ÍNDICE
INTRODUCCIÓN _______________________________________________________2
OBJETIVO _____________________________________________________________ _____________________________________________________________22
UNIDAD 3, CINEMÁTICA DEL CUERPO RÍGIDO _____________________3
3.1 TRASLACIÓN DE CUERPO RÍGIDO _____________________________5
3.2 ROTACIÓN ROTACIÓN ALREDEDOR DE UN EJE FIJO ______________________8
3.3 MOVIMIENTO PLANO GENERAL ______________________________1
BIBLIOGRAFÍA ________________________________________________________22
_________________________________________________________23 _____23 CONCLUSIÓN ____________________________________________________ 1
2
INTRODUCCIÓN
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
OBJETIVO
A6!#'(% A6!#'(%6/$ 6/$ *)%$0(/$ *)%$0(/$ /*/#6#% /*/#6#%*0/$ *0/$ '%(' "% !' CINEMÁTI CINEMÁTICA CA DEL CUERPO CUERPO RÍGIDO '(' '(%*"%( ' '!)!'( !'$ :)%(4'$ ;)% $% %*%('* ")('*0% %! 6/+#6#%*0/ %* 6';)#*'(#' 6%'*#$6/$, '$- /6/ !/$ '(96%0(/$ #*%690#/$ !' %*%(-' 3
6%9*#' ;)% $% (%$%*0'* ")('*0% %! 6/+#6#%*0/ "% !/$ )%(/$ (-#"/$ ;)% /"'6/$ /$%(+'( %* %! 0('$)($/ "% *)%$0(' '((%(' !'/('!. OBJETIVOS PARTICULARES T('0'(%6/$ "% '6!#'( *)%$0(/$ /*/#6#%*0/$ (%'!#4'( !/$ "#+%($/$ (/!%6'$ $/(% )%(/$ (-#"/$ (-#"/$ ;)% /"'6/$ /"'6/$ /$%(+'( )0#!#4'*"/ !/$ !/$ /*%0/$ "% 0(''?/, 0(''?/, %*%(-' '*0#"'" "% 6/+#6#%*0/. P(#*##/ "% 0(''?/ !' %*%(-'. E*%(-' #*0#' %*%(-' /0%*#'!. C'*0#"'" "% 6/+#6#%*0/ !#*%'!. C'*0#"'" "% 6/+#6#%*0/ '*)!'(. I6)!$/ '*0#"'" "% 6/+#6#%*0/. C/;)%$ %*0(#/$.
CINEMÁTICA DEL CUERPO RÍGIDO
L' #*%690#' "%! $7!#"/ (-#"/ %$ )*' '!#'#7* "% !' #*%690#' #*%690#' '! '! 6/+#6#%*0/ "% )* /?%0/ 0(#"#6%*$#/*'! (-#"/ %* %! %$'#/. E! 6/+#6#%*0/ 69$ %*%('! "%! $7!#"/ (-#"/ )%"% /*$#"%('($% /6/ !' $)%(/$##7* "% "/$ 0#/$ "% 6/+#6#%*0/ 9$#/$ "% 0('$!'#7* 0('$!'#7* "% (/0'#7* (/0'#7*..
C/*%0/ "% $7!#"/ (-#"/ E*0%*"%6/$ /( $7!#"/ (-#"/ )* $#$0%6' :-$#/ %* %! ;)% !' "#$0'*#' %*0(% "/$ )*0/$ 6'0%(#'!%$ )'!%$;)#%(' "% %!!'$ %(6'*%% #*+'(#'!% %* %! 0('*$)($/ "%! 0#%6/ 0#%6/.. L/$ )%(/$ $7!#"/$ ;)% 6'*%?'6/$ $% "%:/(6'* $#%6(%, %* 6'/( / 6%*/( ('"/, )'*"/ %$09* $/6%0#"/$ ' !'$ '#/*%$ "% !'$ :)%(4'$ :)%(4'$@@ $#* %6'(/, 4
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;)% %$ %;)#+'!%*0% , ' ;)% !' ('-4 )'"('"' "% )*' /*$0'*0% %$ /0(' /*$0'*0%.
