Chapter 14 Oscillations

Transcription

1 Chapter 4 Oscillations Conceptual Probles 3 n object attached to a spring exhibits siple haronic otion with an aplitude o 4. c. When the object is. c ro the equilibriu position, what percentage o its total echanical energy is in the or o potential energy? (a) One-quarter. (b) One-third. (c) One-hal. (d) Two-thirds. (e) Three-quarters. Picture the Proble The total energy o an object undergoing siple haronic otion is given by E, where is the orce constant and is the tot aplitude o the otion. The potential energy o the oscillator when it is a U x x distance x ro its equilibriu position is ( ). Express the ratio o the potential energy o the object when it is. c ro the equilibriu position to its total energy: U E ( x) tot x x Evaluate this ratio or x. c and 4. c: U ( c) (.c) E ( 4.c) 4 tot and (a) is correct. 7 Two systes each consist o a spring with one end attached to a bloc and the other end attached to a wall. The identical springs are horizontal, and the blocs are supported ro below by a rictionless horizontal table. The blocs are oscillating in siple haronic otions with equal aplitudes. However, the ass o bloc is our ties as large as the ass o bloc. How do their axiu speeds copare? (a) v ax v ax, (b) v ax v ax, (c) v ax v ax, (d) This coparison cannot be done by using the data given. Deterine the Concept The axiu speed o a siple haronic oscillator is the product o its angular requency and its aplitude. The angular requency o a siple haronic oscillator is the square root o the quotient o the orce constant o the spring and the ass o the oscillator. Relate the axiu speed o syste to its orce constant: Relate the axiu speed o syste to its orce constant: v ω ax v ω ax 85

2 86 Chapter 4 Divide the irst o these equations by the second and sipliy to obtain: v v ax ax ecause the systes dier only in the asses attached to the springs: v v ax ax Substituting or and sipliying v ax yields: v 4 ax ( c) is correct. 9 In general physics courses, the ass o the spring in siple haronic otion is usually neglected because its ass is usually uch saller than the ass o the object attached to it. However, this is not always the case. I you neglect the ass o the spring when it is not negligible, how will your calculation o the syste s period, requency and total energy copare to the actual values o these paraeters? Explain. Deterine the Concept Neglecting the ass o the spring, the period o a siple haronic oscillator is given by T π ω π where is the ass o the oscillating syste (spring plus object) and its total energy is given by E. Neglecting the ass o the spring results in your using a value or the ass o the oscillating syste that is saller than its actual value. Hence, your calculated value or the period will be saller than the actual period o the syste and the calculated value or the requency, which is the reciprocal o the period, will be higher than the actual value. ecause the total energy o the oscillating syste depends solely on the aplitude o its otion and on the stiness o the spring, it is independent o the ass o the syste, and so neglecting the ass o the spring would have no ect on your calculation o the syste s total energy. 3 Two ass spring systes and oscillate so that their total echanical energies are equal. I the orce constant o spring is two ties the orce constant o spring, then which expression best relates their aplitudes? (a) /4, (b), (c), (d) Not enough inoration is given to deterine the ratio o the aplitudes. total

3 Oscillations 87 Picture the Proble We can express the energy o each syste using E and, because the energies are equal, equate the and solve or. Express the energy o ass-spring syste in ters o the aplitude o its otion: Express the energy o ass-spring syste in ters o the aplitude o its otion: ecause the energies o the two systes are equal we can equate the to obtain: E E Substitute or and sipliy to obtain: (b) is correct. 7 Two siple pendulus are related as ollows. Pendulu has a length and a bob o ass ; pendulu has a length and a bob o ass. I the requency o is one-third that o, then (a) 3 and 3, (b) 9 and, (c) 9 regardless o the ratio /, (d) 3 regardless o the ratio /. Picture the Proble The requency o a siple pendulu is independent o the ass o its bob and is given by g. π Express the requency o pendulu : Siilarly, the length o pendulu is given by: g π g g Divide the irst o these equations by the second and sipliy to obtain: g g

