2. The acceleration of a simple harmonic oscillator is zero whenever the oscillating object is at the equilibrium position.

Transcription

1 CHAPER : Vibratins and Waes Answers t Questins. he blades in an electric shaer ibrate, apprxiately in SHM. he speaers in a stere syste ibrate, but usually in a ery cplicated way since any ntes are being sunded at the sae tie. A pian string ibrates when struc, in apprxiately SHM. he pistns in a car engine scillate, in apprxiately SHM. he ree end a diing bard scillates ater a dier jups, in apprxiately SHM.. he acceleratin a siple harnic scillatr is zer wheneer the scillating bject is at the equilibriu psitin.. he tin the pistn can be apprxiated as siple harnic. First all, the pistn will hae a cnstant perid while the engine is running at a cnstant speed. he speed the pistn will be zer at the extrees its tin the tp and btt the stre which is the sae as in siple harnic tin. here is a large rce exerted n the pistn at ne extree its tin the cbustin the uel ixture and siple harnic tin has the largest rce at the extrees the tin. Als, as the cranshat es in a circle, its cpnent tin in ne diensin is transerred t the pistn. It is siilar t Fig Since the real spring has ass, the ass that is ing is greater than the ass at the end the spring. Since, a larger ass eans a saller requency. hus the true requency will be saller than the assless spring apprxiatin. And since the true requency is saller, the true perid will be larger than the assless spring apprxiatin. Abut / the ass the spring cntributes t the ttal ass alue. 5. he iu speed is gien by A. Varius cbinatins changing A,, and/r can result in a dubling the iu speed. Fr exaple, i and are ept cnstant, then dubling the aplitude will duble the iu speed. Or, i A and are ept cnstant, then reducing the ass t ne-urth its riginal alue will duble the iu speed. Nte that changing either r will als change the requency the scillatr, since. 6. he scale reading will scillate with daped scillatins abut an equilibriu reading 5.0 g, with an initial aplitude 5.0 g (s the range readings is initially r 0.0 g and 0.0 g). Due t rictin in the spring and scale echanis, the scillatin aplitude will decrease er tie, eentually cing t rest at the 5.0 g ar. 7. he perid a pendulu clc is inersely prprtinal t the square rt g, by Equatin -a, g. When taen t high altitude, the alue g will decrease (by a sall aunt), which eans the perid will increase. I the perid is t lng, the clc is running slw and s will lse tie. 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 69

2 Chapter Vibratins and Waes 8. he tire swing apprxiates a siple pendulu. With a stpwatch, yu can easure the perid g the tire swing, and then sle Equatin -a r the length, ae the water slsh, yu ust shae the water (and the pan) at the natural requency r water waes in the pan. he water then is in resnance, r in a standing wae pattern, and the aplitude scillatin gets large. hat natural requency is deterined by the size the pan saller pans will slsh at higher requencies, crrespnding t shrter waelengths r the standing waes. he perid the shaing ust be the sae as the tie it taes a water wae t ae a rund trip in the pan. 0. Se exaples resnance: Pushing a child n a playgrund swing yu always push at the requency the swing. Seeing a stp sign scillating bac and rth n a windy day. When singing in the shwer, certain ntes will sund uch luder than thers. Utility lines alng the radside can hae a large aplitude due t the wind. Rubbing yur inger n a wineglass and aing it sing. Blwing acrss the tp a bttle. A rattle in a car (see Questin ).. A rattle in a car is ery ten a resnance phenenn. he car itsel ibrates in any pieces, because there are any peridic tins ccurring in the car wheels rtating, pistns ing up and dwn, ales pening and clsing, transissin gears spinning, drieshat spinning, etc. here are als ibratins caused by irregularities in the rad surace as the car is drien, such as hitting a hle in the rad. I there is a lse part, and its natural requency is clse t ne the requencies already ccurring in the car s nral peratin, then that part will hae a larger than usual aplitude scillatin, and it will rattle. his is why se rattles nly ccur at certain speeds when driing.. he requency a siple peridic wae is equal t the requency its surce. he wae is created by the surce ing the wae ediu that is in cntact with the surce. I yu hae ne end a taut string in yur hand, and yu e yur hand with a requency Hz, then the end the string in yur hand will be ing at Hz, because it is in cntact with yur hand. hen thse parts the ediu that yu are ing exert rces n adjacent parts the ediu and cause the t scillate. Since thse tw prtins the ediu stay in cntact with each ther, they als ust be ing with the sae requency. hat can be repeated all alng the ediu, and s the entire wae thrughut the ediu has the sae requency as the surce.. he speed the transerse wae is easuring hw ast the wae disturbance es alng the crd. Fr a unir crd, that speed is cnstant, and depends n the tensin in the crd and the ass density the crd. he speed a tiny piece the crd is easuring hw ast the piece crd es perpendicularly t the crd, as the disturbance passes by. hat speed is nt cnstant i a sinusidal wae is traeling n the crd, the speed each piece the crd will be gien by the speed relatinship a siple harnic scillatr (Equatin -9), which depends n the aplitude the wae, the requency the wae, and the speciic tie bseratin. 4. Fr Equatin -9b, the undaental requency scillatin r a string with bth ends ixed is F. he speed waes n the string is gien by Equatin -,. Cbining 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 70

