Oscillaions PHY 207 - oscillaions - J. Hedberg - 2017 1. Periodic Moion 2. Sinusoidal Moion 3. How do we ge his kind of moion? 4. Posiion - Velociy - cceleraion 5. spring wih vecors 6. he reference circle 7. he reference circle 8. he mah of oscillaions 9. Mah of roaions 10. Frequency of Vibraion 11. Hooke's law 12. 13. Elasic Poenial 14. Pendulums 15. he period of he pendulum 16. phase 17. Equaions of Moion 18. Differenial equaions! 19. he physical pendulum 20. Damping 21. No damping on an oscillaor 22. Yes damping on an oscillaor 23. Large angles 24. Large ngles Periodic Moion Phenomena in naure can repea. his chaper will deal wih moions ha recur in ime. he op graph shows he posiive of an objec wih respec o ime. he objec shown sars a res a posiion +, hen moves a consan velociy owards posiion, hen pauses, hen reurns o posiion + and sars he moion over. his is an example of periodic moion. he lower graph shows an objec undergoing a similar moion, excep for some small differences. Insead of sopping for a long ime, and hen moving a consan velociy, his objec slows down, pauses for an insan, hen speeds up in he opposie direcion, only o sar slowing down again. his ype of moion can be described by a sinusoidal funcion. nd will be he main ype of moion we look a. Sinusoidal Moion he simples case is a sinusoidal moion. Here he graph has a sine or cosine shape. Page 1
posiion x PHY 207 - oscillaions - J. Hedberg - 2017 -x x() = sin() or x() = cos() his ype of moion is called "simple harmonic moion" How do we ge his kind of moion? Simple harmonic moion will resul from a resoring force, like Hooke s Law: F = k x his is an example of a linear resoring force: he ne force is owards he equilibrium posiion and is proporional o he disance from equilibrium. k is a consan known as he 'spring consan' he force is proporional o he displacemen. F = kx Quick Quesion 1 Wha is he velociy and force a poin P? (+x is o he righ) + P a) Velociy is zero; force is o he righ. b) Velociy is zero; force is o he lef. c) Velociy is negaive; force is o he lef. d) Velociy is negaive; force is o he righ. e) Velociy is posiive; force is o he righ. Page 2
PHY 207 - oscillaions - J. Hedberg - 2017 a =0, he moion begins, in his case, a he maximum displacemen. +x a =1, he objec is again a res he maximum displacemen. -x a =/4, he objec is a he equilibrium posiion a =/2, he objec is a res a he maximum displacemen. a =3/4, he objec is moving hrough he equilibrium posiion Posiion - Velociy - cceleraion +x -x +v -v +a -a Quick Quesion 2 Here is he acceleraion of a mass wih respec o ime. which poin(s) is he speed of he mass zero? +a B C D F -a E Quick Quesion 3 which poin(s) is he velociy of he mass he mos negaive? he reference circle Imagine shining a ligh a a roaing able. he shadow cas by he roaing objec would creae a sinusoidal curve in ime. his allows us o always correlae simple harmonic moion wih he displacemen of an objec around a circle. he reference circle + Page 3
shadow lamp he posiion of he shadow on he screen will be given by: Bu, since he angular velociy ω imes he ime, equals he angular displacemen: we can also wrie: x = cos θ ω = θ x = cos ω PHY 207 - oscillaions - J. Hedberg - 2017 he mah of oscillaions Frequency and Period are used o describe a repeaing moion: So far, we have usually alked abou period, ha is how long for one oscillaion. Frequency is probably more useful for many applicaions hough. Frequency is he number of cycles per second. Frequency: 1 Herz is he uni of frequency: Mah of roaions We'll need o be able o express he various characerisics of an oscillaing sysem: he angular velociy, ω was "radians per second" hus, one roaion is 2π radians, so we can see ha he frequency is: Likewise for he period: 1 f = = ω = 2π f ω = 2π 1 f +x -x he displacemen of an objec in SHM is +v -v +a -a x = cos ω he velociy can be obained from a lile geomery, or calculus: v = ω sin ω nd lasly, he acceleraion is given by: a = ω 2 cos ω Page 4
