UNSCH
FISICA II
UNIVERSIDAD NACIONAL NACIONAL DE SAN S AN CRISTÓBAL DE HUAMANGA FACULTAD DE INGENIERÍA DE MINAS GEOLOGIA Y CIVIL ESCUELA DE FORMACIÓN PROFESIONAL PROFESIONAL DE INGENIERÍA CIVIL
PRÁCTICA Nº 8
“MOVIMIENTO ARMONICO AMORTIGUADO” “PENDULO COMPUESTO” ASIGNATURA PROFESOR ALUMNO GRUPO CICLO ACADEMICO FECHA EJECUCIÓN FECHA ENTREGA
: FS 241 - FISICA II : K!"!# $ANAMPA %UISPE : BAUTISTA BAU TISTA PA%UIY PA%UIYAURI& AURI& E#'( Y!)*+', !)*+' , CCOLLANA MEDINA& M'.! A,! : M/#)!+: 0-11 / : 213-I : 1011213 : 312213 AYACUCHO5PERU 213
P6',/ 1
UNSCH
FISICA II
I. TÍTUL TÍTULO: O:
“MOVIMIE “MOVIMIENTO NTO ARMONICO ARMONICO AMORTIGU AMORTIGUADO”. ADO”.
II. OBJET JETIVOS IVOS :
E+).7'/# ! 89''!,)8 7! ., +'+)!/ 8+'/,)!& ./,78 !;'+)!, <.!#=/+ !;)!#,/+ >.! +! 8?8,!, / 89''!,)8@
III. III. FUNDA FUNDAME MENT NTO O TEÓR TEÓRICO ICO : MOVIMIENTO ARMÓNICO AMORTIGUADO
U, 89'' 89''!,) !,)8 8 /#, /#,'8 '8 /8#)' /8#)'./ ./78 78 !+ />.! />.! +'+)! +'+)!/ / 8+'/ 8+'/,)! ,)! 78,7 78,7! ! / /?' ?'). ).7 7 ,8 ?!# ?!#/,! /,!! ! 8,+ 8,+))/,) /,)!& 7!# !#!'! !'!,7 ,78 8 #/7 #/7. ./! /!, ,)! */+) /+)/ >.! >.! ! +'+) +'+)! !/ / +! 7!)' 7!)'!, !,!& !& !,) !,)8,! 8,!++ +. )#/!)8#'/ ,8 )'!,! ')!+ 8+& 7!"'78 / /+ <.!#=/+ 7! #8=/'!,)8 8 / <.!#=/ <.!#=/ 7! #8=/'! #8=/'!,)8 ,)8 8 / <.!#=/ <.!#=/ 7! #!+'+) #!+'+)!, !,'/ '/ 7! !7'8 !7'8 >.! 7'+'?/, !,!#/& ./ !;?#!+', /)!6)'/ !+: − λt cos
X = A e
(ωt + θ )
CASOS DE MOVIMIENTO AMORTIGUADO Ca! "#a$!%&'("a)!
V!/8+ ?#'!#8 / +')./', !, / >.! ! #8=/'!,)8 !+ ?!>.!8@ E,)8 E,)8, ,!+ !+ +! .? .?! ! >.! >.! / <#! <#!.! .!, ,'/ '/ ,/). ,/).#/ #/ 7! 8+' 8+'/ /' ', , / ./7#/78 !+ /8# / ./7#/78 7! / #!+'+)!,'/: W > λ 2
2
0
C./,78 +! .?! !+)/ 8,7'', ! #/7'/,78 !+ ,!/)'98& 8, 8 ./ / #/= ./7#/7/ !+ ., ,!#8 '/',/#'8 ?.#8@ S' 7!,'8+: W − λ & 8+ 9/8#!+ 7! J >.!7/, !, ,!#8+ '/',/#'8+: √ W 2
2
0
A+ ?.!+& / +8.', !+ ! ?#87.)8 7! 78+ .! !+ 8+'/,)!@ E 8)#8 !+ .,/ !;?8,!,'/ 8, !;?8,!,)! ,!/)'98& ?8# 8 >.! !+ 7!#!'!,)!@ E P6',/ 2
UNSCH
FISICA II
#!+.)/78
!+
.,/
<.,',
>.!
8+'/
8,
<#!.!,'/
W − λ 8, /?').7 7!#!'!,)!@ /,./# √ W 2
2
0
L/ ,!/ ,!#/ !, / .#/ /7.,)/ .!+)#/ / !98.', !, ! )'!?8 7! / ?8+'', 7! / /+/@ L/ ,!/ /=. 7! )#/=8+ ',7'/ / !98.', 7! / /?').7 7! / 8+'/',@ Ca! !#%*a$!%&'("a)!
C8,+'7!#!8+ /*8#/ / +')./', !, / >.! ! #8=/'!,)8 !+ . #/,7!:
2
W 0 < λ
2
A*8#/& ! #/7'/,78 !+ ?8+')'98& ?8# 8 >.! /+ 78+ +8.'8,!+ 7! / W − λ & 8+ !./', /#/)!#+)'/ +8, #!/!+@ S' 7!,'8+ √ W 2 0
9/8#!+ 7! J +8, #!/!+ ,!/)'98+@ A$!%&'("a$*+&! A$!%&'("a$*+&! ,%-&',!
