Simple Harmonic Motion

Transcription

1 Sipe Hronic Moion 75 6 Sipe Hronic Moion Periodic nd Oscior (Vibror) Moion () oion, which repe isef over nd over in fer reur inerv of ie is ced periodic oion Revouion of erh round he sun (period one er), Roion of erh bou is por is (period one d), Moion of hour s hnd of coc (period -hour) ec re coon epe of periodic oion. () Oscior or vibror oion is h oion in which bod oves o nd fro or bc nd forh repeed bou fied poin in definie inerv of ie. In such oion, he bod is confined wih in wedefined iis on eiher side of en posiion. Oscior oion is so ced s hronic oion. (i) Coon epes re () he oion of he penduu of w coc he oion of od ched o sprin, when i is pued nd hen reesed. he oion of iquid conined in U-ube when i is copressed once in one ib nd ef o isef. oded piece of wood foin over he surfce of iquid when pressed down nd hen reesed eecues oscior oion. (ii) Hronic osciion is h osciion which cn be epressed in ers of sine hronic funcion (i.e. sine or cosine funcion). pe : sin or cos (iii) Non-hronic osciion is h osciion which cn no be epressed in ers of sine hronic funcion. I is cobinion of wo or ore hn wo hronic osciions. pe : sin b sin. Sipe Hronic Moion () Sipe hronic oion is speci pe of periodic oion, in which price oves o nd fro repeed bou en posiion. () In iner S.H.M. resorin force which is ws direced owrds he en posiion nd whose niude n insn is direc proporion o he dispceen of he price fro he en posiion h insn i.e. Resorin force Dispceen of he price fro en posiion. F F = Where is nown s force consn. Is S.I. uni is Newon/eer nd diension is [M ]. () In sed of srih ine oion, if price or cenre of ss of bod is osciin on s rc of circur ph, hen for nur S.H.M. Resorin orque () nur dispceen () Soe Iporn Definiions () ie period () : I is he es inerv of ie fer which he periodic oion of bod repes isef. S.I. uni of ie period is second. () Frequenc (n) : I is defined s he nuber of osciions eecued b bod per second. S.I uni of frequenc is herz (Hz). () nur Frequenc () : nur frequenc of bod eecuin periodic oion is equ o produc of frequenc of he bod wih fcor. nur frequenc = n

2 75 Sipe Hronic Moion Is uni is rd/sec. () Phse () : Phse of vibrin price n insn is phsic quni, which copee epress he posiion nd direcion of oion, of he price h insn wih respec o is en posiion. In oscior oion he phse of vibrin price is he ruen of sine or cosine funcion invoved o represen he enerised equion of oion of he vibrin price. 0. sin sin( 0 ) here, 0 = phse of vibrin price. 0 = Inii phse or epoch. I is he phse of vibrin price = 0 = 0 sin sin sin n sin( ) (i) sin when he ie is noed fro he insn when he vibrin price is en posiion. (ii) cos when he ie is noed fro he insn when he vibrin price is eree posiion. (iii) sin( ) when he vibrin price is phse edin or in fro he en posiion. () If he projecion of P is en on X-is hen equions of S.H.M. cn be iven s cos ( ) cos cos (n ) + + = sin = sin = cos = cos Fi. 6. () Se phse : wo vibrin price re sid o be in se phse, if he phse difference beween he is n even uipe of or ph difference is n even uipe of ( / ) or ie inerv is n even uipe of ( / ) becuse ie period is equiven o rd or wve enh (). () Opposie phse : When he wo vibrin prices cross heir respecive en posiions he se ie ovin in opposie direcions, hen he phse difference beween he wo vibrin prices is 80o. Opposie phse ens he phse difference beween he price is n odd uipe of (s,, 5, 7..) or he ph difference is n odd uipe of (s,,...) or he ie inerv is n odd uipe of ( / ). () Phse difference : If wo prices perfors S.H.M nd heir equion re sin( ) nd sin( ) hen phse difference ) ( ) Dispceen in S.H.M. ( () he dispceen of price eecuin S.H.M. n insn is defined s he disnce of price fro he en posiion h insn. () Sipe hronic oion is so defined s he projecion of unifor circur oion on n dieer of circe of reference. () If he projecion is en on -is. hen fro he fiure X N O Y Y Fi. 6. P = M X (5) Direcion of dispceen is ws w fro he equiibriu posiion, price eiher is ovin w fro or is coin owrds he equiibriu posiion. Veoci in S.H.M. () Veoci of he price eecuin S.H.M. n insn, is defined s he ie re of chne of is dispceen h insn. () In cse of S.H.M. when oion is considered fro he equiibriu posiion, dispceen sin So v d d cos sin [s sin = /] () en posiion or equiibriu posiion ( = 0 nd = = 0), veoci of price is iu nd i is =. v () eree posiion ( = nd = =/), veoci of osciin price is zero i.e. v = 0. v (5) Fro v v ( ) () v his is he equion of eipse. Hence he rph beween v nd is n eipse. For =, rph beween v nd is circe. Fi. 6.5 (6) Direcion of veoci is eiher owrds or w fro en posiion dependin on he posiion of price. cceerion in S.H.M. Fi. 6. () he cceerion of he price eecuin S.H.M. n insn, is defined s he re of chne of is veoci h insn. So cceerion v (B)

3 Sipe Hronic Moion 75 dv d d d ( cos ) sin [s sin ] () In S.H.M. s cceerion is no consn. So equions of rnsor oion cn no be ppied. Hence () In S.H.M. cceerion is iu eree posiion ( = ). when sin iu i.e.. Fro equion (ii) when. (i) In S.H.M. cceerion is iniu en posiion or Fro equion (i) in 0 when sin 0 i.e. 0 or or. Fro equion (ii) in 0 when 0 (ii) cceerion is ws direced owrds he en posiion nd so is ws opposie o dispceen i.e., Grph beween cceerion () nd dispceen () is + srih ine s shown Sope of he ine = Coprive Sud of Dispceen Veoci nd cceerion () he hree quniies dispceen, veoci nd cceerion show hronic vriion wih ie hvin se period. () he veoci piude is ies he dispceen piude () he cceerion piude is piude / ies he dispceen () In S.H.M. he veoci is hed of dispceen b phse ne (5) In S.H.M. he cceerion is hed of veoci b phse ne / (6) he cceerion is hed of dispceen b phse ne of be 6. : Vrious phsic quniies in S.H.M. differen posiion : Dispceen + O Grph Foru en posiion / Fi. 6.6 sin = 0 eree posiion = F cceerion O Force O ner in S.H.M. or v sin sin( ) or F or sin F in = 0 F in = 0 = F () Poeni ener : his is n ccoun of he dispceen of he price fro is en posiion. (i) he resorin force F = ins which wor hs o be done. Hence poeni ener U is iven b U du dw 0 Fd d where = Poeni ener equiibriu posiion. U0 If U0 = 0 hen U [s / ] (ii) so U sin ( cos ) 0 + U0 [s sin ] Hence poeni ener vries periodic wih doube he frequenc of S.H.M. (iii) Poeni ener iu nd equ o o ener eree posiions U / / when ; / ; (iv) Poeni ener is iniu en posiion U in 0 when 0 ; 0 ; 0 () ineic ener : his is becuse of he veoci of he price ineic ner v ( ) [s v ] Veoci v O / v cos sin( ) v = v in = 0 (i) so cos ( cos ) [s v cos ] Hence ineic ener vries periodic wih doube he frequenc of S.H.M.

