Oscillations. Simple Harmonic Motion The most basic oscillation, with sinusoidal motion, is called simple harmonic motion.

Transcription

1 14 Oscillaions his loudspeae cone geneaes sound waes by oscillaing bac and foh a audio fequencies. Looing head he goal of Chape 14 is o undesand syses ha oscillae wih siple haonic oion. Siple Haonic Moion he os basic oscillaion, wih sinusoidal oion, is called siple haonic oion. he oscillaing ca is an eaple of siple haonic oion. You ll lean how o use he ass and he sping consan o deeine he fequency of oscillaion. Oscillaion In his chape you will lean o: Repesen siple haonic oion boh gaphically and aheaically. Undesand he dynaics of oscillaing syses. Recognize he siilaiies aong any ypes of oscillaing syses. Siple haonic oion has a ey close connecion o unifo cicula oion. You ll lean ha an edge-on iew of unifo cicula oion is none ohe han siple haonic oion. Looing Bac Secion 4.5 Unifo cicula oion Spings Siple haonic oion occus when hee is a linea esoing foce. he siples eaple is a ass on a sping. You will lean how o deeine he peiod of oscillaion. he bounce a he boo of a bungee jup is an ehilaaing eaple of a ass oscillaing on a sping. Looing Bac Secion 1.4 Resoing foces Enegy of Oscillaions If hee is no ficion o ohe dissipaion, hen he echanical enegy of an oscillao is conseed. Conseaion of enegy will be an ipoan ool. he syse oscillaes beween all ineic enegy and all poenial enegy ll ineic Looing Bac Secion 1.5 Elasic poenial enegy Secion 1.6 Enegy diagas ll poenial Pendulus ass swinging a he end of a sing o od is a pendulu. Is oion is anohe eaple of siple haonic oion. he peiod of a pendulu is deeined by he lengh of he sing; neihe he ass no he apliude aes. Consequenly, he pendulu was he basis of ie eeping fo any cenuies. Daping and Resonance If hee s dag o ohe dissipaion, hen he oscillaion uns down. his is called a daped oscillaion. he apliude of a daped oscillaion undegoes eponenial decay. Oscillaions can incease in apliude, soeies daaically, when dien a hei naual oscillaion fequency. his is called esonance. Illusaed Chape Peiews gie an oeiew of he upcoing ideas fo each chape, seing he in cone, eplaining hei uiliy, and ying he o eising nowledge (hough Looing Bac efeences). hese peiews build on he cogniie psychology concep of an adance oganize.

2 378 chape 14. Oscillaions FiguRE 14.1 Eaples of posiion esusie gaphs fo oscillaing syses. Posiion Posiion Posiion he oscillaion aes place aound an equilibiu posiion. he oion is peiodic. One cycle aes ie. his oscillaion is sinusoidal Siple Haonic Moion Objecs o syses of objecs ha undego oscillaoy oion a epeiie oion bac and foh aound an equilibiu posiion ae called oscillaos. FiguRE 14.1 shows posiion-esus-ie gaphs fo hee diffeen oscillaing syses. lhough he shapes of he gaphs ae diffeen, all hese oscillaos hae wo hings in coon: 1. he oscillaion aes place abou an equilibiu posiion, and 2. he oion is peiodic, epeaing a egula ineals of ie. he ie o coplee one full cycle, o one oscillaion, is called he peiod of he oion. Peiod is epesened by he sybol. closely elaed piece of infoaion is he nube of cycles, o oscillaions, copleed pe second. If he peiod is 1 1 s, hen he oscillao can coplee 1 cycles in one second. Conesely, an oscillaion peiod of 1 s allows only 1 1 of a cycle o be copleed pe second. In geneal, seconds pe cycle iplies ha 1/ cycles will be copleed each second. he nube of cycles pe second is called he fequency f of he oscillaion. he elaionship beween fequency and peiod is f = 1 o = 1 f (14.1) Figues ae caefully sealined in deail and colo so sudens focus on he physics fo insance, he objec of inees in echanics. Eplici insucion as annoaions diecly on figues helps sudens o inepe figues and gaphs. BLE 14.1 Unis of fequency Fequency Peiod 1 3 Hz = 1 ilohez = 1 Hz 1 s 1 6 Hz = 1 egahez = 1 MHz 1 s 1 9 Hz = 1 gigahez = 1 GHz 1 ns FiguRE 14.2 pooype siplehaonic oion epeien. (a) i ac Oscillaion he poin on he objec ha is easued he unis of fequency ae hez, abbeiaed Hz, naed in hono of he Gean physicis Heinich Hez, who poduced he fis aificially geneaed adio waes in By definiion, 1 Hz K 1 cycle pe second = 1 s -1 We will fequenly deal wih ey apid oscillaions and ae use of he unis shown in able NOE Uppecase and lowecase lees ae ipoan. 1 MHz is 1 egahez = 1 6 Hz, bu 1 Hz is 1 illihez = 1-3 Hz! Fequency and peiod of a loudspeae cone EMPLE 14.1 Wha is he oscillaion peiod of a loudspeae cone ha ibaes bac and foh 5 ies pe second? SOLE he oscillaion fequency is f = 5 cycles/s = 5 Hz = 5. Hz. he peiod is he inese of he fequency; hence = 1 f = 1 5 Hz = 2. * 1-4 s = 2 s (b) uning poin is easued fo he equilibiu posiion whee he objec would be a es. he oion is sinusoidal, indicaing SHM. he oion is syeical abou he equilibiu posiion. Maiu disance o he lef and o he igh is. syse can oscillae in any ways, bu we will be especially ineesed in he sooh sinusoidal oscillaion (i.e., lie a sine o cosine) of he hid gaph in Figue his sinusoidal oscillaion, he os basic of all oscillaoy oions, is called siple haonic oion, ofen abbeiaed SHM. Le s loo a a gaphical descipion befoe we die ino he aheaics of siple haonic oion. FiguRE 14.2a shows an ai-ac glide aached o a sping. If he glide is pulled ou a few ceniees and eleased, i will oscillae bac and foh on he nealy ficionless ai ac. FiguRE 14.2b shows acual esuls fo an epeien in which a copue was used o easue he glide s posiion 2 ies eey second. his is a posiion-esus-ie gaph ha has been oaed 9 fo is usual oienaion in ode fo he -ais o ach he oion of he glide. he objec s aiu displaceen fo equilibiu is called he apliude of he oion. he objec s posiion oscillaes beween = - and = +. When using a gaph, noice ha he apliude is he disance fo he ais o he aiu, no he disance fo he iniu o he aiu.

3 14.1. Siple Haonic Moion 379 FiguRE 14.3a shows he daa wih he gaph aes in hei noal posiions. You can see ha he apliude in his epeien was =.17, o 17 c. You can also easue he peiod o be = 1.6 s. hus he oscillaion fequency was f = 1/ =.625 Hz. FiguRE 14.3b is a elociy-esus-ie gaph ha he copue poduced by using / o find he slope of he posiion gaph a each poin. he elociy gaph is also sinusoidal, oscillaing beween - a (aiu speed o he lef) and + a (aiu speed o he igh). s he figue shows, he insananeous elociy is zeo a he poins whee = {. hese ae he uning poins in he oion. he aiu speed a is eached as he objec passes hough he equilibiu posiion a =. he elociy is posiie as he objec oes o he igh bu negaie as i oes o he lef. We can as hee ipoan quesions abou his oscillaing syse: 1. How is he aiu speed a elaed o he apliude? 2. How ae he peiod and fequency elaed o he objec s ass, he sping consan, and he apliude? 3. Is he sinusoidal oscillaion a consequence of Newon s laws? ass oscillaing on a sping is he pooype of siple haonic oion. Ou analysis, in which we answe hese quesions, will be of a sping-ass syse. Een so, os of wha we lean will be applicable o ohe ypes of SHM. Kineaics of Siple Haonic Moion FiguRE 14.4 edaws he posiion-esus-ie gaph of Figue 14.3a as a sooh cue. lhough hese ae epiical daa (we don ye hae any heoy of oscillaion) he gaph fo his paicula oion is clealy a cosine funcion. he objec s posiion is () = cos1 2p 2 (14.2) whee he noaion () indicaes ha he posiion is a funcion of ie. Because cos(2p) = cos(), i s easy o see ha he posiion a ie = is he sae as he posiion a =. In ohe wods, his is a cosine funcion wih peiod. Be sue o conince youself ha his funcion agees wih he fie special poins shown in Figue NOE he aguen of he cosine funcion is in adians. ha will be ue houghou his chape. I s especially ipoan o eebe o se you calculao o adian ode befoe woing oscillaion pobles. Leaing i in degee ode will lead o eos. We can wie Equaion 14.2 in wo alenaie fos. Because he oscillaion fequency is f = 1/, we can wie () = cos(2pf) (14.3) Recall fo Chape 4 ha a paicle in cicula oion has an angula elociy ha is elaed o he peiod by = 2p/, whee is in ad/s. Now ha we e defined he fequency f, you can see ha and f ae elaed by (in ad/s) = 2p = 2pf (in Hz) (14.4) In his cone, is called he angula fequency. he posiion can be wien in es of as () = cos (14.5) Equaions 14.2, 14.3, and 14.5 ae equialen ways o wie he posiion of an objec oing in siple haonic oion. FiguRE 14.3 Posiion and elociy gaphs of he epeienal daa. (a) he speed is zeo when. () (b) (/s).7.7 he speed is aiu as he objec passes hough s a a FiguRE 14.4 he posiion esus ie gaph fo siple haonic oion. 1. Sas a 2. Passes hough a 1 4 (s) (s) 5. Reuns o a 2 4. Passes hough a Reaches a 1 2 BLE 14.2 Deiaies of sine and cosine funcions d 1a sin(b + c)2 = +ab cos(b + c) d d 1a cos(b + c)2 = -ab sin(b + c) d NOE paagaphs houghou guide sudens away fo nown peconcepions and aound coon sicing poins and highligh any ah- and ocabulay-elaed issues ha hae been poen o cause difficulies.

