Curvature. Institute of Lifelong Learning, University of Delhi pg. 1

Transcription

1 Dicipline Coure-I Semeer-I Paper: Calculu-I Leon: Leon Developer: Chaianya Kumar College/Deparmen: Deparmen of Mahemaic, Delhi College of r and Commerce, Univeriy of Delhi Iniue of Lifelong Learning, Univeriy of Delhi pg. 1

2 able of Conen: Chaper : 1: Learning Oucome : Inroducion 3: o 3.1: Radiu of o 3.. Formula for o 3.3. n inerpreaion of in dimenional-pace o 3.4. Exercie 4. angenial and ormal Componen of cceleraion o 4.1. pplicaion of Modeling o 4.: Exercie Summary Reference for furher reading 1. Learning Oucome fer you have read hi chaper, you hould be able o Define he curvaure, find he radiu of curvaure, inerpreaion of curvaure in wo dimenional pace, find he angenial and normal componen of acceleraion. Iniue of Lifelong Learning, Univeriy of Delhi pg.

3 . Inroducion he curvaure of a curve a a given poin i a meaure of he rae of change of bending of he curve a ha poin. You can imagine ha he urn on a 5 paie coin indicae a more rapid change of direcion per uni lengh of arc lengh han ha on a one rupee coin. Mahemaically peaking, we ay ha he curvaure a a poin on a 5 paie coin i greaer han ha a he correponding poin on a one rupee coin. ow we give he proper definiion of he curvaure of a curve a a poin. 3. : If C i a pace curve defined by he funcion r (u), hen we have een ha dr du i a vecor in he direcion of angen o C. If he calar u i aken a he arc lengh S meaured from ome fixed poin on C, hen dr i a uni angen vecor o C and i denoed by. he rae a which change wih repec o S i a meaure of he of C and i given by d direcion of d. he a any given poin on C i normal o he curve a ha poin. If i a uni vecor in hi normal direcion, i i called he principal normal o he curve. hen d pecified poin, where i called he of C a he Iniue of Lifelong Learning, Univeriy of Delhi pg. 3

4 3.1. Radiu of :- he quaniy curvaure 1 i called he radiu of Definiion: If C i a mooh cure in wo dimenional pace or hree dimenional pace ha i parameerized by arc lengh hen he curvaure of C, denoed by Greek "kappa" i defined by () d r () Value addiion: lernaive formula for calculaing If r i a mooh curve, hen curvaure i k 1 d v d, where v i he v uni angen vecor and v. d Example 1: Show ha a circle of radiu a cenred a he origin ha conan curvaure 1 a. Soluion: he equaion of circle of radiu a cenred a origin in erm of arc lengh can be wrien a r() a co i a in j, o a a a On differeniaing above equaion w.r r() a in. i aco j a a a a Iniue of Lifelong Learning, Univeriy of Delhi pg. 4

5 i co in a gain differeniae w.r.. a 1 1 r() co. i in. j a a a a i in 1 1 co a a a j a j " 1 1 K() r() co in a a a a in 1 1 a a a a co Hence, circle ha conan curvaure 1 a. Example : Show ha a line in wo dimenional pace or hree dimenional pace ha zero curvaure. Soluion: he equaion of a line in wo dimenional pace and hree dimenional pace in erm of arc lengh i given by r r0 u (1) Where r 0 i he erm in of poin on he line and u i a uni vecor parallel o he line. On differeniaing equaion (1) w.r.. d r d r () r o u O u u r O i co an Iniue of Lifelong Learning, Univeriy of Delhi pg. 5

6 gain Differeniaing w.r.., we ge ' d r d r () u 0 in ce u i con an k() r () Formula for : We ae he heorem which provide wo formula for curvaure in erm of a general parameer. heorem 1: If r () i a mooh vecor-valued funcion in wo dimenional pace or hree dimenional pace, hen for each value of a which r () exi, he k can be expreed a () and (i) () k() () r (ii) 3 r() r k() (3) r Where () i uni angen vecor a. Value ddiion: oe Formula () will be applicable, when be applicable when r () () i known, however formula (3) will and i derivaive are known. Iniue of Lifelong Learning, Univeriy of Delhi pg. 6

7 Example 3: Find he for he circular helix defined by x = a co, y = ain, z = c, where a > 0. Soluion: he radiu vecor for he helix i given by r() x() i y() j z() k aco i ain j c k On differeniaing above w.r.. r() ain i aco j c k gain Differeniaing w.r., we ge r() aco i ain j ow, r() r() i j k ain a co c a co ain o acin i j ac co k a Hence, r a a c in co a c in co r r ac ac a Iniue of Lifelong Learning, Univeriy of Delhi pg. 7

