1. P r o b l e m a s
d e
E s t á t i c a
1 F u e r z a s y M o m e n t o s
2 E q u i l i b r i o d e l p u n t o
3 E q u i l i b r i o d e l s ó l i d o s i n r o z a m i e n t o
4 E q u i l i b r i o d e l s ó l i d o c o n r o z a m i e n t o
5 E q u i l i b r i o d e l s i s t e m a d e s ó l i d o s
6 E n t r a m a d o s y a r m a d u r a s
7 M e c a n i s m o s : p o l e a s , c u ñ a s , t o r n i l l o s
8 M é t o d o d e l o s t r a b a j o s v i r t u a l e s
9 F u e r z a s d i s t r i b u i d a s : c a b l e s y v i g a s
10 C e n t r o s d e g r a v e d a d
3. 2¢¡ ¡ ('¤£¦¥¨§©§© 4 §§
¥
Fuerzas y momentos
!#%$')(10
2 ¡ © 43¨§ %5£3%© © § 43 © §0 £¨163 ©¡871 2
9 © § §©A@ © §¨)1£¨¤©¢£ £¤© ('©5£ © BCDE#FG CCH C¨CIG C¨CH CC PP
×××++=
Q R SI
T
B¨C#U
=
Q
9 © § §©A@ © §¢¤©5£ £¤© ('0©5£ ©
VW
X
Ya`bc¨YedW
X
fbgih5p
q¨fr
VW¨fst#g
Weff
bguh¢p
=
v
=
v
=
wex¨y¤€e‚„ƒ†…ˆ‡5…ˆx†‰‘ex’…e‡5…’“”‡†•—–
˜!™ed ‡•4€y¤“ ™ …’“”‡f‡•!y ™ •4€¨‰ ™hg ‡fy ™ •1x†‰‘ex†…‡5…’“”‡• g ‡f ™ig™ €… ™fg ‡jy ™ •1k€e‡ d4l™ •m–
nop’qpp’qrsit5u#v q†wwv qxiyszneqq{eq¨q |}~|}}~
+=
++=
€mƒ‚e„a…‡†iˆŠ‰Œ‹h ‡5“”‡ d ‰Œ‚¤… ™5d y ™ed ‡•4€y¤“ ™ …’“”‡ g ‡jy ™ • g x•1k€e‡ d4l™ •1‚Ž… g ‚„ ™ig™ •1‡5…y ™ k4‚„€ du™i‘’g™ … g x’‡iy¨‰ƒ g €y„xj“Œ‡iy
”5…e€y‡x–•’€e‡fk—x d ‰ ™ y ™e˜ x d ‚ l x†…’“ ™ y–
™
š—›Aœ
—›ƒ›'ž
Ÿ5›—›ž
α
α
Ÿƒ›ƒ›ž
—›ƒ›'ž
š—›Aœ
¡
™
¢
Ÿ5›—›ž
ƒ›—›£ž
š—›Aœ
4. d x¢¡y„‡5‰ ™ • g ‡¤£m•—“””ƒ“‚‡ ™ –
¥
¦§
¨
©©§
¨
§!
$#%
=
'
==
(0)21436587@96A
(0)21436587@96A
B CDFEHG‰ –#we‡ g ‚I¡¨€QP ™–™ ‡†•u ™ y ™ y ™ •4€¨‰ ™ g ‡’y ™ •mk€e‡ d4l™ •—–¢R ‚ g ‚„‡5… g xŒ‡ya‰ƒ g €y„x g ‡ y ™jd ‡•4€y “ ™ …’“”‡‘•u‡ x¢¡“‚„‡5…e‡TSVU
¥W8X`Y ‰Œ‚ g ‚„‡¢… g x’‡iy”5…e€y‡x–•’€e‡fk—x d ‰ ™ x†…y ™e˜ x d ‚ l x†…’“ ™ y‡•1x¢¡’“‚„‡¢…e‡badc¢e
€mƒ‚¨„ …‡†iˆ ‰gfŠ ‡ƒ“‡ d ‰Œ‚¤… ™¢d ‡iyih ™ y‡x d g ‡iy ‰ƒ g €y„x “ y ™ g ‚ d ‡i‚„ƒ’… g ‡Šy ™ k€e‡ d4l™qpsr •’€e‡ ˜e™ “ •’€e‡ ™ y ‚„ ™5d ™ y
¡y„x•†€‡ g ‡ y ™ k4‚„€ d8™ ™g P €¨…’“ ™ ™¢d8™ •†€e‡ y ™fd ‡•8€y “ ™ …“”‡ g ‡ ™ ‰2¡ ™ •!k€e‡ d8l™ • •u‡ ™ €… ™ k€e‡ d4l™ he‡ d “‚„ ™ y g ‡tWuu8X •i‚
‡iy¨‰ƒ g €y„x g ‡jy ™ k€‡ d4l™vpxw ‡†• g ‡€yuuTX –
€mƒ‚¨„ …‡†iˆ ‰‚ ‡ƒ“”‡ d ‰Œ‚¤… ™5d y ™d ‡•4€y¤“ ™ …’“”‡ g ‡iy •i‚ •4“”‡5‰ ™ g ‡Œk)€e‡ d4l™ • x†…5€ d—d ‡¢…’“”‡• •†€e‡ •u‡ ‚¤… g ‚„ ™ ‡5… y ™ k4‚„€ d8™
™g P €¨…’“ ™ • ™ ¡‚‡‡5… g x’•†€e‡ƒS w…„‡†!ˆ¢‰T’‘8“•”–„˜—d‰‰™’‘8“edq„gfh‰™jik“ml–„n†!fh‰™qo
p
q
r s
r t
r u
r v
wxQy
zxQy
{}|~y
zx~y
“m”–„gˆ€d‘Hf ‚
α
„ƒ—„d‘…†Q†
α
‡ ˆ
‡ s
‡ ‰
‡ Š
‹ t
‹€Œ α Ž
5. o¢¡¤£¦¥¨§©‰ ‰„ —d‰‰ˆ
ˆ
!$#%')(01!32547698$6 § 6 ¥A@9©7£¦CB£DFEHGI@QPRGI 6 3§ 6 DSD 6HTU WV i¤D 61TU YX 8$6 E1£ 8 £a`$R£ 8$6 Bb£DFc` 6 ¥)d1£ID@oRefPR£¦¥A§©¥ 8g6
6 DA§hG7DpiR£BbGI¥ 6 DqD 6 GIr$§© 6 6 Bb£s¥ 6 D)`7Bt§h£¢3§ 6 i 6 Bvu¢ T `7BbGxwI` 6 cAG3¥A@a£yE1GI 6 B 6¨€6 Vso$‚ƒ£D„E1GI@QPRG3 6 3§ 6 D 8g6 B£D„c` 6 ¥)d1£ID
D…GIv†
‡‰ˆ „ †Q—d„o „ (a’‘ ˆdo ‰”“ ‚ ‡ ” „
−
† ‘ o — (a †‰‰0o ‰”“
‡id „
−
d‰0o ‰ (
− •
„o —q“ ‚ ‡xl–„ †Q— ‘ o (
−
†@— ‘ o ”“
‚ƒ£–¥ 6 D)`7B§h£¦3§ 6q6 D¦† ‡ „
Σ
‡˜—3™ so ‰ (
−
— † o ˆ¤“ d “ „ „o ‰T ‚
α
„
−
—
•
†
egfQhpiyjqklym
nIop3qrstnonvu1nn
w
nvunvuyx
==
z
+= {
|}~
egfQhpiyjqklym
6 P$¥ 6 D 6 3§h£HEH©b€I T ¥u1c©bE1£ 8$6 Bb£D‚c` 6 ¥)d1£D
ƒ‰„y…R† #‡Hˆ‰!‹ŠyŒs6 § 6 ¥A@9©R£¢¥Bb£q¥ 6 D)`7Bt§h£¢3§ 6Ž8$6 Bb£Dc` 6 ¥)d1£D‚¥ 6 P$¥ 6 D 6 3§h£ 8 £D 6 B£Žc)© T `$¥…£Ž£ 8€ `$3§h£o Œ £¦¥‰D)`Q@a€ 8 `7BG
i 6 BRu¢ T `7BbGywI` 6 cAG3¥A@a£qE1GI 6 B 6¨€6 Vso
ƒ‰„y…R† #‡Hˆ‰!9‘aŒs6 D…EHGI@QPRGI 6 ¥’`$R£ c` 6 ¥)d1£s‡ 8$6 @a€ 8 `7BbG —sfh‰‰€ 6 8 G7D5E1G3@QPRGI 6 $§ 6 D‡ ˆ iS‡x”C§h£HB 6 D5wI` 6
‡ ˆ cAGI¥A@ 6 EHGI“‡”`$•u¦ T `7BbG 8$6 —d‰† i–wI` 6 D)` 8 ©bc 6 ¥ 6 RE©b£ 8$6 @a€ 8 `7BbGgD “ ˆ˜— “m”™D 6 £š© T `R£HB£ †‰‰‰ qo
Œs6 § 6 ¥A@9©R£¦¥D)`›D˜@a€ 8 `7BbG7D i 6 B7u¦ T `7BG w3` 6 cAGI¥A@a£¦ o
œ@xv
p
žŸ}w@x
¡ |@xv
¡AŸ}xv
¢ x~y
Ÿ}x~y
{xQy
|QxQy
¡ x@x
r
r ˆ
rs
Ÿ}x~y α
6. ›¥…GIr7B 6 @a£ID 8g6¢¡ DA§huy§©E1£o
•
Œs6 Bb£FB 6 i 8$6 B›D 6 RGy£¦PgB©bE1£ 8 £”£B3§ ¥…©bu¦ T `gBbG 8$6 c)©g© 8 GsPRGI¥‚Bb£D˜§ ¥ 6 D‚c` 6 ¥)d1£IDD 6 §¨© 6 6
££
¤¥§¦©¨¥§¦©¨
=
!$#%('0)(1324657)8@9BA3#(DC)E8F8GIH4 #P57)RQ3GS7C1I'0)5TA#24)
UVXW4Y`abVXWcY de fff +=
gh)i2cS79p#(G#(13%S7)i24#Rqsr42IA578t24#P57)E8t24$8@9BA3#(DC)E8
u(vcvcvwx =− yy
€i #( )X1324‚%1s57)E8ƒ'„G#E8t#%(A3)%…S713#E8F8G#iH4'†S7#(13#
‡ˆc‰4’‘“”•–—(‰cˆI˜”•–™‰4ˆ4˜ de === ff
g$h6ikj‚limon‚p
qR# #8G#(1I'0)%S7r1srE t9 S7%)i24#P57)E8t9†A3#XDC)E8
uwv©x3y{z’|}~€s‚ƒ#E8G%q 13#X„A413)…9†A3#(DC)ƒ†„#(1‡248w%q 1#(1I'0#E8k†wˆ‰!Š†@‹Œ'†)57#E8kTA3#†wˆŽ9pDqs#%1†‘Ac1
t(13rEA5’’TA#‡8G#)57)sqS“'0)2’24#5@tX13rEA57’TA3#s9Dpqs)Š†@‹%I1”†•!–57$8Rqsr42A$578i2c#s†wˆI!Œ24#s†@‹¢%(A4q 57)(1—5’)
#57)%S7r16˜š™›‹œ‰žP™Ÿˆ¡ ¢w)5’%(A57)(@#5cqsr42IA57‚24#P57)E8t%q 13#(1I'†#E8‡!578tt(13rEA$578tTA3#i9pDqs)(1s%16†¢
£
£ ¤
£ ¥
α
β ¦¢§ α
¨ α
α ¦¢©Xª¬«§o ® β ¦R¯©°o«©o ® ± ¥ ¦³²D« ´ ± µ ± ¤ ¦³²D« ¨ ±
7. ¢¡ )(B'†S 1 ž¤£¦¥§£©¨¥£©£
g$h6ikj‚limon‚p
qR# #8G#(1I'0)%S7r1srE t9 S7%)i24#P57)E8t9†A3#XDC)E8
g$h6ikj‚limon‚p
qR# #8G#(1I'0)%S7r1srE t9 S7%)i24#P57)E8t9†A3#XDC)E8
!
