5.5 Modeling Harmonic Motion

Transcription

1 44 CHAPTER 5 Trigonomeric Funcions of Real Numbers 77(b)Skech a graph of he funcion d for. (c) Wha happens o he disance d as approaches? (c) From he graph deermine he values of a which he lengh of he shadow equals he man s heigh. To wha ime of da does each of hese values correspond? (d) Explain wha happens o he shadow as he ime approaches 6 P.M. (ha is, as ). d 3 mi 6 f S 56. Lengh of a Shadow On a da when he sun passes direcl overhead a noon, a six-foo-all man cass a shadow of lengh S 6 ` co p ` where S is measured in fee and is he number of hours since 6 A.M. (a) Find he lengh of he shadow a 8: A.M., noon, : P.M., and 5:45 P.M. (b) Skech a graph of he funcion S for. Discover Discussion 57. Reducion Formulas Use he graphs in Figure 5 o explain wh he following formulas are rue. an a x p b co x sec a x p b csc x 5.5 Modeling Harmonic Moion Periodic behavior behavior ha repeas over and over again is common in naure. Perhaps he mos familiar example is he dail rising and seing of he sun, which resuls in he repeiive paern of da, nigh, da, nigh,... Anoher example is he dail variaion of ide levels a he beach, which resuls in he repeiive paern of high ide, low ide, high ide, low ide,... Cerain animal populaions increase and decrease in a predicable periodic paern: A large populaion exhauss he food suppl, which causes he populaion o dwindle; his in urn resuls in a more pleniful food suppl, which makes i possible for he populaion o increase; and he paern hen repeas over and over (see pages ). Oher common examples of periodic behavior involve moion ha is caused b vibraion or oscillaion. A mass suspended from a spring ha has been compressed and hen allowed o vibrae vericall is a simple example. This same back and forh moion also occurs in such diverse phenomena as sound waves, ligh waves, alernaing elecrical curren, and pulsaing sars, o name a few. In his secion we consider he problem of modeling periodic behavior.

2 SECTION 5.5 Modeling Harmonic Moion 443 Modeling Periodic Behavior The rigonomeric funcions are ideall suied for modeling periodic behavior. A glance a he graphs of he sine and cosine funcions, for insance, ells us ha hese funcions hemselves exhibi periodic behavior. Figure shows he graph of sin. If we hink of as ime, we see ha as ime goes on, sin increases and decreases over and over again. Figure shows ha he moion of a vibraing mass on a spring is modeled ver accurael b sin. =ß _ O P (ime) Figure sin Figure Moion of a vibraing spring is modeled b sin. Noice ha he mass reurns o is original posiion over and over again. A ccle is one complee vibraion of an objec, so he mass in Figure complees one ccle of is moion beween O and P. Our observaions abou how he sine and cosine funcions model periodic behavior are summarized in he following box. Simple Harmonic Moion The main difference beween he wo equaions describing simple harmonic moion is he saring poin. A, we ge a sin v # a cos v # a In he firs case he moion sars wih zero displacemen, whereas in he second case he moion sars wih he displacemen a maximum (a he ampliude a). If he equaion describing he displacemen of an objec a ime is hen he objec is in simple harmonic moion. In his case, ampliude a period p v frequenc v p a sin v or a cos v Maximum displacemen of he objec Time required o complee one ccle Number of ccles per uni of ime

3 444 CHAPTER 5 Trigonomeric Funcions of Real Numbers The smbol v is he lowercase Greek leer omega, and n is he leer nu. Noice ha he funcions a sin pn and a cos pn have frequenc n, because pn/p n. Since we can immediael read he frequenc from hese equaions, we ofen wrie equaions of simple harmonic moion in his form. Res posiion Figure 3 _ Figure 4 < = ß 4π 3 > Example A Vibraing Spring The displacemen of a mass suspended b a spring is modeled b he funcion where is measured in inches and in seconds (see Figure 3). (a) Find he ampliude, period, and frequenc of he moion of he mass. (b) Skech he graph of he displacemen of he mass. Soluion (a) From he formulas for ampliude, period, and frequenc, we ge (b) The graph of he displacemen of he mass a ime is shown in Figure 4. An imporan siuaion where simple harmonic moion occurs is in he producion of sound. Sound is produced b a regular variaion in air pressure from he normal pressure. If he pressure varies in simple harmonic moion, hen a pure sound is produced. The one of he sound depends on he frequenc and he loudness depends on he ampliude. Example Vibraions of a Musical Noe A uba plaer plas he noe E and susains he sound for some ime. For a pure E he variaion in pressure from normal air pressure is given b where V is measured in pounds per square inch and in seconds. (a) Find he ampliude, period, and frequenc of V. (b) Skech a graph of V. ampliude a in. period p v sin 4p p 4p s frequenc v 4p Hz p p V. sin 8p (c) If he uba plaer increases he loudness of he noe, how does he equaion for V change? (d) If he plaer is plaing he noe incorrecl and i is a lile fla, how does he equaion for V change?