4 75 Sipe Hronic Moion (ii) ineic ener is iu en posiion nd equ o o ener en posiion. when 0 ; 0 ; 0 (iii) ineic ener is iniu eree posiion. in 0 when ; /, / () o echnic ener : o echnic ener ws reins consn nd i is equ o su of poeni ener nd ineic ener i.e. U ( ) o ener is no posiion funcion. () ner posiion rph ner Fi. 6.7 (i) = 0; U = 0 nd = (ii) = ; U = nd =0 (iii) (iv) ; U nd ; U vere Vue of P.. nd.. he vere vue of poeni ener for copee cce is iven b U vere 0 U d 0 sin ( ) he vere vue of ineic ener for copee cce vere = 0 d 0 cos d hus vere vues of ineic ener nd poeni ener of hronic oscior re equ nd ech equ o hf of he o ener vere U vere. Differeni quion of S.H.M. For S.H.M. (iner) cceerion (Dispceen) d or or d d or 0 [s d For nur S.H.M. = 0 =+ ] o ener () Poeni ener (U) ineic ener () d c nd 0 d c where [s c = Resorin orque consn nd I = Moen I of ineri] How o Find Frequenc nd ie Period of S.H.M. Sep : When price is in is equiibriu posiion, bnce forces cin on i nd oce he equiibriu posiion heic. Sep : Fro he equiibriu posiion, dispce he price sih b dispceen nd find he epression of ne resorin force on i. Sep : r o epress he ne resorin force cin on price s proporion funcion of is dispceen fro en posiion. he fin epression shoud be obined in he for. F Here we pu ve sin s direcion of F is opposie o he dispceen. If be he cceerion of price his dispceen, we hve Sep : Coprin his equion wih he bsic differeni equion of S.H.M. we e or n s is he nur frequenc of he price in S.H.M., is ie period of osciion cn be iven s (i) In differen pes of S.H.M. he quniies nd wi o on in differen fors nd nes. In ener is ced ineri fcor nd is ced sprin fcor. hus Ineri fcor Sprin fcor or n Sprin fcor Ineri fcor (ii) In iner S.H.M. he sprin fcor snds for force per uni dispceen nd ineri fcor for ss of he bod eecuin S.H.M. nd in nur S.H.M. snds for resorin orque per uni nur dispceen nd ineri fcor for oen of ineri of he bod eecuin S.H.M. For iner S.H.M. Sipe Penduu Force/Dispceen Dispcee n cceeri on () n ide sipe penduu consiss of hev poin ss bod (bob) suspended b weihess, ineensibe nd perfec feibe srin fro riid suppor bou which i is free o oscie. () Bu in rei neiher poin ss nor weihess srin eis, so we cn never consruc sipe penduu sric ccordin o he definiion. () Suppose sipe penduu of enh is dispced hrouh s ne fro i s en (veric) posiion. Consider ss of he bob is nd iner dispceen fro en posiion is S P sin O cos Fi. 6.8

5 Sipe Hronic Moion 755 Resorin force cin on he bob F sin or F (When is s sin So ~ rc = Lenh F (Sprin fcor) Ineri fcor Sprin fcor / OP = ) Fcor ffecin ie Period of Sipe Penduu () piude : he period of sipe penduu is independen of piude s on s is oion is sipe hronic. Bu if is no s, sin hen oion wi no rein sipe hronic bu wi becoe oscior. In his siuion if 0 is he piude of oion. ie period sin () Mss of he bob : ie period of sipe penduu is so independen of ss of he bob. his is wh (i) If he soid bob is repced b hoow sphere of se rdius bu differen ss, ie period reins unchned. (ii) If ir is swinin in swin nd noher sis wih her, he ie period reins unchned. () Lenh of he penduu : ie period where is he disnce beween poin of suspension nd cener of ss of bob nd is ced effecive enh. (i) When siin ir on swinin swin snds up, her cener of ss wi o up nd so nd hence wi decrese. (ii) If hoe is de he boo of hoow sphere fu of wer nd wer coes ou sow hrouh he hoe nd ie period is recorded i he sphere is ep, inii nd fin he cener of ss wi be he cener of he sphere. However, s wer drins off he sphere, he cener of ss of he sse wi firs ove down nd hen wi coe up. Due o his nd hence firs increse, reches iu nd hen decreses i i becoes equ o is inii vue. (iii) Differen rphs Fi. 6.9 () ffec of : i.e. s increse decreses. (i) s we o hih bove he erh surfce or we o deep inside he ines he vue of decrese, hence ie period of penduu () increses. (ii) If coc, bsed on sipe penduu is en o hi (or on n oher pne), wi decrese so wi increses nd coc wi becoe sower. (iii) Differen rphs Fi. 6.0 (5) ffec of eperure on ie period : If he bob of sipe penduu is suspended b wire hen effecive enh of penduu wi increse wih he rise of eperure due o which he ie period wi increse. ( ) (If is he rise in eperure, inii 0 enh of wire, = fin enh of wire) So ( ) / i.e. Osciion of Penduu in Differen Siuions () Osciion in iquid : If bob sipe penduu of densi is de o oscie in soe fuid of densi (where <) hen ie period of sipe penduu es incresed. s hrus wi oppose is weih hence or eff eff. V. i.e. eff. V ' eff. eff. hrus () Osciion under he infuence of eecric fied : If bob of ss crries posiive chre q nd penduu is pced in unifor eecric fied of srenh eff. (i) If eecric fied direced veric upwrds. ffecive cceerion q Fi Q hrus Fi. 6.

6 756 Sipe Hronic Moion So q (ii) If eecric fied is veric downwrd hen eff. q q () Penduu in if : If he penduu is suspended fro he ceiin of he if. (i) If he if is res or ovin down wrd /up wrd wih consn veoci. nd n Fi. 6. (ii) If he if is ovin up wrd wih consn cceerion nd n Fi. 6.5 ie period decreses nd frequenc increses (iii) If he if is ovin down wrd wih consn cceerion nd n ie period increse nd frequenc decreses Fi. 6.6 (iv) If he if is ovin down wrd wih cceerion nd n = 0 Fi. 6.7 I ens here wi be no osciion in penduu. Siir is he cse in seie nd he cenre of erh where effecive cceerion becoes zero nd penduu wi sop. () Penduu in n cceered vehice : he ie period of sipe penduu whose poin of suspension ovin horizon wih cceerion Fi Q = In his cse effecive cceerion eff. / nd n ( / ) ( ) If sipe penduu suspended in cr h is ovin wih consn speed v round circe of rdius r. v r Soe Oher pes of Penduu () Infinie enh penduu : If he enh of he penduu is coprbe o he rdius of erh hen (i) If R, hen R (ii) If R( )hen R so R R 6. 0 so 8. 6 inues 0 nd i is he iu ie period which n osciin sipe penduu cn hve R (iii) If R so hour () Second s Penduu : I is h sipe penduu whose ie period of vibrions is wo seconds. Puin = sec nd = 99. c / sec in we e Hence enh of second s penduu is 99. c or ner eer on erh surfce. [s For he oon he enh of he second s penduu wi be /6 eer rh oon ] 6 () Copound penduu : n riid bod suspended fro fied suppor consiues phsic penduu. Consider he siuion when he bod is dispced hrouh s ne. orque on he bod bou O is iven b G O I eff. Fi. 6.9 CM I 0 Fi. 6.8

