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ca | Simpe harmonic motion damped. | =| We know thot the motion of a simple bendulum, Swinging in oly , dies out eventually , because ‘the Qiy dryag and the Frickon oF the svubbort OPpose the motion of the bendulum and dissipate US | | energy gradvally. The pendulum is said +© excuse dumbed oscillations . —|In damped ascillakons , the energy ofthe system is dissibated continuously. — || for small damping , the oscillation remain abproxi- mately peviodic.Tthe dissipoting Forces are gen eyva- lly the Frictonal_ forces. +|| heve oa block of mass m_ connected 2 an elastic sbring of sbring constant kk Osdillates Vertically. if the block {s pushed down _o Ute and yeleased , its angular Freqvency of oscillation is We = | K (nator al Frequency). m. | . pS | suryounding medium (oly) will | exert oa damping foyce On the motion of the bloods and the ole mechanical energy of the block - wo 8 ——~H ——— gbring system will decyease. as tne eneyay Loss Wi, abbeay as heok of the svurvov- > ‘ nding medium and the block alse - force depends on the nature of me >| the damin - medium - eyrrounsin Scanned with CamScanner ifthe block ‘is immersed in a liquid, the Magni de ___ss of daming wit be much greatey_and the dissi potion Of energy much Pastey. the. damping force fs geneyally byoporHonal fo velod- ty of the bon (stokes?Law) and acts obbssite to the divecton of velocity. Ly ee ~oxnyVv ing force Fu = -bY (usvally vatid only yr Small velocity) Lothere the fositve Constant b debends Oh chayac- af Af A y uw ‘ Loa yesistve Force / Aerodynamic oy viscous drag /Dampi- wy [ teyistics of the medium (viscosity » Foy example) and the size and shabe oftne block ,etc. & b is positive damping constant in Kgls. if the mass is bulled down oF pushed vba Ute the yestoring Porce On the block dve to the + Sbving is Fs = —-Kx, Where x is the displacement t= lof the mass from its equilibyium positon - ‘€z Thus , the totol Perce acing onthe mass at any ro Hme t; is Fe -kx - bv: if alt) isthe accelevation of mass of time t, nen by Newtons Law of motion abptied along the dvecion of motions we have ma(t) = —Kx [t) —bvk) => mOtikv +bv=0 Wwe are discussing one-dimensional maton, & Aa kal “= R. ~ a here we have dyopbed the vector notaton because XD »_ —, Using the First and second derivates OF x(t) for vit) and lt) 1 vespectively se have. zs Scanned with CamScanner mechanical enetgy of the vndambed oscil Loy a dambed oscitlakor , the homputude constant but debends on time. - 2 atoy 1s 1720 is not a ~ZbE 3s as a eee Ele): Le(Ae 2) =i kate 2” =} kate > 7 2 2. he total enervqu_of the system decreses exponentially TF With Hme. m smal damping means that the dimen sionless rato bis much less than J. vim Power loss in Dambed Oscillation —Q - factor 4 because of the work done against the non-cons- ervatve forces, the energy Steed in cn ascillatoy C@ntinvosly decreases. the Fradion of enevgy lost in aperiod with vesbect to HS overage vawe Foy thok peviod is ameasure of the quality of the escillator. a ala al arwinlevdce eielal a the less the vawe of Hs Frackion is the move is the ability of the. oscillotoy +o sustain beviadic motion . Q= Aveyage Qnergy stoved in one bertod x2as w! Average less of energy in one period 2 (Fa) 2m if — domping is tess [bs ze us.) Under doambed ] energy of oscillotar = Ct) =) pws pk ; fad = 5 Sma? teto mt Scanned with CamScanner = SMAZWer0 “Rt: = (s-vi at) Be SEAX EL energy ECt)= Fo ep Rt be Power Plt)ede =p fe =b EU) akoM™m m enevay lost in one period =