3 Giancli Physics: Principles with Applicatins, 6 th Editin F these tw relatinships gies. By wrapping the string with wire, the ass the string can be greatly increased withut changing the length r the tensin the string, and thus the string has a lw undaental requency. 5. I yu strie the hrizntal rd ertically, yu will create priarily transerse waes. I yu strie the rd parallel t its length, yu will create priarily lngitudinal waes. 6. Fr Equatin -4b, the speed waes in a gas is gien by B. A decrease in the density due t a teperature increase therere leads t a higher speed sund. We expect the speed sund t increase as teperature increases. 7. (a) Siilar t the discussin in sectin -9 r spherical waes, as a circular wae expands, the circuerence the wae increases. Fr the energy in the wae t be cnsered, as the circuerence increases, the intensity has t decrease. he intensity the wae is prprtinal t the square the aplitude he water waes will decrease in aplitude due t dissipatin energy r iscsity in the water (dissipatie r rictinal energy lss). 8. Assuing the tw waes are in the sae ediu, then they will bth hae the sae speed. Since, the wae with the saller waelength will hae twice the requency the ther wae. Fr Equatin -8, the intensity wae is prprtinal t the square the requency the wae. hus the wae with the shrter waelength will transit 4 ties as uch energy as the ther wae. 9. he requency ust stay the sae because the edia is cntinuus the end ne sectin crd is physically tied t the ther sectin crd. I the end the irst sectin crd is ibrating up and dwn with a gien requency, then since it is attached t the ther sectin crd, the ther sectin ust ibrate at the sae requency. I the tw pieces crd did nt e at the sae requency, they wuld nt stay cnnected, and then the waes wuld nt pass r ne sectin t anther. 0. he string culd be tuched at the lcatin a nde withut disturbing the tin, because the ndes d nt e. A string ibrating in three segents has ndes in additin t the nes at the ends. See the diagra. nde nde. he energy a wae is nt lcalized at ne pint, because the wae is nt lcalized at ne pint, and s t tal abut the energy at a nde being zer is nt really a eaningul stateent. Due t the intererence the waes the ttal energy the ediu particles at the ndes pints is zer, but the energy the ediu is nt zer at pints the ediu that are nt ndes. In act, the antinde pints hae re energy than they wuld hae i nly ne the tw waes were present.. A ajr distinctin between energy transer by particles and energy transer by waes is that particles ust trael in a straight line r ne place t anther in rder t transer energy, but waes can diract arund bstacles. Fr instance, sund can be heard arund a crner, while yu cannt thrw a ball arund a crner. S i a barrier is placed between the surce the energy and the 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 7

4 Chapter Vibratins and Waes lcatin where the energy is being receied, and energy is still receied in spite the barrier, it is a gd indicatin that the energy is being carried by waes. I the placeent the barrier stps the energy transer, it culd be that the energy transer is being carried ut by particles. It culd als be that the energy transer is being carried ut with waes whse waelength is uch saller than the diensins the barrier. Slutins t Prbles. he particle wuld trael ur ties the aplitude: r x A t x 0 t x A t x 0 t x A. S the ttal distance 4A he spring cnstant is the rati applied rce t displaceent. F 80 N 75 N 05 N 5. 0 N x he spring cnstant is und r the rati applied rce t displaceent. F g 68 g 9.8 s 5. 0 N x x 5 0 he requency scillatin is und r the ttal ass and the spring cnstant N.467 Hz.5 Hz 568 g 4. (a) he spring cnstant is und r the rati applied rce t displaceent. F g.7 g 9.80 s 75 N N x x.6 0 he aplitude is the distance pulled dwn r equilibriu, s A.5 0 he requency scillatin is und r the ttal ass and the spring cnstant. 75N.66 Hz.6 Hz.7 g 5. he spring cnstant is the sae regardless what ass is hung r the spring. cnstant.0 Hz 0.60 g 0.8 g.8 Hz 6. he table data is tie psitin shwn, alng with the 0 - A sthed graph. /4 0 0 Eery quarter a / A perid, the ass /4 0 - es r an - A tie / 5/4 0 extree pint t the equilibriu. he graph resebles a csine wae (actually, the ppsite a csine wae) Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 7