4 38 chape 14. Oscillaions FiguRE 14.5 Posiion and elociy gaphs fo siple haonic oion. Posiion Velociy a a () cos () a sin 2 2 Jus as he posiion gaph was clealy a cosine funcion, he elociy gaph shown in FiguRE 14.5 is clealy an upside-down sine funcion wih he sae peiod. he elociy, which is a funcion of ie, can be wien () = - a sin1 2p 2 = - a sin(2pf) = - a sin (14.6) NOE a is he aiu speed and hus is a posiie nube. We deduced Equaion 14.6 fo he epeienal esuls, bu we could equally well find i fo he posiion funcion of Equaion fe all, elociy is he ie deiaie of posiion. able 14.2 on he peious page einds you of he deiaies of he sine and cosine funcions. Using he deiaie of he posiion funcion, we find () = d d = - 2p 1 sin 2p 2 = -2pf sin(2pf) = - sin (14.7) Copaing Equaion 14.7, he aheaical definiion of elociy, o Equaion 14.6, he epiical descipion, we see ha he aiu speed of an oscillaion is a = 2p = 2pf = (14.8) Equaion 14.8 answes he fis quesion we posed aboe, which was how he aiu speed a is elaed o he apliude. No supisingly, he objec has a geae aiu speed if you sech he sping fahe and gie he oscillaion a lage apliude. Woed eaples follow a consisen poble-soling saegy and include caeful eplanaions of he undelying, and ofen unsaed, easoning. syse in siple haonic oion EMPLE 14.2 n ai-ac glide is aached o a sping, pulled 2. c o he igh, and eleased a = s. I aes 15 oscillaions in 1. s. a. Wha is he peiod of oscillaion? b. Wha is he objec s aiu speed? c. Wha ae he posiion and elociy a =.8 s? MODEL n objec oscillaing on a sping is in SHM. SOLE a. he oscillaion fequency is f = 15 oscillaions 1. s = 1.5 oscillaions/s = 1.5 Hz hus he peiod is = 1/f =.667 s. b. he oscillaion apliude is =.2. hus a = 2p = 2p(.2 ).667 s = 1.88 /s c. he objec sas a = + a = s. his is eacly he oscillaion descibed by Equaions 14.2 and he posiion a =.8 s is = cos1 2p 2 1 = (.2 ) cos 2p(.8 s).667 s 2 = (.2 )cos(7.54 ad) =.625 = 6.25 c he elociy a his insan of ie is = - a sin1 2p 2 = - (1.88 /s) sin 1 2p(.8 s).667 s 2 = - (1.88 /s) sin(7.54 ad) = /s = -179 c/s =.8 s, which is slighly oe han one peiod, he objec is 6.25 c o he igh of equilibiu and oing o he lef a 179 c/s. Noice he use of adians in he calculaions. Finding he ie EMPLE 14.3 ass oscillaing in siple haonic oion sas a = and has peiod. wha ie, as a facion of, does he objec fis pass hough = 1 2? SOLE Figue 14.4 showed ha he objec passes hough he equilibiu posiion = a = 1 4. his is one-quae of he oal disance in one-quae of a peiod. You igh epec i o ae 1 8 o each 1 2, bu his is no he case because he SHM gaph is no linea beween = and =. We need o use () = cos(2p/). Fis, we wie he equaion wih = 1 2 : = 2 1 = cos 2p 2 hen we sole fo he ie a which his posiion is eached: = 2p 1 1 cos = p 2p 3 = 1 6 SSESS he oion is slow a he beginning and hen speeds up, so i aes longe o oe fo = o = 1 2 han i does o oe fo = 1 2 o =. Noice ha he answe is independen of he apliude.

5 14.2. Siple Haonic Moion and Cicula Moion 381 Sop o hin 14.1 n objec oes wih siple haonic oion. If he apliude and he peiod ae boh doubled, he objec s aiu speed is a. Quadupled. b. Doubled. c. Unchanged. d. Haled. e. Quaeed Siple Haonic Moion and Cicula Moion he gaphs of Figue 14.5 and he posiion funcion () = cos ae fo an oscillaion in which he objec jus happened o be a = a =. Bu you will ecall ha = is an abiay choice, he insan of ie when you o soeone else sas a sopwach. Wha if you had saed he sopwach when he objec was a = -, o when he objec was soewhee in he iddle of an oscillaion? In ohe wods, wha if he oscillao had diffeen iniial condiions. he posiion gaph would sill show an oscillaion, bu neihe Figue 14.5 no () = cos would descibe he oion coecly. o lean how o descibe he oscillaion fo ohe iniial condiions i will help o un o a opic you sudied in Chape 4 cicula oion. hee s a ey close connecion beween siple haonic oion and cicula oion. Iagine you hae a unable wih a sall ball glued o he edge. FiguRE 14.6a shows how o ae a shadow oie of he ball by pojecing a ligh pas he ball and ono a sceen. he ball s shadow oscillaes bac and foh as he unable oaes. his is ceainly peiodic oion, wih he sae peiod as he unable, bu is i siple haonic oion? o find ou, you could place a eal objec on a eal sping diecly below he shadow, as shown in FiguRE 14.6b. If you did so, and if you adjused he unable o hae he sae peiod as he sping, you would find ha he shadow s oion eacly aches he siple haonic oion of he objec on he sping. Unifo cicula oion pojeced ono one diension is siple haonic oion. o undesand his, conside he paicle in FiguRE I is in unifo cicula oion, oing couneclocwise in a cicle wih adius. s in Chape 4, we can locae he paicle by he angle f easued ccw fo he -ais. Pojecing he ball s shadow ono a sceen in Figue 14.6 is equialen o obseing jus he -coponen of he paicle s oion. Figue 14.7 shows ha he -coponen, when he paicle is a angle f, is Recall ha he paicle s angula elociy, in ad/s, is = df d = cos f (14.9) (14.1) his is he ae a which he angle f is inceasing. If he paicle sas fo f = a =, is angle a a lae ie is siply s f inceases, he paicle s -coponen is f = (14.11) () = cos (14.12) his is idenical o Equaion 14.5 fo he posiion of a ass on a sping! hus he -coponen of a paicle in unifo cicula oion is siple haonic oion. NOE When used o descibe oscillaoy oion, is called he angula fequency ahe han he angula elociy. he angula fequency of an oscillao has he sae nueical alue, in ad/s, as he angula elociy of he coesponding paicle in cicula oion. FiguRE 14.6 pojecion of he cicula oion of a oaing ball aches he siple haonic oion of an objec on a sping. (a) (b) Cicula oion of ball Ligh fo pojeco Shadow unable Ball Oscillaion of ball s shadow Sceen Siple haonic oion of bloc FiguRE 14.7 paicle in unifo cicula oion wih adius and angula elociy. y f cos f cos f Paicle in unifo cicula oion he -coponen of he paicle s posiion descibes he posiion of he ball s shadow. New conceps ae inoduced hough obseaions abou he eal wold and heoies, gounded by aing sense of obseaions. his inducie appoach illusaes how science opeaes, and has been shown o ipoe suden leaning by econciling new ideas wih wha hey aleady now.

6 382 chape 14. Oscillaions nalogy is used houghou he e and figues o consolidae suden undesanding by copaing wih a oe failia concep o siuaion. cup on he unable in a icowae oen oes in a cicle. Bu fo he ouside, you see he cup sliding bac and foh in siple haonic oion! FiguRE 14.8 paicle in unifo cicula oion wih iniial angle f. y f cos f f ngle a ie is f f. Iniial posiion of paicle a cos f he iniial -coponen of he paicle s posiion can be anywhee beween and, depending on f. he naes and unis can be a bi confusing unil you ge used o he. I ay help o noice ha cycle and oscillaion ae no ue unis. Unlie he sandad ee o he sandad iloga, o which you could copae a lengh o a ass, hee is no sandad cycle o which you can copae an oscillaion. Cycles and oscillaions ae si ply couned eens. hus he fequency f has unis of hez, whee 1 Hz = 1 s -1. We ay say cycles pe second jus o be clea, bu he acual unis ae only pe second. he adian is he SI uni of angle. Howee, he adian is a defined uni. Fuhe, is definiion as a aio of wo lenghs (u = s/) aes i a pue nube wihou diensions. s we noed in Chape 4, he uni of angle, be i adians o degees, is eally jus a nae o eind us ha we e dealing wih an angle. he 2p in he equaion = 2pf (and in siila siuaions), which is saed wihou unis, eans 2p ad/cycle. When uliplied by he fequency f in cycles/s, i gies he fequency in ad/s. ha is why, in his cone, is called he angula fequency. NOE Hez is specifically cycles pe second o oscillaions pe second. I is used fo f bu no fo. We ll always be caeful o use ad/s fo, bu you should be awae ha any boos gie he unis of as siply s -1. he Phase Consan Now we e eady o conside he issue of ohe iniial condiions. he paicle in Figue 14.7 saed a f =. his was equialen o an oscillao saing a he fa igh edge, =. FiguRE 14.8 shows a oe geneal siuaion in which he iniial angle f can hae any alue. he angle a a lae ie is hen f = + f (14.13) In his case, he paicle s pojecion ono he -ais a ie is () = cos( + f ) (14.14) If Equaion descibes he paicle s pojecion, hen i us also be he posiion of an oscillao in siple haonic oion. he oscillao s elociy is found by aing he deiaie d/d. he esuling equaions, () = cos( + f ) () = - sin( + f ) = - a sin( + f ) (14.15) ae he wo piay ineaic equaions of siple haonic oion. he quaniy f = + f, which seadily inceases wih ie, is called he phase of he oscillaion. he phase is siply he angle of he cicula-oion paicle whose shadow aches he oscillao. he consan f is called he phase consan. I specifies he iniial condiions of he oscillao. o see wha he phase consan eans, se = in Equaions 14.15: = cos f = - sin f (14.16) he posiion and elociy a = ae he iniial condiions. Diffeen alues of he phase consan coespond o diffeen saing poins on he cicle and hus o diffeen iniial condiions. he pefec cosine funcion of Figue 14.5 and he equaion () = cos ae fo an oscillaion wih f = ad. You can see fo Equaions ha f = ad iplies = and =. ha is, he paicle sas fo es a he poin of aiu displaceen. FiguRE 14.9 illusaes hese ideas by looing a hee alues of he phase consan: f = p/3 ad (6 ), -p/3 ad (-6 ), and p ad (18 ). Noice ha f = p/3 ad and f = -p/3 ad hae he sae saing posiion, = 1 2. his is a popey of he cosine funcion in Equaion Bu hese ae no he sae iniial condiions. In one case he oscillao sas a 1 2 while oing o he igh, in he ohe case i sas a 1 2 while oing o he lef. You can disinguish beween he wo by isualizing he oion.

7 14.2. Siple Haonic Moion and Cicula Moion 383 FiguRE 14.9 Oscillaions descibed by he phase consans f = p/3 ad, -p/3 ad, and p ad. f p/3 ad f p/3 ad f p ad y y y he saing poin of he oscillaion is shown on he cicle and on he gaph. p f p f f p Eensie use is ade of uliple epesenaions placing diffeen epesenaions side by side o help sudens deelop he ey sill of anslaing beween wods, ah, and figues. Essenial o good poble-soling, his sill is oelooed in os physics eboos. he gaphs each hae he sae apliude and peiod. hey ae shifed elaie o he f ad gaphs of Figue 14.5 because hey hae diffeen iniial condiions. a a a a a a ll alues of he phase consan f beween and p ad coespond o a paicle in he uppe half of he cicle and oing o he lef. hus is negaie. ll alues of he phase consan f beween p and 2p ad (o, as hey ae usually saed, beween -p and ad) hae he paicle in he lowe half of he cicle and oing o he igh. hus is posiie. If you e old ha he oscillao is a = 1 2 and oing o he igh a =, hen he phase consan us be f = -p/3 ad, no +p/3 ad. using he iniial condiions EMPLE 14.4 n objec on a sping oscillaes wih a peiod of.8 s and an apliude of 1 c. = s, i is 5. c o he lef of equilibiu and oing o he lef. Wha ae is posiion and diecion of oion a = 2. s? MODEL n objec oscillaing on a sping is in siple haonic oion. SOLE We can find he phase consan f fo he iniial condiion = -5. c = cos f. his condiion gies f = cos = cos = { 2 p ad = {12 3 Because he oscillao is oing o he lef a =, i is in he uppe half of he cicula-oion diaga and us hae a phase consan beween and p ad. hus f is 2 3 p ad. he angula fequency is = 2p = 2p = 7.85 ad/s.8 s hus he objec s posiion a ie = 2. s is () = cos( + f ) = (1 c) cos1 (7.85 ad/s) (2. s) p = (1 c) cos(17.8 ad) = 5. c he objec is now 5. c o he igh of equilibiu. Bu which way is i oing? hee ae wo ways o find ou. he diec way is o calculae he elociy a = 2. s: = - sin( + f ) = +68 c/s he elociy is posiie, so he oion is o he igh. lenaiely, we could noe ha he phase a = 2. s is f = 17.8 ad. Diiding by p, you can see ha f = 17.8 ad = 5.67p ad = (4p p) ad he 4p ad epesens wo coplee eoluions. he ea phase of 1.67p ad falls beween p and 2p ad, so he paicle in he cicula-oion diaga is in he lowe half of he cicle and oing o he igh.

