Learning Objecives Circular and Harmonic Moion (Verical Transformaions: Sine curve) Algebra ; Pre-Calculus Time required: 10 150 min. The sudens will apply combined verical ranslaions and dilaions in he conex of applicaions of he sine funcion. Sudens will be exploring circular and harmonic moion graphically, symbolically, conexually, and hrough compuer simulaion. The examples are simplified in he fac ha ime is essenially ignored, bu heigh and range of moion are explored in-deph. Adjusmens in period lengh and saring poin will be addressed laer in he uni, so he examples can be revisied in ha more auhenic conex. We believe sudens will be more successful a ha poin afer having masered he verical componens of a sinusoidal curve firs. Mah Objecives Undersand verical ranslaions and dilaions as applied o he sine funcion. Specifically, sudens will creae equaions, graphs, and simulaions of sinusoidal curves, given a descripion of a relevan siuaion. Technology Objecives The suden will creae simulaions of verical componens of circular and harmonic moion on GX. Mah Prerequisies The suden mus undersand he sine curve, verical ranslaions, and verical dilaions. The suden mus undersand radian measure for angles of roaion. Technology Prerequisies Basic undersanding of Geomery Expressions, developed hrough earlier lessons in his uni. Maerials A compuer wih Geomery Expressions for each suden or pair of sudens. 008 Salire Sofware Incorporaed Page 1 of 1
Overview for he Teacher Circular and Harmonic Moion (Verical Transformaions: Sine curve) Algebra ; Pre-Calculus Time required: 10 150 min. The lesson sars wih a basic sine curve in he absrac, ying circular moion dynamically o he funcion graph, which is creaed as a locus in GX. Sudens hen do verical ranslaions and dilaions o i, and apply hose ideas o some basic circular moion conexs. Building on ha, sudens modify heir moion simulaions o creae harmonic moion firs conneced o a circular paern, hen o a funcion graph of a sine curve. There is an opional aciviy which asks sudens o firs produce simulaions, hen produce equaions and graphs based on simulaions of oher sudens. In shor, he lesson reinforces and pracices verical dilaions and ranslaions as applied o sine curves, and furher relaes hem o simplified realisic conexs and meanings. 1) This firs exercise reviews he sine curve as a record of verical moion of a poin around he uni circle. I also inroduces he idea of a locus and ses sudens up for he more advanced modeling and simulaions hey will be doing. When sudens consruc he locus, GX will choose values for which mach hose in he variables window by defaul. These can be changed in he locus pop-up. B 1 A (,0) C (,sin()) You may wan o discuss he locaion of key poins wih sudens in erms of fracions of π, since hose inervals aren readily obvious o some sudens, and he axis scale is in inegers. Many of hem will, a leas menally, need o ranslae his o degrees o make good sense of he paern. Of course, you can add more eacher checkpoins o he lesson, bu his one is fairly imporan o make sure sudens are on he righ rack. ) A) y = sin() + 3 B) Remind sudens ha hey mus ype in he * o indicae muliplicaion in GX. Sudens should be able o produce he modified model wih only a couple of changes o values. Encourage hem no o sar over from scrach. B C A (,3) (,3+ sin()) 008 Salire Sofware Incorporaed Page of 1
Circular and Harmonic Moion (Verical Transformaions: Sine curve) Algebra ; Pre-Calculus Time required: 10 150 min. C) A verical dilaion of he funcion corresponds wih a proporional srech of he radius of he circle. D) A verical shif of he funcion corresponds wih a verical shif of he circle. 