^+D0* $ ,#M1< $
W X ?)/1)( $ -0/1“W”<•K 6 <%*>/ &$ a@Y† i–M1< $ ,< & >? $ $ 4
- > &$ (=€ & %
< *>/g,
] X˜— ]m^™, $ š 6 4
< 0* VYYY ZGo
ŒI$ $ )('4,<›,˜(=€ & <%*>/%, i $ *%K 6 <%*/M!< $ ?)/1)(= o
egf3hFiOjGkQlOm
 $ 2 $ , $ !D0-0>€1 6 K?>- &$ *>,‚?
< $ @,
r
rs
Ÿ}x~y
α
r ˆ
X
›U/1H%* $ (=1, &C$ ,)DKy
-o
ŒI$ *>.* $ 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‚ƒ-&/G Q !:„)"!…'V)!,+ƒ†„‡"&wG Q 1-&k†wˆ‰Š†(‹ŒV#%-&k6)!†wˆŽ'F,GI†‘)?
t!r-)#\’6)‡&/#%sG5“’"#(t:!r-)#%’6)!I',FGIŠ†(‹1”†•–#%&4GIH")#%&E?s†wˆ1Œ"I†(‹)"G Q #%—#\
#%5%H$˜š™›‹œ‰ž2™Ÿˆ¡ ¢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 ž
!"$#&%('*),+.-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¢
—^”£žz™=š
¤
‘ ’ “*”–•— ˜2•‚™=š
β
›
α
“¥(¢
— ¦ ¤
β
“œ”–—Ÿž !"~#&%('),€Z:‚[n@
1ƒGqn@6e;52:@([n/
1ƒG/B>:,8@=<_@([n/[n/=8`„8/6G5<_/63!QR@:e
>[n@=<jG/|@=<4q1EQR@
1j…†^‡†&ˆ‰†Š@63=Q^‹@=:Q^<_/
1jA>/=<ED6@21
G/([7Œq1E7B507BmG;>[n5jŽt(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/U1]QRc!Qq&36@;t
!"~#&%(')21BŠ[{3=>5B<_/=8<4/
14/=:;Q@6G5|/=:.[&@0AEq&e
><_@ @6G43–>:;QR@ƒQqn/=:/ G / =@ <4q1EQR@ IŒ7œ[ 52<qne;/=:65fV,[n51/7Q^<_/=7B5w1
G/ƒ[n@
1 AC>/=<ED6@
1vF T VŒF Y /
1]QRc=:B/:B/([8>:;QR507B/6Gqn5lG/j[n51z[&@6G51‰t8! 5 1 7B
m G2>[&51vG/j[n@
1vA>/=<_D6@
11452: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}~ cKe€‚mK[ƒ_…„8e‡†T[ ;=ˆT<‰RŠI A‹[ka]^Œ†TbV_bŽbh†e‡ef‘*fe7_jŽŠ‘’†[‡fhei_j[8“7]^e•”m:[%–Ke•†T[Cf—mTcŠ]^‘˜†T[
“8‘‘_†[7cKe8†Telkp™ >K<‰IK<VUŠšQ› eka]^eœ[Cf—mcŠ]^‘q†[œ“8‘‘Š_j†[7cKeC†elkp™ ;7<dL<nRš ]‚eCf“8‘`ž‘gk[`mK[lk…]Ÿ_jeœ[7cgfeœ€…bŽlmT_jee8†4 ¡mcŠ]^e¢W
£ ‘:k'–KeCf‘_j[k†[6felk“C‘‘_j†[ƒcKe8†elk[lka]^Œ7c¤†e8†‘:k[7c¥`ž[=]Ÿ_‘:kW¦Y'[=]^[7_…`2bVcKe7_'[§fH`ž‘`ž[7cŠ]‚‘¥†[¨o˜_j[lk…—K[C“=]^‘¥†[§f
‘_bŽŠ[ƒc0©«ª0f‘:k`ž‘`ž[7c]^‘:k¬†[po­_[lk…—K[8“7]^‘†[Of‘:k¬[® [k0¯ <H°±<p² W
µ
¶h¼½¹j¾…¹
°
´
¿
À
³
°
¶¸·º¹®»º¹
Š[/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
RTU‚U½IKUU:L
f`ž‘Š`ž[7cŠ]^‘6†[¢fe2€®mK[7_…„8eœ_j[lk…—K[C“=]^‘q†T[Cf@‘_ºbŽŠ[7c6[ka]^Œž†Te8†‘J—:‘_
M = OP ∧ F
< †‘ŠcK†[ž[§f—mc]^‘
—mcŠ]^‘¤“7mKe§f”m:b[ƒ_je †T[6feq_j[8“7]^e#kº‘—K‘_®]‚[ †[ o ‘`že7cK†T‘ [Cft—mc]^‘
|™ >%<?I%<nU«š…< [Cf
_j[lk…—K[C“=]^‘†[Cf:‘_bhŽŠ[7cœ[lk
;=U ™ I
−
—K‘Š_Qf‘:k –K[8“=]^‘Š_j[lkQ†T['fe!Kelk[
fe€‚mK[7_…„8eZ_j[k…—K[8“=]‚‘\†T[fh‘:k¬[® [lk¢¯!# ° # ² %
W $mMk–KeCf‘Š_j[lkQk‘c'&
(+ 9
−
`ž‘Š`ž[7cŠ]^‘ †[ fe €‚mK[7_j„8e
>0š
f—_‘†mK“=]‚‘ž[lk“CeCfe7_Q†[Cf:–K[C“=]^‘_
(*) 9EI:U
[lkZmTc
>KU
,
‡<"
—T_j‘—K‘_“Cb‘cKe'f‘:k«`ž‘`ž[ƒcŠ]^‘:kH†T[
(-, 9U
>KU AED½` ”m:[peC“=]GFKe k ‘HT_j[Zmc
rts=uKvMwhxCyqz2{T{- cœfhep€jbŽlm_jee8†4 ¡mcŠ]^e k[Z_j[ƒ—_j[lk[ƒcŠ]^eimTc\—:e7_@†[Z`ž‘`ž[7cŠ]^‘
ΜC =
—:fe7c:‘ › ‘_bd„8‘c]^eCfªž‘]Ÿ_j‘'—Ke7_@†[Z`ž‘`ž[7c]^‘
G 9%;¡IKU AED½` ”mK[e8“7]GFKe k‘H_[ mTc\—fe7cK‘”mK[€a‘Š_a`že UHI “C‘c
Μ
[Cf › ‘_b„8‘cŠ]^eCf®WƒY'[=]^[ƒ_a`2bVcKe7_@Žl_jŒC€…b“8e7`ž[ƒcŠ]^[[CfT`ž‘`ž[7cŠ]‚‘ _[lk…m:fn]‚e7cŠ]^[ †[e7`K‘k—:e7_j[lk
O P
O Q
JLKNM
.0/1325476985:
fK`ž‘`ž[7cŠ]‚‘;=<M[lk [Cf†[pmc—:e7_¬†[O€‚mK[ƒ_…„8elk †[ >KU?> Ž
k4bn]ŸmKe8†Telk [7cœ[CfT—:fe7c:‘ › ‘_bd„8‘c]^eCf:‘[7cmc—:fe7cK‘'—Keƒ_jeCf[Cfh‘
eCf › ‘Š_bd„8‘cŠ]‚eCfªqk[7—:e7_je8†ek¬mcKe †Kb½ka]^e7c:“Cbe †[Omcœ`ž[=]Ÿ_‘¢DK[Cf:`ž‘Š`ž[7cŠ]^‘!; G [lkH[CfM†T[mcœ—Keƒ_je †[ €®mK[7_…„8elkQ†[ >KU> Ž
kºbn]ŸmKeC†elkt[7c¢[§fM—:fhe7cK‘¢bVcK“CfhbdcKe8†‘0‘2[ƒc2mc0—:feƒcK‘œ—:e7_jeCf[§f‘2e§fM—:fhe7cK‘¢bVcK“CfhbdcKe8†‘ª6k[ƒ—Ke7_je8†TelkHmTcKe'†Kb½ka]^e7c:“Cbe †T[ R ` <
]^eCf “C‘`ž‘¢k[ `mK[lka]½_je [7cJfe €jbŽlm_je'e š W @¦e7_je'€ae8“CbŸfhbn]^e7_
fekjm`že †[\f‘:k%`ž‘`ž[7cŠ]‚‘:kQ†[\f‘:k
†‘:k¬—Keƒ_j[lk < f‘:k@–K[C“=]^‘_j[k
”mK[Of‘:k¬€a‘_…`že7c0k[ › e7c\]^‘`že8†T‘\“C‘c0k…mMk¬†TbV_j[C“8“Cb‘c:[lk—Ke7_jeCfh[Cfelk¬eOfeZ_j[8“=]‚ep†T[bdcŠ]^[7_4kº[8“8“CBb AŠc†T[fh‘:k—:fe7cK‘kW
W
¿
TVU
−
¿
RES
™…e š
@ _‘H:f[7`žek@†T[ ka]^Œ=]‚bh“8eŠW
o k [ƒ—Ke7_je8†Telk
mcKe†Tb½ka]‚e7cK“Cbe gW $m¢`ž‘`ž[7cŠ]‚‘0[kQmTc
o ª
f¦—Ke7_
_j[k…m:fn]^eƒcŠ]^[[lka]‚Œ€a‘_a`že8†T‘0—K‘_ZfelkZ€®mK[7_…„8elk ¨
−
–K[8“=]^‘_ —K[ƒ_a—K[7cK†Kb“7m:feƒ_ eCf¦—:fe7c:‘J†[8€…bVcb†‘0—K‘_ o ª
o < :
— fhe7cK‘J”mK[€…‘_a`že“8‘c fhe › ‘_bd„8‘c]^eCf±mcXŒ7cKŽlm:fh‘
−
€jbŽlm_je- š
θ,
.
