Chapter 14 Oscillations

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

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

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

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

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

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:

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.

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:

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.

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

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.

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

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?

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.

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

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

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 7.5 +. 5 6 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

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 5 + 4. 4 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

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

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.

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

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.

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 )

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.445 4 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

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 8.5.77 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

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 8 6 4 x (c) - -4-6 -8-4 5 6 7 8 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 ()

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). 9 7. 7. 4. 9 6 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.

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

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

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

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

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?

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

468 Chapter 4 Siple Haronic Motion and Sprins 47.4- 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/.4 6.89 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

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 48 5.- 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/ 5..88 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

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/

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 )

47 Chapter 4 5 4.5- 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

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 )

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/.57 4.56 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

Oscillations 475 54 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

476 Chapter 4 the sprin is stretched additionally by the addition of the stone to the ass. (b) he tie required for the bloc to travel fro its lowest point to its hihest point is half its period. (c) When the bloc is at the point of axiu upward displaceent, it is oentarily at rest and the net force actin on it is its weiht. (a) xpress the frequency of the otion in ters of and tot : f () π tot where tot is the total ass suspended fro the sprin. pply F y to the stone when it is at its equilibriu position: Δy Δy Substitute for in equation () to obtain: f π Δy tot Substitute nuerical values and evaluate f: f π (. )( 9.8 /s ) (.5 )(.5 ).997 Hz.Hz (b) he tie to travel fro its lowest point to its hihest point is one-half its period: (c) When the stone is at a point of axiu upward displaceent: t F f.997s net.9 N ( ).5s (.)( 9.8/s ) 56 Referrin to Proble 69, find the axiu aplitude of oscillation at which the pebble will reain in contact with the bloc. Picture the Proble We can use the axiu acceleration of the oscillator to express a ax in ters of,, and. can be deterined fro the equilibriu of the syste when the sprin is stretched additionally by the addition of the stone to the ass. If the stone is to reain in contact with the bloc, the bloc s axiu downward acceleration ust not exceed.

Oscillations 477 xpress the axiu acceleration of the oscillator in ters of its anular frequency and aplitude of the otion: a ω ax Relate ω to the force constant of the sprin and the ass of the blocplus-stone: ω tot Substitute for ω to obtain: pply F y to the stone when it is at its equilibriu position: a ax () tot Δy Δy where Δy is the additional distance the sprin stretched when the stone was placed on the bloc. Substitute for in equation () to obtain: a ax Δy tot Set a ax and solve for ax : Substitute nuerical values and evaluate ax : Δytot ax.5. tot Δy (.5 ) 5 c ax 57 n object of ass. is attached to the top of a vertical sprin that is anchored to the floor. he unstressed lenth of the sprin is 8. c and the lenth of the sprin when the object is in equilibriu is 5. c. When the object is restin at its equilibriu position, it is iven a sharp downward blow with a haer so that its initial speed is. /s. (a) o what axiu heiht above the floor does the object eventually rise? (b) How lon does it tae for the object to reach its axiu heiht for the first tie? (c) Does the sprin ever becoe unstressed? What iniu initial speed ust be iven to the object for the sprin to be unstressed at soe tie? Picture the Proble (a) he axiu heiht above the floor to which the object rises is the su of its initial distance fro the floor and the aplitude of its otion. We can find the aplitude of its otion by relatin it to the object s axiu speed. (b) ecause the object initially travels downward, it will be three-fourths of the way throuh its cycle when it first reaches its axiu

478 Chapter 4 heiht. (c) We can find the iniu initial speed the object would need to be iven in order for the sprin to becoe uncopressed by applyin conservation of echanical enery. (a) Relate h, the axiu heiht above the floor to which the object rises, to the aplitude of its otion: Relate the axiu speed of the object to the anular frequency and aplitude of its otion and solve for the aplitude: y pply F to the object when it is restin at its equilibriu position to obtain: h + 5. c () v ax ω or, because ω, vax () Δ y Δy Substitute for in equation (): v Δy ax v ax Δy Substitutin for in equation () yields: h v Δy ax + 5. c Substitute nuerical values and evaluate h:. h. /s 9.8 /s 6.7 c + 5. c (b) he tie required for the object to reach its axiu heiht the first tie is three-fourths its period: t 4 xpress the period of the otion of the oscillator: π π π Δy Δy Substitute for in the expression for t to obtain: y t Δ π π 4 Δy

Oscillations 479 Substitute nuerical values and evaluate t: t π. 9.8 /s.6 s (c) ecause h < 8. c, the sprin is never uncopressed. Usin conservation of enery and lettin U be zero 5 c above the floor, relate the heiht to which the object rises, Δy, to its initial inetic enery: ΔK + ΔU + ΔUs or, because K f U i, v Δy + Δy i ( ) ( L y ) i ecause y L : Δ v Δy + ( Δy) ( Δy) y i i and v Δy v Δy i i Substitute nuerical values and evaluate v i : ( )(.c) 77c/s v i 9.8/s hat is, the iniu initial speed that ust be iven to the object for the sprin to be uncopressed at soe tie is 77 c/s 58 winch cable has a cross-sectional area of.5 c and a lenth of.5. Youn s odulus for the cable is 5 GN/. 95- enine bloc is hun fro the end of the cable. (a) y what lenth does the cable stretch? (b) reatin the cable as a siple sprin, what is the oscillation frequency of the enine bloc at the end of the cable? Picture the Proble We can relate the elonation of the cable to the load on it usin the definition of Youn s odulus and use the expression for the frequency of a sprin-ass oscillator to find the oscillation frequency of the enine bloc at the end of the wire. (a) Usin the definition of Youn s odulus, relate the elonation of the cable to the applied stress: Y stress strain F F Δ Δ Y M Y Substitute nuerical values and evaluate Δ : Δ ( 95)( 9.8/s )(.5) (.5c )( 5GN/ ).55.

48 Chapter 4 (b) xpress the oscillation frequency of the wire-enine bloc syste: f π eff M xpress the effective sprin constant of the cable: eff F Δ M Δ Substitute for eff to obtain: f π Δ Substitute nuerical values and evaluate f: f π 9.8/s.55 5Hz Siple Pendulu Systes 59 [SSM] Find the lenth of a siple pendulu if its frequency for sall aplitudes is.75 Hz. Picture the Proble he frequency of a siple pendulu depends on its lenth and on the local ravitational field and is iven by f. π L he frequency of a siple pendulu oscillatin with sall aplitude is iven by: f L π L 4π f Substitute nuerical values and L 9.8 /s evaluate L: ( 4π.75 s ) 44 c 6 Find the lenth of a siple pendulu if its period for sall aplitudes is 5. s. Picture the Proble We can deterine the required lenth of the pendulu fro the expression for the period of a siple pendulu. xpress the period of a siple pendulu: L π L 4π Substitute nuerical values and evaluate L: L ( 5.s) ( 9.8/s ) 6. 4π

Oscillations 48 6 What would be the period of the pendulu in Proble 6 if the pendulu were on the oon, where the acceleration due to ravity is one-sixth that on arth? Picture the Proble We can find the period of the pendulu fro π L oon where oon and L 6.. 6 xpress the period of a siple pendulu on the oon: π L oon Substitute nuerical values and π 6. evaluate : ( 9.8/s ) 6 s 6 If the period of a 7.-c-lon siple pendulu is.68 s, what is the value of at the location of the pendulu? Picture the Proble We can find the value of at the location of the pendulu by solvin the equation π L for and evaluatin it for the iven lenth and period. xpress the period of a siple pendulu where the ravitational field is : L 4π L π Substitute nuerical values and evaluate : 4 π (.7) (.68s) 9.79/s 6 siple pendulu set up in the stairwell of a -story buildin consists of a heavy weiht suspended on a 4.--lon wire. What is the period of oscillation? Picture the Proble We can use pendulu. π L to find the period of this xpress the period of a siple pendulu: π L Substitute nuerical values and evaluate : 4. π 9.8/s.7s

48 Chapter 4 64 Show that the total enery of a siple pendulu underoin oscillations of sall aplitude φ (in radians) is Lφ. Hint: Use the approxiation cosφ φ for sall φ. Picture the Proble he fiure shows the siple pendulu at axiu θ anular displaceent φ. he total enery of the siple pendulu is equal to its initial ravitational potential L cos θ enery. We can apply the definition of ravitational potential enery and use the sall-anle approxiation to show that Lφ. h L U xpress the total enery of the siple pendulu at axiu displaceent: Referrin to the diara, express h in ters of L and φ : ax displaceent U h ( φ ) h L Lcosφ L cos Substitutin for h yields: L[ cosφ ] Fro the power series expansion for cosφ, for φ << : cosφ φ Substitute and siplify to obtain: L[ ( φ )] Lφ 65 [SSM] siple pendulu of lenth L is attached to a assive cart that slides without friction down a plane inclined at anle θ with the horizontal, as shown in Fiure 4-8. Find the period of oscillation for sall oscillations of this pendulu. Picture the Proble he cart accelerates down the rap with a constant acceleration of sinθ. his happens because the cart is uch ore assive than the bob, so the otion of the cart is unaffected by the otion of the bob oscillatin bac and forth. he path of the bob is quite coplex in the reference frae of the rap, but in the reference frae ovin with the cart the path of the bob is uch sipler in this frae the bob oves bac and forth alon a circular arc. o solve this proble we first apply Newton s second law (to the bob) in the inertial reference frae of the rap. hen we transfor to the reference frae