#$ %'
%'
(
++
++
=
)1032¤46587@9BADCFEHGPIRQTSPUWVSYXGa`cb©XdfegbGa`ihTdfprq©Gfstvu@wxGaXyq¤SI€QTSYUWVSPXGaX©‚GIƒp…„‡†ˆp‰r‚d@‘8GIƒ’PbGfe“SP`“UfGaX
GaXY”`•G‡I—–˜b©Xd™aXeb‘8Sfq©G‡gPtihj†kIib6IlUfm¤qPb‘nSIfGIi‚oaXpGaXd‘ndD`•G@‘nd@Q@qnmPXfssr…gtvu1d@‘8Qab‘nda`f‘ndDUfdTeXwqx”bqyq©Gz‘ndI
QTSPUWVSPXGaXY‚GI{†|‘nSIv™aXebw‘nSI
α
„ †
α
‰ ’PbGƒeiSP`“UfdaX}QTSPXWpft
)10324~587@9AD€X‚‘ndIƒq¤qndTeYSPXd@‘nGIƒq©Gfb©XDVda`•d@‘nG¦‘nGaV–„VGTq©S`•GTQ3‚d¢Xeb‘nda`jq©G|da`RqIi‚dIj…¦†”‡¦†8ˆ‰†ŠdTQa”‹daX”`•GIje§bGa`ihTdPI
q©G@‘¤UŒqIiUfSHUfm©qYb‘nSjŽt@u1d@‘8Qab‘nda`v‘nd˜`•GPIib‘x‚d¢XY‚Gƒp…t
‘ ’ “B”–•¦— ˜P•‚™aš ›
β
“œ”–¦—Ÿž ‘ ¡ “s¢—””£žz™aš ¤
α
“¥@¢— ¦¦ ¤
§
§ ¨
©‰ª£«
α
β
¬
®
¯
°
±²
³
´
µ¶
µ·
µP¸¹
º
»
8. 6`RS¢¡‘nGaUfdPIq¤G…1I“‚™3§q8QTdYt
£
¤¦¥¨§©
¤V©`•GPIRdaXq©SW‘ndPIegbGa`ihTdIvGaX}QTSYUWVSPXGaX©‚GI{†vI•b©UfdaXq©S IRG€S¢¡Y§q8GaXGj‘nd˜`•GIibw‘x‚daXY§G
!#
++= $
%
¤¦¥¨§©
Š‘'GTQ3‚SP` b©Xwqx‚da`Rq8S G¢X}‘8dƒqq„`•GTQTQ¦qnmPX †vIGaXY§qnq¤S q¤G€‘8dƒegbGa`ihTd€GI
( )
()01 ++−= 2
2
3 d€e§bGa`ihTd€GaX}QTSYUWVSPXGaX©‚GIvGI p5476at98
− @BABCDAFEHG
)10324~587@9APIfŠ‘{Qab'¡Sf`•GaV©`RGIRGaXY§dTq©S|GaXF‘8dHeiq8eb©`•d dTqRQ–b¤XY‚dƒ§qnGaXG q©G da`RqIi‚d sŒUœ‘ SP`qneYGaXTSf†D‘nSIGU©”`•GaUfSwI
q©Gƒ‘ndI egbGa`ihTdIvp „ †Œp ‰ GI“‚™aXfG¢XfG@‘¤V©b©XY‚SHUfGTqqnSlq©Gj‘nSIz‘8dTq©SI‰tV3 SI Ufm©qPb‘8SIvq©Gj‘ndIve§bGa`•hTdIIRSPXk „XW7Ya`cbdY
egfihprq Wtsu`cbwv egfxhyp€ Wƒ‚„`†… egfˆ‡‰’‘a“”‘U•—–P˜d™ueU•gfhei•j‘lkjmXfn“”eU™'“”‘poq‡
rtsauuvwwhxgyqz|{g}~ ™ue€‚mu‘ƒ•…„Ve‡†„‘ Yaˆ„`‰‚Šs e‹‘¢k—“”Œ†„˜†•˜Ž'˜h†'e‡ef‘BfeU•jŽŠ‘’†'‘‡fhei•j‘V“U“”e•”¢mX‘7–ue•†„‘gf—'m„™Š“”‘˜†„‘
“V‘'‘¢•†'‘U™ueV†„elkp™ bu`‰su`†…Šš› e¢k—“”eœ‘gf—'m'™Š“”‘q†'‘œ“V‘'‘Š•j†'‘U™ueg†'elkp™ YU`dv¢`n‚'š “‚egf“V‘¢–ž‘gk‘–¨mu‘lk…“Ÿ•jeœ‘U™gfeœ€…˜Žlm„•je¨eV†R ¡m'™Š“”e¢‡
£ ‘Xk9–uegf‘¢•j‘¢k†'‘Tfelk“g‘'‘¢•j†'‘ƒ™ueV†'elk‘lk—“”ŒU™¤†'eV†'‘Xk‘U™¥–ž‘a“Ÿ•‘Xk‡¦‰9‘a“”‘U•…–P˜†™ueU•9‘§fr–ž‘¢–ž‘U™Š“‚‘¥†'‘¨o˜•j‘lk…—u‘g“a“”‘¥†'‘§f
‘¢•˜ŽŠ‘ƒ™H©«ªHf‘Xk–ž‘¢–ž‘U™'“”‘Xk¬†'‘po•‘lk…—u‘V“U“”‘†'‘yf‘Xk¬‘® ‘¢kH¯ `r°±`p² ‡
³
´
µ
¶¸·º¹®»º¹
¶h¼½¹j¾…¹
¿
À
°
°
´
³
µ
Á
¿ Â
¿ Ã
Ä Å
9. ‡¢¡7eU•®“‚˜†™ ‚„…¤£‚…¦¥§£½su…'…Xv
¥
¨ f–ž‘Š–ž‘U™Š“”‘T†'‘¢feP€®mu‘U•…„Veœ•j‘lk…—u‘g“a“”‘q†„‘gfd‘¢•º˜ŽŠ‘U™T‘¢k—“”Œž†„eV†'‘t—X‘¢•
FM © ∧= OP
` †'‘Š™u†'‘ž‘§f—'m'™'“”‘ ‘lkm„™
—'m'™Š“”‘¤“Umue§f”¢mX˜‘ƒ•je †„‘Tfeq•j‘V“U“”e5kº‘¢—u‘¢•®“‚‘ †'‘ o ‘¢–žeU™u†„‘ ‘gft—'m'™'“”‘|™ b7`cs7`n…«š…` ‘gf –ž‘Š–ž‘U™Š“”‘ †'‘ fe €‚mu‘U•j„Ve
•j‘lk…—u‘g“a“”‘†'‘gfX‘¢•˜hŽŠ‘U™œ‘lk
! Ya… ™ s#
−
b%$')(10Hš
¨ f¦—'•‘'†¢mu“a“‚‘ž‘lk“gegfeU•†'‘gfX–u‘g“a“”‘¢• —u‘Š•f‘Xk –u‘V“a“”‘Š•j‘lk†„‘9fe32uelk‘
,
$‡`40 —„•j‘¢—u‘¢•“g˜‘¢™ue9f‘Xk«–ž‘¢–ž‘ƒ™Š“”‘Xkr†„‘
fe€‚mu‘U•…„Ve•j‘¢k…—u‘V“a“‚‘’†„‘fh‘Xk¬‘® ‘lk¢¯35 ° 5 ² ‡76'mwk–uegf‘Š•j‘lkk‘¢™98
@BA WisX… @C W
−
bu… @ED W)('…
FHG#IQPSRUTWVSX
¨ fu–ž‘¢–ž‘U™Š“‚‘%Ya`w‘lk ‘gf¦†'‘pm'™¨—XeU•¬†'‘y€‚mu‘ƒ•…„Velk †'‘ bu…cb Ž kR˜n“ŸmueV†„elk ‘U™œ‘gf„—XfeU™X‘ › ‘¢•˜d„V‘¢™'“”egfX‘‘U™m'™—XfeU™u‘9—ueƒ•jegf‘gfh‘
egf › ‘Š•˜d„V‘¢™Š“‚egf'ªqk‘U—XeU•jeV†'e¢k¬m'™ue †u˜½k—“”eU™X“g˜e †'‘ym'™œ–ž‘a“Ÿ•‘¢hu‘gfX–ž‘Š–ž‘U™Š“”‘3Y q ‘lkr‘gfw†„‘m'™œ—ueƒ•je †'‘ €®mu‘U•…„Velk†'‘ bu…1b Ž
kº˜n“Ÿmueg†'elkt‘U™¢‘§fw—XfheU™u‘¢˜†™u“gfh˜d™ueV†'‘H‘P‘ƒ™Pm'™H—Xfeƒ™u‘œ—XeU•jegf‘§f‘Pe§fw—XfheU™u‘¢˜†™u“gfh˜d™ueV†'‘ªTk‘ƒ—ueU•jeV†„elkrm„™ue9†u˜½k—“”eU™X“g˜e †„‘ ‚ – `
“”egf “g‘¢–ž‘¢k‘ –¨mu‘lk—“½•je ‘U™tfe €j˜Žlm'•je9e š ‡§d¦eU•je9€—eV“g˜Ÿfh˜n“”eU•fekjm'–že †'‘’f‘Xk7–ž‘¢–ž‘U™Š“‚‘Xk†'‘’f‘Xk†'‘Xk¬—ueƒ•j‘lk ` f‘Xkd–u‘g“a“”‘¢•j‘¢k
”¢mu‘yf‘Xk¬€—‘¢•…–žeU™Hk‘ › eU™’“”‘¢–žeV†„‘’“g‘¢™Hk…mwk¬†„˜†•j‘g“V“g˜‘¢™X‘lk—ueU•jegfh‘gfelk¬eyfe•j‘V“a“‚ep†„‘˜d™Š“”‘U•Rkº‘V“V“g˜feŠ™¨†„‘fh‘Xk—XfeU™u‘¦k‡
™…e š
rtsauuvwwhxgyqzP{„{E¨ ™œfhep€j˜Žlm'•jeeV†R ¡m'™Š“”e k‘•j‘ƒ—'•j‘lk‘ƒ™Š“”eim„™’—XeU•d†'‘–ž‘¢–ž‘U™Š“”‘
Μg =
bu… eih½– ”¢mX‘peg“a“qpue k‘r2„•j‘m'™
—XfeU™X‘ › ‘¢•˜d„V‘¢™'“”egf¢ªž‘¢“Ÿ•j‘9—ueU•d†'‘–ž‘¢–ž‘U™'“”‘
Μ
q W7Y¡su… eih½– ”¢mu‘eV“U“qpue k‘r2'•‘ m„™’—¦feU™u‘”¢mu‘€—‘Š•—–že ('…rs “g‘¢™
‘gf › ‘¢•˜„V‘¢™Š“”egf®‡ƒ‰9‘a“”‘ƒ•—–P˜†™ueU•dŽl•jŒg€…˜“VeU–ž‘ƒ™Š“”‘‘gf„–ž‘¢–ž‘U™Š“‚‘ •‘lk…mXfn“‚eU™Š“”‘ †'‘eU–#2u‘¦k—XeU•j‘lk
tvuxw
y €
y
‚iƒ
¿
„†…
‡
−−−−
¿
10. d •‘r2Xf‘U–že¢kd†„‘ ¨ k—“”Œa“‚˜h“VeŠ‡
YU…
¨ f¦—ueU••j‘¢k…mXfn“”eƒ™Š“”‘¨‘lk—“‚Œ¨€—‘¢•—–žeV†„‘H—u‘¢•felk€®mu‘U•…„Velk o¨ª
−
o k‘ƒ—ueU•jeV†„elkm'™ue†„˜½k—“‚eU™u“g˜e¡ g‡ 6'm¢–ž‘¢–ž‘U™Š“‚‘H‘¢km„™
–u‘V“a“”‘¢• —u‘ƒ•——u‘U™u†u˜“UmXfeƒ• egf¦—XfeU™X‘t†'‘V€…˜†™¦˜†'‘H—u‘¢• o ª
−
o ` —XfheU™u‘t”¢mu‘¨€…‘¢•—–že¨“V‘¢™ fhe › ‘¢•˜d„V‘¢™'“”egf±m'™ˆŒU™uŽlmXfh‘
θ,
€j˜Žlm'•jeE2 š
.
™ 2 š
d¦eU•je “gegf“UmXfheU•rfe †u˜½k—“”eU™X“g˜e¢ ` 2'•jeV„V‘œ†'‘gfX—XeU•d•j‘lkjmXfn“”eU™'“”‘ ` eU—Xfh˜“geU™u†'‘žfhe f‘¡ª¢†'‘§f¦“V‘Xk‘U™X‘¨egf„“Ÿ•˜Œƒ™uŽlmXf‘¤£¦¥¤§•k‘“‚˜‘U™X‘
W ¨ WisX` (¢bwv – f†mu‘gŽŠ‘’‘§f'–ž‘Š–ž‘U™Š“”‘†'‘gf„—ueƒ•ƒ•‘lk…mXfn“‚eU™Š“”‘‘lk
Y W Ya…Xv¢`© e#hw–
d¦eU•je“Vegfh“UmXfeU•d‘gfXŒU™XŽlmXf‘
θ,
eU—¦fh˜“VeU™X†'‘¨feyf‘¡ªP†'‘gf k‘U™u‘egf'“Ÿ•º˜ŒU™uŽlmXfh‘¦£¥§ k‘ “‚˜‘U™u‘”¢mX‘
θ
WFˆ¦¥„`csSs
FHG#IQPSRUTWVSX
r sau„vwwŸxgyˆzq{œ~ ™Xe%2ueU•…•je › ‘Š•˜d„V‘¢™Š“‚egf †'‘ b –i†„‘9feU•ŽŠ‘ž‘lk—“‚Œk‘¢–ž‘U“‚˜†'e9eym'™ue9€‚mu‘U•j„Ve–u‘U•®“®˜“Vegf › eV“g˜e9e 2ue® a‘œ†'‘
Ys e„Ž eU—Xfh˜“geV†'e¢‘U™¤k…m¥‘'“Ÿ•‘U–ž‘¥‡ ‰9‘U–ž‘Xk—“½•jeU•y”¢mu‘H‘lky‘g”¢mX˜n–uegfh‘U™Š“”‘Heœm„™ueP€®mu‘U•…„VeH†'‘ Y¡s e„Ž › eV“g˜hePe 2Xe® ‘
eƒ—Xfh˜“Veg†'e‘U™Hk…mœ‘'“Ÿ•j‘ƒ–ž‘¦£ ªžem'™—ueƒ•†„‘ kº‘U™Š“‚˜†„‘ › ‘¢•jeU•˜h‘’†„‘ b© e„Ž7h½–ˆ‡
y
tWu w
‡
θ
ƒ
!H…
¿
# …
# ¿
# ¿
¿
$ …
11. ‡¢¡7eU•®“‚˜†™ ‚„…¤£‚…¦¥§£½su…'…Xv
Y„Y
Equilibrio del punto
FHG#IQPSRUTWVSX
‘¢™X†„˜“g˜feŠ™¨†„‘p‘g”¢mX˜hfh˜ 2'•˜‘
¡¢£¥¤¦§©¨
¤
¢£¤§©¨
§©¨
£
α+
==
α
!
#$
=α %
'(')0
=
132457698A@CBEDGFIHQPSRTPSUWVYXa`bSU3PAc7d¥bAcfeU3ghPAciVqphgsrcfeutvcfbwgXaUxPAy€yAXfp`ghPucfbw‚ƒr¥PSU…„€bQ†ˆ‡ughPucfbQ‚xXf‰rhUxbwb€g©‘’rh`sRTb”“¥bSUxb
• r¥P–PAc—cfe • r¥P–ghP™˜degfhghPi“¥PjkeqjkP–PS`¥yGr¥PS`sRlUkPgPG`mP • rX9c9Xn—hUkXfeqjoXPAcVqphgrpcfeughP™cfb–‚ƒr¥PSU…„€bq†3r€PjsghPgt¥uhegfwv
xyxz
α
{
|’}o~
{
x y
x z α
|’}o~
12. 7Uke—cfPSVqbjˆgP¢¡3jWR¤£RƒX9y€bsv
¥§¦
¨©§
¡ • rX9clXa—hUkXfeuPS` PAc“hr`sRTe! ¡ • rpX9c9Xa—hUoXfeuPS` PAc“hrh`sRTe#
$%$(' )
0132465(7
)
8924657
@ 2A2
== %CBED(FHGIBEFPD % '§Q'§$ )
0RS24T57
)
U 24657
V
==
W “¥PSUxbG`¥ghe • r¥P€gb
BYX`HG
¦ d˜ba f c BdX(efGgBEe§XhG
¦ dPiSp f c rq ˜˜dPa f c BEe§stG d ¦ dPa f
132457698A@CB Dvuw¡ˆ` PAc Pj • r¥PSVqbIghPIcfb ‚…Xf‰rhUxbPx PAc —pcfe • r¥P ghPq“¥Pjkev1 jkPqVqbS`sRƒXfPG`¥PYPG` P • rX9c9Xa—UkXfeEySr¥bS`¥ge jkP
bS“c9XfyAbrh`¥b ‚ƒr¥PSUx„€b€y‚ƒpehevf PS` PAc “hr`sRTe…„ gPAcijoX j…RTPSVqbCghPvy€bS—cfPj viHwPRTPSUWVYXn`¥bSUwcfbj RTPS` j©Xfe`¥PjuPS` cfej
y€bS—cfPjitYPAc“¥Pjoe†™v
x
{
‡
ˆ
‰ ‘“’o~
”§’o~
BYX9e
BYX`
¥ e3•
–
BEe§X
BEe§s
¦ e3•
13. Av¢¡ bSU RƒXa` ie¤£ƒe a¥£ ¦ eheSp
¥ i
¨©§
¡ • rX9c9Xa—UkXfeuPS` PAc“hrh`sRƒe ! ¡ • rX9clXa—hUkXfeuPS` PAc“hr`sRTe#
¦¢§
¦©¨
©
!