O
S
W
0U
θ
J9K M
™ š
< _je8„8‘œ†[Cf:—:e7_@_j[lkjm:fn]^e7c]^[ < e7—:fhb“Ce7cK†‘žfhe
@¦e7_je “CeCf“7m:fhe7_Hfe †Kb½ka]^e7c:“Cbe
9
9EI:< >ML `
fVmK[CŽŠ‘\[§f`ž‘Š`ž[7cŠ]^‘†[CfT—Keƒ_S_[lk…m:fn]‚e7cŠ]^[[lk
9
;
@¦e7_je“8eCfh“7m:fe7_@[Cf:Œ7c:Žlm:f‘
•k[Z]‚b[7c:[
f[¡ª¢†[§f“8‘:k[7c:‘eCfT]Ÿ_bŒƒcKŽlm:f‘
;=U:L<
ADM`
e7—fhb“8e7c:†‘feOf[¡ª2†[Cf k[7cK‘eCf]Ÿ_ºbŒ7cKŽlm:fh‘
θ,
k[ ]‚b[7cK[”m:[
θ
9.ˆT<?I5I
e Ke7_…_je › ‘Š_bd„8‘cŠ]‚eCf †[ > `E†T['fe7_ŽŠ‘ž[lka]‚Œk‘`ž[7]‚b†e'eOmcKe'€‚mK[7_j„8e–K[7_®]®b“8eCf › e8“Cbe'e Ke® =‘œ†[
r =s uTvMwŸxCyXzq{ œ~ c:
;I ATŽ e7—:fhb“Ce8†e¢[7c¤k…m¥[ Ÿ] _[7`ž‘ W Y'[7`ž‘:ka]½_je7_O”mK[0[lkO[C”m:bn–KeCfh[7cŠ]^[0eœmTcKe2€®mK[7_…„8e0†[ ;¡I ATŽ › e8“Cbhe2e :e® ‘
eƒ—:fhb“8eC†e[7c0k…mœ[ ]Ÿ_j[ƒ`ž‘
ª eZmc—Keƒ_†T[ kº[7cŠ]‚b†T‘ › ‘_je7_bh‘\†T[ > ATŽ%D½`XW
ž
¿
U
.0/1325476985:
¿
¿
U
¿
;7U
W%e7_®]‚bVc
RTU‚U½IKUU:L
Equilibrio del punto
!"#%$'&)(+*-,/.103254625798;:=<>57!2)?%@>)?BA7!CD2)?E8GFDCIH?BAKJL?B>MC:=7N2)OPO):BF<QCD2K?B>MRSH257UTP>3VXWKCD2K?B>3RN:BYZHD7N>M>PC[\HD<I46>^]>57N>
_ H2`2)?a?BA _ H2`CD2cbdZegfhCD2i]2ZjkAGjk2`25<O/H25<I4l7k2g2/<m2 _ H:'?':naD7k:BAGjo:2)?8GFDCHp?BAKCD2c?B>`RSH257UTP>qV!rP2ZjsCD2gtuDegfwv
xZz
xy
|\}o~
α
{
.0/1325476985:
‘c:†Tb“CbBAŠc†T[p[C”m:bhfhb _b‘
x y
|\}o~
{
x z
α
= α = +α
α =
=
;T;
%7kAa?B258G>jXC2!j944S:'OP>Iv
!"#%$'&)(+* ,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^€34q0O‚ƒu€up}.rt0„3.…n80†^‚.ua€‡23€nO0834t.2ˆ‚0Š‰1€ | 7/10Œ‹
u.Œrt02X}€.2^.Ž.2m.O57^€f‚f€†/v€46.2t‚0,4^u€nO€‘2t€‡2^3€n80830X’“„.}./prq€‡20/J‚f0u;}.2u€f42.u”.2t‚4^u”n80†^‚.u(’
—
ž
šB›œ
˜
–
ŸB›œ
™
B›œ
•
_ 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^€34q0O‚ƒu€up}.rt0„3.…n80†^‚.ua€‡23€nO0834t.2ˆ‚0Š‰1€ | 7/10Œ‹
u.Œrt02X}€.2^.Ž.2m.O57^€f‚f€†/v€46.2t‚0,4^u€nO€‘2t€‡2^3€n80830X’“„.}./prq€‡20/J‚f0u;}.2u€f42.u”.2t‚4^u”n80†^‚.u(’
—
ž
ŸB›œ
™
B›œ
š ›œ
ŸB› ›
?(A
’H0/ }€‡2
¢ WWP LWW^V
_a`bcdfeOgihij#" J2 .O‚ .uv57.rt0„3.6‚0 ‰ € | 7/v0„083$ 72X}0#72o†^‚457^.„3. ]W„D 3.Š,.uv4q.up}~…72^€34o0Š} /1.u n80†^‚.u
34^u”3.-.O‚f‚f4^u”n842X}.2^€34^u”.26726,^‚f024&%4/v€('842X}0O‚ ’“„.}./prq€‡20/”‚0uƒ}.2u€4X2.u”.2t‚4^u”n80† ‚.u(’
>
; 7
97
87
<
:7
:7
?
=
9 .2u€‘2 .2 .O‚^n80†^‚f. ?
y
9 .2u€‘2 .2 .O‚^n80†^‚f. L
y
9 .2u€‘2 .2 .O‚^n80†^‚f.Ž¢
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 0830Ž72^4Š‹q.‚†^‚457^.10 ,.uv0 NW-D
’O“„.}./ rq€‡20/”.O‚320‚4/”3.
] W-D 8
_Œ`bcd eOgMh j )
.O57^€f‚f€†/v€4’+ ^
4 u
Ÿ54
α
6
™
–
˜
y
V } |
α !