θ Oscillations 48 ovin with the cart in order to exploit the siplicity of the otion in that frae. Draw the free-body diara for the bob. Let φ denote the anle that the strin aes with the noral to the rap. he forces on the bob are the tension force and the force of ravity: θ φ r L θ+φ r pply Newton s nd law to the bob, labelin the acceleration of the bob relative to the rap a R : + a R he acceleration of the bob relative to the rap is equal to the acceleration of the bob relative to the cart plus the acceleration of the cart relative to the rap: Substitute for a a R in R Rearrane ters and label acr as eff, where eff is the acceleration, relative to the cart, of an object in free fall. (If the tension force is set to zero the bob is in free fall.): o find the anitude of eff, first draw the vector addition diara representin the equation. Recall that eff a CR a CR sin θ: a R a C + a + : + ( + ) CR a C a CR + ( a CR ) ac Label a as CR to obtain eff + eff a C () r r eff eff θ a r CR β θ sin θ

θ 484 Chapter 4 Fro the diara, find the anitude of eff. Use the law of cosines: o find the direction of eff, first redraw the vector addition diara as shown: eff + sin θ ( sinθ)cos β ut cos β sinθ, so eff + sin θ sin θ ( sin θ) cos θ hus eff cosθ δ cosθ β θ sin θ Fro the diara find the direction. Use the law of cosines of eff aain and solve for δ: sin θ and so δ θ + cos θ cosθ cosδ o find an equation for the otion of the bob draw the free-body diara for the forces that appear in equation (). Draw the path of the bob in the reference frae ovin with the cart: θ φ L r θ φ r eff ae the tanential coponents of each vector in equation () in the frae of the cart yields. he tanential coponent of the acceleration is equal to the radius of the circle ties the anular acceleration a t rα ( ): d φ eff sinφ L dt where L is the lenth of the strin and d φ is the anular acceleration of the dt bob. he positive tanential direction is counterclocwise.

Oscillations 485 Rearranin this equation yields: For sall oscillations of the pendulu: d φ L + eff sinφ () dt φ <<and sin φ φ Substitutin for () yields: sinφ in equation d φ L + effφ dt or d φ eff + φ dt L () quation () is the equation of otion for siple haronic otion with anular frequency: ω eff L where ω is the anular frequency of the oscillations (and not the anular speed of the bob). he period of this otion is: π ω π L eff (4) Substitute cosθ for eff in equation (4) to obtain: L π eff π L cosθ Rears: Note that, in the liitin case θ, θ 9,. π L and. s 66 he bob at the end of a siple pendulu of lenth L is released fro rest fro an anle φ. (a) Model the pendulu s otion as siple haronic otion and find its speed as it passes throuh φ by usin the sall anle approxiation. (b) Usin the conservation of enery, find this speed exactly for any anle (not just sall anles). (c) Show that your result fro Part (b) arees with the approxiate answer in Part (a) when φ is sall. (d) Find the difference between the approxiate and exact results for φ. rad and L.. (e) Find the difference between the approxiate and exact results for φ. rad and L..

486 Chapter 4 Picture the Proble he fiure shows the siple pendulu at axiu anular displaceent φ. We can express the anular position of the pendulu s bob in ters of its initial anular position and tie and differentiate this expression to find the axiu speed of the bob. We can use conservation of enery to find an exact value for v ax and the approxiation cosφ φ to show that this value reduces to the forer value for sall φ. Lcos φø φ ø h L U (a) Relate the speed of the pendulu s bob to its anular speed: he anular position of the pendulu as a function of tie is iven by: Differentiate this expression to express the anular speed of the pendulu: Substitute in equation () to obtain: dφ v L () dt φ φ cosωt dφ φω sinωt dt v Lφ ω sinωt v ax sinωt Siplify v ax to obtain: Lφ L vax φ L (b) Use conservation of enery to relate the potential enery of the pendulu at point to its inetic enery at point : ΔK + ΔU or, because K U, K U Substitute for K and U : v h xpress h in ters of L and φ : h L( cosφ ) Substitutin for h yields: v L( cosφ )

Solve for v v ax to obtain: L( cosφ ) ax Oscillations 487 v () (c) For φ << : cosφ φ Substitute in equation () to obtain: L( ) L ax φ φ v in areeent with our result in part (a). (d) xpress the difference in the results fro (a) and (b): Substitute for v ax,a and v ax,b and siplify to obtain: Δ v v ax, v Δv φ a L L ax,b L( cosφ ) ( cosφ ) ( φ ) () Substitute nuerical values and evaluate Δv: Δv ( ) /s ( 9.8 /s )(. ). rad ( cos(. rad) ) (e) valuate equation () for φ. rad and L. : Δv ( ). /s ( 9.8 /s )(. ). rad ( cos(. rad) ) *Physical Pendulus 67 [SSM] thin 5.- dis with a -c radius is free to rotate about a fixed horizontal axis perpendicular to the dis and passin throuh its ri. he dis is displaced slihtly fro equilibriu and released. Find the period of the subsequent siple haronic otion. Picture the Proble he period of this physical pendulu is iven by π I MD where I is the oent of inertia of the thin dis about the fixed horizontal axis passin throuh its ri. We can use the parallel-axis theore to express I in ters of the oent of inertia of the dis about an axis throuh its center of ass and the distance fro its center of ass to its pivot point. xpress the period of a physical pendulu: π I MD

488 Chapter 4 Usin the parallel-axis theore, find the oent of inertia of the thin dis about an axis throuh the pivot point: I Ic + MR MR MR + MR Substitutin for I and siplifyin yields: MR π π MR R Substitute nuerical values and evaluate : (.) π 9.8/s ( ).s 68 circular hoop that has a 5-c radius is hun on a narrow horizontal rod and allowed to swin in the plane of the hoop. What is the period of its oscillation, assuin that the aplitude is sall? Picture the Proble he period of this physical pendulu is iven by π I MD where I is the oent of inertia of the circular hoop about an axis throuh its pivot point. We can use the parallel-axis theore to express I in ters of the oent of inertia of the hoop about an axis throuh its center of ass and the distance fro its center of ass to its pivot point. xpress the period of a physical pendulu: π I MD Usin the parallel-axis theore, find the oent of inertia of the circular hoop about an axis throuh the pivot point: I Ic + MR MR + MR MR Substitute for I and siplify to obtain: MR π π MR R Substitute nuerical values and evaluate : (.5) π 9.8/s.s 69.- plane fiure is suspended at a point c fro its center of ass. When it is oscillatin with sall aplitude, the period of oscillation is.6 s. Find the oent of inertia I about an axis perpendicular to the plane of the fiure throuh the pivot point.

Oscillations 489 Picture the Proble he period of a physical pendulu is iven by π I MD where I is its oent of inertia about an axis throuh its pivot point. We can solve this equation for I and evaluate it usin the iven nuerical data. xpress the period of a physical pendulu: Substitute nuerical values and evaluate I: I I MD π I MD 4π (.)( 9.8/s )(.)(.6s).5 4π 7 You have desined a cat door that consists of a square piece of plywood that is. in. thic and 6. in. on a side, and is hined at its top. o ae sure the cat has enouh tie to et throuh it safely, the door should have a natural period of at least. s. Will your desin wor? If not, explain qualitatively what you would do to ae it eet your requireents. Picture the Proble he pictorial representation shows the cat door, of heiht h and width w, pivoted about an axis throuh a-a. We can use I aa' π D to find the period of the door but first ust find I a a'. he diara also shows a differential strip of heiht dy and ass d a distance y fro the axis of rotation of the door. We can interate the differential expression for the oent of inertia of this strip to deterine the oent of inertia of the door. a h w d a' y dy he period of the cat door is iven by: I a' π a () D where D is the distance fro the center of ass of the door to the axis of rotation.