P6',/ 3
2
UNSCH
FISICA II
A*8#/ )!,!8+ / 8,7'', 2
2
W 0= λ
C./,78 8.##! !+)8& ! #/7'/,78 7! / !./', /#/)!#+)'/ !+ !#8& ?8# 8 >.! !;'+)! .,/ ,'/ +8.', 7! / <8#/ eJt @ D! ,.!98 !+ .,/ <.,', 7!#!'!,)! !, ! )'!?8& !+ 7!'#& / /?').7 7! /+ 8+'/'8,!+ )'!,7! / !#8@ E )'!?8 )?'8 /+8'/78 / /8#)'./'!,)8 !+
L/ .#/ / / 7!#!*/ .!+)#/ / !98.', 7! / /?').7 ?/#/ 9/8#!+ 7! 8+ 8!'!,)!+ 8##!+?8,7'!,)!+ / 8+ )#!+ /+8+@ P.!7! 8"+!#9/#+! >.! ! /8#)'./'!,)8 !+ 6+ !.! ! 8!'!,)! 7! #8=/'!,)8 !+ /8# !, ! #'!, +8"#!/8#)'./78@ E+)8 ?.!7! !,)!,7!#+! 8"+!#9/,78 / P6',/ 4
UNSCH
FISICA II
.#/ >.! .!+)#/ / !98.', 7! / 9!8'7/7 !, /7/ .,8 7! 8+ /+8+@ C./,78 ! /8#)'./'!,)8 !+ #)'8& ! 7.8 7! / 9!8'7/7 /8# / /8# ?/#)! 7! )'!?8@ C88 ! #8=/'!,)8 !+ ?#8?8#'8,/ / / 9!8'7/7& !+ 6+ !.! <.,'8,/, 88 /8#)'./78#!+ +! /.+)/, ?/#/ >.! )#/"/!, !, !+)! #'!, #)'8@
IV. MATERIALES : 1@ C."!)/ 8, /./ 2@ R!+8#)! ?!+/+ 3@ C#8,8!)#8 4@ R!/ #/7./7/ @ B//,=/
RECIPIENTE CON AGUA CARGAS
REGLA METRICA
RESORTE
P6',/
UNSCH
FISICA II
CRONOMETRO BALANA
V. PROCEDIMIENTOS : A. MOVIMIENTO OSCILATORIO AMORTIGUADO . S"/*+)a *0 %*!%&* 1*%&',a0$*+&* 2 /a%a )'&'+&a /*a 34/*a5 $')a 0a *0!+(a,'6+ ,!%%*/!+)'*+&*. L"*(! )*&*%$'+* *0 1a0!% )* 7. O 9 .;,$ Maa 37(5 =.= =.= =.=4 =.=>
D*
a5 Pa%a =.=7( ∆ x =0.025
P8# / ! 7! H88(!
P6',/
UNSCH
FISICA II
F = k 1∆ x mg = k 1∆ x k 1 k 1
= =
mg
∆ x
( 0.01kg ) ( 9.8m / s
2
)
0.025m k 1 = 3.92 N / m
D* $a+*%a '$'0a% /a%a 0a !&%a $aa: * !#&'*+* 0a '("'*+&* &a#0a:
M/+/ ( @1 @2 @3 @4
D!<8#/', =.=; =.=;; =.=8; =.;
H//,78 ! ?#8!7'8 ?/#/ K: k 1+ k 2 + k 3 + k 4 4
= k
K =3.5825 N / m
. I+&a0* *0 *"'/! ,!$! *+ 0a (. .
P6',/ Q
K N 4.? 4.;@ 4.>; 4.>=
UNSCH
FISICA II
4. C"*0("* *+ *0 *&%*$! 0'#%* )*0 %*!%&* "+ /*! )* ;=( 2 $a+&+(a0! &!&a0$*+&* "$*%(')! *+ *0 0-"')! $a%"* 0a +"*1a /!','6+ )* *"'0'#%'!.
>. D*/0a,* 0a $aa $ a,'a a#a! ,"')a+)! "* 0a $aa !,'0* 1*%&',a0$*+&* 2 ,!+ "+ ,%!+!$*&%! $')a *0 &'*$/! /a%a ; !,'0a,'!+* /!% $*+! ,"a&%! 1*,*. R*/'&a /a%a ; 1a0!%* )'<*%*+&* )* a$/0'&")3&a#0a5
P6',/
UNSCH
FISICA II
;. C!+ 0! )a&! !#&*+')! )*&*%$'+* *0 /*%'!)! *+ *0 a'%* 2 0"*(! *+ *0 a("a 0a <%*,"*+,'a a+("0a% 2 )*0 $!1'$'*+&! a$!%&'("a)! 0a ,!+&a+&* )* a$!%&'("a$'*+&!.