4 88 Chapter 4 Substitute or to obtain: 3 9 (c) is correct. 3 Two daped, driven spring-ass oscillating systes have identical asses, driving orces, and daping constants. However, syste s orce constant is our ties syste s orce constant. ssue they are both very wealy daped. How do their resonant requencies copare? (a) ω ω, (b) ω ω, (c) ω ω, (d) ω 4 ω, (e) Their resonant requencies cannot be copared, given the inoration provided. Picture the Proble For very wea daping, the resonant requency o a springass oscillator is the sae as its natural requency and is given by ω, where is the oscillator s ass and is the orce constant o the spring. Express the resonant requency o Syste : ω The resonant requency o Syste is given by: ω Dividing the irst o these equations by the second and sipliying yields: ω ω ecause their asses are the sae: ω ω Substituting or yields: ω ω 4 ( b) is correct. Estiation and pproxiation 5 Estiate the width o a typical grandather clocs cabinet relative to the width o the pendulu bob, presuing the desired otion o the pendulu is siple haronic.

5 Oscillations 89 Picture the Proble I the otion o the pendulu in a grandather cloc is to be siple haronic otion, then its period ust be independent o the angular aplitude o its oscillations. The period o the otion or largeaplitude oscillations is given by Equation 4-3 and we can use this expression to obtain a axiu value or the aplitude o swinging pendulu in the cloc. We can then use this value and an assued value or the length o the pendulu to estiate the width o the grandather clocs cabinet. w θ w aplitude w bob Reerring to the diagra, we see that the iniu width o the cabinet is deterined by the width o the bob and the width required to accoodate the swinging pendulu: Express waplitude in ters o the angular aplitude θ and the length o the pendulu : w w bob + w aplitude and w waplitude + () w w bob bob w sinθ aplitude Substituting or () yields: w aplitude in equation w w bob sinθ + () w bob Equation 4-3 gives us the period o a siple pendulu as a unction o its angular aplitude: T T + sin θ +... I T is to be approxiately equal to T, the second ter in the bracets ust be sall copared to the irst ter. Suppose that: sin θ. 4 Solving or θ yields: sin (.63) 7.5 θ

6 9 Chapter 4 I we assue that the length o a grandather cloc s pendulu is about.5 and that the width o the bob is about c, then equation () yields: w w bob ( ).5 sin Siple Haronic Motion 3 particle o ass begins at rest ro x +5 c and oscillates about its equilibriu position at x with a period o.5 s. Write expressions or (a) the position x as a unction o t, (b) the velocity v x as a unction o t, and (c) the acceleration a x as a unction o t. Picture the Proble The position o the particle as a unction o tie is given by x cos ( ω t +δ ). The velocity and acceleration o the particle can be ound by dierentiating the expression or the position o the particle with respect to tie. (a) The position o the particle is given by: For a particle that begins ro rest ro x +5 c, δ and +5 c. Hence: The angular requency o the particle s otion is: Substituting or ω gives: x cos t x ω T x ( ω +δ ) ( + 5 c) cosωt π s 4.9s ( ) 3 5c cos s 3 t (.5) cos[ ( 4.s ) t] (b) Dierentiate the result in Part (a) with respect to tie to express the velocity o the particle: dx dt ( 5c) s sin s t (. /s) sin[ ( 4.s ) t] v x 3 3