49 Chapter 4 xpress the oent of inertia, about the axis a-a, of the cat door: di a a' y d or, because d ρ dv ρtd ρwtdy, di a a' ρwt y dy Interatin this expression between y and y h yields: I a a' h ρ wt y dy ρwth ecause ρ : I a a' wth h V wht wht Substitutin for D and I a a' in h π π equation () yields: ( h) h Substitute nuerical values and evaluate :.54 c 6. in in π.64 s 9.8/s ( ) hus the door s period is too short. he only way to increase it is to increase the heiht of the door. 7 You are iven a eterstic and ased to drill a narrow hole throuh it so that, when the stic is pivoted about a horizontal axis throuh the hole, the period of the pendulu will be a iniu. Where should you drill the hole? Picture the Proble Let x be the distance of the pivot fro the center of the eter stic, the ass of the eter stic, and L its lenth. We ll express the period of the eter stic as a function of the distance x and then differentiate this expression with respect to x to deterine where the hole should be drilled to iniize the period. xpress the period of a physical pendulu: I π () MD xpress the oent of inertia of the eter stic about an axis throuh its center of ass: I c L

Oscillations 49 Usin the parallel-axis theore, express the oent of inertia of the eter stic about an axis throuh the pivot point: I I c + x L + x Substitute in equation () and siplify to obtain: π L + x x π L + x x he condition for an extree value d of is that L + x. dx x valuate this derivative to obtain: 4x x L L + x x x L Notin that only the positive solution is physically eaninful, solve for x: x L c 8.9c he hole should be drilled at a distance: d 5. c 8.9 c.c fro the center of the eter stic. 7 Fiure 4-9 shows a unifor dis of radius R.8, a ass of 6., and a sall hole a distance d fro the dis s center that can serve as a pivot point. (a) What should be the distance d so that the period of this physical pendulu is.5 s? (b) What should be the distance d so that this physical pendulu will have the shortest possible period? What is this shortest possible period? Picture the Proble (a) Let represent the ass and R the radius of the unifor dis. he dis is a physical pendulu. We ll use the expression π I d for the period of a physical pendulu. o find I we use the parallel-axis theore ( I I c + d ). (b) he period is a iniu when d dx, where, to avoid notational difficulties, we have substituted x for d. (a) xpress the period of a physical pendulu: π I d

49 Chapter 4 Usin the parallel-axis theore ( I I c + d ), relate the oent of inertia about the axis throuh the hole to the oent of inertia I c about the parallel axis throuh the center of ass. Obtain I c fro able 9-: I I c + d R + d Substitutin for I yields: π π R + d d R + d d () Square both sides of this equation, siplify, and substitute nuerical values to obtain: Use the quadratic forula or your raphin calculator to obtain: d 4π or d R d + (.55) d +. d.8 4c he second root, d., is reater than R, so it is too lare to be physically eaninful. (b) he period is related to the distance d by equation (). will be a iniu when ( R + d ) d is a iniu. Set the derivative of this expression equal to zero to find relative axia and inia. We ll replace d with x to avoid the notational challene of differentiatin with respect to d. valuatin d R + x yields: dx x d ( R + d ) d R + d d where we have chaned x bac to d. ( ) Solvin for d yields: d R

Oscillations 49 valuate equation () with d R to obtain an expression for the shortest possible period: R + R π π R R Substitute nuerical values and evaluate : (.8) π 9.8/s.s Rears: We ve shown that d R corresponds to an extree value; that is, to a axiu, a iniu, or an inflection point. o verify that this value of d corresponds to a iniu, we can either () show that d /dx evaluated at x R (where x d) is positive, or () raph as a function of d and note that the raph is a iniu at d R. 7 [SSM] Points P and P on a plane object (Fiure 4-) are distances h and h, respectively, fro the center of ass. he object oscillates with the sae period when it is free to rotate about an axis throuh P and when it is free to rotate about an axis throuh P. oth of these axes are perpendicular to the plane of the object. Show that h + h /(4π), where h h. Picture the Proble We can use the equation for the period of a physical pendulu and the parallel-axis theore to show that h + h /4π. xpress the period of the physical pendulu: π I d Usin the parallel-axis theore, relate the oent of inertia about an axis throuh P to the oent of inertia about an axis throuh the plane s center of ass: I I c + h Substitute for I to obtain: π I + h h c Square both sides of this equation Ic + h and rearrane ters to obtain: 4π h () ecause the period of oscillation is the sae for point P : I h c + h I h c + h

494 Chapter 4 Cobinin lie ters yields: h h I c h ( h ) Provided h h : I c hh Substitute in equation () and hh siplify to obtain: + h 4π h 4π h + h 74 physical pendulu consists of a spherical bob of radius r and ass suspended fro a riid rod of neliible ass as in Fiure 4-. he distance fro the center of the sphere to the point of support is L. When r is uch less than L, such a pendulu is often treated as a siple pendulu of lenth L. + r (a) Show that the period for sall oscillations is iven by where 5L π L / is the period of a siple pendulu of lenth L. (b) Show that when r is uch saller than L, the period can be approxiated by ( + r /5L ). (c) If L. and r. c, find the error in the calculated value when the approxiation is used for the period. How lare ust be the radius of the bob for the error to be. percent? Picture the Proble (a) We can find the period of the physical pendulu in ters of the period of a siple pendulu by startin with π I L and applyin the parallel-axis theore. (b) Perforin a binoial expansion (with r << L) on the radicand of our expression for will lead to ( + r /5L ). (a) xpress the period of the physical pendulu: π I L Usin the parallel-axis theore, relate the oent of inertia of the pendulu about an axis throuh its center of ass to its oent of inertia about an axis throuh its point of support: I I c 5 r + L + L Substitute for I and siplify to obtain: 5 r + L L r r π π + + L 5L 5L

Oscillations 495 r (b) xpandin 5 + L binoially yields: r + 5L r + 5L provided r << L r r + 5L + 8 5L + hiher - order ters Substitute in our result fro Part (a) to obtain: r + 5L (c) xpress the fractional error when the approxiation is used for this pendulu: Δ r + 5L r 5L Substitute nuerical values and evaluate Δ/: Δ (.c) 5( c).8% For an error of.%: r 5L. r L. 5 Substitute the nuerical value of L and evaluate r to obtain: r ( c).5.4c 75 Fiure 4- shows the pendulu of a cloc in your randother s house. he unifor rod of lenth L. has a ass.8. ttached to the rod is a unifor dis of ass M. and radius.5. he cloc is constructed to eep perfect tie if the period of the pendulu is exactly.5 s. (a) What should the distance d be so that the period of this pendulu is.5 s? (b) Suppose that the pendulu cloc loses 5. in/d. o ae sure that your randother won t be late for her quiltin parties, you decide to adjust the cloc bac to its proper period. How far and in what direction should you ove the dis to ensure that the cloc will eep perfect tie? Picture the Proble (a) he period of this physical pendulu is iven by π I MD. We can express its period as a function of the distance d by usin the definition of the center of ass of the pendulu to find D in ters of d and the parallel-axis theore to express I in ters of d. Solvin the resultin

496 Chapter 4 quadratic equation yields d. (b) ecause the cloc is losin 5 inutes per day, one would reposition the dis so that the cloc runs faster; that is, so the pendulu has a shorter period. We can deterine the appropriate correction to ae in the position of the dis by relatin the fractional tie loss to the fractional chane in its position. (a) xpress the period of a physical pendulu: π tot I x c Solvin for I x c yields: I x c tot () 4π xpress the oent of inertia of the physical pendulu, about an axis throuh the pivot point, as a function of d: Substitute nuerical values and evaluate I: Locate the center of ass of the physical pendulu relative to the pivot point: I Ic + Md L + Mr + I + Md (.8 )(.) (. )(.5 ) + (. ) + (. ) d.8 (.8 )(.) + (. ) xc. and x.4. 6d c + d d Substitute in equation () to obtain:.8 +.4 +.6d (. ) d ( 9.8/s )(. ) ( ) 4π.49698 /s () Settin.5 s and solvin for d yields: (b) here are 44 inutes per day. If the cloc loses 5. in per day, then the period of the cloc is related to the perfect period of the cloc by: d.6574 where we have ept ore than three sinificant fiures for use in Part (b). 45 44 45 44 perfect perfect where perfect.5 s.

Substitute nuerical values and evaluate : Substitute.5 s in equation () and solve for d to obtain: d 44 45.44 (.5s).5s Oscillations 497 Substitute.5 s in equation () and solve for d to obtain: xpress the distance the dis needs to be oved upward to correct the period: d'.786 Δd d d'.c.44.786 Daped Oscillations 76.- object oscillates with an initial aplitude of. c. he force constant of the sprin is 4 N/. Find (a) the period, and (b) the total initial enery. (c) If the enery decreases by. percent per period, find the linear dapin constant b and the Q factor. Picture the Proble (a) We can find the period of the oscillator fro π. (b) he total initial enery of the sprin-object syste is iven by. (c) he Q factor can be found fro its definition Q π Δ and the dapin constant fro Q ω. ( ) cycle b (a) he period of the oscillator is iven by: π Substitute nuerical values and evaluate :. π 4 N/.444s (b) Relate the initial enery of the oscillator to its aplitude: Substitute nuerical values and evaluate : ( 4 N/)(. ).8 J (c) Relate the fractional rate at which the enery decreases to the Q value and evaluate Q: Q π ( Δ ) cycle π. 68

498 Chapter 4 xpress the Q value in ters of b: Solve for the dapin constant b: ω Q b ω π π b Q Q π Q Q Substitute nuerical values and evaluate b: b (.)( 4 N/) 68.45/s 77 [SSM] Show that the ratio of the aplitudes for two successive oscillations is constant for a linearly daped oscillator. t τ Picture the Proble he aplitude of the oscillation at tie t is () t e where τ /b is the decay constant. We can express the aplitudes one period apart and then show that their ratio is constant. Relate the aplitude of a iven oscillation pea to the tie at which the pea occurs: xpress the aplitude of the oscillation pea at t t + : xpress the ratio of these consecutive peas: t τ ( ) e t ( t+ ) τ ( + ) e t t ( t) e ( t+ ) ( + ) e t τ τ constant e τ 78 n oscillator has a period of. s. Its aplitude decreases by 5. percent durin each cycle. (a) y how uch does its echanical enery decrease durin each cycle? (b) What is the tie constant τ? (c) What is the Q factor? Picture the Proble (a) We can relate the fractional chane in the enery of the oscillator each cycle to the fractional chane in its aplitude. (b) and (c) oth the Q value and the decay constant τ can be found fro their definitions. (a) Relate the enery of the oscillator to its aplitude:

Oscillations 499 ae the differential of this relationship to obtain: Divide the second of these equations by the first and siplify to obtain: d d d d d pproxiate d and d by Δ and Δ and evaluate Δ/: Δ (5.%).% (b) For sall dapin: Δ τ and τ Δ.s..s (c) he Q factor is iven by: Substitute nuerical values and evaluate Q: π Q ω τ τ π Q.s (.s) 6. 8 79 linearly daped oscillator has a Q factor of. (a) y what fraction does the enery decrease durin each cycle? (b) Use quation 4-4 to find the percentae difference between ω and ω. Hint: Use the approxiation + + for sall x. ( ) x x Picture the Proble We can use the physical interpretation of Q for sall dapin ( ) π Q to find the fractional decrease in the enery of the Δ cycle oscillator each cycle. (a) xpress the fractional decrease in enery each cycle as a function of the Q factor and evaluate Δ : Δ π Q π.4. (b) he percentae difference between ω and ω is iven by: ω' ω ω' ω ω

5 Chapter 4 Usin the definition of the Q factor, use quation 4-4 to express the ratio of ω to ω as a function of Q: ω ω ' b 4 ω 4Q Substitutin for ω' yields: ω' ω ω ω 4 Q Use the approxiation + + ( ) x x for x << to obtain: 4Q 8Q Substitutin for 4 Q siplifyin yields: and ω' ω ω 8Q 8Q Substitute the nuerical value of Q ω' ω and evaluate ω ω' ω ω : 8( ). 8 linearly daped ass sprin syste oscillates at Hz. he tie constant of the syste is. s. t t the aplitude of oscillation is 6. c and the enery of the oscillatin syste is 6 J. (a) What are the aplitudes of oscillation at t. s and t 4. s? (b) How uch enery is dissipated in the first -s interval and in the second -s interval? Picture the Proble he enery of the sprin-and-ass oscillator varies with t τ tie accordin to e and its enery is proportional to the square of the aplitude. % (a) Usin e t τ and solve for the aplitude as a function of tie:, xpress the aplitude of the oscillations as a function of tie: valuate the aplitude when t. s: e iply that Hence e t τ and t τ t 4 s ( 6. c) e e t τ.s 4.s (. s) ( 6.c) e.6c

Oscillations 5 valuate the aplitude when t 4. s: (b) xpress the enery of the syste at t, t. s, and t 4. s: he enery dissipated in the first. s is equal to the neative of the chane in echanical enery: he enery dissipated in the second.-s interval is: 4.s 4.s ( 4. s) ( 6.c) e.c.s ( ) e..s (. s) e e 4..s ( 4. s) e e Δ Δ. s. s 4. s. s. s ( e e ) ( 6 J)( e ) 8 J 4. s. s. s. s ( e e ) ( 6 J) e ( e ) 4 J 8 [SSM] Seisoloists and eophysicists have deterined that the vibratin arth has a resonance period of 54 in and a Q factor of about 4. fter a lare earthquae, arth will rin (continue to vibrate) for up to onths. (a) Find the percentae of the enery of vibration lost to dapin forces durin each cycle. (b) Show that after n periods the vibrational enery is iven by n (.984) n, where is the oriinal enery. (c) If the oriinal enery of vibration of an earthquae is, what is the enery after. d? Picture the Proble (a) We can find the fractional loss of enery per cycle fro the physical interpretation of Q for sall dapin. (b) We will also find a eneral expression for the earth s vibrational enery as a function of the nuber of cycles it has copleted. (c) We can then solve this equation for the earth s vibrational enery after any nuber of days. (a) xpress the fractional chane in enery as a function of Q: Δ π π Q 4.57% (b) xpress the enery of the daped oscillator after one cycle: xpress the enery after two cycles: Δ Δ Δ

5 Chapter 4 Generalizin to n cycles: n Δ n (.57) n (.984) n (c) xpress. d in ters of the nuber of cycles; that is, the nuber of vibrations the earth will have experienced: 4h 6.d.d d h 88 in 54in 5. valuate ( d): 5. ( d) (.984). 4 8 pendulu that is used in your physics laboratory experient has a lenth of 75 c and a copact bob with a ass equal to 5. o start the bob oscillatin, you place a fan next to it that blows a horizontal strea of air on the bob. With the fan on, the bob is in equilibriu when the pendulu is displaced by an anle of 5.º fro the vertical. he speed of the air fro the fan is 7. /s. You turn the fan off, and allow the pendulu to oscillate. (a) ssuin that the dra force due to the air is of the for bv, predict the decay tie constant τ for this pendulu. (b) How lon will it tae for the pendulu s aplitude to reach.º? Picture the Proble he diara shows ) the pendulu bob displaced throuh an anle θ and held in equilibriu by the force exerted on it by the air fro the fan and ) the bob acceleratin, under the influence of ravity, tension force, and dra force, toward its equilibriu position. We can apply Newton s nd law to the bob to obtain the equation of otion of the daped pendulu and then use its solution to find the decay tie constant and the tie required for the aplitude of oscillation to decay to. θ θ l r r bv r r r F r fan (a) pply τ Iα to the pendulu to obtain: d θ sinθ + Fd I dt

Oscillations 5 xpress the oent of inertia of the pendulu about an axis throuh its point of support: Substitute for I and F d to obtain: ecause θ << and v ω dθ/dt: he solution to this second-order hooeneous differential equation with constant coefficients is: pply F a to the bob when it is at its axiu anular displaceent to obtain: I d θ + bv + sinθ dt d θ dθ + b + θ dt dt or d θ dθ + b + θ dt dt ( ω't + δ ) τ θ θ e t cos () where θ is the axiu aplitude, τ /b is the tie constant, and the ω' ω b ω. frequency ( ) F x F fan sinθ and F y cosθ Divide the x equation by the y Ffan sinθ equation to obtain: tanθ cosθ or F fan tanθ When the bob is in equilibriu, the dra force on it equals F fan : bv tanθ τ b v tanθ Substitute nuerical values and τ evaluate τ : ( 9.8/s ) 7. /s 8.6s tan 5. 8.s (b) Fro equation (), the anular aplitude of the otion is iven by: When the aplitude has decreased to. : θ θ. τ e t τ 5. e t t τ or e. ae the natural loarith of both t τ ln sides of the equation to obtain: (.) t τ ln(.)

54 Chapter 4 Substitute the nuerical value of τ and evaluate t: ( 8.6s) ln(.) 6 s t 8 [SSM] You are in chare of onitorin the viscosity of oils at a anufacturin plant and you deterine the viscosity of an oil by usin the followin ethod: he viscosity of a fluid can be easured by deterinin the decay tie of oscillations for an oscillator that has nown properties and operates while iersed in the fluid. s lon as the speed of the oscillator throuh the fluid is relatively sall, so that turbulence is not a factor, the dra force of the fluid on a sphere is proportional to the sphere s speed relative to the fluid: Fd 6πaηv, where η is the viscosity of the fluid and a is the sphere s radius. hus, the constant b is iven by 6 π aη. Suppose your apparatus consists of a stiff sprin that has a force constant equal to 5 N/c and a old sphere (radius 6. c) hanin on the sprin. (a) What is the viscosity of an oil do you easure if the decay tie for this syste is.8 s? (b) What is the Q factor for your syste? Picture the Proble (a) he decay tie for a daped oscillator (with speeddependent dapin) syste is defined as the ratio of the ass of the oscillator to the coefficient of v in the dapin force expression. (b) he Q factor is the product of the resonance frequency and the dapin tie. (a) Fro Fd 6πaηv and F bv it follows that: d, ecause τ b, we can substitute for b to obtain: Substitutin yields: ρv and siplifyin Substitute nuerical values and evaluate η (see able - for the density of old): (b) he Q factor is the product of the resonance frequency and the dapin tie: b b 6πaη η 6π a η 6πaτ ρv η 6πaτ η πa ρ ρ 6πaτ 9τ 4 a (.6 ) ( 9. / ) 5.5 Pa s 9 (.8 s) Q ωτ τ τ τ 4 ρv πa ρ