?(LW uv.2
α
y
NW
y
? ¢ A
r
?8V
¡ /v4†^‚.rt0uJ3.aup}~}€fn80X’
3 4^uŒn80†^‚.u(’
_Œ`bcd eOgMh j J2i.‚ .uv57.rt063.m‚06‰ € | 7/106083*$B72X}0# 72ˆ†^‚457^. 3. ]WW D 3.…,^.uv4q,.23. 3. “ .}./prq€20/^0 .O‚ €2X}./ 20‚4q3. 20O‚f4/1.ua3.6‚0„‰ 7./'80 ,0/v0„57^.„0rm†^4^u nO0 †^‚.ua.up}2t}.2uv4^u F † .O‚-20O‚4/
3.-‚f0u;}.2u€f42.u”.2t‚4^u”n80†^‚.uC,0/v0 y VXWWŽD ’“„0}4O} |
y A& ¢
α
;
:
13254
α
<
=
h 70234 ‚0}.2u€‘2t.2t.O‚ nO0†^‚.%4X/v€ '842X}0O‚ uv.80Ž27 ‚0#.2t.O‚,72X}4 0YnO42n7/ /1.26} /1.u ‰7./'80uƒ‹ ,^0/10-57.-.up}
.2 .857^€f‚ €‡†/v€4tu 7ou 7rt0 %^0Ž3.Šu./Jn8./14’
_
y W
u€.234 s‚f0Ž‰ 7./'80-57.-. $./1n8.-.O‚^n80†^‚.72^€34 0‚,72X}4 ,>.2 .O‚,72X}4 0 ‹ .O‚ 20O‚4/J3 e ’
423€fnO€‘2 | /1~8‰1€n80-3.-.857^€ ‚f€‡†/v€f4
+-,
* #
$%(' " 1325476
= $&%(' ! +α
.
α
y
/0
)
¢ L] L
D
89254
70234H‚f0M}.2u€‘X2K.2K.‚nO0†^‚. , 0
uv.80 27 ‚0 .2
.O‚Œ,72X}4
0
n842n7/p/1.2
} /1.u ‰ 7./'80u„‹
,0/10 57^. .u } .2
.857^€f‚ €‡†/v€4tu 7ou 7rt0 %^0Ž3.Šu./Jn8./14’
_J: y
u€.234
W
‚0-‰7./'80-57.-. $(./1n8.-.O‚^n80†^‚f.Œ72^€3460O‚,72}4 . .2m.‚,72X}4/0 ‹ ƒ.O‚ 20O‚f4/J3 e ’
?]
! "$#&%')(+*-,+&#-."/./-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
:~}
€‚„ƒ
: 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~zPJI€HPOCSFpHPLuIKJ‚ptVGUtVI\GUt;T;ƒ„OCX0JZtVGxSbT;t…FZS†QrS=TAGxE~vwEyFZI†‡CI\IOˆTVS‰‚ptVŠWH JS\S=X‹ŒHPOPQrS0
DIQrIKJpvAt„OCSKJŽTVSYQrIKO9F+tV~OXPI=TuGxSzuTVI=‘PT;SWF’OCE~JMvwS=T;IWFŽIKOwTVEuFŽSKLCE=hEF“hAI=TuqKOCŠWHTVEXPI
Ix”~Hut;TRt„zPJZtVE9
™
cCe
˜
−α
α
[“j
[Ž]
/)/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 t„O e ePc…lCePe
3546 798;:=<?>g@+*,^OATVS ‚ptVŠWH JS SxX‹HPO0QrS I=TCzTVEP”~HCI X I ~ePe
fˆXPI LCIWF+EF+IYvwSO0Q tVIKOuI IOI=”~Hut;T;t zPJZtVEIKOwTVS LCEuFtVG=tV~O
t„OCX tVG=SxXPSzCS ‹E TVSgSxGxG=t;~O X IgTVSg‚ HCIJ/.xS10 SKLT;tVGxSxX S IKO I=T I2PQRJZIKvwEsTRt„zPJIgXPITVSGHCIKJXPSg”~HCILCSWF+S LCE0JNTVS LCE TVIxS
3
DIQ IKJMvAt„OCSJ’IUT0‡CSUTVE~J^XPI TVS
‚ HCIKJ4.xS0
JLK
M
I
T
O/QSR
OPK
N
N
O4QSR
f
g
$!"# h Y R
&
= "!$# %
= "!$# \-]_^`Fabcaed
3546 798;:=<?> @P@5,^O I=TYIWFZ”~HuIKvwS XPI TVS ‚tVŠWHPJS SxX‹ŒH O0QrS=‘kI=TG=SKzuTVI-687ˆIWFMQrq HPOutVXPEmLCE~JFpH I2PQRJIvwE97ˆS HPO
vHCI=T;TVIsGKHWhSsGxE~O9FMQrSKO0Q IAXPIwJZtVŠPt;XPI.AI~F;:s_<~ewf=Rv ?>utNFZIwSLuT;tVGxSAIKO IUT^I2PQ…JIKvwE@7 XPI=TŽGxSKzuTVIHPOCSA‚ HCIKJ/.xS
‡CIKJ Q tVG=S=T9XPI~FZGxIKOCX IKO0QrB
I A ( D
_ CWe f}I=T`F+trFMQrIKvwSIWFMQrqIKOAIx”~Ht;T;t„zPJ+tVEGKHCSOCXPEI=T9qKOCŠWHuTVE _ dPeFEN=DgIQrIKJMvAt OCSKJkTVS
θ
TVE~OCŠPt Q…HCXOCSQRHPJZS=T 'HG XPIUTPvHuI=T;TVI
I
[+K
UWV
X$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 tVŠWHutVIKOPQrIWF X …t FpLuEuF+t Q t„‡CEuF+‘ GxSUTVGKHuTVSJ?T;S Q IKO F+tV0O 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$Y R
h Y R
J Y R
lue
={SKJ Q t„O e ePc…lCePe
l^a
[9JZE~zuTVIKvwS~F^X I ,5FMQrqQ t;GxS0
3Y46 7 8;:U<?>s@ [ SKJZSTVEuF X t…FLCEuF+t Q t ‡CEuF XPI=T L JE~zuTVIvwSSKO0QrIKJ+tVE~JZ‘“GxS=TVGKHTVSKJ VT SQrIO F+tV~O?XPIUT G=SKzuTVI h TVSJIxSxGxGUtV~O
IKOwTVS
SKJMQ tVGKHuT;SxG=tV~OsF+tTVS~F zuKS JMJSWFNFZE0OuE~vwEPŠKOCI=SWF h FpH LCIWFZE IWFŽXPIAalCe f 
l l
={SKJ Q t„O e ePc…lCePe
%'&)(+*-,/.103254#687:9#;=<#;?>@>A;CBEDFCDGIH?9#J1;LK+J=MENNLK+JCOEJPADRQTSUD9#GVXW/YEK5Z[PAJL;?O#D1Q\;3PAD<>AJ=KD\P]P^Y+O#J?>`_^VUa1VbJP
S/VPA;PcWd;1S-aDFCDPAJFeY#JP^W/>`;fJ?9Sb;f_^VbGY+>`;f;KhgiY9IWj;k1l]J?WjJ?>@F=V9E;?>nm;opJ1S#q#;1SbDI>rKJ:Sb;:_dY#J?>^s;ftuO#;?>A;FC;?9IWjJv9#J?>rSb;
<E;?>@>`;J?9wJxY\VUSUV<>yVbDzJ?9CSb;nO#DEPyVba{Vb|9CV9#K+VUa;K;]<#o}Sb;P}>`J;1aa1VbD9EJPpJ?9CSbDEPp;?O#D1Q\D\P~k
Ÿ
t
+N€
 N€
>CE~zPJI TVS zCSKJpJS?SxGQCSKO GKHuSQRJE ‚ HCIKJ/.xSWF ,kT LCIWF+E 3N‘ŽT;SWF OCE~JMvwSUTVIWFIKO TVEFSKLCE=hEuF —~‘ – h{TVS ‚ HCIKJ4.xS
SKLuT;tVG=SxXPS
IKOI=TuI2PQRJZIKvwE ˜ 
DgtVSxŠWJZSKvwS
XPI=T9FZ T;t;XPET;t„zPJI
ž
Ÿ
t
NI€
 NI€
—‚
%

E~OCX t;G=tV~OXPI
Ix”~Ht;T;t„zPJ+tVE
fƒe‚„@…h†‡{ˆ‰L‹Š
f‚
‚ŒiŽ „‡{ˆ‰‘
 N€
%
’I“ „d”b…h•—–™˜1…™˜ Ž „… Œyš – Œ “ Ž †…)„
f
Œ~Ž †n”X›@…~„ “ † Œiœ † “ ”U–~„
t
•ŽE~vwSKOuXPEvwE~vwIKO0QrEF JZIWFpLCIxGKQrEXPI ˜
LCIJSKOCXPE”~HuIxXPS
–†_
– '
−
'
G=EuF eFE _
e
!"#$
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
-,/F3kl]J)WjJv>@F=V9#;?>cJ1SE;1SU;?>`GI;?F=VbJv9IWjDzKJ1S+FeY#J1S/SbJIk
INN
ž
Ÿ
N€
 N€
%
DI<>`J3Sb;L<#;v>@>`;R;a)W #;?9ua?Y#;)W>`D
Ÿ
;?OESUVba1;K;J?9wJ1SEJ W/>AJ?FCD
l
_dY#J?>^s;Pwm ™SpO#JPAD5%

cSb;P9ED>@FC;1SbJPfJv9uSUDEP:;?O#D1Q\DEP——I —‚zQ
Sb;_dY#J?>`s;
k
Vb;G>A;?FC;KJ1S PA|+SUVUKDeSUV<>`J
t
ž
Ÿ
N€
 N€
—‚
%
—
D9#K+VUa1Vb|9wKJJxY\VUSUV<>yVbD
‚

„^…y†
/
ŒiŽ „ {ˆ‰L
%
.1ˆ‰Lƒe‚„^…y†—‡1ˆ‰LƒR
fƒC
Š
ŒiŽ „‡{ˆ‰
—‚
t
0132
’I“ „d”b…h•—–™˜1…™˜ Ž „… Œyš – Œ “ Ž †…)„
Œ Ž †
”X›@…~„ “ † Œiœ1 † “ ”b–)„
—
pDFC;?9EKD]FCDFCJ?9IWjD\P
O#Jv>`;?9#KDzxYEJK;
>AJP^O#Ja?WjDzKJn Š
!