#%$'(
==
c BYX9e BEe§X c )¥0
1!2©3
4
56
0
12©3
4
7
0
1!2©3
4 8@98@A8@B
==
CEDFGIH¢PQ%RTS(UFVQ H
WYX©`badcfegcfeih p WgXqrasWtq@Xuawv'xyv €h p Wtq@‚„ƒ…c¢†ˆ‡†‰h pWtq@‘’awv € “”†rh
•—–˜ ™'dfeygihwjykml Pon¢UFGpDRqQ%FsrtH(uvHxwzy v'†(‡T{ | F(up}~sU P”€Q%RqHy‚ƒu!€up}FrtH„Q%F…nVH¢†”‚F¥u—€‡PQ €nyHVQ%RtF¢Pˆ‚HŠ‰I€ | U%GIHŒ‹
u!FŒrtH¢Pˆ}€FP”FŽF¢PmFyS(U”€f‚f€†%Gv€RTFPt‚HDR”u€ny€‘(Pt€‡P”Q €nVHVQ Hˆ’¢“„F©}FGprq€‡PH¢Gt‚fH¥uY}FP'u!€fR(PF¥u”FPt‚R”u”nVH¢†”‚F¥u@’
•
–
—
˜
™
šf›!œ
f›!œ
ž
Ÿf›!œ
W X¢q
W X©`
“%‡(
¡
W q@‘
W q@‚
¢ ‡(
e‡(
“%‡(
W q@X
14. ¡ GvR(†”‚FrtH(utQ F¡ —up}~©}€fnVHˆ’
c@e
¢¤£¦¥¨§©
•—–˜ ™'dfeygihwjml Pon¢UFGpDRqQ%FsrtH(uvHxwzy v'†(‡T{ | F(up}~sU P”€Q%RqHy‚ƒu!€up}FrtH„Q%F…nVH¢†”‚F¥u—€‡PQ €nyHVQ%RtF¢Pˆ‚HŠ‰I€ | U%GIHŒ‹
u!FŒrtH¢Pˆ}€FP”FŽF¢PmFyS(U”€f‚f€†%Gv€RTFPt‚HDR”u€ny€‘(Pt€‡P”Q €nVHVQ Hˆ’¢“„F©}FGprq€‡PH¢Gt‚fH¥uY}FP'u!€fR(PF¥u”FPt‚R”u”nVH¢†”‚F¥u@’
—
!
™
š ›!œ
f›!œ
ž
Ÿf›!œ
Ÿf› ›#
15. ’¢¡rHG }€‡P ¢ ‡¤£‡%€¥£ v‡%‡”†
cV†
¢¤£ ¥¨§©
W FP'u€‘(P FP Fy‚”nVH†”‚fF c ¦ §©¨ y
W FP'u€‘(P FP Fy‚”nVH†”‚fF v ¦ § y ¢!#
W FP'u€‘(P FP Fy‚”nVH†”‚fFŽ¢ ¦ §%$ y '(
¢¤£ ¥¨§©
) y021 † } |
α 3
c@v‡ uvFP
α
y x¥‡ ¦ ) y c 1 ¢ e r
•—–©˜™'dfeygihij54 tP Fy‚ F¥uvS(UF¢rtH„Q%FT‚H ‰ € | U%GvH„HVQ'6 U%Pˆ}H51U%Po†”‚R%S(U”F„Q%F “%‡„h Q FŠDF(uvRqF(up}~…U%P”€Q%RoHŠ} GIF¥u nVH†”‚F(u
Q%R”u”Q%FEFy‚f‚fR”u”nVR(Pˆ}F¢P”€Q%R”u”FPTU%PTD”‚fHPR87R(Gv€@9VR(Pˆ}Hy‚ ’¢“„F©}F¢Gprq€‡PHG”‚H(uƒ}FP'u!€RˆPF¥u”FPt‚R”u”nVH† ‚F¥u@’
•Œ–©˜ ™'d eygwh j¤A tPzFy‚ F¥uvS(UF¢rtH Q%F ‚H ‰ € | U%GIH HVQB6 U Pˆ}H ‚R uw} GIF¥u nUFGpD”R”uwU%P”€Q R”uwDR(GinVH¢†”‚F¥u F¥u }~PxFP
FyS(U”€f‚f€†%Gv€R'’¤C R”u †”‚R%S(U”F¥uEDu‹GF D”F¥uvHP “ ‡Eh nVHVQ%HŽU%P”RŠ‹qF¤‚†”‚R%S(U”FIH DF¥uvH x¥‡Eh ’y“„F©}FG rq€‡PHG”Fy‚QPH¤‚R(G”Q%F
)
ŸSR
–
T
˜™
α
VU
WU
XU XU
Y U
`
a
b
c
16. ¡ GvR(†”‚FrtH(utQ F¡ —up}~©}€fnVHˆ’
c“
¢¤£ ¥¨§©
h¡ £¢ UHPQ R ‚H}FP'u€‘(PtFPtFy‚ nyH†”‚F©7RˆGv€ 9VR(Pˆ}Hy‚ uvFVHŽP%U ‚H51%F¢PtFy‚D%U%Pˆ}R H‰nyR(PnU%G GIFPT} GIF(u ‰UFG9VH(uƒ‹ D”HGIHES(UFEF¥up}¥¤
FP FVS(U”€f‚ €‡†%Gv€Rtu Uou U rtH 7”HŽQ FŠu!FGtnVFGIR'’
•£¦¨§©¦¨§ y ‡
u!€FPQ R § ©s‚fHŽ‰ UFG9VHES(UFEF 6©FGInVFEFy‚”nVH¢†”‚FU%P”€Q R H¤‚%D U%Pˆ}R DbFP Fy‚ D%U%Pˆ}R H ‹ § Fy‚ PHy‚R(GtQ e§ ’
¢ R(PQ €fny€‘(P | GI~V‰I€nVHEQ%FEFVS(U”€ ‚f€‡†%Gv€fR
¢ UHPQ Rr‚fHw}FP'u!€‘ˆPuF¢PuF¤‚nyH†”‚F D H uvFVH P%U ‚H FP Fy‚ŒD%U%Pˆ}R H nVR(PnU GpGIFP } GIF(u ‰ UFG9VH¥u„‹ DHGIH S(U”F F¥u }¤ F¢P
FVS(U”€f‚ €‡†%Gv€Rtu Uou U rtH 7”HŽQ FŠu!FGtnVFGIR'’
•£¦¨§tX¦¨§ y ‡
u!€FPQ R § ‚HE‰UF¢G9VHES(UFEF 6@F¢GInVFEFy‚”nVH†”‚fFŒU P”€Q%RTHy‚ D%U%P%}R F F¢PmF¤‚%D U%Pˆ}RGH ‹ § ƒFy‚ PHy‚fR(GtQ e§ ’
•Œ–©˜ ™'d eygwh j! tPiF¤‚ F(uvS(UFrtHTQ%Fm‚HT‰ € | U%GIHTHVQB6fU%Pˆ}H51 U%Pˆ†”‚R%S(U”F Q F “ ‡%‡ h Q F…D”F¥uvRqDFPQ F Q F Q R”uŒnVH†”‚F(u@’
“ F©}FGprq€PHG$#”H% Fy‚ €Pˆ}FG PH¤‚RqQ F PHy‚fR(GIF¥u—Q%FT‚H„‰ UFG9VH § DHGvH„S(U”F„Hrm†”R”u nyH†”‚F¥u—F¥up}¥¤Pt}FP'uvR”u p †'% Fy‚EPHy‚R(G
Q FE‚fH¥uY}FP'u!€fR(PF¥u”FPt‚R”u”nVH¢†”‚F¥ugDHGvH § y †ˆ‡%‡Žh ’¢“„H©}R(#y} |
α
y e8£ ¢
)0213421
5
α+
= 687@9
A
6$7@9
B
§ ¨ y ¢ v“ 1 v h
CED
F
G$H
IQPSRUT
α
VWPSRVWPSR
X
IQPSR
α
Y
`a
17. ¢¡¤£¦¥¨§© !#%$('
)¨0
132 465¢879A@#§CBEDC85E¥F4GFGEH9I(QP6R§S 2
TU¥¨§C¥FH9IGVP 2XW 4 2(W 5E¥¨R(P8G W G W ©`Y¨a©`G¨ WS2(Wcb P6¥edf¥¢@#(g©QI4h4GVP8¥iDIG¨§CpE¥q¥¤r(P685E¥¢4¥tsvu(¥q4G W ¥w© W Dx¥E5EG¨§yP8¥F5 2 (485¢87€
€$( b $ƒ‚
≤
s
≤
0 '9„‚
…(†ATU¥¨§S¥‡G¢Peˆ¥¢P 2 §‰s '9‘‚ b P8¥ W ©G¨ W 2 G W G¨iP 2XW 4 2(W 5E¥¨RXP8G W’WS2 “4 W ©©`¥ W 4G‡5EG¤§ 2 ¡•”y“G¢P–rI€© 2 `
5 2 5¨I§—§CG¨A5¨I¥¨©Q§ 2 DIG¨§CpE¥ Wcb PI(GE@ 2 r¥¨§C¥FH9IGFG W ©`GFG¨AGEH9I(6PQR§S 2fW Ih§CG W I(Pg©`¥¤€©`GF¥F4G W G¨§•5EG¨§ 2 ¡
132 465¢879A@#§CBEDC85E¥F4GFGEH9I(QP6R§S 2
s™˜d )Ee¢f ¡g'e‚ h sjikl$X¡ !q‚
snmo 0 '9„‚
FXyp
Xq˜
α
rsctwuvw€xsytwu
z{
=
|
X~}
α
X ˜
€‚„ƒ
18. T~§ 2 R(P8G¨df¥ W 4Ge” W ©`Bw©65E¥€¡
)¢e
¢¡¤£¦¥¨§©¨
”nH9I(6P6 R§S 2 G¨AG¢PrI© 2 ”jH9IX6P6R§c 2 G¨AG¢PrI€© 2
r(P685¢¥¨4 2 P8¥VP8G 4G¢P W G¤ 2fW Ge©8G¨G
!
#%$'
(
)102#
3
=
α
=
α
4 56 7
8192@
A
@CB'D
E
=
α
=
α
F rG¨§C¥¤4 2 H9IGE4¥
α
b €!HG h I ) X¡ e ‚ h PFi ) e#0b $ ‚ h Pe˜d $ ff b $ ‚
QSR'T2UWVYXa`cbedgfehi¢pSq1rgsutwvgi¢pxv2r€yH‚rgƒ…„†s‡p‰ˆ€svgs‡pi¢p’‘”“–•˜—™2d€egfih ‘Wjk•ml2™Hngegfos‡pprquysr€yut†ˆ€i¢pkvwsxˆgt†ƒuyHrs
r€yyqxƒuzu„†s{h|pCs’ƒuvuiah¢ƒuy}pi~z€ts€r€ygƒppr€vusut‚pt†q…t†s’q…tY„Yƒ„ygˆHtt†qxƒ˜„Yt…pƒ†rƒa„eqxi~vwiyps†‡gs’s1yˆ„†ƒ‰‚pt†Š‡r2tƒ’ƒaˆ%‹Œr€y€rƒH
hs'rsutpvet„ygƒutŽ„†ƒ‰rsuyWpCt†~yˆ€sa„uqxƒ1zu„†sa‘€„Yƒ‡p’ygi~twvwƒa„Ys‡pŽsuyw„†iupŽƒuvgiah¢i¢p“hesa„uquygŠ‡r¢„†i¨ˆ€ssx”~rutY„‚t„z€tt†iW
‘ j
•
fg–
α‘”“•
fg—
α
‘Ž“
‘“j
α™ge
− α˜
™
™2e
− α α
19. a¢¡{ƒut t„y¤£2e¦¥ e€™§¥…lge€e©¨
—'™
!#
$$$ %'(§)0 %
1 )¦)
24365
7
24365
8
26345 ==
!#
9A@ •|lg‘ d2d˜v
QSR'T2UWVYXa`cbgdCB D”ye„†ƒ ‚pt†Š‡r2tƒ ƒxˆ%‹r€yHrƒ sa„gz¢„†i€”~rgs ˆ2sE¨~e€efˆˆ€s vgs‡pCipCs‰vwƒ1yH t†suyus s1y¤sa”~rutY„Yt z€tt†i¨suyw„†ƒ vgiup%t†qat†~y
t„ygˆ2t†qaƒxˆ€ƒzgƒ ‹i „†ƒgƒxqxqatY~y ˆ2sg„†ƒg‚ rgs1tGFxƒIH ƒuv¢„Yt†qxƒxˆ2ƒsuy sa„ s'P€‚tsuvwis„‚t„z€tsgˆ€s¨„†ƒq1rgsutˆ€ƒg”~rgsvgƒ‡pCƒ vgiHtx„†ƒ vgi2„†sxƒ
Q1hs' sutwvet„ygƒ1t’s…„H‡gƒ…„†i~t”ˆ€s „†ƒ‚ rgsutRFxƒH
QSR'T2UWVYXa`cb d€dSD”y sa„‰s‡p”~rusuvwƒ ˆ€s „†ƒ ‚t†Š‡r€tƒ ƒxˆ%‹Œr2yHrƒa‘ksa„¦qaƒuzu„†sETVUˆs‡pwrq r€yut†ˆ€imvgi~t¤ppr s'P€‚ts1vwiWUˆƒ r€y
v¤rgsa„Y„†ssqur‡h¢ƒsqxi~yWpwrƒuyH seˆ€swtt†Š€tYˆ€s¢Fes~pYXs•`¨~ewfa¥‚v cbutxpswƒ1vu„Yt†qxƒesuy s…„”s'P€…tsuvwidU ˆ€sa„Žqxƒuzu„†sr€ygƒe‚ rgsutGFxƒ
‡gsut t†qaƒa„Wˆ€s~pqxsuygˆ2suyHrsfe @ •hg‡e f}sa„–pCtrpwrsuvwƒs‡pwrqsuyesx”~r¢tY„Yt„z€tCt†iqurgƒ1ygˆ€isa„WquygŠ‡ru„†i
θ
• d€epixahgs'rsutwvet ygƒutk„†ƒ
„†i~ygŠ€t …rgˆygƒ'‚r€tƒa„ 9rq ˆ€s…„€v¤rusa„Y„†s
s
tvu
w
x
y€u
yGƒ‚„
…‡†
s ˆ6‰ ‚ w
xp
‘Cu
„
’E“•”¢–p—¦˜™—ed
x
f
g
h ‰ ‚
yRƒ‚
20. ‘Wti~zu„†suvwƒ~p”ˆ2s DSpwrq' tYqxƒH
lue
Equilibrio del sólido sin rozamiento
b ¡
¢¤£
¥ £
QSR'T2UWVYXa`cb’d§¦ ‘ ƒutƒ „†i¢p p%t†Š‡rut†suy€rs‡p ˆ2t…ppvuiupCt t„‡giupC‘ qxƒ…„†quru„†ƒ1tc„Yƒ suy pCt†Hy ˆ2sa„gqxƒuzu„†smh „†ƒ tsxƒxqxqatY~y˜s1y „†ƒ
ƒut t†q1ru„†ƒxqatY~y“¢Dk„2vgs~pi¨ˆ€s „†ƒ zgƒutwtƒpsqxiHy pCt†ˆ€s1tƒˆ€s‡ppv€tsxqat†ƒuz¢„†sH
ˆ6‰ ‚
¨
¨
©
„
t ‰ ‚
h ‰ ‚t ‰ ‚
¨
22. ‘Wti~zu„†suvwƒ~p”ˆ2s DSpwrq' tYqxƒH
l2l
!#
Q‰R'T2U VYX…`cbsd¡ ‘ ƒutƒ„†iup ˆ2t…pvgiupCt t ‡giup ˆ€sa„ v2ti~zu„†s1vwƒ¤ƒuyHrsutCt†i~t‘“qxƒa„†qur¢„†ƒut „†ƒrs1y pCt†~ycˆ€s…„ qaƒuzu„†s h „†ƒtsxƒxqxq…t†~y
suyw„†ƒƒutw t†quru„Yƒxqat†~yspCt¢„†ƒ~p zuƒutwtƒ‡pxpiHy£¢ui~vwi€Š¥¤uygsaƒ‡p h ppr vgs‡pi s‡pŽˆ€se—lge f
23. a¢¡{ƒut t„y¤£2e¦¥ e€™§¥…lge€e©¨
l £
!#
bgi~z€ts „†ƒ zgƒutptƒcƒxq'¡ gƒuy quruƒ'‚ti ‚ rgsutGFxƒ‡p£¢ Dk„ vgs‡pCi Qx‘Ž„Yƒ‡p ygi~twvwƒ…„†s‡psuy „†i¢pƒuvgiah¢iup # —~‘ # – h{„†ƒ ‚ rgsutRFxƒ
ƒuvu„Yt†qaƒxˆ€ƒsuysa„us'P€‚tsuvwi ˜
hgt†ƒxŠ‡tƒuvwƒˆ€sa„Wp2„YtYˆ€i¤„Yt„z€ts
¤ i~ygˆ2tYqat†~yˆ€ssx”~r¢tY„Yt„z€tCt†i
•Ži~vwƒuyuˆ€ivwi~vwsuyHri¢p ts‡ppvgsxquri¨ˆ€s ˜ ¥– 9
− ¦¨§
9 qaiup £2epi • e
© vgs1tƒuygˆ€i¨”~rusxˆ€ƒ ¥ –†• !$#%(')0 213#405'6
798A@CBEDGFIHQPSR5TVUXW5Ya`5YcbdbeYgfih)pghqsrcW5tIYvuCtawixxv¨uCtgyit€eh‚„ƒ…h)W5q†ˆ‡G‰iuS‘€etvYcy5hI’YQ€eh)`betauh’€“€”‰Cy5tcb–•”†…—I†˜t€
ƒG†¡€eY€™‡dYIƒE—(h)pgh£€etpe‰5t€”‡Gb–YftcW£ƒ˜Yf•”†˜q‰Cb–YfY(uhgi‰Ws‡jYkIl“tc‡jtcbdpa†WiYcbnmY(optIƒ5q5YIƒ˜hsbrutXƒ˜YX•d‰5tcb”s(Yftuy5YcbeYpgYcWs‡jtvW5tcbrƒ˜Y
`iYcbdb–YtcWwt(x)‰’†…ƒ…†`by†˜hztcWgƒ˜Yny5hi€y†˜—{†˜|)Wg†W5uC†…—(Y(uY“)`5o}ƒ˜Y€}b–t(YI—(—I†˜h)Wit€ptcWgƒ˜hi€pYcy5hI’h’€~k
x)€
t
f!
