Chapter 9: Oscillations

Transcription

1 Chaper 9: Ocillaion Now if hi elecron i diplaced fro i equilibriu poiion, a force ha i direcly proporional o he diplaceen reore i like a pendulu o i poiion of re. Pieer Zeean Objecive 1. Decribe he condiion neceary for iple haronic oion.. Wrie down an appropriae expreion for diplaceen of he for Aco(ω) or Ain(ω) decribing he oion. 3. Apply he relaionhip beween frequency, angular frequency, and period. 4. Deerine he oal energy of an objec undergoing iple haronic oion, and kech graph of kineic and poenial energie a funcion of ie or diplaceen. 5. Idenify he inia, axia, and zero of diplaceen, velociy, and acceleraion for an objec undergoing iple haronic oion. 6. Analyze he iple haronic oion of a pring-block ocillaor. 7. Analyze he iple haronic oion of an ideal pendulu. Chaper 9: Ocillaion 1

2 Siple Haronic Moion When an objec i diplaced, i ay be ubjec o a reoring force, reuling in a periodic ocillaing oion. If he diplaceen i direcly proporional o he linear reoring force, he objec undergoe iple haronic oion (SHM). Exaple of linear reoring force cauing iple haronic oion include a a on a pring, he pendulu on a grandfaher clock, a ree lib ocillaing afer you bruh i walking hrough he wood, a child on a wing, even he vibraion of ao in a olid can be odeled a iple haronic oion. Unifor circular oion alo ha ie o iple haronic oion. Conider an objec oving in a horizonal circle of radiu A a conan angular velociy ω a hown in he diagra below. A any given poin in ie, he x-poiion of he objec can be decribed by x=acoθ, and he y-poiion can be decribed by y=ainθ. A x = Aco y = Ain k x=-a x=0 x=a Now iagine a block on a fricionle urface aached o a wall by a pring. If ha block i diplaced fro i equilibriu poiion by an aoun A, and releaed, i iple haronic oion along he fricionle urface will irror he x-oion of he objec oving in unifor circular oion. Siple haronic oion and unifor circular oion are very cloely relaed, a iple haronic oion decribe one dienion of an objec oving in unifor circular oion! Referring back o he chaper on roaional oion, recall ha he angular diplaceen of an objec θ i equal o he angular velociy uliplied by he ie inerval. You can herefore wrie he x-coordinae of he objec in UCM and SHM a θ=ω x = Acoθ x = Aco(ω) In hi equaion, ω i known a he angular frequency, correponding o he nuber of radian per econd for an objec raveling in unifor circular Chaper 9: Ocillaion

3 oion. You can conver fro angular frequency o frequency and period uing he equaion: π ω= πf = T Therefore a general for decribing he oion of an objec undergoing SHM can be wrien a: x() = Aco(ω +φ) The ybol phi (Φ) refer o he phae angle, or aring poin, of a ine or coine curve. Ue he coine curve if he oion ar a axiu apliude. Ue he ine curve if he oion ar a x=0. If he oion being decribed ar oewhere beween axiu apliude and equilibriu (x=0), you ll have o add in a phae angle coponen. A x() = Aco( ) 0 -A 3 A x() = Ain( ) 0 -A 3 If he axiu diplaceen of an objec undergoing SHM i A, hen he axiu peed of he objec i ωa, which you can find by aking he lope of he poiion-ie curve, and i axiu acceleraion i ω A, which you can find by aking he lope of he velociy-ie curve Q: An ocillaing ye i creaed by a releaing an objec fro a axiu diplaceen of 0. eer. The objec ake 60 coplee ocillaion in one inue. Deerine he objec angular frequency A: ω= πf = π(1 Hz) = π rad Chaper 9: Ocillaion 3

4 9.0 Q: Referring o proble 9.01, deerine he objec poiion a ie =10 econd. ω=π rad A=0. = A: x = Aco(ω) x = (0.)co(π) x = (0.)co(π rad 10) = Q: Referring o proble 9.01, deerine he ie when he objec i a poiion x= A: x = Aco(π) co(π) = x A π = x co 1 A x co co 1 A 0. = = = π π rad Noe: if you arrived a an anwer of roughly 9.55 econd, ake ure your calculaor i funcioning in Radian ode inead of Degree ode for hee rigonoeric funcion. Horizonal Spring-Block Ocillaor A popular deonraion vehicle for iple haronic oion i he pringblock ocillaor. The horizonal pring-block ocillaor coni of a block of a iing on a fricionle urface, aached o a verical wall by a pring of pring conan k, a hown in he diagra below. k x=-a x=0 x=a The block i hen diplaced an aoun A fro i equilibriu poiion and allowed o ocillae back and forh. A he block i on a fricionle urface, in he ideal cenario he block would coninue i periodic oion indefiniely. The angular frequency of he block ocillaion can be deerined fro he pring conan and he a. ω = k 4 Chaper 9: Ocillaion

