15. Bicycle Wheel. Graph of height y (cm) above the axle against time t (s) over a 6-second interval. 15 bike wheel

Transcription

1 15. Biccle Wheel The graph We moun a biccle wheel so ha i is free o roae in a verical plane. In fac, wha works easil is o pu an exension on one of he axles, and ge a suden o sand on one side and hold he wheel sead while anoher suden sands on he oher side and roaes he wheel a leisurel pace. I ie a small sick cross-wise a some poin on he circumference and orien he wheel so he class sees i as a verical axis, and sees he small sick as a horizonal bar ha oscillaes up and down. I ask he suden o roae he wheel a a consan rae (using a finger insered a a spoke hear he axle). I ask he class o focus onl on he sick, and forge everhing else. The objecive is o describe he heigh of he sick as a funcion of ime. I ask hem o draw he graph. Le be he heigh of he sick, and be ime, so we wan he graph of agains. To specif he saring condiions we ake =0 o be he midpoin of he oscillaion (when he sick is level wih he axle) and suppose he sick sars here a =0. And also suppose he sick begins b moving upwards. Mos sudens draw a graph like ha a he righ. Now we need o pu a scale on he graph and ha means aking measuremens. Firs he -scale. We measure he radius of he wheel o be cm, so ha he oscillaion is beween = and =. Now he -scale. We need o know he ime for a complee revoluion of he wheel. How are we o measure ha? The obvious suggesion is o ime one revoluion, bu a beer idea, o reduce error, is o ime a number of hese and divide. We find ha 10 revoluions ake 16 seconds, giving us 1.6 seconds per revoluion. The graph wih scale insered is drawn a he righ. Time is measured in seconds. The moion of he sick is compleel specified b is rajecor over a complee revoluion, because his is simpl repeaed again and again. We sa ha he verical moion of he sick is periodic, wih period 1.6 seconds, and ampliude cm Isn' his jus he sin graph? Didn we jus cover ha? Well ou're righ, excep he variable is insead of θ. So we have o cope wih ha. Graph of heigh (cm) above he axle agains ime (s) over a 6-second inerval. 15 bike wheel 1

2 The search for an equaion Now ha we have a graph, I ask he class o find an equaion for i. Find a formula which will give us a an value of. I give hem a few momens o absorb he quesion, bu no-one moves. The seem perplexed. There's a crucial srucural idea o ge hold of here, and ha is ha he formula we are afer is reall a cascade of wo funcional relaionships (composiion of funcions). I draw on he board he diagram a he righ and ask "wh does change?" Because θ changes! Tha's righ depends on because depends on he angle θ and in roaing he wheel we are making θ depend on. Tha is: is a funcion of θ θ is a funcion of.. The sraeg suggesed b his decomposiion is ha we find expressions for each of hese funcions and hen pu hem ogeher o obain an expression for in erms of. Finding One a a ime. Le s firs r o find in erms of θ. This is a simple maer of using he rigonomer of he riangle: θ = sinθ A suden calls me over. She is confused. We have =0 a θ=0 = sinθ. Check ha his does wha we wan i o do. As θ increases, sinθ oscillaes beween +1 and 1, and so will oscillae beween + and. As required. Now we wan θ in erms of. Ineresingl enough, he class has much more rouble wih his one rouble undersanding exacl wha we mean b saing θ is a funcion of. [Yes, here s a bi of jargon in ha. I s no how ou d sa i "on he sree" is i.] Wha we mean is his. When we sar a an agreed poin θ=0 a =0 and hen roae he wheel a a fixed speed, we are making θ change wih ime, so ha a differen imes we have differen values of θ, and ha makes θ ino a funcion of. Can we find an expression for his funcion. and = a θ=90, so shouldn we have =15 a θ=45? Bu when she plugs θ=45 ino he formula, she ges = sin45 = 21.2 and ha s cerainl no 15. Wha wen wrong? I s an excellen quesion hough i dismaed me a firs. She is making he assumpion ha sinθ is proporional o θ. Bu i ain. I ask her o sud he graph of = sinθ on [0,90]. Wha would i have o look like for her reasoning o be correc? 15 bike wheel 2

3 Finding θ I ge hem o abulae some simple (,θ) values. [Noe ha in he able θ is repored in degrees. In spie of he fac ha we have recenl inroduced radians, he enaciousl hang ono degrees. I don' reall blame hem radians doesn' reall come ino heir own unil he calculus. However, below we will produce he formula for boh measures.] Now wha's he formula ha would give his able? I am me wih silence. The seem o be msified. Tha is an unexpeced and slighl bewildering surprise for me. θ How do I lead hem? I ask hem o find he value of θ which belongs o =0.1. And in a bi, mos of hem repor i as one quarer of 90 which is θ = Now wha have he done o ge his resul? he have assumed ha θ increases in proporion o. If θ increases b 90 over 0.4 seconds, hen over 0.1 seconds i will increase b: And a an ime i will be: θ = 90 Acuall, we d like o wrie his a bi differenl. The number ha has paricular significance is no 0.4 bu 1.6 as ha is he period of he moion, so we d like o see ha in he formula. So we wrie insead: And ha s our formula. 0.4 θ = Is his righ? Does θ in fac increase in proporion o? Yes i does? Wh? because he wheel roaes a a consan rae. Oka, le s pu he wo formulas ogeher. and ha makes: a funcion of θ: = sinθ θ a funcion of : θ = a funcion of : = sin bike wheel 3