Oscillations 55 Substitute nuerical values and evaluate Q: Q 4π 5 N c c ( ) ( ) (.8 s).6 9. / 5 Driven Oscillations and Resonance 84 linearly daped oscillator loses. percent of its enery durin each cycle. (a) What is its Q factor? (b) If its resonance frequency is Hz, what is the width of the resonance curve Δω when the oscillator is driven? Picture the Proble (a) We can use the physical interpretation of Q for sall dapin to find the Q factor for this daped oscillator. (b) he width of the resonance curve depends on the Q factor accordin to Δ ω ω Q. (a) Usin the physical interpretation of Q for sall dapin, relate Q to the fractional loss of enery of the daped oscillator per cycle: Q π ( Δ ) cycle valuate this expression for π Δ. : Q 4. cycle ( ) % (b) Relate the width of the resonance curve to the Q value of the oscillatory syste: Substitute nuerical values and evaluate Δω: ω πf Δ ω Q Q Δω π 4 - ( s ) 6.rad/s 85 Find the resonance frequency for each of the three systes shown in Fiure 4-. Picture the Proble he resonant frequency of a vibratin syste depends on the ass of the syste and on its stiffness constant accordin to f or, in the case of a siple pendulu oscillatin with sallaplitude vibrations, f. π π L

56 Chapter 4 (a) For this sprin-and-ass oscillator we have: f π 4. N/.Hz (b) For this sprin-and-ass oscillator we have: f π 8. N/ 5 Hz (c) For this siple pendulu we have: f π 9.8/s..5Hz 86 daped oscillator loses.5 percent of its enery durin each cycle. (a) How any cycles elapse before half of its oriinal enery is dissipated? (b) What is its Q factor? (c) If the natural frequency is Hz, what is the width of the resonance curve when the oscillator is driven by a sinusoidal force? Picture the Proble (a) We ll find a eneral expression for the daped oscillator s enery as a function of the nuber of cycles it has copleted. We can then solve this equation for the nuber of cycles correspondin to the loss of half the oscillator s enery. (b) he Q factor is related to the fractional enery loss per cycle throuh Δ π Q. (c) he width of the resonance curve is Δ ω ω Q where ω is the oscillator s natural anular frequency. (a) xpress the enery of the daped oscillator after one cycle: xpress the enery after two cycles: Δ Δ Δ Generalizin to n cycles: n Δ n Substitutin nuerical values yields: ( ) n.5. 5 or.5 (.965) n Solvin for n yields: n ln.5 ln.965 9.5 coplete cycles.

Oscillations 57 (b) pply the physical interpretation of Q for sall dapin to obtain: Q Δ π π.5 8 (c) he width of the resonance curve is iven by: Substitute nuerical values and evaluate Δ ω : ( Δ ) ω πf Δω f Q π ( Δ ) ( Hz)(.5).5rad/s Δ ω 87 [SSM].- object oscillates on a sprin of force constant 4 N/. he linear dapin constant has a value of. /s. he syste is driven by a sinusoidal force of axiu value. N and anular frequency. rad/s. (a) What is the aplitude of the oscillations? (b) If the drivin frequency is varied, at what frequency will resonance occur? (c) What is the aplitude of oscillation at resonance? (d) What is the width of the resonance curve Δω? Picture the Proble (a) he aplitude of the daped oscillations is related to the dapin constant, ass of the syste, the aplitude of the drivin force, and F the natural and drivin frequencies throuh. ( ω ω ) + b ω (b) Resonance occurs when ω ω. (c) t resonance, the aplitude of the oscillations is F b ω. (d) he width of the resonance curve is related to the dapin constant and the ass of the syste accordin to Δω b. (a) xpress the aplitude of the oscillations as a function of the drivin frequency: ecause ω : ( ω ω ) + b ω F F ω + b ω Substitute nuerical values and evaluate : 4 N/.. N (. ) (. rad/s) + (. /s) (. rad/s) 4.98 c

58 Chapter 4 (b) Resonance occurs when: ω ω Substitute nuerical values and evaluate ω: ω 4 N/. 4.rad/s 4.4 rad/s (c) he aplitude of the otion at F resonance is iven by: b ω Substitute nuerical values and evaluate : ( ). /s ( 4.4 rad/s ) 5.4c. N (d) he width of the resonance curve is: b./s Δ ω..rad/s 88 Suppose you have the sae apparatus described in Proble 74 and the sae old sphere hanin fro a weaer sprin that has a force constant of only 5. N/c. You have studied the viscosity of ethylene lycol with this device, and found that ethylene lycol has a viscosity value of 9.9 Pa s. Now you decide to drive this syste with an external oscillatin force. (a) If the anitude of the drivin force for the device is. N and the device is driven at resonance, how lare would be the aplitude of the resultin oscillation? (b) If the syste were not driven, but were allowed to oscillate, what percentae of its enery would it lose per cycle? Picture the Proble (a) he aplitude of the steady-state oscillations when the syste is in resonance is iven by F bω. (b) We can relate the fractional enery loss to the Q value of the oscillator. (a) he aplitude of the steady-state oscillations when the syste is in resonance is iven by: ecause b 6πaη, and ω : F bω F F 6πaηω 6πaη

Oscillations 59 Substitutin yields: ρv and siplifyin F 6πaη ρv F 6πaη 4 πa ρ F πη πaρ Substitute nuerical values and evaluate :. N π ( 9.9 Pa s) π (.6 )( 9. / ) ( 5. N/c) 4.5 c (b) his is a very wealy daped syste and so we can relate the fractional enery loss per cycle to the syste s Q value: Q π ( Δ ) cycle ω τ ecause τ : b 6πaη π ( Δ ) 6πaη cycle ω Substitutin for and ω and siplifyin yields: π ( Δ ) cycle ρv 6πaη 4 πa ρ 6πaη a ρ 9η Solve for ( Δ ) cycle to obtain: 9πη Δ cycle a ρ ( ) ) cycle Substitute nuerical values and evaluate ( Δ : ( Δ ) General Probles 9π ( 9.9 Pa s) 4 cycle 5.7 (.6 ) ( 9. / ) 7.5 5. N/c 89 particle s displaceent fro equilibriu is iven by x(t).4 cos(.t + π/4), where x is in eters and t is in seconds. (a) Find the frequency and period of its otion. (b) Find an expression for the speed of the particle as a function of tie. (c) What is its axiu speed?

5 Chapter 4 Picture the Proble (a) he particle s displaceent is of the for x cos ω t + δ. hus, we have.4, ω. rad/s, and δ π/4. We can ( ) find the frequency of the otion fro its anular frequency and the period fro the frequency. (b) he particle s velocity is the tie derivative of its sin ωt +δ. displaceent. (c) he particle s axiu speed occurs when ( ) (a) he particle s displaceent fro equilibriu is of the for x cos ( ω t +δ ). Coparin this to the iven equation we see that: ω. rad/s and so ω. rad/s f π π.48hz.477 Hz he period of the particle s otion is the reciprocal of its frequency:.9 s f.477s.s (b) Differentiate x cos ( ω t +δ ) with respect to tie to obtain an expression for the particle s velocity: v x dx d dt dt ωsin [ cos( ωt + δ )] ( ωt + δ ) Substitutin for, ω, and δ yields: v x π 4 (. rad/s)(.4 ) sin (. rad/s) t + (. /s) sin (. rad/s) t + π 4 (c) he particle s axiu speed sin ωt +δ : occurs when ( ) v x (. /s)( ). /s ax 9 n astronaut arrives at a new planet, and ets out his siple device to deterine the ravitational acceleration there. Prior to his arrival, he noted that the radius of the planet was 755. If his.5--lon pendulu has a period of. s, what is the ass of the planet? Picture the Proble We can apply Newton s nd law and the law of ravity to an object at the surface of the new planet to obtain an expression for the ass of the planet as a function of the acceleration due to ravity at its surface. We can use the period of the astronaut s pendulu to obtain an expression for the acceleration of ravity a at the surface of the new planet. pply Newton s nd law and the law of ravity to an object of ass at the surface of the planet: GM R planet planet a M planet a R planet G

he period of the astronaut s siple pendulu is related to the ravitational field a at the surface of the new planet: L 4π L π a a Oscillations 5 Substitutin for a and siplifyin 4π R yields: M planet G Substitute nuerical values and evaluate M planet : M π ( 755 ) (.5 ) ( 6.67 N / )(. s) planet 4 5 planet.7 9 pendulu cloc eeps perfect tie on arth s surface. In which case will the error be reater: if the cloc is placed in a ine of depth h or if the cloc is elevated to a heiht h? Prove your answer and assue h << R. Picture the Proble ssue that the density of arth ρ is constant and let represent the ass of the cloc. We can decide the question of where the cloc is ore accurate by applyin the law of ravitation to the cloc at a depth h below/above the surface of arth and at arth s surface and expressin the ratios of the acceleration due to ravity below/above the surface of arth to its value at the surface of arth. L xpress the ravitational force actin on the cloc when it is at a depth h in a ine: xpress the ravitational force actin on the cloc at the surface of arth: ' GM' ( R ) h where M is the ass between the location of the cloc and the center of arth. GM R Divide the first of these equations by the second and siplify to obtain: ' GM' ( R h) GM R M' M ( R h) R xpress M : ( ) M' ρ V' πρ 4 R h xpress M : M ρ V 4 πρr

5 Chapter 4 Substitute for M and M to obtain: 4 ' πρ( R h) 4 πρ R ( R h) R Siplifyin and solvin for yields: xpress the ravitational force actin on the cloc when it is at an elevation h: xpress the ravitational force actin on the cloc at the surface of arth: R h h ' R R or h ' () R '' GM ( R + h) GM R Divide the first of these equations by GM the second and siplify to obtain: '' ( R + h) GM R ( R h) R Factorin yields: R fro the denoinator Solve for to obtain: '' h + R h '' + () R Coparin equations () and (), we see that ' is closer to than is ''. hus the error is reater if the cloc is elevated. 9 Fiure 4-4 shows a pendulu of lenth L with a bob of ass M. he bob is attached to a sprin that has a force constant. When the bob is directly below the pendulu support, the sprin is unstressed. (a) Derive an expression for the period of this oscillatin syste for sall-aplitude vibrations. (b) Suppose that M. and L is such that in the absence of the sprin the period is. s. What is the force constant if the period of the oscillatin syste is. s?

