simple and physical pendulums
TRANSCRIPT
-
7/26/2019 Simple and Physical Pendulums
1/29
SIMPLE AND PHYSICAL
PENDULUMSENERGY OF OSCILLATING
SYSTEMSLECTURE 43
-
7/26/2019 Simple and Physical Pendulums
2/29
Lecture Objectives
! I"e#ti$% t&e 'eri(" )#" t&e $re*ue#c% ($s(+e (sci,,)ti#- s%ste+s #)+e,% s'ri#-+)ss. si+',e 'e#"u,u+ )#" '&%sic),'e#"u,u+!/! A#),%0e t&e +(ti(# ($ )# (sci,,)ti#-
s%ste+ i# ter+s ($ t&e )b(ve +e#ti(#e"*u)#tities i#c,u"i#- e#er-%!
-
7/26/2019 Simple and Physical Pendulums
3/29
Again for periodic motion
AmplitudeM)1i+u+ "is',)ce+e#t $r(+ e*ui,ibriu+
PeriodTi+e $(r (#e c%c,e 2
Frequency
Nu+ber ($ c%c,es 'er u#it ti+e 2
Angular frequency2
-
7/26/2019 Simple and Physical Pendulums
4/29
-A
A
Period T = time it take for the bob to!ing from A back to A "or from #A anback to -A$
%
-
7/26/2019 Simple and Physical Pendulums
5/29
-A
A
The time it reach from #A to A i
T&2
%
-
7/26/2019 Simple and Physical Pendulums
6/29
-A
A
'a(imum peed i at (=% "at
time T&)$
%
-
7/26/2019 Simple and Physical Pendulums
7/29
*imple pendulum
,dealied model
Point ma upendedby a male.untretchable tring
/etoring force i equal
to the tangentialcomponent of the netforce
Tenion act to makethe point ma mo0e in
-
7/26/2019 Simple and Physical Pendulums
8/29
*imple pendulum
F(r ) s+),, )#-,e
)''r(1i+)ti(#.
Derive" 'eri(". $re*ue#c% )#" )#-u,)r$re*ue#c%5
-
7/26/2019 Simple and Physical Pendulums
9/29
*ample Problem A imple pendulumFi#" t&e 'eri(" )#" $re*ue#c% ($ ) si+',e'e#"u,u+ !666+ ,(#- )t ) ,(c)ti(# 7&ere - 89!:66+;s/!
*olutionFr(+ t&e "e
-
7/26/2019 Simple and Physical Pendulums
10/29
Phyical pendulum
Any real pendulum thatue an e(tended body
!ith all the maconcentrated at a inglepoint
,mportant quantitieA#-u,)r "is',)ce+e#t
Dist)#ce ($ 'iv(t '(i#t t( t&ece#ter ($ -r)vit%
M(+e#t ($ i#erti) ($ t&e b("%T(t), +)ss
-
7/26/2019 Simple and Physical Pendulums
11/29
Phyical pendulum
F(r ) s+),, )#-,e
)''r(1i+)ti(#.
Period. frequency and angular frequency
-
7/26/2019 Simple and Physical Pendulums
12/29
A#-u,)r $re*ue#
Fre*ue#c%
Peri("
A#-u,)r $re*ue#c%
Fre*ue#c%
Peri("
Phyical Pendulu"mall amplitude
*imple Pendulum"mall amplitude$
-
7/26/2019 Simple and Physical Pendulums
13/29
A 7rec=i#- b),, 7&ic& is ) s(,i" +et),
s'&ere 2I 8 /;> MR/
&)s ) +)ss M )#"r)"ius R! T&e c&)i# c(##ecti#- itssur$)ce $r(+ t&e cr)#e &)s ) ,e#-t&e*u), t( t7ice its r)"ius! Assu+e t&)tt&e )#-,e ($ (sci,,)ti(# is s+),,.c),cu,)te t&e 7rec=i#- b),,?s 'eri(" )s(sci,,)tes! C),cu,)te it?s )#-u,)r$re*ue#c%!
I c)+e i# ,i=e@
-
7/26/2019 Simple and Physical Pendulums
14/29
N(7. c(#si"er ) '(i#t +)ss M c(##ecte"
i#t( ) stri#- ($ ,e#-t& e*u), t( t&rice t&er)"ius ($ t&e 7rec=i#- b),, i# t&e 'revi(us'r(b,e+! C),cu,)te t&e 'eri(" ($ its(sci,,)ti(# )s it +(ves 7it& ) ver% s+),,)#-,e! Deter+i#e t&e r)ti( ($ its 'eri("
7it& t&e 'eri(" ($ t&e 7rec=i#- b),,!
rec=i#- b),, @ #(t )#%+(reB
-
7/26/2019 Simple and Physical Pendulums
15/29
E#er-% i# SHM
-
7/26/2019 Simple and Physical Pendulums
16/29
,n *5'. retoring force i proportionato diplacement
6onider a pring-ma ytem
7on-contant acceleration.
*5' i a imple cae of periodic moti
Again *imple harmonic motion"*5'$
9 i *5'
-
7/26/2019 Simple and Physical Pendulums
17/29
9nergy in *5'
7o non-coner0ati0e force
7egligible ma of the pring
From the coner0ation of energy.
