Thus the force is proportional but opposite to the displacement away from equilibrium.

Transcription

1 Chaper 3 : Siple Haronic Moion Hooe s law saes ha he force (F) eered by an ideal spring is proporional o is elongaion l F= l where is he spring consan. Consider a ass hanging on a he spring. In equilibriu he force of graviy on he ass (-g) is equal and opposie o he force eered by he spring (g= l) One can easure he by ploing l versus g. In SI has unis N/ g - where g 9.8s l If we displace he ass by an aoun fro equilibriu he oal force on he ass is hen: F ( l ) g Thus he force is proporional bu opposie o he displaceen away fro equilibriu.

2 If we release he ass a ie = wha is he displaceen () a soe laer ie laer? Newon s Law ells us d a F or - ; Eqn. () d Observaions abou he resuling oion:. he oion repeas iself; ha is i is periodic; ( ) ( nt ) where n is any ineger and T is he period of he oion where he period T or frequency f=/t is only a funcion of /.. A any ie he force F() and resuling acceleraion a() are proporional o he displaceen. Clearly a() is ie dependen (i.e. no consan). d ( ) a( ) d ( ) 3. Velociy is aiu when = and zero when = aiu. This follows fro energy conservaion. The su of poenial energy (U) and ineic energy (K) us say consan. K( v) v K U U ( ) consan F( ') d' v ' d' consan

3 Soluion o Eqn. We can show eperienally ha he soluion of Eqn wih he iniial condiion v()= and ()=A a = is ( ) A cos ; Eqn where angular frequency f and f is he frequency T If he vecor A roaes a an angular frequency such ha hen he projecion of A on he -ais (poin P) has he following for ()=Acos. Eperien shows ha his funcion closely aches he oscillaing oion of he ass on a spring. The shadow on he screen (poin P) oves wih velociy d A van sin sin Asin ; Eqn 3 d T whereas he acceleraion of poin P on he ais d d a rad cos v A an cos Acos ; Eqn. 4

4 Insering Eqns 4 and ino Eqn yields d ( ) d Acos or ( ); f Eqn Acos T ( ) Acos Unis of are radians/s (or rad/s) whereas for he frequency f has unis s -, Herz or Hz. Noe he frequency (period of he oion) does no depend on he apliude of vibraion. Chec if his is rue.

5 Acos A increases fro o o 3 ass increases fro o o 3. Chec eperienally if T increases fro o o 3

6 Phase angle I is possible ha a ie = he apliude is no a is aiu and he velociy is herefore non zero. Why? This siuaion is described by including an iniial phase angle ino he soluion ( ) Acos( ); Eqn 5 Noe he period and apliude of he funcion are unchanged. The phase angle siply produces an offse on he ie ais = - i follows ha he velociy and acceleraion are hen given by v( ) a( ) d d d d A sin( A cos( ) )

7 Show ha ( ) Acos( d is also soluion of - ; Eqn. () d Consider for eaple an oscillaion wih phase = /3 ) ( ) Acos( / 3); ( ) A/ v( ) d d A sin( / 3); v() 3 A? a( ) d d A cos( / 3); a() A /

8 Noe any SHM is described by 3 paraeers, A and Given hese you can deerine he posiion, velociy or acceleraion a any poin in ie. Alernaively one can deerine soe or all of he paraeers given specific inforaion abou he posiion, velociy or acceleraion a paricular ie. Each piece of inforaion provides an equaion relaing he 3 paraeers. For eaple, suppose you now =(/) / Given he iniial velociy is v and iniial posiion is hen you can calculae A and as follows: v o Asin ; o Acos v Asin Acos an or arcan v and v A [sin cos ] A Suppose you now he iniial acceleraion is a are he apliude and phase? and iniial velociy is wha

9 Eaple: Suppose a.5g ass is aached o a horizonal spring. The spring is pulled on wih a force of 6N and his causes a displaceen of =.3. Then he ass is released. (a) Wha are he velociy and posiion of he ass s laer? (b) Suppose if a = he velociy is./s and posiion is -. wha are he posiion and velociy s laer

10 Energy in SHM Reurn o he hanging ass on a spring. The oal energy is he su of ineic plus poenial energy. K U U K v ( l ( l l l A sin ) g ; l A ; ) l l A g g A independen of cos sin ie A sin

11 Siple pendulu Suppose a poin ass is suspended on a sring of lengh L. A ie = he ass is displaced by an aoun or he angle away fro equilibriu. Then here is a resoring force F hus g sin One can easure g g L g ( g L g ; independen of L f) L ; or g L! Wha happens when is no sall?