shadow PHY 207 - oscillaions - J. Hedberg - 2017 Quick Quesion 4 If an objec has a velociy given by: v() = B sin ω wha will is maximum speed be? a) v max = B b) v max = 2B c) v max = 2B d) v max = Bω e) Canno say since he funcion depends on ime Quick Quesion 5 n objec moves wih simple harmonic moion. If he ampliude and he period are boh doubled, he objec s maximum speed is: a) quarered. b) halved. c) unchanged. d) doubled. e) quadrupled. Example Problem #1: mass oscillaing on a spring (in SHM) sars a x = and oscillaes wih a period. wha ime (in erms of ) does he objec pass hrough x = /2 for he firs ime? Frequency of Vibraion Saring wih Hooke's Law for he force on a spring hese springs mass sysems have differen frequencies of vibraion F = kx Since Newon's second law says ha F = ma, we can hen wrie: k( cos ω) = m( ω 2 cos ω) Le's solve his for ω: ω = k m Hooke's law he spring consan k ells us 'how siff he spring is' he unis of he spring consan are Newons per meer (N/m) Page 5
Quick Quesion 6 Below is ploed he posiion vs. ime for hree ball & spring sysems. Which one has he smalles spring consan? + a) b) B c) C d) = B = C e) Canno be deermined PHY 207 - oscillaions - J. Hedberg - 2017 B + C + E oal Mechanical 1 1 = m + + 2 v2 I 2 ω2 ranslaional Kineic Roaional Kineic mgh Graviaional poenial E oal Mechanical 1 1 = m + + + 2 v2 I 2 ω2 mgh ranslaional Kineic Roaional Kineic Graviaional poenial 1 k 2 x2 Elasic Poenial Elasic Poenial he elasic poenial energy of a spring mass sysem ells us how much energy is sored in he spring. PE = 1 k 2 x2 When he displacemen is zero from he equilibrium, hen no poenial energy exiss in he sysem. We can see he ransformaion from poenial o kineic and back again in a simple harmonic oscillaor he oal energy remains consan: 1 1 E = m + k 2 v2 2 x2 Page 6
poenial he same value mass has zero velociy a endpoins (x = or, he urning poins); all energy is poenial a hose poins PHY 207 - oscillaions - J. Hedberg - 2017 E kineic mass has greaes speed a equilibrium (x = 0); all energy is kineic x = 0 + E he same value poenial kineic x = 0 + hus, le's solve for 1 1 k = m 2 2 2 v2 max v max v max k = m Bu, v max = ω hus, ω = k m 1 k m f = or = 2π 2π m k he frequency and period are deermined by he physical properies of he oscillaor. ll ha maers is he mass of he objec and he spring consan. he frequency and period do no depend on he ampliude. Small and large oscillaions have he same frequency. Whenever we have his condiion, of energy rading off beween kineic and poenial, we'll ge an oscillaory behaviour. Some are easier o describe han ohers. Pendulums cable of lengh L. simple pendulum consiss of a mass m aached o a fricionless pivo by a Page 7
PHY 207 - oscillaions - J. Hedberg - 2017 L m s For small angles, he angenial force on he pendulum bob will be given by: F ang = mg sin θ When θ is very small ( θ <.2 radians), sin θ = θ. F ang mg = s L his is essenially a resoring force, jus like we had for he mass/spring sysem. mg F = kx F = s L Now k mg, hus we could replace our frequencies of he spring mass L sysems wih: 1 k 1 g f spring-mass = f pendulum = 2π m 2π L he period of he pendulum 1 Since f f = 2π g, we can easily wrie he period of a pendulum. his is how L long i akes o do one oscillaion: pend L = 2π g his is only rue for small oscillaions. Bu, we can sill do a lo wih small oscillaions. Quick Quesion 7 Wha is he period for his pendulum, of lengh 140 cm? Page 8
Quick Quesion 8 When he pendulum is a he lowes poin, wha can you say abou is acceleraion? (Consider all possible acceleraions!) a) he oal acceleraion is zero b) he oal acceleraion is no zero PHY 207 - oscillaions - J. Hedberg - 2017 phase +x -x +x -x he phase angle ϕ essenially allows us o indicae wha he iniial displacemen is: 0, or some nonzero value. hese wo plos are boh sinusoidal. he op plo we recognize as a x = cos(ω) plo. he boom would be a familiar x = sin(ω) plo. Boh however, can be described by including a phase facor ϕ: x() = cos(ω + ϕ) Page 9