5 Giancli Physics: Principles with Applicatins, 6 th Editin 7. he relatinship between requency, ass, and spring cnstant is (a) Hz.5 0 g N 0.6 N N 4.8 Hz g 8. he spring cnstant is the sae regardless what ass is attached t the spring. cnstant g 0.60 Hz g 0.88 Hz g 0.68 g 0.60 Hz 0.59 g 0.88 Hz 0.60 Hz 9. (a) At equilibriu, the elcity is its iu.. A A A Hz s.5 s Fr Equatin (-5), we ind the elcity at any psitin. x s.565 s.6 s A 0. (d) E 0.60 g.45 s.80j.8 J ttal Since the bject has a iu displaceent at t = 0, the psitin will be described by the csine unctin. x 0. cs.0 Hz t x 0. cs 6.0 t 0. he relatinship between the elcity and the psitin a SHO is gien by Equatin (-5). Set that expressin equal t hal the iu speed, and sle r the displaceent. x A x A x A x A x A 0.866A 4 4. Since F x a r an bject attached t a spring, the acceleratin is prprtinal t the displaceent (althugh in the ppsite directin), as a x. hus the acceleratin will hae hal its iu alue where the displaceent has hal its iu alue, at x 0. he spring cnstant can be und r the stretch distance crrespnding t the weight suspended n the spring. F g.6 g 9.80 s 8.5 N x x 0.5 Ater being stretched urther and released, the ass will scillate. It taes ne-quarter a perid r the ass t e r the iu displaceent t the equilibriu psitin. 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 7

6 Chapter Vibratins and Waes g 0.8s 8.5N. (a) he ttal energy an bject in SHM is cnstant. When the psitin is at the aplitude, the speed is zer. Use that relatinship t ind the aplitude. E x A tt A x.0 g 80 N 0.55 s Again use cnseratin energy. he energy is all inetic energy when the bject has its iu elcity. E x A tt A 80 N s 0.58 s.0 g 4. he spring cnstant is und r the rati applied rce t displaceent. F 80.0 N N x 0.00 Assuing that there are n dissipatie rces acting n the ball, the elastic ptential energy in the laded psitin will bece inetic energy the ball. E E x x i 5. (a) he wr dne t cpress a spring is stred as ptential energy. W.0J W x 46.7 N 4. 0 N x N s 0.80 g he distance that the spring was cpressed beces the aplitude its tin. he iu acceleratin is gien by a A. Sle this r the ass N 0.. g. g a A A a 5 s 6. he general r the tin is x Acs t 0.45cs 6.40t. (a) he aplitude is A x (d) he requency is und by he ttal energy is gien by 6.40 s 6.40 s.09 Hz.0 Hz E A ttal 0.60 g 6.40 s J.5 J he ptential energy is gien by 0.60 g 6.40 s 0.0.J. J ptential E x x 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 74

7 Giancli Physics: Principles with Applicatins, 6 th Editin he inetic energy is gien by E E E.488 J. J.77 J.4 J inetic ttal ptential 7. I the energy the SHO is hal ptential and hal inetic, then the ptential energy is hal the ttal energy. he ttal energy is the ptential energy when the displaceent has the alue the aplitude. E E x A x A 0.707A pt tt 8. I the requencies and asses are the sae, then the spring cnstants r the tw ibratins are the sae. he ttal energy is gien by the iu ptential energy. A E A A E A A A 9. (a) he general equatin r SHM is Equatin (-8c), y Acs t. Fr the pupin, t y 0.8 cs s (d) he tie t return bac t the equilibriu psitin is ne-quarter a perid. t 0.65 s 0.6 s 4 4 he iu speed is gien by the angular requency ties the aplitude. A A s 0.65 s he iu acceleratin is gien by s a A A 0.65 s he iu acceleratin is irst attained at the release pint the pupin.. 0. Cnsider the irst ree-bdy diagra r the blc while it is at equilibriu, s that the net rce is zer. Newtn s nd law r ertical rces, chsing up as psitie, gies this. F F F g 0 F F g y A B A B Nw cnsider the secnd ree-bdy diagra, in which the blc is displaced a distance x r the equilibriu pint. Each upward rce will hae increased by an aunt x, since x 0. Again write Newtn s nd law r ertical rces. F F F F g F x F x g x F F g x y net A B A B A B his is the general r a restring rce that prduces SHM, with an eectie spring cnstant. hus the requency ibratin is as llws. eectie F A FB g FA FB g x 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 75