3) A) A verical dilaion of magniude 5, and a verical ranslaion of unis (up). B) y = 5sin(x) + C) 10 8 A (-5,) 5 B C (,+5 sin()) -10-8 Harmonic Moion: Sudens can quickly see how harmonic and circular moion correspond. We highly recommend doing he following more deailed demonsraion model for he class. Some sudens will find his more visually inuiive for he paern described. Plo a poin direcly above your poin F E G C, consrain is posiion, and connec 1 (,1) i o C wih a line segmen. Make his segmen a dashed line o 10 represen he spring by highlighing i, righ clicking, and using 8 properies/line syle. Draw a horizonal line segmen which A (-5,) goes hrough ha poin o represen 5 he board. B C a Draw a small circle wih cener on H (,+5 sin()) poin C, and consrain is radius. While his adds nohing mahemaically o he drawing, when you animae i, i looks like a weigh suspended from a spring beneah a -10-8 board. Sudens may wan o duplicae his; exploring how o do so independenly may help hem o become more familiar wih he sofware if hey have exra ime. 008 Salire Sofware Incorporaed Page 3 of 1
Circular and Harmonic Moion (Verical Transformaions: Sine curve) Algebra ; Pre-Calculus Time required: 10 150 min. ) One hing o wach for is now we are ranslaing he sine curve down, insead of up. Also, he oal range of moion is 1 inches or unis, which makes a dilaion of magniude. Sudens will need o ype in a range of values for : eiher 0 hrough.8 or 0 hrough 1.5. A) Y = *sin()-10 B) A verical ranslaion of 10(down en unis) and a verical dilaion of scale facor. C) The coordinaes of he animaed poin should be some negaive consan for x (or x0wih a negaive value in he variables window), and *sin()-10 for y. D) The whole GX screen is given below. Sudens will only have he funcion copied, no he poins or coordinaes. A (,-10+ sin()) B (,-10+ sin()) -8-10 -1-1 -1-18 5) A) y = 3.5sin() 8 B) A verical dilaion w/ a scale facor 3.5, and a verical shif of 8(8 unis down). C) The coordinaes of he animaed poin should be some negaive consan for x (or x0wih a negaive value in he variables window), and 3.5*sin()-8 for y. D) The whole GX screen is given below. Sudens will only have he funcion copied, no he poins or coordinaes 008 Salire Sofware Incorporaed Page of 1
Circular and Harmonic Moion (Verical Transformaions: Sine curve) Algebra ; Pre-Calculus Time required: 10 150 min. A (,-8+3.5 sin()) -8-10 (,-8+3.5 sin()) B -1 Classmae Simulaion Challenge: Obviously, answers will vary. This aciviy is opional, bu may be a moivaing way for sudens o ge some pracice done. I can be expanded o have sudens aemp more han wo of heir classmaes models, or possibly made ino a game or cones. This all depends on ime, compuer access, suden ineres, and he insrucor s classroom managemen syle. A similar aciviy will follow he laer lesson and incorporae phase shifs and period changes. Pracice: Sudens should do independen pracice wriing equaions, graphs, and verbal siuaions. The Classmae Challenge is a good way o ge his pracice sared. Since such exercises are fairly sandard and easy o obain, we don include a full se here. Some ideas for eacher-creaed exercises are given below. In all cases, he period mus remain.8, and he objec in quesion mus sar ou a he saring poin of he naural sine curve. These consrains limi he range of he examples and force hem o be somewha conrived, bu ha will be largely remedied in he laer lesson. A Ferris wheel has a radius of 5 fee and is cenered 55 fee above he ground. Find he heigh of Tom afer seconds. (Modify he numbers as many imes as you like, jus make sure your heigh is always greaer han your radius.) Use he gear wheel of a machine, and se he problem up like he Ferris wheel (bu wih smaller values and/or differen unis). This can be se up as a large, indusrial machine, like in he lesson, or a small, precision machine, which migh beer allow for including decimal values for he dilaions and ranslaion. Harmonic Moion: Weigh on a spring hanging from a fixed poin. This one is a lile rickier, since he ime has o sar when he weigh is a is poin of equilibrium, and on is way up: A weigh hangs on he end of a spring below a fixed board. A res, i is 7 inches below he board. Someone sreches i down o 1 inches below, and les go. Assume he spring preserves all he energy from graviy, so he moion coninues up and down indefiniely. Le ime sar when he weigh is 7 inches below he board and on is way up. Give he equaion and he graph of he weigh s heigh as a funcion of ime. Again, modify he numbers as many imes as you wan. Give an equaion, and have sudens wrie a semi-plausible siuaion ha maches i. E.g. y = 10sin() + 13 A Ferris wheel wih radius 10 meers is cenered 13 meers above he ground. The equaion gives he heigh of a person afer seconds. 008 Salire Sofware Incorporaed Page 5 of 1
