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7/21/2019 Mov Amortiguado - Pendulo Compuesto

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UNSCH FISICA II

UNIVERSIDAD NACIONAL DE SAN CRISTBAL DE HUAMANGA

FACULTAD DE INGENIERA DE MINAS GEOLOGIA Y CIVILESCUELA DE FORMACIN PROFESIONAL DE INGENIERA CIVIL

PRCTICA N 8

MOVIMIENTO ARMONICO AMORTIGUADOPENDULO COMPUESTO

ASIGNATURA : FS 241 - FISICA IIPROFESOR : K !"!# $ANAMPA %UISPEALUMNO : BAUTISTA PA%UIYAURI& E#'( Y! )*+',

CCOLLANA MEDINA& M' .! A, !

GRUPO : M/#)!+: 0-11 /CICLO ACADEMICO : 2 13-IFECHA EJECUCIN : 10 11 2 13FECHA ENTREGA : 3 12 2 13

AYACUCHO5PERU2 13

P6 ',/ 1


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UNSCH FISICA II

I. TTULO: MOVIMIENTO ARMONICO AMORTIGUADO.

II. OBJETIVOS :

E+).7'/# ! 89' '!,)8 7! ., +'+)! / 8+ ' /,)!& ./,78 !;'+)!,.! +! 8?8,!, / 89' '!,)8@

III. FUNDAMENTO TERICO :

MOVIMIENTO ARMNICO AMORTIGUADO

U, 89' '!,)8 /# ,' 8 / 8#)' ./78 !+ />.! +'+)! / 8+ ' /,)!78,7! / / ? ').7 ,8 ?!# /,! ! 8,+)/,)!& 7! #! '!,78#/7./ !,)! */+)/ >.! ! +'+)! / +! 7!)'!,!& !,)8, !+ +.

)#/ ! )8#'/ ,8 )'!,! ')!+ 8+& 7!"'78 / /+


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UNSCH FISICA II#!+. )/78 !+ .,/


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UNSCH FISICA II

A*8#/ )!,! 8+ / 8,7' ' ,

W 02 = 2

C./,78 8 .##! !+)8& ! #/7' /,78 7! / ! ./ ' , /#/ )!# +)' / !+!#8& ?8# 8 >.! !;'+)! .,/ ,' / +8 . ' , 7! /

P6 ',/ 4


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UNSCH FISICA II.#/ >.! .!+)#/ / !98 . ' , 7! / 9! 8 '7/7 !, /7/ .,8 7! 8+/+8+@ C./,78 ! / 8#)' ./ '!,)8 !+ # )' 8& ! 7. 8 7! /

9! 8 '7/7 / 8# / / 8# ?/#)! 7! )'! ?8@ C8 8 ! #8=/ '!,)8 !+?#8?8# '8,/ / / 9! 8 '7/7& !+ 6+ !

RECIPIENTECON AGUA

CARGAS

RESORTE REGLAMETRICA


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UNSCH FISICA II

V. PROCEDIMIENTOS :

A. MOVIMIENTO OSCILATORIO AMORTIGUADO

. S" /*+)a *0 %* !%&* 1*%&',a0$*+&* 2 /a%a )' &'+&a /* a34/* a 5 $')a 0a *0!+(a,'6+ ,!%%* /!+)'*+&*. L"*(!)*&*%$'+* *0 1a0!% )* 7.O 9 .;,$

Ma a 37(5 D*

CRONOMETRO

BALAN A


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UNSCH FISICA II

( )( )

m N k m

smkg k

xmg

k

xk mg

xk F

/92.3025.0

/8.901.0

1

2

1

1

1

1

=

=

=

==

D* $a+*%a '$'0a% /a%a 0a !&%a $a a : * !#&'*+* 0a'("'*+&* &a#0a:

M/+/ ( D!