{ˆ‰
Š„@…h†
−
∆
ZE
Z ŒiŽ „ ‡1ˆ1‰
'
&
=
Z
ƒ‹Š
ŒiŽ „ ‡1ˆ1‰="N
% $ !#"
M
, ,
1k$;?> WdV9
 N dN M#NNE
%[&)(+* ,U.1032C4
PAJ
;vO#D1Q\;
J?9
PyD<>`J
 :
7:9E;—<#;?>^>`;—B#DIFCDGIH?9#J;]KJ
P`Y‹J +W/>`J?FCD
Y9#;
KJ—O#JPAD]Q3SUD9#GVXW/YEKCZJP@Wj;:;v> WdVba?Y\Sb;K;]J?93P^YJ W/>AJ?FCD
P^YO#Jv>`_^Vba1VUJ
SUVjPA;
Wd;1SwaDIFCD‹PAJFeY#JP@W/>A;
J?9
Sb;
_^VUGY>`;
Ÿ
Q
;1Khg Y9Wj;Ik
l]J)WjJ?>^F=V9#;?>pSb;>`J;aa{Vb|9wJ?9CSb;;?> WdVUa?YESb;a{Vb|9k
Ÿ
1‰
1ˆ1‰
DI<>`J—Sb;n<#;?>@>A;;1a)W #;?9
l
W/>AJPp_dY#J?>^s;P[m ™S+O#JPAD]%

Sb;n9#DI>@FC;1SEJ?9wJ1SE;?O#D1Q\D
Vb;G>A;?FC;zKJ1ScPA|+SUVUKDRSUV<>`Jkr;zaD9#K#Vba1Vb|99#J1aJPA;?>yVb;]O#;?>`;zxY#J]Y9
QLSb;n>`J;a1a1Vb|9wJ?9
PA|#SUVbKD3PADFCJ)W VbKDL;—W/>AJP
PAJaD> WdJ?9wJ?9zY9zF=VP^FCD]OY9IWdD DCPAJ;v9
J?9wJxYEVUS/V<>AVbDzJPpxYEJSU;P W/>`JPp_dY#Jv>^s;P
—‚
Ÿ
k
_ Y#J?>^s;P[JP@WdJ
OE;?>`;1SbJ{Sb;Ppo
ž
1{‰
Ÿ
1ˆ‰
—‚
%
D9#K+VUa1Vb|9wKJJxY\VUSUV<>yVbD
(
"$#&% 0 - '
!
= "$#&% *,+-/.
) = +α
α
%
α
’I“ „d”b…h•—–™˜1…™˜ Ž „… Œyš – Œ “ Ž †…)„
—‚
Œ~Ž †n”X›@…~„ “ † Œiœ † “ ”U–~„
‡1ˆ‰
pDFC;?9EKD]FCDFCJ?9IWjD\P
O#Jv>`;?9#KDzxYEJK;
>AJP^O#Ja?WjDzKJ
‚
M})
‚„@…h†
{‰ Z
Ÿ
α
2
rf
 Mc) M
−
Z
Š
ŒiŽ „{‰L
N
M-
[P@W)WdVUa;Ik
>AD<ESbJ?FC;PcK+J
)+*-,/.1032547698: ;<=?>=@A@B=DC!EFGEIHKJ<L5=DMLNE )PORQSE<!HITVUXWY9Z[LN\U]=?L<^L`_W T3Q3Ta>I@bTSE9L<9WI<=?c5=-d!TSYI=`Y
NLF
TSLNbeAJ@TSc5=fQ3TgN=hYILi@=`Y/TSEfjkU]=lQmc`EFGEnNLiFhWLNAUX@B=hL<oQS=pe\T3HWI@B=p=`YrqsWI<KU]=KtIuL-U]L@\F
Ta<=@+L5Q1d=5QSEK@vY/L5Qxw<HW!QSE
YILyL`_W!T3QXTa>I@TSE α NbT!Z!z7{5jyt
ª
©
α
¨
DI<>`J—Sb;n<#;?>@>A;;1a)W #;?9
l
W/>AJPp_dY#J?>^s;P[m
™S+O#JPAD]%

Sb;n9#D>@FC;{S#Jv9eJ{S#;vO#D1Q\D
Vb;G>A;?FC;zKJ1ScPA|+SUVUKDRSUV<>`Jkr;zaD9#K#Vba1Vb|99#J1aJPA;?>yVb;]O#;?>`;zxY#J]Y9
J?9wJxYEVUS/V<>AVbDzJPpxYEJSU;P W/>`JPp_dY#Jv>^s;P
f'QLSb;n9#D>^FC;1SEJ?9
PA|#SUVbKD3PADFCJ)W VbKDL;—W/>AJP
PAJaD> WdJ?9wJ?9zY9zF=VP^FCD]OY9IWdD DCPAJ;v9
 k
_ Y#J?>^s;P[JP@WdJ
OE;?>`;1SbJ{Sb;Ppo
¥
¦
£
}i~
¤
α
α
¨
D9#K+VUa1Vb|9wKJJxY\VUSUV<>yVbD
α
‡ †
-‰ ŠŒ‹
€
)
|
α
=
ˆ
‚-ƒK„
‡
‚ƒ„ … 
α
=
α
KŽa‘S’r“•”—–5’—–5˜™’'šb›”'šsŽa˜œ’-
š'˜œVžA’'ŸŽ œšs¡`¢ŒœŽ 3”'
}i~
α
pDFC;?9EKD]FCDFCJ?9IWjD\P
€
)
α
α
§
α
©
§
O#Jv>`;?9#KDzxYEJK;
>AJP^O#Ja?WjDzKJ
Ÿ
a1DEPcM
α
α
! M
α −
aDEP
α −
α −
Š
=
α
α
"N
"$#%'& #'(
ME
5t =@ U‘Ta< {
)+*-,/.10325476G83 ;<!= >=Œ@A@B=CEFGEIHKJ<!L`=Y/LiQSEK<HITVUXWY Z O MLNbEf) LNAU‘w WI< TSYI=ME@—WI<E YILpN\W1N L54IUg@BLFGE!N =•WI<
M=N=`Y/E@ _KWL M/WL`YILDYILNbQXT6!`=@ NbTa< @BE.!`=F
TSLŒ<KU]E ME@ WI<=DHW+S= dL@ U‘TSc5=5Q‘t- =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 =Œ<hUX@BLN e‘WL@"!`=N$#
%—QMLNEi)&/QS= <!E@AFG=5Q L<GLlQ =ME5O E €vO QS= @BL`=`c5c5T('<GL<*) YTa@TSHITSY/=
ML@AMLŒ<Y/TScW QS=@AFGL<IU]Ly=yQ3= HW+S=Kt
uiTS=`H@=FG=pYIL5Q N,'/Q3T3YIE[Q3Ta>[email protected] p
= c`E<YTSc5T('< <L5c`LN=@bTS=M=@B=p_WLWI< N,'Q3TSYIE7NEFGL-U TSYIE =•UX@LNve WL@"!`=N+LNAU‘L
L<fL`_W!T3QXTa>I@TSEpLN _W!LyQ3=NmUX@BLN e‘WLŒ@"!`=N NyL c`E@ U‘L<fL<pWI<pF
TgN\FGEMIWI<KU‘E0/mEGNL`=Œ<iM!=@B=5QSLlQS=N21
i
g
h
¦ E<Y/T3c5T('<fYILyL`_W T3Q3Ta>I@bTSE
α
d*e
f
b c
j
d e
α
f
< ;
5> ?@
α
=
=
ACBED α
=
<
9:
F.GIHJ(KMLONQPRKQPRS
HTKUWV
NUXGISCY
K5H
USCY$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
(+(‚ ::': b„ƒq
?wGx…c,B+€
!&8NA.?,A.'z€,.Ju
?B…u' " ::': by† b‡
&G€'B%ˆ.,.+c'…‰.B'aiŠ†u‹)ˆ.,.+c'…‰.B=…‚ :': bŒ)-A8
&.@'x|A.‡CŽƒ
?@!‰N!A.Š,.&.'
