Chapter 14 Oscillations

Transcription

1 Chapter 4 Oscillations Conceptual Probles rue or false: (a) For a siple haronic oscillator, the period is proportional to the square of the aplitude. (b) For a siple haronic oscillator, the frequency does not depend on the aplitude. (c) If the net force on a particle underoin one-diensional otion is proportional to, and oppositely directed fro, the displaceent fro equilibriu, the otion is siple haronic. (a) False. In siple haronic otion, the period is independent of the aplitude. (b) rue. In siple haronic otion, the frequency is the reciprocal of the period which, in turn, is independent of the aplitude. (c) rue. his is the condition for siple haronic otion If the aplitude of a siple haronic oscillator is tripled, by what factor is the enery chaned? Deterine the Concept he enery of a siple haronic oscillator varies as the square of the aplitude of its otion. Hence, triplin the aplitude increases the enery by a factor of 9. [SSM] n object attached to a sprin exhibits siple haronic otion with an aplitude of 4. c. When the object is. c fro the equilibriu position, what percentae of its total echanical enery is in the for of potential enery? (a) One-quarter. (b) One-third. (c) One-half. (d) wo-thirds. (e) hree-quarters. Picture the Proble he total enery of an object underoin siple haronic otion is iven by tot, where is the force constant and is the aplitude of the otion. he potential enery of the oscillator when it is a distance x fro its equilibriu position is U ( x) x. xpress the ratio of the potential enery of the object when it is. c fro the equilibriu position to its total enery: U ( x) tot x x 45

2 46 Chapter 4 valuate this ratio for x. c and 4. c: U ( c) (.c) ( 4.c) 4 tot and (a) is correct. 4 n object attached to a sprin exhibits siple haronic otion with an aplitude of. c. How far fro equilibriu will the object be when the syste s potential enery is equal to its inetic enery? (a) 5. c. (b) 7.7 c. (c) 9. c. (d) he distance can t be deterined fro the data iven. Deterine the Concept ecause the object s total enery is the su of its inetic and potential eneries, when its potential enery equals its inetic enery, its potential enery (and its inetic enery) equals one-half its total enery. quate the object s potential enery to one-half its total enery: U s total Substitutin for U s and total yields: x ( ) x Substitute the nuerical value of and evaluate x to obtain:. c x 7.7 c is correct. and ( b) 5 wo identical systes each consist of a sprin with one end attached to a bloc and the other end attached to a wall. he sprins are horizontal, and the blocs are supported fro below by a frictionless horizontal table. he blocs are oscillatin in siple haronic otions such that the aplitude of the otion of bloc is four ties as lare as the aplitude of the otion of bloc. How do their axiu speeds copare? (a) v ax v ax, (b) v ax v ax, (c) v 4v, (d) his coparison cannot be done by usin the data iven. ax ax Deterine the Concept he axiu speed of a siple haronic oscillator is the product of its anular frequency and its aplitude. he anular frequency of a siple haronic oscillator is the square root of the quotient of the force constant of the sprin and the ass of the oscillator. Relate the axiu speed of syste to the aplitude of its otion: v ax ω Relate the axiu speed of syste to the aplitude of its otion: v ax ω

3 Oscillations 47 Divide the first of these equations by the second to obtain: v v ax ax ω ω ecause the systes differ only in v ax aplitude, ω ω, and: v ax Substitutin for and siplifyin yields: v v 4 ax ax ( c) is correct. 4 v 4v ax ax 6 wo systes each consist of a sprin with one end attached to a bloc and the other end attached to a wall. he sprins are horizontal, and the blocs are supported fro below by a frictionless horizontal table. he blocs are oscillatin in siple haronic otions with equal aplitudes. However, the force constant of sprin is four ties as lare as the force constant of sprin. How do their axiu speeds copare? (a) v ax v ax, (b) v ax v ax, (c) v 4v, (d) his coparison cannot be done by usin the data iven. ax ax Deterine the Concept he axiu speed of a siple haronic oscillator is the product of its anular frequency and its aplitude. he anular frequency of a siple haronic oscillator is the square root of the quotient of the force constant of the sprin and the ass of the oscillator. Relate the axiu speed of syste to its force constant: Relate the axiu speed of syste to its force constant: v ω ax ax v ω Divide the first of these equations by the second and siplify to obtain: v v ax ax ecause the systes differ only in their force constants: v v ax ax

4 48 Chapter 4 Substitutin for and siplifyin yields: v v 4 ax ax ( b) is correct. v v ax ax 7 [SSM] wo systes each consist of a sprin with one end attached to a bloc and the other end attached to a wall. he identical sprins are horizontal, and the blocs are supported fro below by a frictionless horizontal table. he blocs are oscillatin in siple haronic otions with equal aplitudes. However, the ass of bloc is four ties as lare as the ass of bloc. How do their axiu speeds copare? (a) v ax v ax, (b) v ax v ax, (c) v ax v ax, (d) his coparison cannot be done by usin the data iven. Deterine the Concept he axiu speed of a siple haronic oscillator is the product of its anular frequency and its aplitude. he anular frequency of a siple haronic oscillator is the square root of the quotient of the force constant of the sprin and the ass of the oscillator. Relate the axiu speed of syste to its force constant: v ω ax Relate the axiu speed of syste to its force constant: v ω ax Divide the first of these equations by the second and siplify to obtain: v v ax ax ecause the systes differ only in the asses attached to the sprins: v v ax ax Substitutin for and siplifyin v ax yields: v ax v 4 ax ( c) is correct. v ax 8 wo systes each consist of a sprin with one end attached to a bloc and the other end attached to a wall. he identical sprins are horizontal, and the blocs are supported fro below by a frictionless horizontal table. he blocs are