8 384 chape 14. Oscillaions NOE he inese-cosine funcion cos -1 is a wo alued funcion. You calculao euns a single alue, an angle beween ad and p ad. Bu he negaie of his angle is also a soluion. s Eaple 14.4 deonsaes, you us use addiional infoaion o choose beween he. Sop o hin quesions a he end of a secion allow sudens o quicly chec hei undesanding. Using poweful aning-as and gaphical echniques, hey ae designed o efficienly pobe ey isconcepions and encouage acie eading. (nswes ae poided a he end of he chape.) Sop o hin 14.2 he figue shows fou oscillaos a =. Which one has he phase consan f = p/4 ad? (a) (b) (c) (d) () FiguRE 14.1 he enegy is ansfoed beween ineic enegy and poenial enegy as he objec oscillaes, bu he echanical enegy E = K + U doesn change. (a) Enegy is ansfoed beween ineic and poenial, bu he oal echanical enegy E doesn change. E Enegy in Siple Haonic Moion We e begun o deelop he aheaical language of siple haonic oion, bu hus fa we haen included any physics. We e ade no enion of he ass of he objec o he sping consan of he sping. n enegy analysis, using he ools of Chapes 1 and 11, is a good saing place. FiguRE 14.1a shows an objec oscillaing on a sping, ou pooype of siple haonic oion. Now we ll specify ha he objec has ass, he sping has sping consan, and he oion aes place on a ficionless suface. You leaned in Chape 1 ha he elasic poenial enegy when he objec is a posiion is U s = 1 2 ( )2, whee = - e is he displaceen fo he equilibiu posiion e. In his chape we ll always use a coodinae syse in which e =, aing =. hee s no chance fo confusion wih gaiaional poenial enegy, so we can oi he subscip s and wie he elasic poenial enegy as (b) Enegy Poenialenegy cue U = (14.17) hus he echanical enegy of an objec oscillaing on a sping is E = K + U = (14.18) E uning poin oal enegy line Enegy hee is puely ineic. uning poin FiguRE 14.1b is an enegy diaga, showing he poenial-enegy cue U = as a paabola. Recall ha a paicle oscillaes beween he uning poins whee he oal enegy line E cosses he poenial-enegy cue. he lef uning poin is a = -, and he igh uning poin is a = +. o go beyond hese poins would equie a negaie ineic enegy, which is physically ipossible. You can see ha he paicle has puely poenial enegy a and puely ineic enegy as i passes hough he equilibiu poin a. aiu displaceen, wih = { and =, he enegy is Enegy hee is puely poenial. E(a = {) = U = (14.19) =, whee = { a, he enegy is E(a = ) = K = 1 2 ( a) 2 (14.2)

9 14.3. Enegy in Siple Haonic Moion 385 he syse s echanical enegy is conseed because he suface is ficionless and hee ae no eenal foces, so he enegy a aiu displaceen and he enegy a aiu speed, Equaions and 14.2, us be equal. ha is 1 2 ( a) 2 = (14.21) hus he aiu speed is elaed o he apliude by a = (14.22) his is a elaionship based on he physics of he siuaion. Ealie, using ineaics, we found ha a = 2p = 2pf = (14.23) Copaing Equaions and 14.23, we see ha fequency and peiod of an oscillaing sping ae deeined by he sping consan and he objec s ass : = f = 1 2p = 2p (14.24) hese hee epessions ae eally only one equaion. hey say he sae hing, bu each epesses i in slighly diffeen es. Equaions ae he answe o he second quesion we posed a he beginning of he chape, whee we ased how he peiod and fequency ae elaed o he objec s ass, he sping consan, and he apliude. I is pehaps supising, bu he peiod and fequency do no depend on he apliude. sall oscillaion and a lage oscillaion hae he sae peiod. Because enegy is conseed, we can cobine Equaions 14.18, 14.19, and 14.2 o wie E = = = 1 2 ( a) 2 (conseaion of enegy) (14.25) FiguRE Kineic enegy, poenial enegy, and he oal echanical enegy fo siple haonic oion. Enegy he oal echanical enegy E is consan. Poenial enegy Kineic enegy ny pai of hese epessions ay be useful, depending on he nown infoaion. Fo eaple, you can use he apliude o find he speed a any poin by cobining he fis and second epessions fo E. he speed a posiion is = B (2-2 ) = (14.26) FiguRE shows gaphically how he ineic and poenial enegy change wih ie. hey boh oscillae bu eain posiie because and ae squaed. Enegy is coninuously being ansfoed bac and foh beween he ineic enegy of he oing bloc and he soed poenial enegy of he sping, bu hei su eains consan. Noice ha K and U boh oscillae wice each peiod; ae sue you undesand why. Posiion using conseaion of enegy EMPLE g bloc on a sping is pulled a disance of 2 c and eleased. he subsequen oscillaions ae easued o hae a peiod of.8 s. a. wha posiion o posiions is he bloc s speed 1. /s? b. Wha is he sping consan? MODEL he oion is SHM. Enegy is conseed. SOLE a. he bloc sas fo he poin of aiu displaceen, whee E = U = a lae ie, when he posiion is and he speed is, enegy conseaion equies = Soling fo, we find = 2-2 = 2 - B B whee we used / = 2 fo Equaion he angula fequency is easily found fo he peiod: = 2p/ = 7.85 ad/s. hus = (.2 ) 2 1. /s - B ad/s 2 2 = {.15 = {15 c hee ae wo posiions because he bloc has his speed on eihe side of equilibiu. b. lhough pa a did no equie ha we now he sping consan, i is saighfowad o find fo Equaion 14.24: = 2p = 4p2 2 = 4p2 (.5 g) (.8 s) 2 = 31 N/

10 386 chape 14. Oscillaions Sop o hin 14.3 he fou spings shown hee hae been copessed fo hei equilibiu posiion a = c. When eleased, he aached ass will sa o oscillae. Ran in ode, fo highes o lowes, he aiu speeds of he asses. (a) (b) (c) (d) (c) 14.4 he Dynaics of Siple Haonic Moion Ou analysis hus fa has been based on he epeienal obseaion ha he oscillaion of a sping loos sinusoidal. I s ie o show ha Newon s second law pedics sinusoidal oion. oion diaga will help us isualize he objec s acceleaion. FiguRE shows one cycle of he oion, sepaaing oion o he lef and oion o he igh o ae he diaga clea. s you can see, he objec s elociy is lage as i passes hough he u equilibiu poin a =, bu is no changing a ha poin. cceleaion easues he change of he elociy; hence a u = u a =. FiguRE Posiion and acceleaion gaphs fo an oscillaing sping. We e chosen f =. Posiion cceleaion a a a 2 a a 2 when FiguRE Moion diaga of siple haonic oion. he lef and igh oions ae sepaaed eically fo claiy bu eally occu along he sae line. Sae poin a 2 a a o he lef a a o he igh Equilibiu a a a Sae poin In conas, he elociy is changing apidly a he uning poins. he igh uning poin, changes fo a igh-poining eco o a lef-poining eco. hus he u acceleaion a u a he igh uning poin is lage and o he lef. In one-diensional oion, he acceleaion coponen a has a lage negaie alue a he igh uning poin. Siilaly, he acceleaion a u a he lef uning poin is lage and o he igh. Consequenly, a has a lage posiie alue a he lef uning poin. Ou oion-diaga analysis suggess ha he acceleaion a is os posiie when he displaceen is os negaie, os negaie when he displaceen is a aiu, and zeo when =. his is confied by aing he deiaie of he elociy: a a 2 a in 2 when a = d d = d d ( - sin ) = -2 cos (14.27) hen gaphing i. FiguRE shows he posiion gaph ha we saed wih in Figue 14.4 and he coesponding acceleaion gaph. Copaing he wo, you can see ha he acceleaion

11 14.4. he Dynaics of Siple Haonic Moion 387 gaph loos lie an upside-down posiion gaph. In fac, because = cos, Equaion fo he acceleaion can be wien a = - 2 (14.28) ha is, he acceleaion is popoional o he negaie of he displaceen. he acceleaion is, indeed, os posiie when he displaceen is os negaie and is os negaie when he displaceen is os posiie. Recall ha he acceleaion is elaed o he ne foce by Newon s second law. Conside again ou pooype ass on a sping, shown in FiguRE his is he siples possible oscillaion, wih no disacions due o ficion o gaiaional foces. We will assue he sping iself o be assless. s you leaned in Chape 1, he sping foce is gien by Hooe s law: FiguRE he pooype of siple haonic oion: a ass oscillaing on a hoizonal sping wihou ficion. Sping consan 2 F sp Oscillaion (F sp ) = - (14.29) he inus sign indicaes ha he sping foce is a esoing foce, a foce ha always poins bac owad he equilibiu posiion. If we place he oigin of he coodinae syse a he equilibiu posiion, as we e done houghou his chape, hen = and Hooe s law is siply (F sp ) = -. he -coponen of Newon s second law fo he objec aached o he sping is (F ne ) = (F sp ) = - = a (14.3) Equaion 14.3 is easily eaanged o ead a = - (14.31) You can see ha Equaion is idenical o Equaion if he syse oscillaes wih angula fequency = 1/. We peiously found his epession fo fo an enegy analysis. Ou epeienal obseaion ha he acceleaion is popoional o he negaie of he displaceen is eacly wha Hooe s law would lead us o epec. ha s he good news. he bad news is ha a is no a consan. s he objec s posiion changes, so does he acceleaion. Nealy all of ou ineaic ools hae been based on consan acceleaion. We can use hose ools o analyze oscillaions, so we us go bac o he ey definiion of acceleaion: a = d d = d 2 d 2 cceleaion is he second deiaie of posiion wih espec o ie. If we use his definiion in Equaion 14.31, i becoes d 2 d = - (equaion of oion fo a ass on a sping) (14.32) 2 Equaion 14.32, which is called he equaion of oion, is a second-ode diffeenial equaion. Unlie ohe equaions we e deal wih, Equaion canno be soled by diec inegaion. We ll need o ae a diffeen appoach. Soling he Equaion of Moion he soluion o an algebaic equaion such as 2 = 4 is a nube. he soluion o a diffeenial equaion is a funcion. he in Equaion is eally (), he posiion as a funcion of ie. he soluion o his equaion is a funcion () whose second deiaie is he funcion iself uliplied by ( -/). One ipoan popey of diffeenial equaions ha you will lean abou in ah is ha he soluions ae unique. ha is, hee is only one soluion o Equaion ha saisfies he iniial condiions. If we wee able o guess a soluion, he uniqueness popey would ell us ha we had found he only soluion. ha igh see a ahe

12 388 chape 14. Oscillaions sange way o sole equaions, bu in fac diffeenial equaions ae fequenly soled by using you nowledge of wha he soluion needs o loo lie o guess an appopiae funcion. Le us gie i a y! We now fo epeienal eidence ha he oscillaoy oion of a sping appeas o be sinusoidal. Le us guess ha he soluion o Equaion should hae he funcional fo () = cos( + f ) (14.33) n opical echnique called inefeoey eeals he bell lie ibaions of a wine glass. whee,, and f ae unspecified consans ha we can adjus o any alues ha igh be necessay o saisfy he diffeenial equaion. If you wee o guess ha a soluion o he algebaic equaion 2 = 4 is = 2, you would eify you guess by subsiuing i ino he oiginal equaion o see if i wos. We need o do he sae hing hee: Subsiue ou guess fo () ino Equaion o see if, fo an appopiae choice of he hee consans, i wos. o do so, we need he second deiaie of (). ha is saighfowad: () = cos( + f ) d d = - sin( + f ) (14.34) d 2 d 2 = -2 cos( + f ) If we now subsiue he fis and hid of Equaions ino Equaion 14.32, we find - 2 cos( + f ) = - cos( + f ) (14.35) Equaion will be ue a all insans of ie if and only if 2 = /. hee do no see o be any esicions on he wo consans and f hey ae deeined by he iniial condiions. So we hae found by guessing! ha he soluion o he equaion of oion fo a ass oscillaing on a sping is () = cos( + f ) (14.36) whee he angula fequency = 2pf = B is deeined by he ass and he sping consan. (14.37) NOE Once again we see ha he oscillaion fequency is independen of he apliude. Equaions and see soewha anicliacic because we e been using hese esuls fo he las seeal pages. Bu eep in ind ha we had been assuing = cos siply because he epeienal obseaions looed lie a cosine funcion. We e now jusified ha assupion by showing ha Equaion eally is he soluion o Newon s second law fo a ass on a sping. he heoy of oscillaion, based on Hooe s law fo a sping and Newon s second law, is in good ageeen wih he epeienal obseaions. his conclusion gies an affiaie answe o he las of he hee quesions ha we ased ealy in he chape, which was whehe he sinusoidal oscillaion of SHM is a consequence of Newon s laws. nalyzing an oscillao EMPLE 14.6 = s, a 5 g bloc oscillaing on a sping is obseed oing o he igh a = 15 c. I eaches a aiu displaceen of 25 c a =.3 s. a. Daw a posiion-esus-ie gaph fo one cycle of he oion. b. wha ies duing he fis cycle does he ass pass hough = 2 c?