7 Sprin consn () Sipe Hronic Moion 757 Br sin (i) where = disnce beween poin of suspension nd cenre of ss of he bod. If I be he M.I. of he bod bou O. hen I (ii) d d Fro (i) nd (ii), we e I sin s nd re d d opposie direced d since is ver s d I d Coprin wih he equion. we e d I so I I I (Pre is heore) c (where = rdius of rion) eff = ffecive enh of penduu = Disnce beween poin of eff suspension nd cenre of ss. be 6. : Soe coon phsic penduu Bod, ie period (v) Sprin consn depend upon rdius nd enh of he wire used in sprin. (vi) he sprin consn is inverse proporion o he sprin enh. Fi. 6.0 ension Lenhof sprin h ens if he enh of sprin is hved hen is force consn becoes doube. (vii) When sprin of enh is cu in wo pieces of enh nd such h n. If he consn of sprin is hen sprin consn of firs pr ( n ) n Sprin consn of second pr nd rio of sprin consn Sprin Penduu Lenh of he sprin () ( n ) n poin ss suspended fro ss ess sprin or pced on fricioness horizon pne ched wih sprin (fi.) consiues iner hronic sprin penduu Rin Disc Sprin Sse R R When sprin is sreched or copressed fro is nor posiion ( = 0) b s disnce, hen resorin force is produced in he sprin becuse i obes Hoo s w i.e. F F where is ced sprin consn. R R (i) I s S.I. uni Newon/ere, C.G.S uni Dne/c nd diension is [M ] (ii) cu is esure of he siffness/sofness of he sprin. (iii) For ssess sprin consn resorin esic force is se ever where (iv) When sprin copressed or sreched hen wor done is sored in he for of esic poeni ener in i. ie period nd Frequenc Fi. 6. Ineri fcor Sprin fcor n () ie period of sprin penduu depends on he ss suspended or n i.e. reer he ss reer wi be he ineri nd so esser wi be he frequenc of osciion nd reer wi be he ie period. () he ie period depends on he force consn of he sprin i.e. or n () ie of sprin penduu is independen of cceerion due o rvi. h is wh coc bsed on sprin penduu wi eep proper ie ever where on hi or oon or in seie nd ie period of sprin penduu wi no chne inside iquid if dpin effecs re neeced.

8 758 Sipe Hronic Moion () Mssive sprin : If he sprin hs ss M nd ss is M suspended fro i, effecive ss is iven b e ff. Hence eff (5) Reduced ss : If wo sses of ss nd re conneced b sprin nd de o oscie on horizon surfce, he reduced ss is r iven b so h r r Fi. 6. (6) If sprin penduu, osciin in veric pne is de o oscie on horizon surfce, (or on incined pne) ie period wi rein unchned. (7) quiibriu posiion for sprin in horizon pin is he posiion of nur enh of sprin s weih is bnced b recion. Whie in cse of veric oion equiibriu posiion wi be 0 wih 0 Fi. 6. If he srech in veric oded sprin is 0 hen for equiibriu of ss, So h 0 i.e. 0 ie period does no depends on becuse on wih, wi so o 0 chne in such w h reins consn 0 Osciion of Sprin Cobinion () Series cobinion : If wo sprins of sprin consns nd re joined in series s shown hen Fi. 6. (i) In series cobinion equ forces cs on sprin bu eension in sprins re differen. (ii) Sprin consns of cobinion s s R (iii) If n sprins of differen force consn re conneced in series hvin force consn,... respecive hen,... S If sprin hve se sprin consn hen (iv) ie period of cobinion S n S ( ) () Pre cobinion : If he sprins re conneced in pre s shown Fi. 6.5 (i) In pre cobinion differen forces cs on differen sprins bu eension in sprins re se (ii) Sprin consns of cobinion P (iii) If n sprins of differen force consn re conneced in pre hvin force consn,... respecive hen P, If sprin hve se sprin consn hen (iv) ie period of cobinion Vrious Forue of S.H.M. P P n P ( ) () S.H.M. of iquid in U ube : If iquid of densi conined in veric U ube perfors S.H.M. in is wo ibs. hen ie period L h where L = o enh of iquid coun, h = Heih of undisurbed iquid in ech ib (L=h) Fi. 6.6 () S.H.M. of foin cinder : If is he enh of cinder dippin in iquid hen ie period Fi. 6.7 () S.H.M. of s b roin down in hei-spheric bow R r h R Fi. 6.8

9 Sipe Hronic Moion 759 R = Rdius of he bow r =Rdius of he b () S.H.M. of pison in cinder Mh P M = ss of he pison = re of cross secion h = heih of cinder P = pressure in cinder (5) S.H.M. of bod in unne du on n chord of erh R = 8.6 inues Fi. 6.0 (6) orsion penduu : In orsion penduu n objec is suspended fro wire. If such wire is wised, due o esici i eer resorin oque = C. In his cse ie period is iven b I C where I = Moen of ineri disc r C = orsion consn of wire = = Moduus of esici of wire nd r = Rdius of wire (7) Loniudin osciions of n esic wire : Wire/srin pued disnce nd ef. I eecues oniudin osciions. Resorin force F Y Y = Youn s oduus = re of cross-secion Hence Y Free, Dped, Forced nd Minined Osciions () Free osciion (i) he osciion of price wih funden frequenc under he infuence of resorin force re defined s free osciions (ii) he piude, frequenc nd ener of osciion reins consn P h Gs Fi. 6.9 Wire Fi. 6. Fi. 6. R Wire M Disc (iii) Frequenc of free osciion is ced nur frequenc becuse i depends upon he nure nd srucure of he bod. () Dped osciion (i) he osciion of bod whose piude oes on decresin wih ie re defined s dped osciion (ii) In hese osciion he piude of osciion decreses eponeni due o dpin forces ie fricion force, viscous force, hsersis ec. (iii) Due o decrese in piude he ener of he oscior so oes on decresin eponeni Fi. 6. (iv) he force produces resisnce o he osciion is ced dpin force. If he veoci of oscior is v hen Dupin force F d bv, b = dpin consn (v) Resun force on dped oscior is iven b d d F FR Fd v b 0 d d (vi) Dispceen of dped oscior is iven b b/ e sin( ) where nur frequenc of he dped oscior = 0 0 ( b/) he piude decreses coninuous wih ie ccordin o e ( b/) (vii) For dped oscior if he dpin is s hen he echnic ener decreses eponeni wih ie s () Forced osciion b e / Fi. 6. (i) he osciion in which bod oscies under he infuence of n eern periodic force re nown s forced osciion (ii) he piude of oscior decrese due o dpin forces bu on ccoun of he ener ined fro he eern source i reins consn. (iii) Resonnce : When he frequenc of eern force is equ o he nur frequenc of he oscior. hen his se is nown s he se of resonnce. nd his frequenc is nown s resonn frequenc. (iv) Whie swinin in swin if ou pp push periodic b pressin our fee ins he round, ou find h no on he osciions cn now be inined bu he piude cn so be incresed. Under his condiion he swin hs forced or driven osciion.