5 Oscillations 49 oscillatin in siple haronic otions with equal aplitudes. However, the ass of bloc is four ties as lare as the ass of bloc. How do the anitudes of their axiu acceleration copare? (a) a ax a ax, (b) a ax a ax, (c) a ax a ax, (d) a ax 4 a ax, (e) his coparison cannot be done by usin the data iven. Deterine the Concept he axiu acceleration of a siple haronic oscillator is the product of the square of its anular frequency and its aplitude. he anular frequency of a siple haronic oscillator is the square root of the quotient of the force constant of the sprin and the ass of the oscillator. Relate the axiu acceleration of syste to its force constant: Relate the axiu acceleration of syste to its force constant: a ax ω a ax ω Divide the first of these equations by the second and siplify to obtain: a a ax,ax ecause the systes differ only in the asses attached to the sprins: a a ax ax Substitutin for and siplifyin yields: a a ax ax ( d ) 4 is correct. 4 a ax 4 a ax 9 [SSM] In eneral physics courses, the ass of the sprin in siple haronic otion is usually nelected because its ass is usually uch saller than the ass of the object attached to it. However, this is not always the case. If you nelect the ass of the sprin when it is not neliible, how will your calculation of the syste s period, frequency and total enery copare to the actual values of these paraeters? xplain. Deterine the Concept Nelectin the ass of the sprin, the period of a siple haronic oscillator is iven by π ω π where is the ass of the oscillatin syste (sprin plus object) and its total enery is iven by. total

6 44 Chapter 4 Nelectin the ass of the sprin results in your usin a value for the ass of the oscillatin syste that is saller than its actual value. Hence your calculated value for the period will be saller than the actual period of the syste. ecause ω, nelectin the ass of the sprin will result in your usin a value for the ass of the oscillatin syste that is saller than its actual value. Hence your calculated value for the frequency of the syste will be larer than the actual frequency of the syste. ecause the total enery of the oscillatin syste is the su of its potential and inetic eneries, inorin the ass of the sprin will cause your calculation of the syste s inetic enery to be too sall and, hence, your calculation of the total enery to be too sall. wo ass sprin systes oscillate with periods and. If and the systes sprins have identical force constants, it follows that the systes asses are related by (a) 4, (b), (c) /, (d) /4. Picture the Proble We can use π to express the periods of the two ass-sprin systes in ters of their force constants. Dividin one of the equations by the other will allow us to express in ters of. xpress the period of syste : π 4π Relate the ass of syste to its period: 4π Divide the first of these equations by the second and siplify to obtain: 4π 4π ecause the force constants of the two systes are the sae:

7 Oscillations 44 Substitutin for and siplifyin yields: 4 ( a) is correct. 4 wo ass sprin systes oscillate at frequencies f and f. If f f and the systes sprins have identical force constants, it follows that the systes asses are related by (a) 4, (b), (c), (d) 4. Picture the Proble We can use f to express the frequencies of the π two ass-sprin systes in ters of their asses. Dividin one of the equations by the other will allow us to express in ters of. xpress the frequency of ass-sprin syste as a function of its ass: f π xpress the frequency of asssprin syste as a function of its ass: f π Divide the second of these equations by the first to obtain: f f Solve for : f f f (d) is correct. f 4 wo ass sprin systes and oscillate so that their total echanical eneries are equal. If, which expression best relates their aplitudes? (a) /4, (b), (c), (d) Not enouh inforation is iven to deterine the ratio of the aplitudes. Picture the Proble We can relate the eneries of the two ass-sprin systes throuh either or ω and investiate the relationship between their aplitudes by equatin the expressions, substitutin for, and expressin in ters of.

8 44 Chapter 4 xpress the enery of ass-sprin syste : ω xpress the enery of ass-sprin syste : ω Divide the first of these equations by ω the second to obtain: ω Substitute for and siplify: Solve for : ω ω ω ω ω ω Without nowin how ω and ω, or and, are related, we cannot siplify this expression further. (d) is correct. [SSM] wo ass sprin systes and oscillate so that their total echanical eneries are equal. If the force constant of sprin is two ties the force constant of sprin, then which expression best relates their aplitudes? (a) /4, (b), (c), (d) Not enouh inforation is iven to deterine the ratio of the aplitudes. Picture the Proble We can express the enery of each syste usin and, because the eneries are equal, equate the and solve for. xpress the enery of ass-sprin syste in ters of the aplitude of its otion: xpress the enery of ass-sprin syste in ters of the aplitude of its otion: ecause the eneries of the two systes are equal we can equate the to obtain:

9 Oscillations 44 Substitute for and siplify to obtain: (b) is correct. 4 he lenth of the strin or wire supportin a pendulu bob increases slihtly when the teperature of the strin or wire is raised. How does this affect a cloc operated by a siple pendulu? Deterine the Concept he period of a siple pendulu depends on the square root of the lenth of the pendulu. Increasin the lenth of the pendulu will lenthen its period and, hence, the cloc will run slow. 5 lap hanin fro the ceilin of the club car in a train oscillates with period when the train is at rest. he period will be (atch left and riht coluns). reater than when. he train oves horizontally at constant velocity.. less than when. he train rounds a curve at constant speed.. equal to when C. he train clibs a hill at constant speed. D. he train oes over the crest of a hill at constant speed. Deterine the Concept he period of the lap varies inversely with the square root of the effective value of the local ravitational field. -. he period will be reater than when the train rounds a curve of radius R with speed v. -D. he period will be less than when the train oes over the crest of a hill of radius of curvature R with constant speed. -. he period will be equal to when the train oves horizontally with constant velocity.

10 444 Chapter 4 6 wo siple pendulus are related as follows. Pendulu has a lenth L and a bob of ass ; pendulu has a lenth L and a bob of ass. If the period of is twice that of, then (a) L L and, (b) L 4L and, (c) L 4L whatever the ratio /, (d) L L whatever the ratio /. Picture the Proble he period of a siple pendulu is independent of the ass of its bob and is iven by π L. xpress the period of pendulu : π L xpress the period of pendulu : π L Divide the first of these equations by the second and solve for L /L : L L Substitute for and solve for L to obtain: L 4 L (c) is correct. L 7 [SSM] wo siple pendulus are related as follows. Pendulu has a lenth L and a bob of ass ; pendulu has a lenth L and a bob of ass. If the frequency of is one-third that of, then (a) L L and, (b) L 9L and, (c) L 9L reardless of the ratio /, (d) L L reardless of the ratio /. Picture the Proble he frequency of a siple pendulu is independent of the ass of its bob and is iven by f L. π xpress the frequency of pendulu : f L π L 4π f Siilarly, the lenth of pendulu is iven by: L 4π f

11 Oscillations 445 Divide the first of these equations by the second and siplify to obtain: L L 4π f 4π f f f f f Substitute for f to obtain: L L f f (c) is correct. 9 L 9L 8 wo siple pendulus are related as follows. Pendulu has a lenth L and a bob of ass ; pendulu has a lenth L a bob of ass. hey have the sae period. he only thin different between their otions is that the aplitude of s otion is twice that of s otion, then (a) L L and, (b) L L and, (c) L L whatever the ratio /, (d) L L whatever the ratio /. Picture the Proble he period of a siple pendulu is independent of the ass of its bob and is iven by π L. For sall aplitudes, the period is independent of the aplitude. xpress the period of pendulu : π L xpress the period of pendulu : π L Divide the first of these equations by the second and solve for L /L : L L ecause their periods are the sae: L L (c) is correct. L L 9 rue or false: (a) he echanical enery of a daped, undriven oscillator decreases exponentially with tie. (b) Resonance for a daped, driven oscillator occurs when the drivin frequency exactly equals the natural frequency.