7 Oscillations 9 (c) Dierentiate the result in Part (b) with respect to tie to express the acceleration o the particle: a x dv dt x ( 5c) s cos s t ( 4.4 /s ) cos[ ( 4.s ) t] The position o a particle is given by x.5 cos πt, where x is in eters and t is in seconds. (a) Find the axiu speed and axiu acceleration o the particle. (b) Find the speed and acceleration o the particle when x.5. Picture the Proble The position o the particle is given by x cosω t, where.5 and ω π rad/s. The velocity is the tie derivative o the position and the acceleration is the tie derivative o the velocity. (a) The velocity is the tie derivative o the position and the acceleration is the tie derivative o the acceleration: The axiu value o sinωt is + and the iniu value o 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 The axiu value o cosωt is + and the iniu value o cosωt is : a ax ω 5/s (.5)( π s ) (b) Use the Pythagorean identity sin ωt + cos ωt to eliinate t ro the equations or x and v: v ω + x v ω x Substitute nuerical values and evaluate v (.5 ): Substitute x or cosωt in the equation or a to obtain: Substitute nuerical values and evaluate a: v (.5 ) ( π rad/s) (.5 ) (.5 ) 6.3 /s a ω x a π ( rad/s) (.5 ) 5 /s

8 9 Chapter 4 Siple Haronic Motion as Related to Circular Motion 39 particle oves at a constant speed o 8 c/s in a circle o radius 4 c centered at the origin. (a) Find the requency and period o the x coponent o its position. (b) Write an expression or the x coponent o its position as a unction o tie t, assuing that the particle is located on the +y-axis at tie t. Picture the Proble We can ind the period o the otion ro the tie required or the particle to travel copletely around the circle. The requency o the otion is the reciprocal o its period and the x-coponent o the particle s x cos ω t + δ. We can use the initial position o the position is given by ( ) particle to deterine the phase constant δ. (a) Use the deinition o speed to ind the period o the otion: T (.4) πr π v.8 /s s ecause the requency and the period are reciprocals o each other: T 3.4s.3 Hz (b) Express the x coponent o the position o the particle: The initial condition on the particle s position is: Substitute in the expression or x to obtain: ( ω t + δ ) cos( π + δ ) x cos t () x ( ) cos ( ) cosδ π δ Substitute or, ω, and δ in equation x ( 4c) cos ( ) () to obtain:.s t + Energy in Siple Haronic Motion π 43.5-g object on a rictionless horizontal surace oscillates at the end o a spring o orce constant 5 N/. The object s axiu speed is 7. c/s. (a) What is the syste s total echanical energy? (b) What is the aplitude o the otion?

9 Oscillations 93 Picture the Proble The total echanical energy o the oscillating object can be expressed in ters o its inetic energy as it passes through its equilibriu position: E v. Its total energy is also given by E. We can tot ax equate these expressions to obtain an expression or. tot (a) Express the total echanical energy o the object in ters o its axiu inetic energy: Substitute nuerical values and evaluate E: E v E ax (.5g)(.7/s).368J.3675J (b) Express the total echanical energy o the object in ters o the aplitude o its otion: E tot E tot Substitute nuerical values and evaluate : (.3675J) 5 N/ 3.83c Siple Haronic Motion and Springs g object on a rictionless horizontal surace is attached to one end o a horizontal spring, oscillates with an aplitude o c and a requency o.4 Hz. (a) What is the orce constant o the spring? (b) What is the period o the otion? (c) What is the axiu speed o the object? (d) What is the axiu acceleration o the object? Picture the Proble (a) The angular requency o the otion is related to the orce constant o the spring through ω. (b) The period o the otion is the reciprocal o its requency. (c) and (d) The axiu speed and acceleration o an object executing siple haronic otion are v ax ω and a ax ω, respectively. (a) Relate the angular requency o the otion to the orce constant o the spring: Substitute nuerical values to obtain: ω ω (.4s ) ( 3.g).68N/ 68 N/