4 SECTION 5.5 Modeling Harmonic Moion 445. =. ß 8π Soluion (a) From he formulas for ampliude, period, and frequenc, we ge ampliude.. _. Figure 5 (s) period p 8p 4 frequenc 8p p (b) The graph of V is shown in Figure 5. 4 (c) If he plaer increases he loudness he ampliude increases. So he number. is replaced b a larger number. (d) If he noe is fla, hen he frequenc is decreased. Thus, he coefficien of is less han 8p. Example 3 Modeling a Vibraing Spring 4 cm A mass is suspended from a spring. The spring is compressed a disance of 4 cm and hen released. I is observed ha he mass reurns o he compressed posiion afer 3 s. (a) Find a funcion ha models he displacemen of he mass. (b) Skech he graph of he displacemen of he mass. Res posiion Soluion (a) The moion of he mass is given b one of he equaions for simple harmonic moion. The ampliude of he moion is 4 cm. Since his ampliude is reached a ime, an appropriae funcion ha models he displacemen is of he form a cos v Since he period is p 3, we can find v from he following equaion: period p v 4 =4 ç 6π 3 p v Period 3 v 6p Solve for v So, he moion of he mass is modeled b he funcion 4 cos 6p _4 Figure 6 where is he displacemen from he res posiion a ime. Noice ha when, he displacemen is 4, as we expec. (b) The graph of he displacemen of he mass a ime is shown in Figure 6.

5 446 CHAPTER 5 Trigonomeric Funcions of Real Numbers In general, he sine or cosine funcions represening harmonic moion ma be shifed horizonall or vericall. In his case, he equaions ake he form a sinv c b or a cosv c b The verical shif b indicaes ha he variaion occurs around an average value b. The horizonal shif c indicaes he posiion of he objec a. (See Figure 7.) b+a =a ßÓÒ(-c)Ô+b b+a =a çóò(-c)ô+b b b b-a b-a c π c+ Ò c π c+ Ò Figure 7 (a) (b) Example 4 Modeling he Brighness of a Variable Sar A variable sar is one whose brighness alernael increases and decreases. For he variable sar Dela Cephei, he ime beween periods of maximum brighness is 5.4 das. The average brighness (or magniude) of he sar is 4., and is brighness varies b.35 magniude. (a) Find a funcion ha models he brighness of Dela Cephei as a funcion of ime. (b) Skech a graph of he brighness of Dela Cephei as a funcion of ime. Soluion (a) Le s find a funcion in he form a cosv c b The ampliude is he maximum variaion from average brighness, so he ampliude is a.35 magniude. We are given ha he period is 5.4 das, so v p Since he brighness varies from an average value of 4. magniudes, he graph is shifed upward b b 4.. If we ake o be a ime when he sar is a maximum brighness, here is no horizonal shif, so c (because a cosine curve achieves is maximum a ). Thus, he funcion we wan is.35 cos.6 4. Figure (das) where is he number of das from a ime when he sar is a maximum brighness. (b) The graph is skeched in Figure 8.

6 SECTION 5.5 Modeling Harmonic Moion 447 The number of hours of daligh varies hroughou he course of a ear. In he Norhern Hemisphere, he longes da is June, and he shores is December. The average lengh of daligh is h, and he variaion from his average depends on he laiude. (For example, Fairbanks, Alaska, experiences more han h of daligh on he longes da and less han4honheshores da!) The graph in Figure 9 shows he number of hours of daligh a differen imes of he ear for various laiudes. I s apparen from he graph ha he variaion in hours of daligh is simple harmonic Hours * N 5* N 4* N 3* N * N Figure 9 Graph of he lengh of daligh from March hrough December a various laiudes Mar. Apr. Ma June Jul Aug. Sep. Oc. Nov. Dec. Source: Lucia C. Harrison, Daligh, Twiligh, Darkness and Time (New York: Silver, Burde, 935), page 4. Example 5 Modeling he Number of Hours of Daligh In Philadelphia (4 N laiude), he longes da of he ear has 4 h 5 min of daligh and he shores da has 9 h min of daligh. (a) Find a funcion L ha models he lengh of daligh as a funcion of, he number of das from Januar. (b) An asronomer needs a leas hours of darkness for a long exposure asronomical phoograph. On wha das of he ear are such long exposures possible? Soluion (a) We need o find a funcion in he form a sinv c b whose graph is he 4 N laiude curve in Figure 9. From he informaion given, we see ha he ampliude is a A B.83 h Since here are 365 das in a ear, he period is 365, so v p.7 365