8 Chapter Vibratins and Waes. he equatin tin is x 0.8sin 6.50t Asin t. (a) he aplitude is A x 0.8. he requency is und by 6.50 s 6.50 s.0 Hz he perid is the reciprcal the requency..0 Hz s. (d) he ttal energy is gien by (e) E A ttal 0.00 g 6.50 s J 0.9 J. he ptential energy is gien by E x x 0.00 g 6.50 s J 5. 0 J ptential. he inetic energy is gien by E E E 0.95J 0.05J J 0.86 J. inetic ttal ptential () x() tie (sec). (a) Fr A, the aplitude is A.5. Fr B, the aplitude is A.5. A B Fr A, the requency is cycle eery 4.0 secnds, s 0.5 Hz. Fr B, the requency is A cycle eery.0 secnds, s 0.50 Hz. B Fr C, the perid is 4.0 s. Fr B, the perid is.0 s A B (d) Object A has a displaceent 0 when t 0, s it is a sine unctin. x A sin t x.5 sin t A A A A Object B has a iu displaceent when t 0, s it is a csine unctin. x A cs t x.5 cs t B B B B. (a) Find the perid and requency r the ass and the spring cnstant g 4 N s s.04 Hz he initial speed is the iu speed, and that can be used t ind the aplitude. A A.96 s g 4 N 0. he iu acceleratin can be und r the ass, spring cnstant, and aplitude a A 0. 4 N g 7.9 s 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 76

9 Giancli Physics: Principles with Applicatins, 6 th Editin (d) (e) Because the ass started at the equilibriu psitin x = 0, the psitin unctin will be prprtinal t the sine unctin. x 0. sin.04 Hz t x 0. sin 4.08 t he iu energy is the inetic energy that the bject has when at the equilibriu psitin. E g.96 s. J 4. We assue that dwnward is the psitie directin tin. Fr this tin, we hae 05 N, A 0.80, 0.60 g and 05 N 0.60 g 4.50 rad s. (a) Since the ass has a zer displaceent and a psitie elcity at t = 0, the equatin is a sine unctin. y t 0.80 sin 4.rad s t he perid scillatin is gien by 0.845s. he spring will hae 4.5 rad s its iu extensin at ties gien by the llwing. t n s n 0.8 s, n 0,,, 4 he spring will hae its iniu extensin at ties gien by the llwing. t n.8 0 s n 0.8 s, n 0,,, in 4 5. I the blc is displaced a distance x t the right in the diagra, then spring # will exert a rce F x, in the ppsite directin t x. iewise, spring # will exert a rce F x, in the sae directin as F. hus the net rce n the blc is F F F x x x. he eectie spring cnstant is thus, and the perid is gien by. 6. he energy the scillatr will be cnsered ater the cllisin. hus E A M A M his speed is the speed that the blc and bullet hae iediately ater the cllisin. inear entu in ne diensin will hae been cnsered during the cllisin, and s the initial speed the bullet can be und. p p M bere ater M A M g N s.5 0 g g 7. he perid the juper s tin is r the perid and the juper s ass. 8.0 s 8 cycles g 4.7N 4N 4.75 s 4.75 s. he spring cnstant can then be und 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 77

10 Chapter Vibratins and Waes he stretch the bungee crd needs t pride a rce equal t the weight the juper when he is at the equilibriu pint. g 65.0 g 9.80 s x g x N hus the unstretched bungee crd ust be s 8. (a) he perid is gien by.7s cycle. 6 cycles he requency is gien by 6 cycles 0.60 Hz. 60 s 9. he perid a pendulu is gien by g. Sle r the length using a perid.0 secnds. g.0 s 9.8 s g he perid a pendulu is gien by pendulu bth n Mars and n Earth. g Mars g g Mars Earth g g Earth Mars Earth Mars Earth 0.80 s. s 0.7 g g Earth Mars g. he length is assued t be the sae r the. he perid a pendulu is gien by g (a) g.8s 9.8 s I the pendulu is in ree all, there is n tensin in the string supprting the pendulu bb, and s n restring rce t cause scillatins. hus there will be n perid the pendulu will nt scillate and s n perid can be deined.. (a) he requency can be und r the length the pendulu, and the acceleratin due t graity. g 9.80 s Hz 0.57 Hz ind the speed at the lwest pint, use the cnseratin energy relating the lwest pint t the release pint the pendulu. ae the lwest pint t be the zer leel graitatinal ptential energy. E E KE PE KE PE tp btt tp tp btt btt 0 g cs 0 btt h cs 0 cs 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 78

11 Giancli Physics: Principles with Applicatins, 6 th Editin btt g cs 9.80 s cs s he ttal energy can be und r the inetic energy at the btt the tin. E ttal btt 0.65 g 0.57 s J. here are 4 h 60 in h 60s in 86, 400 s in a day. he clc shuld ae ne cycle in exactly tw secnds (a tic and a tc ), and s the clc shuld ae 4,00 cycles per day. Ater ne day, the clc in questin is 0 secnds slw, which eans that it has ade 5 less cycles than required r precise tieeeping. hus the clc is nly aing 4,85 cycles in a day. Accrdingly, the perid the clc ust be decreased by a actr 4,85 4, 00. 4,85 4,85 g g new ld new ld 4, 00 4, 00 4,85 4, new ld 4, 00 4, 00 hus the pendulu shuld be shrtened by Use energy cnseratin t relate the ptential energy at the iu height the pendulu t the inetic energy at the lwest pint the swing. ae the lwest pint t be the zer lcatin r graitatinal ptential energy. See the diagra. E E KE PE KE PE tp btt tp tp btt btt 0 gh gh g cs h cs 0 cs 5. he equatin tin r an bject in SHM that has the iu displaceent at t 0 is gien by x Acs t. Fr a pendulu, x and s x A, where ust be easured in radians. hus the equatin r the pendulu s angular displaceent is cs t cs t I bth sides the equatin are ultiplied by 80 rad, then the angles can be easured in degrees. hus the angular displaceent the pendulu can be written as belw. Please nte that the arguent the csine unctin is still in radians. cs t 5 cs 5.0 t (a) t 0.5 s 5 cs t.6 s 5 cs (here the tie is exactly 4 perids) t 500 s 5 cs (here the tie is exactly 50 perids) 6. he wae speed is gien by. he perid is.0 secnds, and the waelength is s. s 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 79