10 760 Sipe Hronic Moion (v) In forced osciion, frequenc of dped oscior is equ o he frequenc of eern force. (vi) Suppose n eern drivin force is represened b F() = cos F0 d he oion of price under cobined cion of () Resorin force ( ) Dpin force ( bv) nd Drivin force F() is iven b bv F cos d b d d d F cos 0 he souion of his equion ives sin( d ) wih piude 0 F / 0 0 ( ) ( b/ ) d nd 0 d 0 ( 0 ) n b / where 0 = Nur frequenc of oscior. (vii) piude resonnce : he piude of forced oscior depends upon he frequenc d of eern force. When d, he piude is iu bu no infinie becuse of presence of dpin force. he corresponds frequenc is ced resonn frequenc ( 0 ). 0 F 0 / 0 Fi. 6.5 (viii) ner resonnce : 0, oscior bsorbs iu ineic ener fro he drivin force sse his se is ced ener resonnce. resonnce he veoci of driven oscior is in phse wih he drivin er. he shrpness of he resonnce of driven oscior depends on he dpin. In he driven oscior, he power inpu of he drivin er in iu resonnce. () Minined osciion : he osciion in which he oss of oscior is copensed b he suppin ener fro n eern source re nown s inined osciion. Super Posiion of S.H.M s (Lissjous Fiures) If wo S.H.M's c in perpendicur direcions, hen heir resun oion is in he for of srih ine or circe or prbo ec. dependin on he frequenc rio of he wo S.H.M. nd inii phse difference. hese fiures re ced Lissjous fiures. Le he equions of wo uu perpendicur S.H.M's of se frequenc be Neiibe dpin Low dpin Hih dpin hen he ener equion of Lissjou's fiure cn be obined s cos sin For = 0 : 0 0 his is srih ine psses hrouh oriin nd i's sope is. re Phse diff.() be 6. : Lissjou's fiures in oher condiions quion 0 (wih ) Obique eipse Obique eipse Srih ine Fiure For he frequenc rio : he wo perpendicur S.H.M's : sin( ) nd sin Differen Lissjou's fiures s foows = (Circe) Fi. 6.6 (ipse) sin nd sin( ) = 0,, Fiure of eih = /, / Doube prbo = / Prbo

11 Sipe Hronic Moion 76 If sin nd b cos re wo S.H.M. hen b he superiposiion of hese wo S.H.M. we e sin b cos sin( ) his is so he = 5/, 7/ Doube prbo Fi. 6.7 = / Prbo equion of S.H.M.; where b nd n ( b / ) In he bsence of resisive force he wor done b sipe penduu in one copee osciion is zero If is he nur piude of penduu hen Heih rises b he bob h = ( cos) Veoci en posiion Suppose bod of ss vibre sepre wih wo differen sprins (of sprin consns nd ) wih ie period nd respecive. nd If he se bod vibres wih series cobinion of hese wo sprins hen for he sse ie period If he se bod vibres wih pre cobinion of hese wo sprins hen ie period of he sse he penduu coc runs sow due o increse in is ie period wheres i becoes fs due o decrese in ie period. If infinie sprin wih force consn,,, 8... respecive re conneced in series. he effecive force consn of he sprin wi be /. Percene chne in ie period wih nd. If is consn nd enh vries b n%. hen % chne in ie n period If is consn nd vries b n%. hen % chne in ie period n (Vid on for s percene chne s 5%). Suppose sprin of force consn oscies wih ie period. If i is divided in o n equ prs hen sprin consn of ech pr wi becoe n nd ie period of osciion of ech pr wi becoe. n If hese n prs conneced in pre hen he sse becoes ' n ff n. So ie period of e If price perfors S.H.M. whose veoci is v disnce fro en posiion nd veoci v disnce v v ; v v v v v v ; v v v v ( cos) Wor done in dispceen W U ( cos).. en posiion en ( cos) ension in he srin of penduu en posiion : v () =( cos ) ere posiion : = cos Dispceen of S.H.M. nd Phse. he phse of price eecuin sipe hronic oion is when i hs [MP P 985] () Miu veoci Miu ener Miu cceerion Miu dispceen. price srs S.H.M. fro he en posiion. Is piude is nd ie period is. he ie when is speed is hf of he iu speed, is dispceen is () cos (-cos) [Hrn C 996; CBS PM 996; MH C 00]. he piude nd he periodic ie of S.H.M. re 5c nd 6sec respecive. disnce of.5c w fro he en posiion, he phse wi be () 5 / / / / 6. wo equions of wo S.H.M. re sin( ) nd b cos( ). he phse difference beween he wo is v h B

12 76 Sipe Hronic Moion () [MP PM 985]

13 5. he piude nd he ie period in S.H.M. is 0.5 c nd 0. sec respecive. If he inii phse is / rdin, hen he equion of S.H.M. wi be () 0.5 sin5 0.5 sin 0.5 sin cos 5 6. he equion of S.H.M. is sin( n ), hen is phse ie is [DPM 00] () n n Sipe Hronic Moion price is osciin ccordin o he equion X 7 cos 0.5, 6. price in S.H.M. is described b he dispceen funcion where is in second. he poin oves fro he posiion of ( ) cos( ). If he inii ( 0) posiion of he price equiibriu o iu dispceen in ie [CPM 989] is c nd is inii veoci is c/ s. he nur frequenc of ().0 sec.0 sec he price is rd / s, hen i s piude is.0 sec 0.5 sec 8. sipe hronic oscior hs n piude nd ie period. () c c he ie required b i o rve fro = o = / is[cbs PM 99; SCR 996; BHU 997] c.5 c () / 6 / / / 9. Which of he foowin epressions represen sipe hronic oion [Rooree 999] () sin( ) B cos( ) n( ) sin cos price is vibrin wih sipe hronic oion wih period of.00 0 sec nd iu speed of /s. he iu dispceen of he price is () [MU (Med.) 999] () None of hese. he phse ( ie ) of price in sipe hronic oion es [MU (n.) 999] () On he posiion of he price ie On he direcion of oion of he price ie Boh he posiion nd direcion of oion of he price ie Neiher he posiion of he price nor is direcion of oion ie [CBS PM 99] () Wve ovin hrouh srin fied boh ends. price is ovin wih consn nur veoci on he rh spinnin bou is own is circuference of circe. Which of he foowin seens is rue [MU (n.) 999] () he price so ovin eecues S.H.M. he projecion of he price on n one of he dieers eecues S.H.M. he projecion of he price on n of he dieers eecues S.H.M.. B bouncin beween wo riid veric ws Price ovin in circe wih unifor speed price is ovin in circe wih unifor speed. Is oion is[cpm 978; CB () Periodic nd sipe hronic Periodic bu no sipe hronic None of he bove. sipe hronic oion is represened b F ( ) 0 sin(0 0.5). he piude of he S.H.M. is () = 0 = 0 = 0 = 5 [DPM 998; CBS PM 000; MH C 00] 5. Which of he foowin equion does no represen sipe hronic oion [er (Med.) 00]. price is eecuin sipe hronic oion wih period of None of he bove seconds nd piude ere. he shores ie i es o rech. wo sipe hronic oions re represened b he equions poin fro is en posiion in seconds is [MC (Med.) 000] 0. sin00 nd 0. cos. he phse () / difference of he veoci of price wih respec o he veoci of price is [I 005] /8 /6 () sin cos sin b cos n 7. price eecues sipe hronic oion of ie period. Find he ie en b he price o o direc fro is en posiion o hf he piude [UPS 00] () / / / 8 / 8. price eecuin sipe hronic oion on -is hs is oion described b he equion sin( ) B. he piude of he sipe hronic oion is B + B B [Oriss J 00] 9. price eecuin S.H.M. of piude c nd = sec. he ie en b i o ove fro posiive eree posiion o hf he piude is [BHU 995] () sec / sec / sec / sec 0. Which one of he foowin is sipe hronic oion periodic