10 94 Chapter 4 (b) Relate the period o the otion to its requency: T.47s.4s.4s (c) The axiu speed o the object is given by: v ω π ax Substitute nuerical values and evaluate v ax : v ax π (.4s )(.).5/s.5/s (d) The axiu acceleration o the object is given by: a ax ω Substitute nuerical values and evaluate a ax : a ax (.4s ) (.) 3/s 4 π Siple Pendulu Systes 59 Find the length o a siple pendulu i its requency or sall aplitudes is.75 Hz. Picture the Proble The requency o a siple pendulu depends on its length g and on the local gravitational ield and is given by. π The requency o a siple pendulu oscillating with sall aplitude is given by: g g π Substitute nuerical values and 9.8 /s evaluate : (.75 s ) 44 c 65 siple pendulu o length is attached to a assive cart that slides without riction down a plane inclined at angle θ with the horizontal, as shown in Figure 4-8. Find the period o oscillation or sall oscillations o this pendulu. Picture the Proble The cart accelerates down the rap with a constant acceleration o gsinθ. This happens because the cart is uch ore assive than the bob, so the otion o the cart is unaected by the otion o the bob oscillating bac and orth. The path o the bob is quite coplex in the reerence

11 θ Oscillations 95 rae o the rap, but in the reerence rae oving with the cart the path o the bob is uch sipler in this rae the bob oves bac and orth along a circular arc. To solve this proble we irst apply Newton s second law (to the bob) in the inertial reerence rae o the rap. Then we transor to the reerence rae oving with the cart in order to exploit the siplicity o the otion in that rae. Draw the ree-body diagra or the bob. et φ denote the angle that the string aes with the noral to the rap. The orces on the bob are the tension orce and the orce o gravity: θ φ T r θ+φ g r pply Newton s second law to the bob, labeling the acceleration o the bob relative to the rap a r R : The acceleration o the bob relative to the rap is equal to the acceleration o the bob relative to the cart plus the acceleration o the cart relative to the rap: Substitute or a r r r r in T g ar R r r Rearrange ters and label g acr as g r, where is the acceleration, relative to the cart, o an object in ree all. (I the tension orce is set to zero the bob is in ree all.): r r r T + g a r a r R r a C R r + a + : T + g ( + ) r CR r r a C a CR r r r r T + g a CR a abel g r a as CR g r to obtain r r r T + g a C () ( ) g r C r

12 θ 96 Chapter 4 To ind the agnitude o g r, irst draw the vector addition diagra representing the equation r r r g g. Recall that a CR a CR g sin θ: g r g r g g Fro the diagra, ind the agnitude o g r. Use the law o cosines to obtain: To ind the direction o g r, irst redraw the vector addition diagra as shown: θ a r CR β gsin θ θ g g + g sin θ g( gsinθ)cos β ut cos β sinθ, so g g + g sin θ g sin θ g ( sin θ) g cos θ Thus g gcosθ δ gcosθ g β θ gsin θ Fro the diagra ind the direction o g r. Use the law o cosines again and solve or δ: g sin θ g and so δ θ + g cos θ g cosθ cosδ To ind an equation or the otion o the bob draw the ree-body diagra or the orces that appear in equation (). Draw the path o the bob in the reerence rae oving with the cart: θ φ T r θ φ g r

13 Oscillations 97 Tae the tangential coponents o each vector in equation () in the rae o the cart yields. The tangential coponent o the acceleration is equal to the radius o the circle ties the angular acceleration a t rα : ( ) Rearranging this equation yields: For sall oscillations o the pendulu: d φ g sinφ dt where is the length o the string and d φ is the angular acceleration o the dt bob. The positive tangential direction is counterclocwise. d φ + g sinφ () dt φ << and sin φ φ Substituting or () yields: sinφ in equation d φ + gφ dt or d φ g + φ dt (3) Equation (3) is the equation o otion or siple haronic otion with angular requency: ω g where ω is the angular requency o the oscillations (and not the angular speed o the bob). The period o this otion is: T π ω π g (4) Substitute g cosθ or g in equation (4) to obtain: T π g π g cosθ Rears: Note that, in the liiting case θ, T. *Physical Pendulus T π g. s θ 9, 67 thin 5.-g dis with a -c radius is ree to rotate about a ixed horizontal axis perpendicular to the dis and passing through its ri. The dis is displaced slightly ro equilibriu and released. Find the period o the subsequent siple haronic otion.

