19. Oscillations. Objectives. By Liew Sau Poh. Outcomes. Outcomes. Periodic Motion Characteristics of SHM. Position VS.
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1 9. Oscillions By iew Su Poh Ojecies 9. Chrcerisics of siple hronic oion 9. Kineics of siple hronic oion 9.3 Enery in siple hronic oion 9.4 Syses in siple hronic oion 9.5 Dpe oscillions 9.6 Force oscillions n Resonnce Oucoes ) efine siple hronic oion y ens of he equion = ) Show h = o sin s soluion of = c) erie n use he forul = ( ) ) escrie, wih rphicl illusrions, he riion in isplceen, elociy n ccelerion wih ie e) escrie, wih rphicl illusrions, he riion in elociy n ccelerion wih isplceen f) erie n use he epressions for ineic enery n poenil enery ) escrie, wih rphicl illusrions, he riion in ineic enery n poenil enery wih ie n isplceen 3 Oucoes h) erie n use epressions for he perios of oscillions for sprin-ss n siple penulu syses i) escrie he chnes in pliue n enery for pe oscillin syse j) isinuish eween uner pin, criicl pin n oer pin ) isinuish eween free oscillions n force oscillions l) se he coniions for resonnce o occur 4 9. Chrcerisics of SHM his ype of oion is he os persie oion in he unierse. ll os oscille uner hronic oion. We cn oel his oion wih liner resorin force. Perioic Moion Moion h repes in reulr pern oer n oer in is clle perioic oion. Siple hronic oion is specific ype of perioic oion h hs siple sine or cosine we shpe. 5 6 Posiion VS. ie rph Wh is he siple heicl for of SHM oion? he isplceen of he oscillin ss ries sinusoilly s funcion of ie. Here Oscillin ss on Sprin Perioic Moion Siple Hronic Moion 7 8
2 9 he resorin force of n iel sprin is ien y: F = - where is he sprin consn n is he isplceen of he sprin fro is unsrine lenh. he inus sin inices h he resorin force lwys poins in opposie irecion o he isplceen of he sprin. Siple Hronic Moion When here is resorin force, F = -, siple hronic oion occurs. 9. Kineics of SHM Siple Hronic Moion F s = - (SHM) occurs when he force cin on oy is proporionl o he F s = isplceen of he oy fro soe equiliriu F s = + posiion (e. sprin or penulu). - = 9. Kineics of SHM = - (/) f we ry = cos(w+f) s soluion o his equion, we oin: So 3 sin ( SHM ) cos 9.3 Enery in SHM Equion for Siple Hronic Moion 9. Kineics of SHM - F s = - = F s = F s = + When he loc che o he sprin (lef) is isplce sll isnce fro equiliriu, he sprin eers resorin force which is proporionl o he isplceen: 9.3 Enery in SHM ol Enery = Kineic Enery + Poenil Enery E = K + U 4 K U 9.3 Enery in SHM Moion K K sin sin U U cos cos E consn rne of oion E U consn 5 6 urnin poin urnin poin
3 KE n PE Conersion pliue F s = F s = - KE U KE U cos cos pliue is he niue of he iu isplceen. F s = = KE U 8 Perio, Frequency, f For ny ojec in siple hronic oion, he ie require o coplee one cycle is he perio. he frequency f of he siple hronic oion is he nuer of cycles of he oion per secon. f 9 Enery of he Siple Hronic Oscillor For isplceen = cos (w+f), we cn sy h ineic enery, KE is: PE E E E ol ol ol KE PE. KE cos sin Poenil enery (elsic) PE is: sin cos hus, ol enery is proporionl o pliue. Enery rnsfer / KE, U = correspons o he sreche sprin. 3 / - KE, U + = correspons o equiliriu posiion of sprin. nulr Frequency 4 Since / or f / ( resorin force )
4 9.4 Syses in SHM. Penulus. he Siple Penulu. he Physicl Penulu 3.. SHM & Unifor Circulr Moion 3. Dpe SHM 4. Force Oscillions & Resonnce Griionl Penulu Siple Penulu: o of ss hun on n unsrechle ssless srin of lenh. 5 6 Siple Penulu he Siple Penulu 7 SHM for sll F sin ccelerion ~ - isplceen SHM F 8 sin Bu Coprin = (sin ) in r penulu lein ril of in: Physicl Penulu rii oy pioe ou poin oher hn is cener of ss (co). SHM for sll h F sin h F Pio Cener of Mss ccelerion ~ - isplceen SHM h 9 3 quic eho o esure h he orsionl Penulu Siple Hronic Moion orsion Penulu: 33 Sprin: 34 ny Oscillin Syse: ineri sprininess h
5 SHM & Unifor Circulr Moion he projecion of poin oin in unifor circulr oion on ieer of he circle in which he oion occurs eecues SHM. SHM & Unifor Circulr Moion he reference poin of rius. he projecion of on ieer of he circle eecues SHM. he eecuion of unifor circulr oion escries SHM. 35 hp://posiron.ps.uci.eu/~iry/usic/hl/eos/siplehronicmoion/circul r.hl SHM & Unifor Circulr Moion. he projecion of on ieer of he circle eecues SHM. c o s 36 UC rine Physics of Music Siple Hronic Moion pple Deonsrions rius = n le SHM & Unifor Circulr Moion he projecion of poin oin in unifor circulr oion on ieer of he circle in which he oion occurs eecues SHM. Mesureens of he nle eween Clliso n Jupier: Glileo (6) () () () plne s in c o s rius = 37 cos 38 erh Equions of Moion (SHM) Displceen-ie Grph = cos = - sin = cos = - cos = ± ( - ).5 = - [he efiniion] Velociy-ie Grph ccelerion-ie Grph = sin = cos 4 4
![Velociy-Displceen Grph ccelerion-displceen Grph = ± ).5 = [he efiniion] - - 43 44 Phse Relionship Free oscillions When syse oscilles wihou eernl forces cin on i, he syse is in free oscillion.](/docs-images/81/84410158/images/6-2.jpg "he pliue of oscillion is consn, which will no rop. Displceen, ie 45 46-9.")