14 () 76 Sipe Hronic Moion 6. wo prices re eecuin S.H.M. he equion of heir oion re 0 sin, 5 sin. Wh is he rio of heir piude [DC 996] () : : 5 : None of hese. he periodic ie of bod eecuin sipe hronic oion is sec. fer how uch inerv fro ie = 0, is dispceen wi be hf of is piude [BHU 998] () 6 sec sec 8 6 sec sec 5. sse ehibiin S.H.M. us possess [C 99] () Ineri on sici s we s ineri sici, ineri nd n eern force sici on 6. If sin nd cos, hen wh is he phse 6 difference beween he wo wves [RP 996] () / / 6 / Veoci of Sipe Hronic Moion. sipe penduu perfors sipe hronic oion bou X = 0 wih n piude nd ie period. he speed of he penduu X wi be [MP PM 987] () [MP PM 99] 5 () 6. bod is eecuin sipe hronic oion wih n nur frequenc rd / s. he veoci of he bod dispceen, when he piude of oion is 60, is 0. [Pb. If C sipe 996; Pb. penduu PM 997; oscies FMC 998; wih n piude of 50 nd CPM 999] ie period of sec, hen is iu veoci is [IIMS 998; MH C 000; DPM 000] () 0 /s 60 / s () 0.0 / s 0.5 / s / s 0 / s. bod of ss 5 is eecuin S.H.M. bou poin wih piude 0 c. Is iu veoci is 00 c/sec. Is veoci wi be 50 c/sec disnce [CPM 976] () sipe hronic oscior hs period of 0.0 sec nd n piude of 0.. he niude of he veoci in he cenre of osciion is [JIPMR 997] () sec 5. price eecues S.H.M. wih period of 6 second nd piude of c. Is iu speed in c/sec is () / [IIMS 98] 6. price is eecuin S.H.M. If is piude is nd periodic ie seconds, hen he iu veoci of he price wi be () / s / s / s / s 7. S.H.M. hs piude nd ie period. he iu veoci wi be () [MP PM 985; CPM 997; UPS 999] 8. bod is eecuin S.H.M. When is dispceen fro he en posiion is c nd 5 c, he correspondin veoci of he bod is 0 c/sec nd 8 c/sec. hen he ie period of he bod is [CPM 99; MP P () sec / sec sec / sec 9. price hs sipe hronic oion. he equion of is oion is 5 sin, where is is dispceen. If he 6 dispceen of he price is unis, hen i veoci is 0.8 / s 0.6 / s. If he dispceen of price eecuin SHM is iven b 0.0 sin(0 0.6) in ere, hen he frequenc nd iu veoci of he price is [FMC 998] () 5 Hz, 66 / s 58 Hz, / s 5 Hz, 66 / s 5 Hz, / s

15 . he iu veoci nd he iu cceerion of bod ovin in sipe hronic oscior re hen nur veoci wi be () rd/sec rd/sec / s nd / s. [Pb. PM 998; MH C 999, 00] 0.5 rd/sec rd/sec. If price under S.H.M. hs ie period 0. sec nd piude 0 (). I hs iu veoci / s 5 / s 0 / s 6 None of hese [RP 000]. price eecuin sipe hronic oion hs n piude of 6 c. Is cceerion disnce of c fro he en posiion is 8 c / s. he iu speed of he price is [MC (n.) 000] () 8 c/s 6 c/s c/s c/s 5. price eecues sipe hronic oion wih n piude of c. he en posiion he veoci of he price is 0 c/s. he disnce of he price fro he en posiion when is speed becoes 5 c/s is () c 5 c ( ) c ( 5) c [MC (Med.) 000] 6. wo prices P nd Q sr fro oriin nd eecue Sipe Hronic Moion on X-is wih se piude bu wih periods seconds nd 6 seconds respecive. he rio of he veociies of P nd Q when he ee is () : : : : [MC 00] 7. price is perforin sipe hronic oion wih piude nd nur veoci. he rio of iu veoci o iu cceerion is [er (Med.) 00] 9. he veoci of price perforin sipe hronic oion, when i psses hrouh is en posiion is () Infini [MH C (Med.) 00; BCC 00] Zero Miniu Miu 0. he veoci of price in sipe hronic oion dispceen fro en posiion is () Sipe Hronic Moion 765 [BCC 00; RPM 00]. price is eecuin he oion cos( ). he iu veoci of he price is () cos sin None of hese [BHU 00; CPM 00]. price eecuin sipe hronic oion wih piude of 0.. cerin insn when is dispceen is 0.0, is cceerion is 0.5 /s. he iu veoci of he price is (in /s) () he piude of price eecuin SHM is c. he en posiion he speed of he price is 6 c/sec. he disnce of he price fro he en posiion which he speed of he price becoes 8 c / s, wi be () c c c c [Pb. P 00]. he iu veoci of sipe hronic oion represened b sin00 is iven b 6 () [BCC 005] 5. he dispceen equion of price is sin cos. he piude nd iu veoci wi be respecive () 5, 0,,, 6. Veoci en posiion of price eecuin S.H.M. is v, he veoci of he price disnce equ o hf of he piude () v v () / v v 8. he nur veociies of hree bodies in sipe hronic oion re 7. he insnneous dispceen of sipe penduu oscior is,, wih heir respecive piudes s,,. If he hree bodies hve se ss nd veoci, hen [BHU 00] iven b cos. Is speed wi be iu ie () () cceerion of Sipe Hronic Moion. Which of he foowin is necessr nd sufficien condiion for S.H.M. [NCR 97]

16 766 Sipe Hronic Moion () Consn period Consn cceerion Proporioni beween cceerion nd dispceen fro equiibriu posiion Proporioni beween resorin force nd dispceen fro equiibriu posiion. If hoe is bored on he dieer of he erh nd sone is dropped ino hoe [CPM 98] () he sone reches he cenre of he erh nd sops here he sone reches he oher side of he erh nd sops here he sone eecues sipe hronic oion bou he cenre of he erh he sone reches he oher side of he erh nd escpes ino spce. he cceerion of price in S.H.M. is [MP PM 99] () ws zero ws consn Miu he eree posiion Miu he equiibriu posiion. he dispceen of price ovin in S.H.M. n insn is iven b sin. he cceerion fer ie is (where is he ie period) [MP P 98] () 5. he piude of price eecuin S.H.M. wih frequenc of 60 Hz is 0.0. he iu vue of he cceerion of he price is () / sec / sec [DPM 998; CBS PM 999; FMC 00; Pb. PM 00; Pb. P 00, 0; CPM 99, 95, 0; RPM 005; MP PM 005] / sec 88 / sec 6. s bod of ss 0.0 is eecuin S.H.M. of piude.0 nd period 0.0 sec. he iu force cin on i is () N 00. N N 76. N 7. bod eecuin sipe hronic oion hs iu cceerion equ o o is () eres / sec nd iu veoci equ 6 eres / sec. he piude of he sipe hronic oion eres 0 9 eres [MP PM 995; DPM 00; RP 00; Pb. P 00] 6 9 eres eres 8. For price eecuin sipe hronic oion, which of he foowin seens is no correc [MP PM 997; IIMS 999; er PM 005] () he o ener of he price ws reins he se he resorin force of ws direced owrds fied poin he resorin force is iu he eree posiions he cceerion of he price is iu he equiibriu posiion 9. price of ss 0 rs is eecuin sipe hronic oion wih n piude of 0.5 nd periodic ie of ( / 5) seconds. he iu vue of he force cin on he price is[mp P 999; MP PM () 5 N.5 N 5 N 0.5 N 0. he dispceen of n osciin price vries wih ie (in seconds) ccordin o he equion = sin. he iu cceerion of he price is pproie () 5.c / s.8c / s.6c / s 0.6c / s. price ovin on he -is eecues sipe hronic oion, hen he force cin on i is iven b () ep ( ) Where nd re posiive consns cos () [CBS PM 99]. bod is vibrin in sipe hronic oion wih n piude of 0.06 nd frequenc of 5 Hz. he veoci nd cceerion of bod is [FMC 999] () 5.65 / s nd 6.8/ s nd 8.9/ s nd 9.8/ s nd 5. 0 / s / s 8. 0 / s / s. price eecues hronic oion wih n nur veoci nd iu cceerion of.5 rd/sec nd 7.5 /s respecive. he piude of osciion is () [IIMS 999; Pb. P 999]. 0.0 boc oscies bc nd forh on horizon surfce. Is dispceen fro he oriin is iven b: ( 0c)cos[( 0rd/ s) /rd]. Wh is he iu cceerion eperienced b he boc [MU (n.) 000] () 0 / s 0 / s 5. In S.H.M. iu cceerion is () piude cceerion is consn 0 / s 0 / s quiibriu None of hese [RP 00; BVP 00]

