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1/17

FISICA II FIC

UNIVERSIDAD NACIONAL

ACULTAD DE INGENIERIA CIVILESCUELA PRO ESIONAL ING. CIVIL

PRÁCTICA DE LABORATORIO Nº 02

CURSO

: FISICA II

TEMA : M DULO DE RIGIDEZ DE UN MATERIALÓ

DOCENTE : Msc. VAZQUEZ GARCIA OPTACIANO

ALUMNO : ESPADA CRISTOBAL Elindo Maik.

CODIGO : 122.!".#$%

Huaraz 2014

I. OBJETIVOS.


2/17

FISICA II FIC

1.1. D!r"#$ar %a c&$s!a$! %'s!#ca ( u$ rs&r! )&r "*!&(& (#$'"#c&.1.2. Ca%cu%ar % "+(u%& ( r#,#(z (% -#%& ( u$ rs&r! -%#c(a%.

II. MATERIALES A UTILIZAR.

• U$ rs&r! -%#c(a%.• U$ rs&r! u$#rsa% c&$ (&s ar#%%as ( -#rr& / u$a $uz.• U$a r,%a ,ra(ua(a $ "#%"!r&s.• U$ r$#r cu/a s$s##%#(a( s 0.0 "".• U$ "#cr+"!r& cu/a s$s##%#(a( s 0.01 "".• Psas ra$ura(as / )&r!a )sas.• U$a a%a$za.• U$ cr&$&"!r&.• U$ $#% ( uru3as.

III. MARCO TEÓRICO Y CONCEPTUAL .


3/17

FISICA II FIC

3.1. Vibracio!" #ibr!" "i a$or%i&'a$i!%o (! )ar%*c'#a".

U$ "*!&(& )ara ca%cu%ar %a c&$s!a$! %'s!#ca + 5 ( u$ rs&r! % "*!&(& (#$'"#c& % 6uc&")r$( u$ "&#"#$!& ar"+$#c& s#")% )ara "&s!rar s!&7 c&$s#(ra"&s u$ cur)& (

"asa $ sus)$(#(& ( u$ rs&r! !a% c&"& s "us!ra $ %a F#,. 1

S# s (s)%aza a% cur)& u$a (#s!a$c#a /" a )ar!#r ( %a )&s#c#+$ ( 6u#%#r#& s!'!#c& /%u,& s % su%!a s#$ %&c#(a( #$#c#a%7 % cur)& s "&r' -ac#a arr#a / -ac#a aa3&ra%#za$(& u$ M.A.S. ( a")%#!u( /".

Para (!r"#$ar % )r#&(& ( &sc#%ac#+$ (% cur)& $7 s a)%#ca %a s,u$(a %/ ( N8!&$$ u$a )&s#c#+$ ar#!rar#a ,7 s!& s9

P&r &!r& %a(& % cur)& s!a $ %a )&s#c#+$ ( 6u#%#r#& s!'!#c&7 %a s,u$(a %/ ( $8!&$7s scr#.

( )2...

0

0

mg k

k mg

F

st

st

y

=

=−

=↓∑

δ

δ

R")%aza$(& %a cuac#+$ 25 $ 157 rsu%!a9


4/17

FISICA II FIC

( ):...02

2

=+ ym

k

dt

yd

Hac#$(& ;&2"7 %a cuac#+$ :57 )u( scr##rs $ %a ?&r"a s#,u#$!.

@a cuac#+$ 45 c&$s!#!u/ %a cuac#+$ (#?r$c#a% 6u (scr# % "&#"#$!& ar"+$#c&s#")% / su s&%uc#+$ !#$ %a ?&r"a.

( ) ( )...0 φ += t w sen y y

m

D&$( /" s %a a")%#!u( (% M.A.S 80 s %a ?rcu$c#a a$,u%ar / φ % '$,u%& ( (s?asa3. E% )r#&(& ( &sc#%ac#+$ ( %a )ar!cu%a s9

( )B...2 k mT π =

S# s c&$s#(ra %a "asa ?c!#a (% rs&r! "r? 57 %a cuac#+$ s scr# ( %a ?&r"a9

( )( )C...2

k

mmT

rt +

= π

S# s !raza u$a ,ra?#ca T- s. $7 %a #s!$c#a ( $r s % "&!#& )&r % cua% %a cura $& )asa )&r % &r#,$. @a cuac#+$ 5 s!a%c u$ "(#& c+"& -a%%ar % a%&r ( %a c&$s!a$!%'s!#ca ( u$ rs&r! )&r % "*!&(& (#$'"#c&.

3.-. L!, (! /oo+!.