6 Velociy-Displceen Grph ccelerion-displceen Grph = ± ).5 = [he efiniion] Phse Relionship Free oscillions When syse oscilles wihou eernl forces cin on i, he syse is in free oscillion. he pliue of oscillion is consn, which will no rop. Displceen, ie Dpe Oscillions n ny rel syses, nonconserie forces re presen his is no loner n iel syse (he ype we he el wih so fr) Fricion is coon nonconserie force n his cse, he echnicl enery of he syse iinishes in ie, he oion is si o e pe 9.5 Dpe Oscillions Dpe hronic oion is hronic oion wih fricionl or r force. f he pin is oifies he unpe oscillion Dpe SHM SHM in which ech oscillion is reuce y n eernl force. F Dpe SHM Fne Resorin Force SHM F Dpin Force n opposie irecion o elociy Does neie wor Reuces he echnicl enery D ifferenil equion 49 5
7 9.5 Dpe Oscillions rph for pe oscillion he pliue ecreses wih ie he lue she lines represen he enelope of he oion 9.5 Dpe Oscillion One eple of pe oion occurs when n ojec is che o sprin n suere in iscous liqui he rerin force cn e epresse s R = - where is consn n is clle he pin coefficien Dpe Oscillion Howeer, if he pin is lre, i no loner reseles SHM ll. : unerpin: here re few sll oscillions efore he oscillor coes o res Dpe Oscillion B: criicl pin: his is he fses wy o e o equiliriu. C: oerpin: he syse is slowe so uch h i es lon ie o e o equiliriu Dpe Oscillion here re syses where pin is unwne, such s clocs n wches. hen here re syses in which i is wne, n ofen nees o e s close o criicl pin s possile, such s uooile shoc sorers n erhque proecion for uilins. 9.5 Dpe Oscillion n Orer Hooeneous iner Differenil Equion: Soluion of Differenil Equion: ( ) e cos where: = SHM Dpe Oscillions uo Shoc sorers ( ) e cos 57 sll pin he nurl frequency " criiclly pe" " oer pe " Eponenil soluion o he DE 58 ypicl uooile shoc sorers re esine o prouce slihly uner-pe oion
8 9.6 Force Oscillions & Resonnce Force oscillions occur when here is perioic riin force. his force y or y no he he se perio s he nurl frequency of he syse. f he frequency is he se s he nurl frequency, he pliue ecoes quie lre. his is clle resonnce. 9.6 Force Oscillions & Resonnce is possile o copense for he loss of enery in pe syse y pplyin n eernl force he pliue of he oion reins consn if he enery inpu per cycle ecly equls he ecrese in echnicl enery in ech cycle h resuls fro resisie forces Force Oscillions & Resonnce fer riin force on n iniilly sionry ojec eins o c, he pliue of he oscillion will increse fer sufficienly lon perio of ie, E riin = E los o inernl hen sey-se coniion is reche he oscillions will procee wih consn pliue Force Oscillions & Resonnce When he frequency of he riin force is ner o he nurl frequency (» ) n increse in pliue occurs his ric increse in he pliue is clle resonnce he nurl frequency is lso clle he resonnce frequency of he syse 9.6 Force Oscillions & Resonnce he shrpness of he resonn pe epens on he pin. f he pin is sll (), i cn e quie shrp; if he pin is lrer (B), i is less shrp. Eernl frequency f ie pin, resonnce cn e wne or unwne. Musicl insruens n V/rio receiers 6 epen on i. 9.6 Force Oscillions & Resonnce Ech oscillion is rien y n eernl force o inin oion in he presence of pin: F cos w = riin frequency Force Oscillions & Resonnce Ech oscillion is rien y n eernl force o inin oion in he presence of pin. 65 n Orer nhooeneous iner Differenil Equion: F cos 9.6 Force Oscillions & Resonnce 66 n Orer Hooeneous iner Differenil Equion: Sey-Se Soluion of Differenil Equion: ( ) cos where: n F F cos w = nurl frequency w = riin frequency
9 9.6 Force Oscillions & Resonnce he nurl frequency, w, is he frequency of oscillion when here is no eernl riin force or pin. F less pin ore pin w = nurl frequency w = riin frequency When w = w resonnce occurs! Sop he SHM cuse y wins on hihrise uilin 4 on weih oune on sprin on hih floor of he Ciicorp uilin in New Yor. he weih is force o oscille he se frequency s he uilin u 9 erees ou of 7 phse. 69 Sury OSCON Free Dpe Displceen, Force Oscillions n Resonnce less pin 7 ie ore pin
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