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UNSCH FISICA II

;. C!+ 0! )a&! !#&*+')! )*&*%$'+* *0 /*%'!)! *+ *0 a'%*2 0"*(! *+ *0 a("a 0a @11 3@4

;. . D*&*%$'+* 0a


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UNSCH FISICA II

E+ . ,! !+/#'8 )!,!# !, .!,)/ / .! ?/#/ 7'.! *! 8+ !7'78 !, ! /"8#/)8#'8&

!, )!8# / 8, / +' .'!,)!


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12/37

UNSCH FISICA II

O# *%1a,'6+:

E 8! '!,)! 7! / 8#)' ./ '!,)8 !+ >.'!, +! !, /# /# 7!7'+ ',.'# /+ / ? ').7!+ !, /7/ 8+ ' / ' ,@ P.7' ,78+!!, 8,)#/# ., ?#8 !7'8 7! / 8,+)/,)! 7! / 8#)' ./ '!,)8: "

0.45716

@. Pa%a "+a a$/0'&") '+','a0 )* ;,$ )*&*%$'+* 0a *,"a,'6+)*0 $!1'$'*+&! a%$6+',! a$!%&'("a)! 2 !#%* *0 $' $!/a/*0 $'0'$*&%a)! (%a "* A3&5 2 3&5.

cuandow 02> 2

2 w02= i w i esunacantidad imaginaria

x= e t [B e i w it +C e i wi t ]

aciendoB = A2

e i ! C = A2

e i

obtenemos

x= A e t [ ei (wi t +)+e i(wi t + )2 ]P/#/ ,.!+)#/ ?#6 )' / .+/#! 8+ = 0

x= A e t

[ ei (w i t )+e i (wi t )

2

]O#&*+*$! X = A e t cos (wi t + )

Da&! 4.;8 ;NK$

M/+/ ( F#! .!, '/ ,/).#/ C8! '!,)! 7!/ 8#)' ./ '!,)8

P6 ',/ 12


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UNSCH FISICA II

m= 0.05 x0= 0.05

4646.8

05.0/5825.3

0

0

0

=

=

=

W

kg m N

W

mk

W

= b2 m

= 0.457160.1

= 4.5716

La


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UNSCH FISICA II

X =( 5 )e 4.5716 t cos [(7.1239 )t ]

Ta#0a& 3 5 3&5=. 3@1 2=. 2@ 33=.4 1@2 Q=.> @ 22=.; @ Q=.@ @321=. @2 3

=.8 @12 4=.? @ 12@ 13

. @ 324

. @ 2

.4 @ 120

.> @ 2

.; @ 2

.@ @ 33

. @ 21

.8 @ 13I+&*%/%*&a,'!+:

S! ?.!7! /?#! '/# >.! / !7'7/ >.! ! )'! ?8 /. !,)! // ? ').7 7'+ ',. ! */+)/ ,/ !,)! 7!)!,!#+! !, +. ?.,)8 7!!>.' '"#'8 !, / ?#/ )' / !+)8 +. !7! 7!"'78 / / !;'+)!, '/7! 8! '!,)! 7! / 8#)' ./ '!,)8 ?!#8 !, / )!8#'/ 8 8 +!)#/)/ 7! .,/


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UNSCH FISICA II

D8,7!

!+ ! .! +! . ? / ! /+8 7! 8+ ' / '8,!+ / 8#)' ./7/+& +! 7!"!

. ? '# >.!

22

0 >& / >.! !, 8+ 7! 6+ /+8+ / ,8 !;'+)!,

8+ ' / '8,!+@

L.! 8& ?/#/ 7!)!# ',/# 8+ 9/ 8#!+ 7! / 8,+)/,)! 7! #!+)'). ' ,:

>

>

>

>

>

mbk

m

bm

m

k m

m

b

m

k

m

b

m

k

4

44

2

2

2

2

2

2

2

22

0

. U+ /*0!&6+ )* !0)a)! $a%, a a0 /a ! a 0! 0a%(! )* "+,a$'+!. P!% " * 0* !%)*+a %!$/*% 0a ,"a+)! ,%"Qa+ "+

/"*+&*E, ?#' !# . /# / %* !+a+,'a !+ ., .! +! ?#87. !./,78 ., .!#?8 /?/= 7! 9'"#/# !+ +8 !)'78 / / / ' , 7! .,/