17 6. price is eecuin sipe hronic oion wih n piude of 0.0 ere nd frequenc 50 Hz. he iu cceerion of he price is [MP P 00] () 00 / s 00 / s Sipe Hronic Moion / s 00 / s () c c 7. cceerion of price, eecuin SHM, i s en posiion is[mh C (Med.) 00] c c () Infini Vries. For price eecuin sipe hronic oion, he ineic ener Miu Zero is iven b o cos. he iu vue of poeni 8. Which one of he foowin seens is rue for he speed v nd ener is [CPM 98] he cceerion of price eecuin sipe hronic oion [CBS PM 00] () 0 () When v is iu, is iu Vue of is zero, whever be he vue of v 0 When v is zero, is zero Zero No obinbe When v is iu, is zero 5. he poeni ener of price wih dispceen X is U(X). he 9. Wh is he iu cceerion of he price doin he SHM oion is sipe hronic, when ( is posiive consn) X sin where is in c [DC 00] () U U X () c c / s / s c / s c 0. price eecues iner sipe hronic oion wih n piude of c. When he price is c fro he en posiion he niude of is veoci is equ o h of is cceerion. hen is ie period in seconds is [er P 005] (). In sipe hronic oion, he rio of cceerion of he price o is dispceen n ie is esure of () Sprin consn (nur frequenc) / s nur frequenc Resorin force ner of Sipe Hronic Moion [UPS 00]. he o ener of price eecuin S.H.M. is proporion o [CPM 97, 78; MC 99; RP 999; MP PM 00; Pb. PM 00; MH C 00] () Dispceen fro equiibriu posiion Frequenc of osciion Veoci in equiibriu posiion Squre of piude of oion. price eecues sipe hronic oion on srih ine wih n piude. he poeni ener is iu when he dispceen is [CPM 98] () Zero. price is vibrin in sipe hronic oion wih n piude of c. wh dispceen fro he equiibriu posiion, is is ener hf poeni nd hf ineic[ncr 98; MNR 995; RPM 995; DC 000; UPS 000] U U X 6. he ineic ener nd poeni ener of price eecuin sipe hronic oion wi be equ, when dispceen (piude = ) is () [MP PM 987; CPM 990; DPM 996; MH C 997, 99; FMC 999; CPM 000] 7. he o ener of he bod eecuin S.H.M. is. hen he ineic ener when he dispceen is hf of he piude, is () [RPM 99, 96; CBS PM 995; JIPMR 00] 8. he poeni ener of price eecuin S.H.M. is.5 J, when is dispceen is hf of piude. he o ener of he price be () 8 J 0 J J.5 J 9. he nur veoci nd he piude of sipe penduu is nd respecive. dispceen X fro he en posiion if is ineic ener is nd poeni ener is V, hen he rio of o V is [CBS PM 99] () X /( X ) X /( X ) ( X )/ X ( X ) / X 0. When he poeni ener of price eecuin sipe hronic oion is one-fourh of is iu vue durin he osciion, he dispceen of he price fro he equiibriu posiion in ers of is piude is [CBS PM 99; MC (n.) 995;

18 768 Sipe Hronic Moion () / / / / MP PM 99, 000; MP P 995, 96, 00]. price of ss 0 is describin S.H.M. on srih ine wih period of sec nd piude of 0 c. Is ineic ener when (),,,,,, v i is 5 c fro is equiibriu posiion is 9. he o [MP ener PM 996] of price eecuin S.H.M. is 80 J. Wh is he () 7.5 ers.75 ers poeni ener when he price is disnce of / of piude fro he en posiion 75 ers 0.75 ers. When he dispceen is hf he piude, he rio of poeni ener o he o ener is () [CPM 999; JIPMR 000; er P 00]. he P.. of price eecuin SHM disnce fro is equiibriu posiion is () 8 ( ) Zero [Rooree 99; CPM 997; RPM 999]. veric ss-sprin sse eecues sipe hronic osciions wih period of s. quni of his sse which ehibis sipe hronic vriion wih period of s is M [SCR 998] () Veoci Poeni ener Phse difference beween cceerion nd dispceen Difference beween ineic ener nd poeni ener 5. For n S.H.M., piude is 6 c. If insnneous poeni ener is hf he o ener hen disnce of price fro is en posiion is [RP 000] () c 5.8 c. c 6 c 6. bod of ss is eecuin sipe hronic oion. Is dispceen seconds is iven b 6 sin( 00 /). Is iu ineic ener is () 6 J 8 J [MC (n.) 000] J 6 J 7. price is eecuin sipe hronic oion wih frequenc f. he frequenc which is ineic ener chne ino poeni ener is [MP P 000] () f/ f f f 8. here is bod hvin ss nd perforin S.H.M. wih piude. here is resorin force F, where is he dispceen. he o ener of bod depends upon () 60 J 0 J 0 J 5 J 0. In sipe hronic oscior, he en posiion [CBS PM 00] [er (n.) 00] () ineic ener is iniu, poeni ener is iu Boh ineic nd poeni eneries re iu ineic ener is iu, poeni ener is iniu Boh ineic nd poeni eneries re iniu [I 00]. Dispceen beween iu poeni ener posiion nd iu ineic ener posiion for price eecuin S.H.M. is () +. When ss M is ched o he sprin of force consn, hen he sprin sreches b. If he ss oscies wih piude, wh wi be iu poeni ener sored in he sprin () M. he poeni ener of sipe hronic oscior when he price is hf w o is end poin is (where is he o ener) () 8. bod eecues sipe hronic oion. he poeni ener (P..), he ineic ener (..) nd o ener (..) re esured s funcion of dispceen. Which of he foowin seens is rue [I 00] () P.. is iu when = 0.. is iu when = 0.. is zero when = 0.. is iu when is iu 5. If <> nd <U> denoe he vere ineic nd he vere poeni eneries respecive of ss describin sipe hronic oion, over one period, hen he correc reion is () <> = <U> <> = <U> <> = <U> <>= <U> 6. he o ener of price, eecuin sipe hronic oion is ()