PHY 207 - oscillaions - J. Hedberg - 2017 Quick Quesion 9 + Which of he following funcions describes his SHM curve (period of moion = )? 2π a) x() = sin( ) 2π 2π 2π 2π 2π b) x() = cos( ) c) x() = sin( + π) π 2 π 2 π 2 d) x() = cos( + ) e) x() = sin( ) f) x() = cos( ) Quick Quesion 10 + Which graph porrays he following funcion? π x() = sin(ω ) 2 B + C + + D Example Problem #2: Make a skech of he following funcion: 2π 3π x() = 4 cos( ) 4 2 Quick Quesion 11 grandfaher clock, which uses a pendulum o keep accurae ime, is adjused a sea level. he clock is hen aken o an aliude of several kilomeers. How will he clock behave in is new locaion? a) he clock will run slow. b) he clock will run fas. c) he clock will run he same as i did a sea level. Page 10
Equaions of Moion PHY 207 - oscillaions - J. Hedberg - 2017 We saw ha he force from a spring on a mass was given by: We can rewrie his a lile bi: Bu, we know ha ime. a x F sp = kx = m a x k = x m a x is really jus he second derivaive of x wih respec o. so, puing his ogeher: a x dv = x d = 2 x d d 2 his is he equaion of moion for a spring mass sysem - meaning, i predics he moion for all ime based on a few physical properies. Bu, how do we solve his? Differenial equaions! rying o solve his equaion won' work wih regular algebra. Insead, x has o be anoher funcion of ime. We'll ry his one o sar: x() = cos(ω + ϕ) he firs derivaive of his is: dx = ẋ = ω sin(ω + ϕ) d nd he second derivaive is: d 2 x = ẍ = ω 2 cos(ω + ϕ) d 2 he physical pendulum d 2 x k = x d 2 m d 2 x k = x d 2 m If ω =, hen we can see ha k m his is indeed a soluion o he above equaion of moion: ω 2 k cos(ω + ϕ) = cos(ω + ϕ) m n imporan hing o noe: cceleraion is no consan. COM pivo poin h COM In he case where we have somehing more complicaed han a ball and a rope, we'll have o use our roaional moion echniques a lile more in deph. his is he physical pendulum. he period for such an objec would be given by: I = 2π mgh I h Page 11
where, I is he momen of ineria for he objec, and h is he disance from he cener of mass o he pivo poin. PHY 207 - oscillaions - J. Hedberg - 2017 Example Problem #3: Derive his relaionship: I = 2π mgh Example Problem #4: Skech y = e, y = e, y = e /1, y = e /10. Damping n exponenial decay funcion: x() = e /τ consan for his decaying curve. Now, τ is no orque, bu will be called he ime x( = τ) = e 1 he ime consan describes how quickly he exponenial funcion approaches zero. ime consan.37 e his way. Differen ime consans will effec he graphs in τ =.5 τ = 1 τ = 2 he ime consan, τ, can be defined: "he ime where he value ges o 1/e of he iniial value, or abou a hird (.37)" No damping on an oscillaor n oscillaor ha is no damped, and loses no energy will have a moion like his: x() = sin( ) Page 12
2π x() = sin( ) PHY 207 - oscillaions - J. Hedberg - 2017 Yes damping on an oscillaor However, if we damp an oscillaor, hen we'll see a decay in he ampliude of he oscillaions. x() = e 2π τ sin( ) hese oscillaion have he same period, and frequency, and. However, he τ which dicaes he damping is differen. x() = e 2π τ sin( ) Here, he differen ime consans are given by: τ = {.5, 1, 2} he mah of he damp he damped oscillaor consiss, a he mos fundameal level, of a spring, mass and damping force. We can say he damping force is proporional o he velociy of he sysem: hus, in our classic sum of forces approach: Which, in a more formal mah approach looks like: F y Which, from diff eq. can be solved by: F d = bv kx = ma m ẍ + b ẋ + kx = 0 = bv x() = e b/2m cos( ω + ϕ) Damping mpliude 1.0 0.5 0.0-0.5-1.0 0 2 4 6 8 10 ime x() = e b/2m cos( ω + ϕ) he damping leads o a differen ω ω ω k b = 2 m 4m 2 Driving a resonaor ll he oscillaors in he real world will have some kind of damping force: Page 13