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UNSCH FISICA IIXC./,78 ., ?! 8) , 7! +8 7/78+ !+)6 /# */,78 9/ / #.=/# .,?.!,)!& #8 ?! / /# */@ D! ,8 */ !# 8& 8+ 8 ?!+ >.! 7/, //# */# ?87# /, )!,!# 8 ?8,!,)!+ 8, .,/

E$EMPLOS DE OSCILACIONES AMORTIGUADAS:

- U, ! ! ? 8 7! ., 8+ ' /78# / 8#)' ./78 !+ ., .!#?8 .,'78 /., #!+8#)! +. !# '78 !, ., >.'78 9'+ 8+8

- %. !+ / +.+?!,+' ,Z L/ +.+?!,+' , 7! ., 9!* . 8 )'!,!8 8 8" !)'98 /"+8#"!# /+ '##! . /#'7/7!+ 7! )!##!,8 +8"#! !>.! +! 7!+? /=/ /,)'!,!, /+ #.!7/+ !, 8,)/ )8 8, !+.! 8 ?/#/ ?#8?8# '8,/# ., /7! ./78 ,'9! 8,


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UNSCH FISICA II

- U, ,' 8 !, ., 8 . ?'8 ?.!7! /,)!,!#+! !, 89' '!,)8!7'/,)! ! ?. 8,!+ /?#8?'/7/ !,)! +', #8,'=/78+@ L/

/ ? ').7 7! 89' '!,)8 ?!# /,! ! 8,+)/,)! +' ! +. ','+)#87! !,!# / ?8# ' 8 7! 89' '!,)8 ' ./ / !;/ )/ !,)! /#!7. ' , !, !,!# / ! 6,' / !, /7/ ' 8 >.! #!+. )! 7! /+.',/+ 9'"#/, ?8#>.! )'!,!, ?'!=/+ !, #8)/ ' , >.!,8 !+)6, ?!#.' '"#/7/+@ S' +! +. !)/ .,/ 6>.',/7! !+)/+ / .,/ !+)#. ).#/ >.! ?.!7! 9'"#/#& 7' */ !+)#. ).#/ +!8,9'!#)! !, ., +'+)! / .! ?.!7! ',' '/# +.89' '!,)8 ?8# / / ' , 7! / 6>.',/@

- P.!7! #8 ?!#+! ., 9/+8 8, "/ 8 / 8#)' ./ '!,)8 !7'/,)!.,/ 8,7/ +8,8#/ ',)!,+/ 8, .,/


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UNSCH FISICA II

2

21

kA E t =

E+ / !,!# / ',' '/' S' / / ? ').7 7'+ ',. ! !, [ 7.#/,)! /7/ ' 8& /

/ ? ').7 !, ! +! .,78 ' 8 +!#6:

A x

x

A

10095

%95 _ __________

%100 _ __________

=

x

E+ / ,.!9/ / ? ').7@

/

( ) ( ) 222

20975.0

21

400361

121

10095

21

21

kAkA Ak Ak E =

=

=

E ( )

20975.021

kA

!+ / 9/#'/ ' , 7! / !,!# /

" E .! !+ !)'! ?8 >.! )/#7/ / !,!# / !, 7'+ ',.'# !, ! .! 8& " !+ ?!>.! 8 !+ #/,7!& !8+ ' /78# ?!#7!#6 .,/


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UNSCH FISICA II

( )( )748.30

234.64

2===

QT

VII. OBSERVACIONES CONCLUSIONES :

OBSERVACIONES :

L/+ /+/+ >.! )8 / 8+ /+ 8,+'7!#/ 8+ 8 8 ?.,)./ !+& / >.!?/#/ >.! +! 7! !+)/ 8,7' ' , +! ,! !+')/ >.! / 8, ').7 7! /.!#7/ )'!,! >.! )!,!# .,/ 8, ').7 / 8#& / >.! 7! !+/ .! +', / ?#!+!, '/ 7!