'ow'|*
i " :':': b ‹.ŠBc,.=
Š.-Bo‘
“
”ˆ•–
™
š
—!˜M–
’
`)acbed/f?g\h
d
"#
o Jx& :: " :':
fijkGlmˆnqprC&MJo '=,.
@‡A.ˆ
@!C'…A'&q€,.J?o y=cA.|B(& (>-
@+A'C_A'&!'
j ˆ
A
M…&G€
!&yo =ƒ …o& M&y
zJJ
A
oo
& h )‹ …&G€&8w)xzNA.‡z…,'A'
,)
oM
?+=‚ :':': b Œ! j A.o+%o,.+€Cw)x/'ooNA.CŠ,'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$7DhO
—
‚
$.!‰
ˆ.!‰
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
*$H5
7R$@'
—
¯
‚
}
.!‰
ˆ.!‰
r {
r z
~ !*7
C7Dc!*$'E')a!:>7
5
7R$U7D
r {
ƒ zu˜ ƒ {E™šœ›EGž)Ÿ Š
ˆ.!‰
ƒ {f ¢¡3™£Gž)Ÿ Š‡¤
r z
…
¥&¦ ™N§Dšœ¨Z©xªšxª¡3™£šq U«3©q ¦ ¡›3š™
q¡›T§9¬šq™ ¦ ›! ¢­)®^› ¦ §
©q™
}
€!"(*$"!"'&8F>2e'32K1')C8FJ*$'=‚
Œ 1'^@*$Ja!:(')*$ƒ { ŠŽ3$yR‘/
’
ƒE{„<
−
…‡†
<(C(2ˆ.!‰+Š‹.
B?Š”“•–/
:
O |8N7R ˆ.N.I,.$.(•
(
s @ '5bC$')BK7DCG7D'&8F' *$' @&V)X7
'&8F„XgR(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!:'`¤bŠ ƒ
µ
wx5$]5D! *$'xƒ '32^•!$yR“f/ y&*$'E*$!*'fa&:')*$E'5$]5D&
µ
Š?.$yhŽ
$"PxZK@Z&"$#&%')\_‘*'5DBK7D#3:$@)*œL¢:$&8F *$'u"32`Š "!#`4+5
!#$798;:(* <hŠ?,Xy>2'
"&8N7D'^'H'?')a!:>7
5
7R$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 *$'X"32"%xŠ„ !#O>WA'8F'X7SHY )[
*7R$:¢Lb'5\*7
)#3@"+*$'G5=2c5
7D* 5
7R$@'+*'5Dd [EC)5DC:>5Db5D"8F'h2œ7Dc! *$'5xC)>5D' d C)[Z5D-BN:'KV)-*$'
@&V)X7D'^&8F-''5 14> d>*[m2U7 '5 14>2'AC)!h2œ7D*$'@J5
7I2Uy>*$')*!:(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$U7D
ˆ
!(5D'"!2
*'=w288N7
C)&O
mFnFoqpfr3sutvAwhx$y'zGx"z|{~}1V€rvw‚
™ š‘›Vœ
˜
ƒ
• –
o„pZ…€r†vAw‚GxyzGx"z†{‡}ˆr3st_vAwA‚
‰DŠ r5‹CsuŒ_ŽAsŽA€rs…… Š €ts‘r
…€th‹‡’3s…r Š t&V“”t Š ‹=…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$0I^;=0_Y`0Iba
8caEde;C0>@!&YE!&GD2[0>"f;C!G00I!90J8g"b;=
2[104=B&<`;C'46@A@A4CB&<"_;ChHh!`0I@@;CG\";i%?D!j!j"%'kTGHS#&%94=;=46H"4C!1l
ˆ$,
l K2546
/ H3I>T!DY`!"?NMOP "$;CQ%4C?%":SRV%"D2[UT '>Y`0 a 8 ;C!&?"4‡2 %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;ˆ@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$y'zGx"z { }1V€rvut‚
‚
~x€
µ
 z
‚ ƒ
‰DŠ r5‹CsuŒ_ŽAsŽA€rs…… Š €ts‘r
…€th‹‡’3s…r Š t&V“”t Š ‹=…r
… x€
„
m n xqpZ…€rvst‚qo$yzx'z { }ˆrstvut‚
{|
!
%&(')$*'!+,-/.)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¿0À”ÁÂ)Ã[Ä ‡Å…0†„†V‡ˆ,)©$·!Ƈ‚«ˆ‚ª•)‚’ º ¡¢,‰ˆ)†
Zǘ’‚Z‰•)0¡Y‰[’€,‚ª‡«•„‰‡)¢†V‡)…0„†š‡)‰ˆ$[)·!’