Oscillations 5 Picture the Proble he fiure shows this syste when it has an anular displaceent θ. he period of the syste is related to its anular frequency accordin to π/ω. We can find the equation of otion of the syste by applyin Newton s nd law. y writin this equation in ters of θ and usin a sall-anle approxiation, we ll find an expression for ω that we can use to express. θ L r xˆ i θ y M r θ x (a) he period of the syste in ters of its anular frequency is iven by: π () ω pply F a to the bob: Fx x sin θ Ma and F y cosθ M x liinate between the two equations to obtain: Notin that x Lθ and d θ a x Lα L, eliinate the dt variable x in favor of θ : For θ <<, tanθ θ : x M tan θ Ma x d θ ML Lθ M tanθ dt d θ ML Lθ Mθ dt ( L + M)θ or d θ θ ω θ + dt M L where ω + M L

54 Chapter 4 Substitute in equation () to obtain: π M + L (b) When (no sprin),. s, and M. we have:. s π () L With the sprin present and. s we have:. s π + L () Solvin equations () and () siultaneously yields: 9.6 N/ 9 [SSM] bloc that has a ass equal to is supported fro below by a frictionless horizontal surface. he bloc, which is attached to the end of a horizontal sprin with a force constant, oscillates with an aplitude. When the sprin is at its reatest extension and the bloc is instantaneously at rest, a second bloc of ass is placed on top of it. (a) What is the sallest value for the coefficient of static friction μ s such that the second object does not slip on the first? (b) xplain how the total echanical enery, the aplitude, the anular frequency ω, and the period of the syste are affected by the placin of on, assuin that the coefficient of friction is reat enouh to prevent slippae. Picture the Proble pplyin Newton s nd law to the first object as it is about to slip will allow us to express μ s in ters of the axiu acceleration of the syste which, in turn, depends on the aplitude and anular frequency of the oscillatory otion. (a) pply Fx ax to the second object as it is about to slip: pply Fy to the second object: Use fs, ax μsfn to eliinate f s, ax and Fn between the two equations and solve for μ s : f s, ax aax F n s μ a ax a μ ax s

Oscillations 55 Relate the axiu acceleration of the oscillator to its aplitude and anular frequency and substitute for ω : a ax ω + Finally, substitute for a ax to obtain: μ s ( + ) (b) is unchaned. is unchaned because. ω is reduced and is increased by increasin the total ass of the syste. 94 - box hans fro the ceilin of a roosuspended fro a sprin with a force constant of 5 N/. he unstressed lenth of the sprin is.5. (a) Find the equilibriu position of the box. (b) n identical sprin is stretched and attached to the ceilin and box and is parallel with the first sprin. Find the frequency of the oscillations when the box is released. (c) What is the new equilibriu position of the box once it coes to rest? Picture the Proble he diara shows the box hanin fro the stretched sprin and the free-body diara when the box is in equilibriu. We can apply F y to the box to derive an expression for x. In (b) and (c), we can proceed siilarly to obtain expressions for the effective force constant, the new equilibriu position of the box, and frequency of oscillations when the box is released. x x y (x x ) (a) pply Fy obtain: to the box to Substitute nuerical values and evaluate x: ( x x ) x + x x ( )( 9.8/s ) 5 N/.46 +.5

56 Chapter 4 (b) Draw the free-body diara for the bloc with the two sprins exertin equal upward forces on it: y (x x ) (x x ) x x pply Fy obtain: to the box to ( x x ) + ( x x ) or ( x x ) () eff eff where When the box is displaced fro this equilibriu position and released, its otion is siple haronic otion and its frequency is iven by: ω eff Substitute nuerical values and evaluate ω: ω ( 5 N/).6 rad/s (c) Solve equation () for x to obtain: Substitute nuerical values and evaluate x: x + x x ( )( 9.8/s ).48 ( 5 N/) +.5 95 he acceleration due to ravity varies with eoraphical location because of arth s rotation and because arth is not exactly spherical. his was first discovered in the seventeenth century, when it was noted that a pendulu cloc carefully adjusted to eep correct tie in Paris lost about 9 s/d near the equator. (a) Show by usin the differential approxiation that a sall chane in the acceleration of ravity Δ produces a sall chane in the period Δ of a pendulu iven by Δ / Δ /. (b) How lare a chane in is needed to account for a 9 s/d chane in the period?

Oscillations 57 Picture the Proble We ll differentiate the expression for the period of siple L pendulu π with respect to, separate the variables, and use a Δ Δ differential approxiation to establish that. (a) xpress the period of a siple pendulu in ters of its lenth and the local value of the acceleration due to ravity: π L Differentiate this expression with respect to to obtain: Separate the variables to obtain: d d d d d [ π L ] d π L For Δ << we can approxiate d and d by Δ and Δ: Δ Δ (b) Solve the equation in Part (a) for Δ: Δ Δ Substitute nuerical values and evaluate Δ for a 9 s/d chane in the period: s d h ( ) 9.c/s Δ 9.8/s d 4h 6s 96 sall bloc that has a ass equal to rests on a piston that is vibratin vertically with siple haronic otion described by the forula y sin ωt. (a) Show that the bloc will leave the piston if ω >. (b) If ω and 5 c, at what tie will the bloc leave the piston? Picture the Proble If the displaceent of the bloc is y sin ωt, its acceleration is a ω sinωt. (a) t axiu upward extension, the bloc is oentarily at rest. Its downward acceleration is. he downward acceleration of the piston is ω. herefore, if ω >, the bloc will separate fro the piston.

58 Chapter 4 (b) xpress the acceleration of the sall bloc: a ω sinωt For ω and 5 c: a sinωt Solvin for t yields: t sin ω sin Substitute nuerical values and t.5 sin evaluate t: ( 9.8/s ) 4s 97 [SSM] Show that for the situations in Fiure 4-5a and Fiure 4-5b the object oscillates with a frequency f ( / π ) eff /, where eff is iven by (a) eff +, and (b) / eff / + /. Hint: Find the anitude of the net force F on the object for a sall displaceent x and write F eff x. Note that in Part (b) the sprins stretch by different aounts, the su of which is x. Picture the Proble Choose a coordinate syste in which the +x direction is to the riht and assue that the object is displaced to the riht. In case (a), note that the two sprins undero the sae displaceent whereas in (b) they experience the sae force. (a) xpress the net force actin on the object: F net x x where eff + ( + ) x x eff (b) xpress the force actin on each sprin and solve for x : F x x x x xpress the total extension of the sprins: x + x F eff Solvin for eff yields: eff F x + x x x + x x x + x + ae the reciprocal of both sides of the equation to obtain: eff +

Oscillations 59 98 Durin an earthquae, a floor oscillates horizontally in approxiately siple haronic otion. ssue it oscillates at a sinle frequency with a period of.8 s. (a) fter the earthquae, you are in chare of exainin the video of the floor otion and discover that a box on the floor started to slip when the aplitude reached c. Fro your data, deterine the coefficient of static friction between the box and the floor. (b) If the coefficient of friction between the box and floor were.4, what would be the axiu aplitude of vibration before the box would slip? Picture the Proble pplyin Newton s nd law to the box as it is about to slip will allow us to express μ s in ters of the axiu acceleration of the platfor which, in turn, depends on the aplitude and anular frequency of the oscillatory otion. (a) pply F x ax to the box as it is about to slip: f s, ax aax pply F y to the box: F n Use fs, ax μsfn to eliinate f s, ax and Fn between the two equations: Relate the axiu acceleration of the oscillator to its aplitude and anular frequency: Substitute for a ax in the expression for μ s : μs a ax and a ω ax ω 4π μ s a μ ax s Substitute nuerical values and evaluate μ s : μ 4π (.) s (.8s) ( 9.8/s ).6 (b) Solve the equation derived above μs μs ax for ax : ω 4π Substitute nuerical values and evaluate ax : ax (.4)( 9.8/s )(.8s) 6.4c 4π

5 Chapter 4 99 If we attach two blocs of asses and to either end of a sprin of force constant and set the into oscillation by releasin the fro rest with the sprin stretched, show that the oscillation frequency is iven by ω (/μ) /, where μ /( + ) is the reduced ass of the syste. Picture the Proble he pictorial representation shows the two blocs connected by the sprin and displaced fro their equilibriu positions. We can apply Newton s nd law to each of these coupled oscillators and solve the resultin equations siultaneously to obtain the equation of otion of the coupled oscillators. We can then copare this equation and its solution to the equation of otion of the siple haronic oscillator and its solution to show that the oscillation frequency is ω ( μ) where μ /( + ). x x x F F n (x x ) (x x ) x pply F a to the bloc whose ass is and solve for its acceleration: pply F a to the bloc whose ass is and solve for its acceleration: d x dt ( x x ) a or a d x dt x ( x ) ( x x ) a or a d x dt x ( x ) d x dt Subtract the first equation fro the second to obtain: d ( x x ) dt d x dt where x x x + x he reduced ass of the syste is: μ + μ +