D! / )/"/ /,)!#'8#: M/+/ ( =.= =.= =.=4 =.=>
D!<8#/', @2 @ @ @11
;.. D*&*%$'+* 0a <%*,"*+,'a +a&"%a0 Wo
=
k m
cuando ⇒ m
= 0.05kg
k
Wo
=
Wo
=
Wo
= 8.4646
m 3.5825 N / m 0.05kg
C!$*+&a%'!:
P6',/ 0
K N 3@02 3@ 3@4 3@4
UNSCH
FISICA II
E+ . ,!!+/#'8 )!,!# !, .!,)/ / <#!.!,'/ ,/).#/ ?.!+)8 >.! !+)! 9/8# !+ '?8#)/,)! ?/#/ *//# / <#!.!,'/ 7! /8#)'./78@
•
;.. D*&*%$'+* *0 /*%'!)! *+ *0 a("a t =
2 π w❑
$aa9=.=; 7'0!(%a$! T P/#/ 1 8+'/'8,!+
A 3,$5
== ==; == ==; ==4
)1 @ @Q @2 @0 @
)2 @2 @4 @ @01 @Q
)3 @Q2 @Q @Q4 @0 @Q4
T ?#8!7'8 8.8 8.8 8.8=44 8.? 8.8
T 18+'/'8, T @ @ @33 @02 @Q
O#*%1a,'6+: P87!8+ 8"+!#9/# >.! ?/#/ 7'
E ?!#'878 !, ! /./ !+ ! >.! *!8+ !7'78 !, ! /"8#/)8#'8&
•
!, )!8#/ 8, / +'.'!,)! <8#./ ;.4. D*&*%$'+* a$!%&'("a)! t =
0a
t =
2 π
w❑
<%*,"*+,'a
88 )*0
2 π w❑
w❑=
2 π
t
a$/0'&") ==
/+/@( T @ P6',/ 1
Q@14
w i=√ w0
2
− β
2
$!1'$'*+&!
UNSCH
FISICA II ==; == ==; ==4
@ @33 @02 @Q
Q@14 Q@13Q Q@44 Q@1
;.>. D*&*%$'+* 0a ,!+&a+&* )* a$!%&'("a$'*+&! 2
b − W = m 2m 2 k b 2 W = − m 2m 2 b = k − W 2 2m m k
b 2m
=
b = 2m
k m
− W 2
k m
− W 2 $aa9=.=;7(
94.;8;NK$
' .>= .>= .4 .=>> .;@
b =2 m
b =2 ( 0.05 )
b =2 ( 0.05 )
b =2 ( 0.05 )
b =2 ( 0.05 )
b =2 ( 0.05 )
√ √ √ √ √
3.5825 0.05 3.5825 0.05 3.5825 0.05 3.5825 0.05 3.5825 0.05
√
k 2 2 −i wi m
−7.140
2
b =0.4546
−7.140
2
b =0.4546
−7.137
2
b =0.4551
−7.044
2
b =0.4694
−7.156
2
b =0.4521
P6',/ 11
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. 2
cuandow 0 > λ
2
√ λ − w0 =i wi esunacantidad imaginaria 2
2
− λt
x =e
[B e
+ C e−i w t ]
i w i t
i
A iθ A iθ aciendo B = e ! C = e 2 2 obtenemos
x = A e
[
i (w i t + θ)
e
− λt
+ e−i( w t +θ ) i
2
]
P/#/ ,.!+)#/ ?#6)'/ .+/#!8+ x = A e
[
i (w i t )
e
− λt
+ e−i ( w t ) i
2
θ= 0
]
O#&*+*$! − λt
X = A e
cos ( wi t + θ )
Da&!4.;8;NK$
M/+/ (
F#!.!,'/ ,/).#/
P6',/ 12
C8!'!,)! 7! /8#)'./'!,)8
UNSCH
FISICA II λ =
W 0
m=0.05
x 0=0.05
b 2m
k
=
m 3.5825 N / m
W 0
=
W 0
= 8.4646
λ =
0.45716 0.1
0.05kg
λ =4.5716
La <%*,"*+,'a a+("0a% )* 0a !,'0a,'6+ a$!%&'("a)a w i *
√ λ − w0 =i wi 2
2
2
2
2
w i=√ w0
2
− λ +w =wi 0
w i=√ 8.4646 − 4.5716 2
2
2
λ − w0 = i wi
2
2
− λ
2
2
w i=( 7.1239 ) rad / s
La /!','6+ '+','a0 = )*&*%$'+a+ 0a a$/0'&") A 2 0a
. Pa%a &9=
x 0= Acos ( θ )
0.05
= 0.05 cosθ
θ= ( 0 " )
La *,"a,'6+ )* 0a !,'0a,'6+ a$!%&'("a)a * − λt
X = A e
cos ( w i t )
P6',/ 13
UNSCH
FISICA II
− 4.5716 t
X =(5 ) e
cos [ ( 7.1239 ) t ]
Ta#0a & 35 =. =. =.4 =.> =.; =.@ =. =.8 =.? . . .4 .> .; .@ . .8
3&5 3@12 2@33 1@2Q @22 @Q @321 @23 @124 @12 @13 @324 @2 @120 @2 @2 @33 @21 @13
I+&*%/%*&a,'!+: •
•
S! ?.!7! /?#!'/# >.! / !7'7/ >.! ! )'!?8 /.!,)! / /?').7 7'+',.! */+)/ ,/!,)! 7!)!,!#+! !, +. ?.,)8 7! !>.''"#'8 !, / ?#/)'/ !+)8 +.!7! 7!"'78 / / !;'+)!,'/ 7! 8!'!,)! 7! /8#)'./'!,)8 ?!#8 !, / )!8#'/ 88 +! )#/)/ 7! .,/ <.,'!, “!” ,.,7/ !/#W / +!# !#8@
E )'!?8 /.!,)/ 7! /,!#/ !;?!,'/ )/,"'!, 7! <8#/ +!,8'7/@
VI. CUESTIONARIO:
P6',/ 14
UNSCH
FISICA II
. La !,'0a,'!+* a$!%&'("a)a !,"%%'%+ /a%a ,"a0"'*% 1a0!% )* 7 2 # */0'"*
.