§)‚ªŽ!©‰jÈ$È$É+…!‡Ê„
‰3)!†š‘‡!‹‰0„¬)~)‚j‚‡)…‚‡!‹W,‰j‚(‡…‡)ˆ†…0†
‡)‚’Nˆ$‚^©-®†V©«†
‚(‡!‹[†V‡$©«†š‡)‚‡!‹‚^€‰_Ë|[„
‰j‰0……0†
¸‡
ˆ$‚ª„
‰jŽ)‚(/‘‰j§)‚ª„
‚j‚Ë|‚,…0‚j‚0„Y…‰€„‚^)†š‘!‡!‹‰0„)$‡Y†
ˆ$‰3‰0„…0†„†š‡)ˆ,1‚‡Ê’/1•)‰‹‚k’/$•)‚?†
Ì¬ ¶ ‚‹‚©«†š‡)‰‚0„$®)‰0„

ˆ$‚0„…$‚Ž,†
…0†
‚‡$‹‚3ˆ$‚™,!‘‰©«†
‚(‡!‹ ¬
µ
Ð
Ñ
Ò
Ï
Í
qÎ
tYu]vxw1y3z|{1}
)~ €,‚ƒ‚0„^…0†„†V‡ˆ,Š‰…‹WŒ‰‡ŠŽ)‚(/‘‰-’”“N‚0„•)‚-’?—–˜ˆ$‚0„™…0†„†š‡)ˆ,›x„‰œŽ‚/‘‰ž)†š‘!‡!‹‰0„Ÿ ˆ$‚0„™…‰€„
‚¢¡£„‰
,‚-’/„V‹‰‡!‹‚^¤¥j‚(‡‚0„)•$$‡!‹]ˆ$‚j‰•)0¡Y¦o›§)‚j‚-’¨„‰*’/$©‰jˆ$‚ª„
‰™‡)!©‰0„¡ˆ‚3„‰^Ž_)‚/‘‰3ˆ$‚™,!‘‰(©«†
‚‡!‹¬(­N‰3Ž)‚,‘‰
ˆ$‚™,!‘‰©«†
‚(‡!‹k‹†
‚‡‚ª’,*®)‰H„
¯©°±Y†V©²Ž³-´
¥
µ µ
¶ †
‰0·-,‰©‰3ˆ$‚0„’¸„W†
ˆ$]„†V€$‚¹¡«…‡)ˆ†…0†
¸‡[ˆ$‚3‚§Y†„†V€$?†
U¬
Ñ
Í
Ó
Ó
ÙÚÜÛ
qÎ
Ô Õ
Ô Õ
Ø Õ
ÖW×
Í
µ
qÎ
mÞÝß?à^s!ámZn-p
Íâ
Ñ
"$#
¬Š‰_‹†V‡ 
¡ž‚-’/‹°1‰•)0¡Y‰ˆ«‚‡
º»¼½¾¿0À”Á«Â „ €„
$§!)‚*)©$·!Ƈ‚¢ˆ$‚]„
‰1Ž/†
·-$,‰1‰ˆÜËܐ$‡!‹‰ª‹_†
‚‡)‚*$‡¢•)‚-’?ʈ‚
µ
$‡)‰•)‰,‚ˆÅ®)‚_‹†…‰0„¬ „&…$‚Ž/†…0†
‚‡!‹‚ˆ$‚[,!‘‰©«†
‚(‡!‹ž‚(‡!‹W,‚«‰©]€)‰-’*’/$•)‚Ž/†
…0†
‚’3‚-’
´$› ¬ „+€„
$§!)‚¢’‚
‚‡)…)‚(‡!‹W,‰Š‚‡i‚§Y†„†V€$?†
ހ)‰_Ë|˜„
‰Š‰…0…0†
¸‡iˆ$‚ „
‰ŠŽ)‚(/‘‰Ÿ ‹‰0„Z…© ’‚¹©])‚-’/‹Wµ
,‰Š‚‡ „
‰ŠŽ/†
·-$‰‰0ˆ:Ëܐ$‡$‹‰!¬
¶ ‚‹‚/©«†V‡)‰‚0„Y†V‡!‹‚_®)‰0„
1ˆ$‚®)‰0„,‚-’&ˆ$‚^Ÿƒ•)‰‰^§!)‚3‚0„€„
$§‚ª’?‚©‰(‡!‹‚‡)·!‰3‚‡[‚§!†„†V€†
U¬
É
Ÿ
tYu]vxw1y3z|{1}
‡ …‡)ˆ)†
…0†
‡‚-’Nˆ$‚^©-®†V©«†‚‡!‹[†V‡©«†V‡)‚‡!‹‚0›„
‰^‚‰……0†¸‡‚‡«„
‰^•)‰(,‚ˆ]Ž/©‰^…0‡„
‰™‡)©‰0„$‡[°‡·-„
 ‹‰0„§‚
θ
‹‰‡ ´ ´$›o‚-’&ˆ$‚…H†V ´|È$É?¬
θ
µ
θ
~)€,‚«‚H„+€„$§)‚ʉ…‹WŒ‰‡ƒ Ž)‚(/‘‰-’[“ ‚0„•)‚-’?”– › „‰Ž)‚(/‘‰[Ÿ ¡—„
‰[,‚-’/„V‹‰‡!‹‚[¤ ‚‡ƒ‚0„‰(•)0¡Y› §)‚«‚-’j„‰
/’ $©‰ˆ‚«„
‰[‡)©‰0„ ¡ ˆ$‚„‰[Ž)‚,‘‰[ˆ$‚1,$‘‰©«†
‚‡!‹U¬ ‡œ… ‡)ˆ)†
…0†
‡‚-’^ˆ$‚1©®†V©«†
‚‡!‹ž†V‡$©«†V‡‚‡!‹‚0› „
‰[Ž_)‚/‘‰[ˆ‚
,!‘‰©«†
‚(‡!‹k‹†
‚‡‚ª’,*®)‰H„
¯©°±Y†V©²Ž³-´
¬
µ µ
¶ †
‰0·-,‰©‰3ˆ$‚0„’¸„W†
ˆ$]„†V€$‚¹¡«…‡)ˆ†…0†
¸‡[ˆ$‚™©-®†V©«†‚‡!‹]†V‡$©«†š‡)‚‡!‹‚™)‰0…0†
‰3‰€)‰_Ë
}
¤!
-
Ÿ
¤
³
#
#
)+*
Ÿ,#
º
.
Î
â
qÎ
º
nÎ
‡)ˆ)†
…0†
¸‡[ˆ$‚™©-®†š©«†
‚‡!‹]†V‡©«†V‡)‚‡!‹‚‰…0†
‰3‰?†V€)‰
}/'
)
)1*
qÎ
Î
¤
'
) *
n Î
Ÿ0'
â
k•Y„†
…‰‡ˆ$]„
‰j„
‚Ì¡«ˆ$‚0„’‚‡)1‰3‰©]€)Y’
³
º
nÎ
‹ †
°‡)·-Y„
’&§)‚ˆ‰
´ %&
"$# µ ≤
"
≤
"('
´ÅÈ$È
µ

–€„
‚©‰’ˆ‚
’‹°‹†…‰!¬
º»¼½¾¿0À”Á Â Ä ‡)‰Ê€)‰‰©$·!Ƈ)‚0‰Êˆ‚•‚-’ º ¡„
‡·$† ‹W)ˆ’?‚ʉ(•)0¡Y‰«•)Z’/—‚±$‹ ,‚©[¦ ’€$,‚«$‡
’/)‚0„
)?†š‘‡!‹‰H„U·!’›Y…‚Ž/†
…0†‚‡!‹‚*ˆ$‚j,$‘‰©«†
‚‡!‹ ›]¡ ’/Ê‚±‹W,‚© ‚-’‹°j$‡†ˆ$‰ª$‡Ê…‰(€„
‚0›§)‚ª•)‰-’‰
•)+$‡)‰^•„
‚‰0›‚0„Y…)‰H„U„
‚j‚_Ë|‚(,…‚™$‡)‰3Ž)‚(/‘‰^Ÿ—§)‚™©‰µ‡!‹†‚‡)‚j„
‰™€)‰‰^‚(‡[„‰•’?†
…0†¸‡†V‡)ˆ†
…0‰ˆ$‰3‚‡Ê’?† ‹W‰…0†
¸‡
ˆ$‚™©-®†V©«†
‚‡$‹]†V‡$©«†V‡)‚(‡!‹‚!¬ ¶ ‚‹‚©«†š‡)‰‚0„$®)‰0„
ˆ$‚ ‚‡[Ž‡)…0†
¸‡[ˆ$‚ ¡ ¬
µ
α β
α
Ÿ
¦
β
tYu]vxw1y3z|{1}
~)€$,‚[„‰Z€)‰,‰]‰0…‹WŒ)‰‡ž…()‰‹W,Ŏ_)‚/‘‰-’*“
Ž)‚/‘‰^Ÿ—‰•Y„†
…‰ˆ‰^‚(‡*¬
„•‚-’ º ›„‰k‡©‰0„ }›„
‰1Ž‚/‘‰1ˆ$‚*,!‘‰©«†‚‡!‹¢Ž³Y´
µ µ
*¡
„‰
¶ †
‰·-,‰(©‰^ˆ‚0„’¸„†
ˆ1„W†V€$,‚
Ñ
}
¤
³
-
α−β
º
(
α
β
‡)ˆ†…0†
¸‡[ˆ$‚3‚§Y†„†V€$?†

" !