19 Independen of / 7. he ineic ener of price eecuin S.H.M. is 6 J when i is is en posiion. If he ss of he price is 0., hen wh is he iu veoci of he price () 5 / s 5 / s 0 / s 0 / s [MH C 00] 8. Consider he foowin seens. he o ener of price eecuin sipe hronic oion depends on is () piude () Period () Dispceen Of hese seens [RPM 00; BCC 005] () () nd () re correc () nd () re correc () nd () re correc (), () nd () re correc 9. price srs sipe hronic oion fro he en posiion. Is piude is nd o ener. one insn is ineic ener is /. Is dispceen h insn is () / / / / [er P 005] 0. price eecues sipe hronic oion wih frequenc f. he frequenc wih which is ineic ener oscies is [II J 97, 87; Mnip M 995; MP P 997; DC 997; DC 999; UPS 000; RP 00; RPM 00; BHU 005] Sipe Hronic Moion 769 () f / f () 8.6 inues f f. inues d. he piude of price eecuin SHM is de hree-fourh Wi no rech he oher end eepin is ie period consn. Is o ener wi be [RPM 00] (). he iu speed of price eecuin S.H.M. is / s nd is iu cceerion is.57 / sec. he ie period of he price wi be [DPM 00] 9 None of hese 6 () sec.57 sec. price of ss is hnin veric b n ide sprin of force consn. If he ss is de o oscie veric, is o ener is [CPM 978; RP 999] () Miu eree posiion Miu en posiion Miniu en posiion Se posiion. bod is ovin in roo wih veoci of 0 / s perpendicur o he wo ws sepred b 5 eers. here is no fricion nd he coisions wih he ws re esic. he oion of he bod is [MP PM 999] () No periodic Periodic bu no sipe hronic Periodic nd sipe hronic Periodic wih vribe ie period. bod is eecuin Sipe Hronic Moion. dispceen is poeni ener is nd dispceen is poeni ener is. he poeni ener dispceen ( ) is [MC 0 () ie Period nd Frequenc. price oves such h is cceerion is iven b b, where is he dispceen fro equiibriu posiion nd b is consn. he period of osciion is () b b [NCR 98; CPM 99; MP PM 99; b b MNR 995; UPS 000] d. he equion of oion of price is 0, where is d posiive consn. he ie period of he oion is iven b (). unne hs been du hrouh he cenre of he erh nd b is reesed in i. I wi rech he oher end of he unne fer.57 sec sec 5. he oion of price eecuin S.H.M. is iven b 0.0sin00 (.05), where is in eres nd ie is in seconds. he ie period is [CPM 990] () 0.0 sec 0. sec 0.0 sec 0. sec 6. he ineic ener of price eecuin S.H.M. is 6 J when i is in is en posiion. If he piude of osciions is 5 c nd he ss of he price is 5., he ie period of is osciion is () sec sec 5 0 sec 5 sec [Hrn C 996; FMC 998]

20 770 Sipe Hronic Moion 7. he cceerion of price perforin S.H.M. is c / sec 6. sipe hronic wve hvin n piude nd ie period is represened b he equion 5 sin( ). hen he vue of disnce of c fro he en posiion. Is ie period is [MP P 996; MP PM 997] piude () in () nd ie period () in second re [Pb. P 00] () 0.5 sec.0 sec () 0, 5, 8..0 sec. sec 0, 5, o e he frequenc doube of n oscior, we hve o 7. price eecuin sipe hronic oion of piude 5 c hs [CPM 999] iu speed of. c/s. he frequenc of is osciion is () Doube he ss Hf he ss () Hz Hz Hz Hz Qudrupe he ss 8. he dispceen (in eres) of price perforin sipe hronic oion is reed o ie (in seconds) s Reduce he ss o one-fourh 9. Wh is consn in S.H.M. [UPS 999] 0.05cos. he frequenc of he oion wi be () Resorin force ineic ener () 0.5 Hz.0 Hz Poeni ener Periodic ie.5 Hz.0 Hz 0. If sipe hronic oscior hs o dispceen of 0.0 nd cceerion equ o.0s n ie, he nur frequenc of he oscior is equ o Sipe Penduu () 0 rd s 00 rd s [CBS PM 99; RPM 996] 0. rd s rd s. he equion of sipe hronic oion is X 0. cos( ) where X nd re in nd sec. he frequenc of oion is [er (n.) 00] () / 0.7/ 000 /. Mr he wron seen [MP PM 00] () S.H.M. s hve fied ie period oion hvin se ie period re S.H.M. In S.H.M. o ener is proporion o squre of piude Phse consn of S.H.M. depends upon inii condiions. price in SHM is described b he dispceen equion ( ) cos( ). If he inii ( = 0) posiion of he price is c nd is inii veoci is c/s, wh is is piude? he nur frequenc of he price is s () c c 0.0cos he frequenc of he oion wi be [UPS 00] () 0.5 Hz.0 Hz Hz Hz [DPM 00]. he period of sipe penduu is doubed, when () Is enh is doubed [CPM 97; MNR 980; FMC 995; Pb. P/PM 00] he ss of he bob is doubed Is enh is de four ies he ss of he bob nd he enh of he penduu re doubed. he period of osciion of sipe penduu of consn enh erh surfce is. Is period inside ine is () Greer hn qu o Less hn [CPM 97; DPM 00] Cnno be copred. sipe penduu is de of bod which is hoow sphere coninin ercur suspended b ens of wire. If ie ercur is drined off, he period of penduu wi () Reins unchned Increse Decrese Becoe erric [NCR 97; BHU 979] c.5 c. penduu suspended fro he ceiin of rin hs period,. price eecues SHM in ine c on. Is veoci when when he rin is res. When he rin is cceerin wih unifor cceerion, he period of osciion wi[ncr 980; CPM 997] pssin hrouh he cenre of ine is c/s. he period wi be [Pb. P 000] () Increse Decrese ().07 s.07 s Rein unffeced Becoe infinie.07 s 0.07 s 5. he ss nd dieer of pne re wice hose of erh. he 5. he dispceen (in ere) of price in, sipe hronic period of osciion of penduu on his pne wi be (If i is oion is reed o ie (in seconds) s second's penduu on erh) () sec sec sec sec [II 97; DC 00]