12 446 Chapter 4 (c) If the Q factor of a daped oscillator is hih, then its resonance curve will be narrow. (d) he decay tie τ for a sprin-ass oscillator with linear dapin is independent of its ass. (e) he Q factor for a driven sprin-ass oscillator with linear dapin is independent of its ass. (a) rue. ecause the enery of an oscillator is proportional to the square of its aplitude, and the aplitude of a daped, undriven oscillator decreases exponentially with tie, so does its enery. (b) False. For a daped driven oscillator, the resonant frequency ω is iven b by, ω' ω where ω is the natural frequency of the oscillator. ω (c) rue. he ratio of the width of a resonance curve to the resonant frequency equals the reciprocal of the Q factor ( Δω ω Q ). Hence, the larer Q is, the narrower the resonance curve. (d) False. he decay tie for a daped but undriven sprin-ass oscillator is directly proportional to its ass. (e) rue. Fro Δω ω Q one can see that Q is independent of. wo daped sprin-ass oscillatin systes have identical sprin and dapin constants. However, syste s ass is four ties syste s. How do their decay ties copare? (a) τ 4τ, (b) τ τ, (c) τ τ, (d) heir decay ties cannot be copared, iven the inforation provided. Picture the Proble he decay tie τ of a daped oscillator is related to the ass of the oscillator and the dapin constant b for the otion accordin to τ b. xpress the decay tie of Syste : he decay tie for Syste is iven by: τ τ b b Dividin the first of these equations by the second and siplifyin yields: τ τ b b b b

13 Oscillations 447 ecause their dapin constants are the sae: τ τ Substitutin for yields: τ τ 4 ( a) is correct. 4 τ 4τ wo daped sprin-ass oscillatin systes have identical sprin constants and decay ties. However, syste s ass is syste s ass. hey are tweaed into oscillation and their decay ties are easured to be the sae. How do their dapin constants, b, copare? (a) b 4b, (b) b b, (c) b b, (d) b b, (e) heir decay ties cannot be copared, iven the inforation provided. Picture the Proble he decay tie τ of a daped oscillator is related to the ass of the oscillator and the dapin constant b for the otion accordin to τ b. xpress the dapin constant of Syste : he dapin constant for Syste is iven by: Dividin the first of these equations by the second and siplifyin yields: ecause their decay ties are the sae: b τ b τ b b b b τ τ τ τ Substitutin for yields: b b ( b) is correct. b b wo daped, driven sprin-ass oscillatin systes have identical drivin forces as well as identical sprin and dapin constants. However, the ass of syste is four ties the ass of syste. ssue both systes are very wealy daped. How do their resonant frequencies copare?

14 448 Chapter 4 (a) ω ω, (b) ω ω, (c) ω ω, (d) ω 4 ω, (e) heir resonant frequencies cannot be copared, iven the inforation provided. Picture the Proble For very wea dapin, the resonant frequency of a sprinass oscillator is the sae as its natural frequency and is iven by ω, where is the oscillator s ass and is the force constant of the sprin. xpress the resonant frequency of Syste : ω he resonant frequency of Syste is iven by: ω Dividin the first of these equations by the second and siplifyin yields: ω ω ecause their force constants are the sae: ω ω Substitutin for yields: ω ω ( c) 4 is correct. ω ω [SSM] wo daped, driven sprin-ass oscillatin systes have identical asses, drivin forces, and dapin constants. However, syste s force constant is four ties syste s force constant. ssue they are both very wealy daped. How do their resonant frequencies copare? (a) ω ω, (b) ω ω, (c) ω ω, (d) ω 4 ω, (e) heir resonant frequencies cannot be copared, iven the inforation provided. Picture the Proble For very wea dapin, the resonant frequency of a sprinass oscillator is the sae as its natural frequency and is iven by ω, where is the oscillator s ass and is the force constant of the sprin.

15 Oscillations 449 xpress the resonant frequency of Syste : ω he resonant frequency of Syste is iven by: ω Dividin the first of these equations by the second and siplifyin yields: ω ω ecause their asses are the sae: ω ω Substitutin for yields: ω ω 4 ( b) is correct. ω ω 4 wo daped, driven siple-pendulu systes have identical asses, drivin forces, and dapin constants. However, syste s lenth is four ties syste s lenth. ssue they are both very wealy daped. How do their resonant frequencies copare? (a) ω ω, (b) ω ω, (c) ω ω, (d) ω 4 ω, (e) heir resonant frequencies cannot be copared, iven the inforation provided. Picture the Proble For very wea dapin, the resonant frequency of a siple pendulu is the sae as its natural frequency and is iven by ω L, where L is the lenth of the siple pendulu and is the ravitational field. xpress the resonant frequency of Syste : ω L he resonant frequency of Syste is iven by: ω L

16 45 Chapter 4 Dividin the first of these equations by the second and siplifyin yields: ω ω L L L L Substitutin for L yields: ω ω ( c) L 4L is correct. ω ω stiation and pproxiation 5 [SSM] stiate the width of a typical randfather clocs cabinet relative to the width of the pendulu bob, presuin the desired otion of the pendulu is siple haronic. Picture the Proble If the otion of the pendulu in a randfather cloc is to be siple haronic otion, then its period ust be independent of the anular aplitude of its oscillations. he period of the otion for lareaplitude oscillations is iven by quation 4- and we can use this expression to obtain a axiu value for the aplitude of swinin pendulu in the cloc. We can then use this value and an assued value for the lenth of the pendulu to estiate the width of the randfather clocs cabinet. w θ w aplitude L w bob Referrin to the diara, we see that the iniu width of the cabinet is deterined by the width of the bob and the width required to accoodate the swinin pendulu: w w bob + w aplitude and w waplitude + () w w bob bob