à
Ñ
α
}
º
$#&%'
"
-
³
©‰‡ˆ$*©©‚‡!‹Y’¯‚-’/•)‚…‹1ˆ$‚
3•)‚(,‰‡)ˆ$1§‚ˆ$‰
´"
α
…0’
µ =
α −
´
α
β
´
µ
"
’‚‡
α − β β

¬Š‰_‹†V‡ 
¡«„‡)·$† ‹Wˆ ’‚™©‰‡!‹†‚‡)‚3‚‡[‚§†W„†V€$†Z‰(•)0¡Y‰ˆ$‰
º»¼½¾¿0À”ÁZÂ*Ä ‡)‰^€)‰/,‰^)©$·!Æ(‡)‚‰jˆ$‚^•)‚’´
•)3’/ž‚±‹W,‚©k¦ ’!€$,‚Z$‡)‰Z•‰,‚ˆ«®)‚‹†
…‰0„ /)·!’‰ ’/µ ”‚±‹W,‚© ‚-’‹°*$‡Y†
ˆ$¢‰*$‡”…‰€„
‚1Ž/† Ë|¢‰]„
‰*•)‰,‚ˆ
‚‡ ‚0„N•$$‡$‹
 ›˜…-¡Y‰”„‡)·$† ‹Wˆ ‚-’ › & §)‚ÅŽ/©‰Å…‡¹„
‰Ê•)‰(,‚ˆ $‡ °‡·-„
 ˆ$‚ ɬ ¶ ‚‹‚/©«†V‡)‰|“¯‚0„
°‡)·-„
 ›Z„‰‹‚‡U’?†
¸!‡]ˆ‚0„…‰€„
‚x¡Ê„
‰3Ž)‚,‘‰^ˆ‚!‘‰©«†
‚‡$‹U¬
α
s0sHÎ
¦
α
tYu]vxw1y3z|{1}
°0„
…Y„
1ˆ$‚0„°‡)·-„
α
¬
Ù|Ù|Û
= α
ÌÛ
− α
ß?à
%
α
m
Í
*
Î
¦
­N‰1€)‰/,‰[Ž/©‰[…‡œ„
‰Z•‰,‚ˆÊ$‡ž°(‡)·-„
ň$‚]É¡ƒ„
‰*‹‚(‡U’?†
¸‡žˆ‚0„¯…‰€„‚]Ž©‰]…!‡œ„
‰]ˆ†V,‚0……0†
¸‡žˆ‚Z€)‰,‰Z‡
°‡)·-„
1ˆ$‚ Hɬ
¶ †
‰·-,‰(©‰]ˆ$‚0„&’¸„W†
ˆ$”„†V€,‚ª¡ …‡)ˆ†…0†
¸‡žˆ$‚]‚0§†„†š€$†
U¬~)€$,‚]„
‰*€)‰(,‰1‰…‹WŒ‰‡”ÊŽ_)‚/‘‰-’:›U‚0„ •)‚-’ º › „‰
‹‚‡U’?†¸‡ ˆ$‚H„…‰€Y„
‚ ¡ „
‰1,‚‰0……0†
¸‡ž¤ ‚‡œ‚0„‰•)0¡Y*¦ §)‚[‚-’j„
‰Ê’/$©‰ˆ$‚„
‰[Ž_)‚/‘‰[ˆ‚1,!‘‰©«†‚‡!‹-,› ˆ)†V†
·$†
ˆ‰
)‰…0†
‰3‰/†V€)‰0›k©‰-’N„
‰™‡)!©‰0„¬
&
–€„
‚©‰’ˆ‚
’‹°‹†…‰!¬
Ù|Ù|Û
º
º
,/!03#"$&24%(50'),1"+687*&0"59%,.1- %,/0"1
¤
ϕ
¤
=
=
ϕ − ϕ Ù|Ù|Û
©‰‡)ˆ$*©©‚(‡!‹’+,‚-’,•)‚…‹1ˆ$‚¦˜’‚3€!‹_†
‚‡)‚j„
‰‹‚‡’?†
¸‡[ˆ$‚0„…‰(€„
‚
C
<
A
:
=@D1?
=1>@?
;
´”
µ
B
$~ U’/‹† ‹W-¡Y‚(‡)ˆ$1¡”•)‚(,‰‡)ˆ$”’‚ª‹†
‚‡‚*‚0„U®)‰0„
! ˆ$‚H„ °‡·-„

´ ÌÉx¬ ‡)$…H†
ˆ$‚H„°‡)·-Y„
¢’‚*€!‹†‚‡)‚k‚H„
ϕ
®)‰0„
ˆ$‚j„
‰™,‚‰0……0†
¸‡Y›Ç¹´ ¬(­N‰3Ž)‚(/‘‰3ˆ$‚™,!‘‰©«†
‚(‡!‹1‚-’ ’/1•$,0¡Y‚0……0†
¸‡Z®)‚‹†
…‰0„
µ
µ

E q u ilib r io d e l s is te m a d e s ó lid o s
B,+.-/1032465879;:;=<>?=@BADCE>FHG'IG=JGKLC BNMPOBCEQ3@ISRGKDUT'
BV@OB'[email protected] :
MYKLG'JCMZK[CE<?G
@HIKLV\CWO3KLC!IG]?IBCZ^_MHJGKLCW`baUOV\CEFHCbKLG=F!@GKD3?'GHOB'ADBQ@L]Idce@UfghOIUKLV\CHKLG'JCM;KLCf<?G=@
K?@JG?A?BVBG%iCELkj\CEUOlOmUKL;MnKD@oV\Gf3?CP@BICYWOB!>BV@Om\VBpEYFHGIBfGq@oV\<OmGfr8GUG>Fb=OB8UGsK[BpE
IGOV\<OBG#MPOB=?G\\VVBpEWGHOB'
BV@OBVBpEt
;
^
u
:
!#"%$'&)(%*
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

ž
—
—š™
¥§¦D¨L©%ª«¬ª­D¨A©®¯®°D±ª²ª£³´yª¦
›œ
α
˜
›
Ÿ ¡£¢.¤
–
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
[&\‚%]
ƒ(TxU
„ DE CFEoB%l†…)ElrILC‡j‰ˆG g jkIŠH&B%l q I&B%w grq%g B9Gsv|BoH&BhCFI&zEDj g BDGsv‹I lrE‰m g vŒK)E qg nLGfBm0H&BjkI… g j g BDGsv|I g G&j g G)BDGsv|BtŽlrE"w‹K)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)B‘B%l&…)E%lrILC`H&BtlrE{G)IsCjkE%l
\
M’”“T&TxU
µ
•T 
\
„ C7I p lrBDjkELm`H(Bšmvv gq Es
dfeCg>hjilk)monqp>rtsuwv'xwyWz+{}|#uwv~y9xwy#{}|q€#w‚'ƒ„{†…‡wy{}|#‡>zy9…u‰ˆvW‡‹Š/Œu‰Ž‹|#|‘Wuwy)|‰~>y9j’)“ƒ”~>yx‹y{}|ˆv)Š‹•.|#|
{}yq…wy#{–ˆzvtyWu—Š/vt˜†™/…z}vv.~$šH…uˆv›1œ‰yCˆyWz–Q™:uwvWz'wv.ž6ŠlvŸ˜ …wyWz¡.vŸ~yz|¡.vWQ™/yWuˆ |yWuQy)Š¢{†…wy)Š/|•W…wvWuw~>|Š/vxwyz${}|#uwv
{}ytyuw•W…wyWuˆzvxwvWzv.~>vtvT‚£7¤q¥~y)Š¦yC§ˆzy|¨£©{$™©y)Š¦vWxw|1ª«|QyWuo¬­{}yt•)|#uS{4™/~>yWzvTŠl™®{}|«¯P‡wž°{4™jy)Š©•.|y.˜™/•)™/yWuˆy~y
z|¡.vWQ™/yuˆ|yWu¨±ª²¬³y{ µ ‚£«µ‰~>yCˆyWz–Q™:uwvWz5Š/v~>™®{–ˆvuw•)™/v¶W§«™7v'·'vŸŠlv¸#…wy'x…wy)~yt{†…‡‹™:z6Š/v'xwyWz${4|#uwvt{4™7u
´
¸#…wyy)ŠˆvW‡‹ŠlŒ#u~y{4Šl™l•.y›
ÉÄËÊfÆ
Â
¬
Ç+ÄÈÆ
¨
Ã#Ä Å9Æ
—
