Motion on a Curve and Curvature

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Edward Goodwin
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1 Moion on Cue nd Cuue his uni is bsed on Secions 9. & 9.3, Chpe 9. All ssigned edings nd execises e fom he exbook Objecies: Mke cein h you cn define, nd use in conex, he ems, conceps nd fomuls lised below: 1. define cuiline moion in ems of ecos.. sole poblems eling o elociy, speed, cceleion, mgniude of he cceleion nd ngul momenum 3. sole iniil lue poblems 4. find he cuue of plne cue nd spce cue 5. find he dius of cuue of plne cue gien poin. 6. sole pcicl poblems Reding: Red Secions 9. & 9.3, pges Execises: Complee poblems 1

2 Peequisies Befoe sing his Secion you should be fmili wih he concep of scl nd eco funcions. be fmili wih diffeeniing nd ineging eco funcions 3. be fmili wih cues nd ngen eco Discussion in his secion: Fis we pesen some exmples of moion on cue Nex we discuss he eco funcions of single ible hen wih some exmples we will show how eco funcions help in sudying moion of cue

3 Moion on Cue he posiion of moing picle cn lwys be wien s eco funcion. x i y j z k In mechnics i cn be shown h: Velociy: 1 s ime deiie of he posiion eco Acceleion: d nd ime deiie of he posiion eco Exmple: he posiion eco of picle ime is gien by d Find he elociy, speed nd cceleion sec. b Find he elociy in he diecion u ij-1k 3

4 Moion on Cue: Obseions Speed mgniude of elociy speed d Relion beween cceleion, elociy, nd posiion disnce elociy cceleion posiion elociy 4

5 Engineeing nd Physics Exmples Cuiline Moion in he Plne A pojecile is lunched n ngleθ wih n iniil elociy nd n iniil heigh s j. Find he speed nd posiion ny imeinsn. s cosθi sinθj Giionlcceleion : gj 5

6 Engineeing nd Physics Exmples A wh ngle should he iplne wih speed 18 mph elese he supply pck o hi he ge? 6

7 Engineeing nd Physics Exmples Moemen of plnes ound sun F F Mm k u,, d dp m m u / Moemen of sellies ound gound 7

8 Exmples Exmple Gph he ce of moing picle wih posiion eco cos i sin j 3k Soluion: x y 4, z 3 Cicle in Plne 8

9 Exmple: Gph he picle posiion of moing picle gien by z y x 4; sin cos k j i Cicul Helix 9 9

9 9 Exmple: Gph he picle posiion of moing picle gien by z y x 4; sin cos k j i Cicul Helix z y x π / π / π π π / π / π π π / π / π π π / π / π π π / π / 3 z 4 y x,

10 Lengh of Spce Cue s: If is smooh funcion, hen he lengh of he smooh cue ced by is gien by wih s b b ' [ f < f, ' ] g, h [ g' ] >, hen [ h' ] b In mechnics when is he posiion eco, s is he moed disnce by picle. 1

11 11 Engineeing nd Physics Exmples Exmple: Velociy nd Acceleion. nd nd he ecos Gph he cue defined by.5 moing picle is gien by he posiion of k j i Pbol.5,, y x z y x k j i k j i i k j i P4,,5. Soluion:

12 Engineeing nd Physics Exmples Exmple: Cenipel Acceleion cosωi sinωj sin cos ω ωi ω ωj ω cos ωi sinω j ω, i.e., is in he opposie diecion of wih, / 1

13 Soluion: Engineeing nd Physics Exmples Exmple: Cuiline Moion in he Plne A pojecile is lunched wih n iniil elociy cos sin θ i θ j nd n iniil heigh : s s. j Find he speed nd posiion ny imeinsn. Giionl cceleion : gj, gj gj c 1 Since cosθ i sinθ j. c cos i sin 1 θ θ j 13

14 14 Exmple Con. j i sin θ cosθ g s g 1 y x sin θ, cosθ j i j c j c j i sin 1 cos sin 1 cos s g s s g θ θ θ θ Q H/W: Find he nge nd mximum heigh.

15 Engineeing nd Physics Exmples Exmple P9.-17Pojecile A pojecile is fied fom cnnon diecly ge h is dopped fom es simulneously s he cnon is fied. Show h he pojecile will sike he ge midi. x o,y o p, 15

16 Exmple Con. P, θ x o,y o 16

17 Exmple Con. P x o,y o θ Wih A h ime, he eicl componen of p is is which equls o he eicl componen of. 17

18 d d Impon Remk ds, cn we sy d ds b, s ds s 1 ds s? u Yes du O, ime e of disnce speed d d d s, cn we sy? No his issue is discussed in he nex secion

19 9.3 Cuue nd Componens of Acceleion Fo gien cue C: Uni ngen Veco: ngen: Uni ngen: Also: ds hus: d ds s c lengh 19

20 Definiion Cuue of smooh cue: d d ' κ ds ds ' Cuue Fs chnges Rdius of Cuue: ρ 1/κ κ depends on he speed. Fo gien cuue, he speed of moing c cn be ecommended o oid skidding. Slow chnges

21 Exmple: Cuue of Cicle Find he cuue of cicle wih dius. Soluion: cosi sinj nd sini cosj sini cosj nd cosi sinj 1 κ 1

22 Acceleion: ngenil nd Noml Uni Vecos Suppose picle moes on smooh cue C Since nd e pllel & 1 nd Also: d 1 d d is noml o C Uni noml: N d / d / ' Exmple: <, N '? > 1, < 1, > / 1

23 3 Acceleion: ngenil nd Noml Componens N N N d κ Recll: d d d d κ κ κ Q o Q o N / Qd o

24 Acceleion: N κ Ohe foms: Using he definiions of do nd coss poducs of &, one cn show h he ngenil nd noml N componens of e: N N d d N k κ 3 3 4

25 he Binoml he uni eco B x N which is pependicul o he plne conining nd N, is clled he binoml o C. B N, N, nd B: Moing ihedl Plne of nd N: Oscilling plne Plne of N nd B: Noml plne Plne of nd B: Recifying plne Exmple: <, > N <, > / '?, B 1 N? d & N 5

() t. () t r () t or v. ( t) () () ( ) = ( ) or ( ) () () () t or dv () () Section 10.4 Motion in Space: Velocity and Acceleration

() t. () t r () t or v. ( t) () () ( ) = ( ) or ( ) () () () t or dv () () Section 10.4 Motion in Space: Velocity and Acceleration Secion 1.4 Moion in Spce: Velociy nd Acceleion We e going o dive lile deepe ino somehing we ve ledy inoduced, nmely () nd (). Discuss wih you neighbo he elionships beween posiion, velociy nd cceleion you