Es!a %/ s!a%c 6u s# s a)%#ca u$a car,a !r$a a#a% a u$ cur)&7 % s?urz& s(#rc!a"$! )r&)&rc#&$a% a %a (?&r"ac#+$ u$#!ar#a7 s#$(& %a c&$s!a$! ( )r&)&rc#&$a%#(a(% MODU@O E@STICO & RIGIDEZ7 s#")r / cua$(& $& s s&r)as % %"#! (

)r&)&rc#&$a%#(a(7 s!& s9

( )H...∈= E nσ

D&$(9 σ s % s?urz& $&r"a%7 ∈ s %a (?&r"ac#+$ u$#!ar#a7 E s % "&(u%& ( %as!#c#(a(.S# %a car,a a)%#ca(a a% cur)& s !a$,$c#a%7 s!a )r&(uc#r' (?&r"ac#&$s a$,u%ars7 $

s!as c&$(#c#&$s %a %/ ( H&& s!a%c.

( )J...γ τ G=

D&$(9 τ s % s?urz& c&$s!a$!7 0 s % "&(u%& ( r#,#(z / %a (?&r"ac#+$ u$#!ar#a )&rc#za%%a.


5/17

FISICA II FIC

3.3. Tor"i.

@%'"s !&rs#+$ a %a (?&r"ac#+$ 6u )r#"$!a u$a arra ?#3a )&r u$& ( sus !r"&s /% &!r& s&"!#(& a u$ )ar M


6/17

FISICA II FIC

( )10... pt I L

G M

θ =

D&$(9 I) s % "&"$!& ( #$rc#a )&%ar ( %a scc#+$ !ra$srsa% c#rcu%ar c&$ rs)c!& a u$3 6u )asa )&r su c$!r&7 θ s % '$,u%& ( ,#r&7 L %a %&$,#!u( ( %a arra / 0 % "+(u%& (r#,#(z.

3.5. R!"or%!" /!#icoi(a#!"

@a F#,. 4a7 r)rs$!a u$ rs&r! -%#c(a% ( s)#ras crra(as s!#ra(as a3& %a acc#+$ (u$a ?urza a#a% P. E% rs&r! s!' ?&r"a(& )&r u$ a%a"r ( ra(#& r7 $r&%%a(& $ ?&r"a (-*%#c ( ra(#& R . @a )$(#$! ( s!a -*%#c s )6uLa ( !a% "a$ra 6u )&("&sc&$s#(rar c&$ as!a$! a)r&#"ac#+$ 6u ca(a s)#ra s!a s#!ua(a $ u$ )%a$& )r)$(#cu%ara% 3 (% rs&r!.

Para (!r"#$ar %&s s?urz&s )r&(uc#(&s )&r %a ?urza P7 s -ac u$ c&r! a% rs&r! )&r u$ascc#+$ ""7 / s (!r"#$a %as ?urzas rs#s!$!s $csar#as )ara % 6u#%#r#& ( u$a ( %as

)&rc#&$s s)ara(as )&r s!a scc#+$. Ds)u*s s a$a%#za %a (#s!r#uc#+$ ( s?urz&s. @a F#,.4 "us!ra % (#a,ra"a ( cur)& %#r ( %a )ar! su)r#&r (% rs&r!7 )ara 6u % rs&r!s!* $ 6u#%#r#&7 $ %a scc#+$ ""7 (r' ac!uar u$a ?urza ( c&r! P7 / u$ "&"$!& M%6 PR .

E% s?urz& c&r!a$! "'#"& s )r&(uc $ %a )ar! #$!r$a (% rs&r! / #$ )rsa(&

)&r9

( )11...2

12

:

21

−=

+=+=

r

r

r

R P

I

r M

A

P

r

p

t r

π τ

τ τ τ

E$ %&s rs&r!s $ %&s 6u % a%&r ( r s )6uL& c&")ara(& c&$ % a%&r ( R7 %a raz+$.

02 = Rr 7 $!&$cs9


7/17

FISICA II FIC

( )12...2

:r

R P r

π τ =

Fig. "a. Resorte #eli$oidal sometido Fig. "%. D.C.L. del la se$$i&n m'ma $arga a!ial

3.7. E#o&aci (! ' R!"or%!

@a %&$,ac#+$ (% rs&r! ( s)#ras crra(ass,$ su 3 )u( (!r"#$ars c&$ su?#c#$!

)rc#s#+$7 ")%a$(& %a !&ra ( %a !&rs#+$. @a F#,.B7 r)rs$!a u$ %"$!& #$?#$#!a"$! )6uL& (%a%a"r (% rs&r! a#s%a(& c&"& u$ cur)& %#r( %&$,#!u( (@.

( )1:...α Rd dL =

D&$( R s % ra(#& (% rs&r!7 r)rs$!a(& )&r

OS $ %a ?#,ura / s % '$,u%& c$!ra% $ S ( (L.