5 Knowing he angular frequency, i quie raighforward hen o calculae he period of ocillaion for he pring-block ocillaor. T S = π ω ω= k T S = π k Noe ha he period of ocillaion depend only on he a of he block and he pring conan. There i no dependency on he agniude of he diplaceen of he block Q: A 5-kg block i aached o a 000 N/ pring a hown and diplaced a diance of 8 c fro i equilibriu poiion before being releaed. 000 N/ 5 kg x=-8 c x=0 x=8 c Deerine he period of ocillaion, he frequency, and he angular frequency for he block. 5kg 9.04 A: T = π = π = S k 000 N 1 1 f = = = 3.18Hz T ω= πf = π(3.18 Hz) = 0 rad 9.05 Q: Rank he following horizonal pring-block ocillaor reing on fricionle urface in er of heir period, fro longe o hore. 500 N/ 10 kg A 50 N/ 7 kg B x=-1 c x=0 x=1 c x=-5 c x=0 x=5 c 000 N/ kg C 1000 N/ 5 kg D x=-8 c x=0 x=8 c x=-15 c x=0 x=15 c 9.05 A: B, A, D, C Chaper 9: Ocillaion 5

6 I alo inereing o look a he energy of he pring-block ocillaor while i undergoing iple haronic oion. Becaue he urface i fricionle, he oal energy of he ye reain conan. However, here i a coninual ranfer of kineic energy ino elaic poenial energy and back. When he block i a i equilibriu poiion, here i no elaic poenial energy ored in he pring, herefore all of he energy of he block i kineic. The block ha achieved i axiu peed. A hi poiion, here i alo no ne force on he block, herefore he block acceleraion i zero. When he block i a i axiu apliude poiion, all of i energy i ored in he pring a elaic poenial energy. For an inan i kineic energy i zero, herefore i velociy i zero. Furher, a hi poiion, he pring exhibi a axiu force on he block, providing he axiu acceleraion. Le ake a look a hi graphically by exaining a pring-block ocillaor a variou poin in i periodic pah. -x x C A B Graph of Relaed Phyical Quaniie X V U K F a 6 Chaper 9: Ocillaion

7 A B C Diplaceen (x) 0 X -X Velociy (v) ax 0 0 Poenial Energy (U) 0 ax ax Kineic Energy (K) ax 0 0 Force (F) 0 -ax ax Acceleraion (a) 0 -ax ax Of coure, hrough he enire ie inerval, he oal echanical energy of he pring-block ocillaor reain conan. In realiy, oe energy i ypically lo o fricion or oher non-conervaive force. Over ie, he apliude of he ocillaion eadily decreae. Thi i known a daping, or daped haronic oion Q: A -kg block i aached o a pring. A force of 0 newon reche he pring o a diplaceen of 50 c. Find: A) he pring conan B) he oal energy C) he peed of he block a he equilibriu poiion D) he peed of he block a x=30 c E) he peed of he block a x=-40 c F) he acceleraion a he equilibriu poiion G) he agniude of he acceleraion a a x=50 c H) he ne force on he block a he equilibriu poiion I) he ne force a x=5 c J) he poiion where he kineic energy i equal o he poenial energy. F 0N 9.06 A: A) k = = = x N 1 1 B) E = U = kx = (40 )(0.5 ) = 5J T S N E (5 J ) C) 1 T U = K = v = E v= = =. S T kg 1 1 D) E = U + K = kx + v E kx = v T S T v = E kx (5 J) (40 N T )(0.3 ) = kg = 1.8 Chaper 9: Ocillaion 7