P8#
8 ./
?87!8+
#!?#!+!,)/#
/ )#',8
− bv = −b x
88
b
#!+?8,+/"! 7! /8#)'./'!,)8& 78,7!
!+ .,/ 8,+)/,)! .
− b x
b<0
8,+)/,)! 7! /8#)'./'!,)8 .,/ <.!#=/ 8, )!,7#/ ! '+8 +!,)'78 >.! / <.!#=/ #!.?!#/78#/& ,8 ! 8?.!+)8 / +)/@ b>0
b=0
Y 7!"! .?'#+! >.! / >.! +' / ,8 !;'+)'#/ <.!#=/ 7! /8#)'./'!,)8 / ./ */! >.! / /?').7 7'+',./@ E, 8,.+', ?/#/ >.! *// 8+'/'8,!+ /8#)'./7/+ +! 7!"! .?'# >.!
b>0
@
S/"!8+ >.!:
P6',/ 1
UNSCH
FISICA II
ω 0
β
D8,7! !+ !
!+ / ?.+/', ?#8?'/ 8
P/#/ >.! +! .?/ ! /+8 7! 8+'/'8,!+ /8#)'./7/+& +! 7!"! ω 0
.?'# >.! 8+'/'8,!+@
2
> β
2
& / >.! !, 8+ 7!6+ /+8+ / ,8 !;'+)!,
L.!8& ?/#/ 7!)!#',/# 8+ 9/8#!+ 7! / 8,+)/,)! 7! #!+)').',: ω 0
2
> β
2
2
b > m 2m k b b k > m m > ⇒ m 4m m 4m b k > m 4 k
2
2
2
2
2
. U+ /*0!&6+ )* !0)a)! $a%,a a0 /a! a 0! 0a%(! )* "+ ,a$'+!. P!% " * 0* !%)*+a %!$/*% 0a ,"a+)! ,%"Qa+ "+ /"*+&*
E, ?#'!# ./# / %*!+a+,'a !+ ., .! +! ?#87.! ./,78 ., .!#?8 /?/= 7! 9'"#/# !+ +8!)'78 / / /', 7! .,/ <.!#=/ ?!#'7'/& .8 ?!#'878 7! 9'"#/', 8','7! 8, ! ?!#'878 7! 9'"#/', /#/)!#+)'8 7! 7'*8 .!#?8@ E, !+)/+ '#.,+)/,'/+ ! .!#?8 9'"#/& /.!,)/,78 7! <8#/ ?#8#!+'9/ / /?').7 7! 89''!,)8 )#/+ /7/ .,/ 7! /+ /)./'8,!+ +.!+'9/+ 7! / <.!#=/@ P6',/ 1
UNSCH
FISICA II
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
UNSCH
FISICA II
XC./,78 ., ?!8), 7! +87/78+ !+)6 /#*/,78 9/ / #.=/# ., ?.!,)!& #8?! / /#*/@ D! ,8 */!#8& 8+ 8?!+ >.! 7/, / /#*/# ?87#/, )!,!# 8?8,!,)!+ 8, .,/ <#!.!,'/ './ / /.,/ 7! /+ <#!.!,'/+ /#/)!#+)'/+ 7! ?.!,)!@ A #8?!# / /#*/ !9')/, >.! *// #!+8,/,'/@
4. B%'+)* **$/0! )* !,'0a,'!+* a$!%&'("a)a 2 )* !,'0a,'!+*
E$EMPLOS DE OSCILACIONES AMORTIGUADAS: - U, !!?8 7! ., 8+'/78# /8#)'./78 !+ ., .!#?8 .,'78 / ., #!+8#)! +.!#'78 !, ., >.'78 9'+8+8
- %. !+ / +.+?!,+',Z L/ +.+?!,+', 7! ., 9!*.8 )'!,! 88 8"!)'98 /"+8#"!# /+ '##!./#'7/7!+ 7! )!##!,8 +8"#! ! >.! +! 7!+?/=/ /,)'!,!, /+ #.!7/+ !, 8,)/)8 8, ! +.!8 ?/#/ ?#8?8#'8,/# ., /7!./78 ,'9! 8,<8#) +!.#'7/7 7! /#*/@
E$EMPLOS DE OSCILACIONES FORADAS: - C88 8.##! !, ! /+8 7! / <.!#=/ /?'/7/ / ., /.)89' ?8# 8+ .!!+ 7! +. +.+?!,+', ./,78 ! 9!*.8 9/ ?8# .,/ /=/7/ 8, "/*!+ P6',/ 1
UNSCH
FISICA II
- U, ,'8 !, ., 8.?'8 ?.!7! /,)!,!#+! !, 89''!,)8 !7'/,)! “!?.8,!+” /?#8?'/7/!,)! +',#8,'=/78+@ L/ /?').7 7! 89''!,)8 ?!#/,!! 8,+)/,)! +' ! +.','+)#8 7! !,!#/ ?8# '8 7! 89''!,)8 '.// !;/)/!,)! / #!7.', !, !,!#/ !6,'/ !, /7/ '8 >.! #!+.)! 7! /+ <.!#=/+ #!+'+)'9/+@