Substitute to obtain: d x x dt μ Oscillations 5 () Copare this equation to the equation of the siple haronic oscillator: d x dt x he solution to this equation is: x x cos( ω t +δ ) where ω ecause of the siilarity of the two differential equations, the solution to equation () ust be: ( ω + δ ) x x cos t where ω and μ μ + In one of your cheistry labs you deterine that one of the vibrational odes of the HCl olecule has a frequency of 8.969 Hz. Usin the result of Proble 99, find the effective sprin constant between the H ato and the Cl ato in the HCl olecule. Picture the Proble We can use ( ) ω μ and μ /( + ) fro Proble 99 to find the sprin constant for the HCl olecule. Use the result of Proble 99 to relate the oscillation frequency to the force constant and reduced ass of the HCl olecule: ω μω μ xpress the reduced ass of the HCl olecule: μ + Substitute for μ to obtain: xpress the asses of the hydroen and Cl atos: ω + au.67 7 and 5.45 au 5.9 6 Substitute nuerical values and evaluate : 7 6 - (.67 )( 5.9 )( 8.969 s ).N/ 7 6.67 + 5.9

5 Chapter 4 If a hydroen ato in HCl were replaced by a deuteriu ato (forin DCl) in Proble, what would be the new vibration frequency of the olecule? Deuteriu consists of proton and neutron. Picture the Proble In Proble, we derived an expression for the oscillation frequency of a sprin-and-two-bloc syste as a function of the force constant of the sprin and the reduced ass of the two blocs. We can solve this proble, assuin that the "sprin constant" does not chane, by usin the result of Proble and the reduced ass of a deuteriu ato and a Cl ato in the equation for the oscillation frequency. Use the result of Proble to relate the oscillation frequency to the force constant and reduced ass of the DCl olecule: xpress the reduced ass of the DCl olecule: ω μ μ + he asses of the deuteriu and Cl atos are: au.4 7 and 5.45 au 5.9 6 Substitute nuerical values and evaluate ω: ω 6 (.4 )( 5.9 ).4 7.N/ 6.44 rad/s 7 + 5.9 6 bloc of ass on a horizontal table is attached to a sprin of force constant, as shown in Fiure 4-6. he coefficient of inetic friction between the bloc and the table is μ. he sprin is unstressed if the bloc is at the oriin (x ), and the +x direction is to the riht. he sprin is stretched a distance, where > μ, and the bloc is released. (a) pply Newton s second law to the bloc to obtain an equation for its acceleration d x/dt for the first half-cycle, durin which the bloc is ovin to the left. Show that the resultin equation can be written as d x' dt ω x', where ω and x x x, with μ μ ω x. (b) Repeat Part (a) for the second half-cycle as the bloc oves to the riht, and show that d x'' dt ω x'', where x x + x and x has the sae value. (c) Use a spreadsheet prora to raph the first 5 half-cycles for x. Describe the otion, if any, after the fifth half-cycle. Picture the Proble he pictorial representation shows the bloc ovin fro riht to left with an instantaneous displaceent x fro its equilibriu position.

Oscillations 5 he free-body diara shows the forces actin on the bloc durin the half-cycles that it oves fro riht to left. When the bloc is ovin fro left to riht, the directions of the inetic friction force and the restorin force exerted by the sprin are reversed. We can apply Newton s nd law to these otions to obtain the equations iven in the proble stateents and then use their solutions to plot the raph called for in (c). x y F n x f x (a) pply F x ax to the bloc while it is ovin to the left to obtain: d x f x dt Usin f μ Fn μ, eliinate f in the equation of otion: d x dt or d x dt x + μ μ x Let μ x d x dt or to obtain: ( x x ) d x' x' ω x' dt provided x x x and μ μ x ω he solution to the equation of otion is: ( ω + δ ) x' x ' cos t and its derivative is v' ω x ' sin ωt + δ ( )

54 Chapter 4 he initial conditions are: x' ( ) x x and v' ( ) pply these conditions to obtain: x' ( ) x ' cosδ x x and v' ω x ' sinδ ( ) Solve these equations siultaneously to obtain: δ and x' x x and x' ( x x ) t or cosω ( x x ) cos t x x ω + () (b) pply F a to the bloc while it is ovin to the riht to obtain: d x f x dt Usin f μ Fn μ, eliinate f in the equation of otion: d x dt or d x dt x μ μ x + Let μ x d x dt or to obtain: ( x + x ) d x" x" ω x" dt provided x x + x and μ μ x. ω he solution to the equation of otion is: ( ω + δ ) x" x " cos t and its derivative is v" ω x " sin ωt + δ ( ) he initial conditions are: x" ( ) x + x and v" ( ) pply these conditions to obtain: x" ( ) x " cosδ x + x and v" ω x " sinδ ( )

Oscillations 55 Solve these equations siultaneously to obtain: δ and x " x + x and ( x + x ) t x" cosω or ( x + x ) cos t x x ω () (c) spreadsheet prora to calculate the position of the oscillator as a function of tie (equations () and ()) is shown below. he constants used in the position functions (x and s were used for siplicity) and the forulas used to calculate the positions are shown in the table. fter each half-period, one ust copute a new aplitude for the oscillation, usin the final value of the position fro the last half-period. Cell Content/Forula lebraic For x C7 C6 +. t + Δt x cosπ t + x D7 (S($D$6+$$))*COS(PI()*C7)$$ x x cosπ x D7 (S($D$6$$))*COS(PI()*C7)+$$ x x cosπ x D7 (S($D$6+$$))*COS(PI()*C7)$$ x x cosπ x D7 ($$$$)*COS(PI()*C7)+$$ ( ) D47 ($D$46$$)*COS(PI()*C47)+$$ ( x x ) cosπ t + x C D x 4 t x 5 (s) () 6.. 7. 9.56 8. 8.8 9. 6.9.4.78 5 4.7.4 54 4.8.9 55 4.9.5 56 5.. he followin raph was plotted usin the data fro coluns C (t) and D (x). Note that the otion of the bloc ceases after five half - cycles.

56 Chapter 4 8 6 4 x () - -4-6 -8-4 5 t (s) Fiure 4-7 shows a unifor solid half-cylinder of ass M and radius R restin on a horizontal surface. If one side of this cylinder is pushed down slihtly and then released, the half-cylinder will oscillate about its equilibriu position. Deterine the period of this oscillation. Picture the Proble he diara shows the half-cylinder displaced fro its equilibriu position throuh an anle θ. he frequency of its otion will be found by expressin the echanical enery in ters of θ and dθ/dt. For sall θ dθ we will obtain an equation of the for κθ + I. Differentiatin both dt d θ dθ sides of this equation with respect to tie will lead to κθ I dt +, an dt equation that ust be valid at all ties. ecause the situation of interest to us d θ requires that dθ/dt is not always equal to zero, we have κθ + I or dt d θ κ + θ, the equation of siple haronic otion with ω κ I. We ll dt I 4R show that the distance fro O to the center of ass D, is iven by D, and π let the distance fro the contact point C to the center of ass be r. Finally, we ll tae the potential enery to be zero where θ is zero and assue that there is no slippin.

Oscillations 57 O h R θ D r C pply conservation of enery to obtain: U + K M ( h D) + I C dθ dt () Fro able 9-, the oent of inertia of a solid cylinder about an axis perpendicular to its face and throuh its center is iven by: xpress the oent of inertia of the half-cylinder about the sae axis: Use the parallel-axis theore to relate I c to I : I, solid cylinder ( M ) R MR where M is the ass of the halfcylinder. [ MR ] MR I, half cylinder I I + Ic MD Substitute for I and solve for I c : I c I D M MR D M pply the parallel-axis theore a second tie to obtain an expression for I C : I C MR D M + Mr M R D + r () pply the law of cosines to obtain: r R + D RDcosθ

Chapter 4 58 Substitute for r in equation () to obtain: + + θ θ cos cos C R D MR RD D R D R M I Substitute for h and I C in equation (): ( ) cos cos + dt d R D MR MD θ θ θ Use the sall anle approxiation cos θ θ to obtain: [ ] + dt d R D MR MD θ θ θ ecause θ <<, we can nelect the θ in the square bracets to obtain: + dt d R D MR MD θ θ Differentiatin both sides with respect to tie and siplifyin yields: + θ θ D dt d R D R, or + θ θ R D R D dt d, the equation of siple haronic otion with R D R D ω. ()