8 a c a 4 a c a Hence, k() 3 r r r a a c a a c a c 3 Which i independen o herefore, helix ha conan curvaure. Value ddiion: oe (1) if c = 0, he helix reduce o a circle of radiu a and i curvaure reduce o 1 a (ii) If a = 0, he helix become he z axi and i curvaure reduce o 0. Example 4 : Find he curvaure of he ellipe a he end poin of he major and minor axe defined by r co i 3in j, 0 and ue a graphing uiliy o generae he graph of k(). r i j ok Soluion: Given, co 3in On differeniaing above w.r.. we have r i j in 3co Iniue of Lifelong Learning, Univeriy of Delhi pg. 8

9 gain differeniae w.r.., we ge r i j co 3in ow, r() r() i j k in 3co o co 3in o i o j o k 6in 6co 6k Hence, r () co 3in 4co 9in r() r () 6 6 We know ha k() 3 r r r 6 4in 9co 3 (4) Iniue of Lifelong Learning, Univeriy of Delhi pg. 9

10 Since, he end poin of he minor axi are (, 0) and (-, 0) which correpond o = 0 and = repecively Figure 1 Subiuing hee value in equaion (4), we ge k k 6 6 k o 9 7 4in 0 9co k 7 9 4in 9co 3 Since he end poin of he major axi are (0, 3)and (0, -3) which correpond o and 3 repecively. Subiuing hee value in (4), we ge 6 6 k k in 9co 4 Iniue of Lifelong Learning, Univeriy of Delhi pg. 10

11 6 6 k k in 3 9co 3 4 Figure 3.3. n inerpreaion of in dimenional-pace: ueful geomeric inerpreaion of curvaure in wo dimenional pace can be obained by conidering he angel meaured couner clock wie from he direcion of poiive x axi o he uni angen vecor. We can expre in erm of a follow i j co in On differeniaing above equaion w.r.. d ini co j d d d d. d Iniue of Lifelong Learning, Univeriy of Delhi pg. 11

12 Figure 3 We known ha d d d k() d d in co d Which ell u ha curvaure in -dimenional pace can be expreed a he magniude of he rae of change of wih repec o S. Value addiion : Formula Summary d (1) k() r () k() r (3) k() 3 r r r (4) Formula for calculaing : If r i a mooh curve, hen Iniue of Lifelong Learning, Univeriy of Delhi pg. 1

13 d principal uni normal i d, d d where v i he uni v angen vecor Exercie I: Find curvaure k() for he following curve 1. in 1 co r i j k 3 3 r 1 i j, Find, and curvaure for he following curve r 3in i 4co j 4 k 3. 1 r i j k, r e co i e in j k 5. r 6 in i 6 co j 5 k 6. r i j 7. 3 r i j 8. 4co in r e i e j 3 9. Iniue of Lifelong Learning, Univeriy of Delhi pg. 13

14 r co i in j, 0 r i j k 11. 4co 4in 1 1 r i j k r i j k coh inh r i j k r co in i in co j 3k angenial and ormal Componen of cceleraion: paricle i moving along a mooh curve, o find componen of i acceleraion along he angen and he normal o he curve a any inan. hi Queion i anwered by he following heorem. heorem : angenial and ormal Componen of cceleraion n objec moving along a mooh curve ha velociy v and acceleraion v and d d k d d and i he are lengh along he rajecory. Proof: In order o derive he formula for Iniue of Lifelong Learning, Univeriy of Delhi pg. 14

15 d d Since he derivaive of () i orhogonal o We know ha d v d d d d By produc Rule of Differeniaion, we ge d d d d d d d d d d d d d d d d d k in ce k d d Value ddiion: Formula for angenial and ormal componen of cceleraion he acceleraion of a moving objec can be wrien a Where, d v i he angenial componen of acceleraion d k v d i he normal componen of acceleraion. Iniue of Lifelong Learning, Univeriy of Delhi pg. 15

16 Figure 4: Componen of acceleraion he following heorem provide alernaive formula for and in erm of derivaive r and r of he poiion vecor r (). heorem 3: Formula for he componen of acceleraion Le r () be he poiion vecor of an objec moving along a mooh curve C. hen he angenial and normal componen of he objec' acceleraion are given by r. r r and r r r Proof : Le be he angel beween r and r. Since he acceleraion i given by r Uing formula, r. r r r co Iniue of Lifelong Learning, Univeriy of Delhi pg. 16

17 We have co r co r r. r r Uing formula, in = rr r r, we have in r in r r r r Value ddiion : lernaive formula of calculaing normal componen of acceleraion Example 5: Find he angenial and normal componen of he acceleraion r i j k of an objec ha move wih poiion vecor 3 r i j k Soluion: Given 3 Iniue of Lifelong Learning, Univeriy of Delhi pg. 17