„ EDCFE"lrE @)Im grqg nLG g G)H rg q E%H&ElrEtv|E p lEtG)I‰BmvBDG q IsG)H grq%g ILG)Bm‡H&BtjkI… g j g BDGsv|I g G&j g GBDGsv|Bs`G B%l*E9@)I%ŽdIkB9G
\
EH&BDjm H&BtlrE{G)ILCjkEl)E q v )EtlrEAw‹K)B9CzEAH&B{CFIszEDj g B9Gsv|IofH 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(lˆisILG)I"H&BAw‹K)BDCFzEm m7BAH&BHLK q B g G&jkBH g EDv|EDjkBDGsv‹BxBls…)ElrILC`H&BtlrEAw‹K)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 G‰B%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 ˆ ™7u>‚ ‚®w‚‚‹µ
œty Š/v ˜†™/…z}v {}y ˆ ™/yWuwy Š/|‹{wv)Š/|#zy#{ xwyzvWuw~| ¸#…wy.~>v xwvWz}v Š/v uw|#z–v)ŠyWucy1Š
= =
vWxw|)ª«|±ƒ ´ «µC£/µ ƒo›9œ‰yŠlv y.•W…‹v.•)™/Œ#u Cž {4yfˆ ™/yWuwy ˜ …wyWz†¡.v~y9z|¡.vWQ™/yWuˆ | % "$#" ! ›
=
œty)Š¦xw|>'Š &/|#uw|²~yq˜ …wyWz†¡.v{{4y‰|‡ˆ ™/yWu‹yT™:uy.~>™/vWˆvWyWuˆ yTŠlvu‹|#z–v)Š¢yWu)(
/0,1,32
, +
+,- * ›87«™
6 5 =
− . 6 4 =
y)Š«•.|y.˜™/•)™/yWuˆy ~y'z|¡.vWQ™lyWuˆ|qyWuy)Šj{†…‹y)Š/|Ty{
‚>£3«µ£y)Š wv)Š/|#z¶C§S™7|T~yŠ/v˜ …wyWz†¡.v~y9z|¡.vWQ™/yWuˆ |‰y#{ ‹‚‚£7€
ƒ ªPŠ/v9xwyWz${}|uwv9x…wy.~y{}y)…‹™7z5{†…‡‹™lyWuw~|txw|#z<Š/vµˆv´ ‡‹Š/v{4™7u¸#…wyy{†ˆv~y{4Šl™l•.y›
h 9<; uoŠ/vŸx‹|‹{4™/•)™lŒ#u²Š'&7Q™¿ˆy)£«Š/vˆvW‡‹Šlvy{–ˆ ¶yWu {4™¿ˆ…wv)•)™/Œ#u—~y|‹™:Q™/yWuˆ|P™7uQ™7u‹yWuˆy)£#ªoŠlv{˜ …‹yWz†¡.v{°~yz|¡.vWQ™/yWuˆ |
:
ˆ ™/yWuwyu {…=wv1Š/|#z¶C§«™7|w£ µ xw|zŠ/vuw|#z†v)Š ›?>6vP˜ …wyWz¡.vQ~yz|¡.vQ™/yWuˆ| yWuA@ ˆ ™/yWu‹yPŠ/vP~>™7z}y.•.•)™/Œu ~y—Š/vˆvW‡‹Šlv
~>™7z}™/™l~v9Žwv.•)™/vvWz†z}™7‡wv›
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)Š/vWuw~>|‰v•.yz|qŠ/v{†…v~y9|#yWuˆ |‹{ zy{†xwy)•Cˆ|T~y( £w{}yˆ ™/yWu‹y
n
» ·
gml
p
=
v j
o
rFs:t
u
gmh
k
+
q
α
o
n
=j
p
rFs:t
gml
α
− j
=
¡
¢
£
Ÿ
+
˜ ’ = ™›š œ •W—
’ •0’
™›š0œ W• — ‘ ‘
·
k
µ ¹
gih
œtyŸŠ/vŸŠ/yHªQ~y)Šj{}yuw|TvWx‹Šl™/•)v.~vv)Šˆz}™/¶uw…‹Š/|T~y˜ …wyz†¡.v{<˜–|#z–v)~|txw|#z<Š/|‹{wy.•Cˆ |#zy{ dxw ½zyJ{i| £
•W…wyWuˆ v'¸…wy †Q‡Oˆ ‚ = :€„ƒ… ‰ ‚ = ~ 8€D } Š3‹
ŒFŽ
¤
¼
º
θ
f
µ¹
{ ªTˆyu‹™/yWuw~>|‰yu
=
ž
‹8¥§¦U¨D¦'‹©?Œ ª
\«m¬
Ž­Z«3®eŒ¥¯¥$Œ¦U¨°Œ ª
Œ
¡
Ÿ
—
˜  – – 
‘
=
¥
=
ž
¡
Ÿ =
•
Ÿ
+
” 
ž
Ÿ =
’“’ \
 
‘

±0²„³1´Tµ
Entramados y armaduras
“
®$«:­ZŒ µ Ž3¥ \Œ ¥§¦0¦U¨OŽ
º!"$#&%('*)
Á
ª Z­ Ž=Œ3¥§¦'®§‹+ ¦'‹®eŽ Œ ­ZŽ-,e¨/.‹®eŽ ² \Œ0¦QŒ ® µ ¨ ª
Ž ®10 3Ž 2 Z­ Ž¥ ®eŒOŽ33¨Z«3ª
Œ¥Œ ª ­°Ž Ž ® ¦ ¨/ ‹:­ZŽ$F¨/43ª
© ­ZŽ
Á ¿BFÁ Ã B*C B Ã
i
ª
«3® µ ŽF­?ŒªzŒF­?Ž65
«F©?«87:9
;2 ­ZŽ¥<,U‹
Œ®>=OŽ¥ Š3‹
ŒYŽ30¦@?
Ž ªŒ ªz­ZŽ¥A
Ž ® ®e
Ž ¥
7 7
© ¥›¨¦'®eŽ6:ŽEDWŽ ªŽ¦'®eŽ$3F¨/43ª
«TŽFO« µ 5®eŒ¥›¨43ªG
±
*NM
.
L
O
±
J
µ
*NM
.
H
I
K
´Lµ
´Tµ
´Lµ
´Tµ
Ž ® ¦U¨Dª
³
º!"$#&%
+*;
' K
Œ0¦QŒ ® µ ¨Dª
Ž ® ­ZŽ ,U‹:Œ >® =OŽ 1
Š ‹
Œ Ž3 ¦ ?
Ž Œª
¨Dª
\¨/FŽ ª
«z¥›¨¦'®eŽ
ŽEDWŽªŽ¦'®$Ž33F¨/41ª« F
Ž O« µ \
5 ®eŒ¥›¨/41ªG
±
OŽO\ŽA‹ª
Ž Œ ­°Ž¥ Ž®§®eŽ¥ Œ ­ZŽ Ž ® µ ŽO1‹®eŽ ŽF D ‹ª¦QŽ ²
µ
*
H
I
µ
±
*
*
J
µ
K
µ
»
±
®$«:­ZŒ µ Ž3¥ \Œ ¥§¦0¦U¨OŽ
º!"$#&%
K
0Œ ¦QŒ ® µ ¨Dª
Ž ®i­ZŽ ,U‹:Œ >® =OŽzŠ1‹
ŒzŽ3 ¦ ?
ŽzŒª OŽO\Ž ‹ª
Ž\Œz­ZŽ¥:Ž ®§®eŽ¥YŒz­ZŽŽ® µ ŽO3‹®$ŽŒz­ZŽ , ¨/.‹®eŽ
ŽO D ‹\ª1¦QŽ^© ¥ 1
¨ ¦ ®eŽ6
ŽED0Ž ªŽ¦'®eŽ33¨/43ª« Ž F
« µ 5®eŒ¥ ¨/43ªG
Ž ® ¦U¨Dª
³
+*;
H
O
´ µ
I
*
µ
³
µ
K
J
µ
»
³
®$«:­ZŒ µ Ž3¥ \Œ ¥§¦0¦U¨OŽ
´
Ž ® ¦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
Descargar