13 14.5. Veical Oscillaions 389 MODEL he oion is siple haonic oion. SOLE a. he posiion equaion of he bloc is () = cos( + f ). We now ha he apliude is =.25 and ha =.15. Fo hese wo pieces of infoaion we obain he phase consan: f = cos = cos-1 (.6) = {.927 ad he objec is iniially oing o he igh, which ells us ha he phase consan us be beween -p and ad. hus f = ad. he bloc eaches is aiu displaceen a = a ie =.3 s. ha insan of ie a = = cos( + f ) his can be ue only if cos( + f ) = 1, which equies + f =. hus = -f = - ( ad).3 s = 3.9 ad/s Now ha we now, i is saighfowad o copue he peiod: = 2p = 2. s FiguRE gaphs () = (25 c) cos( ), whee is in s, fo = s o = 2. s. b. Fo = cos( + f ), he ie a which he ass eaches posiion = 2 c is FiguRE Posiion esus ie gaph fo he oscillao of Eaple (c) 2. s (s) = cos f c = 3.9 ad/s 1 1 cos-1 25 c ad =.51 s calculao euns only one alue of cos -1, in he ange o p ad, bu we noed ealie ha cos -1 acually has wo alues. Indeed, you can see in Figue ha hee ae wo ies a which he ass passes = 2 c. Because hey ae syeical on eihe side of =.3 s, when =, he fis poin is (.51 s -.3 s) =.21 s befoe he aiu. hus he ass passes hough = 2 c a =.9 s and again a =.51 s. Sop o hin 14.4 his is he posiion gaph of a ass on a sping. Wha can you say abou he elociy and he foce a he insan indicaed by he dashed line? a. Velociy is posiie; foce is o he igh. b. Velociy is negaie; foce is o he igh. c. Velociy is zeo; foce is o he igh. d. Velociy is posiie; foce is o he lef. e. Velociy is negaie; foce is o he lef. f. Velociy is zeo; foce is o he lef. g. Velociy and foce ae boh zeo eical Oscillaions We hae focused ou analysis on a hoizonally oscillaing sping. Bu he ypical deonsaion you ll see in class is a ass bobbing up and down on a sping hung eically fo a suppo. Is i safe o assue ha a eical oscillaion has he sae aheaical descipion as a hoizonal oscillaion? O does he addiional foce of gaiy change he oion? Le us loo a his oe caefully. FiguRE shows a bloc of ass hanging fo a sping of sping consan. n ipoan fac o noice is ha he equilibiu posiion of he bloc is no whee he sping is a is unseched lengh. he equilibiu posiion of he bloc, whee i hangs oionless, he sping has seched by L. Finding L is a saic-equilibiu poble in which he upwad sping foce balances he downwad gaiaional foce on he bloc. he y-coponen of he sping foce is gien by Hooe s law: (F sp ) y = - y = + L (14.38) FiguRE Gaiy seches he sping. Unseched sping he bloc hanging a es has seched he sping by L. L F sp F G Eensie use is ade of uliple epesenaions placing diffeen epesenaions side by side o help sudens deelop he ey sill of anslaing beween wods, ah, and figues. Essenial o good poble-soling, his sill is oelooed in os physics eboos.

14 39 chape 14. Oscillaions Sping seched by L Hand-dawn seches ae incopoaed ino selec woed eaples o poide a clea odel of wha sudens should daw duing hei own poble soling. FiguRE he bloc oscillaes aound he equilibiu posiion. Bloc s equilibiu posiion y Sping seched by L y F sp F G F ne Oscillaion aound he equilibiu posiion is syeical. Equaion aes a disincion beween L, which is siply a disance and is a posiie nube, and he displaceen y. he bloc is displaced downwad, so y = - L. Newon s fis law fo he bloc in equilibiu is fo which we can find (F ne ) y = (F sp ) y + (F G ) y = L - g = (14.39) L = g (14.4) his is he disance he sping seches when he bloc is aached o i. Le he bloc oscillae aound his equilibiu posiion, as shown in FiguRE We e now placed he oigin of he y-ais a he bloc s equilibiu posiion in ode o be consisen wih ou analyses of oscillaions houghou his chape. If he bloc oes upwad, as he figue shows, he sping ges shoe copaed o is equilibiu lengh, bu he sping is sill seched copaed o is unseched lengh in Figue When he bloc is a posiion y, he sping is seched by an aoun L - y and hence ees an upwad sping foce F sp = ( L - y). he ne foce on he bloc a his poin is (F ne ) y = (F sp ) y + (F G ) y = ( L - y) - g = ( L - g) - y (14.41) Bu L - g is zeo, fo Equaion 14.4, so he ne foce on he bloc is siply (F ne ) y = -y (14.42) Equaion fo eical oscillaions is eacly he sae as Equaion 14.3 fo hoizonal oscillaions, whee we found (F ne ) = -. ha is, he esoing foce fo eical oscillaions is idenical o he esoing foce fo hoizonal oscillaions. he ole of gaiy is o deeine whee he equilibiu posiion is, bu i doesn affec he oscillaoy oion aound he equilibiu posiion. Because he ne foce is he sae, Newon s second law has eacly he sae oscillaoy soluion: y() = cos( + f ) (14.43) wih, again, = 2/. he eical oscillaions of a ass on a sping ae he sae siple haonic oion as hose of a bloc on a hoizonal sping. his is an ipoan finding because i was no obious ha he oion would sill be siple haonic oion when gaiy was included. Bungee oscillaions EMPLE 14.7 n 83 g suden hangs fo a bungee cod wih sping consan 27 N/. he suden is pulled down o a poin whee he cod is 5. longe han is unseched lengh, hen eleased. Whee is he suden, and wha is his elociy 2. s lae? MODEL bungee cod can be odeled as a sping. Veical oscillaions on he bungee cod ae SHM. isulize FiguRE shows he siuaion. SOLE lhough he cod is seched by 5. when he suden is eleased, his is no he apliude of he oscillaion. Oscillaions occu aound he equilibiu posiion, so we hae o begin by finding he equilibiu poin whee he suden hangs oionless. he cod sech a equilibiu is gien by Equaion 14.4: L = g = 3. Seching he cod 5. pulls he suden 2. below he equilibiu poin, so = 2.. ha is, he suden oscillaes wih apliude = 2. abou a poin 3. beneah he bungee FiguRE suden on a bungee cod oscillaes abou he equilibiu posiion. cod s oiginal end poin. he suden s posiion as a funcion of ie, as easued fo he equilibiu posiion, is y() = (2. ) cos( + f ) he bungee cod is odeled as a sping.

15 14.6. he Pendulu 391 whee = 2/ = 1.8 ad/s. he iniial condiion y = cos f = - equies he phase consan o be f = p ad. = 2. s he suden s posiion and elociy ae y = (2. ) cos1 (1.8 ad/s) (2. s) + p ad2 = 1.8 y = - sin( + f ) = -1.6 /s he suden is 1.8 aboe he equilibiu posiion, o 1.2 below he oiginal end of he cod. Because his elociy is negaie, he s passed hough he highes poin and is heading down he Pendulu Now le s loo a anohe ey coon oscillao: a pendulu. FiguRE 14.19a shows a ass aached o a sing of lengh L and fee o swing bac and foh. he pendulu s posiion can be descibed by he ac of lengh s, which is zeo when he pendulu hangs saigh down. Because angles ae easued ccw, s and u ae posiie when he pendulu is o he igh of cene, negaie when i is o he lef. u wo foces ae acing on he ass: he sing ension and gaiy F u G. I will be conenien o epea wha we did in ou sudy of cicula oion: Diide he foces ino angenial coponens, paallel o he oion, and adial coponens paallel o he sing. hese ae shown on he fee-body diaga of FiguRE 14.19b. Newon s second law fo he angenial coponen, paallel o he oion, is (F ne ) = a F = (F G ) = -g sin u = a (14.44) Using a = d 2 s/d 2 fo acceleaion aound he cicle, and noing ha he ass cancels, we can wie Equaion as d 2 s = -g sin u (14.45) 2 d his is he equaion of oion fo an oscillaing pendulu. he sine funcion aes his equaion oe coplicaed han he equaion of oion fo an oscillaing sping. he Sall-ngle ppoiaion Suppose we esic he pendulu s oscillaions o sall angles of less han abou 1. his esicion allows us o ae use of an ineesing and ipoan piece of geoey. FiguRE 14.2 shows an angle u and a cicula ac of lengh s = u. igh iangle has been consuced by dopping a pependicula fo he op of he ac o he ais. he heigh of he iangle is h = sin u. Suppose ha he angle u is sall. In ha case hee is ey lile diffeence beween h and s. If h s, hen sin u u. I follows ha sin u u (u in adians) he esul ha sin u u fo sall angles is called he sall-angle appoiaion. We can siilaly noe ha l fo sall angles. Because l = cos u, i follows ha cos u 1. Finally, we can ae he aio of sine and cosine o find an u sin u u. able 14.3 suaizes he sall-angle appoiaion. We will hae ohe occasions o use he sall-angle appoiaion houghou he eainde of his e. NOE he sall-angle appoiaion is alid only if angle u is in adians! How sall does u hae o be o jusify using he sall-angle appoiaion? I s easy o use you calculao o find ha he sall-angle appoiaion is good o hee FiguRE he oion of a pendulu. (a) (b) u and s ae negaie on he lef. Cene of cicle angenial ais he gaiaional foce has a angenial coponen g sin u. u u u u and s ae posiie on he igh. L s c lengh (F G ) l cos u F G he ension has only a adial coponen. u (F G ) FiguRE 14.2 he geoeical basis of he sall angle appoiaion. h sin u s u BLE 14.3 Sall angle appoiaions. u us be in adians. sin u u an u sin u u cos u 1