Circular and Harmonic Moion (Verical Transformaions: Sine curve) Algebra ; Pre-Calculus Time required: 10 150 min. Give a graph, and have sudens produce he equaion and/or a verbal siuaion. These can be creaed easily in GX wih he funcion ool. Then use Edi/Copy Drawing, open a Word documen, and Edi/Pase Special. Pase i as an enhanced meafile, and you won lose any special characers. You can crop, resize, and forma he diagram in Word. 008 Salire Sofware Incorporaed Page of 1
Name: Dae: Circular and Harmonic Moion As you may know, he sine funcion and is corresponding graphic curve are used o model almos all ypes of circular moion. They can also be used o model a similar paern called harmonic moion. In his lesson, you will invesigae hose paerns, and learn o represen hem wih equaions, funcion graphs, verbal descripions, and dynamic simulaions using compuer sofware. For now, we will use a simplified version of he paerns in which he cycle repeas iself every approximaely.8 ( π ) seconds or minues, and he moion always sars a he naural saring poin of he sine funcion. Laer in he uni, we ll learn how o deal wih changes in hese wo componens, and reurn o re-invesigae our models. 1) Model he basic sine curve: As you recall, sin(θ ) corresponds o he y-coordinae of a poin on he uni circle, roaed θ radians counerclockwise from he posiive x-axis. Open a GX file, and draw a circle anywhere wih poin A a is cener and B on he circle; consrain is radius o be one. For hese models, we are no going o be concerned wih he x-coordinae on he uni circle, or cos(θ ). Because of his, we can move he circle o he lef of he origin. Consrain he cener o a poin lef of he origin, such as (, 0). Draw in line segmen AB, and consrain he angle i makes wih he x-axis o be. The angle of roaion is, which in our simplified models also represens he ime poin B has been in moion (due o he cycle being.8 seconds, as menioned above.) Animae he drawing; show wo revoluions of poin B. To do his, selec in he Variables ool panel, and ype in 0 o 1.5 in he boxes a he boom, hen click on he play buon. Now you are going o le he x-axis represen ime, and creae a funcion of he heigh of poin B a various imes. Draw poin C anywhere, hen consrain is coordinaes o (, sin()). Reanimae he drawing. To see all he possible poin C s a once, we creae a locus. A locus is a se of all poins ha mee specific crieria (like having he coordinaes we saed.) Click on poin C and hen consruc locus. Reanimae he poin. Wha have you creaed? Have your eacher check your drawing and animaion, and iniial here: ) Modify he sine curve. A his poin, you know how o indicae a verical dilaion and ranslaion in a funcion equaion. A) Give he equaion for y = sin(x) afer a verical dilaion of and a verical ranslaion of 3: B) Wha change o he circle corresponds wih a verical srech of he funcion? C) Wha change o he circle corresponds wih a verical shif of he funcion? 008 Salire Sofware Incorporaed
B) Now modify your circle and he coordinaes of poin C o represen he changes you indicaed in par A. The animaion of your new model should sill work. Make sure he circle, poin C, he locus (funcion graph), and he animaion all sill mach each oher. Skech your resul below. 3) Now consider a gear wheel ha is par of an indusrial machine. I has a radius of 5 fee, and is mouned fee above he floor. (Again, assume a period of.8 seconds, and a saring poin direcly righ of cener.) We wan o know he heigh of a paricular poin on he wheel as a funcion of ime. A) Wha ransformaions of he paren funcion y = sin(x) mus be made o give a funcion equaion for his siuaion? B) Wrie he funcion equaion for he given siuaion. 008 Salire Sofware Incorporaed