f‚
7
!fƒe‚„d…h†‡{ˆ(‰v#‹Š
‚ŒiŽ „‡{ˆ(‰#‘1
’s“ „d”˜…h•—–™˜I…™˜ Ž „… Œyš – Œ “ Ž †…A„
Œ~Ž †n”ˆ›d…~„ “ † Œiœ( † “ ”…–~„
7
ž
Ÿ
xs€ xs€
t
—‚
!
Ÿ
Cx)€ x)€
t
24. beh)`iƒ˜tcpgY)€™uCt¢¡‘€d‡¤£A‡d†…—(Ysk
w¦¥
§©¨
hs`b–tQƒ˜Yv`5Yvbdb–Y‚Y(—A‡!5YcWu—c‰5YA‡¡b–h •d‰5tcb”s(Y)€wm#¡™ƒpy5t)€ehS7 ™ƒ˜Y)€Wih)bdpgYIƒ˜t)€ftvWuƒ…hi€XYcy5hI’hi€——!s —‚z ƒ˜Y£•d‰5tcb–s(Y
Ycyiƒ…†˜—IY(uYtcWwtIƒit%$‡Gbetcpgh Ÿ k
l †˜Y(qbeYcpgYutIƒ €e|Cƒ…†…uheƒ…†`b–t
h)W5uC†…—I†˜|)Wwutt(x)‰’†…ƒ…†`by†˜h
'ph)pgYcWiuh“pgh)pgtcWs‡jh’€ bet€”y5t(—c‡jhzutn
−
! ŒiŽ „ ‡IˆI‰ ƒ‹Š)(¨ ŒiŽ „ ‡IˆI‰a#4x
0 y5tvb–YcW5uhzx)‰it(uY !$# ( Š 13#1( Š¨„d…h†2{ˆ(‰2
∆
i# 35467=8
9
7‘8A@CB D…FIHQPSRA@VUXW5Ya`5Ycb”b–Yaf5h)pghqsrcWit(Y£ut xxv utay5t€eh‚ ƒ˜h)W5q†ˆ‡¡‰5uS‘€etvYcy5hI’YQ€eh)`Cb–tauChi€“€–‰y5tcb–•–†˜—I†˜t€
ƒ…†¡€eY€ ‡dYIƒ —(h)pghS€et pe‰it€d‡Gb–YetvWSƒ˜Ye•”†˜q‰beYeY(uhg ‰CWs‡jYsk t pgYcWs‡d†…tcW5tetcWQt(x)‰’†…ƒ…†`by†˜ha`5Y g~h‚ƒ˜YzY(—I—I†˜|)WQx)‰5teƒ˜tzt gAtcb–—(t
‰Wzpe‰5tIƒ…ƒ˜tn‰W’†˜uhzYX€”‰wt%$‡Gb–tvpghX ut—(hsWE€d‡jYcW‡jtCB# 'sxxEDGpQkl“tA‡jtvbdpa†W5Ycb™tIƒiYIƒ…Ycb–qsYcpa†˜tvWs‡jhzutIƒCpe‰5tIƒGƒ˜tsk
Ÿ
x)€ x)€
7
ž
!fƒg1 „”…y†F2Iˆ(‰vƒe‚„”…y†—‡Iˆ(‰vƒ‚# Š
1 ŒiŽ „G2{ˆ(‰v# —‚ ŒiŽ „‡{ˆ(‰
’s“ „d”˜…h•—–™˜I…™˜ Ž „… Œyš – Œ “ Ž †…A„
Œ Ž † ”ˆ›d…~„ “ † ŒiœI † “ ”˜–A„
t
—!
‚
7
HIQP
7
ž
Ÿ
x)€ x)€
t
—‚
—!
25. Ik¢¡6Ycb ‡d†W x Ddx¤£ D¡w5xxi'
wE'
§©¨
hs`b–t—ƒ˜Yn`5YcbdbeYYI—A‡ !5YcW ‡Gbet€p•d‰5tcb”s(Y)€‘m ¡™ƒCy5t)€eh“7 ƒ˜YnW5hsbdpgYIƒitcWwtIƒiYcy5hI’h —‚ vƒ˜Ynb–t(Y(—I—I†˜|)WwtcW Ÿ k
l †˜Y(qbeYcpgYzutIƒ™€e|Cƒ…†…uh‚ƒ…†`b–tk¦¥rYz—(h)W5u5†˜—I†˜|)W£W5tI—(t€eYcby†˜Y“y5Ycb–Yzx)‰5t“‰W €e|5ƒ…†˜uhQ€eh)pgtA‡ †˜uhvY—‡Gbet€ • ‰5tcb”s(Y€‘t€d‡dt
tcWwt(x)‰i†…ƒG†`be†˜hzt€px)‰itƒ…Y€ ‡Gb–t€p•d‰5tvb”s(Y€ €et—(h)b ‡dtcWwtcWz‰Wzpa†¡€”pgh“y‰Ws‡dh¨§ hg€et(YvW yiYcb–YIƒ˜t{ƒ˜Y€po
h)W5uC†…—I†˜|)Wwutt(x)‰’†…ƒ…†`by†˜h
'ph)pgYcWiuh“pgh)pgtcWs‡jh’€ bet€”y5t(—c‡jhzut Ÿ ‚„d…h†©¢{‰
−
Š)(¨ ŒiŽ „{‰v# x
0 y5tvb–YcW5uhzx)‰it(uY ‚# w}%A0
α
# P !r!f# w™%A w
7‘8A@CB D…FIHQPgR$# UXWiY—`5Ycb”b–Y—f5hspghqsrcW5t(Y“ut ¦£X ut—y5t)€eh“Qƒ…h)W5q†ˆ‡G‰iugt)€d‡jYXYvb ‡d†˜—c‰’ƒ˜Y(uY“tcWQ€”‰£t%$‡Gbetcpgh Ÿ
€et Yvy5hI’Y tcW €–‰‹t%$C‡Gb–tcpgh €yh)`b–t ‰W5Y €”‰y5tvb–•”†˜—I†…t ƒ…†j€eY ‡dYIƒw—(hspgh‹€etpe‰5t€d‡GbeY tcW ƒ˜Y •”†…q‰b–Y YIuhg ‰W‡jYsk
l“tA‡jtcb”pa†W5Ycbpƒ˜Yb–t(Y(—(—{†˜|)WwtcWgƒ˜YYcb ‡d†…—c‰iƒ˜Y(—{†˜|)Wk
‡Iˆ(‰
% !
—‚
7
α
=
α+
= ')(0 1
2 3
46587
9
46587
@
ABDCEGFH E
’s“ „d”˜…h•—–™˜I…™˜ Ž „… Œyš – Œ “ Ž †…A„
Œ~Ž †n”ˆ›d…~„ “ † Œiœ( † “ ”…–~„
7
ž
Ÿ
2Iˆ(‰
% !
—‚
Ÿ
2IˆI‰
I‰
I{‰
α
26. beh)`iƒ˜tcpgY)€™uCt¢¡‘€d‡¤£A‡d†…—(Ysk
wi
§©¨
hs`b–t—ƒ˜Yn`5YcbdbeYYI—A‡ !5YcW ‡Gbet€p•d‰5tcb”s(Y)€‘m ¡™ƒCy5t€eh“7 ƒ˜YnW5h)bdpgY{ƒ5tvWet{ƒ5Yvy5hI’h f!9vƒ˜YnW5h)b”pgYIƒitcW ¡ k
l †˜Y(qbeYcpgYzutIƒ™€e|Cƒ…†…uh‚ƒ…†`b–tk¦¥rYz—(h)W5u5†˜—I†˜|)W£W5tI—(t€eYcby†˜Y“y5Ycb–Yzx)‰5t“‰W €e|5ƒ…†˜uhQ€eh)pgtA‡ †˜uhvY—‡Gbet€ • ‰5tcb”s(Y€‘t€d‡dt
tcWwt(x)‰i†…ƒG†`be†˜hzt€px)‰itƒ…Y€ ‡Gb–t€p•d‰5tvb”s(Y€ €et—(h)b ‡dtcWwtcWz‰Wzpa†¡€”pgh“y‰Ws‡dh¨§ hg€et(YvW yiYcb–YIƒ˜t{ƒ˜Y€po
h)W5uC†…—I†˜|)Wwutt(x)‰’†…ƒ…†`by†˜h
'ph)pgYcWiuh“pgh)pgtcWs‡jh’€ bet€”y5t(—c‡jhzut Ÿ w ! —(hi€
α −
Š
α
¢¤£¦¥§
¨© #4x
0 y5tvb–YcW5uhzx)‰it(uY #
—Ihi€™w
α
# ! α #$% ' =−α−α (0)21(0)31 α 465!79859@
ACBEDGFIHQPSRUTWV¦X Y`¦acb¦a!dedfahg3i0pqisrut!`¦vSahw¦v¤xi A€y‚ƒi0`3rs„†…ˆ‡¦‰W‘v¤x’…“acv!`”v–•0‡2„QQ„—bsd˜„ƒiWv!`W‡s`¦ac™SaEd3„ƒ‰sa–‰
xv!p„ƒv¤x˜eet!d„ƒ™SafQ„gxah‰svida–‰G„ƒifjk…“alm™–i0pqinxviph‡¦v0xe…ˆdfahv!`oƒape’„Qr¤‡sdfapa–‰rqs‡s`u…“autsu¡vE…“v!d’p„—`¦a!dCvSId¦aSƒiudv‰GvSxw!`¦r¤‡3ƒi
‰svyv–•0‡3„Qˆ„—bsd„ƒi α
x˜„33zU{Sjyt
| α
}i~
€
A
α
=
α
=
α ‚!ƒ0„‚Eƒu„ …
† ‡ˆ‡
‰EŠŒ‹
uŽ—‘ƒ’r“•”—–S’—–S˜¤™’9š˜›¤”9šsŽ—˜!œ¤’E
š9˜!œ†že’9ŸŽ œ0šs¡–¢Œœ¤Ž Q”9
α
A
£
¤
¥
}i~
€
¦
α
α
α
§
§
¨
α
α
©
α
¨
ª
©
27. St¢¡ a!d …‘„—` {¤£¦¥§£©¨¥£©£
!#%$')(
0 iubsdfvƒa•b¦a!dedfaa–™E…21¦aŒ`h…ˆdfv¤x e‘‡¦v!d43–a0x657—¦w¦v¤xiiA#8Gƒa `3i0depqaS2v!`qvl2a!w¦iSy2i €vy ƒa dfv–a–™S™S„@90`qv!`BA ‰¦„—d„ƒrs„ƒ‰Ga
w¦v!dew¦vŒ`¦‰G„ƒ™!‡2ƒa!depqv!`s…“vyayQa r¤‡Cƒaut
ui„ƒa–r¤da!pqap‰svS xD9GQ„Q‰si‘Q„—bsdfvstFE ap™–i0`¦‰¦„ƒ™S„@90` `¦vS™–v¤xa!d˜„ƒa¡w¦a!dfap•0‡¦v¡‡s` xD9¦Q„ƒ‰siUxi0pqvE… „ƒ‰si a•…ˆdv¤xve ‡¦v!d43–a¤xCv¤xe…‘v
v!`fv–•0‡3„Qˆ„—bsd„ƒipv¤x •0‡3vyQa¤xm…ˆdfv¤x e‘‡¦vŒd43–a¤x xvy™–i0d …‘v!`fv!`p‡s`pp„gx’pqi¡ws‡s`u…‘iHGmiqxv–aŒ`iw3a!dfaSƒvlƒa¤xPI
¦ i0`¦‰G„Q™S„@90`f‰svyv–•0‡2„QQ„—bsd˜„ƒi
ACBEDGFIHQPSRUTqVQ Y`3a b¦aŒdedfag¦i0pqisrut!`3v–a¡‰Gviƒiu`¦rs„†…ˆ‡¦‰ 2y w¦v¤x˜ifA v¤xe…‘w ‡s`2„ƒ‰saw¦i0d—‡s`¦i ‰svpx’‡Ix vSRs…gdfv!pqi3x a•‡s`
w¦a¤xa–‰Gi0d •u‡¦v wG‡¦v–‰svh‰sv¤x˜ˆ„T3–a!d x˜„—` dfiF3–a!p„ƒvŒ`u…“i w¦i0d ‡s`¦ahr¤‡Cƒa d¦v!d …‘„ƒ™SaS‘t'E akb¦a!dedacxv a!w¦iSy2acxi0bsdv ‡s`¦a
x’‡sw¦v!de’„ƒ™S„ƒv ™l„QUC—`¦‰ud„ƒ™–a Q„gx˜a ‰Gvoda–‰G„ƒi jyt 0 „ Qa Qi0`¦rs„†…ˆ‡3‰ ‰Gv Qaob3a!dedfa v¤xq{SjV8v‰sv!…“v!dep„—`3a!dfvS w!`¦r¤‡3ƒi α
‰sv
v–•0‡3„Qˆ„—bsd„ƒiIt
W X
Y `a`
bSc¢d =
α
=
α egfih
pFqsr§t@uwvyx€‚u€‚ƒr„u)…‡†x)…ˆqsƒg‰uSr
…)ƒg‰6t‘’u)r“q”‰i…ˆ•—–¢‰q”tUx)r
˜H™
dBe
f
α
f
g
h
i
˜ ™
d e
j
α
α
k
l
m
n
o
α
28. ¢¡¤£¦¥¨§©¦© !#%$('0)132
4¢5
67£¦8¨9£@£¦©83%£AB¡¤©CEDF©1)%£G9©6i H e j i
− I α
PCQFR
S
TVU WYX
`@©a§cbE'dCe9¡fg¤© ('0©8F© hiqp
=α
rtsqu
v DF©w¡f8F9£Gx¦e¨©19