16 392 chape 14. Oscillaions significan figues, an eo of.1%, up o angles of.1 ad ( 5 ). In pacice, we will use he appoiaion up o abou 1, bu fo angles any lage i apidly loses alidiy and poduces unaccepable esuls. If we esic he pendulu o u 6 1, we can use sin u u. In ha case, Equaion fo he ne foce on he ass is (F ne ) = -g sin u -gu = - g L s whee, in he las sep, we used he fac ha angle u is elaed o he ac lengh by u = s/l. hen he equaion of oion becoes d 2 s d 2 = - g L s (14.46) he pendulu cloc has been used fo hundeds of yeas. his is eacly he sae as Equaion fo a ass oscillaing on a sping. he naes ae diffeen, wih eplaced by s and / by g/l, bu ha does no ae i a diffeen equaion. Because we now he soluion o he sping poble, we can iediaely wie he soluion o he pendulu poble jus by changing aiables and consans: s() = cos( + f ) o u() = u a cos( + f ) (14.47) he angula fequency = 2pf = g L (14.48) is deeined by he lengh of he sing. he pendulu is ineesing in ha he fequency, and hence he peiod, is independen of he ass. I depends only on he lengh of he pendulu. he apliude and he phase consan f ae deeined by he iniial condiions, jus as hey wee fo an oscillaing sping. Daa-based Eaples help sudens wih he sill of dawing conclusions fo laboaoy daa. Designed o suppleen lab-based insucion, hese eaples also help sudens in geneal wih aheaical easoning, gaphical inepeaion, and assessen of esuls. he aiu angle of a pendulu EMPLE g ass on a 3-c-long sing oscillaes as a pendulu. I has a speed of.25 /s as i passes hough he lowes poin. Wha aiu angle does he pendulu each? MODEL ssue ha he angle eains sall, in which case he oion is siple haonic oion. SOLE he angula fequency of he pendulu is g = B L = 9.8 /s 2 B.3 he gaiee = 5.72 ad/s EMPLE 14.9 Deposis of ineals and oe can ale he local alue of he feefall acceleaion because hey end o be dense han suounding ocs. Geologiss use a gaiee an insuen ha accuaely easues he local fee-fall acceleaion o seach fo oe deposis. One of he siples gaiees is a pendulu. o achiee he highes accuacy, a sopwach is used o ie 1 oscillaions of a pendulu of diffeen lenghs. one locaion in he field, a geologis aes he following easueens: he speed a he lowes poin is a =, so he apliude is = s a = a.25 /s = 5.72 ad/s =.437 he aiu angle, a he aiu ac lengh s a, is u a = s a L = =.146 ad = 8.3 SSESS Because he aiu angle is less han 1, ou analysis based on he sall-angle appoiaion is easonable. Wha is he local alue of g? Lengh () ie (s)

17 14.6. he Pendulu 393 MODEL ssue he oscillaion angle is sall, in which case he oion is siple haonic oion wih a peiod independen of he ass of he pendulu. Because he daa ae nown o fou significan figues ({1 on he lengh and {.1 s on he iing, boh of which ae easily achieable), we epec o deeine g o fou significan figues. SOLE Fo Equaion 14.48, using f = 1/, we find 2 = 1 2p L g 2 2 = 4p2 g L ha is, he squae of a pendulu s peiod is popoional o is lengh. Consequenly, a gaph of 2 esus L should be a saigh line passing hough he oigin wih slope 4p 2 /g. We can use he epeienally easued slope o deeine g. FiguRE is a gaph of he daa, wih he peiod found by diiding he easued ie by 1. s epeced, he gaph is a saigh line passing hough he oigin. he slope of he bes-fi line is 4.21 s 2 /. Consequenly, g = 4p2 slope = 4p 2 = s 2 /s2 / SSESS he fac ha he gaph is linea and passes hough he oigin confis ou odel of he siuaion. Had his no been he FiguRE Gaph of he squae of he pendulu s peiod esus is lengh. 2 (s 2 ) y Bes-fi line L () case, we would hae had o conclude eihe ha ou odel of he pendulu as a siple, sall-angle pendulu was no alid o ha ou easueens wee bad. his is an ipoan eason fo haing uliple daa poins ahe han using only one lengh. he Condiions fo Siple Haonic Moion You can begin o see how, in a sense, we hae soled all siple-haonic-oion pobles once we hae soled he poble of he hoizonal sping. he esoing foce of a sping, F sp = -, is diecly popoional o he displaceen fo equilibiu. he pendulu s esoing foce, in he sall-angle appoiaion, is diecly popoional o he displaceen s. esoing foce ha is diecly popoional o he displaceen fo equilibiu is called a linea esoing foce. Fo any linea esoing foce, he equaion of oion is idenical o he sping equaion (ohe han pehaps using diffeen sybols). Consequenly, any syse wih a linea esoing foce will undego siple haonic oion aound he equilibiu posiion. his is why an oscillaing sping is he pooype of SHM. Eeyhing ha we lean abou an oscillaing sping can be applied o he oscillaions of any ohe linea esoing foce, anging fo he ibaion of aiplane wings o he oion of elecons in elecic cicuis. Le s suaize his infoaion wih a acics Bo. CiCS B O 14.1 idenifying and analyzing siple haonic oion 1 If he ne foce acing on a paicle is a linea esoing foce, he oion will be siple haonic oion aound he equilibiu posiion. 2 he posiion as a funcion of ie is () = cos( + f ). he elociy as a funcion of ie is () = - sin( + f ). he aiu speed is a =. he equaions ae gien hee in es of, bu hey can be wien in es of y, u, o soe ohe paaee if he siuaion calls fo i. 3 he apliude and he phase consan f ae deeined by he iniial condiions hough = cos f and = - sin f. 4 he angula fequency (and hence he peiod = 2p/) depends on he physics of he paicula siuaion. Bu does no depend on o f. 5 Mechanical enegy is conseed. hus = = 1 2 ( a) 2. Enegy conseaion poides a elaionship beween posiion and elociy ha is independen of ie. Eecises 7 12, acics Boes gie sep-bysep pocedues fo deeloping specific sills (dawing fee-body diagas, using ay acing, ec.).

18 394 chape 14. Oscillaions FiguRE physical pendulu. Moen a of gaiaional oque u d l Mg Disance fo pio o cene of ass he Physical Pendulu ass on a sing is ofen called a siple pendulu. Bu you can also ae a pendulu fo any solid objec ha swings bac and foh on a pio unde he influence of gaiy. his is called a physical pendulu. FiguRE shows a physical pendulu of ass M fo which he disance beween he pio and he cene of ass is l. he oen a of he gaiaional foce acing a he cene of ass is d = l sin u, so he gaiaional oque is = -Mgd = -Mgl sin u he oque is negaie because, fo posiie u, i s causing a clocwise oaion. If we esic he angle o being sall (u 6 1 ), as we did fo he siple pendulu, we can use he sall-angle appoiaion o wie = -Mglu (14.49) Gaiy causes a linea esoing oque on he pendulu ha is, he oque is diecly popoional o he angula displaceen u so we epec he physical pendulu o undego SHM. Fo Chape 12, Newon s second law fo oaional oion is a = d 2 u d = 2 I whee I is he objec s oen of ineia abou he pio poin. Using Equaion fo he oque, we find d 2 u d 2 = -Mgl u (14.5) I Copaison wih Equaion shows ha his is again he SHM equaion of oion, his ie wih angula fequency Mgl = 2pf = (14.51) B I I appeas ha he fequency depends on he ass of he pendulu, bu ecall ha he oen of ineia is diecly popoional o M. hus M cancels and he fequency of a physical pendulu, lie ha of a siple pendulu, is independen of ass. Life-science and bioengineeing woed eaples and applicaions focus on he physics of life-science siuaions in ode o see he needs of life-science sudens aing a calculus-based physics class. swinging leg as a pendulu EMPLE 14.1 suden in a bioechanics lab easues he lengh of his leg, fo hip o heel, o be.9. Wha is he fequency of he pendulu oion of he suden s leg? Wha is he peiod? MODEL We can odel a huan leg easonably well as a od of unifo coss secion, pioed a one end (he hip) o fo a physical pendulu. he cene of ass of a unifo leg is a he idpoin, so l = L/2. SOLE he oen of ineia of a od pioed abou one end is I = 1 3 ML 2, so he pendulu fequency is f = 1 Mgl = 1 Mg(L/2) 2p B I 2p B ML 2 /3 = 1 3g =.64 Hz 2p B 2L he coesponding peiod is = 1/f = 1.6 s. Noice ha we didn need o now he ass. SSESS s you wal, you legs do swing as physical pendulus as you bing he fowad. he fequency is fied by he lengh of you legs and hei disibuion of ass; i doesn depend on apliude. Consequenly, you don incease you waling speed by aing oe apid seps changing he fequency is difficul. You siply ae longe sides, changing he apliude bu no he fequency. Sop o hin 14.5 One peson swings on a swing and finds ha he peiod is 3. s. second peson of equal ass joins hi. Wih wo people swinging, he peiod is a. 6. s b. 73. s bu no necessaily 6. s c. 3. s d. 63. s bu no necessaily 1.5 s e. 1.5 s f. Can ell wihou nowing he lengh

19 14.7. Daped Oscillaions Daped Oscillaions pendulu lef o iself gadually slows down and sops. he sound of a inging bell gadually dies away. ll eal oscillaos do un down soe ey slowly bu ohes quie quicly as ficion o ohe dissipaie foces ansfo hei echanical enegy ino he heal enegy of he oscillao and is enionen. n oscillaion ha uns down and sops is called a daped oscillaion. hee ae any possible easons fo he dissipaion of enegy, such as ai esisance, ficion, and inenal foces wihin a eal sping as i flees. he foces inoled in dissipaion ae cople, bu a siple linea dag odel gies a quie accuae descipion of os daped oscillaions. ha is, we ll assue a dag foce ha depends linealy on he elociy as D u u = -b (odel of he dag foce) (14.52) whee he inus sign is he aheaical saeen ha he foce is always opposie in diecion o he elociy in ode o slow he objec. he daping consan b depends in a coplicaed way on he shape of he objec and on he iscosiy of he ai o ohe ediu in which he paicle oes. he daping consan plays he sae ole in ou odel of ai esisance ha he coefficien of ficion does in ou odel of ficion. he unis of b need o be such ha hey will gie unis of foce when uliplied by unis of elociy. s you can confi, hese unis ae g/s. alue b = g/s coesponds o he liiing case of no esisance, in which case he echanical enegy is conseed. ypical alue of b fo a sping o a pendulu in ai is.1 g/s. Objecs oing in a liquid can hae significanly lage alues of b. FiguRE shows a ass oscillaing on a sping in he pesence of a dag foce. Wih he dag included, Newon s second law is (F ne ) = (F sp ) + D = - - b = a (14.53) Using = d/d and a = d 2 /d 2, we can wie Equaion as d 2 d + b d 2 d + = (14.54) Equaion is he equaion of oion of a daped oscillao. If you copae i o Equaion 14.32, he equaion of oion fo a bloc on a ficionless suface, you ll see ha i diffes by he inclusion of he e inoling d/d. Equaion is anohe second-ode diffeenial equaion. We will siply asse (and, as a hoewo poble, you can confi) ha he soluion is he shoc absobes in cas and ucs ae heaily daped spings. he ehicle s eical oion, afe hiing a oc o a pohole, is a daped oscillaion. FiguRE n oscillaing ass in he pesence of a dag foce. Sping consan F sp D () = e -b/2 cos( + f ) (daped oscillao) (14.55) whee he angula fequency is gien by = B - b2 4 = 2 B 2 - b2 4 (14.56) 2 Hee = 2/ is he angula fequency of an undaped oscillao (b = ). he consan e is he base of naual logaihs, so e -b/2 is an eponenial funcion. Because e = 1, Equaion educes o ou peious soluion, () = cos( + f ), when b =. his aes sense and gies us confidence in Equaion lighly daped syse, which oscillaes any ies befoe sopping, is one fo which b/2 V. In ha case, is a good appoiaion. ha is, ligh daping does no affec he oscillaion fequency. FiguRE is a gaph of he posiion () fo a lighly daped oscillao, as gien by Equaion Noice ha he e e -b/2, which is shown by he dashed line, FiguRE Posiion esus ie gaph fo a daped oscillao. is he iniial apliude. he enelope of he apliude decays eponenially: a e b/2