R=O under dumbed blam 22 < VITE ox Ry, 2'/e overdamped blam SWe =>R/ar > We oy 8%my >be crittcol damped blam= Wo=> R/aL =Wire oR P= Ye Forced Oscillation uhen a Sustem (such as a system bendvlum oy a block attached to sbying) is disblaced From is equilibrium position And yveleased: i+ oscillates with is naturel Frequenvy Loe ond the the osciVates 7 called free ae oscillattons all Free oscillating eventvally die out because of the evey present dambing Lorces Scanned with CamScanner ee om «< Boos never 7 cori hss ORNS U 5 eon Se Howeyer an exteynol agency can 7 ed forced A these oscillations, these -cat ave call Oy driven oscillations we consider the case when Ane frequenGy. external Force is tself periodic »with & a CEN A Frequency We Cotled driven frequency - tne most important fact of Forced Pre Pevio- | dic oscilaton is. that the syStem Oscitlates -§ not with its nove Frequency & (We) but at the Frequeny Uk _of the external agency ; the’ Free oscillatfons die ovtdve +o dambing the most familiar example of forced oscillation iS wh- i+ en a child ino garden swing periodically presses lis Feet against the ground (oy someone else bevie dicatly gives the child @_pbush)+0 maintain the oscillations. 4 the oscillation in a system can be indefinitely maintained by sveblying enevgy cont inveusly . \n mechanical System +Ms can be done by subj ec- ting i+ to an exteynal Force Which itself has haymanic me de bendence. BM ANAIA AA laAlaAA An_interesting Phenomenon occurs ig the frequen joy of the external Ce es) & Source & is equal oy Nearly to he natural Frequency of the sustem.. The ambl itude Of oscillaHon is found to dram ole 9 : iney many Folds insuch coses. this = ED x is called yeson ‘ Subbose Gn external force F (+ ance L ) of amblitvde Ee that varies perio dically with fime -peried applica * \ a) Pa Scanned with CamScanner gee 4 ton @ = -Vo Vo ux ove volncidy © = t=O , moment when we g uae ge) Wa Xe disblacement of Particle” of _F “all apbly fhe periodic Force ) a amplitude of the force OScillator_debends on the a I Languiay) frequency of the driving force — we consider these #00 cases. CE | ra <4 a (a) || Small_dambing, ayiving Frequency fay Prom Nat- = | ucal Frequency: & i lin case , wab << m lwo to?) £66226 Wy foy any reason valve of b , then A= Fe = Fo Jim? (1o27-679)4 wo dd Lode 2ere This makes it Cleary thot the maximum possible amplitude for agiven diving Prequeny is governed by the driving frequency and the dambing 1 and is never 9° the phe nomenon of inareaseg in amblinde when the dyiv- ing Force is closed tv the natural frequency of the osc- “ator 1s called yesonance. Scanned with CamScanner 3—4tuhen p 050 ofey ——— Pure plitude, veduced tof yo el be ide ce ——|__ | Sin ere ieee combleses 160 osdilotiens ts of initial wowe, csnet ’ of amblitude When it Completes 260 cee .——G A bortide of mass mis otached to a string [an dihos a naturel Ong - Frequency ti0 - An extey- Nal Force €(4) broportional 45 ws wt (us 4 we) g is Obplied to the GUlloKon The time disblacement OF oscillation Gilt be broporhones bo’ ut ava s)_| $)_-m 2 2 cal US "= 1n(s%, 152) £2 (vo,*+42) wo, tHu37 L=A ws (wat +6) Az Fo b-O=> Ax Te alin? (us. sa?) 4 7st Ui ew?) Q | Ampiutude of dom bed oscittotor decveases 46 On k- V ; - ‘ . Tes _ i original! mognitde is 55. In onctney 10s it Wit, decrease t to K HMes , its originch mognituse , where ot equals. 7 - & 0-7 O- O-F oO. BA 24. 