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 ,(#