20 396 chape 14. Oscillaions FiguRE Seeal oscillaion enelopes, coesponding o diffeen alues of he daping consan b. Fo ass 1. g pliude Enegy is conseed if hee is no daping. salle b causes less daping. b g/s b.3 g/s b.1 g/s (s) b.3 g/s Enelope fo Figue lage b causes he oscillaions o dap oe quicly. acs as a slowly aying apliude: a () = e -b/2 (14.57) whee is he iniial apliude, a =. he oscillaion eeps buping up agains his line, slowly dying ou wih ie. slowly changing line ha poides a bode o a apid oscillaion is called he enelope of he oscillaions. In his case, he oscillaions hae an eponenially decaying enelope. Mae sue you sudy Figue long enough o see how boh he oscillaions and he decaying apliude ae elaed o Equaion Changing he aoun of daping, by changing he alue of b, affecs how quicly he oscillaions decay. FiguRE shows jus he enelope a () fo seeal oscillaos ha ae idenical ecep fo he alue of he daping consan b. (You need o iagine a apid oscillaion wihin each enelope, as in Figue ) Inceasing b causes he oscillaions o dap oe quicly, while deceasing b aes he las longe. aheaical aside Eponenial decay Eponenial decay occus in a as nube of physical syses of ipoance in science and engineeing. Mechanical ibaions, elecic cicuis, and nuclea adioaciiy all ehibi eponenial decay. he nube e = p is he base of naual logaihs in he sae way ha 1 is he base of odinay logaihs. I aises naually in calculus fo he inegal du 3 u = ln u his inegal which shows up in he analysis of any physical syses fequenly leads o soluions of he fo u = e -/ = ep( -/ ) whee ep is he eponenial funcion. e 1 u u sas a. u decays o 37% of is iniial alue a. u decays o 13% of is iniial alue a 2. gaph of u illusaes wha we ean by eponenial decay. I sas wih u = a = (because e = 1) and hen seadily decays, asypoically appoaching zeo. he quaniy is called he decay consan. When =, u = e -1 =.37. When = 2, u = e -2 =.13. guens of funcions us be pue nubes, wihou unis. ha is, we can ealuae e -2, bu e -2 g aes no sense. If / is a pue nube, which i us be, hen he decay consan us hae he sae unis as. If epesens posiion, hen is a lengh; if epesens ie, hen is a ie ineal. In a specific siuaion, is ofen called he decay lengh o he decay ie. I is he lengh o ie in which he quaniy decays o 37% of is iniial alue. No ae wha he pocess is o wha u epesens, a quaniy ha decays eponenially decays o 37% of is iniial alue when one decay consan has passed. hus eponenial decay is a uniesal behaio. Eey ie you ee a new syse ha ehibis eponenial decay, is behaio will be eacly he sae as eey ohe eponenial decay. he decay cue always loos eacly lie he figue shown hee. Once you e leaned he popeies of eponenial decay, you ll iediaely now how o apply his nowledge o a new siuaion. e 2 2 Enegy in Daped Syses When consideing he oscillao s echanical enegy, i is useful o define he ie consan (also called he decay ie) o be = b (14.58) Because b has unis of g/s, has unis of seconds. Wih his definiion, we can wie he oscillaion apliude as a () = e -/2.

21 14.7. Daped Oscillaions 397 Because of he dag foce, he echanical enegy is no longe conseed. any paicula ie we can copue he echanical enegy fo E() = 1 2 ( a) 2 = 1 2 (e -/2 ) 2 = e -/ = E e -/ (14.59) whee E = is he iniial enegy a = and whee we used (z ) 2 = z 2. In ohe wods, he oscillao s echanical enegy decays eponenially wih ie consan. s FiguRE shows, he ie consan is he aoun of ie needed fo he enegy o decay o e -1, o 37%, of is iniial alue. We say ha he ie consan easues he chaaceisic ie duing which he enegy of he oscillaion is dissipaed. Roughly wo-hids of he iniial enegy is gone afe one ie consan has elapsed, and nealy 9% has dissipaed afe wo ie consans hae gone by. Fo pacical puposes, we can spea of he ie consan as he lifeie of he oscillaion abou how long i lass. Maheaically, hee is nee a ie when he oscillaion is oe. he decay appoaches zeo asypoically, bu i nee ges hee in any finie ie. he bes we can do is define a chaaceisic ie when he oion is alos oe, and ha is wha he ie consan does. FiguRE Eponenial decay of he echanical enegy of an oscillao..37e Enegy E.13E he oscillao sas wih enegy E. he enegy has deceased o 37% of is iniial alue a. he enegy has deceased o 13% of is iniial alue a 2. 2 daped pendulu EMPLE g ass swings on a 6-c-sing as a pendulu. he apliude is obseed o decay o half is iniial alue afe 35. s. a. Wha is he ie consan fo his oscillao? b. wha ie will he enegy hae decayed o half is iniial alue? MODEL he oion is a daped oscillaion. SOLE a. he iniial apliude a = is a =. = 35. s he apliude is a = 1 2. he apliude of oscillaion a ie is gien by Equaion 14.57: a () = e -b/2 = e -/2 In his case, 1 -(35. s)/2 = e 2 Noice ha we do no need o now iself because i cancels ou. o sole fo, we ae he naual logaih of boh sides of he equaion: ln = -ln 2 = ln e -(35. s)/2 = s 2 his is easily eaanged o gie = 35. s 2 ln 2 = 25.2 s If desied, we could now deeine he daping consan o be b = / =.2 g/s. b. he enegy a ie is gien by E() = E e -/ he ie a which an eponenial decay is educed o 1 2 E, half is iniial alue, has a special nae. I is called he half-life and gien he sybol 1/2. he concep of he half-life is widely used in applicaions such as adioacie decay. o elae 1/2 o, we fis wie E(a = 1/2 ) = 1 2 E = E e - 1/2 / he E cancels, giing 1 2 = e - 1/2/ gain, we ae he naual logaih of boh sides: ln = -ln 2 = ln e -1/2/ = - 1/2 / Finally, we sole fo 1/2 : 1/2 = ln 2 =.693 his esul ha 1/2 is 69% of is alid fo any eponenial decay. In his paicula poble, half he enegy is gone a 1/2 = (.693) (25.2 s) = 17.5 s SSESS he oscillao loses enegy fase han i loses apliude. his is wha we should epec because he enegy depends on he squae of he apliude. Sop o hin 14.6 Ran in ode, fo lages o salles, he ie consans a o d of he decays shown in he figue. ll he gaphs hae he sae scale. E E E E (a) (b) (c) (d)

22 398 chape 14. Oscillaions 14.8 Dien Oscillaions and Resonance FiguRE he esponse cue shows he apliude of a dien oscillao a fequencies nea is naual fequency of 2. Hz. pliude he oscillaion has aiu apliude when f e f. his is esonance. he oscillaion has only a sall apliude when f e diffes subsanially fo f. f e (Hz) his is he naual fequency. FiguRE he esonance apliude becoes highe and naowe as he daping consan deceases. pliude f 2. Hz b.8 g/s b.2 g/s b.8 g/s lighly daped syse has a ey all and ey naow esponse cue. heaily daped syse has lile esponse. f e (Hz) hus fa we hae focused on he fee oscillaions of an isolaed syse. Soe iniial disubance displaces he syse fo equilibiu, and i hen oscillaes feely unil is enegy is dissipaed. hese ae ey ipoan siuaions, bu hey do no ehaus he possibiliies. nohe ipoan siuaion is an oscillao ha is subjeced o a peiodic eenal foce. Is oion is called a dien oscillaion. siple eaple of a dien oscillaion is pushing a child on a swing, whee you push is a peiodic eenal foce applied o he swing. oe cople eaple is a ca diing oe a seies of equally spaced bups. Each bup causes a peiodic upwad foce on he ca s shoc absobes, which ae big, heaily daped spings. he elecoagneic coil on he bac of a loudspeae cone poides a peiodic agneic foce o die he cone bac and foh, causing i o send ou sound waes. i ubulence oing acoss he wings of an aicaf can ee peiodic foces on he wings and ohe aeodynaic sufaces, causing he o ibae if hey ae no popely designed. s hese eaples sugges, dien oscillaions hae any ipoan applicaions. Howee, dien oscillaions ae a aheaically cople subjec. We will siply hin a soe of he esuls, saing he deails fo oe adanced classes. Conside an oscillaing syse ha, when lef o iself, oscillaes a a fequency f. We will call his he naual fequency of he oscillao. he naual fequency fo a ass on a sping is 1/ /2p, bu i igh be gien by soe ohe epession fo anohe ype of oscillao. Regadless of he epession, f is siply he fequency of he syse if i is displaced fo equilibiu and eleased. Suppose ha his syse is subjeced o a peiodic eenal foce of fequency f e. his fequency, which is called he diing fequency, is copleely independen of he oscillao s naual fequency f. Soebody o soehing in he enionen selecs he fequency f e of he eenal foce, causing he foce o push on he syse f e ies eey second. lhough i is possible o sole Newon s second law wih an eenal diing foce, we will be conen o loo a a gaphical epesenaion of he soluion. he os ipoan esul is ha he oscillaion apliude depends ey sensiiely on he fequency f e of he diing foce. he esponse o he diing fequency is shown in FiguRE fo a syse wih = 1. g, a naual fequency f = 2. Hz, and a daping consan b =.2 g/s. his gaph of apliude esus diing fequency, called he esponse cue, occus in any diffeen applicaions. When he diing fequency is subsanially diffeen fo he oscillao s naual fequency, a he igh and lef edges of Figue 14.27, he syse oscillaes bu he apliude is ey sall. he syse siply does no espond well o a diing fequency ha diffes uch fo f. s he diing fequency ges close and close o he naual fequency, he apliude of he oscillaion ises daaically. fe all, f is he fequency a which he syse wans o oscillae, so i is quie happy o espond o a diing fequency nea f. Hence he apliude eaches a aiu when he diing fequency eacly aches he syse s naual fequency: f e = f. he apliude can becoe eceedingly lage when he fequencies ach, especially if he daping consan is ey sall. FiguRE shows he sae oscillao wih hee diffeen alues of he daping consan. hee s ey lile esponse if he daping consan is inceased o.8 g/s, bu he apliude fo f e = f becoes ey lage when he daping consan is educed o.8 g/s. his lage-apliude esponse o a diing foce whose fequency aches he naual fequency of he syse is a phenoenon called esonance. he condiion fo esonance is f e = f (esonance condiion) (14.6) Wihin he cone of dien oscillaions, he naual fequency f is ofen called he esonance fequency. n ipoan feaue of Figue is how he apliude and widh of he esonance depend on he daping consan. heaily daped syse esponds faily