12 Chapter Vibratins and Waes 7. he distance between wae crests is the waelength the wae. 4 s 6 Hz. 8. ind the waelength, use s.00 0 s AM: AM: 90 t Hz 600 0Hz FM: s.00 0 s.4.78 FM:.78 t Hz 08 0 Hz he elastic and bul duli are taen r able 9- in chapter 9. he densities are taen r able 0- in chapter N (a) Fr water: B.4 0 s.00 0 g Fr granite: E 45 0 N g 4. 0 s Fr steel: E 00 0 N g 5. 0 s 40. he speed a lngitudinal wae in a slid is gien by E. Call the density the less dense aterial, and the density the re dense aterial. he less dense aterial will hae the higher speed, since the speed is inersely prprtinal t the square rt the density. E E.4 4. ind the tie r a pulse t trael r ne end the crd t the ther, the elcity the pulse n the crd ust be nwn. Fr a crd under tensin, we hae x F x t t F 8 50 N 0.65 g s F. 4. (a) he speed the pulse is gien by x s 78 s t 6 s he tensin is related t the speed the pulse by cable can be und r its lue and density. F. he ass per unit length the 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 80

13 Giancli Physics: Principles with Applicatins, 6 th Editin d g.78 g V d F F 77.5 s.78g 8. 0 N 4. he speed the water wae is gien by B, where B is the bul dulus water, r able 9-, and is the density sea water, r able 0-. he wae traels twice the depth the cean during the elapsed tie. 9 t t B.0s.0 0 N t.05 0 g (a) Bth waes trael the sae distance, s x t t. We let the saller speed be, and the larger speed be. he slwer wae will tae lnger t arrie, and s t is re than t. t t.0in t 0 s t 0 s t t 5.5 s 0 s 0 s 0 s 8.5 s 5.5 s x t 8.5 s 0 s.9 0 his is nt enugh inratin t deterine the epicenter. All that is nwn is the distance the epicenter r the seisic statin. he directin is nt nwn, s the epicenter lies n a circle radius.9 0 r the seisic statin. Readings r at least tw ther seisic statins are needed t deterine the epicenter s psitin. 45. We assue that the earthquae wae is ing the grund ertically, since it is a transerse wae. An bject sitting n the grund will then be ing with SHM, due t the tw rces n it the nral rce upwards r the grund and the weight dwnwards due t graity. I the bject lses cntact with the grund, then the nral rce will be zer, and the nly rce n the bject will be its weight. I the nly rce is the weight, then the bject will hae an acceleratin g dwnwards. hus the liiting cnditin r beginning t lse cntact with the grund is when the iu acceleratin caused by the wae is greater than g. Any larger dwnward acceleratin and the grund wuld all quicer than the bject. he iu acceleratin is related t the aplitude and the requency as llws. g 9.8 s a A g A g Hz (a) Assue that the earthquae waes spread ut spherically r the surce. Under thse cnditins, Eq. (-6b) applies, stating that intensity is inersely prprtinal t the square the distance r the surce the wae. I I he intensity is prprtinal t the square the aplitude, and s the aplitude is inersely prprtinal t the distance r the surce the wae. A A Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 8

14 Chapter Vibratins and Waes 47. (a) Assuing spherically syetric waes, the intensity will be inersely prprtinal t the square the distance r the surce. hus Ir will be cnstant. I r I r I near near ar ar r 6 48 ar 9 9 I.0 0 W W W near ar rnear he pwer passing thrugh an area is the intensity ties the area. P IA W W 48. Fr Equatin (-8), i the speed, ediu density, and requency the tw waes are the sae, then the intensity is prprtinal t the square the aplitude. I I E E A A A A.4 he re energetic wae has the larger aplitude. 49. Fr Equatin (-8), i the speed, ediu density, and requency the tw waes are the sae, then the intensity is prprtinal t the square the aplitude. I I P P A A A A.7 he re energetic wae has the larger aplitude. 50. he bug es in SHM as the wae passes. he iu KE a particle in SHM is the ttal energy, which is gien by E A. Cpare the tw KE ia. KE A A KE A A ttal.5 c.0 c (a) he energy is all inetic energy at the ent when the string has n displaceent. here is n elastic ptential energy at that ent. Each piece the string has speed but n displaceent. 5. he requencies the harnics a string that is ixed at bth ends are gien by n, and s n the irst ur harnics are 440 Hz, 880 Hz, 0 Hz, 760 Hz he undaental requency the ull string is gien by 94 Hz. I the length is uningered reduced t / its current alue, and the elcity waes n the string is nt changed, then the new requency will be Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 8