21 Sipe Hronic Moion sipe penduu is se up in roe which oves o he rih wih n cceerion on horizon pne. hen he hred of he. he bob of sipe penduu of ss nd o ener wi hve iu iner oenu equ o penduu in he en posiion es n ne wih he veric [CPM 98] [MP PM 986] () n n n n in he forwrd direcion in he bcwrd direcion in he bcwrd direcion in he forwrd direcion 7. Which of he foowin seens is no rue? In he cse of / 6 sipe penduu for s piudes he period of osciion is [NCR 98] () Direc proporion o squre roo of he enh of he penduu Inverse proporion o he squre roo of he cceerion due o rvi Dependen on he ss, size nd eri of he bob Independen of he piude 8. he ie period of second's penduu is sec. he spheric bob () he se Incresed b /5 which is ep fro inside hs ss of 50. his is now repced b noher soid bob of se rdius bu hvin differen Decresed b / ies None of he bove ss of 00. he new ie period wi be 6. he enh [NCR of 97] sipe penduu is incresed b %. Is ie period () sec sec wi [MP P 99; RP 00] sec 8 sec () Increse b % Increse b 0.5% 9. n esures he period of sipe penduu inside sionr if nd finds i o be sec. If he if cceeres upwrds wih n 7. sipe penduu wih bob of ss oscies fro o C cceerion /, hen he period of he penduu wi be [NCR nd 990; bc BHU o 00] such h PB is H. If he cceerion due o rvi is, hen he veoci of he bob s i psses hrouh B is () [CBS PM 995; DPM 995; Pb. PM 996] Penduu P C (). he enh of he second penduu on he surfce of erh is. he enh of seconds penduu on he surfce of oon, where is /6h vue of on he surfce of erh, is () / [CPM 97]. If he enh of second's penduu is decresed b %, how n seconds i wi ose per d [CPM 99] () 97 sec 7 sec 77 sec 86 sec 5. he period of sipe penduu is esured s in sionr if. If he if oves upwrds wih n cceerion of 5, he period wi be [MNR 979] Decrese b 0.5% Increse b % 5 Lif H 5 0. sipe penduu is suspended fro he roof of roe which oves in horizon direcion wih n cceerion, hen he ie period is iven b, where is equ o 8. Idenif correc seen on he foowin [BHU 997] (). second's penduu is pced in spce boror orbiin round he erh heih R, where R is he rdius of he erh. he ie period of he penduu is () Zero sec sec Infinie [CPM 989; RPM 995] B () H H H Zero [Mnip M 995] () he reer he ss of penduu bob, he shorer is is frequenc of osciion sipe penduu wih bob of ss M swins wih n o nur piude of 0. When is nur piude is o 0, he ension in he srin is ess hn o Mcos 0. s he enh of sipe penduu is incresed, he iu veoci of is bob durin is osciion wi so decreses he frcion chne in he ie period of penduu on chnin he eperure is independen of he enh of he penduu

22 77 Sipe Hronic Moion 9. he bob of penduu of enh is pued side fro is equiibriu posiion hrouh n ne nd hen reesed. he bob wi hen pss hrouh is equiibriu posiion wih speed v, where v equs [Hrn C 996] () ( sin ) ( cos) ( cos) ( sin ) 0. sipe penduu eecuin S.H.M. is fin free on wih he suppor. hen () Is periodic ie decreses Is periodic ie increses I does no oscie None of hese. penduu bob hs speed of /s is owes posiion. he penduu is 0.5 on. he speed of he bob, when he enh es n ne of () o 60 o he veric, wi be (If / s / s / s / s. he ie period of sipe penduu is sec. If is enh is incresed ies, hen is period becoes () 6 sec 8 sec sec sec Zero 0 / s ) [MP P 996] [CBS PM 999; DPM 999]. If he e bob of sipe penduu is repced b wooden bob, hen is ie period wi [IIMS 998, 99] () Increse Decrese Rein he se Firs increse hen decrese. In sipe penduu, he period of osciion is reed o enh of he penduu s [MC (Med.) 995] () consn consn consn consn 5. penduu hs ie period. If i is en on o noher pne. In seconds penduu, ss of bob is 0. If i is repced b 90 hvin cceerion due o rvi hf nd ss 9 ies h of he ss. hen is ie period wi erh hen is ie period on he oher pne wi be [CM Bihr 995] [Oriss PM 00] () / 6. sipe penduu is eecuin sipe hronic oion wih ie period. If he enh of he penduu is incresed b %, he percene increse in he ie period of he penduu of incresed enh is [BHU 99, 96; Pb. PM 995; FMC 00; IIMS 00; I 00] () 0% % 0% 50% 7. If he enh of sipe penduu is incresed b 00%, hen he ie period wi be incresed b [RPM 999] () 00% 00% 00% 00% 8. he enh of seconds penduu is [RP 000] () 99.8 c 00 c 99 c None of hese 9. he ie period of sipe penduu in if descendin wih consn cceerion is [DC 998; MP PM 00] () Infinie 0. chipnzee swinin on swin in siin posiion, snds up sudden, he ie period wi () Becoe infinie Increse [C (n./med.) 000; I 00; DPM 00] Rein se Decrese. he cceerion due o rvi pce is /sec. hen he ie period of sipe penduu of enh one ere is () sec sec sec sec. pe oscied wih ie period. Sudden, noher pe pu on he firs pe, hen ie period [I 00] () Wi decrese Wi be se Wi increse None of hese. sipe penduu of enh hs brss bob ched is ower end. Is period is. If see bob of se size, hvin densi ies h of brss, repces he brss bob nd is enh is chned so h period becoes, hen new enh is () () sec sec sec sec 5. he ie period of sipe penduu when i is de o oscie on he surfce of oon [J & C 00] () Increses Reins unchned Decreses Becoes infinie

23 6. sipe penduu is ched o he roof of if. If ie period of osciion, when he if is sionr is. hen frequenc of osciion, when he if fs free, wi be () Zero / None of hese [DC 00] 7. sipe penduu, suspended fro he ceiin of sionr vn, hs ie period. If he vn srs ovin wih unifor veoci he period of he penduu wi be () Less hn Greer hn qu o Unchned [RPM 00] 8. If he enh of he sipe penduu is incresed b %, hen wh is he chne in ie period of penduu () % 0% % % [MH C 00; UPS 005] 9. o show h sipe penduu eecues sipe hronic oion, i is necessr o ssue h [CPM 00] () Lenh of he penduu is s Mss of he penduu is s Sipe Hronic Moion 77 piude of osciion is s [BHU 005] cceerion due o rvi is s () / 0. he heih of swin chnes durin is oion fro 0. o.5. he iniu veoci of bo who swins in his swin is 9. sipe [CPM penduu 997] is en fro he equor o he poe. Is period [ () 5. / s. / s.95 / s Zero. he piude of n osciin sipe penduu is 0c nd is period is sec. Is speed fer sec fer i psses is equiibriu posiion, is () Zero 0.57 / s 0. / s 0. / s. sipe penduu consisin of b of ss ied o hred of enh is de o swin on circur rc of ne in veric pne. he end of his rc, noher b of ss is pced res. he oenu rnsferred o his b res b he swinin b is [NCR 977] () Zero. sipe penduu hns fro he ceiin of cr. If he cr cceeres wih unifor cceerion, he frequenc of he sipe penduu wi [Pb. PM 000] () Increse Becoe infinie Decrese Rein consn. he periodic ie of sipe penduu of enh nd piude c is 5 seconds. If he piude is de c, is periodic ie in seconds wi be [MP PM 985] () he rio of frequencies of wo penduus re :, hen heir enh re in rio [DC 005] () / / / 9 9 / 6. wo penduus bein o swin siuneous. If he rio of he frequenc of osciions of he wo is 7 : 8, hen he rio of enhs of he wo penduus wi be () 7 : 8 8 : 7 9 : 6 6 : 9 [J & C 005] 7. sipe penduu hnin fro he ceiin of sionr if hs ie period. When he if oves downwrd wih consn veoci, he ie period is, hen () is infini [Oriss J 005] 8. If he enh of penduu is de 9 ies nd ss of he bob is de ies hen he vue of ie period becoes () Decreses Increses Reins he se Decreses nd hen increses 50. penduu of enh if P. When i reches Q, i osses 0% of is o ener due o ir resisnce. he veoci Q is () 6 /sec /sec /sec 8 /sec Q 5. here is sipe penduu hnin fro he ceiin of if. When he if is snd si, he ie period of he penduu is. If he resun cceerion becoes /, hen he new ie period of he penduu is [DC 00] () he period of sipe penduu esured inside sionr if is found o be. If he if srs cceerin upwrds wih cceerion of /, hen he ie period of he penduu is [RPM 000; DPM () P