23 Challenge Eaple 399 lile, een a esonance, bu i esponds o a wide ange of diing fequencies. Vey lighly daped syses can each ecepionally high apliudes, bu noice ha he ange of fequencies o which he syse esponds becoes naowe and naowe as b deceases. his allows us o undesand why a few singes can bea cysal gobles bu no inepensie, eeyday glasses. n inepensie glass gies a hud when apped, bu a fine cysal goble ings fo seeal seconds. In physics es, he goble has a uch longe ie consan han he glass. ha, in un, iplies ha he goble is ey lighly daped while he odinay glass is heaily daped (because he inenal foces wihin he glass ae no hose of a high-qualiy cysal sucue). he singe causes a sound wae o ipinge on he goble, eeing a sall diing foce a he fequency of he noe she is singing. If he singe s fequency aches he naual fequency of he goble esonance! Only he lighly daped goble, lie he op cue in Figue 14.28, can each apliudes lage enough o shae. he esicion, hough, is ha is naual fequency has o be ached ey pecisely. he sound also has o be ey loud. singe o usical insuen can shae a cysal goble by aching he goble s naual oscillaion fequency. swinging pendulu CHLLENgE EMPLE pendulu consiss of a assless, igid od wih a ass a one end. he ohe end is pioed on a ficionless pio so ha he od can oae in a coplee cicle. he pendulu is ineed, wih he ass diecly aboe he pio poin, hen eleased. he speed of he ass as i passes hough he lowes poin is 5. /s. If he pendulu lae undegoes sall-apliude oscillaions a he boo of he ac, wha will is fequency be? MODEL his is a siple pendulu because he od is assless. Howee, ou analysis of a pendulu used he sall-angle appoiaion. I applies only o he sall-apliude oscillaions a he end, no o he pendulu swinging down fo he ineed posiion. Founaely, enegy is conseed houghou, so we can analyze he big swing using conseaion of echanical enegy. isulize FiguRE is a picoial epesenaion of he pendulu swinging down fo he ineed posiion. he pendulu lengh is L, so he iniial heigh is 2L. FiguRE Befoe and afe picoial epesenaion of he pendulu swinging down fo an ineed posiion. SOLE he fequency of a siple pendulu is f = 1g/L /2p. We e no gien L, bu we can find i by analyzing he pendulu s swing down fo an ineed posiion. Mechanical enegy is conseed, and he only poenial enegy is gaiaional poenial enegy. Conseaion of echanical enegy K f + U gf = K i + U gi, wih U g = gy, is 1 2 f 2 + gy f = 1 2 i 2 + gy i he ass cancels, which is good since we don now i, and wo es ae zeo. hus Soling fo L, we find 1 2 f 2 = g(2l) = 2gL L = 2 f 4g = (5. /s)2 4(9.8 /s 2 ) =.638 Now we can calculae he fequency: f = 1 g 2p B L = 1 2p B 9.8 /s =.62 Hz SSESS he fequency coesponds o a peiod of abou 1.5 s, which sees easonable. consisen 4-sep appoach poides a poble-soling faewo houghou he boo (and all suppleens): sudens lean he ipoance of aing assupions (in he MODEL sep), gaheing infoaion, and aing seches (in he VISULIZE sep) befoe eaing he poble aheaically (SOLVE) and hen analyzing hei esul (SSESS). Challenge Eaples illusae how o inegae uliple conceps and use oe sophisicaed easoning in poble-soling, ensuing an opial ange of woed eaples fo sudens o sudy in pepaaion fo hoewo pobles.

24 4 chape 14. Oscillaions SuMMRy he goal of Chape 14 has been o undesand syses ha oscillae wih siple haonic oion. geneal Pinciples Dynaics Enegy Unique and ciically acclaied isual chape suaies consolidae undesanding by poiding each concep in wods, ah, and figues and oganizing hese ino a eical hieachy fo Geneal Pinciples (op) o pplicaions (boo). SHM occus when a linea esoing foce acs o eun a syse o an equilibiu posiion. Hoizonal sping (F ne ) = - Veical sping he oigin is a he equilibiu posiion L = g/. (F ne ) y = -y Boh: = B Pendulu (F ne ) = - 1 g L 2 s = B g L = 2p B = 2p B L g y s L If hee is no ficion o dissipaion, ineic and poenial enegy ae alenaely ansfoed ino each ohe, bu he oal echanical enegy E = K + U is conseed. E = = 1 2 ( a) 2 = In a daped syse, he enegy decays eponenially E = E e -/ whee is he ie consan. E E.37E ll ineic ll poenial ipoan Conceps Siple haonic oion (SHM) is a sinusoidal oscillaion wih peiod and apliude. Fequency f = 1 ngula fequency = 2pf = 2p Posiion () = cos( + f ) = cos1 2p + f 2 Velociy () = - a sin( + f ) wih aiu speed a = cceleaion a () = - 2 () = - 2 cos( + f ) SHM is he pojecion ono he -ais of unifo cicula oion. f = + f is he phase he posiion a ie is () = cos f = cos( + f ) he phase consan f deeines he iniial condiions: = cos f = - sin f y f f cos f cos f pplicaions Resonance When a syse is dien by a peiodic eenal foce, i esponds wih a lage-apliude oscillaion if f e f, whee f is he syse s naual oscillaion fequency, o esonan fequency. pliude f f e Daping If hee is a dag foce D u u = -b, whee b is he daping consan, hen (fo lighly daped syses) () = e -b/2 cos( + f ) he ie consan fo enegy loss is = /b.

25 Concepual Quesions 41 es and Noaion oscillaoy oion oscillao peiod, fequency, f hez, Hz siple haonic oion, SHM apliude, angula fequency, phase, f phase consan, f esoing foce equaion of oion sall-angle appoiaion linea esoing foce daped oscillaion daping consan, b enelope ie consan, half-life, 1/2 dien oscillaion naual fequency, f diing fequency, f e esponse cue esonance esonance fequency, f C ONCEPu L QuESi ONS 1. bloc oscillaing on a sping has peiod = 2 s. Wha is he peiod if: a. he bloc s ass is doubled? Eplain. Noe ha you do no now he alue of eihe o, so do no assue any paicula alues fo he. he equied analysis inoles hining abou aios. b. he alue of he sping consan is quadupled? c. he oscillaion apliude is doubled while and ae unchanged? 2. pendulu on Plane X, whee he alue of g is unnown, oscillaes wih a peiod = 2 s. Wha is he peiod of his pendulu if: a. Is ass is doubled? Eplain. Noe ha you do no now he alue of, L, o g, so do no assue any specific alues. he equied analysis inoles hining abou aios. b. Is lengh is doubled? c. Is oscillaion apliude is doubled? 3. FiguRE Q14.3 shows a posiion- (c) esus-ie gaph fo a paicle in 1 SHM. Wha ae (a) he apliude, (b) he angula fequency, and (s) (c) he phase consan f? Eplain. 2 4 FiguRE Q Equaion saes ha = 1 2 ( a ) 2. Wha does his ean? Wie a couple of senences eplaining how o inepe his equaion. 5. bloc oscillaing on a sping has an apliude of 2 c. Wha will he apliude be if he oal enegy is doubled? Eplain. 6. bloc oscillaing on a sping has a aiu speed of 2 c/s. Wha will he bloc s aiu speed be if he oal enegy is doubled? Eplain. 7. FiguRE Q14.7 shows a posiion-esus-ie gaph fo a paicle in SHM. a. Wha is he phase consan f? Eplain. b. Wha is he phase of he paicle a each of he hee nubeed poins on he gaph? FiguRE Q FiguRE Q14.8 shows a elociy-esus-ie gaph fo a paicle in SHM. a. Wha is he phase consan f? Eplain. b. Wha is he phase of he paicle a each of he hee nubeed poins on he gaph? a a FiguRE Q FiguRE Q14.9 shows he poenial-enegy diaga and he oal enegy line of a paicle oscillaing on a sping. a. Wha is he sping s equilibiu lengh? b. Whee ae he uning poins of he oion? Eplain. c. Wha is he paicle s aiu ineic enegy? d. Wha will be he uning poins if he paicle s oal enegy is doubled? Enegy (J) 2 PE FiguRE Q Suppose he daping consan b of an oscillao inceases. a. Is he ediu oe esisie o less esisie? b. Do he oscillaions dap ou oe quicly o less quicly? c. Is he ie consan inceased o deceased? 11. a. Descibe he diffeence beween and. Don jus nae he; say wha is diffeen abou he physical conceps hey epesen. b. Descibe he diffeence beween and 1/ Wha is he diffeence beween he diing fequency and he naual fequency of an oscillao? 2 E (c) Concepual Quesions equie caeful easoning and can be used fo goup discussions o indiidual wo.

26 42 chape 14. Oscillaions he end-of-chape pobles ae aed by sudens o show difficuly leel wih he aiey epanded o include oe eal-wold, challenging, and eplicily calculus-based pobles. Eecises (fo each secion) allow sudens o build hei sills and confidence wih saighfowad, one-sep quesions. Pobles labeled Eecises Secion 14.1 Siple Haonic Moion EE cise s and PoblE s inegae aeial fo ealie chapes. 1. When a guia sing plays he noe, he sing ibaes a 44 Hz. Wha is he peiod of he ibaion? 2. n ai-ac glide aached o a sping oscillaes beween he 1 c a and he 6 c a on he ac. he glide coplees 1 oscillaions in 33 s. Wha ae he (a) peiod, (b) fequency, (c) angula fequency, (d) apliude, and (e) aiu speed of he glide? 3. n ai-ac glide is aached o a sping. he glide is pulled o he igh and eleased fo es a = s. I hen oscillaes wih a peiod of 2. s and a aiu speed of 4 c/s. a. Wha is he apliude of he oscillaion? b. Wha is he glide s posiion a =.25 s? Secion 14.2 Siple Haonic Moion and Cicula Moion 4. Wha ae he (a) apliude, (b) fequency, and (c) phase consan of he oscillaion shown in FiguE E14.4? FiguE E14.4 (c) Wha ae he (a) apliude, (b) fequency, and (c) phase consan of he oscillaion shown in FiguE E14.5? FiguE E (c) n objec in siple haonic oion has an apliude of 4. c, a fequency of 2. Hz, and a phase consan of 2p/3 ad. Daw a posiion gaph showing wo cycles of he oion. 7. n objec in siple haonic oion has an apliude of 8. c, a fequency of.25 Hz, and a phase consan of -p/2 ad. Daw a posiion gaph showing wo cycles of he oion. 8. n objec in siple haonic oion has apliude 4. c and fequency 4. Hz, and a = s i passes hough he equilibiu poin oing o he igh. Wie he funcion () ha descibes he objec s posiion. 9. n objec in siple haonic oion has apliude 8. c and fequency.5 Hz. = s i has is os negaie posiion. Wie he funcion () ha descibes he objec s posiion. 1. n ai-ac glide aached o a sping oscillaes wih a peiod of 1.5 s. = s he glide is 5. c lef of he equilibiu posiion and oing o he igh a 36.3 c/s. a. Wha is he phase consan? b. Wha is he phase a = s,.5 s, 1. s, and 1.5 s? (s) (s) Secion 14.3 Enegy in Siple Haonic Moion Secion 14.4 he Dynaics of Siple Haonic Moion 11. bloc aached o a sping wih unnown sping consan oscillaes wih a peiod of 2. s. Wha is he peiod if a. he ass is doubled? b. he ass is haled? c. he apliude is doubled? d. he sping consan is doubled? Pas a o d ae independen quesions, each efeing o he iniial siuaion g ai-ac glide is aached o a sping. he glide is pushed in 1 c and eleased. suden wih a sopwach finds ha 1 oscillaions ae 12. s. Wha is he sping consan? g ass aached o a hoizonal sping oscillaes a a fequency of 2. Hz. = s, he ass is a = 5. c and has = -3 c/s. Deeine: a. he peiod. b. he angula fequency. c. he apliude. d. he phase consan. e. he aiu speed. f. he aiu acceleaion. g. he oal enegy. h. he posiion a =.4 s. 14. he posiion of a 5 g oscillaing ass is gien by () = (2. c) cos(1 - p/4), whee is in s. Deeine: a. he apliude. b. he peiod. c. he sping consan. d. he phase consan. e. he iniial condiions. f. he aiu speed. g. he oal enegy. h. he elociy a =.4 s g bloc is aached o a sping wih sping consan 16 N/. While he bloc is siing a es, a suden his i wih a hae and alos insananeously gies i a speed of 4 c/s. Wha ae a. he apliude of he subsequen oscillaions? b. he bloc s speed a he poin whee = 1 2? Secion 14.5 Veical Oscillaions 16. sping is hanging fo he ceiling. aching a 5 g physics boo o he sping causes i o sech 2 c in ode o coe o equilibiu. a. Wha is he sping consan? b. Fo equilibiu, he boo is pulled down 1 c and eleased. Wha is he peiod of oscillaion? c. Wha is he boo s aiu speed? 17. sping wih sping consan 15 N/ hangs fo he ceiling. ball is aached o he sping and allowed o coe o es. I is hen pulled down 6. c and eleased. If he ball aes 3 oscillaions in 2 s, wha ae is (a) ass and (b) aiu speed? 18. sping is hung fo he ceiling. When a bloc is aached o is end, i seches 2. c befoe eaching is new equilibiu lengh. he bloc is hen pulled down slighly and eleased. Wha is he fequency of oscillaion? Secion 14.6 he Pendulu 19. ass on a sing of unnown lengh oscillaes as a pendulu wih a peiod of 4. s. Wha is he peiod if a. he ass is doubled?