15 Giancli Physics: Principles with Applicatins, 6 th Editin ingered uningered 94 Hz 44 Hz 54. Fur lps is the standing wae pattern r the 4 th harnic, with a requency gien by 4 80 Hz. hus 70 Hz, 40 Hz, 0 Hz and 50 Hz 4 5 are all ther resnant requencies. 55. Adjacent ndes are separated by a hal-waelength, as exainatin Figure -40 will shw. 9 s x nde 475Hz 56. Since n, tw successie ertnes dier by the undaental requency, as shwn belw. n n n 50 Hz 80 Hz 70 Hz n n 57. he speed waes n the string is gien by equatin (-), a string with bth ends ixed are gien by equatin (-9b), n F. he resnant requencies n ib, where is the length ib the prtin that is actually ibrating. Cbining these relatinships allws the requencies t be calculated. n F 50 N Hz n g 0.90 ib Hz 87.Hz S the three requencies are 90 Hz, 580 Hz, 870 Hz, t signiicant igures. 58. Fr Equatin (-9b), n n, we see that the requency is prprtinal t the wae speed n the stretched string. Fr equatin (-), F, we see that the wae speed is prprtinal t the square rt the tensin. hus the requency is prprtinal t the square rt the tensin. F 00 Hz F F F 0.95 F F 05 Hz hus the tensin shuld be decreased by 4.8%. 59. he string ust ibrate in a standing wae pattern t hae a certain nuber lps. he requency the standing waes will all be 60 Hz, the sae as the ibratr. hat requency is als expressed n by Equatin (-9b),. he speed waes n the string is gien by Equatin (-), n 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 8

16 Chapter Vibratins and Waes F. he tensin in the string will be the sae as the weight the asses hung r the end F the string, g. Cbining these relatinships gies an expressin r the asses hung r the end the string. n n F n g 4 n (a) n n g Hz.9 0 g.89 g. g 9.80 s 5.89 g 4.89 g g 5. 0 g 60. he tensin in the string is the weight the hanging ass, F string can be und by F g waelength waes created n the string will thus be gien by g. he speed waes n the, and the requency is gien as 60 Hz. he g g 9.80 s Hz.9 0 g he length the string ust be an integer ultiple hal the waelength r there t be ndes at bth ends and thus r a standing wae. hus,,,, and s n. his gies 0.7, 0.75,.,.49 as the pssible lengths, and s there are 4 standing wae patterns that ay be achieed. 6. Fr the descriptin the water s behair, there is an anti-nde at each end the tub, and a nde in the iddle. hus ne waelength is twice the tube length Hz. s tub 6. he speed in the secnd ediu can be und r the law reractin, Equatin (-0). sin sin sin s 6. s sin sin sin he angle reractin can be und r the law reractin, Equatin (-0). sin. s sin sin sin sin sin.8 s 64. he angle reractin can be und r the law reractin, Equatin (-0). he relatie elcities can be und r the relatinship gien in the prble. 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 84

17 Giancli Physics: Principles with Applicatins, 6 th Editin sin sin sin 5 sin sin sin he angle reractin can be und r the law reractin, Equatin (-0). he relatie elcities can be und r Equatin (-4a). sin E SG SG water sin E SG SG water SG.6 sin sin sin sin SG he errr is allwed due t diractin the waes. I the waes are incident at the edge the dish, they can still diract int the dish i the relatinship is satisied. rad I the waelength is lnger than that, there will nt be uch diractin, but shadwing instead. 67. he unusual decrease water crrespnds t a trugh in Figure -4. he crest r pea the wae is then ne-hal waelength distant. he pea is 5 away, traeling at 750 /hr. x 5 60 in x t t 0 in 750 hr hr 68. Apply the cnseratin echanical energy t the car, calling cnditin # t be bere the cllisin and cnditin # t be ater the cllisin. Assue that all the inetic energy the car is cnerted t ptential energy stred in the buper. We nw that x 0 and 0. E E x x x x 500 g N. s Cnsider the cnseratin energy r the persn. Call the unstretched psitin the ire net the zer lcatin r bth elastic ptential energy and graitatinal ptential energy. he aunt stretch the ire net is gien by x, easured psitiely in the dwnward directin. he ertical displaceent r graitatinal ptential energy is gien by the ariable y, easured psitiely r the upward directin. Calculate the spring cnstant by cnsering energy between the windw height and the lwest lcatin the persn. he persn has n inetic energy at either lcatin. E E gy gy x (a) tp btt tp btt btt y y tp 8. g 65 g 9.8 s.0 0 N x btt 4 btt. I the persn were t lie n the ire net, they wuld stretch the net an aunt such that the upward rce the net wuld be equal t their weight. 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 85