12 U ube c Consider a colun of waer of lengh l=.5 and area a in an open U ube. A ie = he waer is saionary and displaced by an aoun =c. Wha is he level on he righ hand side s laer? waer

13 Daped Haronic Oscillaor In all real oscillaors he oal energy U+K does no reain consan due o soe resisance o he oion which leads o dissipaion of he energy. Ofen his daping force is proporional o bu opposie in direcion o he velociy. d d d b or using F a b ; d d d F oal or where and b bd d d d or is he undaped frequency is he daping rae d d The soluion o Eqn * has 3 has a differen funcional fors depending on how big he agniude of he daping rae (/ ). d d Eqn *. Underdaped oion when or b<() /. The oion is a siilar o an undaped oscillaor. ( ) Ae cos( ' ' 4 )

14 Noe he apliude of he oscillaions decreases eponenially. Thus he oal energy of he oscillaor a he peas is jus he poenial energy since K=. - p/ - p/ E( p ) A e Ee ; p nt / are he ies when he apliude pea. Also he presence of daping reduces he oscillaion frequency and reaches zero when ' 4 A any insan E de d v dv v d d d v( a ) bv Thus he insananeous rae of energy loss varies during he period sin. However if he aoun of energy los per period is sall copared o he oal energy a ha ie hen we can replace v by is average value over one period which is jus equal o v E / where E is he average oal energy for ha cycle d E E b / Thus E E e ; noe his is consisen wih he above d epression for he energy a he pea ies.

15 Criical daping: In his circusance here is no oscillaion and he / soluion of Eqn * is jus a single eponenial. Verify ( ) Ae saifies Eqn *. This is useful if you wan he oscillaor o reurn o equilibriu as quicly as possible. For eaple shoc absorbers in a car provide daping of he springs so ha afer a bup he poenial energy in he springs is dissipaed as quicly as possible. Overdaping: If he daping rae hen he reurn o equilibriu is very slow and in general is characerized by a su of wo eponenials a a Ce Ce resuling in fro he iaginary arguen o he cosine funcion since when ' is an iaginary nuber

16 Iagine an oscillaor in a very viscous ediu such as olasses such ha / >> or b>>() /. Then show ha he approiae soluion o Eqn * is Ae A / b which ay also be derived by assuing he acceleraion er is zero. Noe he ass of he objec does no affec he oion in his case!

17 Driven oscillaor and resonance Consider a ass on a spring wih a naural frequency / Now suppose on eers a ie varying force o he ass a a differen frequency Wha does he oion loo lie? To approiae his one ae our ass on a spring and ove he poin of aachen up and down wih an apliude y a a frequency as shown on he righ.. Wha happens when is far away y( ) F y ( y( )) cos y F cos cos Wha happens when approaches? To be ore quaniaive we can neglec any daping so ha he equaion of oion is d d F cos ; Eqn **

18 Since he force is varying a a frequency i is reasonable o epec a soluion =Acos. Subsiuing his ino Eqn ** we ge F / F / A A F / ; or A / Thus () F / cos is a soluion. A Noe he negaive apliude for is equivalen o a phase shif of A cos( ); for for Noe he apliude of vibraion diverges when he driving frequency aches he naural frequency of vibraion. This iplies ha he oal energy in he oscillaor increases as approaches. Where does he energy coe fro?

19 Fro wha you saw in he deonsraion do you hin here igh be anoher soluion? Wha happens if a = he ass is saionary bu displaced fro equilibriu bu you sill have he driving force on?

20 Driven haronic oscillaor wih daping Wha do you hin will happen if we include daping? d d d d F cos As before if we sar he ass in he equilibriu posiion wih no velociy and hen urn on he driving force we epec an oscillaion a he driving frequency ( ) Acos( ) ; where A F / Noe if he firs er in he square roo is uch greaer han he second (far away fro resonance) hen he soluion is he sae as for an undaped oscillaor. As before he oscillaion is in phase wih he driving frequency for < ( =) whereas hey are 8 degrees ou of phase for >.

21 As he daping rae becoes saller he resonance is ore sharply peaed and higher. The apliude is siilar o he undaped case. b=.() / or Below he resonance oscillaion and driving frequency are in phase. On resonance he oscillaiona nd driving force are 9 degrees ou of phase and well above he resonance hey are 8 degree ou of phase.

22 Suppose he driving frequency is well above he naural frequency. Wha do hin will happen if insead of saring he ass off in equilibriu we sar if off ou of equilibriu. Wha will he soluion loo lie? The Q-facor of an oscillaor is defined as o he energy los per cycle ies he raio of energy relaive Q E E T Ee E / e / T

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