7 448 CHAPTER 5 Trigonomeric Funcions of Real Numbers = = Figure Since he average lengh of daligh is h, he graph is shifed upward b, so b. Since he curve aains he average value () on March, he 8h da of he ear, he curve is shifed 8 unis o he righ. Thus, c 8. So a funcion ha models he number of hours of daligh is.83 sin.7 8 where is he number of das from Januar. (b) A da has 4 h, so h of nigh correspond o 3 h of daligh. So we need o solve he inequali 3. To solve his inequali graphicall, we graph.83 sin.7 8 and 3 on he same graph. From he graph in Figure we see ha here are fewer han 3 h of daligh beween da (Januar ) and da (April ) and from da 4 (Augus 9) o da 365 (December 3). Anoher siuaion where simple harmonic moion occurs is in alernaing curren (AC) generaors. Alernaing curren is produced when an armaure roaes abou is axis in a magneic field. Figure represens a simple version of such a generaor. As he wire passes hrough he magneic field, a volage E is generaed in he wire. I can be shown ha he volage generaed is given b E E cos v where E is he maximum volage produced (which depends on he srengh of he magneic field) and v/ p is he number of revoluions per second of he armaure (he frequenc). Magnes N S Wh do we sa ha household curren is V when he maximum volage produced is 55 V? From he smmer of he cosine funcion, we see ha he average volage produced is zero. This average value would be he same for all AC generaors and so gives no informaion abou he volage generaed. To obain a more informaive measure of volage, engineers use he roo-mean-square (rms) mehod. I can be shown ha he rms volage is / imes he maximum volage. So, for household curren he rms volage is 55 V Figure Example 6 Wire Modeling Alernaing Curren Ordinar -V household alernaing curren varies from 55 V o 55 V wih a frequenc of 6 Hz (ccles per second). Find an equaion ha describes his variaion in volage. Soluion The variaion in volage is simple harmonic. Since he frequenc is 6 ccles per second, we have v 6 or v p p Le s ake o be a ime when he volage is 55 V. Then E a cos v 55 cos p

8 SECTION 5.5 Modeling Harmonic Moion 449 Damped Harmonic Moion The spring in Figure on page 443 is assumed o oscillae in a fricionless environmen. In his hpoheical case, he ampliude of he oscillaion will no change. In he presence of fricion, however, he moion of he spring evenuall dies down ; ha is, he ampliude of he moion decreases wih ime. Moion of his pe is called damped harmonic moion. Damped Harmonic Moion (a) Harmonic moion: =ß 8π a()=e If he equaion describing he displacemen of an objec a ime is ke c sin v or ke c cos v c hen he objec is in damped harmonic moion. The consan c is he damping consan, k is he iniial ampliude, and p/v is he period.* _a()=_e Damped harmonic moion is simpl harmonic moion for which he ampliude is governed b he funcion a ke c. Figure shows he difference beween harmonic moion and damped harmonic moion. (b) Damped harmonic moion: =e ß 8π Figure Hz is he abbreviaion for herz. One herz is one ccle per second. Example 7 Modeling Damped Harmonic Moion Two mass-spring ssems are experiencing damped harmonic moion, boh a.5 ccles per second, and boh wih an iniial maximum displacemen of cm. The firs has a damping consan of.5 and he second has a damping consan of.. (a) Find funcions of he form g ke c cos v o model he moion in each case. (b) Graph he wo funcions ou found in par (a). How do he differ? Soluion (a) A ime, he displacemen is cm. Thus g ke c # cosv # k, and so k. Also, he frequenc is f.5 Hz, and since v pf (see page 443), we ge v p.5 p. Using he given damping consans, we find ha he moions of he wo springs are given b he funcions g e.5 cos p and g e. cos p (b) The funcions g and g are graphed in Figure 3. From he graphs we see ha in he firs case (where he damping consan is larger) he moion dies down quickl, whereas in he second case, percepible moion coninues much longer. _ 5 _ 5 Figure 3 _ g ()= e.5 ç π _ g ()= e. ç π *In he case of damped harmonic moion, he erm quasi-period is ofen used insead of period because he moion is no acuall periodic i diminishes wih ime. However, we will coninue o use he erm period o avoid confusion.