14 98 Chapter 4 Picture the Proble The period o this physical pendulu is given by T π I MgD where I is the oent o inertia o the thin dis about the ixed horizontal axis passing through its ri. We can use the parallel-axis theore to express I in ters o the oent o inertia o the dis about an axis through its center o ass and the distance ro its center o ass to its pivot point. Express the period o a physical pendulu: T π I MgD Using the parallel-axis theore, ind the oent o inertia o the thin dis about an axis through the pivot point: I I 3 c + MR MR MR + MR Substituting or I and sipliying yields: T 3 MR π π MgR 3R g Substitute nuerical values and evaluate T: T (.) 3 π 9.8/s ( ).s 73 Points P and P on a plane object (Figure 4-3) are distances h and h, respectively, ro the center o ass. The object oscillates with the sae period T when it is ree to rotate about an axis through P and when it is ree to rotate about an axis through P. oth o these axes are perpendicular to the plane o the object. Show that h + h gt /(), where h h. Picture the Proble We can use the equation or the period o a physical pendulu and the parallel-axis theore to show that h + h gt /. Express the period o the physical pendulu: T π I gd Using the parallel-axis theore, relate the oent o inertia about an axis through P to the oent o inertia about an axis through the plane s center o ass: I I c + h Substitute or I to obtain: T π I + h gh c

15 Oscillations 99 Square both sides o this equation gt Ic + h and rearrange ters to obtain: h () ecause the period o oscillation is the sae or point P : I h c + h I h c + h Cobining lie ters yields: h h I c h ( h ) Provided h h : I c hh Substitute in equation () and gt h h + h sipliy to obtain: h Solving or h + h gives: Daped Oscillations h + h gt 77 Show that the ratio o the aplitudes or two successive oscillations is constant or a linearly daped oscillator. t τ Picture the Proble The aplitude o 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 o a given oscillation pea to the tie at which the pea occurs: Express the aplitude o the oscillation pea at t t + T: Express the ratio o these consecutive peas: t τ ( ) e t ( t+ T ) τ ( + T ) e t t ( t) e ( t+ T ) ( + T ) e t τ τ constant e T τ

16 3 Chapter 4 8 Seisologists and geophysicists have deterined that the vibrating Earth has a resonance period o 54 in and a Q actor o about 4. ter a large earthquae, Earth will ring (continue to vibrate) or up to onths. (a) Find the percentage o the energy o vibration lost to daping orces during each cycle. ( ) n E (b) Show that ater n periods the vibrational energy is given by E n.984, where E is the original energy. (c) I the original energy o vibration o an earthquae is E, what is the energy ater. d? Picture the Proble (a) We can ind the ractional loss o energy per cycle ro the physical interpretation o Q or sall daping. (b) We will also ind a general expression or Earth s vibrational energy as a unction o the nuber o cycles it has copleted. (c) We can then solve this equation or Earth s vibrational energy ater any nuber o days. (a) Express the ractional change in energy as a unction o Q: ΔE E π π Q 4.57% (b) Express the energy o the daped oscillator ater one cycle: ΔE E E E Express the energy ater two cycles: ΔE ΔE E E E E E Generalizing to n cycles: E n ΔE E E n E (.57) n E (.984) n (c) Express. d in ters o the nuber o cycles; that is, the nuber o vibrations Earth will have experienced: 4h 6.d.d d h T 88 in 54in 53.3T Evaluate E( d): 53.3 E ( d) E(.9843). 43E

