One of the common descriptions of curvilinear motion uses path variables, which are measurements made along the tangent t and normal n to the path of

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2 Oe of he commo descipios of cuilie moio uses ph ibles, which e mesuemes mde log he ge d oml o he ph of he picles. d e wo ohogol xes cosideed sepely fo eey is of moio.

3 These coodies poide ul descipio fo cuilie moio d e fequely he mos diec d coeie coodies o use; hey moe log he ph wih he picle. The posiie diecio fo y posiio is lwys ke owd he cee of cuue of he ph. Theefoe o posiio eco is equied.

4 We ow use he coodies d o descibe he elociy d cceleio. Fo his pupose, we ioduce ui ecos e i he -diecio d e i he -diecio. Duig diffeeil iceme of ime d, he picle moes diffeeil disce ds log he cue fom A o A: ds= db whee b is i dis.

5 I is uecessy o coside he diffeeil chge i bewee A d A fo diffeeil ime peiod. ds d db d b b e

6 The mgiude of elociy: ds d b db d db b d Theefoe, he elociy i eco fom is e b e

7 d The cceleio of he picle is defied s, d d we obsee h he cceleio is eco which eflecs boh he chge i mgiude d he chge i diecio of elociy. We ow diffeeie he elociy pplyig he odiy ule fo he diffeeiio of he poduc of scl d eco d ge d d d e d e e

8 e The ui eco ow hs ozeo deiie becuse lhough is mgiude sys he sme, is diecio chges. As he picle moes fom poi A o A, he ui eco e becomes. e de e The ecoil diffeece is: e e de e

9 e To fid we lyze he chge i duig diffeeil iceme of moio s he picle moes fom A o A. e I he limi, he mgiude of will be de e db db. Fo diffeeil ime iel, he diecio of be cosideed pepedicul o. Theefoe, he diecio of d de e d e e c c be cosideed he sme s h of. de e e db e e

10 The deiie of de db e e will be obied s, o de e db By diidig o d de d db e d e be b e e b e, b

11 e e e e e e e e e e e e e e b b b e e e e

12 Hee, s b b b b b b b (Chge i diecio of he elociy) (Chge i mgiude of he elociy)

13 Rdius of Cuue (Eğilik Yıçpı): If jecoy is gie s y=f(x), he dius of cuue c be deemied wih he followig fomul: 1 dy dx d y dx 3/ The bsolue lue sig is used o guee h will lwys be posiie +.

14 The elociy ecos A d A' e dw o s sme poi i ode o expli he chge i he mgiude d diecio of he eloicy eco d how hese chges geee he oml d geil compoes of he cceleio eco. The chge i elociy eco is d d i idices he diecio of he cceleio eco. Whe d is diided io wo compoes oe oml d he ohe ge o he elociy, he oml compoe is defied s d d i he limi i will be equl o he c legh obied by oig he elociy eco wih mgiude fo gle of db. d db The oml compoe of cceleio will be equl o he ime deiie of d. d d b

15 The geil compoe of is d. Is mgiude is equl o d o he chge i he legh of he elociy eco. d The geil compoe of cceleio will be equl o he ime deiie of d d d d d s

16 The oml compoe of cceleio is lwys dieced owd he cee of cuue. The geil compoe of cceleio, o he ohe hd, will be i he posiie -diecio of moio if speed is icesig d i he egie - diecio if he speed is decesig. Figue shows he schemic epeseios of he iio i he cceleio eco fo picle moig fom A o B wih () icesig speed d (b) decesig speed. A iflecio poi o he cue, he oml cceleio goes o zeo sice becomes ifiie. 0

17 Cicul moio is impo specil cse of ple cuilie moio whee he dius of cuue becomes he cos dius of he cicle d he gle b is eplced by he gle mesued fom y coeie dil efeece o OP. d d (Agul elociy) = = cos d d (Agul cceleio)

18 1. Six cceleio ecos e show fo he c whose elociy eco is dieced fowd. Fo ech cceleio eco descibe i wods he iseous moio of he c.

19 . A flywheel oes wih ible elociy. A cei ime, he geil cceleio of poi A o he flywheel is 1 m/s, while he oml cceleio of poi B o he flywheel is 0.6 m/s. Deemie he elociy of poi A d he ol cceleio of poi B. A 00 mm B 150 mm

20 3. A bsebll plye eleses bll wih he iiil codiios show i he figue. Deemie he dius of cuue of he jecoy () jus fe elese d (b) he pex. Fo ech cse, compue he ime e of chge of he speed.

21 4. A mble olls dow chue which is be i he shpe of pbol. Is equio is gie s y (m) y=x 6x + 9 [m] The mble psses poi A wih speed of 3 m/s, which is icesig he e of 5 m/s. Deemie he oml d geil compoes d, of he cceleio of he mble s i psses poi A. Also deemie he gle bewee he elociy eco d he cceleio eco poi A. y=f(x) A x=5 m x (m)

22 5. The pi P is cosied o moe i he sloed guides which moe igh gles o oe ohe. A he is epeseed, A hs elociy o he igh of 0. m/s which is decesig he e of 0.75 m/s ech secod. A he sme ime, B is moig dow wih elociy of 0.15 m/s which is decesig he e of 0.5 m/s ech secod. Fo his is, deemie he dius of cuue of he ph followed by P.

23 6. A bll is how hoizolly fom he op of 50 m cliff A wih speed of 15 m/s d lds poi C. Becuse of sog hoizol wid he bll hs cos cceleio i he egie x-diecio. Deemie he dius of cuue of he ph of he bll B whee is jecoy mkes gle of 45 o wih he hoizol. Neglec y effec of i esisce i he eicl diecio.

24 7. I he desig of cool mechism, he eicl sloed guide is moig wih cos elociy x = 150 mm/s duig he iel of is moio fom x = -80 mm o x = +80 mm. Fo he is whe x = 60 mm, clcule he - d -compoes of cceleio of he pi P, which is cofied o moe i he pbolic gooe. Fom hese esuls, deemie he dius of cuue of he ph his posiio.

1. Six acceleration vectors are shown for the car whose velocity vector is directed forward. For each acceleration vector describe in words the

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