Ka3& %a acc#+$ (% "&"$!& ( !&rs#+$7 M%7 %ra(#& Oa ( %a scc#+$ !ra$srsa% (% a%a"r ,#rar' -as!a &cu)ar Ob. E% )u$!& ( a)%#cac#+$( %a ?urza O )u$!& C5 (sc$(r' r!#ca%"$! %a (#s!a$c#a C!.

( )14...c&s β cd ce ≅

C&"& % '$,u%& (θ s )6uL& % arc& c( )u( c&$s#(rars c&"& u$a rc!a )r)$(#cu%ara OC7 c&$ %& 6u %a cu. 145 s scr#.

( )1...c&sβ cd ce =

D

P

P R Pr

P

M!

de

c

S

o

dL

β

dα

dθ

Mt

β

Fig. ( De)orma$i&n de n resorte#eli$oidal


8/17

FISICA II FIC

D %a ,r'?#ca s &sra 6u9θ d occd .≅ / 7c&s oc Rocos ==β c&$ %& 6u %a c. 15 s scr#9

( )( ) ( )

( )1B...θ δ

θ

Rd ced

oc Rd occe

==

=

D&$( ( s % '$,u%& (θ !&rs#+$ c&rrs)&$(#$! a% %"$!& (@.

T$#$(& $ cu$!a %a c. 1057 s! '$,u%& $ ?u$c#+$ (% "&"$!& !&rs&r s scr#9

( )1C.... p

t

I G

dL M d =θ

R")%aza$(& %a c. 15 $ 1B57 rsu%!a9

( )( )

( )1H...

.

2

p

p p

t

GI

dL PRd

GI

dL PR R

I G

dL M Rd

=

=

=

δ

δ

@a (#s!a$c#a r!#ca% c


9/17

FISICA II FIC

4

:

4

:

.

...4

2

...2

r G

N R P

r G

N R P π

π

π δ =

=

2:5

@a c. 2:5 $&s )r"#! (!r"#$ar )r#"$!a%"$! % "&(u%& ( r#,#(z ( u$ rs&r!s#")r 6u s c&$&zca9 N < $"r& ( s)#ras7 = < c&$s!a$! (% rs&r!7 R < ra(#& (%rs&r! / r < ra(#& (% a%a"r.

IV. METODOLO0IA.

2.1. Para (!%!r$iar #a co"%a%! !#8"%ica (!# r!"or%!:

a. Ar"ar % 6u#)& !a% c&"& s "us!ra $ %a ?#,. 17 sus)$(#$(& % rs&r! (% s&)&r!

-&r#z&$!a%.

4

:

...4r

Rk N G =


10/17

FISICA II FIC

. M(#r %a %&$,#!u( @&5 (% rs&r! s#$ (?&r"ar.

c. C&%&car % )s& P1 $ % !r"& %#r (% rs&r! / %%ar%& %$!a"$! -as!a %a )&s#c#+$

( 6u#%#r#& s!'!#c&.

(. @%ar % s#s!"a rs&r!)sa ( %a )&s#c#+$ ( 6u#%#r#& 91 -as!a %a )&s#c#+$ 9-

)r&(uc#$(& as u$ s!#ra"#$!& 9 $!r 2 a : c".

. S&%!ar / (3ar &sc#%ar % s#s!"a.

?. A c&$!#$uac#+$ "(#r c&$ % cr&$&"!r& %a (urac#+$ ( u$as 10 &sc#%ac#&$s. A$&!ar sus

a%&rs $ %a Ta%a I.

,. Ca%cu%ar % )r#&(& ( &sc#%ac#+$.

-. R)!#r !&(&s %&s )as&s ( a -as!a & )ara %as ("'s )sas / a$+!s sus rs)c!#&s

a%&rs $ %a Ta%a I

Tab#a I (a%o" , c8#c'#o" )ara 9a##ar +

Nº MASA

(g)

Tiempo (s) Tiempoprom.(t)

Periodo(T)

T2

(s2)1 2 3 4 5

1 76.2 5.02 5.02 4.99 5.00 5.016 5.016 0.5016 0.2516

2 101.2 5.89 5.88 5.87 5.91 5.93 5.896 0.5896 0.376

3 126.2 7.75 7.76 7.72 7.80 7.81 7.768 0.7768 0.603

4 151.2 8.55 8.56 8.53 8.57 8.56 8.554 0.8554 0.732

5 176.2 9.04 9.08 9.06 9.07 9.08 9.066 0.9066 0.822

2.-. Para ca#c'#ar !# Mo('#o (! Ri&i(!; (!# R!"or%!.

a. C&$ % r$#r />& "#cr+"!r& "(#r 12 cs % (#'"!r& (% rs&r!. A$+!s sus

a%&rs $ %a Ta%a II.