18 Differeniaing above w.r.. 3 r v i j k gain differeniae w.r. r i j 6 ow. 3 6 r r i k r r v i j k 6 6 Hence, he componen of acceleraion are 3 r. r v r v rr v 4 r v Iniue of Lifelong Learning, Univeriy of Delhi pg. 18

19 Example 6: n objec move wih poiion vecor r co, in, angenial and normal componen of acceleraion. Find Soluion: Given r co, in, On differeniaing w.r... we ge in,co,1 v r gain differeniae w.r... we ge r co, in, 0 d v in co 1 0 d co in 0 v 1 1 o 1 Hence, he acceleraion i normal o he curve pplicaion of Modeling: In order o compue he angenial and normal componen of acceleraion and, we will illurae ome applicaion. ccording o ewon' econd law of moion Iniue of Lifelong Learning, Univeriy of Delhi pg. 19

20 F m where i he acceleraion ince, F m m m F F where, d F m and F mk d d Example 7: endency of a vehicle o kid car weighing,700 lb make a urn on a fla road while raveling a 56 f/ec (abou 38 ml/h). If he radiu of he urn i 70 f. How much fricional force i required o keep he car from kidding? Soluion : Given, w =,700 lb g = 3 f/, = 70f d = 56 f/ We know ha m = W g k = he fricional force required o keep he car from kidding i given by Iniue of Lifelong Learning, Univeriy of Delhi pg. 0

21 F mk d b heorem 4: cceleraion of an objec wih conan peed he acceleraion of an objec moving wih conan peed i alway orhogonal o he direcion of moion. Proof: ince an objec ha conan peed r i conan By heorem, r i perpendicular o r Bu r i in he direcion of he objec' moion i orhogonal o he direcion of moion Example 8: Period of a aellie n arificial aellie ravel a conan peed in a able circular orbi 0,000 km above he earh' urface. How long doe i ake for he aellie o make one complee circui of he earh? Soluion : Since earh i a phere of radiu 6,440 k.m. heigh R 6,440 0,000 6,440 km Radiou of earh Iniue of Lifelong Learning, Univeriy of Delhi pg. 1

22 Le m and v be aellie' ma and peed hu for abiliy, we have mv R GmM R Where M i earh' ma and G i graviaion conan Bu GM = 398,600 km 3 / and By ubiuing R = 6,440, We ge GM 398, 600 v R 6, approximaely Le be he ime required for he aellie o make one complee circui of he earh R v 6, , econ approximaely = 11hr 53 min (approximaely) Example 9: Find he acceleraion calar componen, wihou finding and, wrie he acceleraion of he moion r in co i co in j, 0 in he form. Soluion : Given in co co in r i j dr v i j d co co in in in co Iniue of Lifelong Learning, Univeriy of Delhi pg.

23 in i co j v co in We know ha d v d 1 dv i j d in co co in 1 We know ha Exercie-II: Find he angenial and normal componen of he objec' acceleraion for he following curve. 5 1 r co i 1 co j in k r i j k. in co in 3. in co r a i a j a k Iniue of Lifelong Learning, Univeriy of Delhi pg. 3

24 r 1 3 i j 3 k r 3co, in 6. r in, co 7. r i e j r i j 8. Wrie in he form for he following curve a he green value of wihou finding and r 1 i j k, 1 9. r co i in j k, r i 1 j 1 k, r e co i e in j e k, 0 1. For he following problem he peed v of a moving objec i given. Find, and he angenial and normal componen of acceleraion a he indicaed ime 13. v 5 3 ; 1 Iniue of Lifelong Learning, Univeriy of Delhi pg. 4

25 14. v 1 ; v in co ; v e ; 0 Summary: In hi leon, we have emphaize on he following Definiion of curvaure, how o find he radiu of curvaure, inerpreaion of curvaure in wo dimenional and hree dimenional pace, o find he angenial and normal componen of acceleraion. Reference for furher reading: 1. M.J. Srau, G.L. Bradley and K.J. Smih, Calculu (3rd Ediion), Dorling Kinderley (India) Pv. Ld. (Pearon Educaion), Delhi, H. non, I. Biven and S. Davi (7h Ediion), Johan Wiley and ong (ia) Pv. Ld. Singapore, M.D. Weir, J. Ha, F.R. Giordano, homa' Calculu (11h Ediion) Pearon Educaion in Souh ia. 4. homa and Finney, Calculu (9h Ediion), Pearon Educaion in Souh ia. Iniue of Lifelong Learning, Univeriy of Delhi pg. 5