.,/


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UNSCH FISICA II

?!#7'7/ !+ /"+8#"'7/ ?8# ! !7'8 '# .,7/,)! 8 #/7'/7/ 7! / .,//,!#/@

L/ /,)'7/7

!+ 8,8 '7/ 8 8 )'! ?8 7! #! / / ' ,& )'! ?8,! !+/#'8 ?/#/ >.! / !,!# / 7!+?. + 7! /7/ ?!# 878 +! #!7.= // 1 ! +. 9/ 8# / 8 '!,=8 7! ?!# 878 @

VIII. BIBLIOGRAFA :

F +' / II& H. "!#)8 L! 9/ N/9!#8+& E7')8#'/ M8+*!#/& +! .,7/!7' ' ,@F'+' / I& S!# / R/ 8,7& E7')8#'/ M G#/ H' &%.',)/ !7' ' ,@*))?: ' / !+@ 88 !@ 8 @?! ' #!+Z' .# *))?: @)!;)8+ '!,)' 8+@ 8 ' / !,!+ >.' ' / !,!# '/-#!+8,/, '/@ '[3D#!+8,/, '/[2B7![2B!,!# '/[2 . [3D1[2 * [3D!+[2 +/[3DN

*))?: !"?/ !+@. @!+ .+!#+ # / .,/ L/ .,/1 4@?7< *))?: ' / !+@ 88 !@ 8 @?! ' #!+Z' .# *))?: !;/@.,,!@!7.@/# 7!?/# /#!/+ +' / ! ! ) / ,! TEORIA 8+ ' / '8,!+


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UNSCH FISICA IIE+)/ ! ./ ' , ,8 /#/ )!#'=/ / ., MAS@ +.?8,'!,78 >.! /+8+ ' / '8,!+ +8, 7! ?!>.! /+ / ? ').7!+& ?87! 8+ +.?8,!# >.!+!, ] `] 7! 878 >.! / ! ./ ' , 7! 89' '!,)8 !+@

$+ %gd &

= 0

D! 8+)#/,78 >.! ! 89' '!,)8 /, . /# 8+ ' /)8#'8 !+/# ,' 8 +' ? ! 8,


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UNSCH FISICA II

2 2 20 0CG I I md I mR md = + = +

& R

+! 7!,8 ',/ #/7'8 7! '#8& ?/#/ .,/

9/#' / !, 78,7!

22

12

d R =

+'!,78d

/ 8, ').7 7! / 9/#' /@

E ?!#'878 +! !+ #'"!

P6 ',/ 2
http://www.sc.ehu.es/sbweb/fisica/solido/din_rotacion/inercia/inercia.htm#Momento%20de%20inercia%20de%20una%20varillahttp://www.sc.ehu.es/sbweb/fisica/solido/din_rotacion/inercia/inercia.htm#Momento%20de%20inercia%20de%20una%20varilla


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2 2

2 R d

T gd

+=

D! ' ./ /,!#/ +! ?.!7! #!?#!+!,)/# ?/#/

./ >.'!# .!#?8 # '78@ C./,78 +! #!?#!+!,)/T

!,


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4. D*&*%$'+* *0 /*%'!)! )*0 / +)"0! ,!$/"* &!a0%*)*)!% )* O &!$a+)! *0 &'*$/! /a%a =! ,'0a,'!+* /!% 0! $*+! 4 1*,* .

>. M')a 0a )' &a+,'a ) "* a2 *+&%* *0 ,*+&%! )*" /*+ '6+ 2 *0 ,*+&%! )* (%a1*)a).