27 Eecises and Pobles 43 b. he sing lengh is doubled? c. he sing lengh is haled? d. he apliude is doubled? Pas a o d ae independen quesions, each efeing o he iniial siuaion g ball is ied o a sing. I is pulled o an angle of 8. and eleased o swing as a pendulu. suden wih a sopwach finds ha 1 oscillaions ae 12 s. How long is he sing? 21. Wha is he peiod of a 1.--long pendulu on (a) he eah and (b) Venus? 22. Wha is he lengh of a pendulu whose peiod on he oon aches he peiod of a 2.--long pendulu on he eah? 23. sonaus on he fis ip o Mas ae along a pendulu ha has a peiod on eah of 1.5 s. he peiod on Mas uns ou o be 2.45 s. Wha is he fee-fall acceleaion on Mas? 24. unifo seel ba swings fo a pio a one end wih a peiod of 1.2 s. How long is he ba? Secion 14.7 Daped Oscillaions Secion 14.8 Dien Oscillaions and Resonance g spide is dangling a he end of a sil head. You can ae he spide bounce up and down on he head by apping lighly on his fee wih a pencil. You soon discoe ha you can gie he spide he lages apliude on his lile bungee cod if you ap eacly once eey second. Wha is he sping consan of he sil head? 26. he apliude of an oscillao deceases o 36.8% of is iniial alue in 1. s. Wha is he alue of he ie consan? 27. Sech a posiion gaph fo = s o = 1 s of a daped oscillao haing a fequency of 1. Hz and a ie consan of 4. s. 28. In a science useu, a 11 g bass pendulu bob swings a he end of a 15.--long wie. he pendulu is saed a eacly 8: a.. eey oning by pulling i 1.5 o he side and eleasing i. Because of is copac shape and sooh suface, he pendulu s daping consan is only.1 g/s. eacly 12: noon, how any oscillaions will he pendulu hae copleed and wha is is apliude? 29. Vision is blued if he head is ibaed a 29 Hz because he BIO ibaions ae esonan wih he naual fequency of he eyeball in is soce. If he ass of he eyeball is 7.5 g, a ypical alue, wha is he effecie sping consan of he usculaue ha holds he eyeball in he soce? Pobles 3. FiguRE P14.3 is he elociy-esus-ie gaph of a paicle in siple haonic oion. a. Wha is he apliude of he oscillaion? b. Wha is he phase consan? c. Wha is he posiion a = s? FiguRE P14.3 (c/s) (s) FiguRE P14.31 is he posiion-esus-ie gaph of a paicle in siple haonic oion. a. Wha is he phase consan? b. Wha is he elociy a = s? c. Wha is a? (c) FiguRE P14.31 (s) y B (s) he wo gaphs in FiguRE P14.32 ae fo wo diffeen eical ass-sping syses. If boh syses hae he sae ass, wha is he aio / B of hei sping consans? 33. n objec in SHM oscillaes wih a peiod of 4. s and an apliude of 1 c. How long does he objec ae o oe fo =. c o = 6. c? g bloc oscillaes on a sping wih sping consan 2 N/. = s he bloc is 2 c o he igh of he equilibiu posiion and oing o he lef a a speed of 1 c/s. Deeine (a) he peiod and (b) he apliude. 35. sonaus in space canno weigh heseles by sanding on a BIO bahoo scale. Insead, hey deeine hei ass by oscillaing on a lage sping. Suppose an asonau aaches one end of a lage sping o he bel and he ohe end o a hoo on he wall of he space capsule. fellow asonau hen pulls he away fo he wall and eleases he. he sping s lengh as a funcion of ie is shown in FiguRE P a. Wha is he ass if he sping consan is 24 N/? b. Wha is he speed when he sping s lengh is 1.2? FiguRE P14.35 L () he oion of a paicle is gien by () = (25 c)cos(1), whee is in s. wha ie is he ineic enegy wice he poenial enegy? 37. a. When he displaceen of a ass on a sping is 1 2, wha facion of he enegy is ineic enegy and wha facion is poenial enegy? b. wha displaceen, as a facion of, is he enegy half ineic and half poenial? 38. Fo a paicle in siple haonic oion, show ha a = (p/2) ag whee ag is he aeage speed duing one cycle of he oion g bloc aached o a sping wih sping consan 2.5 N/ oscillaes hoizonally on a ficionless able. Is elociy is 2 c/s when = -5. c. a. Wha is he apliude of oscillaion? b. Wha is he bloc s aiu acceleaion? c. Wha is he bloc s posiion when he acceleaion is aiu? d. Wha is he speed of he bloc when = 3. c? 6 FiguRE P14.32 (s) Pobles (spanning conceps fo he whole chape), equie in-deph easoning and planning, and allow sudens o pacice hei poble-soling saegies. Cone-ich pobles equie sudens o siplify and odel oe cople eal-wold siuaions. Specifically labeled pobles inegae conceps fo uliple peious chapes.

28 44 chape 14. Oscillaions Bio pobles ae se in lifescience, bioengineeing, o bioedical cones. Daa-based pobles allow sudens o pacice dawing conclusions fo daa (as deonsaed in he new daabased eaples in he e). 4. bloc on a sping is pulled o he igh and eleased a = s. I passes = 3. c a =.685 s, and i passes = -3. c a =.886 s. a. Wha is he angula fequency? b. Wha is he apliude? Hin: cos(p - u) = -cos u g oscillao has a speed of 95.4 c/s when is displaceen is 3. c and 71.4 c/s when is displaceen is 6. c. Wha is he oscillao s aiu speed? 42. n ulasonic ansduce, of he ype used in edical ulasound iaging, is a ey hin dis ( =.1 g) dien bac and BIO foh in SHM a 1. MHz by an elecoagneic coil. a. he aiu esoing foce ha can be applied o he dis wihou beaing i is 4, N. Wha is he aiu oscillaion apliude ha won upue he dis? b. Wha is he dis s aiu speed a his apliude? g bloc hangs fo a sping wih sping consan 2 N/. he bloc is pulled down 5. c fo he equilibiu posiion and gien an iniial elociy of 1. /s bac owad equilibiu. Wha ae he (a) fequency, (b) apliude, and (c) oal echanical enegy of he oion? 44. You lab insuco has ased you o easue a sping consan using a dynaic ehod leing i oscillae ahe han a saic ehod of seching i. You and you lab pane suspend he sping fo a hoo, hang diffeen asses on he lowe end, and sa he oscillaing. One of you uses a ee sic o easue he apliude, he ohe uses a sopwach o ie 1 oscillaions. You daa ae as follows: Mass (g) pliude (c) ie (s) Use he bes-fi line of an appopiae gaph o deeine he sping consan g bloc hangs fo a sping wih sping consan 1 N/. = s he bloc is 2 c below he equilibiu poin and oing upwad wih a speed of 1 c/s. Wha ae he bloc s a. Oscillaion fequency? b. Disance fo equilibiu when he speed is 5 c/s? c. Disance fo equilibiu a = 1. s? 46. sping wih sping consan is suspended eically fo a suppo and a ass is aached. he ass is held a he poin whee he sping is no seched. hen he ass is eleased and begins o oscillae. he lowes poin in he oscillaion is 2 c below he poin whee he ass was eleased. Wha is he oscillaion fequency? 47. While gocey shopping, you pu seeal apples in he sping scale in he poduce depaen. he scale eads 2 N, and you use you ule (which you always cay wih you) o discoe ha he pan goes down 9. c when he apples ae added. If you ap he boo of he apple-filled pan o ae i bounce up and down a lile, wha is is oscillaion fequency? Ignoe he ass of he pan. 48. copac ca has a ass of 12 g. ssue ha he ca has one sping on each wheel, ha he spings ae idenical, and ha he ass is equally disibued oe he fou spings. a. Wha is he sping consan of each sping if he epy ca bounces up and down 2. ies each second? b. Wha will be he ca s oscillaion fequency while caying fou 7 g passenges? 49. he wo blocs in FiguRE P14.49 oscillae on a ficionless suface wih a peiod of 1.5 s. he uppe bloc jus begins o slip when he apliude is inceased o 4 c. Wha is he coefficien of saic ficion beween he wo blocs? FiguRE P I has ecenly becoe possible o weigh DN olecules BIO by easuing he influence of hei ass on a nano-oscillao. FiguRE P14.5 shows a hin ecangula canilee eched ou of silicon (densiy 23 g/ 3 ) wih a sall gold do a he end. If pulled down and eleased, he end of he canilee ibaes wih siple haonic oion, oing up and down lie a diing boad afe a jup. When bahed wih DN olecules whose ends hae been odified o bind wih gold, one o oe olecules ay aach o he gold do. he addiion of hei ass causes a ey sligh bu easuable decease in he oscillaion fequency. FiguRE P n 4 n hicness = 1 n ibaing canilee of ass M can be odeled as a bloc of ass 1 3 M aached o a sping. (he faco of 1 3 aises fo he oen of ineia of a ba pioed a one end.) Neihe he ass no he sping consan can be deeined ey accuaely pehaps o only wo significan figues bu he oscillaion fequency can be easued wih ey high pecision siply by couning he oscillaions. In one epeien, he canilee was iniially ibaing a eacly 12 MHz. achen of a DN olecule caused he fequency o decease by 5 Hz. Wha was he ass of he DN? 51. I is said ha Galileo discoeed a basic pinciple of he pendulu ha he peiod is independen of he apliude by using his pulse o ie he peiod of swinging laps in he cahedal as hey swayed in he beeze. Suppose ha one oscillaion of a swinging lap aes 5.5 s. a. How long is he lap chain? b. Wha aiu speed does he lap hae if is aiu angle fo eical is 3.? g ass on a 1.--long sing is pulled 8. o one side and eleased. How long does i ae fo he pendulu o each 4. on he opposie side? 53. Oanguans can oe by bachiaion, swinging lie a pendulu beneah successie handholds. If an oanguan has as ha BIO ae.9 long and epeaedly swings o a 2 angle, aing one swing afe anohe, esiae is speed of fowad oion in /s. While his is soewha beyond he ange of alidiy of he sallangle appoiaion, he sandad esuls fo a pendulu ae adequae fo aing an esiae.