Oscillations 59 D is the y coordinate of the center of ass of the seicircular dis shown. surface eleent of area d is shown in the diara. ecause the dis is a continuous object, we ll use Mr rd to find y c D. c xpress the coordinates of the center of ass of the seicircular dis: xpress y as a function of r and θ : xpress d in ters of r and θ : Substitute and evaluate D: x c y c D y r sinθ by syetry. d r dθ dr D yσ d M R π σ r sinθ dθ dr R σ M M σ R M r dr xpress M as a function of r and θ : M σ half σπr dis Substitutin for M and siplifyin D σ R yields: ( σπr ) 4 R π Substitute for D in equation () and siplify to obtain: ω 4 π 8 π R 8 9π 6 R he period of the otion is iven by: Substitutin for ω and siplifyin yields: π ω 9π 6 R π 7. 78 8 R

5 Chapter 4 4 straiht tunnel is du throuh arth as shown in Fiure 4-8. ssue that the walls of the tunnel are frictionless. (a) he ravitational force exerted by arth on a particle of ass at a distance r fro the center of arth when r < R is F r GM / R ( )r, where M is the ass of arth and R is its radius. Show that the net force on a particle of ass at a distance x fro the iddle of the tunnel is iven by F x GM / R ( )x, and that the otion of the particle is therefore siple haronic otion. (b) Show that the period of the otion is independent of the lenth of the tunnel and is iven by π R. (c) Find its nuerical value in inutes. Picture the Proble he net force actin on the particle as it oves in the tunnel is the x-coponent of the ravitational force actin on it. We can find the period of the particle fro the anular frequency of its otion. We can apply Newton s nd law to the particle in order to express ω in ters of the radius of arth and the acceleration due to ravity at the surface of arth. (a) Fro the fiure we see that: F x GM Fr sinθ R GM R x x r r ecause this force is a linear restorin force, the otion of the particle is siple haronic otion. (b) xpress the period of the particle as a function of its anular frequency: pply Fx ax to the particle: Solvin for a yields: π () ω GM x a R GM a R x ω x GM where ω R Use GM R to siplify ω: ω R R R

Oscillations 5 Substitute in equation () to obtain: π R π R (c) Substitute nuerical values and evaluate : π 6 6.7 9.8/s 5.6 s 84.4 in 5 [SSM] In this proble, derive the expression for the averae power delivered by a drivin force to a driven oscillator (Fiure 4-9). (a) Show that the instantaneous power input of the drivin force is iven by P Fv ωf cos ωt sin( ωt δ). (b) Use the identity sin(θ θ ) sin θ cos θ cos θ sin θ to show that the equation in (a) can be written as P ωf sinδ cos ωt ωf cosδ cosωt sinωt (c) Show that the averae value of the second ter in your result for (b) over one or ore periods is zero, and that therefore P av ωf sin δ. (d) Fro quation 4-56 for tan δ, construct a riht trianle in which the side opposite the anle δ is bω and the side adjacent is ( ω ω ), and use this trianle to show that bω sin δ bω. ( ω ω ) + b ω F (e) Use your result for Part (d) to eliinate ω fro your result for Part (c) so that the averae power input can be written as P av F b sin δ bω F ( ω ω ) + b ω. Picture the Proble We can follow the step-by-step instructions provided in the proble stateent to obtain the desired results. (a) xpress the averae power delivered by a drivin force to a driven oscillator: P F v Fv cosθ or, because θ is, P Fv xpress F as a function of tie: F F cosωt xpress the position of the driven oscillator as a function of tie: x cos t ( ω δ )

5 Chapter 4 Differentiate this expression with respect to tie to express the velocity of the oscillator as a function of tie: Substitute to express the averae power delivered to the driven oscillator: v ω sin t P ( ω δ ) ( F cosωt) [ ω sin( ωt δ )] ωf cosωt sin( ωt δ ) (b) xpand sin ( ωt δ ) to obtain: sin( ωt δ ) sinωt cosδ cosωt sinδ Substitute in your result fro (a) and siplify to obtain: (c) Interate sinθ cosθ over one period to deterine sin θ cosθ : ( ) P ωf cosωt sin ωt cosδ cosωt sin δ ωf sin δ cos ωt ωf cosδ cosωt sin ωt π sinθ cosθ sin cos θ θdθ π sin θ π π Interate cos θ over one period to deterine cos θ : cos θ π π π π π cos θdθ π π ( π + ) ( + cosθ ) dθ + π dθ cosθ dθ Substitute and siplify to express P av : P av ωf ωf ωf ωf sinδ cos ωt cosδ cosωt sinωt sinδ ωf sinδ cosδ ( )

Oscillations 5 (d) Construct a trianle that is consistent with bω tanδ ( ω ω ) : Usin the trianle, express sinδ: sinδ ( ω ω ) + b ω bω Usin quation 4-56, reduce this expression to the sipler for: sinδ bω F (e) Solve bω F sinδ for ω: ω sinδ F b Substitute in the expression for P av to eliinate ω: P av F sin b δ Substitute for obtain: sinδ fro (d) to P av bω F ( ) ω ω + b ω 6 In this proble, you are to use the result of Proble 5 to derive quation 4-5. t resonance, the denoinator of the fraction in bracets in Proble 5(e) is b ω and P av has its axiu value. For a sharp resonance, the variation in ω in the nuerator in this equation can be nelected. hen the power input will be half its axiu value at the values of ω, for which the denoinator is b ω. (a) Show that ω then satisfies ( ω ω ) ( ω + ω ) b ω. (b) Usin the approxiation ω + ω ω, show that ω ω ±b/. (c) xpress b in ters of Q. (d) Cobine the results of (b) and (c) to show that there are two values of ω for which the power input is half that at resonance and that they are iven by ω ω ω Q and ω ω ω Q herefore, ω ω Δω ω / Q, which is equivalent to quation 4-5. Picture the Proble We can follow the step-by-step instructions iven in the proble stateent to derive the iven results.

54 Chapter 4 (a) xpress the condition on the denoinator of quation 4-56 when the power input is half its axiu value: Factor the difference of two squares to obtain: ( ω ω ) + b ω b ω and, for a sharp resonance, or ( ω ω ) b ω [( ω ω)( ω + ω) ] b ω ( ω ω) ( ω + ω) b ω (b) Use the approxiation ω + ω ω to obtain: Solvin for ω ω yields: (c) Usin its definition, express Q: (d) Substitute for b in equation () to obtain: ( ω ω) ( ω ) b ω b ω ω ± () ω Q ω b b Q ω ω ω ω ± ω ω ± Q Q xpress the two values of ω: ω ω + ω + andω Q ω ω Q Rears: Note that the width of the resonance at half-power is Δω ω+ ω ω Q, in areeent with quation 4-5. 7 he Morse potential, which is often used to odel interatoic forces, can be written in the for U() r D e β ( r r ) ( ), where r is the distance between the two atoic nuclei. (a) Usin a spreadsheet prora or raphin calculator, ae a raph of the Morse potential usin D 5. ev, β. n, and r.75 n. (b) Deterine the equilibriu separation and sprin constant for sall displaceents fro equilibriu for the Morse potential. (c) Deterine an expression for the oscillation frequency for a hoonuclear diatoic olecule (that is, two of the sae atos), where the atos each have ass. Picture the Proble We can find the equilibriu separation for the Morse potential by settin du/dr and solvin for r. he second derivative of U will ive the "sprin constant" for sall displaceents fro equilibriu. In (c), we

Oscillations 55 can use ω μ, where is our result fro (b) and μ is the reduced ass of a hoonuclear diatoic olecule, to find the oscillation frequency of the olecule. (a) spreadsheet prora to calculate the Morse potential as a function of r is shown below. he constants and cell forulas used to calculate the potential are shown in the table. Cell Content/Forula lebraic For 5 D. β C9 C8 +. r + Δr D8 $$*(XP($$*(C8$$)))^ ( rr ) D e C D D 5 ev β. n r.75 n 4 5 6 r U(r) 7 (n) (ev) 8..95 9..967..676..444.4.69 5.7 4.87676 6.8 4.8799 7.9 4.8856 8. 4.889 9. 4.8868 [ ]

56 Chapter 4 he raph shown below was plotted usin the data fro coluns C (r) and D (U(r)). U (ev).7.6.5.4......5..5..5. r (n) (b) Differentiate the Morse potential with respect to r to obtain: his derivative is equal to zero for extrea: du dr d dr βd β ( rr ) { D[ e ] } β [ ] ( rr e ) β rr ) [ e ] ( βd r r valuate the second derivative of d U d β ( rr ) { βd[ e ]} U(r) to obtain: dr dr β De β ( rr ) valuate this derivative at r r : d U dr r r β D () Recall that the potential function for a siple haronic oscillator is: Differentiate this expression twice to obtain: y coparison with equation () we have: U d U dx x β D

Oscillations 57 (c) xpress the oscillation frequency of the diatoic olecule: ω μ where μ is the reduced ass of the olecule. xpress the reduced ass of the μ hoonuclear diatoic olecule: + Substitute for ω and siplify to obtain: ω β D β D Rears: n alternative approach in (b) is to expand the Morse potential in a aylor series U( r) U( r ) + ( r r ) U' ( r ) + ( r r ) U'' ( r ) + hiher order ters! to obtain U(r) β D( r r ). Coparin this expression to the enery of a sprin-and-ass oscillator we see that, as was obtained above, β D.

58 Chapter 4