17 Oscillations 45 xpress w aplitude in ters of the anular aplitude θ and the lenth of the pendulu L: w Lsinθ aplitude Substitutin for w aplitude in equation w Lsinθ + () () yields: wbob wbob quation 4- ives us the period of a siple pendulu as a function of its anular aplitude: + sin θ +... If is to be approxiately equal to, the second ter in the bracets ust be sall copared to the first ter. Suppose that: sin θ. 4 Solvin for θ yields: θ sin (.6) 7.5 If we assue that the lenth of a randfather cloc s pendulu is about.5 and that the width of the bob is about c, then equation () yields: w w bob ( ).5 sin sall punchin ba for boxin worouts is approxiately the size and weiht of a person s head and is suspended fro a very short rope or chain. stiate the natural frequency of oscillations of such a punchin ba. Picture the Proble For the purposes of this estiation, odel the punchin ba as a sphere of radius R and assue that the spindle about which it rotates to be.5 ties the radius of the sphere. he natural frequency of oscillations of this MD physical pendulu is iven by f ω where M is the ass of the π I pendulu, D is the distance fro the point of support to the center of ass of the punchin ba, and I is its oent of inertia about an axis throuh the spindle fro which it is supported and about which it swivels. xpress the natural frequency of oscillation of the punchin ba: MD f () π I spindle

18 45 Chapter 4 Fro the parallel-axis theore we have: I I + Mh spindle c where h.5r +.5R R Substitutin for I c and h yields: ( ) I MR M R MR spindle Substitute for I spindle in equation () to obtain: f π ( R) M 4.4MR π.r ssue that the radius of the punchin ba is c, substitute nuerical values and evaluate f : f π 9.8 /s. (. ) Hz 7 For a child on a swin, the aplitude drops by a factor of /e in about eiht periods if no additional echanical enery is iven to the syste. stiate the Q factor for this syste. Picture the Proble he Q factor for this syste is related to the decay constant τ throuh Q ω τ πτ and the aplitude of the child s daped otion t τ varies with tie accordin to e. We can set the ratio of two displaceents separated by eiht periods equal to /e to deterine τ in ters of. xpress Q as a function of τ : he aplitude of the oscillations varies with tie accordin to: he aplitude after eiht periods is: πτ Q ω τ () e 8 t τ e ( t+ 8 ) τ ( t+ 8 ) xpress and siplify the ratio 8 /: e t e τ 8 4 τ e τ Set this ratio equal to /e and solve for τ : e 4 τ e τ 4 Substitute in equation () and evaluate Q: π Q ( 4 ) 8π 8 (a) stiate the natural period of oscillation for swinin your ars as you wal, when your hands are epty. (b) Now estiate the natural period of

19 Oscillations 45 oscillation when you are carryin a heavy briefcase. (c) Observe other people as they wal. Do your estiates see reasonable? Picture the Proble ssue that an averae lenth for an ar is about 8 c, and that it can be treated as a unifor rod, pivoted at one end. We can use the expression for the period of a physical pendulu to derive an expression for the period of the swinin ar. When carryin a heavy briefcase, the ass is concentrated ostly at the end of the rod (that is, in the briefcase), so we can treat the ar-plus-briefcase syste as a siple pendulu. (a) xpress the period of a unifor rod pivoted at one end: xpress the oent of inertia of a rod about an axis throuh its end: π I MD where I is the oent of inertia of the stic about an axis throuh one end, M is the ass of the stic, and D ( L/) is the distance fro the end of the stic to its center of ass. I ML Substitute the values for I and D in the expression for and siplify to ML π π obtain: M( L) L Substitute nuerical values and evaluate : (.8) π 9.8/s ( ).5s (b) xpress the period of a siple pendulu: ' π L' where L is slihtly loner than the ar lenth due to the size of the briefcase. ssuin L., evaluate the period of the siple pendulu: '. π 9.8/s.s (c) Fro observation of people as they wal, these estiates see reasonable. Siple Haronic Motion 9 he position of a particle is iven by x (7. c) cos 6πt, where t is in seconds. What are (a) the frequency, (b) the period, and (c) the aplitude of the

20 454 Chapter 4 particle s otion? (d) What is the first tie after t that the particle is at its equilibriu position? In what direction is it ovin at that tie? Picture the Proble he position of the particle is iven by x cos ( ω t +δ ) where is the aplitude of the otion, ω is the anular frequency, and δ is a phase constant. he frequency of the otion is iven by f ω π and the period of the otion is the reciprocal of its frequency. (a) Use the definition of ω to deterine f: f ω π 6π s π.hz (b) valuate the reciprocal of the frequency: f.hz.s (c) Copare x (7. c) cos 6π t to x cos ω t + δ to conclude that: ( ) 7.c (d) xpress the condition that ust be satisfied when the particle is at its equilibriu position: cos ωt π ω t t π ω Substitutin for ω yields: π t ( 6π ).8s Differentiate x to find v(t): v d dt [( 7.c) cos6π t] ( 4π c/s) sin 6π t valuate v(.8 s): v (.8s) ( 4π c/s) sin 6π (.8s) < ecause v <, the particle is ovin in the neative direction at t.8 s. What is the phase constant δ in ( ω t +δ ) x cos (quation 4-4) if the position of the oscillatin particle at tie t is (a), (b), (c), (d) /? Picture the Proble he initial position of the oscillatin particle is related to the aplitude and phase constant of the otion by x cosδ where δ < π.