. C&$ % r$#r />& "#cr+"!r& "(#r 12 cs % (#'"!r& (% -#%& (% rs&r! $

(#?r$!s )&s#c#&$s. A$&!ar sus rs)c!#&s a%&rs $ %a Ta%a II.


11/17

FISICA II FIC

c. C&$!ar % $"r& ( s)#ras 6u )&s % rs&r!. A$&!ar s! a%&r $ %a Ta%a II.

Tab#a II. Da%o" , c8#c'#o" )ara 9a##ar 0

1 - 3 2 5 7 < = > 1? 11 1-

Dc$.4 1.4 1.0 1.1 1.4 1.1 1.0 1.4B 1.44 1.0 1.4 1.0 1.0

($$.4 0.B 0.B0 0.B 0.B0 0.B 0.B0 0.B 0.B0 0.B 0.B 0.B0 0.B

N J J J J J J J J J J J J

V. CUESTIONARIO .

5.1. Co #o" (a%o" (! #a Tab#a I , #a !c'aci


12/17

FISICA II FIC

Sa"&s 6u %a cuac#+$ ( %a rc!a 9 , 6 a b.

Gra(a( 9 J.1">s2

Ac&"&(& c&$$#$! 9 T#2

< a ."#Para $us!r& cas& 9

( )

( ) ( )∑ ∑∑ ∑ ∑ ∑

−

−=

2222

2222

ii

iiiii

T T n

T mT T ma 7

( ) ( )∑ ∑∑ ∑∑−

−=

2222

22

ii

iiii

T T n

mT mT nb 7


13/17

FISICA II FIC

!"A#$%& Nº 01P'"$&& ' &S%$A%$&N *s MASA

0.0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.00 0.05 0.10 0.15 0.20 0.25 0.30

M+s+ (g)

P e r

i o d o d e & s , i - + , i . / ( s

2 )

D %a cuac#&$ 5 !$"&s 6u9( )

k

mmT rt

+= π 2

D&$( ( )2

2

+=T

mmk rf π

MASA(Kg)

Periodo(T)

(!"#)

0.050 0.351 40.315

0.100 0.433 36.994

0.150 0.493 36.662

0.200 0.560 34.636

0.250 0.578 38.467

P"&M'$& 37.415

E$!&$cs % a%&r ( %a c&$s!a$!


14/17

FISICA II FIC

5.-. co #o" (a%o" (! #a Tab#a II , !# a#or (! + ob%!i(o@ 9a##ar !# $o('#o (! ri&i(!;(!# r!"or%! 04 '%i#i;a(o #a !c'aci -34@ co "' r!")!c%io !rror ab"o#'%o ,)orc!%'a#.

$(#) d(#)1 0%0152 0%00093

2 0%0151 0%00090

3 0%0151 0%00094

4 0%0152 0%00093

5 0%0152 0%00092

6 0%0151 0%00093

7 0%0151 0%00094

8 0%0151 0%00090

9 0%0151 0%00091

10 0%0152 0%00094

11 0%0151 0%00090

12 0%0151 0%00094

0%015133 0%000923

Nus!r&s (a!&s s&$9

! 77

& 0%007105

r 0%000462

' 2670412631%185690

Pa x x

x

r

Rk N G

10

44

::

4

:

1001.J510B2.4

51011.C521C.:C5CC4...4===

−

−

Ca%cu%& ( rr&rs )ara % "+(u%& ( r#,#(z9

& r

0.00714 0.00047

0.00710 0.00045

0.00708 0.000470.00714 0.00047

0.00714 0.00046

0.00709 0.00047

0.00708 0.00047

0.00710 0.00045

0.00710 0.00046

0.00713 0.00047

0.00710 0.00045

0.00708 0.00047

7.1103 4.6204

•4

:

10B0.4

1011.C

−

−

=

=

xdr

xdR


15/17

FISICA II FIC

•1:

4

2

10H42.:

.12

x R

G

r

N R

R

G

=∂∂

=∂∂

•14

:

10HH:.C

.1B

xr

G

r

N R

r

G

−=∂∂

−=∂∂

•10

10J4H.H xG

dr r

GdR

R

GG

−=∆

∂

∂+

∂

∂=∆

E% rr&r as&%u!& s9 :


16/17


17/17

FISICA II FIC

1. GO@DEMKERG7 . Fs#ca G$ra% / E)r#"$!a%7 V&% I E(#!.

I$!ra"r#ca$a S.A. M*#c& 1J2.

2. A%&$s& F#$$ Fs#ca ,$ra%.

:. Ha%%#(a/ Rs$#c Fs#ca ,$ra% !&"& I.5

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