18 Chapter Vibratins and Waes g 65 g 9.8 s F x g x N ind the aunt stretch gien a starting height 5, again use cnseratin energy. Nte that ybtt x, and there is n inetic energy at the tp r btt psitins. g g E E gy gy x x x y 0 tp btt tp btt tp x 65 g 9.8 s 65 g 9.8 s x N.0 0 N x 0.065x.7 0 x.5,.458 his is a quadratic equatin. he slutin is the psitie rt, since the net ust be belw the unstretched psitin. he result is Cnsider energy cnseratin r the ass er the range tin r letting g (the highest pint) t the lwest pint. he ass alls the sae distance that the spring is stretched, and has n KE at either endpint. Call the lwest pint the zer graitatinal ptential energy. he ariable x represents the aunt that the spring is stretched r the equilibriu psitin. E E tp btt gy x gy x tp tp tp btt btt btt g g 0 gh H H H g 9.8s. Hz H 0. x = 0 y = H x = H y = 0 7. (a) Fr cnseratin energy, the initial inetic energy the car will all be changed int elastic ptential energy by cpressing the spring. E E x x x s g.89 0 N.8 0 N x 5.0 he car will be in cntact with the spring r hal a perid, as it es r the equilibriu lcatin t iu displaceent and bac t equilibriu. 950 g 0.7s.89 0 N 4 7. he requency at which the water is being shaen is abut Hz. he slshing cee is in a standing wae de, with anti-ndes at each edge the cup. he cup diaeter is thus a hal-waelength, r 6 c. he wae speed can be calculated r the requency and the waelength. 6 c Hz 6c s 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 86

19 Giancli Physics: Principles with Applicatins, 6 th Editin 7. Relatie t the ixed needle psitin, the ripples are ing with a linear elcity gien by re in s in 60 s re his speed is the speed the ripple waes ing past the needle. he requency the waes is 0.7 s 0 Hz he equatin tin is x 0.650cs 7.40t Acs t. (a) he aplitude is A (d) he requency is gien by he ttal energy is gien by 7.40 rad s 7.40 rad s.77 Hz.8 Hz rad.00 g 7.40 rad s J. J ttal E A A. he ptential energy is und by PE x x.00 g 7.40 rad s J.70 J. he inetic energy is und by KE E PE.6 J.70 J 9.4 J. ttal g 75. he requency a siple pendulu is gien by. he pendulu is accelerating ertically which is equialent t increasing (r decreasing) the acceleratin due t graity by the acceleratin the pendulu. (a) g a.50g g new g a 0.5g g new 76. he rce the an s weight causes the rat t sin, and that causes the water t put a larger upward rce n the rat. his extra buyant rce is a restring rce, because it is in the ppsite directin the rce put n the rat by the an. his is analgus t pulling dwn n a ass-spring syste that is in equilibriu, by applying an extra rce. hen when the an steps, the restring rce pushes upward n the rat, and thus the rat water syste acts lie a spring, with a spring cnstant und as llws. F 75 g 9.8 s N x (a) he requency ibratin is deterined by the spring cnstant and the ass the rat. n N.455Hz.5 Hz 0 g As explained in the text, r a ertical spring the graitatinal ptential energy can be ignred i the displaceent is easured r the scillatr s equilibriu psitin. he ttal energy is thus N J 5 J ttal E A. 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 87

20 Chapter Vibratins and Waes 77. (a) he ertnes are gien by n, n,,4 (d) n G : 9 Hz 784 Hz 9 Hz 80 Hz A : 440 Hz 880 Hz 440 Hz 0 Hz I the tw strings hae the sae length, they hae the sae waelength. he requency dierence is then due t a dierence in wae speed caused by dierent asses r the strings. F G 440 G G G A G A.6 9 A A A F G A G A I the tw strings hae the sae ass per unit length and the sae tensin, then the wae speed n bth strings is the sae. he requency dierence is then due t a dierence in waelength. Fr the undaental, the waelength is twice the length the string. 440 G G A A G A. 9 A A G G A G I the tw strings hae the sae length, they hae the sae waelength. he requency dierence is then due t a dierence in wae speed caused by dierent tensins r the strings. F G F F F F F G G G G G G A A A A A AA A (a) Since the crd is nt accelerating t the let r right, the tensin in the crd ust be the sae eerywhere. hus the tensin is the sae in the tw parts the crd. he speed dierence will then be due t the dierent ass densities the tw parts the crd. et the sybl represent the ass per unit length each part the crd. F H H F he waelength rati is und as llws. H H H H H he tw requencies ust be the sae r the crd t reain cntinuus at the bundary. I the tw parts the crd scillate at dierent requencies, the crd cannt stay in ne piece, because the tw parts wuld be ut phase with each ther at arius ties. Since, we see that, and s the waelength is greater in the lighter crd. H H 79. (a) he iu speed is gien by A 64 Hz s. he iu acceleratin is gien by a 4 A 4 64 Hz s. 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 88