21 Oscillations 455 (a) For x : cos δ cos ( ) δ π π, (b) For x : cosδ δ cos ( ) π (c) For x : cosδ δ cos () (d) When x /: cosδ cos π δ [SSM] particle of ass beins at rest fro x +5 c and oscillates about its equilibriu position at x with a period of.5 s. Write expressions for (a) the position x as a function of t, (b) the velocity v x as a function of t, and (c) the acceleration a x as a function of t. Picture the Proble he position of the particle as a function of tie is iven by x cos ( ω t +δ ). Its velocity as a function of tie is v x ω sin ( ωt + δ ) and its acceleration is a x ω cos( ωt + δ ). he initial position and velocity ive us two equations fro which to deterine the aplitude and phase constantδ. (a) xpress the position, velocity, and acceleration of the particle as a function of t: Find the anular frequency of the particle s otion: Relate the initial position and velocity to the aplitude and phase constant: Divide the equation for v by the equation for x to eliinate : Solvin for δ yields: ( ω +δ ) ( ω δ ) cos( ω δ ) x cos t () v x ω sin t + () a x ω t + ω π 4π s 4.9s x cosδ and v ω sinδ v x ωsinδ cosδ ω tanδ v tan δ tan xω () xω

22 456 Chapter 4 Substitute in equation () to obtain: x ( 5c) 4π cos s t (.5) cos[ ( 4.s ) t] (b) Substitute in equation () to obtain: v x ( 5c) 4π s 4π sin s (. /s) sin[ ( 4.s ) t] t (c) Substitute in equation () to obtain: a x ( 5c) 4π s ( 4.4 /s ) cos[ ( 4.s ) t] 4π cos s Find (a) the axiu speed and (b) the axiu acceleration of the particle in Proble. (c) What is the first tie that the particle is at x and ovin to the riht? Picture the Proble he axiu speed and axiu acceleration of the particle in are iven by v ax ω and a ax ω. he particle s position is iven by x cos ( ω t +δ ) where 7. c, ω 6π s, and δ, and its v ω sin ωt + δ. velocity is iven by ( ) (a) xpress v ax in ters of and ω: v ω ( 7.c)( 6π s ) ax./s (b) xpress a ax in ters of and ω: a ω ( 7.c)( 6π s ) ax 5π c/s 5/s t 4π c/s (c) When x : ω t cos ( ) cos t π π ω, π valuate v for ω t : π v ω sin ω hat is, the particle is ovin to the left.

23 Oscillations 457 π valuate v for ω t : ω π v sin ω hat is, the particle is ovin to the riht. π Solve ω t for t to obtain: t π ω π ( 6π s ).5s Wor Proble with the particle initially at x 5 c and ovin with velocity v +5 c/s. Picture the Proble he position of the particle as a function of tie is iven by x cos ( ω t +δ ). Its velocity as a function of tie is iven by v ω sin ( ωt + δ ) and its acceleration by a ω cos( ωt + δ ). he initial position and velocity ive us two equations fro which to deterine the aplitude and phase constant δ. (a) xpress the position, velocity, and acceleration of the particle as functions of t: Find the anular frequency of the particle s otion: Relate the initial position and velocity to the aplitude and phase constant: Divide these equations to eliinate : Solvin for δ yields: ( ω +δ ) ( ω δ ) cos( ω δ ) x cos t () v x ω sin t + () a x ω t + ω π 4π s 4.9s x cosδ and v ω sinδ v x δ ωsinδ cosδ tan v xω ω tanδ () Substitute nuerical values and 5 c/s δ tan.445rad evaluate δ: ( 5c)( 4.9 s )

24 458 Chapter 4 Use either the x or v equation (x is 5c used here) to find the aplitude: cosδ cos(.445 rad) Substitute in equation () to obtain: x (.8) cos[ ( 4.s ) t.45] (b) Substitute nuerical values in equation () to obtain: x 7.7 c v x 4π 4π ( 7.7 c) s sin s t.445 (. /s) sin[ ( 4.s ) t.45] (c) Substitute nuerical values in equation () to obtain: a x ( 7.7 c) 4π s 4π cos s ( 4.9 /s ) cos[ ( 4.s ) t.45] t he period of a particle that is oscillatin in siple haronic otion is 8. s and its aplitude is c. t t it is at its equilibriu position. Find the distance it travels durin the intervals (a) t to t. s, (b) t. s to t 4. s, (c) t to t. s, and (d) t. s to t. s. Picture the Proble he position of the particle as a function of tie is iven by x cos ( ω t +δ ). We re iven the aplitude of the otion and can use the initial position of the particle to deterine the phase constant δ. Once we ve deterined these quantities, we can express the distance traveled Δx durin any interval of tie. xpress the position of the particle as a function of t: Find the anular frequency of the particle s otion: Relate the initial position of the particle to the aplitude and phase constant: Solve for δ: ( c) cos( ω + δ ) x t () ω π π π x cosδ 8.s s 4 x δ cos cos π

25 Oscillations 459 Substitute in equation () to obtain: ( ) + s 4 cos c π π t x xpress the distance the particle travels in ters of t f and t i : ( ) ( ) ( ) Δ s 4 cos s 4 cos c s 4 cos c s 4 cos c i f i f π π π π π π π π t t t t x (a) valuate Δx for t f. s, t i s: ( ) ( ) ( ) c s 4 cos.s s 4 cos c Δ + + π π π π x (b) valuate Δx for t f 4. s, t i. s: ( ) ( ) ( ) c.s s 4 cos 4.s s 4 cos c Δ + + π π π π x (c) valuate Δx for t f. s, t i : ( ) ( ) ( ) ( ) } { c c s 4 cos.s s 4 cos c Δ + + π π π π x (d) valuate Δx for t f. s, t i. s: ( ) ( ) ( ) c.5.s s 4 cos.s s 4 cos c Δ + + π π π π x 5 he period of a particle oscillatin in siple haronic otion is 8. s. t t, the particle is at rest at x c. (a) Setch x as a function of t. (b) Find the distance traveled in the first, second, third, and fourth second after t. Picture the Proble he position of the particle as a function of tie is iven by ( ) ( ) ω + δ t x cos c. We can deterine the anular frequency ω fro the

26 46 Chapter 4 period of the otion and the phase constant δ fro the initial position and velocity. Once we ve deterined these quantities, we can express the distance traveled Δx durin any interval of tie. xpress the position of the particle as a function of t: Find the anular frequency of the particle s otion: Find the phase constant of the otion: Substitute in equation () to obtain: 4 π (a) raph of x ( c) cos s t ( c) cos( ω + δ ) x t () π π π ω s 8.s 4 δ tan v tan xω x follows: π ( c) cos s t 4 xω x (c) t (s) (b) xpress the distance the particle travels in ters of t f and t i : Δx ( c) cos s t ( c) ( c) π f 4 π cos s tf 4 π cos s 4 π cos s 4 ti ti ()