;. R*/'&a /a%a @ 1a0!%* )'


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N 7 T 2 8+ ' / '8,!+ T +T T T4 T3/%!$*)'

!51 @3 31@14 31@2Q 31@2 31@23 1@ 22 @314 31@ 3 @00 31@ 4 31@ 1 1@ 13 @2 3 3 @ 3 @0Q 31@ 2 3 @0 1@ 4

4 @212 32@1Q 32@ 4 32@ Q 32@13 1@ Q@1 1 34@11 34@2 34@30 34@2 1@Q13@11 30@11 30@ 30@ 3 30@ Q 1@0 4

S! ?.!7! ,8)/# >.! '!,)#/+ / 8# +!/ / 7'+)/, '/ !,)#! !?.,)8 7! +.+?!,+' , ! !,)#8 7! #/9!7/7& ! ?!#'878 7!8+ ' / ' , +! */ ! 6+ 8#)8& +!/ ./,78 / 7'+)/, '/ )'!,7/ /!#8 ! ?!#'878 7! 8+ ' / ' , )!,7!#/ / ', ,')8& / ./ +! ?.!7!

7!7. '# 7! / < # . / 7! ?!#'878 7! 8+ ' / ' , ?/#/ ? ,7. 88 ?.!+)8 >.! +! !;)#/ 8 /,)!#'8# !,)! # = 2 & %gd7!+?! / 8+ !,


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S! .' 8+ ! '+ 8 ?#8 !7' '!,)8 ?/#/ /+ +' .'!,)!+ 7'+)/, '/+ >.!


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@3 =.=@? @

?=. ?8

4 =.=>>?>>

=. ?

=.= ;? =. >8;?

=.= = =. 4 =?

P!% " )W

A ?/#)'# 7! )!8#! / 7! S)!',!# 8+ 9/ 8#!+ 7! & (&cg(md +!9/, / #! / '8,/# / .,/


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@ 2 @ 4 @ @ @1 @12 @14

@

@1@1

@2

@2

@3< ; 1@22; b @12Ra 1

(%a ,a IX 1 )W

)W

IX

S*( + 0! %* "0&a)! )*0 E ,*0:

&cg= 0.1167 kg'm

% = 1.2209 kg

M &!)! )* 0! $-+'$! ,"a)%a)! .

) = mX +b & = &cg+ %d

Hallamos:

m

y

b

=

2'2'

'''2''

)( x xn y x x x y

b

=

2'2'

''''

)( x xn y x y xn

m

Se .+/#6 ! +' .'!,)! ./7#8:

P6 ',/ 31


32/37

N! 9)W 9 I W@1332

2@2

0@ 3Q31 @ 1QQ4

0@ 0

0@23Q

Q@ 2342

3@ 0Q2

124 @ 01

0

@10 2

2

@ 13Q @ 4Q

43> @ 44044

@1Q210

@ QQ30

@ 2 2

; @ 2 021

@140

@ 31

@ Q10

@ @ 121 @1320

@ 1 03

@ 144

Y =.484?;;

. @8;

=.=8 @4>

=.=4;=?4

m=6 (0.087634 1 ) (0.383955 ) (1.16875 )

6 (0.035093 ) (0.383955 )

m= % = 1.22048 kg

b= (1.16875 ) (0.035093 ) (0.383955 ) (0.087634 )6 (0.035093 ) (0.383955 )

b= &cg= 0.11669 kg'm

Va0!% &*6%',!:

P6 ',/ 32


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E 8 !,)8 7! ',!# '/ 7! !,)#8 7! #/9!7/7 7! .,/ "/##/)! #' / !,)! !+:

&cg= %d 12

&cg=(1.267 kg) (1.05 m)

12

&cg= 0.116406 kg'm

S! )'!,! ! +' .'!,)! ./7#8 8, 8+ 9/ 8#!+ ?#6 )' 8+: 7/)8+ 7!E; ! ,' 8+ ./7#/78+& /+ 8 8 )/ "' , 7! 9/ 8#/,/ )' 8@

Da&! )*0E ,*0 M-+'$!,"a)%a)! Va0!%*a+a0-&',!Ma a 3$57( 1.2209 1.22048 1.267

&cg 7(.$W 0.1167 0.11669 0.116406

S! ?.!7! ,8)/# .,/ 9/#'/ ' , +' ,' /)'9/ !,)#! 8+ 8 !,)8+7! ',!# '/ 7! !,)#8 7! #/9!7/7 >.! +! !;)#/ !#8, 8, 8+7/)8+ 7! E; ! ! )878 7! ,' 8+ ./7#/78+& #!+?! )8 /8+ 9/ 8#!+ /,/ )' 8+@ A+ 8 8 )/ "' , .,/ 9/#'/ ' , !, //+/ 7! / "/##/@ E+)8 7!"'78 7'9!#+8+