17 Oscillations 3 83 You are in charge o onitoring the viscosity o oils at a anuacturing plant and you deterine the viscosity o an oil by using the ollowing ethod: The viscosity o a luid can be easured by deterining the decay tie o oscillations or an oscillator that has nown properties and operates while iersed in the luid. s long as the speed o the oscillator through the luid is relatively sall, so that turbulence is not a actor, the drag orce o the luid on a sphere is proportional to the sphere s speed relative to the luid: Fd 6πaηv, where η is the viscosity o the luid and a is the sphere s radius. Thus, the constant b is given by 6 π aη. Suppose your apparatus consists o a sti spring that has a orce constant equal to 35 N/c and a gold sphere (radius 6. c) hanging on the spring. (a) What is the viscosity o an oil do you easure i the decay tie or this syste is.8 s? (b) What is the Q actor or your syste? Picture the Proble (a) The decay tie or a daped oscillator (with speeddependent daping) syste is deined as the ratio o the ass o the oscillator to the coicient o v in the daping orce expression. (b) The Q actor is the product o the resonance requency and the daping tie. (a) Fro Fd 6πaηv and F bv it ollows that: d, ecause τ b, we can substitute or b to obtain: Substituting yields: ρv and sipliying Substitute nuerical values and evaluate η (see Table 3- or the density o gold): (b) The Q actor is the product o the resonance requency and the daping tie: b b 6πaη η 6π a η 6πaτ ρv η 6πaτ η 3 πa ρ ρ 6πaτ 9τ 4 3 a 3 3 (.6 ) ( 9.3 g/ ) 5.5 Pa s 9 (.8 s) Q ωτ τ τ τ 4 3 ρv πa ρ 3 Substitute nuerical values and evaluate Q: Q 3 35 N c c ( ) ( ) (.8 s) g/ Driven Oscillations and Resonance

18 3 Chapter g object oscillates on a spring o orce constant 4 N/. The linear daping constant has a value o. g/s. The syste is driven by a sinusoidal orce o axiu value. N and angular requency. rad/s. (a) What is the aplitude o the oscillations? (b) I the driving requency is varied, at what requency will resonance occur? (c) What is the aplitude o oscillation at resonance? (d) What is the width o the resonance curve Δω? Picture the Proble (a) The aplitude o the daped oscillations is related to the daping constant, ass o the syste, the aplitude o the driving orce, and F the natural and driving requencies through. ( ω ω ) + b ω (b) Resonance occurs when ω ω. (c) t resonance, the aplitude o the oscillations is F b ω. (d) The width o the resonance curve is related to the daping constant and the ass o the syste according to Δω b. (a) Express the aplitude o the oscillations as a unction o the driving requency: ecause ω : ( ω ω ) + b ω F F ω + b ω Substitute nuerical values and evaluate : 4 N/. g. N (. g) (. rad/s) + (. g/s) (. rad/s) 4.98 c (b) Resonance occurs when: ω ω Substitute nuerical values and evaluate ω: ω 4 N/. g 4.rad/s 4.4 rad/s

19 Oscillations 33 (c) The aplitude o the otion at F resonance is given by: b ω Substitute nuerical values and evaluate : ( ).g/s ( 4.4rad/s ) 35.4c. N (d) The width o the resonance curve is: b. g/s Δ ω. g. rad/s General Probles 93 bloc that has a ass equal to is supported ro below by a rictionless horizontal surace. The bloc, which is attached to the end o a horizontal spring with a orce constant, oscillates with an aplitude. When the spring is at its greatest extension and the bloc is instantaneously at rest, a second bloc o ass is placed on top o it. (a) What is the sallest value or the coicient o static riction μ s such that the second object does not slip on the irst? (b) Explain how the total echanical energy E, the aplitude, the angular requency ω, and the period T o the syste are aected by the placing o on, assuing that the coicient o riction is great enough to prevent slippage. Picture the Proble pplying Newton s second law to the irst object as it is about to slip will allow us to express μ s in ters o the axiu acceleration o the syste which, in turn, depends on the aplitude and angular requency o the oscillatory otion. (a) pply Fx ax to the second object as it is about to slip: s, ax aax pply Fy to the second F n g object: Use and s, ax μsfn to eliinate s, ax between the two equations F n and solve or μs: μ g a s ax μ a ax s g