27 Oscillations 46 Substitute nuerical values in equation () and evaluate Δx in each of the iven tie intervals to obtain: t f t i Δx (s) (s) (c) Military specifications often call for electronic devices to be able to withstand accelerations of up to ( 98. /s ). o ae sure that your copany s products eet this specification, your anaer has told you to use a shain table, which can vibrate a device at controlled and adjustable frequencies and aplitudes. If a device is placed on the table and ade to oscillate at an aplitude of.5 c, what should you adjust the frequency to in order to test for copliance with the ilitary specification? Picture the Proble We can use the expression for the axiu acceleration of an oscillator to relate the ilitary specification to the copliance frequency. xpress the axiu acceleration of an oscillator: xpress the relationship between the anular frequency and the frequency of the vibrations: a ω ax ω πf Substitute for ω to obtain: aax 4π f f π a ax Substitute nuerical values and evaluate f: f π 98./s.5 Hz 7 [SSM] he position of a particle is iven by x.5 cos πt, where x is in eters and t is in seconds. (a) Find the axiu speed and axiu acceleration of the particle. (b) Find the speed and acceleration of the particle when x.5. Picture the Proble he position of the particle is iven by x cosω t, where.5 and ω π rad/s. he velocity is the tie derivative of the position and the acceleration is the tie derivative of the velocity.

28 46 Chapter 4 (a) he velocity is the tie derivative of the position and the acceleration is the tie derivative of the acceleration: he axiu value of sinωt is + and the iniu value of sinωt is. and ω are positive constants: dx x cosω t v ωsinω t dt dv and a ω cosω t dt v (.5)( π s ) 7.9/s ax ω he axiu value of cosωt is + and the iniu value of cosωt is : a ax ω 5/s (.5)( π s ) (b) Use the Pythaorean identity sin ωt + cos ωt to eliinate t fro the equations for x and v: v ω + x v ω x Substitute nuerical values and evaluate v (.5 ): v (.5 ) ( π rad/s) (.5 ) (.5 ) 6. /s Substitute x for cosωt in the equation for a to obtain: Substitute nuerical values and evaluate a: a ω x a π ( rad/s) (.5 ) 5 /s 8 (a) Show that cos(ωt + δ) can be written as s sin(ωt) + c cos(ωt), and deterine s and c in ters of and δ. (b) Relate c and s to the initial position and velocity of a particle underoin siple haronic otion. Picture the Proble We can use the forula for the cosine of the su of two anles to write x cos(ωt + δ) in the desired for. We can then evaluate x and dx/dt at t to relate c and s to the initial position and velocity of a particle underoin siple haronic otion. (a) pply the trionoetric identity cos ωt + δ cosωt cosδ sinωt sin to obtain: ( ) δ x cos( ωt + δ ) [ cosωt cosδ sinωt sinδ ] sinδ sinωt + cosδ cosωt sinωt + cosωt s provided sinδ and cosδ s c c

29 Oscillations 46 (b) When t : valuate dx/dt: x ( ) cosδ v dx d s c dt dt ω cosωt ω sin ωt s c [ sin ωt + cosωt] c valuate v() to obtain: v( ) ω s ω sinδ Siple Haronic Motion as Related to Circular Motion 9 [SSM] particle oves at a constant speed of 8 c/s in a circle of radius 4 c centered at the oriin. (a) Find the frequency and period of the x coponent of its position. (b) Write an expression for the x coponent of its position as a function of tie t, assuin that the particle is located on the +y-axis at tie t. Picture the Proble We can find the period of the otion fro the tie required for the particle to travel copletely around the circle. he frequency of the otion is the reciprocal of its period and the x-coponent of the particle s position is iven by x cos ( ω t + δ ). We can use the initial position of the particle to deterine the phase constant δ. (a) Use the definition of speed to find the period of the otion: (.4) πr π v.8/s.4.s ecause the frequency and the period are reciprocals of each other: f.4s. Hz (b) xpress the x coponent of the position of the particle: he initial condition on the particle s position is: Substitute in the expression for x to obtain: ( ω t + δ ) cos( π + δ ) x cos ft () x ( ) cos ( ) cosδ π δ Substitute for, ω, and δ in equation () to obtain: x ( 4c) cos (.s ) t + π

30 464 Chapter 4 4 particle oves in a 5-c-radius circle centered at the oriin and copletes. rev every. s. (a) Find the speed of the particle. (b) Find its anular speed ω. (c) Write an equation for the x coponent of the position of the particle as a function of tie t, assuin that the particle is on the x axis at tie t. Picture the Proble We can find the period of the otion fro the tie required for the particle to travel copletely around the circle. he anular frequency of the otion is π ties the reciprocal of its period and the x-coponent of the particle s position is iven by x cos ( ω t + δ ). We can use the initial position of the particle to deterine the phase constantδ. (a) Use the definition of speed to express and evaluate the speed of the particle: ( 5c) πr π v.s c/s (b) he anular speed of the particle is: (c) xpress the x coponent of the position of the particle: he initial condition on the particle s position is: Substitutin for x in equation () yields: π π π ω rad/s. s ( ω + δ ) x cos t () x ( ) cosδ δ cos ( ) π Substitute for, ω, and δ in equation π () to obtain: x ( 5 c) cos s t + π nery in Siple Haronic Motion 4.4- object on a frictionless horizontal surface is attached to one end of a horizontal sprin of force constant 4.5 N/. he other end of the sprin is held stationary. he sprin is stretched c fro equilibriu and released. Find the syste s total echanical enery. Picture the Proble he total echanical enery of the object is iven by, where is the aplitude of the object s otion. tot

31 Oscillations 465 he total echanical enery of the syste is iven by: Substitute nuerical values and evaluate tot : tot ( 4.5N/)(.) J tot 4 Find the total enery of a syste consistin of a.- object on a frictionless horizontal surface oscillatin with an aplitude of c and a frequency of.4 Hz at the end of a horizontal sprin. Picture the Proble he total enery of an oscillatin object can be expressed in ters of its inetic enery as it passes throuh its equilibriu position: tot vax. Its axiu speed, in turn, can be expressed in ters of its anular frequency and the aplitude of its otion. xpress the total enery of the object in ters of its axiu inetic enery: he axiu speed v ax of the oscillatin object is iven by: v v ax ω πf ax Substitute for ax v to obtain: ( ) π f π f Substitute nuerical values and evaluate : (.)(.) π (.4s ).4J 4 [SSM].5- object on a frictionless horizontal surface oscillates at the end of a sprin of force constant 5 N/. he object s axiu speed is 7. c/s. (a) What is the syste s total echanical enery? (b) What is the aplitude of the otion? Picture the Proble he total echanical enery of the oscillatin object can be expressed in ters of its inetic enery as it passes throuh its equilibriu position: tot v ax. Its total enery is also iven by tot. We can equate these expressions to obtain an expression for. (a) xpress the total echanical enery of the object in ters of its axiu inetic enery: v ax