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Pa%a 0! )a&! )*0 E ,*0:

* =|+# +,+# | x 100

* - 2 . =|0.116406 kg'm 0.1167 kg'm 0.116406 kg'm | x 100 * - 2 . =| 0.000294 kg'm 0.116406 kg'm | x 100 * - 2 . = | 0.002525643 | x100

P!% $ &!)! )* $-+'$! ,"a)%a)! :

* =|+# +,+# | x 100

* - 2 . =|0.116406 kg'm 0.11669 kg'm 0.116406 kg'm | x 100 * - 2 . =| 0.000284 kg'm 0.116406 kg'm | x 100 * - 2 . = | 0.0024397368 | x 100

S! ?.!7! /?#! '/# ., !##8# ,' 8 !,)#! 8+ 9/ 8#!+ ?/ )' 8+ ! 9/ 8# /,/ )' 8 ?#87. )8 7! ., ".!, #! '+)#8 7! 8+ 7/)8+?!#'878 7! 8+ ' / ' , 7! / "/##/ / 7'+)',)/+ 7'+)/, '/+ ?8#!,7! !+ ., )878 . ! /= ?/#/ !;)#/!# ! 8 !,)8 7!',!# '/ 7! !,)#8 7! #/9!7/7 7! 7'+)',)8+ /)!#'/ !+@

P6 ',/ 34

* - 2 . = 0.2525

* - 2 . = 0.2440


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VI. CUESTIONARIO:

. C!+ ')*%* "+ / +)"0! ,!$/"* &! )!+)* 0a $a a *"+a *


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# =2 m 0 2 + 25 mr 2

mg0 = 2 02 + 25 r 2

g0

VII. OBSERVACIONES CONCLUSIONES:OBSERVACIONES

E+ ' ?8#)/,)! #! '+)#/# 8+ 7/)8+ ?!#'878 7! 8+ ' / ' , 7!? ,7. 8 8 ?.!+)8 7! / ! 8# /,!#/ ?/#/ / ',8#/# 8+!##8#!+@

E, / ?#6 )' / ,8 +! )8 !, .!,)/ 8+ !.! 7'+ ',. !, /+ 7'+)/, '/+ 7! ?.,)8 7!+.+?!,+' , ! !,)#8 7! #/9!7/7 ! ?!#'878 7!8+ ' / ' , +! ', #! !,)/& ?8# !+8 +' !+ >.! / 7'+)/, '/)'!,7! / !#8 ! ?!#'878 7! 8+ ' / ' , )'!,7! / ', ,')8/.,>.! +!/ /+' )' '8@

CONCLUSIONES : E ?!#'878 7! 8+ ' / ' , 7! ., ? ,7. 8 8 ?.!+)8 7!?!,7!

7! 8 !,)8 7! ',!# '/ 7! .!#?8 )8 / !, .!,)/ / .! ! !# ! / 8# #!+'+)!, '/ / '#8@ C8 8 +! /?#! '/

!, / +' .'!,)! .! ! ?!#'878 !, ., ? ,7. 8 8 ?.!+)8 !+

?#6 )' / !,)! ',7!?!,7'!,)! 7! / / ? ').7 8,+'7!#/,78 !6, . 8 7! ', ',/ ' , !,8#!+ / 1 d

P6 ',/ 3


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VIII. BIBLIOGRAFIA:

Fsica II, Humberto Leyva Naveros, Editorial Moshera,segunda edicin.

Fisica I, er!ay "aymond, Editorial Mc #ra! Hill,$uintaedicin.

Fsica, Francis %. ears, Mar& %. 'emas(ui, Editorial Addison %esley )ublising com*any, +ercera edicin.

Fsica, +omo I, "esni& Holiday.

P6 ' / 3Q