( ) y
€
‚
=+αα ƒ(„¢…ƒ(„¢… † α ‡‰ˆ’‘ “’ˆ•”
–A—™˜edGfcg’h d
f!ijk¢l0monqpsrFt j o§)e¨§0¡@©o§uD¨©C¤£v$wq'x£sx¦eF©yD9eF©19©v¤£¦DF£¦¡#%¡z§{©C#|¡#e¨)|e9¡f{9©{§qbf'dCe9¡f}~'!§%©w8q€'¦8
$wA'x@9©o§¢)o¥¨§©@©C@‚ XX9X ˜„ƒq§c$wq'x…)1£¦™D¡f©C€'38Vx¦eF©cDeF©19©z€£¦DF£¦¡t(¡u§c¥¨¡#¡f…©¦u9© 4 XX9X ˜y2¦1† ˜‡£
)1£¦8q€'09©¡f¡7©ˆ§FDF©C¤£™9©…§0‰¥F¡#¡f92—iŠ†u‹¨'A©ˆ§FDF©C¤£™9©…§0‰¥F¡#¡fa©¦…‚ X9X ˜ŒAEeV§£¦8Fd9'x|eF‡gŽƒ§c)1¡¤d3‰x3eF©ŠDF©8F9©
9©o§¨©w9|¡¤©£Bi©¦ 4 X9X9X ˜ ¦‹F©Š¡f£¦™DF©a§Š¥F¡E¡fo‘
’
“
”ˆ•1–
—3˜w–
™
š
29. o2¢¡ ¡t('x8¤£
X¦¥(X¨§©¥
4 X9X
4 §
–A—™˜edGfcg’h d
f ijkql0mˆnqprg8)1w¡t%©o§ 9©aDF©¦¤£
d‡)eF©ˆ§d3g9©…e98q€£¦DF£¦¡t(©c%o§ )1£¦£y¤©a™eF©¦#|¡f@©8 §@bE'dCe9¡¤g1!•e983%92
j ˆ§)e¨§w¡!§0…(©8q€'38y©o§ )1¥¨§©aƒ §…¡¤©11)1)o'0¦8 ©w8y§z¡tt')e¨§o)o'¦8 h 2A‹¨' §…%©8q€'0¦8V$wA'xzx¦eF©‡¤©z§0©…D9e¨©19©
DA§0')1¡o§¨)1w¥¨§©c©Ca‚ X9X9X ˜Œ3 j eF$o§¨©C7©o§DF©C€£g$wA'x£G9©o§¨)1¡#%©o§¨x¦eF©g¤©ŠD9eF©o9©c)1£§d3¡©8%i‰‘
™
“
’ #
I
$ g
4
30. ¢¡¤£¦¥¨§©¦© !#%$('0)132
£
X
Equilibrio del sólido con rozamiento
¢¡¤£¦¥¨§©¦!#%$#'()%0#132416587'@9ACB69ED#F6FEGHB¦94I@9QP)1CRTS13#56UWVYX@B`aP)9E'I@1RbcP)13$6d9EB61bPePdX6I#9'dfdUhgUh9QPiVpqS
gA13241cP)9!2rX#9QP(Vsdt9'cShtfuUh5QX6dtABwvwX68Vx8y8€4PuX6I@9'dfuUhgqUh9rU‚#gSUƒ#AB6t9QP„SUsP…iRSh!0#13…Uƒ†A138VpqS¨uX#581@P)8y3‡e9¤Vp9'(2ˆUƒ#'‰
A‘’9S¨“#Sh13¢B69rShtfxX#9”u†AtB69!d18†A'2ˆUh9”8Vp1E9'!•–I#'d!24'8Vx9'#9'aSh!$#'udt9'9A—3X@USYU‚$6)Uh1%9'Sh˜I#1@P…UhgqUh™3U‚#B¦UhgAB64d
$#‘e9S@gA169AfuUgUh9'8Vx9fB¦9a)18†A'2ˆUh9'6Vp1!2ˆg‚@U‚241!I#')f9qS#9—3X@USUƒ$6)Uh1hy
ibjrkmltnioqptr
3s t#13$¦d9!Shu$#'()˜Ag¤Vsv#'%g”X#¤VYd1ˆfxX#9'd†AQPi‰6wxSbI#93P)1r my@SQP#18(24Sh9QPm9'ES1@P’'I@1Rb1@P rz y r{ RCShuf|X#9'u†A˜}
B69„d18†A'2ˆUh9”8Vp1t9'„•ey
‡eUhA5Q)'24iB69S¨P)™¦SUB61rSU‚$6d9
~ 13#B¦UgUh™3B69i9A—3XbUSU‚$6…Uh1
€1324'@B61!2413249'8Vp1bPe)9QPuI#9Ag'Vp1tB69a‚ ƒi{„`
− …‡†
`@g1@Pˆ¦F3‰CŠ‹F
Œ I#9”d'#B61t—3X@9AB6ƒ { ŠŽQ6y‚‘G ’ fcŠ”“#•–G
‚
—
6F3‰ˆ¦F3‰
ˆ¦F3‰
}
r z
r {
ƒ zu˜ ƒ {i™(šœ›iqžAŸ Š
…
ƒ {f ¢¡Q™£qžAŸ Š‡¤
¥8¦ ™x§hšœ¨©xªšxª¡Q™£šq …«Q©q ¦ ¡'›Qš¤™
q¡'›„§W¬(šq™ ¦ ›3 ¢A®”› ¦ §©q™
¯
‚
—
¦F3‰ˆ#F3‰}
r {
r z
31. y¢¡ '|VxU‚ ˆ¦F¤£xF¦¥§£sD#F6F@•
ˆ©¨
¥@s 'd 9S˜g169AfuUhgqUh9'8Vp9 B69 d18†A'2ˆU9'8Vp1„2ˆg‚@U‚241#yfS $#'ud 9QPuV 9' 9QP(VpB61„B69 241Q“@U‚2ˆUh9”8Vp1 Uƒ62ˆU‚#9'6Vp9 R–S
gA13(d93PuI#13#B#Uh9'8Vp9ifxX@9'u†AiB69„d18†A'2ˆU9'8Vp1t9QPSh„2bUƒ24y6S‚X#9581P…9fg”X62rI@Sh9i—3X#9–¤§˜Š
µ
ƒ
wxS6“#Sh13 B69xƒ 9QP ¨”•36y‚“fG y8B69iB613#B¦9f—8X#9AB6i9S6“#Sh18
µ
ŠcF6y!hŽ
ibjrkmltnioqptr
‡eUh5Qd'24EB69S¢P)™¦SYUhB61 SU‚$6)98y t#13$¦d9CShˆ$#'(dEAg'VYv#'#! fxX#9'd†AQP4‰eS1@PI#9QP)1@P‹B69CShˆ$#'()tRT9S$@Sh1¦—3X#9y S
Vp9'hP…U™3%B69qS¨gA'$bSh9„RSd9QPdX@SWVp'6Vp9 $ 9'%9ShI6X68Vx14B69!'I#1Rb1%„y@—3X#9!9QPmShrPuX624!B69!Shu#13(24qS¦RCB69eSh˜f|X#9'u†AuB¦9
d18†A'2ˆUh9”8Vp1hy
~ 13#B¦UgUh™3B69i9A—3XbUSU‚$6…Uh1
¢¡¤£¦¥¨§©64€x$#'ud0#182416587'#9A’•'‘B¦9!Sh!fuUh5QX6d!ABœv¢X68Vp)( B69u24QP)10–Š2!43¦5–RCS13#56UWVYX@B `hŠcD2ˆybP)9
24'8VxUh9”#9r9'c9A—3XbUSU‚$6…Uh1C'I#1RbB6r9'c9S£$#13)B69tB69!X6P)13I#13(Vp9 Fby •r2 B¦9PdXC956VYd9'241u• RC249AB#Uh'8Vp9!X6
gA'$@Sh9ˆX6bUhB61 cPuX 96VYd9”2416'yh‡!9Sx96Vsd9'241• I#9'#B69ˆX6 g'X@9'(I#1 B69ˆ24QP)047xŠ„23¦5yb‡e9¤Vp9'(2ˆUƒ#'r‰ A‘
B¦U‚$6X¢v¤'˜9S’B¦UA5Qd'24CB69qSaP)™¦SUhB¦1 SU‚$6d9CB¦9Sh%$#'()d $#‘igAShg'XbSh'˜Sh4Vp9'hPœUh™3 B69SxgA'$bSh9 d gA‘ShEfxX#9'u†AEB69
d18†A'2ˆUh9”8Vp1E9'9S 'I#1Rb1 dbB¦‘mP…U 9S 'I#1Rb1P)9egA13hPœUhB69'dtSUsP…1¦ybB69AB3X@gU‚„P…U£95bUsP(Vp9””“#Sh13d93PmB6980 R90 7 I#'d
—3X#9Sh„$#'()uP…9a24”8Vp9'#58i9'9A—8X@USU‚$¦)Uh1t9'4Sh„I#1@P…UgUh™34U‚#B#UhgAAB66y
•
'
%
@)ACB
7
$
D
EGF
}
H
@)ACB
I
P
Q @)ACB
@RACB
SUT
32. ¨)13$@Sh9'243P B¦9aw¢P(V¤VxUgA8y
ˆ6D
€1324'@B61!2413249'8Vp1bPe)9QPuI#9Ag'Vp1tB69 ~
¢¡¤£¥§¦©¨
¢¡
¢¡
+=
!
$# %')(10 2436587 %¤29(
@AB¢B
A
AC
D==
EF
G47IHQPSRTPU7VW%'0 293X%10`Ya(bH©%¤c740 24($%¤d17 (¤deRgfh0i29%$Ya(pde380 29Rgd¢Rgf80`24%$%¤587qRUYaRTr4sRa3`H©%tPuRg%'0 %
vw
x
y w€
==
‚
GWRƒ0q3…„q(†V‡s3hˆ¤('‰Rg%'04P‘39’“Hs%¤('0”d'7q(eYg%H6Hs%¤(10”Ya(H•‰–(Hs(˜— V™—ideYg(Hbd¤38‰i# 380q%'0hP‘%Hb24%fYg3qH$# %Hs3WHtVg24%˜Ya(
P‘%'0IH©Raf80`%'0–Yg($29RT)%¤dedeRgf80`24%•Yg(hr ('§)(h0 3–Hs%$d¤('0 d¤%¢Yg('0q’4Yi7 %¤ch3j0 3j#47 %¤29%k„q('r %'ƒ%¤587WRaYaRTr4©Rg3Il
mpnpoqpfrQsut$vewhx6y9zqx4z|{~}I†€rv¢w¤‚
ƒ o„p…€¢r†vew¤‚qx y zqx4z†{‡}ˆrQs©t•vewe‚
‰hŠ rS‹gsuŒ•ŽesŽe€rs…©… Š €'ts‘r
…€'th‹‡’Qs…r Š t8†“¤”1t Š ‹a…r
• –
—
˜ {
˜
™ š‘›†œ
v¢w¤‚
v¢w¤‚
33. l¢¡ ('uPSRT0¤£¦¥¨§©¥§¥¥
£¦£
!$#%('0)1
2 Rg(ec)('‰–( 24%eY•Hsf9YaRg293 YaRir4)%hl G 3hr4)% Yg( rq('Q)( (¤d‘P43q('06587S7 %'§ˆ¤(8H@9 %eY # %Hs3 24% Yg( r (1Q)(V 24%eY rWYg34587 %¢’Ya(
P‘%'0IH©Raf8024%eYˆd¤('rqYg%hV Yg($)%H)7qY‡P‘('04P‘%(ACB•%'0%¢Yq#4740hPS3`24%•('# 3¢VW3ED ’4587 %•%8HYg(jH§74‰–(•24%bYa(p0 38Q‰–(eYhV˜24%bYa(F7S7 %'§ˆ¤(•29%
)3hˆ¤('‰Rg%10hP‘3Il¢G(H7u7 %'§ˆ¤($24%h)3hˆ¤(1‰Rg%'0hP‘3 PSRa%'0 % H§7PI (eYg38 ‰RQ@SIRT‰–3T7VUW
µ X
B
Yt380 2 RgdeRgf80`24%$%¤5h7qRaYaRTr9sRg3
`badc¦egfihqpsrutv G ( r ('Qs(b„ 3h‰–34cxw'0 %¤($y$€ 24%6Yg(7)Rgc74)(X(¤2ƒ‚†740hP‘(…„ 29%b‰–(H©( — V Yg380 c4R‡P 7 2‡†U’Hs% ‰–('0hPSRa%'0 %X%'0