32 466 Chapter 4 Substitute nuerical values and evaluate : (.5 )(.7 /s).68j.675j (b) xpress the total echanical enery of the object in ters of the aplitude of its otion: tot tot Substitute nuerical values and evaluate : (.675J) 5 N/.8c 44.- object on a frictionless horizontal surface is oscillatin on the end of a sprin that has a force constant equal to. N/ and a total echanical enery of.9 J. (a) What is the aplitude of the otion? (b) What is the axiu speed? Picture the Proble he total echanical enery of the oscillatin object can be expressed in ters of its inetic enery as it passes throuh its equilibriu position: tot vax. Its total enery is also iven by tot. We can solve the latter equation to find and solve the forer equation for v ax. (a) xpress the total echanical enery of the object as a function of the aplitude of its otion: tot tot Substitute nuerical values and evaluate : (.9J) N/.c (b) xpress the total echanical enery of the object in ters of its axiu speed: v v tot ax ax tot Substitute nuerical values and evaluate v ax : v (.9J). ax 77 c/s 45 n object on a frictionless horizontal surface oscillates at the end of a sprin with an aplitude of 4.5 c. Its total echanical enery is.4 J. What is the force constant of the sprin?

33 Oscillations 467 Picture the Proble he total echanical enery of the object is iven by. We can solve this equation for the force constant and substitute the tot nuerical data to deterine its value. xpress the total echanical enery of the oscillator as a function of the aplitude of its otion: tot tot Substitute nuerical values and evaluate : (.4J) (.45).4 N/ 46.- object on a frictionless horizontal surface oscillates at the end of a sprin with an aplitude of 8. c. Its axiu acceleration is.5 /s. Find the total echanical enery. Picture the Proble he total echanical enery of the syste is the su of the potential and inetic eneries. hat is, tot x + v. Newton s nd law relates the acceleration to the displaceent. hat is, x a. In addition, when x, v. Use these equations to solve tot in ters of the iven paraeters, and a ax. he total echanical enery is the su of the potential and inetic eneries. We don t now so we need an equation relatin to one or ore of the iven paraeters: he force exerted by the sprin equals the ass of the object ultiplied by its acceleration: tot x + v a x a x When x, a a ax. hus, aax aax Substitute to obtain: a ax tot x + v When x, v. Substitute to obtain: a ax tot + a ax Substitute nuerical values and evaluate tot : tot (.)(.5/s )(.8 ).4J

34 468 Chapter 4 Siple Haronic Motion and Sprins object on a frictionless horizontal surface is attached to a horizontal sprin that has a force constant 4.5 N/. he sprin is stretched c fro equilibriu and released. What are (a) the frequency of the otion, (b) the period, (c) the aplitude, (d) the axiu speed, and (e) the axiu acceleration? (f) When does the object first reach its equilibriu position? What is its acceleration at this tie? Picture the Proble he frequency of the object s otion is iven by f and its period is the reciprocal of f. he axiu velocity and π acceleration of an object executin siple haronic otion are v ω a ω, respectively. ax ax and (a) he frequency of the otion is iven by: f π Substitute nuerical values and evaluate f: f π 4.5N/ Hz 6.9Hz (b) he period of the otion to is the reciprocal of its frequency:.45 s f 6.89s.5s (c) ecause the object is released fro rest after the sprin to which it is attached is stretched c: c (d) he object s axiu speed is iven by: v ω πf ax Substitute nuerical values and evaluate v ax : v ax π ( 6.89s )(.) 4./s 4./s (e) he object s axiu acceleration is iven by: a ax ω ωvax πfv ax

35 Substitute nuerical values and evaluate a ax : a ax π ( 6.89s )( 4./s).9 /s Oscillations 469 (f) he object first reaches its equilibriu when: t (.45s) 6s 4 4 ecause the resultant force actin on the object as it passes throuh its equilibriu position is zero, the acceleration of the object is: a object on a frictionless horizontal surface is attached to one end of a horizontal sprin that has a force constant 7 N/. he sprin is stretched 8. c fro equilibriu and released. What are (a) the frequency of the otion, (b) the period, (c) the aplitude, (d) the axiu speed, and (e) the axiu acceleration? (f) When does the object first reach its equilibriu position? What is its acceleration at this tie? Picture the Proble he frequency of the object s otion is iven by f and its period is the reciprocal of f. he axiu speed and π acceleration of an object executin siple haronic otion are v ω a ω, respectively. ax ax and (a) he frequency of the otion is iven by: f π Substitute nuerical values and evaluate f: f π.88hz 7 N/ Hz (b) he period of the otion is the reciprocal of its frequency: f.88s.5 s.5s (c) ecause the object is released fro rest after the sprin to which it is attached is stretched 8. c: 8.c

36 47 Chapter 4 (d) he object s axiu speed is iven by: v ω πf ax Substitute nuerical values and evaluate v ax : v ax π (.88s )(.8 ).9465 /s.947 /s (e) he object s axiu acceleration is iven by: a ax ω ω vax πfv ax Substitute nuerical values and evaluate a ax : a ax π (.88s )(.9465/s). /s (f) he object first reaches its equilibriu when: t (.5s).s 4 4 ecause the resultant force actin on the object as it passes throuh its equilibriu point is zero, the acceleration of the object is a. 49 [SSM].- object on a frictionless horizontal surface is attached to one end of a horizontal sprin, oscillates with an aplitude of c and a frequency of.4 Hz. (a) What is the force constant of the sprin? (b) What is the period of the otion? (c) What is the axiu speed of the object? (d) What is the axiu acceleration of the object? Picture the Proble (a) he anular frequency of the otion is related to the force constant of the sprin throuh ω. (b) he period of the otion is the reciprocal of its frequency. (c) and (d) he axiu speed and acceleration of an object executin siple haronic otion are v ax ω and a ax ω, respectively. (a) Relate the anular frequency of the otion to the force constant of the sprin: Substitute nuerical values to obtain: ω ω 4π f 4π (.4s ) (.).68N/ 68 N/

