Damped Oscillatory Motion

–.2.2

y y₁₁== ee^–^– x^x

y₂₂== ––ee^–^– x^x y

y₃₃== ee^–^– x^xsinsin x x

Figure 57 Figure 57

EXAMPLE 2

EXAMPLE 2 Analyzing Harmonic MotionAnalyzing Harmonic Motion Suppose that an object oscillates according to

Suppose that an object oscillates according to the modelthe model

where

where t t is is in in seconds seconds and and is is in in feet. feet. Analyze Analyze the the motion.motion.

Solution

Solution The motion is harmonic because the model is of the formThe motion is harmonic because the model is of the form Because

Because the the object object oscillates oscillates 88 ft ft in in either either direction direction from from itsits starting

starting point. point. The The period period is is the the time,time, in in seconds,seconds, it it takes takes for for one one com- com-plete oscillation

plete oscillation. The frequency i. The frequency is the reciprocal of s the reciprocal of the period,the period, so the objectso the object completes

completes oscillation oscillation per per sec.sec.

Now try Exercise 15.

Damped Oscillatory Motion

In the examIn the example of the streple of the stretched sprtched spring,ing, wewe disregard the effect of friction. Friction causes the amplitude of the motion to disregard the effect of friction. Friction causes the amplitude of the motion to di-minish gradually until the wei

minish gradually until the weight comes to rest. In this situatight comes to rest. In this situation,on, we say that thewe say that the motion has been

motion has been damped damped by the force of friction. Most oscillatory motions areby the force of friction. Most oscillatory motions are damped,

damped, and the decrease in amplitude follows the pattern of exponentiand the decrease in amplitude follows the pattern of exponential decay.al decay.

A typical example of

A typical example of damped oscillatory motiondamped oscillatory motion is provided by the functionis provided by the function de

deﬁﬁned byned by

Figure

Figure 57 57 shows shows how how the the graph graph of of is is bounded bounded above above by by thethe graph

graph of of and and below below by by the the graph graph of of The The damped damped motionmotion curve dips below the

curve dips below the x x-axis -axis at at but but stays stays above above the the graph graph of of FigureFigure 5858 shows

shows a a traditional traditional graph graph of of along along with with the the graph graph of of

Figure 58 Figure 58

Shock absorbers are put on an automobile in order to damp oscillatory Shock absorbers are put on an automobile in order to damp oscillatory mo-tion. Instead of oscillating up and down for a long while after hitting a bump or tion. Instead of oscillating up and down for a long while after hitting a bump or pothole,

pothole, the oscillations of tthe oscillations of the car are quickly damped out fhe car are quickly damped out for a smoother ride.or a smoother ride.

Now try Exercise 21.

– –11

1 1

t t y

y y= sin= sin t t

s(( t t) =) = ee^–^– t^tsinsin t t y

y==ee^–^–^t^t

y y == ––ee^–^{– t}^t

2 2



y^^sinsin t t ..



t t



^^ ee^^^t^tsinsin t t ,,

y y22..

x x^^ ^^

y22 ee^^^x^x..

y11 ee^^^x^x

y33 ee^^^x^xsinsin x x s

 

t t

 

^^ ee^^^t^tsinsint t ..

3 3 2

2^^

 

.48.48

2 2^^

 

2.12.1 a

a ^^ 8,8, a

a sinsin^^t t ..



t t



^^



t t





t t



^^ ^{8 sin 3}^{8 sin 3t}t ,,

6.5

6.5 Harmonic MotionHarmonic Motion 591591

1 1.. ((aa)) (b)

(b) ; The weight is neither; The weight is neither moving upward nor downward. At moving upward nor downward. At the motion of the weight is the motion of the weight is changing from up to down.

changing from up to down.

2 2.. ((aa)) (b)

(b) ; upward; upward 3

3.. ((aa)) (b)

(b) ; upward; upward 4

4.. ((aa)) (b)

(b) ; downward; downward 5.

^^5 cos5 cos44^^ 3 3 t t t

t ^^1,1, s s

 

 

^^22

 

t t

 

^^2 cos2 cos44^^t t

6.5 6.5 Exercises Exercises

(Modeling) Springs

(Modeling) Springs Suppose Suppose that that a a weight weight on on a a spring spring has has initial initial position position and and period P.

period P.

(a)

(a) Find Find a a function function s s given given by by that that models models the the displacement displacement of of thethe weight.

weight.

(b)

(b) Evaluate Evaluate . . Is Is the the weight weight moving moving upward, upward, downward, downward, or or neither neither when when ?? Support your results graphically or numerically.

Support your results graphically or numerically.

1. in.; in.; secsec 2.2. in.; in.; secsec 3.

3. in.; in.; secsec 4.4. in.; in.; secsec

(Modeling) Music

(Modeling) Music A note on the piano has given frequency F. Suppose the maximum A note on the piano has given frequency F. Suppose the maximum displacement

displacement at at the the center center of of the the piano piano wire wire is is given given by by . . Find Find constants constants a a and and soso that

that the the equation equation models models this this displacement. displacement. Graph Graph s s in in the the viewing viewing win- win-dow

dow by by ..

5. ;; 6.6. ;;

7. ;; 8.8. ;;

(Modeling)

(Modeling) Solve each problem. See Examples 1 and 2.Solve each problem. See Examples 1 and 2.

9. SpringSpring An object is attached to a coiled spring, as in Figure 55. It is pulled downAn object is attached to a coiled spring, as in Figure 55. It is pulled down a distance of 4 units from its equilibrium positio

a distance of 4 units from its equilibrium position, and then released.n, and then released. The time forThe time for one complete oscillation is 3 sec.

one complete oscillation is 3 sec.