- M.*/+ 6>.',/+ 9'"#/, ?8#>.! )'!,!, ?'!=/+ !, #8)/', >.! ,8 !+)6, ?!#.''"#/7/+@ S' +! +.!)/ .,/ 6>.',/ 7! !+)/+ / .,/ !+)#.).#/ >.! ?.!7! 9'"#/#& 7'*/ !+)#.).#/ +! 8,9'!#)! !, ., +'+)!/ <8#=/78 >.! ?.!7! ',''/# +. 89''!,)8 ?8# / /', 7! / 6>.',/@ - P.!7! #8?!#+! ., 9/+8 8, "/8 /8#)'./'!,)8 !7'/,)! .,/ 8,7/ +8,8#/ ',)!,+/ 8, .,/ <#!.!,'/ './ 8 . ?#;'/ / / <#!.!,'/ ,/).#/ 7! 9'"#/', 7! '+8@ >. U+ !,'0a)!% &'*+* "+ /*%'!)! )* 4 . " a$/0'&") )'$'+"2* *+ "+ ; )"%a+&* ,a)a ,',0! 3#5 ,"a0 * 0a
τ
,!+&a+&* )* &'*$/! 2 *0
S! )'!,!, 8+ 7/)8+: T = 3 s
P6',/ 10
UNSCH E t
1
=
2
FISICA II
kA2
E+ / !,!#/ ',''/
'
S' / /?').7 7'+',.! !, [ 7.#/,)! /7/ '8& /?').7 !, ! +!.,78 '8 +!#6:
/
A __________ _ 100% x __________ _ 95% x =
95 100
A
x
“ ” E+ / ,.!9/ /?').7@
/
2
95 1 361 = 1 ( 0.0975) kA ∆ E = k ( A) − k A = kA 1 − 2 2 100 2 400 2 1
2
∆ E
1
2
1 ( 0.0975) kA2 2
2
!+ / 9/#'/', 7! / !,!#/
τ
" E .! !+ ! )'!?8 >.! )/#7/ / !,!#/ !, 7'+',.'# !, ! .!8& " !+ ?!>.!8 τ !+ #/,7!& ! 8+'/78# ?!#7!#6 .,/ <#/', . ?!>.!/ 7! !,!#/ 7.#/,)! .,/ 8+'/',@
1 kA 2 2
Q = 2π
E
∆ E
= 2π
1 ( 0.0975) kA 2
= 64.4
2
QT 2π
= τ
P6',/ 2
UNSCH
FISICA II τ =
QT 2π
=
( 64.4 )( 3) 2π
= 30.748
VII. OBSERVACIONES CONCLUSIONES : OBSERVACIONES :
•
L/+ /+/+ >.! )8/8+ /+ 8,+'7!#/8+ 88 ?.,)./!+& / >.! ?/#/ >.! +! 7! !+)/ 8,7'', +! ,!!+')/ >.! / 8,').7 7! / .!#7/ )'!,! >.! )!,!# .,/ 8,').7 /8#& / >.! 7! !+/ <8#/ *0 1!0"$*+ )*0 ,"*%/! +! '+"2*. L8+ 878+ 7! 8+'/', !, ?/#)'./# 8+ 7!"!8+ 7! #!+89!# )8/,78 !, .!,)/ 8+ !!+ 7! +'!)#/@ CONCLUSIONES :
•
•
•
•
S8"#! 8+ 89''!,)8+ '"#!+ 8,.'8+ >.! +', / ?#!+!,'/ 7! .,/ <.!#=/ '?.+8#/& /+ /?').7!+ 7! / 9!8'7/7 9/, 7!#!'!,78& 7! './ /,!#/ / <.!#=/ #!+)/.#/78#/@ E, ! /+8 7! 8+'/78# /8#)'./78& / /?').7 7!#!! !;?8,!,'/!,)! 8, ! )'!?8& / !,!#/ 7'+',.!& 7!+#'"! .,/ )#/!)8#'/ !, <8#/ 7! !+?'#/ !, ! !+?/'8 7! /+ .'!#! /8# !,!#/ ./,78 / <#!.!,'/ !;)!#'8# 8','7! 8, / ?#8?'/ 7! 8+'/78# #!+8,/,'/@ L/+ .#9/+ 7! #!+8,/,'/ +8, )/,)8 6+ !+)#!*/+ ./,)8 !,8# +!/ ! 8!'!,)! 7! /8#)'./',@ D! *!*8& +! ?.!7! 7!8+)#/# >.! / /?').7 ∆ω ≅ 2b / m
•
∆ω
/ / /).#/ !7'/ 7! /+ .#9/+ )8/ ., 9/8#:
C88 / /?').7 7! 8+'/78# /8#)'./78 7'+',.! 8, ! )'!?8& / !,!#/ 7! / ?/#)./ )/"', 8 */!@ L/ !,!#/ P6',/ 21
UNSCH
FISICA II
?!#7'7/ !+ /"+8#"'7/ ?8# ! !7'8 '#.,7/,)! 8 #/7'/7/ 7! /.,/ /,!#/@
τ
•
L/ /,)'7/7 !+ 8,8'7/ 88 )'!?8 7! #!//',& )'!?8 ,!!+/#'8 ?/#/ >.! / !,!#/ 7!+?.+ 7! /7/ ?!#878 +! #!7.=/ / 1! +. 9/8# / 8'!,=8 7! ?!#878@
VIII. BIBLIOGRAFÍA :
I. II. •
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
TITULO: “ PENDULO COMPUESTO” OBJETIVO:
D!)!#',/# ! 8!,)8 7! ',!#'/ 7! !,)#8 7! #/9!7/7 7! .,/ "/##/@
III.