9 45 CHAPTER 5 Trigonomeric Funcions of Real Numbers As he preceding example indicaes, he larger he damping consan c, he quicker he oscillaion dies down. When a guiar sring is plucked and hen allowed o vibrae freel, a poin on ha sring undergoes damped harmonic moion. We hear he damping of he moion as he sound produced b he vibraion of he sring fades. How fas he damping of he sring occurs (as measured b he size of he consan c) is a proper of he size of he sring and he maerial i is made of. Anoher example of damped harmonic moion is he moion ha a shock absorber on a car undergoes when he car his a bump in he road. In his case, he shock absorber is engineered o damp he moion as quickl as possible (large c) and o have he frequenc as small as possible (small v). On he oher hand, he sound produced b a uba plaer plaing a noe is undamped as long as he plaer can mainain he loudness of he noe. The elecromagneic waves ha produce ligh move in simple harmonic moion ha is no damped. Example 8 A Vibraing Violin Sring The G-sring on a violin is pulled a disance of.5 cm above is res posiion, hen released and allowed o vibrae. The damping consan c for his sring is deermined o be.4. Suppose ha he noe produced is a pure G (frequenc Hz). Find an equaion ha describes he moion of he poin a which he sring was plucked. Soluion Le P be he poin a which he sring was plucked. We will find a funcion f ha gives he disance a ime of he poin P from is original res posiion. Since he maximum displacemen occurs a, we find an equaion in he form ke c cos v From his equaion, we see ha f k. Bu we know ha he original displacemen of he sring is.5 cm. Thus, k.5. Since he frequenc of he vibraion is, we have v pf p 4p. Finall, since we know ha he damping consan is.4, we ge f.5e.4 cos 4p Example 9 Ripples on a Pond A sone is dropped in a calm lake, causing waves o form. The up-and-down moion of a poin on he surface of he waer is modeled b damped harmonic moion. A some ime he ampliude of he wave is measured, and s laer i is found ha he ampliude has dropped o of his value. Find he damping consan c. Soluion The ampliude is governed b he coefficien ke c in he equaions for damped harmonic moion. Thus, he ampliude a ime is ke c, and s laer, i is ke c. So, because he laer value is he earlier value, we have ke c ke c We now solve his equaion for c. Canceling k and using he Laws of Exponens, we ge e c # e c e c e c e c Cancel e c Take reciprocals

10 SECTION 5.5 Modeling Harmonic Moion 45 Taking he naural logarihm of each side gives c ln c ln.3. Thus, he damping consan is c Exercises 8 The given funcion models he displacemen of an objec moving in simple harmonic moion. (a) Find he ampliude, period, and frequenc of he moion. (b) Skech a graph of he displacemen of he objec over one complee period.. sin cos sin cos a.5 p 6. 3 sin..4 3 b 7. 5 cosa 3 3 4B 8..6 sin.8 9 Find a funcion ha models he simple harmonic moion having he given properies. Assume ha he displacemen is zero a ime. 9. ampliude cm, period 3 s. ampliude 4 f, period min. ampliude 6 in., frequenc 5/p Hz. ampliude. m, frequenc.5 Hz 3 6 Find a funcion ha models he simple harmonic moion having he given properies. Assume ha he displacemen is a is maximum a ime. 3. ampliude 6 f, period.5 min 4. ampliude 35 cm, period 8 s 5. ampliude.4 m, frequenc 75 Hz 6. ampliude 6.5 in., frequenc 6 Hz 7 4 An iniial ampliude k, damping consan c, and frequenc f or period p are given. (Recall ha frequenc and period are relaed b he equaion f /p.) (a) Find a funcion ha models he damped harmonic moion. Use a funcion of he form ke c cos v in Exercises 7, and of he form ke c sin v in Exercises 4. (b) Graph he funcion. 7. k, c.5, f 3 8. k 5, c.5, f.6 3 cos 9. k, c.5, p 4. k.75, c 3, p 3p. k 7, c, p p/6. k, c, p 3. k.3, c., f 4. k, c., f 8 Applicaions 5. A Bobbing Cork A cork floaing in a lake is bobbing in simple harmonic moion. Is displacemen above he boom of he lake is modeled b. cos p 8 where is measured in meers and is measured in minues. (a) Find he frequenc of he moion of he cork. (b) Skech a graph of. (c) Find he maximum displacemen of he cork above he lake boom. 6. FM Radio Signals The carrier wave for an FM radio signal is modeled b he funcion a sinp9.5 7 where is measured in seconds. Find he period and frequenc of he carrier wave. 7. Predaor Populaion Model In a predaor/pre model (see page 43), he predaor populaion is modeled b he funcion 9 cos 8 where is measured in ears. (a) Wha is he maximum populaion? (b) Find he lengh of ime beween successive periods of maximum populaion. 8. Blood Pressure Each ime our hear beas, our blood pressure increases, hen decreases as he hear ress beween beas. A cerain person s blood pressure is modeled b he funcion p 5 5 sin6p