2 q ||determined underdambped , overdombed or critixcal 1, dompe d- 1) d2e__ 4 4d 43% =O vey _|maainory dt “Os dt? ~ ul _d2z +4 +47 =0 Criticod, — dt > dt . : Scanned with CamScanner —— & & ; 1A ovtemobilé of mas$ 2600 og Nou are riding ta pal iberwalles = ————© Assuming thet you ore examining the ie) . a cyspereion. Sst aan ri akon charactevisties OF tS oa : ir The suskension sags Sem when *ne ens rath & H |] automobile is placed on it: Also. the omblitad ay é J . zi oscillation decyeases by S07. Ww7ing on comp p 1 escillakon- Estimate the values of 1 the spring constant K ond —€ tne damping constant bfw the sbying and shock abseyrbey system sf one wheel cassuming nak —; each wheel supports Tso ks- > 4k (i5em) =300f09 Kz Sx\0% Ulm G | 507”. => T= Tin2=2x | mM ay bb =1844-6Kg | 6 Ap 2 (Hla) 04 =2x| M14 J ax Sec N / b [ok : 2( 750) 0-7 = 22/456 € b J 5 0000 € Cc 2(0-24) J156 [50000 =b-ta x lOolS xsl 30- 2x 2x =b= 200/150. 2KR s b= 1344-6 Ka. see ~ (6) | i ¥ st in dombed _cstillatton after S gscillodion Amblitu de % becomes [-_of initigl then Rnd amblide after : (5 such oscillation = ’ P Scanned with CamScanner Transformer Prim ory coil Secondary coi) Spr Cin ® + ayer Cin Gp c Np ds = Be Ns As tin te €g = & Ng Pin = lp fe Cow : 1 Poot zis Ss Cffidienw = Po. too : n7/. Pin Step vb Steb down, Nb Ns Ne Sb Xe Cp <és Peete ip_> is. \ \ : NcTease inveltage decrease in Voltage 19 In_a@_dyans * former rotio of no. of turns ts 8:1 if brimayy Voltage is 46¢ (60 vol. then Goa in Se Condoyy vo tage. Vp = Ne Vs Ns Scanned with CamScanner AD Qh ideal transformer has inbut bower (Oo oPwost- yote of te foyns @:1 and current in _brimary is 25 amp then Pind potential of secondary csil. brimory Q5a Pin= topoow 1 Sp =P =tQQq00 _ “OV | | t 25 Ss =M C= {00 i €r Np g if] Agee eal) } | 400 6g i y {i (e] Loutbet bower of oa transformer fs_Q000 wok, has . : Lefticiency Qo. resistance of primary coil is (2 F) | won loss ig 100 Wok and rodio of trns Sil then ? {| col culate+ > #1 | input Power _ H QO- Pe yloow% y= j Pin ; % Pin = Po x 100 /. => A000 * (00 > Lue) a0 > fi = 16000 2, bvimory voltage in 900 then colwlote curvent in i : wh ist - aT mory_ cil __ | byimory = — yy | (9009 _ =\OA: ] (900 i > -_ Scanned with CamScanner 94,945 36,10 one | qg- 103 — So 105 Waves bei. Woves on Sting — Wowe is a _distuybance Whith brobogate Unear i : Momentum enevgy or infomation , with out 7 transfer _of material of medium q Classification of Waves ! \Ii Classification on the basis of medium | All mechanical wowe medium is necessarily vequived ex- WWE on _ String , Sound wave, Wowe on Svurf- ace of Woter ,Sbying wowe etc. non mechanical wave: also called EM wove ( electromagnetic Uwowe) medium is not neces- sayvily yeqgvived. ex- Light wove Classification on the basis of brobogaton of energy. Progyessive wone: energy is bTopogated in arn un boonded medium. stakionayy wove: energy Of Wwowe In © _bovr- ded medivm Te PS w STEN SOR Classification on the basis of diemerting of energy's = bropogohon wy jw Scanned with CamScanner | | | | | | | | | | < (=D: wowe on String a BiIl2-D: wove on surface of woter, On Surface —{, of tabla G C|]/3-D: sound towwe, , Lighk wowe 4 Clossification on the basis. of Particle & Transverse wave: Vibvyation of Pavtide is LY 40 divection _& of Probogahion, ex: wave on String. light Wwowe, | — —— & we | ACTS yas nor a § | Xd O0P NE a Vols — € Pressure wowe or longitvdinol wove. vor js []* to Doe c A DOP B i ms ag ‘ ex: sound wore € note: € mechanical woe will be transverse or longitudinal dep- € ending upon nate of medium and mode excitation. € in_Styings , Mechanical waves are always transverse vohen string is Undey +enton. ¢ in Uiquids and gasses machanicol waves are r os finds always longivdinal de becovse Pluses can not rd Sustain Shear Stroin. [tangential to the water Su face) * \In_ liquids Mechanical wowes moybe transverse only d on_ surface of Uguid due to surface tention € in solids me i ‘ = Chan col wow es Maybe transverse - e longitdinal oy both. ; ro Li Scanned with CamScanner 