C) Modify your model in GX o mach he siuaion. You may need o move your circle farher o he lef. Skech your resul below. 1 10 8-10 -8 Harmonic Moion: A closely relaed paern is called harmonic moion. To quickly ge a picure of his, delee your locus from problem 3, and change he x-coordinae consrain on poin C o a consan (like ). Run he animaion again. You can creae a coninuously repeaing paern by changing he animaion mode nex o he conrol buons in he variables ool panel o a double arrow. This simulaes he acion of a weigh hanging from a spring below a board. The resing posiion of he weigh is poin of equilibrium is a inches above he floor. The lowes poin i reaches is 1 inch above he floor, and is maximum heigh is 11 inches above he floor. The spring and he force of graviy counerac each oher in a coninuous back-and-forh paern. This paern would coninue indefiniely assuming he spring can capure and release 100% of he energy from graviy. ) A weigh is suspended beneah a able by a spring. Is resing posiion poin of equilibrium is 10 inches below he able. I is sreched o 1 inches below, and springs up o inches below, hen repeas is moion in a harmonic paern. Assume i complees a full cycle in.8 seconds, and ime sars when i is 10 inches below he able, on is way up. A) Wrie he equaion for his funcion, wih he ableop as a reference poin/heigh of zero: B) Wha ransformaions mus be done o he paren funcion y = sin() in order o produce his new funcion? C) Leing he x-axis represen he able, creae a simulaion of he weigh s moion in a new GX file and animae i. Wha did you inpu as he coordinae of he poin o accomplish his? 008 Salire Sofware Incorporaed
D) Make a funcion graph by ploing a poin wih he coordinaes (, your funcion equaion), and consrucing is locus. Copy your funcion graph. -8-10 -1-1 -1-18 5) A weigh is suspended beneah a able by a spring. Is resing posiion poin of equilibrium is 8 inches below he able. I is sreched o 11.5 inches below, and springs up o.5 inches below, hen repeas is moion in a harmonic paern. Assume i complees a full cycle in.8 seconds, and ime sars when i is 8 inches below he able, on is way up. A) Wrie he equaion for his funcion: B) Wha ransformaions mus be done o he paren funcion y = sin() in order o produce his new funcion? C) Leing he x-axis represen he able, creae a simulaion of he weigh s moion in a new GX file. Place i somewhere o he lef of he y-axis, and animae i. Wha did you inpu as he coordinae of he poin o accomplish his? -8 D) Make a funcion graph and copy i here: -10-1 008 Salire Sofware Incorporaed
Classmae Simulaion Challenge You are going o creae a simulaion of eiher circular or harmonic moion on GX, and one of your classmaes is going o deermine he equaion and he graph from only he moion you creae. 1) Creae your own simulaion. You decide if you wan o do circular or harmonic moion. Build your consrucion in GX. For oday, all simulaions will have a cycle of.8 seconds, and begin a he saring poin of he sine funcion. Also, make sure i will fi on he grid below. Wrie an equaion for he heigh of your poin as a funcion of ime, and skech he corresponding graph below. (This is he answer key.) Equaion: Graph: 10 8 ) Now hide all he elemens of your diagram excep he poin ha is moving. To do his, highligh each objec, righ click, and selec Hide. All ha should be visible on your screen is a coordinae grid and a poin. 008 Salire Sofware Incorporaed
3) Move o a classmae s saion and examine his or her model. Change he value of as much as you need o by scrolling, animaing, or yping in values, bu do no reveal any of he consrucions or consrains. Wrie he equaion, and skech he corresponding graph. Equaion: 10 8 ) Repea a a differen work saion: Equaion: 10 8 008 Salire Sofware Incorporaed