8 E) v = E kx (5 J) (40 N T )(0.4 ) = kg F) a x=0, F=0, herefore acceleraion = 0 = 1.3 G) = = = kx ( F a kx a = 40 N )(0.5 ) = 10 kg H) A equilibriu, here i no pring diplaceen, o ne force i zero. I) F = kx= ( 40 )(0.5 ) = 10N N E E E T 5J 1 T T J) K = U = kx = x = = = 0.35 S k 40 Noe ha he poin where kineic energy and poenial energy are equal i NOT halfway beween he equilibriu and axiu diplaceen poin. N Spring Cobinaion* I i alo poible o aach ore han one pring o an objec. In hee cae, analye can be iplified coniderably by reaing he cobinaion of pring a a ingle pring wih an equivalen pring conan. For pring in parallel, calculae an equivalen pring conan for he ye by aring wih Hooke Law, recognizing ha diplaceen i he ae for boh pring. k1 k x=-a x=0 x=a F = k x+ k x= ( k + k ) x= k x k = k + k eq eq For pring in erie, you will again calculae an equivalen pring conan for he ye, beginning he analyi by realizing he force on each pring u be he ae according o Newon 3rd Law of Moion. k1 k x=-a x=0 x=a 8 Chaper 9: Ocillaion

9 k F = k x = k x x = k x Nex, recognizing ha he oal diplaceen i equal o he u of he diplaceen of he pring, you can cobine he equaion and olve for an equivalen pring conan. x 1 = k x k F = k eq (x 1 + x ) 1 k F = k F= k eq x k + x x 1 k k x = k eq x +1 k 1 k = k k +1 eq k 1 1 = k eq k 1 k Q: Rank he pring block ocillaor in he diagra below fro highe o lowe in er of: I) equivalen pring conan II) period of ocillaion 10 N/ A 0 N/ 5 N/ 15 N/ 3kg 6kg B -5 c x=0 5 c -6 c x=0 6 c C 15 N/ 15 N/ kg 10 N/ 10 N/ 4kg D -8 c x=0 8 c -7 c x=0 7 c 9.07 A: I) B, D, C, A II) A, C, B, D Verical Spring-Block Ocillaor Spring-block ocillaor can alo be e up verically a hown in he diagra. y=-a k F = ky y=yeq y=a +y g Chaper 9: Ocillaion 9

10 Sar your analyi by drawing a Free Body Diagra for he block, noing ha graviy pull he a down, while he force of he pring provide he upward force. Call down he poiive y-direcion. A i equilibriu poiion, y=y eq. You can hen wrie a Newon nd Law Equaion for he block and olve for y eq. equilibriu F ney = g ky = a y g ky = 0 y eq = g k Once he ye ha eled a equilibriu, you can diplace he a by pulling i oe aoun o eiher +A or lifing i an aoun -A. The new ye can be analyzed a follow: F ney = g k( y eq + A) = g ky eq ka g ky eq =0 F ney = ka Thi i he ae analyi you would do for a horizonal pring ye wih pring conan k diplaced an aoun A fro i equilibriu poiion. Thi ean, in hor, ha o analyze a verical pring ye, all you do i find he new equilibriu poiion of he ye, aking ino accoun he effec of graviy, hen rea i a a ye wih only he pring force o deal wih, ocillaing around he new equilibriu poin. No need o coninue o deal wih he force of graviy! 9.08 Q: A -kg block aached o an unreched pring of pring conan k=00 N/ a hown in he diagra below i releaed fro re. I) Deerine he period of he block ocillaion. II) Wha i he axiu diplaceen of he block fro i equilibriu while undergoing iple haronic oion? kg 00 N/ kg 9.08 A: I) T = π = π = 0.63 S k 00 N II) The graviaional poenial energy a he block aring poin, gδy, u equal he elaic poenial energy ored in he pring a i lowe poin. Ue hi o olve for Δy. U g =U S gδy = 1 kδy Δy = g k If Δy i he oal diplaceen of he block fro i highe poin o i lowe poin, he axiu diplaceen of he block fro i equilibriu poin, A, u be half of Δy. A = Δy = g k = (kg)(9.8 ) = N 30 Chaper 9: Ocillaion

11 9.09 Q: A 5-kg block i aached o a verical pring (k=500 N/). Afer he block coe o re, i i pulled down 3 c and releaed. I) Wha i he period of ocillaion? II) Wha i he axiu diplaceen of he pring fro i iniial unrained poiion? 5kg 9.09 A: I) T = π = π = 0.63 S k 500 N II) Fir deerine he diplaceen of he pring when he block i hanging and a re. g (5 kg)(9.8 ) F = 0 kd = g d = = = 0.1 ney k 500 Then he block i pulled down 3 c, o he axiu diplaceen u be he diplaceen while he block i a re in addiion o he 3 c he block i pulled down. y = = 0.13 ax N Ideal Pendulu Ideal Pendulu provide anoher deonraion vehicle for iple haronic oion. Conider a a aached o a ligh ring ha wing wihou fricion abou he verical equilibriu poiion. A he a ravel along i pah, energy i coninuouly ranferred beween graviaional poenial energy and kineic energy. The reoring force in he cae of he ideal pendulu i provided by graviy. The angular frequency of he ideal pendulu, for all angle of hea, i given by: ω = g l Chaper 9: Ocillaion 31