FUNDAMENTO TEORICO:
PENDULO COMPUESTO: P6',/ 22
UNSCH
FISICA II
PENDULO COMPUESTO:
U, ?,7.8 8?.!+)8 8 <+'8 !+ ., .!#?8 #'78 >.! ?.!7! 8+'/# '"#!!,)! / "/8 / /', 7! / #/9!7/7& /#!7!78# 7! ., !! *8#'=8,)/ 8 O >.! ,8 ?/+/ ?8# +. !,)#8 7! #/9!7/7 CG@ P8# )/,)8 +!/ O ! !! *8#'=8,)/ C ! !,)#8 7! /+/ 7! .!#?8& ./,78 / ,!/ OC */! ., 6,.8 ] 8, / 9!#)'/& +! !,!#/ ., )8#>.! #!+)/.#/78# /#!7!78# 7! !! /)./,)! +8"#! ! .!#?8 !+)8+ +! ?#87.!, '/!,)! 7/78 >.! 8+ )8#>.!+ /"'/, 7! 7'#!', ?/./)',/!,)! / !7'7/ >.! 8+'/ 7! /78 /78 ! ./ 8 8,7.! / +. ?8+'', 7! !>.''"#'8@ E )8#>.! !+ T - M7+!, ( Ө ) 78,7! 7: !+ / 7'+)/,'/ OC !,)#! ! !! = ! !,)#8 7! /+/ C@
S' I !+ ! 8!,)8 7! ',!#'/ 7! .!#?8 /#!7!78# 7! !! “O” ^_ !+ / /!!#/', /,./#@ A?'/,78 / +!.,7/ ! 7! ,!)8, ?/#/ / #8)/', +! )'!,!@
# =¿−¿
∑¿
M7@+!,] ! ./ 8##!+?8,7! / 8!,)8 #!+)/.#/78# ?/#/ ., 7!+?/=/'!,)8 /,./# ] A?'/,78 / +!.,7/ ! 7! ,!)8, I _ −¿ M7+!,] %gd $+ senӨ =0 & ₀
P6',/ 23
UNSCH
FISICA II
E+)/ !./', ,8 /#/)!#'=/ / ., MAS@ +.?8,'!,78 >.! /+ 8+'/'8,!+ +8, 7! ?!>.!/+ /?').7!+& ?87!8+ +.?8,!# >.! +!,] `] 7! 878 >.! / !./', 7! 89''!,)8 !+@ %gd $+ & ₀
Ө= 0
D!8+)#/,78 >.! ! 89''!,)8 /,./# 8+'/)8#'8 !+ /#,'8 +'?! 8, <#!.!,'/ ,/).#/ 7! 8+'/', 7/7/ ?8# ω$=
√
%gd & ₀
I.//,78 8, / <#!.!,'/ ,/).#/ 7! 8+'/', 7!+?!/,78 1 ω$= 2 π #
!, T?!#878
E,)8,!+ ! ?!#878 7! ?,7.8 <+'8 ?/#/ ?!>.!/+ /?').7!+ 7! 8+'/', !+)6 7/78 ?8# / !;?#!+', +'.'!,)!:
√
# =2 π
& ₀ %gd
P8# ! )!8#!/ 7! S)!',!#
√
# =2 π
& cg+ %d ² %gd
E*$/0! /a%a "+a 1a%'00a
P8# ! )!8#!/ 7! S)!',!#: I 0
= I CG + md
I CG
2
=
1 12
md 2
I CG
D8,7! !+ ! 8!,)8 7! ',!#'/ 7! / "/##/ 8, #!+?!)8 / +. !,)#8 7! #/9!7/7@ P6',/ 24
UNSCH I 0
FISICA II
= I CG + md 2 ⇒ I 0 = mR 2 + md 2 R
2
=
9/#'/ !, 78,7!
&
R
+! 7!,8',/ #/7'8 7! '#8& ?/#/ .,/
d 2 12
+'!,78
d
/ 8,').7 7! / 9/#'/@
E ?!#'878 +! !+#'"!
P6',/ 2
R 2 + d 2
T = 2π
gd
D! './ /,!#/ +! ?.!7! #!?#!+!,)/# ?/#/ ./>.'!# .!#?8 #'78@ C./,78 +! #!?#!+!,)/
T
!, <.,', 7! d = 0 d & !+ 7!'#& & ! ?!#'878 //,=/ ., 9/8# ',,')8 ?/#/ ./,78 8','7! ! !,)#8 7! /+/ 8, ! !,)#8 7! 8+'/', O@ L/ .#9/ ?#!+!,)/ ., ,'8 ?/#/ ., '!#)8 9/8# 7! ?.!7! /./# 7!#'9/,78 IV.
T
#!+?!)8 7!
d
d
>.! +!
! './/,78 / !#8@
MATERIALES:
V. PROCEDIMIENTO . D*&*%$'+* 0a $aa 2 $')a 0a 0!+('&") D* 0a #a%%a. 3F'(.5
M/+/( 1@2Q .