20 34 Chapter 4 Relate the axiu acceleration o the oscillator to its aplitude and angular requency and substitute or ω : a ax ω + Finally, substitute or a ax to obtain: μ s ( + )g (b) is unchanged. E is unchanged because E. ω is reduced and T is increased by increasing the total ass o the syste. 97 [SSM] Show that or the situations in Figure 4-35a and Figure 4-35b the object oscillates with a requency ( / π ) /, where is given by (a) +, and (b) / / + /. Hint: Find the agnitude o the net orce F on the object or a sall displaceent x and write F x. Note that in Part (b) the springs stretch by dierent aounts, the su o which is x. Picture the Proble Choose a coordinate syste in which the +x direction is to the right and assue that the object is displaced to the right. In case (a), note that the two springs undergo the sae displaceent whereas in (b) they experience the sae orce. (a) Express the net orce acting on the object: (b) Express the orce acting on each spring and solve or x : F net x x where + F ( + ) x x x x x x Express the total extension o the springs: x + x F Solving or yields: Tae the reciprocal o both sides o the equation to obtain: F x + x x x + x + x x + x +

21 Oscillations 35 5 [SSM] In this proble, derive the expression or the average power delivered by a driving orce to a driven oscillator (Figure 4-39). (a) Show that the instantaneous power input o the driving orce is given 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 average value o the second ter in your result or (b) over one or ore periods is zero, and that thereore P av ωf sin δ. (d) Fro Equation 4-56 or tan δ, construct a right triangle in which the side opposite the angle δ is bω and the side adjacent is ( ω ω ), and use this triangle to show that bω sin δ. ω ω F ( ) + b ω bω (e) Use your result or Part (d) to eliinate ω ro your result or Part (c) so that the average power input can be written as P av F b sin δ bω F ( ω ω ) + b ω. Picture the Proble We can ollow the step-by-step instructions provided in the proble stateent to obtain the desired results. (a) Express the average power delivered by a driving orce to a driven oscillator: r P F v r Fv cosθ or, because θ is, P Fv Express F as a unction o tie: F F cosωt Express the position o the driven oscillator as a unction o tie: Dierentiate this expression with respect to tie to express the velocity o the oscillator as a unction o tie: Substitute to express the average power delivered to the driven oscillator: x cos t ( ω δ ) v ω sin t P ( ω δ ) ( F cosωt) [ ω sin( ωt δ )] ωf cosωt sin( ωt δ )

22 36 Chapter 4 (b) Expand sin ( ωt δ ) to obtain: sin( ωt δ ) sinωt cosδ cosωt sinδ Substitute in your result ro (a) and sipliy to obtain: (c) Integrate sinθ cosθ over one period to deterine sin θ cosθ : Integrate cos θ over one period to deterine cos θ : ωf ( ) P ωf cosωt sin ωt cosδ cosωt sin δ sin δ cos ωt ωf cosδ cosωt sin ωt π sinθ cosθ sin cos θ θdθ π cos θ π sin θ π π π π π cos θdθ π π ( π + ) π ( + cosθ ) dθ + π dθ cosθ dθ Substitute and sipliy to express P av : P av ωf ωf cosδ cosωt sinωt ωf sinδ ωf cosδ ωf sinδ cos ωt sinδ ( ) (d) Construct a triangle that is consistent with bω tanδ ( ω ω ) : Using the triangle, express sinδ: sinδ ( ω ω ) + b ω bω

23 Oscillations 37 Using Equation 4-56, reduce this expression to the sipler or: sinδ bω F (e) Solve bω F sinδ or ω: ω sinδ F b Substitute in the expression or P av to eliinate ω: P av F sin b δ Substitute or obtain: sinδ ro (d) to P av bω F ( ) ω ω + b ω