12 You can hen find he period of he pendulu for all angle of hea: T P = π ω ω= g l T P = π l g Noice ha he period of he pendulu i dependen only upon he lengh of he pendulu and he graviaional field rengh... here i no a dependance! 9.10 Q: A grandfaher clock i deigned uch ha each wing (or half-period) of he pendulu ake one econd. How long i he pendulu in a grandfaher clock? l gt (9.8 )( ) 9.10 A: T = π l = = = 1 P g 4π 4π 9.11 Q: Wha i he period of a grandfaher clock on he oon, where he acceleraion due o graviy on he urface i roughly one-ixh ha of Earh? l A: T = π = π = 4.9 P 1 g (9.8 ) Q: Rank he following pendulu of unifor a deniy fro highe o lowe frequency. A B C D 9.1 A: D, A, B, C 3 Chaper 9: Ocillaion

13 Uilizing he geoeric repreenaion of he pendulu previouly developed in queion 5.7, you can conruc a diagra deailing he energy and force acing on he pendulu a variou poin in i parabolic pah. g g in U g = 0 K ax = g U g = g y U g = gl(1 co ) K = 0 y A he highe poin, a hown on he lef, he a i being pulled back oward i equilibriu poiion by graviy. Specifically, he coponen of graviy along he a pah, ginθ. A he equilibriu poiion, he graviaional poenial energy i a a iniu, and he kineic energy of he a i a a axiu. A he highe poin, a hown on he righ, all he energy i graviaional poenial energy again. Uing he law of conervaion of energy, olving for he axiu velociy of he a a i lowe poiion i quie raighforward. 1 K = U v = gl(1 co θ) v= gl(1 coθ g Furher, fro hi ae graph you can creae a graph of kineic energy, graviaional poenial energy, and oal energy a a poiion of he a along he x-axi. E E o K U g The agniude of he energy below he parabolic line repreen he graviaional poenial energy of he a, while above he line he kineic energy i repreened. The oal echanical energy reain conan hroughou he enire pah of he pendulu. x Chaper 9: Ocillaion 33

14 9.13 Q: The period of an ideal pendulu i T. If he a of he pendulu i ripled while i lengh i quadrupled, wha i he new period of he pendulu? A) 0.5 T B) T C) T D) 4T 9.13 A: C) T 9.14 Q: Which of he following are rue for an ideal pendulu coniing of a a ocillaing back and forh on a ligh ring? (Chooe all ha apply.) A) The kineic energy i alway equal o he poenial energy. B) The axiu force on he pendulu occur when he pendulu ha i axiu kineic energy. C) The acceleraion of he pendulu i zero when he a i a i lowe poin. D) The angular acceleraion of he a reain conan A: C) The acceleraion of he pendulu i zero when he a i a i lowe poin Q: A pendulu of lengh 0 c and a 1 kg i diplaced an angle of 10 degree fro he verical. Wha i he axiu peed of he pendulu? 9.15 A: v= gl(1 co θ) = (9.8 )(0. )(1 co10 ) = Q: A pendulu of lengh 0.5 and a 5 kg i diplaced an angle of 14 degree fro he verical. Wha i he peed of he pendulu when i angle fro he verical i 7 degree? A: U = U + K gl(1 co14 ) = gl(1 co7 ) + v gop g boo boo v= gl(co7 co14 ) = Chaper 9: Ocillaion

15 Te Your Underanding 1. Decribe a lea wo ehod of deerine he pring conan of a pring. Which would you expec o provide ore accurae daa? Explain.. Deign an experien o eaure he acceleraion due o graviy uing a pendulu. Wha equipen would you require? Wha daa would you collec? Wha calculaion would you ake? How would you iniize error? 3. An ideal pendulu i diplaced by an angle hea fro he verical. Fill in he following graph decribing he oion of he pendulu in iilar fahion a hoe decribing he pring-block ocillaor. C A B Graph of Relaed Phyical Quaniie X V U K F a Chaper 9: Ocillaion 35

16 4. A uden eaure he a of a block uing a pring-block ocillaor by eauring he period for he block aached o a variey of pring wih known pring conan. Wha would a graph of he period v. pring conan look like? How could you ue ha graph o deerine he a of he block? 5. How could you double he axiu peed of a 40-kg child on a playground wing? 36 Chaper 9: Ocillaion