%¤587WRaYaRTr4©Rg3 ('# 3eVW(¤29(˜%10‡%¢Yƒrq38)24%ˆD 24%–740gH©38# 38uPS%e’P‘(eY 587 %E‰PYW‘†’§“ V‡‰–%¤29Rg('04P‘%`790 d¤(1rqYg%–740qRg243 ( H§7
%@S4P )%'‰–3”€$l 2 %eYh%@S4PUs%'‰–3•y # %'0q24% 790 d'7q%'Q# 3 24% ‰–(8Hs( —idbW–5g—–l 2 %‘P‘%'Q‰Ri0 ('—9 %eY˜I (¢Yg38 ‰u™i0qRT‰–3g29%eY
d¤34%q7§RgdeRg%10hP‘%$24%h)3hˆ¤('‰Ra%'0hP‘3
µ
# (')($587 %•Yg(hr (1Q)( Hs%h‰–('0hP‘%10 ch($%'0`%¤587qRaYURTr4sRg36%'0–Yg(h# 3WH©RgdeRgfh0`Ri0 29Rgd¤(e24(hl
d
e
f g…hji
klknm ioi
prq
m n oqpfrQsutFs¨ttx6y9zqx4z { }I†€rvut'‚
µ
m n xqp…€¢rvs¨t'‚qo6y z x9z { }ˆr§s©t$vut'‚
‰hŠ rS‹gsuŒ•ŽesŽe€rs…©… Š €'ts‘r
…€'th‹‡’Qs…r Š t8†“¤”1t Š ‹a…r
y
€
D
w hxi
` d
`
y
1 z
{ | }
~…x€
z
‚ ƒ
‚
„
~0x€
{ |
… x€
~0x€
34. ¢¡¤£¦¥¨§©¦© !#%$('0)132
465
78£¦@9A6£B£¦©93(£¨C¡D©EGFA©H)%£I6©QP
R¦STU#VXWR
Y
` S
acb
`
` S
R d e
f
g
f
g
f
+= hiEpAqhirp¨q
s FA©¡D@9A6£It¦uA©1
vxwy €¦‚ ƒ
=
„ u…#('†‡uEˆ‰©9A£©@9‘§0 F¡¤'†©¡DQ©1)@uA1)H'’39“©Q§0”)H£¦9A')r'’¦9‘6©Q©1t¦u¨'‡§0'†¥6¡¤'0£‘c©!•'©9A©
–8—˜™ dEedEf
=
gihD'†9AH§†©@93%©Q6©j§k¤©1lEu69¨6Q©1)uA1)r'’¦9‘6©Q©1t¦u¨'‡§0'†¥6¡¤'0£
µ monEprq@s
t‰u“vxwIyQz|{I}
~A¦€D‚ƒ‚H„”…H†0„0††‡¨ˆ¦DŠ‰1…‹‡Œ¨‰‡ŠŽ(A‚@G‘1‰E’”“x‚H„!•A‚E’c—–˜ˆ6‚H„™…H†0„0†š‡Aˆ¦D›x„0‰œŽ(¨‚G‘1‰žA¦¤†š‘13‡3‹%‰H„ Ÿ ˆ6‚H„™…1‰€¨„‚¢¡£„0‰
D‚E’G¨„†‹%‰‡3‹%‚”¤ ¥j‚@‡‚H„A•66‡3‹(“ˆ6‚j‰•AH¡‰ ¦o›§¦A‚j‚E’¨„0‰B’G6©‰jˆ6‚ª„‰™‡A3#©‰H„¦¡ˆ‚Q„0‰”Ž•A‚G‘1‰Qˆ6‚™D3‘1‰@©«†‚‡3‹%¢¬@x‰QŽ(A‚D‘1‰
ˆ6‚™D3‘1‰©«†‚@‡3‹%k‹(†‚‡¨‚ª’DB®A‰r„¦¯©°±‰††©²Ž(³E´
µ µ
¥
¶ †‰H·ED‰©‰Qˆ6‚H„¢’¤¸„‡†ˆ6“„0††€6¤‚¹¡«…1¦‡Aˆ†0…H†¸¦‡‘ˆ6‚Q‚1§¦‰†0„0††€6c†…¬
º!»¼½¢¾0¿HÀ”ÁÂAÃ‘Ä ‡Å…H†0„0††‡¨ˆ¦DA¦©6·3Ƈ¨‚1«ˆ‚ª•A‚¦’¤ º ¡¢D‰1ˆA†Ç˜’¤‚‰•AH¡‰‰‘’¤¦€D‚ª‡«•¨„0‰‡A¢††‡A…H„0†š‡A‰1ˆ6‘#A·3¨’¤
§¦A‚ªŽ#3#©‰jÈ6È6ÉC…13‡Ê„‰QA3¤†š‘1¦‡3‹(‰H„(¬A~A‚j‚‡A…¨‚‡3‹‡D‰j‚@‡…1¦‡Aˆ†0…H†¦‡A‚¦’xˆ6‚”©E®¨††©«†‚@‡3‹%‘††‡6©«†š‡A‚‡3‹%‚”€¨‰•Ë|‘„‰j‰H…1…H†¸¦‡
ˆ6‚ª„‰jŽ(A‚@G‘1‰j§¦A‚ª„‚j‚#Ë|‚D…H‚j‚H„‰…1‰€¨„0‚”A¦¤†š‘13‡3‹%‰H„A6‡‰†ˆ6‰Q‰H„¨…H†0„0†š‡Aˆ¦DI‚‡Ê’GI•A‰#‹%‚k’G6•A‚c†¦Ì¬ ¶ ‚‹%‚#©«†š‡A‰‚H„6®A‰H„¦
ˆ6‚H„¨…16‚1ŽD†…H†‚‡6‹%‚Qˆ6‚™D3‘1‰©«†‚@‡3‹%
µ
¬
Í
q1Î
Ï
Ð Ñ
Ò
Ñ
Ó
Ô ÕÍ
q1Î Ñ
Ó
Ô Õ Ö‡×
Ø Õ
Í
q1Î
ÙÚÜÛ
µ mÞÝ0ßcà”s3á!mnEp
Í1â
35. ¬¢¡Š‰•‹(††‡ ¤£¦¥§£©¨¥£©£
t‰u“vxwIyQz|{I}
‡…1¦‡AˆA†…H†¦‡¨‚E’xˆ6‚”©E®¨††©«†0‚‡3‹%‘††‡©«††‡A‚‡3‹(‚H›„‰”¤‚1‰1…1…H†0¸¦‡‚‡«„‰”•A‰@D‚1ˆ“Ž#¦G©‰”…H¦‡„‰™‡A¦#©‰H„6‡‘°‡¨·E¨„
θ
‹%‰H„¨§¦¨‚
‹%‰‡
θ
´
µ
´£6› o‚E’8ˆ6‚1…r††
θ
´!|È6Éc¬
~A¦€D‚«‚r„C€¨„06§¦A‚ʉ1…‹‡Œ¨‰‡ƒ Ž(A‚@G‘1‰E’‘“ ‚H„•A‚E’c”– › „0‰Ž(A‚@G‘1‰‘Ÿ ¡—„‰‘D‚E’G¨„†‹%‰‡3‹%‚‘¤ ‚‡ƒ‚H„‰@•AH¡‰› §¦A‚«‚E’j„0‰
’G6©‰ˆ‚«„‰‘‡A¦#©‰H„ ¡ ˆ6‚„0‰‘Ž(A‚D‘1‰‘ˆ6‚ID6‘1‰©«†‚‡3‹(…¬ ‡œ…1¦‡AˆA†…H†¦‡¨‚E’”ˆ6‚I©¦®¨††©«†‚‡3‹(ž††‡6©«††‡¨‚‡3‹%‚H› „‰‘Ž•A‚G‘1‰‘ˆ‚
D3‘1‰©«†‚@‡3‹%k‹(†‚‡¨‚ª’DB®A‰r„¦¯©°±‰††©²Ž(³E´
µ µ
¬
¶ †‰H·ED‰©‰Qˆ6‚H„¢’¤¸„‡†ˆ6“„0††€6¤‚¹¡«…1¦‡Aˆ†0…H†¸¦‡‘ˆ6‚™©E®¨††©«†0‚‡3‹%“††‡6©«†š‡A‚‡3‹%‚™A‰H…H†‰Q‰€A‰•Ë
#!¦‡AˆA†…H†¸¦‡‘ˆ6‚™©E®¨†š©«†‚‡3‹%“††‡©«††‡A‚‡3‹(‚ ¨‰1…H†‰Q‰#c††€A‰
$k•‰„0†…1‰‡¨ˆ6“„‰j„‚Ì¡«ˆ6‚H„¢’¤‚‡AI‰Q‰©“€A‰’ ‹ ¤†°‡A·E‰„¨’8§¦A‚1ˆ‰
º!»¼½¢¾0¿HÀ”Á«Â% „ €¨„6§3A‚BA¦©6·3Ƈ¨‚1¢ˆ6‚“„‰IŽG†·E6D‰I‰1ˆÜËÜ6‡3‹(‰ª‹•†‚‡A‚B6‡¢•A‚E’cʈ‚'!(£©£
µ
¡ž‚E’G‹%°I‰•AH¡‰‰1ˆ«‚‡
6‡A‰•A‰D‚1ˆÅ®A‚•‹(†0…1‰H„(¬ „8…16‚1ŽG†0…H†‚‡3‹(‚ˆ6‚‘D3‘1‰©«†‚@‡3‹%ž‚@‡3‹‡D‚«‰©“€A‰E’B’G6•A‚¤ŽG†…H†‚¦’Q‚E’
µ
´)£6›0 ¬ „C€¨„6§3A‚¢’¤‚
‚‡A…A‚@‡3‹‡D‰Š‚‡i‚1§¦‰†0„0††€6c†Þ€A‰•Ë|˜„‰Š‰1…H…H†¸¦‡iˆ6‚ „‰ŠŽ(A‚@G‘1‰Ÿ ‹%‰H„…1¦© ’¤‚¹©“A‚E’G‹‡D‰Š‚‡ „‰ŠŽG†·E6¤‰‰HˆXËÜ6‡6‹%‰3¬
¶ ‚‹%‚G©«††‡A‰‚H„‰††‡3‹(‚•®A‰H„Iˆ6‚ ®A‰H„0¦D‚E’8ˆ6‚”Ÿƒ•A‰¤‰”§3A‚Q‚H„€¨„6§¦¨‚ª’c‚ ©‰@‡3‹%‚‡A·3‰Q‚‡‘‚1§3¨†0„0††€¤†…¬
Ÿ
1©£¦É
º
¤32 Ÿ
465 ´!71¤8©1
µ ≤
4
≤
4@9 ´ÅÈ©¤6È
µ
} 5
º
ACB
Î
ŸD5
â
n1Î
E ³ F
q1Î
¤ 5
}G9
ŸH9 º
â
n1Î
E ³AIB
ÎA
q1Î
A
n
B
Î
¤ 9
36. –¢¤¦€¨„‚©‰¦’ˆ‚ ’#‹%°‹(†0…1‰3¬
1
t‰u“vxwIyQz|{I}
~A¦€6D‚‘„0‰€A‰#D‰“‰H…‹‡ŒA‰‡ž…@A‰‹‡DÅŽ•A‚G‘1‰E’B“ „¢•¨‚E’¤ º ›¢„0‰k‡¨¦#©‰H„ }¡ ¦›¢„‰IŽ(¨‚G‘1‰Iˆ6‚BD3‘1‰©«†0‚‡3‹%¢Ž%³‰´
µ µ
B¡ „0‰
Ž(A‚G‘1‰”Ÿ—‰•‰„0†…1‰1ˆ‰”‚@‡£¢B¬
¶ †‰1·ED‰@©‰”ˆ‚H„¢’¤¸„0†ˆI„‡††€6D‚
#!¦‡Aˆ†0…H†¸¦‡‘ˆ6‚Q‚1§¦‰†0„0††€6c†
¤ ¦©‰‡¨ˆ6B©¦©‚‡3‹%‰’¯¤‚E’G•A‚1…‹%Iˆ6‚ $ ¥§¦©¨¨…H¨’
β
´ 4 ¨ ’¤‚‡ α − β
Q•A‚@D‰‡Aˆ6I§¦¨‚1ˆ6‰
β−α
=µ !
º!»¼½¢¾0¿HÀ”Á Â$# Ä ‡A‰Ê€A‰#¤‰¨¦©6·3ƇA‚H‰Êˆ‚•¨‚E’¤ º ¡„¦‡¨·6† ‹‡Aˆ%¨!’c‚ʉ@•AH¡‰‰«•A¦’G—‚±6‹ D‚©‘¦ ’¤¦€6D‚«6‡
’GA‚H„A¦c†š‘1¦‡3‹%‰r„…#¨·3¨’¤›‰…1‚1ŽG†…H†0‚‡3‹%‚Bˆ6‚jD6‘1‰©«†‚‡3‹(
µ
›“¡ ’GÊ‚±‹‡D‚©£ ‚E’#‹(°j6‡¨†0ˆ6‰ª6‡Ê…1‰@€¨„‚H›¨§¦A‚ª•A‰E’¤‰
•A¦C6‡A‰”•¨„‚1‰H›‚H„‰…A‰r„…„‚j‚•Ë|‚@D…1‚™6‡A‰QŽ(A‚@G‘1‰”Ÿ—§¦A‚™©‰‡3‹(†0‚‡A‚j„‰™€A‰#¤‰”‚@‡‘„0‰ •¨¨’c†…H†0¸¦‡††‡Aˆ†…H‰1ˆ6‰Q‚‡Ê’c† ‹‡¨‰1…H†¸¦‡
ˆ6‚™©E®¨††©«†‚‡6‹%“††‡6©«††‡A‚@‡3‹%‚3¬ ¶ ‚‹%‚#©«†š‡A‰‚H„6®A‰H„¦ˆ6‚
µ
‚‡‘Ž(‡A…H†¸¦‡‘ˆ6‚
α
¡
β
¬
Ÿ
α
β
¦
' )( 41032 à α
´4¥
4658790 α
´
µ
'
E ³
}¡ ºα
Ñ
º
@
$
¢
β
E ³
Ñ
})
α
α − β
¤
38. –¢¤¦€¨„‚©‰¦’ˆ‚ ’#‹%°‹(†0…1‰3¬
¤ ¦©‰‡Aˆ6B©¦©‚@‡3‹%¨’CD‚E’D•A‚1…‹%Iˆ6‚¦˜’¤‚Q¦€3‹•†‚‡A‚j„‰ ‹%‚‡¢’c†¸¦‡‘ˆ6‚H„¨…1‰@€¨„‚
¡ ¢ ´” !
µ
~6…’G‹(† ‹‡E¡‰‚@‡Aˆ6I¡”¦•A‚@D‰‡Aˆ6”’¤‚ª‹(†‚‡¨‚B‚H„…®A‰H„3 ˆ6‚r„ °‡¨·E¨„
ϕ
´ !¤!ÌÉx¬ #!¦‡A6…r†ˆ6‚r„¢°‡A·E‰„¢’¤‚B¦€3‹(†0‚‡A‚k‚r„
®A‰H„¦ˆ6‚j„‰™D‚1‰H…1…H†¸¦‡‰›1ǹ´'1
µ
¬@x‰QŽ(A‚@G‘1‰Qˆ6‚™D3‘1‰©«†‚@‡3‹%I‚E’ ’GI•6DH¡‰‚H…1…H†¸¦‡®A‚#‹(†…1‰H„
£¥¤ 1 !
µ
Ù|Ù|Û
º
¢
¤
¢
Ù|Ù|Û
º
¤ϕ
¦§©¨!#$# ϕ−
==
ϕ
%'
(©)103254#687@9A4CB84DFE©7D)1GH©4I0DGHQPR)SHDITVUH©)S2W7I0
X
Y
`
aIbdc
e
f
g
adhIc
39. ¢¡¤£¦¥¨§© !#%$('
!