21 Giancli Physics: Principles with Applicatins, 6 th Editin 80. Fr the pebble t lse cntact with the bard eans that there is n nral rce the bard n the pebble. I there is n nral rce n the pebble, then the nly rce n the pebble is the rce graity, and the acceleratin the pebble will be g dwnward, the acceleratin due t graity. his is the iu dwnward acceleratin that the pebble can hae. hus i the bard s dwnward acceleratin exceeds g, then the pebble will lse cntact. he iu acceleratin and the aplitude are related by a A. 4 g 9.8 s a 4 A g A Hz 8. Fr a resnant cnditin, the ree end the string will be an antinde, and the ixed end the string will be a nde. he iniu distance r a nde t an antinde is 4. Other wae patterns that it the bundary cnditins a nde at ne end and an antinde at the ther end include 4, 5 4,. See the diagras. he general relatinship is n 4, n,,,. Sling r the waelength gies 0 4, n,,, n n = n = 0 n = he perid a pendulu is gien by (a) g.000 s 9.79 s Austin Austin 4 4 g.000 s s Paris Paris Paris Austin g, and s the length is g.00 s.6 s Mn Mn g he spring, riginally length l 0, will be stretched dwnward t a new equilibriu length when the ass is hung n it. he aunt dwnward stretch l is und r setting the spring rce 0 upward n the ass equal t the weight the ass: l g l g. he length 0 0 the pendulu is then l g 0. he perid the ertical scillatins is gien by, while the perid the pendulu scillatins is gien by er pen cpare the perids the tw tins. g. Nw 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 89

22 Chapter Vibratins and Waes l g g pen 0 l g l 0 0 g g er pen er, by a actr l 0 g 84. Blc stays n tp blc M (executing SHM relatie t the grund) withut slipping due t static rictin. he iu static rictinal rce n is F g. his rictinal rce causes r s blc t accelerate, s a g a g s s. hus r the blcs t stay in cntact withut slipping, the iu acceleratin blc M is als a g s. But an bject in SHM has a iu acceleratin gien by a A Mttal A. Equate these tw expressins r the iu acceleratin. g s s a A g A M s M 0 N 6.5 g 0.4 ttal 85. he speed the pulses is und r the tensin and ass per unit length the wire. F 55 N s 0. g 0.0 he ttal distance traeled by the tw pulses will be the length the wire. he secnd pulse has a shrter tie trael than the irst pulse, by 0.0 s. d d t t t t.00 0 t d s t s s s s he tw pulses eet 6.44 r the end where the irst pulse riginated. 86. Fr the penny t stay n the blc at all ties eans that there will be a nral rce n the penny r the blc, exerted upward. I dwn is taen t be the psitie directin, then the net rce n the penny is F g F a. Sling r the agnitude the nral rce gies F g net N N a. his expressin is always psitie i the acceleratin is upwards (a < 0 ), and s there is n pssibility the penny lsing cntact while accelerating upwards. But i a dwnward acceleratin were t be larger than g, then the nral rce wuld g t zer, since the nral rce cannt switch directins F 0. hus the liiting cnditin is a g. his is the iu alue r the acceleratin. N dwn Fr SHM, we als nw that a A A A. Equate these tw alues r the M M acceleratin. Mg a A g A M 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 90

23 Giancli Physics: Principles with Applicatins, 6 th Editin 87. he car n the end the cable prduces tensin in the cable, and stretches the cable accrding t F Equatin (9-4),, where E is Yung s dulus. Rearrange this equatin t see that EA EA the tensin rce is prprtinal t the aunt stretch, F, and s the eectie spring EA cnstant is. he perid the buncing can be und r the spring cnstant and the ass n the end the cable. 00 g 0.40 s EA N. 0 GA 88. Fr Equatin (9-6) and Figure (9-c), the restring rce n the tp the Jell-O is F, and is in the ppsite directin t the displaceent the tp r the equilibriu cnditin. hus GA the spring cnstant r the restring rce is. I yu were t l at a layer Jell-O clser t the base, the displaceent wuld be less, but s wuld the restring rce in prprtin, and s we estiate all the Jell-O as haing the sae spring cnstant. he requency ibratin can be deterined r the spring cnstant and the ass the Jell-O. GA GA G V A 50N 00 g Hz 005 Pearsn Educatin, Inc., Upper Saddle Rier, NJ. All rights resered. his aterial is prtected under all cpyright laws as they 9