fricion, drag, hermal losses. If we wan his pendulum o keep swinging, we need o add energy o accoun of he los energy. PHY 207 - oscillaions - J. Hedberg - 2017 In order o keep hings oscillaing, we have o drive he oscillaions. driven oscillaion has an applied force which also oscillaes a some frequency. here will be a naural frequency of oscillaion. In he case of he pendulum, ha's. If he oscillaor is driven wih a maching frequency, hen we have he maximum ampliude of displacemen. (Nearly everyhing has a naural resonance frequency - buildings, bridges, wine glasses) Large angles 1 g f 0 = 2π L Our previous analysis relied on he approximaion ha sin(θ) θ. Wha if we have larger values of θ where his is no rue? Large ngles If insead we have o solve his equaion: d 2 θ mg α = = sin θ d 2 L We'll need fancier mah. We'll end up wih somehing ha looks like his: L 1 11 173 22931 1319183 = 2π (1 + + + + + +... ) g 16 θ2 0 3072 θ4 0 737280 θ6 0 1321205760 θ8 0 951268147200 θ10 0 (1) Example Problem #5: 2 kg mass is aached o a spring and sreched 4 cm in he posiive x direcion. = 0, he mass is released and he sysem begins o oscillae a a frequency of 2 Herz. 1. Wrie equaions ha describe he posiion, velociy, and acceleraion of he mass. 2. Skech 3 plos ha show hese hree quaniies as funcions of ime. 3. Compile a able of imporan values: f,, ω,, v max, oal energy, k Page 14
k. PHY 207 - oscillaions - J. Hedberg - 2017 Example Problem #6: fla, square piece of meal wih sides equal o 0.5 meers is hung from a pivo a one corner. If he square oscillaes abou he pivo by jus a few degrees, wha is he period of oscillaion? Example Problem #7: Wha is he phase consan ( ϕ 0 ) for his oscillaing paricle? he posiion is described by: x() = cos(ω + ϕ 0 ) Vmax velociy -Vmax =0 Example Problem #8: [J] 15 Poenial 10 5 oal 0 12 16 20 24 28 x [cm] 1. Wha is he equilibrium lengh of he spring? 2. Where are he minimum and maximum exensions? (i.e. he Page 15
urning poins) 3. Wha is he paricle's maximum kineic energy? 4. If he oal energy of he paricle is doubled, where will he urning poins be? PHY 207 - oscillaions - J. Hedberg - 2017 Example Problem #9: ball of mass m oscillaes on a spring wih spring consan k = 200N/m. he ball's posiion is given by: x =.350m cos(15.0) wih measured in seconds. 1. Find he ampliude of he moion 2. Find he frequency of he moion 3. Wha is he mass 4. Wha is he oal energy 5. ha is he maximum speed of he ball? Example Problem #10: On a cruise ship in he ocean, he waves make he boa rock up and down. passenger experiences a verical moion of ampliude 1 m wih a period of 15 s. (a) Wha is he maximum acceleraion his passenger feels? (b) Wha fracion of g is his? Example Problem #11: 200 g air-rack glider is aached o a spring. he glider is pushed 10 cm agains he spring, hen released. 10 oscillaions are found o ake 12 seconds. Wha is he spring consan of his spring? For a spring mass sysem, we solved his: using his: d 2 x k = x d 2 m x() = cos(ω + ϕ) Since ω = 2πf: 1 k f spring-mass = 2π m nd so, we said anyhing ha has an equaion of moion ha looks like his: d 2 x = Ox d 2 can be considered a harmonic oscillaor. O is some physical characerisics of he sysem in quesion (mass, graviy, springs, ec). Page 16
For he simple (small angle, poin mass) pendulum: PHY 207 - oscillaions - J. Hedberg - 2017 O = g L because he resoring force was found o be: mg F = s = mgθ L hus, we can express our equaions of moion: and will lead o: F = ma = m x = mlθ ml θ = mgθ g θ = θ L and a frequency of oscillaion will be given by: f = 1 g 2π L For he physical pendulum, we'll need o hink in erms of orque: Rearranging: ( h is he disance from he pivo poin o he cener of mass, I is he momen of ineria abou he pivo poin.) So, in he equaion of moion: α = τ = r F = Iα rf I h mgθ I h mg θ = θ I h mg hus, our O in his siuaion is:, and based on he previous soluions, we I can say ha he frequency of oscillaion for his sysem will be: 1 f = = 2π O 1 h mg 2π I Page 17