L8,').7 1@
S"/*+)a 0a #a%%a )* "+ /"+&! O 3,",'00a !/!%&*5 2 (a0! !,'0a%. P6',/ 2
4. D*&*%$'+* *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!%* )'<*%*+&* )* ). @. Pa%a ,a)a ,a! ,a0,"0* *0 $!$*+&! )* '+*%,'a I )* 0a #a%%a a0%*)*)!% )*0 ** "* /aa /!% *0 /"+&! )* "/*+'6+. . Ha(a "+a &a#0a )* 1a0!%* )* I 2 ). (%a"* I ,!+ ) 2 *(+ 0 )*&*%$'+* *0 $!$*+&! )* '+*%,'a )*0 ,*+&%! )* (%a1*)a). C!$/a%* " %*"0&a)! ,!+ *0 1a0!% a+a0-&',!.
V.
DATOS EPERIMENTALES:
S! )'!,! ! +'.'!,)! ./7#8 78,7! +! .!+)#/ ! ?!#'878 7! 8+'/', ?/#/ ., ?,7.8 +'?! !7'7/ / .,/ 7'+)/,'/ 7 >.! */ !,)#! ! !,)#8 7! +.+?!,+', ! !,)#8 7! #/9!7/7@ P6',/ 2Q
N
7
1 2 3 4
T 2 8+'/'8,!+ T T T4
@3 @314 @23 @212 @11 @11
31@14 31@ 3@ 32@1Q 34@11 30@11
31@2Q 3@00 3@0Q 32@4 34@2 30@
T +
31@2 31@4 31@2 32@Q 34@30 30@3
T3/%!$*)' !5 31@23 31@1 3@0 32@13 34@2 30@Q
1@2 1@1 1@4 1@Q 1@Q13 1@04
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+'/', ?/#/ ?,7.8 8?.!+)8
>.!
+!
!;)#/8
/,)!#'8#!,)!
7!+?!/8+ !, <.,', 7! 8!,)8 7! ',!#'/ “I & = ₒ
( )
# ² 2 π
ₒ
&
ₒ
:
( %gd )
=
( ) # 2 π
( %gd )
²
(
)
1.562 seg ² 2 π
& =0.28009 ₒ
& ₀ %gd
M8!,)8 7! ',!#'/ ?/#/ .,/ 7'+)/,'/ @3 !,)#! ! !,)#8 7! +.+?!,+', ! !,)#8 7! #/9!7/7:
•
&
₀
√
# =2 π
( 1.267 kg ) ( 9.8 m / seg ² ) ( 0.365 m)
(@a
P6',/ 2
S!.'8+ ! '+8 ?#8!7''!,)8 ?/#/ /+ +'.'!,)!+ 7'+)/,'/+ >.!
7
T +
I=
1
=.4@;
.;@
2
=.4>
.;;
3
=.@4
.;>8
4
=.
.@=
=.@
.4
=.=
.?;>
=.8= =? =.4 ; =.?8 =. ? =.>8 ;? =.4 =?
C8, 8+ #!+.)/78+ 8")!,'78+ +! 8##8"8#/ ! 8,!?)8 7! 8!,)8 7! ',!#'/ / ./ /,'!+)/ >.!: '!,)#/+ /8# +!/ / 7'+)/,'/ !,)#! ! ?.,)8 7! +.+?!,+', ! !,)#8 7! #/9!7/7& /8# +!#6 ! 8!,)8 7! ',!#'/ !, !+! ?.,)8 8 +!/ +! )'!,! /8# #!+'+)!,'/ / '#8 !, !+! ?.,)8 7! +.+?!,+',@ A7!6+ +! ?.!7! ,8)/# !+)/ 7!,'', 8, ! )!8#!/ 7! S)!',!# & = &cg + md ² ₒ
E ./ !;?#!+/ >.! ! 8!,)8 7! ',!#'/ #!+?!)8 / ., ?.,)8 >.! ,8 +!/ ! !,)#8 7! #/9!7/7 +'!?#! +!#6 /8# / !+)!@ P/#/ */!# / #6/ 7! I= 8, 7a +! )'!,! / +'.'!,)! )/"/ N 1 2
) ( m )
I= ( kg'm ² )
=.44 =.8==? ; =.=?8;? =.4; P6',/ 20
3 4
@ =.=@?@ ? =.=>>?> > =.=;? =.==
=.?8 =.? =.>8;? =.4=?