11 45 CHAPTER 5 Trigonomeric Funcions of Real Numbers where p is he pressure in mmhg a ime, measured in minues. (a) Find he ampliude, period, and frequenc of p. (b) Skech a graph of p. (c) If a person is exercising, his hear beas faser. How does his affec he period and frequenc of p? 9. Spring Mass Ssem A mass aached o a spring is moving up and down in simple harmonic moion. The graph gives is displacemen d from equilibrium a ime. Express he funcion d in he form d a sin v. mean sea level. Skech a graph ha shows he level of he ides over a -hour period. 3. Spring Mass Ssem A mass suspended from a spring is pulled down a disance of f from is res posiion, as shown in he figure. The mass is released a ime and allowed o oscillae. If he mass reurns o his posiion afer s, find an equaion ha describes is moion. d() Res posiion f 3. Tides The graph shows he variaion of he waer level relaive o mean sea level in Commencemen Ba a Tacoma, Washingon, for a paricular 4-hour period. Assuming ha his variaion is modeled b simple harmonic moion, find an equaion of he form a sin v ha describes he variaion in waer level as a funcion of he number of hours afer midnigh. Mean sea level (fee) 6 _6 _ MIDNIGHT A.M. P.M. (ime) MIDNIGHT 3. Tides The Ba of Fund in Nova Scoia has he highes ides in he world. In one -hour period he waer sars a mean sea level, rises o f above, drops o f below, hen reurns o mean sea level. Assuming ha he moion of he ides is simple harmonic, find an equaion ha describes he heigh of he ide in he Ba of Fund above 33. Spring Mass Ssem A mass is suspended on a spring. The spring is compressed so ha he mass is locaed 5 cm above is res posiion. The mass is released a ime and allowed o oscillae. I is observed ha he mass reaches is lowes poin s afer i is released. Find an equaion ha describes he moion of he mass. 34. Spring Mass Ssem The frequenc of oscillaion of an objec suspended on a spring depends on he siffness k of he spring (called he spring consan) and he mass m of he objec. If he spring is compressed a disance a and hen allowed o oscillae, is displacemen is given b f a cos k/m (a) A -g mass is suspended from a spring wih siffness k 3. If he spring is compressed a disance 5 cm and hen released, find he equaion ha describes he oscillaion of he spring. (b) Find a general formula for he frequenc (in erms of k and m). (c) How is he frequenc affeced if he mass is increased? Is he oscillaion faser or slower? (d) How is he frequenc affeced if a siffer spring is used (larger k)? Is he oscillaion faser or slower? 35. Ferris Wheel A ferris wheel has a radius of m, and he boom of he wheel passes m above he ground. If he ferris wheel makes one complee revoluion ever s, find an