E q u i l i b r i o d e l s i s t e m a d e s ó l i d o s
)1032547698A@7B
fDCFEGIHQP¢RTSVUWYXY¥a`¥¤§b§c¥ad(fehgi§bprq9sqatq#uvg fxw€yfgi©Qdsƒ‚1q#uh©…„9¥¨§©f†¤d(yf¥9q¨€uhd‡qFˆ©Q§vq¨prg X w‰uvq9¥¨tg w1¥u‘gi`§cq
drs%uv†’g‡yQ%uvg3sq“§c¥¢sfg”•wrtq#uvg‡–˜—#©…¥ y1†’giprg˜uvqap3dq#uh©Q§c¥9q¨ryf¥9ehf#d§v¥“¥¢sd™ed©…¥f¡¤ghy(s…uv†’gruvq9¥¨tg¢w1¥Yuvgf`§cqad¥
ucdtq¨§cecf†¢fq7i(gi§vkj’gi©…¥¢ylym…uv¥Ywnuhdo†’q¤f©Q§cg€d1fsg‰¥‡yf¥3¥¨§b©f†¨d(ym¥’†¢fpi‰prq¢sf¥¨f©qqdo†’¥¨`(ymqf¡rVq¨©…q¨§bp˜(¥¨§ayf¥V©…q¨su‘fpi
sq¢y(†’¥¨`(yfq5w€yf¥a§cq’¥’†¢†¢fpi‡q¨ryf¥9¥¨§©f†¨d(yf¥¢†¢fpit¡
Y
X
”
u
40. s§vgi`(yfq¨pr¥iulsqDg ub©…„F©m†’¥f¡
¡
)1032547698A@7B
f CFEGsHmP¢RTS£¢¥¤‡XY‰†¢mym(si§cgrsq tq#u‘g§¦¦w€§c¥’smgV” u‘qYq¨†¤dq¨f©Q§v¥Yq¨ q©¨id(myQ`§vfg˜q¨ ym¥ tg(udf†¢fpi‰sm†’¥’s¥Vq¨ yf¥
ehf#d§c¥¥’se™edf©¥f¡q†’gisu‘fsq¨§v¥¨id(q ©…gs¥#u yf¥#u uhdtq¨§vehf†¢fqiu uvgf ym…uv¥#uA¡7rVq¨©…q¨§bp˜(¥¨§ yf¥ §vq’¥’†’†¢mpi q¤ yf¥
¥¨§©f†¤d(yf¥’†¢mpiYw˜yf¥D©…q¤Iu‘fpi‡sq y†¢¥¨`(yfqf¡
f CFEGsHmP¢RTS¢g ysqFˆ©%§cq¨prg sq d¥ `¥¤§b§c¥ (g3igiprg!¨q’¥VsqVyfgf ©Qds#ow€t(q#uvg%$('#0) u‘q¥¨tg¢w1¥7uvgi`§cq
d¥Vt¥¨§cq¢s21q¤§©f†’¥ y yQ%uv¥ woq¤‰q ystdf©…g u — uvgi`§cqVd uvq¨p˜f†¢Qymsi§vg‡©¥¨p3`(3¨oym…uvg sqV§c¥’sfg3”54n— ˜p wTtq#uvg
6 ¡sghyDu‘q¨p˜f†¢myQsi§cg•qiub©…„‰†’gymg†’¥’sg u‘gi`§cq€d(¥xuhdtq¨§vehf†¢fq€i(gi§vkj’gi©…¥¢y §bdfg1uv¥¢— ©¥¢yh†’giprgnuvq71q q¨nyf¥ ehf#d§c¥
¥’sd™edf©…¥¢— q#uh©…¥¨sg q¢y †’gi ™ df©…g q¨•q©¨id(Qym`§vmgI¡1rVqF©…q¤§bp˜¥¨§8 ¥©9 r7`d ™F¥¨§9q y sm¥’#§c¥¨pr¥€sq¢y5uvpymmsg yQ`§cq˜sq˜yf¥
`¥¨§b§c¥“w‰sqF©q¨§bp˜¥¤§ yf¥tg(u‘f† fpi sq¢ys†¢q¨f©Q§cg˜sqY#§v¥@1q’s¥’s˜sqVym¥ `¥¨§h§c¥8A`B9 yf¥iu §cq’¥¢†’†¢fgi(q#u q¨‰yfg(uh¥¤tg¢w1g(u X w
u A(†C9 yf¥Vedq¨§hj’¥Vsq §vgfj’¥¨p˜fq¨©…g7¨id(qY¥’†F©ED¥7uvgf`§cqq¢ys†¢mym(si§cgFAsG9hu‘Iq¢y1tq#uvg‡sq¢y uvq¨p˜m†¢mym(si§cg‡q#u ¡ 0)r—GH©I d„¢y
q#u q¢y¥1¥¢yfgi§ pP(kprgqsq
µ
t(¥¨§c¥Q¨idq9q¢ysu‘%uh©…q¨pr¥9q#ub©…q9q¨‡q©¨fd(mym`§vfgGR
X
u
S
Y
PCT©U
VXWYU
”
`
a
X
u
”
—f'9p
41. ¢¡¤£¦¥¨§© !#%$('
¡¡
)1032547698A@7B
S£¢ gi`§cqryf¥7`(¥¨§b§c¥‡¥¢†F© D¥¨ D(f†’¥¤prq¨f©…q¥¤dCFP§¦ed(q¨§hj’¥#u‘— yf¥#uagi§bpr¥¢ymq#u“q¨ X w u w q¢y tqiuvg7$ sq˜yf¥7`¥¨§h§c¥f¡©¨h¥
†’gism†¢fpiIP§’P¦eS(CmS€sqqq©¨id(Qym`§vmg˜q#u ¨idq7yf¥#uh©Q§cqiu5ed(q¨§hj’¥#u“uvqq†’gi§©…q¤ q¨€d€tdf©gI¡£¨ ¥#u gi§bpr¥ yfq#u q¨ q¨ X w
u uvq9†’gi§©…¥¤3q¤ a —ydq’fgVtgf§ sf†¨igVtd©…g7q¢ytq#uvg7sq yf¥a`¥¨§h§c¥f¡
rqf¥’#§v¥¨pr¥9sq¢ysuvpymmsg3ym`§cq
rVqVyf¥Vehf#d§v¥7uvqVsq’sid†¢q%¨idqqyf¥Ys%ub©…¥¨(†¢f¥ X u q#u — ¡ p —1yd(q’fgrq¢yt„¨#d(ymgrq#u 4 —3'££!‘¡¨ ¥Ys%ub©…¥¤†¢f¥ u `
q#u u ` 4 — $#V©%4o— $$I'9pT¡ ghy(†’q¤f©Q§cg7sq9#§c¥@1q¢s¥’sruvq9q¨†¨d(q¨f©Q§c¥9¥ —©i$1' p sq¢y(qFˆ©%§cq¨prg X sq yf¥a`¥¨§b§c¥¡
G¢
I gism†¢fpi‡sq9q©¨id1mym`§‘fg7sq yf¥a`¥¨§b§c¥ ¡
')( 4 $0#V©…1 42 )
'43 4 5687©7
9§@©A
B
=
ϕ
C¢
f
D
u
φ
a
B 3
B (
φ
—f'9p
— Yp
B (
B 3
f
φ
B 3
φ
E
a
BF
–
–
B F
E
φ
B 3
42. s§vgi`(yfq¨pr¥iulsqDg ub©…„F©m†’¥f¡
¡ $
rVq yf¥9†’gism†¢fpi‡sq9q©¨id1mym`§‘fguvqD©mq¨q 5¡©7A£¢¥¤¦
=ϕ= §
¨
© ¢
¥¨§c¥ q¢y 1¥¢yfgf§hpPk(prg sq¢y †’gq¢ehf†¢fq¤f©…q7sqV§cgfj’¥¨p˜mq¨f©…g—tyf¥ru‘ ©Qd¥¢†¢fpiTq#uDsqVprg 1(p˜mq¨f©…g‰p˜kq¨f©…q¡ ¨ ¥7edq¨§cj’¥
sq §cgj’¥¨p˜fq¨f©g7©fq¤q7uhd71¥¢yfgi§ pr„Fˆ1prg¨idqVq#u5f#d(¥¢ys¥˜t(gi§ yf¥gi§bpr¥ y¡ ¥¨§c¥Vq¢y 1¥¢ymgi§hs¥¢sg‡sq 6 —q¢y 1¥¢yfgi§ sq
yf¥agi§bpr¥¢yrq#u ' ! †¢g(u 4 $ )T¡
rVq#%$ 4 #)' (' uvqD©fq¨q 24 —f'3¡
)1032547698A@7B
)10¥2436587@9ACB%DFE#G(H@I8PQIRGTS%UWVCSYXa`bXdceVgfih(UWp@S4IqV#rceXspt`TV@hupavbVYHwpsptvTpyx€VceV%vYUeXa‚YGTpƒc„‚Y`TXtUW…„IqH@I8Xa†TV%U£Iˆ‡wV%G‰‘p’Pb‰‘p@P
HwV%“V”ceX“ƒ‚TXdc•‰QUepX–G”Pqp—…„Iq˜d‚YUWp—pwS™xd‚4G‰‘pegfhGic„‚jIRG‰‘X–U£IqV%U†Tp€hjSYVbckXdc£…•X–UWpdclSYXgUWpwSTIqVnmkho`TX%ceVgpqHwpwSYp(‚4GTpe
r#X¥‰‘X–U„“CIRGTp–UhX@P4`TXdc£VkS4X@PbH@I8P8IRGbS%UWV#`Tp–UWpts%‚bXuX%c•‰‘XvGTVswY‚TX@Pqs‚TXe@xyVYSYpdcyPqpdczc„‚4`TX–UW…„I8H@IqXdczceXtHwV%G6c£IqSYX–Uep–GP8I{cepdc¥e
m
m
43. @e¢¡jp–Uy‰ IRG¤£¦¥¨§©¥§¥¥
£
!#%$')(10'2
)10¥2436587@9A B4365TV%v4UWXg‚YG `uPqp–GTV IˆGTH@P8IRGbpwSYV s%‚bX…•V%U•“p(‚4G87–GT˜d‚bPqV S4X9£¦¥4@tHwV%GoPqp(†TVUeIˆ‡wV%G‰ p@P ceX c£I ‰ATp–G SYVbc
vbPqVYs%‚bXdc#SYX(`TXdc£VbcCBCDFE ¥¥¥HG hIBQP)ESR¥¥¥TG UWXdc„`TX@H¥‰ I wTp–“XtG‰‘Xe!U#“ƒvTVbcsXdc•‰V7tG ‚4GbIqSYVbca“XwSTIqp–G‰‘XC‚YG
Hwp–vbPqXYeW VbctHwV4Xw…„IqH@I8X–G‰‘XdcaSYXƒUeV‡wp–“CIqX–GY‰‘V—X–GY‰QUWXCP8VbcvvbP8VYs%‚TXdc1h X@P!`bPqp–GTV IRGTH@PQIRGTpwSYV ceV%G
µ
D EX¥Y h
µ
P EX¥Ya`Y
UWXdc„`TX@H¥‰ I wTp–“XtG‰‘Xe r#X¥‰‘X–U„“CIRGTp–UCPqp ‰ X–G c£IcbG SYX@PtHwp–vuPqXhPqpdcC… ‚TXtU„‡wpdcCSYX UWVY‡wp–“CIqX–G‰ V”s‚TXnpwH¥‰dATptG c£V%vYUWX PqVbc
vbPqVYs%‚bXdc€e
B D
egfgh
B P
44. 6UeV%vbPqX–“p%chS4X f1c•‰V7¥‰ I8Hwpe
¦
!#%$')(10'2
)10¥2436587@9AB4BCEsGTpkvTp–U„UWpk‚YGbIq…„V%U•“XƒSYXk`TX%ceV¢¡£¡¥¤R G h PqV%GT˜YI ‰{‚TS§¦ Xdc•‰©7kptUy‰ IqH–‚uPqplX–G c„‚X¥¨Y‰QUWXt“V© h …•V%U•“p
‚YG 7–GT˜d‚bP8VƒSYX £¦¥4@ HwV%GCPqpu†bV%UeIˆ‡wV%GY‰‘p@P e¦5Y‚X¥¨Y‰QUeX–“V ceXap–`TV’hup#ceV%vYUWXP8p#c„‚Y`TX–Ue…„IqH@IqXPQI{cepaSYXv‚YGlvbPqV4s%‚TXtSYXv`TXdceV
£ ` G Yzc£I ‰Q‚bpwSYVoceV%v4UWX(‚YGTp c„‚4`TX–UW…„I8H@IqXIRGTH’P8IRGTpwS4p(U•‚T˜Vbceps‚TX…•V%U•“p(‚ G87–GT˜d‚bPqV S4X ¦`g@tHwV%GoPqp(†TVUeIˆ‡wV%G‰ p@P e
r#X¥‰‘X–U„“CIRGTp–U Pqp UWXwp@HwH@Icb%GqX–G Pqpp–U•‰ IqH–‚bP8pwH@Icb%G hqX’PHwV4Xw…„IqH@I8X–G‰‘X SYX UWV‡wp–“CIqXtG‰‘V “RGbIR“V `Tp–UWp s%‚TX †Tp hup
Xws%‚bI8PQIRvYUeIqV6e
©
h
egf h
45. @e¢¡jp–Uy‰ IRG¤£¦¥¨§©¥§¥¥
!#%$')(10'2
)10¥2436587@9A B¡ gE#GTpXdc£Hwp@PqX–UepSYXF`TXdceVp ceXFpt`TV@hupFX–G—‚YGTpl`TptUWXwS(wTX–U•‰ IqHwp@P P8I{cepsh c£V%vYUWXl‚4G c„‚TX@P8Vg†TV%UeIq‡wV%G‰‘p@P
U•‚T˜VbceV6e f PHwVYXw…„I8H@IqX–G‰ X SYX UWV‡wpt“CIqX–G‰‘ViX%c
µ
E ¥Y RTe Wyp S4I{c„‰‘p–GTH@I8p X–G‰QUWX `bX@PqSYp£¢bVbcCXdcCSYX £¦¥jHt“e!EsGTp
`TX–U™ceVGTp SYXn`bXdceV fp%ceH@IqX–GbSYXn`TV%UgPqp XdceH@p@PqX–UWpYe r#X–‰‘X–U•“CIRGbp–UF†Tpdc„‰‘p s%‚TX `TX@PqSYp¤¢TV”`Y‚TXwS4Xoc„‚YvbIRUgc™IRG s%‚TX Pqp
XdceHwp@P8X–UWpsceXtHwp@Iq˜pYe
46. 6UeV%vbPqX–“p%chS4X f1c•‰V7¥‰ I8Hwpe
R
!#%$')(10'2
A¡
α
=+α ¢¤£¦¥
§¦¨©
f PqX¥wTptGTSYVlp@PbH–‚TpwSUWpwSYV¦Y@hHwV“V
3!
)10¥2436587@9A B#gE#G ceX–“CI8H@I8P8IRGbS%UWV—S4Xƒ`TXdc£V%$ EXR¥ G h UWp@S4IqVCr Y!c£Xp–`TV’hupgceV%v4UWXƒ‚YGbp(c„‚Y`TXtUW…„IqH@I8Xl†TV%UeIq‡wV%G‰‘p@P
U•‚T˜VbcepYeW pƒvTp–U„UWpƒ†TV“VY˜'–GTXwpCSYXƒ`bXdceV B E ¡ ¥¥ G h PqV%GT˜YIR‰Q‚TS –r Y Xdc„‰V7Fp–Uy‰yIqH–‚bPqp@SYpFX–G cW‚ X¥¨Y‰QUeX–“V h ceX
p–`TV@hupceV%v4UWXƒPqpc„‚Y`TX–Ue…„IqH@IqXƒP8I‘ceplSYX@PhceX–“CIqH’I8P8IRGTSUWVgX–Gg‚YGg`Y‚YG‰‘V)( s%‚TXkS4I{c„‰‘pur SYX@PgX¥¨Y‰{UWX–“V©ke%f P!c£I{c•‰‘Xt“pƒceX
“p–G‰ IqXtGTXX–G Xws%‚uI8P8IRvYU£IqV X–GoPqp(`TVuc£IqH@IcbGoIRGTS4IqH@pwSYpe r#X¥‰‘X–U„“CIRGTp–U10!p32aX’P wTp@PqV%UsSYX’PF7–GT˜d‚bPqV
α 4
v52tS4IRv4‚ x€p–UX@P
S4Iqpw˜dUWpt“p#SYX’P c bTP8IqSYVgP8IRvYUeX#SYXlPqpsvTp–U•Uep
4
H32 X@PbwTp@P8V%U SYXkPqpaGbV%U•“p@P X–G6(ohSYX#P8paUWXwp@HwH@Icb%GgX–GgP8psp–Uy‰yIqH–‚bPqp@H@Icb%G
4
S#21Pqpƒ… ‚TXtU„‡wpƒSYXkUWV‡wpt“CIqX–G‰‘V c£V%vYUWXlX@P H@I8PQIRGTS%UWVƒh X@P6wTp@PqV%Uz“RGuIR“VgSYX@P HwVYXw…WIqH@IqX–GY‰‘XlSYX#UWV‡wp–“CIqXtG‰‘V
µ
`Tp–UWplX@P
Xws%‚bI8PQIRvYUeIqV6e
©
(
r
7 E £¦Y ¡1 r
α
α
(
α
r
8@9
A
−
9CBED
α F
8HGPIB
α
QR SUT
α VXWCF
α
Y`Pab
c
de f QR S T
α −
QgRS
α −
e F
f
α
E £¦Y @
hpi
q
rts
α
uwvPx€y‚
β
q
h i
r s
ƒXI S¡„U… G …‡†ˆSH„ D‰DPˆ‘ …‡’ …€“”•… I
de f QR S T
α −
QgRS
α −
e F
f
47. @e¢¡jp–Uy‰ IRG¤£¦¥¨§©¥§¥¥
¤
H32r#XaPqptHwVGTS4IqH@Iab%GFSYXtXws%‚bI8PQIRvYUeIqVceX ‰ I8X–GTX
¡£¢¥¤ ¦¥§¨©¥¢¤§!!