P!% " )W
A ?/#)'# 7! )!8#!/ 7! S)!',!# 8+ 9/8#!+ 7! & (&cg(md ² +! 9/, / #!/'8,/# / .,/ <.,', ',!/ 7! / +'.'!,)! /,!#/: ₒ
& = &cg + %d ²
) =mX + b
ₒ
D8,7! +! ?.!7! 7!7.'# >.!: L/ 8,+)/,)! b >.! !+ / 7'+)/,'/ !,)#! ! ?.,)8 7! • ',)!#+!', 7! / #!)/ 8, ! !! Y& ! ?.,)8 8#'!,& !+ ! Icg ” 8!,)8 7! ',!#'/ #!+?!)8 / +. !,)#8 7! #/9!7/7@ L/ ?!,7'!,)! 7! / <.,', ',!/ m !+ !>.'9/!,)!!,)! • / / /+/ M 7! .!#?8 #'78@ G%,a *+&%* *0 $!$*+&! )* '+*%,'a I %*/*,&! a 0! /"+&! )* "/*+'6+ ,!+ 0a )'&a+,'a a0 ,"a)%a)! “ d²” )* */a%a,'6+ )* *&a a,'a *0 ,*+&%! )* (%a1*)a) ₒ
P6',/ 3
(%a,a IX 1 )W @3 <; 1@22; b @12 Ra 1
@2 @2
IX
@1 @1 @
@2
@4
@
@
@1
@12
)W S*(+ 0! %*"0&a)! )*0 E,*0: &cg =0.1167 kg'm ² % =1.2209 kg
M&!)! )* 0! $-+'$! ,"a)%a)!. ) =mX + b
m Hallamos:
b
=
∑ y
∑ x
'2
n∑ x
'2
Se .+/#6
'
& = &cg + %d ² ₒ
b y
− ∑ x ∑ x y − (∑ x ) '
'
'
2
'
m=
n∑ x ' y ' − ∑ x ' ∑ y ' n∑ x '
! +'.'!,)! ./7#8: P6',/ 31
2
− (∑ x ) '
2
@14
N! 4 > ; @ Y
m=
(
9)W @1332 2 @0 0 @01 0 @440 44 @20 21 @121 =.484? ;;
9 Iₒ @2 0 @23Q Q @102 2 @1Q21 0 @14 0 @132 0 .@8 ;
@3Q31 @2342 3 @13Q @QQ3 0 @3 1 @10 3 =.=8@4 >
)−( 0.383955 ) ( 1.16875 ) 6 ( 0.035093 ) − ( 0.383955 ) ²
6 0.087634 1
m= % =¿ 1.22048 kg
b=
( 1.16875 ) ( 0.035093 )− ( 0.383955 ) ( 0.087634 ) 6 ( 0.035093 ) −( 0.383955 ) ²
b = &cg =0.11669 kg'm ²
Va0!% &*6%',!: P6',/ 32
W @1QQ4 0 @0Q2 12 @4Q 43 @22 @Q 10 @14 4 =.=4;= ?4
E 8!,)8 7! ',!#'/ 7! !,)#8 7! #/9!7/7 7! .,/ "/##/ )!#'/!,)! !+: %d ² &cg = 12
&cg =
( 1.267 kg ) (1.05 m ) ² 12
&cg =0.116406 kg'm ²
S! )'!,! ! +'.'!,)! ./7#8 8, 8+ 9/8#!+ ?#6)'8+: 7/)8+ 7! E;! ,'8+ ./7#/78+& /+ 88 )/"', 7! 9/8# /,/)'8@
Maa 3$57( &cg
Da&! )*0 E,*0
M-+'$! ,"a)%a)!
1.2209
1.22048
0.1167
0.11669
7(.$W
Va0!%* a+a0-&',!
1.267
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+ 88 )/"', .,/ 9/#'/', !, / /+/ 7! / "/##/@ E+)8 7!"'78 7'9!#+8+
S' +! 8,+'7!#/ / 9/8# /,/)'8 88 9/8# )!#'8 V) 0.116406 (@2
P6',/ 33
Pa%a 0! )a&! )*0 E,*0:
|
* =
| − =|
* - 2 .=
* - 2 .
|
+# −+, x 100 +#
|
0.116406 kg'm ² −0.1167 kg'm ² x 100 0.116406 kg'm ²
|
0.000294 kg'm ² x 100 0.116406 kg'm ²
* - 2 .=|−0.002525643| x 100
* - 2 .=0.2525
P!% $&!)! )* $-+'$! ,"a)%a)!:
|
* =
| − =|
* - 2 .=
* - 2 .
|
+# −+, x 100 +#
|
0.116406 kg'm ² −0.11669 kg'm ² x 100 0.116406 kg'm ²
|
0.000284 kg'm ² x 100 0.116406 kg'm ²
* - 2 .=|− 0.0024397368 | x 100
* - 2 .=0.2440
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
VI.
CUESTIONARIO:
. C!+')*%* "+ /+)"0! ,!$/"*&! )!+)* 0a $aa * "+a *<*%a )* %a)'! % 2 0a 0!+('&") )*0 /+)"0! L. a00* " /*%'!)! 2 a/%!'$* /a%a %ZZL.
T!,'!,78 !, 8,+'7!#/', ! ?!#'878 7! 8+'/', ?/#/ ., ?,7.8 8?.!+)8 ?/#/ 6,.8+ ,'8+ !,8#!+ / 1c D/78 ?8#
P8# ! )!8#!/ 7! S)!',!#
√
# =2 π
√
& ₀ & cg+ %d ² # = 2 π %gd %gd
C!+')*%*$! *0 (%a,!
R*$/0aQa+)! /!% 0! 1a0!%* 2a ,!+! ,!+!,')! & c g
2 %
/
2
5
Y / 7'+)/,'/ !,)#! ! ?.,)8 7! +.+?!,+', ! !,)#8 7! #/9!7/7 +! 8,+'7!#/ +88 / 7'+)/,'/ 7! ?,7.8 “L” ?8#>.! !+ /8# !, 8,+'7!#/', 8, ! #/7'8 7! / !+
P6',/ 3
√
# =2 π
2
m0 +
2 5
mr
mg0
2
√
=2 π
2
2
0 + r
2
5
g0
VII. OBSERVACIONES CONCLUSIONES: OBSERVACIONES •
•
•
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