12 SECTION 5.5 Modeling Harmonic Moion 453 equaion ha gives he heigh above he ground of a person on he ferris wheel as a funcion of ime. second (rps). If he maximum volage produced is 3 V, find an equaion ha describes his variaion in volage. Wha is he rms volage? (See Example 6 and he margin noe adjacen o i.) m m 4. Biological Clocks Circadian rhhms are biological processes ha oscillae wih a period of approximael 4 hours. Tha is, a circadian rhhm is an inernal dail biological clock. Blood pressure appears o follow such a rhhm. For a cerain individual he average resing blood pressure varies from a maximum of mmhg a : P.M. o a minimum of 8 mmhg a : A.M. Find a sine funcion of he form f a sinv c b 36. Clock Pendulum The pendulum in a grandfaher clock makes one complee swing ever s. The maximum angle ha he pendulum makes wih respec o is res posiion is. We know from phsical principles ha he angle u beween he pendulum and is res posiion changes in simple harmonic fashion. Find an equaion ha describes he size of he angle u as a funcion of ime. (Take o be a ime when he pendulum is verical.) 37. Variable Sars The variable sar Zea Gemini has a period of das. The average brighness of he sar is 3.8 magniudes, and he maximum variaion from he average is. magniude. Assuming ha he variaion in brighness is simple harmonic, find an equaion ha gives he brighness of he sar as a funcion of ime. 38. Variable Sars Asronomers believe ha he radius of a variable sar increases and decreases wih he brighness of he sar. The variable sar Dela Cephei (Example 4) has an average radius of million miles and changes b a maximum of.5 million miles from his average during a single pulsaion. Find an equaion ha describes he radius of his sar as a funcion of ime. 39. Elecric Generaor The armaure in an elecric generaor is roaing a he rae of revoluions per Blood pressure (mmhg) ha models he blood pressure a ime, measured in hours from midnigh AM 6 AM PM 6 PM AM 6 AM 4. Elecric Generaor The graph shows an oscilloscope reading of he variaion in volage of an AC curren produced b a simple generaor. (a) Find he maximum volage produced. (b) Find he frequenc (ccles per second) of he generaor. (c) How man revoluions per second does he armaure in he generaor make? (d) Find a formula ha describes he variaion in volage as a funcion of ime. (vols) 5 _5. ) (s)

13 454 CHAPTER 5 Trigonomeric Funcions of Real Numbers 4. Doppler Effec When a car wih is horn blowing drives b an observer, he pich of he horn seems higher as i approaches and lower as i recedes (see he figure). This phenomenon is called he Doppler effec. If he sound source is moving a speed relaive o he observer and if he speed of sound is, hen he perceived frequenc f is relaed o he acual frequenc f as follows: f f a b We choose he minus sign if he source is moving oward he observer and he plus sign if i is moving awa. Suppose ha a car drives a f/s pas a woman sanding on he shoulder of a highwa, blowing is horn, which has a frequenc of 5 Hz. Assume ha he speed of sound is 3 f/s. (This is he speed in dr air a 7 F.) (a) Wha are he frequencies of he sounds ha he woman hears as he car approaches her and as i moves awa from her? (b) Le A be he ampliude of he sound. Find funcions of he form A sin v ha model he perceived sound as he car approaches he woman and as i recedes. 43. Moion of a Building A srong gus of wind srikes a all building, causing i o swa back and forh in damped harmonic moion. The frequenc of he oscillaion is.5 ccle per second and he damping consan is c.9. Find an equaion ha describes he moion of he building. (Assume k and ake o be he insan when he gus of wind srikes he building.) 44. Shock Absorber When a car his a cerain bump on he road, a shock absorber on he car is compressed a disance of 6 in., hen released (see he figure). The shock absorber vibraes in damped harmonic moion wih a frequenc of ccles per second. The damping consan for his paricular shock absorber is.8. (a) Find an equaion ha describes he displacemen of he shock absorber from is res posiion as a funcion of ime. Take o be he insan ha he shock absorber is released. (b) How long does i ake for he ampliude of he vibraion o decrease o.5 in? 45. Tuning Fork A uning fork is sruck and oscillaes in damped harmonic moion. The ampliude of he moion is measured, and 3 s laer i is found ha he ampliude has dropped o 4 of his value. Find he damping consan c for his uning fork. 46. Guiar Sring A guiar sring is pulled a poin P a disance of 3 cm above is res posiion. I is hen released and vibraes in damped harmonic moion wih a frequenc of 65 ccles per second. Afer s, i is observed ha he ampliude of he vibraion a poin P is.6 cm. (a) Find he damping consan c. (b) Find an equaion ha describes he posiion of poin P above is res posiion as a funcion of ime. Take o be he insan ha he sring is released. 5 Review Concep Check. (a) Wha is he uni circle? (b) Use a diagram o explain wha is mean b he erminal poin deermined b a real number. (c) Wha is he reference number associaed wih? (d) If is a real number and Px, is he erminal poin deermined b, wrie equaions ha define sin, cos, an, co, sec, and csc.