#$
β−
==
β
%'
)(10!2 7’9A4367vDF7£576 A85 ( 2@9 7 0 5w2A9 3 ( 9B5£C8D@9 (12 A 0FEHG ˜d‚Tp@Pqp–GbSYVlpuH@X–UWVƒPqVbc!“V%“X–GY‰‘Vbc!UWXdc„`bXwH¥‰‘VlSYX ceX ‰ IqX–GbX
IPQ RSUTWV
=α+− X
Y`Y a b
cedAfhgpiWqAr8s4tvuAf£r@i wyx€ƒ‚v„†… ‡
β ˆ‰A‘8’‘v“
‡ ”–•€
‘8—
…
˜d™fehg i£i¥gpihjkier8f7lBmUnAl g r8sol gp gpfer8f7lBmUfhj grqg l g qArsgps
e ftlri q svqAr g1q7g nvqur8fef£tvu g l gp g g skmUfHv g fhqAfetvu)fel1ixwyuAfhgz£ier8f{gpssz£iWj g fhqsv|s4f¥m~}€wyxymUfhqu
‘8’1‚7“
ƒ@„x…
„ ihgpiof7l†…Ailrsvg‡j‰ˆq g jksŠr8f7l q s8f7w grq7g fWqsv|for8fhgps8z£ihj g fhqsv‹s
’
lri‰m g vŒuAi qg nvqff¥mHr8fjks¥… g j g fhqsv|s g q8j g qAfhqsv|ftŽlri4w‹uAfhgpz£i
r8f{gpssz£ihj g fWqsv|s q u8jod)l1f}€
µ
wu
e ftlri q svqAr g1q7g nvqur8fef£tvu g l gp g g skmUfHv g fhqAf‘f7l8…Ai7lrsvg–r8ftlri{qAssgjki7l
’
w’”“„8„x…
µ
•„
’
–
—
˜
—š™
›
›œ
ž
Ÿ ¡£¢F¤
¥§¦h¨v©7ª«¬ªh¨e©®¯®°h±¥ª²ª£³´yª¦
α
48. „ gUs p lrfhjkivm–r@f¡ šmv£¢v g1q is
¤
‚
¥§¦©¨¡ —
„ ihgpi4lri dAs)m grqg nvq g qAr grq i7r8ilritv|i p l1itqAs‰f¥mv£¢fhq q ssqAr grq7g svqAf¥m‡r8ftjks¥… g j g fhqsv|s g q8j g q)fhqsv|fs –q f7lBiWdAs7ŽdskfWq!
’
i£r8fhj#¢¥m r8ftlri{qAsvgjkilAi q v%$Aitlriew‹uAfWgz£ier8f{gpssz£ihj g fWqsv|sofr g g g i g r@i¡)i q7g itlrier8fhgUf q Ais
')(103254607¡098A@CBADCEFBG(18IH¡BG(QPF4R@S8A@TBU0WVX0FPABX0 Y`HFBGa2bHc5HW8I@deI@64£fg0gD
h
e f7l@dAs@l1ˆ1issvqAs4r8few‹uAfhgpz£i¥m mUfer8f£rvu q f g q8jkf£r g ihv|ihjkfhqsv‹fxfls…Ailrsvg–r8ftlriew‹uAfhgpz£ixr@fHgUssz£ihj g fhq8v|s4fhquf7lBmuAf7lrs
α= iprqstu v “3w
x itqAssgjki7lBfWq€y wsvgjki q svq lrie…Afhgyv grq i7l)u8q¢WqAi¥u)lrs
α
tvuAfyfvm f7l)j g mpjksutvuAfywsvgjki lriev|i p lri q svq‰f7l muAf7lrsƒ‚Ar8f lrsdm
r8iv|s)m mUfur8f£rsu q f mu …Ai7lrsvg
’ α
‘8’1‚¥—v“
e fklri mpu8jkiur8f4jkssjkfhqsv|s)m{gpf¥mdAf q v|s r8f„! g i¥uAi7l–i q fhgps mUfus p v g fhqAfkl1i
f q uAi qg nvq
α+= …‡†5ˆ‰ ‘9’“‘S”•‘9–—™˜
dfeggbhjilkAmonqpbrts)uwv9xwy‡zC{}|5uwv~§yWxwy5{}|q€5w‚9ƒ„{†…§‡wy){}|5‡bzyW…§u‰ˆ£v‡‡‹ŠGŒu‰Ž‹|5#|§‘‡uwyA|‰~byWj’A“ƒ”~by¡x‹y {}|)ˆ£vAŠ‹•F|5#|
{}yq…wy5{–ˆ%zvty‡u—ŠGvt˜†™G …§z}v)vF~6šr…§uˆ£v§›Iœ‰ygˆ£y‡z–™Xuwv‡z9wvFžTŠlvŸ˜ …wy‡z¡FvŸ~§y©z|§¡Fv‡™Gy‡uˆ |y‡uyAŠ¢{†…wyAŠG|#•‡…wv‡uw~b|#ŠGv©xwyz6{}|5uwv
{}ytyuw•‡…wy‡u§ˆ%zv)xwv‡zvF~bvtv„‚§£U¤q¥~§yAŠ¦yg§§ˆ%zy#|©¨£©{6™©yAŠ¦v‡xw|Iª«|y‡uo¬{}yt•A|5uƒ{R™G~by‡zv„Šl™®{}|«¯€‡wž°{R™jyAŠ©•F|§yF˜™G•A™Gy‡u§ˆ£y)~§y
z|¡Fv‡™Gyuˆ£|y‡u¨±ª²¬³y { µ ´ ‚§£1«µ‰~bygˆ£y‡z–™Xuwv‡zSŠGv)~b™®{–ˆ£vuw•A™Gv©#¶‡§«™U#v9·9vŸŠlv¸5…wy9x§…wyA~§yt{†…§‡‹™XzTŠGv9xwy‡z6{R|5uwvt{R™Uu
¸5…wy©yAŠ§ˆ£v‡‡‹ŠlŒ5u~§y {RŠl™l•Fy›
α
¹
d
º »
º ¼
½
¾
¬
‚b£¿¤)
º »
d
½
º ¼
¨
À
Á
Â
Ã5ÄÅWÆ
¨
Â
¬
ÇCÄ1ÈÆ
É ÄËÊfÆ
49. A›¢¡ v‡z ˆ ™Uu¤£b‚¦¥ ‚¨§©¥®w‚§‚‹µ
Ҥ
œty ŠGv ˜†™G …§z}v {}y ˆ ™Gy‡uwy ŠG|‹{wvAŠG|5zy5{ ¦
!
== $# %
xwyzv‡uw~§| ¸5…wyF~bv xwv‡z}v ŠGv uw|5z–#vAŠ)y‡u™yIŠ
v‡xw|Aª«|±ƒ'
´ «µ)(g£Gµ ƒo›Wœ‰y©Šlv yF•‡…‹vF•A™GŒ5u102(gž {Ryfˆ ™Gy‡uwy ˜ …wy‡z†¡Fv©~§yWz|§¡Fv‡™Gy‡uˆ | 3465©4
=
7 ›
œtyAŠ¦xw|bŠ98G|5uw|²~§yq˜ …wy‡z†¡Fv {{Ry‰|‡ˆ ™Gy‡u‹y„™Xu§#yF~b™Gv‡ˆ£v‡#y‡uˆ y„Šlv)u‹|5z–#vAŠ¢y‡uA@ BC©D¨EC
F
DGHDIDQP
=−= RS TT
›VU«™
yAŠ«•F|§yF˜™G•A™Gy‡u§ˆ£y ~§y9z|¡Fv‡™ly‡uˆ£|qy‡uyAŠj{†…‹yAŠG|„y { µ ´ ‚b£Q«µ£yAŠ¨wvAŠG|5z#¶g§ƒ™U#|„~§yŠGv©˜ …wy‡z†¡Fv©~§yWz|§¡Fv‡™Gy‡uˆ |‰y5{ ‹‚§‚§£U€
ƒ ª€ŠGvWxwy‡z6{}|uwvWx§…wyF~§y){}yA …‹™UzS{†…§‡‹™ly‡uw~§|txw|5z`ŠGv¡ˆ£v‡‹ŠGv){R™Uu¸5…wy©y {†ˆ£v©~§y {RŠl™l•Fy›
hXW`Y uoŠGvŸx‹|‹{R™G•A™lŒ5u²Š98U™¿ˆ£yA£«ŠGvˆ£v‡‡‹Šlv)y {–ˆ ¶)y‡u {R™¿ˆ%…wvA•A™GŒ5u—~§y#|©‹™X™Gy‡uˆ£|€™Uu§™Uu‹y‡uˆ£yA£5ªoŠlv {˜ …‹y‡z†¡Fv {°~§yz|§¡Fv‡™Gy‡uˆ |
ˆ ™Gy‡uwyu {…awvIŠG|5z©#¶g§«™U#|w£ µ
xw|zŠGv#uw|5z†#vAŠ ›cbTv€˜ …wy‡z¡Fv~§y#z|¡Fv™Gy‡uˆ£| y‡ued ˆ ™Gy‡u‹y€ŠGv€~b™Uz}yF•F•A™GŒu ~§y—ŠGvˆ£v‡‡‹Šlv
~b™Uz}™G§™l~§vWŽwvF•A™Gv©v‡z†z}™U‡wv›
Áfhgpicqrg¨stgvuXwyxb·H€¨xf‚uXƒ„x…fh†Qq‡wˆuw‰xg'‘g’†’xg “”ƒ¨x• i–ƒQ—ƒvuwI˜I™¨w¢qed¢g5·
f …‹vAŠGv‡uw~b|‰v©•Fyz|qŠGv){†…§#v©~§yW#|5#y‡uˆ |‹{ zy {†xwyA•gˆ£|„~§y!@ £w{}y¡ˆ ™Gy‡u‹y
gih
j
k
gml
j
n
ogmhkojgmln p
p −
α
=
q
α+= rpsXt
rpsXtuv
œtyŸŠGvŸŠGyrª~§yAŠj{}yuw|„v‡x‹Šl™G•AvF~§v©vAŠ§ˆ%z}™G¶uw …‹ŠG|„~§y©˜ …wyz†¡Fv {`˜–|5z–#vA~§|txw|5z`ŠG|‹{$wyF•gˆ |5zy { dxw ½zyt{i| £ { ª„ˆ£yu‹™Gy‡uw~b|‰yu
•‡…wy‡uˆ v9¸…wy }~2V€h‚X€„ƒ)…‚†‡yˆ
====
‰
==== ŠQ‹Œp¨Ž
’
‘
’¨“¨’
”•
–¨–
‘
—˜
•‡—’
‘
•H’™›š2œ•‡—
‘
˜¨’™›šHœ =
ž
+
=
ž
=
ž+
= Ÿ
Ÿ
ŸŸ
Ÿ
¡
¡
¢£¡
¤
‹V¥§¦…¨h¦9‹©©cŒ2ª’«m¬Ž¦«Q®eŒ©¥¯¥6Œ!¦…¨°Œ2ªŒ ¥ = ±H²„³I´„µ
Entramados y armaduras
¶ · ¸
µ ¹
·
º
» ·
» ¼
½
θ
¾ ·
¾ ¼
θ
¶ ¼ ¸
µ ¹
¼
α
¶ ·
¿
À
» ·
º
½
» ¼
Á
Â
¶ ¼
Ã
50.
®6«¢¡XŒ µ ŽQ¥ ’Œ¤£ ¥§¦¦¥H¦…¨¨§yŽ©
!¨#%$
º')(1032¨46587@9BA
£ ª ŽaŒQ¥§¦9®§‹C§2¦9‹¨®eŽ ¨Œ ŽEDe¨GF©‹¨®eŽ ² ’ŒH¦Œ2® µ ¨ ªŽ2®IH ŽQP Ž©¥ ®eŒyŽQ§Q§¦¨«QªŒ©¥'Œ2ª °Ž Ž2® ¦ ¨G§2‹XŽ6§p¨GRQª
Á
© Ž
ª«Q® µ ŽpcŒ¢ªzŒpcŽTS«p©c«VUXW¡YP Ž©¥`D…‹Œ¢®bayŽ©¥ ŠQ‹Œ‰ŽQ§H¦dcŽ2ª1Œ2ªzŽ©¥e¡Ž2® ®eŽ©¥
Ái¿ fpÁ Ã fBg
U
f
U
Ã
© ¥›¨¨¦9®eŽT¡XŽih‡Ž2ª1Ž!¦9®eŽ6§Q§p¨GRQª
«„Žp§y« µ S¨®eŒ©¥›¨¨RQªq©
r s
t
u
£
v
±
©
1Bxw
F
y
µ
´vµ ´„µ ´vµ ´„µ
±
©
1Bxw
F
±
©
1Bxw
F
51.
©¢¡ Ž2® ¦…¨hª ³
¤£ ¦¥§£©¨CBY
±
! »
º')(1032¨46587 9¦# u
ŒH¦Œ2® µ ¨hªŽ2® Ž D…‹XŒ2®bayŽ ŠI‹Œ ŽQ§2¦ cŽ Œ¢ª §yŽy’Že‹¨ªŽ ¨Œ °Ž©¥ ¡Ž¢®§®eŽ©¥ ¨Œ Ž Ž2® µ ŽyI‹¨®eŽ Žp h ‹¨ª¨¦Ž ²
¨hª’¨G§pŽ2ª¨«z¥›¨¨¦9®eŽ¡Žih‡Ž¢ª'Ž!¦9®6ŽQ§Q§p¨GRIª'« Žp§y« µ S’®eŒ©¥›¨GRIªq©
©
B1%$
r
s
t
u
£
±
©
BB'$
±
¨
µ
µ
µ
µ
52.
®6«¢¡XŒ µ ŽQ¥ ’Œ¤£ ¥§¦¦¥H¦…¨¨§yŽ©
§¨
º')(1032¨46587¡ £¢ u
ŒH¦Œ2® µ ¨hªŽ2®iŽ D…‹XŒ2®bayŽzŠI‹ŒzŽQ§2¦ cŽzŒ¢ª §yŽy’Ž ‹¨ªŽ1’ŒzŽ©¥¤¡XŽ2®§®eŽ©¥‰¨ŒzŽ1Ž¢® µ ŽyQ‹¨®6Ž1¨ŒzŽ D ¨GF©‹¨®eŽ
Žy h ‹’ªI¦Ž”© ¥ ¨I¦ ®eŽT¡ŽihHŽ2ª1Ž!¦9®eŽQ§Q§¦¨GRQª1« Ž §p« µ S¨®eŒ©¥ ¨GRQªq©
53.
©¢¡ Ž2® ¦…¨hª ³
¤£ ¦¥§£©¨CBY
³
! »
s
t u
r
´ ©
1B'$
y
µ
µ
¨
µ
³ µ
55.
©¢¡ Ž2® ¦…¨hª ³
¤£ ¦¥§£©¨CBY
1
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