L' /$# /$## #7* 7* "%! "%! $7!# $7!#"/ "/ /* /* (%$ (%$% %0/ 0/ '! $# $#$0 $0%6 %6'' "% %? %?%$ %$ //( //("% "%*' *'"/ "/$$ ;)%" ;)%"'' %(:%0'6%* %(:%0'6%*0% 0% "%0%(6#*'" "%0%(6#*'"'' $# /*/%6/$ /*/%6/$ !' /$##7* /$##7* "% 0(%$ )'!%$;)#%(' )'!%$;)#%(' "% $)$ )*0/$, */ '!#*%'"/$, /6/ !/$ )*0/$ 1, 2 3 ;)% $% #*"#'* %* !' F#)(' 1. P'(' %$ %$% %# #:# :#' '(( !' /$# /$## #7* 7* "% ' '"' "' )*/ )*/ "% %! %!!/ !/$$ $% *%% *%%$# $#0'* 0'* 0(%$ 0(%$ '(9 '(96% 6%0( 0(/$ /$ / //("%*'"'$@ "% 6/"/ ;)% %* 0/0'! *%%$#0'6/$, ''(%*0%6%*0%, *)%+% '(96%0(/$ / //("%*'"'$ '(' %$%#:#'( !' /$##7* "%! $7!#"/ %* %! %$'#/. L/$ 0(%$ )*0/$ ;)% %6/$ 0/6'"/ /6/ (%:%(%*#' %$09* !#'"/$ /( !'$ /*"##/*%$ "% (##"%4 %(%$'"'$ /( 1@ %$0/ %$, 0(%$ %)'#/*%$
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
C/*"##7* #*%690#' "% (##"%4 P'(' P'(' "%$( "%$(# ##( #( %! 6/+# 6/+#6# 6#%* %*0/ 0/ "% )* $7!# $7!#"/ "/ (-# (-#"/ "/ "%% "%%((-'6 '6/$ /$ "%$ "%$(# (## #(( %! 6/+#6#%*0/ "% '"' )*/ "% !/$ )*0/$ / '(0-)!'$ 6'0%(#'!%$ ;)% !/ /*$0#0)%*. L' $#0)'#7* )%"% '(%%(*/$ "%6'$#'"/ /6!#'"' %(/, ':/(0)*'"'6%*0%, !' 5
(/#' /*"##7* "% (##"%4 #6/*% #%(0'$ (%$0(##/*%$ '! 6/+#6#%*0/ "% !/$ "#$0#*0/$ )*0/$ 6'0%(#'!%$ "%! $7!#"/, "% 6/"/ ;)% !' $#0)'#7* $% $#6!#:#' %*/(6%6%*0%. P'(' '"' '(%?' "% )*0/$ %(0%*%#%*0%$ '! $7!#"/ (-#"/, !' =P #,P ?> /( %?%6!/, /"%6/$ %$(##( !' /*"##7* %/60(#' "% (##"%4, %$0/ %$, !' %. 1.1, ;)% "%(#+'"' /* (%$%0/ '! 0#%6/ */$ /*")% '
=1> /"%6/$ %$(##( %* !' :/(6'
;)% 0'6#*
=2> "/*"% ( ##?? +#? (%(%$%*0'*, (%$%0#+'6%*0%, %! +%0/( "% /$##7* !' +%!/#"'" "% !' '(0-)!' P # /* (%$%0/ ' !' P ?. L' %. 2 %(%$' )* (%$)!0'"/ #6/(0'*0% '! */ $%( *)!/$ *#*)*/ "% !/$ +%0/(%$ ;)% #*0%(+#%*%* %* %! (/")0/ %$'!'(, '* "% $%( %(%*"#)!'(%$ %*0(% $-. D#/ "% /0(/ 6/"/ todo vector con sus extremos fijos en el sólido sólido rígido rígido =' ;)% %! ( ##?? %$ +9!#"/ '(' )'!;)#%( '( "% )*0/$ perpen endi dicu cula larr a su deri deriva vada da con con /*$0# /*$0#0) 0)%*0 %*0%$ %$ "%! $7!#"/ $7!#"/>> es perp respecto al tiempo =#.%., ' +#?>. L' %. 2 )%"% %$(##($% %* !' :/(6' =3>
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=> %)'#7* ;)% %(%$' !' #)'!"'" %*0(% !'$ (/%#/*%$ "% !'$ +%!/#"'"%$ "% !/$ )*0/$ P # P ? $/(% !' (%0' ;)% !/$ )*%. E$0% (%$)!0'"/ /*$0#0)% !' /*"##7* #*%690#' "% (##"%4 ;)% (##"%4 ;)% $% %*)*#' '$- Las velocidades de los puntos alineados pertenecientes al sólido rígido dan la misma proyección sobre la recta que los une.