4 Analsis of he formula I s worh doing a lile "analsis" of his algebraic form. Ever number has a meaning. Ampliude The, which is he radius of he wheel, is he ampliude of he moion. Tha s where he word amplifier and amp come from. Because if his were a sound wave insead of a roaing wheel, he magniude of he oscillaion would deermine he loudness of he sound. Turning up he amplifier means increasing he ampliude of he waves. Period The 1.6 is he ime i akes he wheel o make one revoluion and is he period of he oscillaion. When increases b 1.6, he expression in he round brackes will increase b 360, and ha will make he sin funcion reurn o where i was. Degrees and radians. Finall, he 360 is clearl here because we are measuring angles in degrees. If we were using radians, hen an increase in of 1.6 would have o change wha was in he round brackes b 2π, and so we'd have o replace 360 b 2π. Thus, we ge wo forms of he -equaion, depending on how we measure angles. Degree form: = sin Radian form: = sin 2π = sin Sinx can mean wo differen hings depending on wheher x is measured in radians or degrees. Thus when calculaing rig funcions, make sure ou know which meaning is inended, and check ha our calculaor is se righ. This ambigui provides wo differen forms for ever equaion. In calculus he convenion is ha radians are used. In fac, he calculus allows us o see wh he radian is he "naural" measure. Wha would he formulas be for oher wheels urning a oher consan raes? If we le A be he radius of he wheel, and p be he period, hen, Degree form: = A sin 360. p Radian form: = A sin 2 π. p A -A p 15 bike wheel 4

5 Problems 1. The heigh (cm) of a sick aached o he rim of a roaing wheel a an ime (s) is given b he formula: = 25 sin 2π where he angle is measured in radians. 0.5 (a) Wha is he period and ampliude of he moion? (b) Draw he graph of agains showing wo full periods. Be sure o include a scale on boh axes. (c) Calculae a =0.1 s? (d) How high is he sick afer exacl 1 minue? (e) Using he sin 1 buon on our calculaor, find he firs wo imes a which he sick has heigh 20 cm. Beside each, sae wheher he sick is moving up or down. Illusrae hese on our graph. (f) Find he firs ime a which he sick has heigh 10 cm and is moving up. (g) How man imes does he sick find iself a heigh 20 cm in he firs seconds? Illusrae hese on our graph and calculae hem exacl using he sin 1 buon on our calculaor. (h) Wha would he formula be if we waned o calculae he sin in degree mode? 2. The heigh (cm) of a sick aached o he rim of a roaing wheel a an ime (s) is given b he formula: = 25sin(40) where he angle is measured in degrees. (a) Wha is he period and ampliude of he moion? (b) Calculae a =20 s? (c) Using he sin 1 buon on our calculaor, find he firs ime a which he sick has heigh 10 cm and is moving up. (d) Wha would he formula be if we waned o calculae he sin in radian mode? 3. The heigh (cm) of a sick aached o he rim of a roaing wheel a an ime (s) is given b he formula: = 25 sin4 where he angle is measured in radians. (a) Wha is he period and ampliude of he moion? (b) Draw he graph of agains showing wo full periods. Be sure o include a scale on boh axes. (c) Calculae a =0.1 s? (d) Using he sin 1 buon on our calculaor, find he firs ime a which =+10. (e) Wha would he formula be if we waned o calculae he sin in degree mode? 4. A sick is aached o he rim of a wheel roaing a a consan rae, and a =0 i has heigh =0 above he axle of he wheel and is moving upwards. The firs wo imes ha akes he value 100 are a =4 and a =8. Find he ampliude and period of he moion. Use he smmer properies of he graph o guide our hinking. 5. In he following quesions repor boh he degree and he radian form of he formula. (a) Suppose he biccle wheel of his secion was of radius 20cm, and urning a 60 per second. Wha hen would be he formula for in erms of? Again we sar wih =0 a =0, and suppose he sick begins b moving upwards. (b) Same wheel as in (a) bu wih a differen saring configuraion. Sar wih =0 a =0, bu suppose he sick begins b moving downwards. Sar wih a rough skech of he graph, and hen find he formula. 15 bike wheel 5

6 6. The arm OA has lengh 1 and is roaing counerclockwise abou O wih a period of 40 seconds. A he same ime a spherical rabbi runs a consan speed back and forh beween O and A wih period 10 seconds. Tha is i akes he rabbi 10 seconds o go from O o A and back o O again. Suppose a =0, he rabbi sars a O and he arm sars horizonal wih A o he righ of O. (a) Wha is he period of he enire moion? (b) Make a skech inside he circle of he rajecor of he rabbi over one complee period. (c) Take he origin of he coordinae ssem o be a O. Find equaions for he x and coordinaes of he rabbi as funcions of ime. O P A 7. In he diagram a he righ he big circle has cenre a he origin and radius 2, and he small circle has cenre a (3,0) and radius 1. Thus he small circle is angen o he big circle a he poin (2,0). Now here s wha happens. The small circle roaes counerclockwise and ravels wihou slipping around he big circle. Suppose i akes exacl 4 seconds o make a complee revoluion, so ha a =4 i is back exacl as shown in he picure. The problem is o find he locus of he poin P on he small circle which sars a (4,0). Specificall, find he x and -coordinaes of P as funcions of ime. (a) Sar b geing a feeling for wha happens. Examine he configuraion a he values = 1,2,3,4, ec. How man imes has he small circle roaed in raveling compleel around he big circle? Draw an approximaion o he pah ha P follows. (b) Le α be he angle from he cenre of he big circle hrough which he poin of conac has roaed and le β be he angle from he cenre of he small circle hrough which he poin P has roaed. Find formulas for x and in erms of α and β. Then find β in erms of α. Finall find β in erms of and ou should have our formula. P x 15 bike wheel 6