37 Oscillations 47 (b) Relate the period of the otion to its frequency:.47s f.4s.4s (c) he axiu speed of the object is iven by: v ω πf ax Substitute nuerical values and evaluate v ax : v ax π (.4s )(.).5/s.5/s (d) he axiu acceleration of the object is iven by: a ax ω 4π f Substitute nuerical values and evaluate a ax : a ax π (.4s ) (.) /s 4 5 n 85.- person steps into a car of ass 4, causin it to sin.5 c on its sprins. If started into vertical oscillation, and assuin no dapin, at what frequency will the car and passener vibrate on these sprins? Picture the Proble We can find the frequency of vibration of the car-andpassener syste usin f, where M is the total ass of the syste. π M he force constant of the sprin can be deterined fro the copressin force and the aount of copression. xpress the frequency of the carand-passener syste: f π M he force constant is iven by: F Δx Δx where is the person s ass. Substitute for in the expression for f to obtain: f π MΔx Substitute nuerical values and evaluate f: f π.6hz ( 85. )( 9.8/s ) ( 485)(.5 )

38 47 Chapter object with an aplitude of.8 c oscillates on a horizontal sprin. he object s axiu acceleration is 6. /s. Find (a) the force constant of the sprin, (b) the frequency of the object, and (c) the period of the otion of the object. Picture the Proble (a) We can relate the force constant to the axiu acceleration by eliinatin ω between ω and a ax ω. (b) We can find the frequency f of the otion by substitutin a ax / for in f. π (c) he period of the otion is the reciprocal of its frequency. ssue that friction is neliible. (a) Relate the anular frequency of the otion to the force constant and the ass of the oscillator: Relate the object s axiu acceleration to its anular frequency and aplitude and solve for the square of the anular frequency: Substitute for ω to obtain: Substitute nuerical values and evaluate : ω ω a ax ω a ax a ax ω () ( 4.5)( 6./s ).8N/.8 (b) Replace ω in equation () by πf and solve for f to obtain: f π a ax Substitute nuerical values and evaluate f: f π 6. /s.8 4.6Hz 4.6 Hz (c) he period of the otion is the reciprocal of its frequency: f 4.6s.4s 5 n object of ass is suspended fro a vertical sprin of force constant 8 N/. When the object is pulled down.5 c fro equilibriu and released fro rest, the object oscillates at 5.5 Hz. (a) Find. (b) Find the aount the sprin is stretched fro its unstressed lenth when the object is in

39 Oscillations 47 equilibriu. (c) Write expressions for the displaceent x, the velocity v x, and the acceleration a x as functions of tie t. Picture the Proble Choose a coordinate syste in which upward is the +y direction. We can find the ass of the object usin ω. We can apply a condition for translational equilibriu to the object when it is at its equilibriu position to deterine the aount the sprin has stretched fro its natural lenth. Finally, we can use the initial conditions to deterine and δ and express x(t) and then differentiate this expression to obtain v x (t) and a x (t). (a) xpress the anular frequency of the syste in ters of the ass of the object fastened to the vertical sprin and solve for the ass of the object: xpress ω in ters of f: Substitute for ω to obtain: ω ω π ω 4 f π 4 f Substitute nuerical values and 8 N/ evaluate : 4π ( 5.5s ).5.57 (b) Lettin Δx represent the aount the sprin is stretched fro its natural lenth when the object is in equilibriu, apply F y to the object when it is in equilibriu: Solve for to obtain: Δx Δx 4π f x 4π f Δ Substitute nuerical values and 9.8/s Δx evaluate Δx: ( 4π 5.5s ) 8. (c) xpress the position of the object as a function of tie: x cos t ( ω + δ )

40 474 Chapter 4 Use the initial conditions x.5 c and v to find δ: δ v tan ω tan x () π valuate ω: ω 8 N/ rad/s Substitute to obtain: x (.5c) cos[ ( 4.56rad/s) t + π ] (.5c) cos[ ( 4.6rad/s) t] Differentiate x(t) to obtain v x : v x ( 86.9c/s) sin[ ( 4.56 rad/s) t] ( 86.4c/s) sin[ ( 4.6rad/s) t] [ t] [ ] Differentiate v(t) to obtain a x : a x ( 9.86 /s ) cos ( 4.56 rad/s) ( 9.9 /s ) cos ( 4.6rad/s) t 5 n object is hun on the end of a vertical sprin and is released fro rest with the sprin unstressed. If the object falls.4 c before first coin to rest, find the period of the resultin oscillatory otion. Picture the Proble Let the syste include the object and the sprin. hen, the net external force actin on the syste is zero. Choose i and apply the conservation of echanical enery to the syste. xpress the period of the otion in ters of its anular frequency: pply conservation of enery to the syste: π () ω i f U + U sprin Substitutin for U and U sprin yields: ( Δ ) Δx + x ω Δx Substitutin for ω in equation () yields: π Δx π Δx Substitute nuerical values and.4 c π evaluate : ( 9.8/s ).6s

41 Oscillations suitcase of ass is hun fro two bunee cords, as shown in Fiure 4-7. ach cord is stretched 5. c when the suitcase is in equilibriu. If the suitcase is pulled down a little and released, what will be its oscillation frequency? Picture the Proble he diara shows the stretched bunee cords supportin the suitcase under equilibriu conditions. We can use eff f to express the frequency π M of the suitcase in ters of the effective sprin constant eff and apply the condition for translational equilibriu to the suitcase to find eff. x M y x x M xpress the frequency of the suitcase oscillator: f eff () π M y pply F to the suitcase to obtain: x + x M or x M or eff x M where eff eff M x Substitute for eff in equation () to obtain: f π x Substitute nuerical values and evaluate f: f π 9.8/s.5.Hz 55.- bloc is suspended fro a sprin. When a sall pebble of ass is placed on the bloc, the sprin stretches an additional 5. c. With the pebble on the bloc, the sprin oscillates with an aplitude of c. (a) What is the frequency of the otion? (b) How lon does the bloc tae to travel fro its lowest point to its hihest point? (c) What is the net force on the pebble when it is at the point of axiu upward displaceent? Picture the Proble (a) he frequency of the otion of the stone and bloc depends on the force constant of the sprin and the ass of the stone plus bloc. he force constant can be deterined fro the equilibriu of the syste when