M'*#:#%$0'6%*0%, !' condición cinemática de rigidez %(%$' %(%$' !' #6/$##!#"'" "% ;)% $% 6/"#:#;)% !' "#$0'*#' %*0(% "/$ )*0/$ )'!%$;)#%(' "%! $7!#"/ %* %! 0('*$)($/ "%! 6/+#6#%*0/ 6/+#6#%*0/ "% $0%, ' ;)% '! $%( $#%6(% $#%6(% $)$ +%!/#"'" +%!/#"'"%$ %$ #)'!%$ %* !' (%0' ;)% !/$ )*%, %$ #6/$#!% ;)% '!)*/ $% '%(;)% '! /0(/. E! 6/+# 6/+#6# 6#%* %*0/ 0/ 69$ 69$ %*% %*%(' ('!! "%! "%! $7!# $7!#"/ "/ (-# (-#"/ "/ )%" )%"%% /*$ /*$#" #"%( %('( '($% $% /6/ /6/ !' $)%(/$##7* "% "/$ 0#/$ "% 6/+#6#%*0/ 9$#/$ "% traslación "% rotación.
3.1 TRASLACIÓN DE CUERPO RÍGIDO M/+#6#%*0/ "% 0('$!'#7* E! 6/+#6#%*0/ "% 0('$!'#7* %$ %! 69$ $%*#!!/ ;)% )%"% (%'!#4'( %! sólido rígido . D%$"% )* )*0/ "% +#$0' %/60(#/, !/ /"%6/$ "%:#*#( "%! 6/"/ $#)#%*0%
6
Se dice que un $7!#"/ encuen entr tra a anim animad ado o de $7!#"/ (-#"/ (-#"/ se encu un 6/+# cuand do tod todo seg segmento ento 6/+#6# 6#%* %*0/ 0/ "% 0('$ 0('$!' !'# #7* 7* cuan rect rectil ilín íneo eo defi defini nido do por por dos dos punt puntos os de aqué aquéll perm perman anec ece e paralelo a si mismo mismo en el transcurso del movimiento. movimiento.
C/*$#"%(%6/$ )* $7!#"/ (-#"/ '*#6'"/ "% )* movimiento de traslación, /6/ $% 6)%$0(' %* !' F#)(' . E* +#(0)" "% condición ión geomét geométric rica a de rigide rigidez z , %! +%0 !' condic +%0/( /( ( ##?? ( #H( ? "%% 6'*0%*%( /*$0'*0% $) 67")!/ %* %! 0('*$)($/ "% )'!;)#%( 6/+#6#%*0/ , '"%69$, %* +#(0)" "% !' "%:#*##7* %/60(#' movimiento to de traslació traslación n, 0'6#* ' "% 6'*0%*%( /*$0'*0% $) "#(%#7*@ "%! movimien %*0/*%$, $#%*"/ )* +%0/( /*$0'*0%, $% )%"% %$(##( =1>
"%(#+'*"/ /* (%$%0/ '! 0#%6/
=2> /*$0#0)%*"/ %$0' #)'!"'" !' /*"##7* #*%690#' "%! movimiento de traslación, %$0/ %$ odos los puntos de un $7!#"/ (-#"/ animado animado de un movimiento movimiento de traslación traslación tienen! en cada instante! la misma velocidad.
E$' +%!/#"'", /6* ' 0/"/$ !/$ )*0/$ "%! $7!#"/, (%#% %! */6(% "% velocidad de traslación "%! $7!#"/ "%% $%( /*$#"%('"' /6/ )* +%0/( !#(%. !#(%. L'$ 6#$6'$ una vez vez /*$#" /*$#"%(' %('#/ #/*%$ *%$ )%"%* )%"%* '!#' '!#'($% ($% ' !' '%!%('#7* '%!%('#7*.. E* /*$%) /*$%)%* %*#', #', una definido el movimiento de un punto cualquiera del sólido rígido que se traslada! tenemos definido el movimiento del sólido .
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velocidad ,
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Si un sólido rígido está animado de varios movimientos simult simultáne áneos os que que origin originan an veloci velocidad dades es v&! v'! ... en un punto genérico $ del sólido! la velocidad resultante v de ese punto genérico es la suma vector torial de las velo veloci cida dade des s que que le corr corres espo pond nde e en cada cada uno uno de los los movimientos componentes por separado.
O0(' :/(6' "% %*)*#'( %! (#*##/ "% $)%(/$##7* %$ !' $#)#%*0% Si un sólido rígido está animado de varios movimientos simultáneos! para cada uno de los cuales se cumple la condición condición cinemática cinemática de rigidez! rigidez! el movimien movimiento to resultante resultante también cumple cumple esa condición.
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(l resultado de la superposición de dos o más rotaciones simultáneas cuyos ejes concurren en un punto es igual a otra rotación cuyo eje pasa por dic%o punto y cuya velocidad angular es la suma )vecto )vectori rial al** de las las velo veloci cida dade des s angu angula lare res s corr corres espo pond ndie ient ntes es a las las rota rotaci cion ones es componentes.
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