Again peed at any point
-
7/26/2019 Simple and Physical Pendulums
18/29
Againpeed at any point
M)1i+u+ s'ee" (ccurs )t t&e e*ui,ib'(siti(#.
-
7/26/2019 Simple and Physical Pendulums
19/29
*ample Problem 0elocity. acceleration and energy
I# t&e (sci,,)ti(# setu' 7it& = 8 /66N;+. + 8 6!>6(sci,,)ti#- +)ss is re,e)se" $r(+ rest )t 18 A 8 6!62)Fi#" t&e +)1i+u+ )#" +i#i+u+ ve,(cities )tt)i
(sci,,)ti#- b("%!2bC(+'ute t&e +)1i+u+ )cce,er)ti(#!2cDeter+i#e t&e ve,(cit% )#" )cce,er)ti(# 7&e# t&
+(ve" &),$7)% t( t&e ce#ter $r(+ its (ri-i#), '(s2"Fi#" t&e t(t), e#er-%. '(te#ti), e#er-% )#" =i#et
)t t&is '(siti(#!
-
7/26/2019 Simple and Physical Pendulums
20/29
2) T&e ve,(cit% v1)t )#% "is',)ce+e#t
1 is5
T&e +)1i+u+ ve,(cit% (ccurs 7&e# t&e b("% is+(vi#- t( t&e ri-&t t&r(u-& t&e e*ui,ibriu+ '(siti(7&ere 1 8 65
T&e +i#i+u+ 2+(st #e-)tive ve,(cit% (ccurs 7&et&e b("% is +(vi#- t( t&e ,e$t t&r(u-& 1 8 6 7it&v),ue v+)18 6!46+;s
-
7/26/2019 Simple and Physical Pendulums
21/29
2b T&e +)1i+u+ 2+(st '(sitive )cce,er)ti(#(ccurs )t t&e +(st #e-)tive v),ue ($ 1 1 8 At&ere$(re si#ce5
T&e +i#i+u+ 2+(st #e-)tive )cce,er)ti(# is:!6+;s/ (ccurri#- )t 1 8 A 8 6!6/6+!
-
7/26/2019 Simple and Physical Pendulums
22/29
2c H),$7)% t( t&e ce#ter $r(+ t&e i#iti), '(siti(# 1 A;/ 8 6!66+ t&e ve,(cit% is5
Ne-)tive s*u)re r((t is c&(se# bec)use t&e b("is +(vi#- $r(+ 1 8 A t(7)r" 1 8 6! T&e)cce,er)ti(# )t t&is '(i#t is5
ve,(cit% )#" )cce,er)ti(# &)ve t&e s)+e si-#t&ere$(re t&e s'ee" is i#cre)si#-
-
7/26/2019 Simple and Physical Pendulums
23/29
2" T&e t(t), e#er-% )t ),, '(i#ts "uri#- t&e+(ti(# is5
T&e '(te#ti), e#er-% )t 1 8 A;/ is5
T&e =i#etic e#er-% )t 1 8 A;/ is5
-
7/26/2019 Simple and Physical Pendulums
24/29
Gr)'&s ($ E. )#" U versus "is',)ce+e#t i# SHM! Tve,(cit% ($ t&e b("% is #(t c(#st)#t. s( t&ese i+)-est&e b("% )t e*u),,% s')ce" '(siti(#s )re #(t e*u),,%
s')ce" i# ti+e!
-
7/26/2019 Simple and Physical Pendulums
25/29
C(#si"er ) S'ri#- +)ss s%ste+! A b,(c=
($ +)ss mis c(##ecte" i#t( ),i-&t7ei-&t s'ri#- ($ c(#st)#t k!C),cu,)te t&e +)1i+u+ "is',)ce+e#t ($t&e b,(c= i$ )t t&e '(siti(# 1 $r(+e*ui,ibriu+. t&e b,(c=?s i#etic E#er-% is
equalt( its P(te#ti), E#er-%! 2E1'ress i#ter+s ($ t&e -ive# *u)#tities
E1ercise
-
7/26/2019 Simple and Physical Pendulums
26/29
;ui
; i
-
7/26/2019 Simple and Physical Pendulums
27/29
Si+',e 'e#"u,u+ (# E)rt&2-E89!:6+;s
/
Si+',e 'e#"u,u+ (#M)rs2-M83!+;s/
2Hi#t5 T)=e t&e r)ti( TM;TEt( e,i+i#)te
L
;ui
-
7/26/2019 Simple and Physical Pendulums
28/29
C(#si"er ) S'ri#- +)ss s%ste+! A b,(c=
($ +)ss mis c(##ecte" i#t( ),i-&t7ei-&t s'ri#- ($ c(#st)#t k 7it&)+',itu"e ($ (sci,,)ti(# A! H(7 $)r $r(+t&e e*ui,ibriu+ '(siti(# is t&e b,(c=7&e# its i#etic E#er-% is equalt( its
P(te#ti), E#er-%! 2E1'ress i# ter+s ($ t&e-ive# *u)#tities
-
7/26/2019 Simple and Physical Pendulums
29/29
Si+',e 'e#"u,u+ (# E)rt&2-E89!:6+;s
/
Si+',e 'e#"u,u+ (#M)rs2-M83!+;s
/
Gr)vit% is 7e)=er i# M)rs. s( t&e 'eri(" is ,(#