(a)

(a) Give an equation that models the position of the object at Give an equation that models the position of the object at timetime t t ..

(b)

(b) Determine Determine the the position position at at sec.sec.

(c)

10.

10. SpringSpring Repeat Exercise 9, but assume that the object is pulled down 6 units andRepeat Exercise 9, but assume that the object is pulled down 6 units and the time for one complete oscillation is 4 sec.

the time for one complete oscillation is 4 sec.

11.

11. Particle Movement Particle Movement Write the equation and then determine the amplitude, period,Write the equation and then determine the amplitude, period, and frequency of the simple harmonic motion of a particle moving uniformly around and frequency of the simple harmonic motion of a particle moving uniformly around a circle of radius 2

a circle of radius 2 units, with angular speedunits, with angular speed (a)

(a) 2 radians per sec2 radians per sec (b)(b) 4 radians per sec.4 radians per sec.

12.

12. Pendulum Pendulum What are the periodWhat are the period PPand frequencyand frequencyT T of oscillation of a pendulum of of oscillation of a pendulum of length ft?

length ft? Hint: Hint: , where, where L Lis the length of the is the length of the pendulum in feet andpendulum in feet and PP is in seconds.

is in seconds.

13.

13. Pendulum Pendulum In Exercise 12, how long should the pendulum be to In Exercise 12, how long should the pendulum be to have period 1 sec?have period 1 sec?

14.

14. SpringSpring The formula for the up and down motion of a weight on a spring isThe formula for the up and down motion of a weight on a spring is given by

given by

If the spring constant

If the spring constantk k is 4, what massis 4, what massmmmust be used to produce a period of must be used to produce a period of 1 sec?1 sec?

15.

15. SpringSpring The height attained by a weight attached to a spring set in motion isThe height attained by a weight attached to a spring set in motion is

inches after

inches after t t seconds.seconds.

(a)

(a) Find the maximum height that the weight rises above the equilibrium positionFind the maximum height that the weight rises above the equilibrium position of

of ..

(b)

(b) When does the weightWhen does the weight ﬁﬁrst rst reach reach its its maximum maximum height, height, if if ??

(c)

t t 00 y

1 1 2 2

t t ^^1.251.25



.3,.3.3,.3





0,0, .05.05



^ss



t t

^^^a^acoscos^^t t





592

592 ^{CHAPTER 6}^{CHAPTER 6} The Circular Functions and Their Graphs The Circular Functions and Their Graphs

(b) ; amplitude: 2;; amplitude: 2;

period:

period: ; ; frequency:frequency:

12.

12. period: period: ; ; frequency:frequency:

period: ; ; frequency:frequency:

(b)

(d)approximately 4; Afterapproximately 4; After 1.3 sec, the weight is about 4 in.

1.3 sec, the weight is about 4 in.

above the equilibrium position.

(d)approximately 2; Afterapproximately 2; After 1.466 sec, the weight is about 2 in.

1.466 sec, the weight is about 2 in.

above the equilibrium position.

1 They are the same.

They are the same. 22.22. forfor and

massmmattached to it is stretched and then allowed to come to rest.attached to it is stretched and then allowed to come to rest.

(a)

(a) If the spIf the spring is strering is stretched tched ft and relft and released, what eased, what are the are the amplitude, amplitude, period, andperiod, and frequency of the r

frequency of the r esulting oscillatory motion?esulting oscillatory motion?

(b)

(b) What is the equation of the motion?What is the equation of the motion?

17.

17. SpringSpring The position of a weight attached to a spring isThe position of a weight attached to a spring is

inches after

inches aftert t seconds.seconds.

(a)

(a) What is the maximum height that the weight rises above the equilibriumWhat is the maximum height that the weight rises above the equilibrium position?

position?

(b)

(b) What are the frequency and period?What are the frequency and period?

(c)

(d)

(d) Calculate Calculate and and interpret interpret ..

18.

18. SpringSpring The position of a weight attached to a spring isThe position of a weight attached to a spring is

inches after

inches aftert t seconds.seconds.

(a)

(a) What is the maximum height that the weight rises above the equilibriumWhat is the maximum height that the weight rises above the equilibrium position?

position?

(b)

(b) What are the frequency and period?What are the frequency and period?

(c)

(d)

(d) Calculate Calculate and and interpret interpret ..

19.

19. SpringSpring A weight attached to a spring is pulled down 3 in. below the equilibriumA weight attached to a spring is pulled down 3 in. below the equilibrium position.

position.

(a)

(a) Assuming Assuming that ththat the frequency e frequency is is cycles pcycles per sec, er sec, determine determine a model a model that githat givesves the position of the weight at time

the position of the weight at time t t seconds.seconds.

(b)

(b) What is the period?What is the period?

20.

20. SpringSpring A weight attached to a spring is pulled down 2 in. below the equilibriumA weight attached to a spring is pulled down 2 in. below the equilibrium position.

position.

(a)

(a) Assuming thAssuming that the period at the period is is sec, determine sec, determine a model that a model that gives the pogives the position of sition of the weight at time

the weight at timet t seconds.seconds.

(b)

(b) What is the frequency?What is the frequency?

Use

Use a a graphing graphing calculator calculator to to graph graph , , , , and and in in the the view- view-ing

ing window window by by ..

21.

21. Find theFind thet t -intercepts o-intercepts of the gf the graph of raph of . Explain . Explain the relatiothe relationship of nship of these intthese interceptsercepts with the

CHAPTER 6

CHAPTER 6 Summary Summary 593593

Chapter 6 Summary

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