Path (space curve) Osculating plane

Transcription

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2 Fo th cuilin motion of pticl in spc th fomuls did fo pln cuilin motion still lid. But th my b n infinit numb of nomls fo tngnt dwn to spc cu. Whn th t nd t ' unit ctos mod to sm oigin by kping thi ointtions th sm, shdd pln such s shown in th figu is dtmind. Th cuilin pth potion in th icinity of P is lso nclosd within this pln. This pln obtind in clos icinity of P is nmd s th osculting pln. Pth (spc cu) Osculting pln

3 Pth (spc cu) Osculting pln is lso contind within this pln just s th n locity nd ccltion ctos t P. In od to dscib th othogonl unit cto systm t P in noml nd tngntil coodints, thid unit cto is dfind s follows nd it is nmd s th binoml unit cto. b t n

4 Th ccltion dscibd in noml nd tngntil coodints will h no binoml componnt. t t ρ n Pth (spc cu) Osculting pln

5 As in th cs of pln motion, th ccltion hs componnt t tngnt to th pth du to th chng in mgnitud of th locity nd componnt n / ρ noml to th cu du to th chng in diction of th locity. As bfo, ρ is th dius of cutu of th pth t th point in qustion nd is msud in th osculting pln.

6 Th dimnsionl Ctsin Coodint Systm is simil to two dimnsions, only th z coodint nd its two tim ditis ddd to th two dimnsionl xpssions. Th unit cto in z diction is. Th position cto, locity cto nd ccltion cto dfind s follows: k x y

7 Position Vcto xi yj xi zk Vlocity Vcto yj xi zk Accltion Vcto yj zk x y Not tht in th dimnsions th position cto in two dimnsions is plcd by.

8 Only th z coodint nd its two tim ditis ddd to th two dimnsionl Pol Coodint xpssions. Th unit cto in z diction is gin. k k is constnt in tms of both mgnitud nd diction. Th nd componnts of th position, locity nd ccltion ctos th sm s in Pol Coodints, th z componnts th sm s in Ctsin Coodints.

9 z z,, zk zk VELOCITY VECTO POSITION VECTO ( ) ( ) z z z z,, zk ACCELEATION VECTO

10 Th Sphicl Coodint systm is dfind by th dil distnc nd two ngls nd. Th distnc fom th pln is dscibd by th ngl ; tht is, is th ngl btwn th position cto nd th pln.

11 Th unit ctos, nd ppndicul to ch oth nd thi positi dictions in th diction of th incs in ltd coodints. Th unit cto points th diction in which incss nd nd stys constnt. Th oth two unit ctos dscibd in simil wy.

12 , cos, ACCELEATION VECTO VELOCITY VECTO POSITION VECTO ( ) ( ) sin cos 1 sin cos cos dt d dt d ACCELEATION VECTO

13 Th Sphicl Coodint Systm is commonly usd in monitoing th motions of plns nd spccft by d tcking quipmnt nd dfining th positions nd motions of obotic ms.

14 1. Th pticl P mos long th spc cu nd hs locity m/s fo th instnt shown. At th sm instnt th pticl hs n ccltion whos mgnitud is 8 m/s. clcult th dius of cutu ρ of th pth fo this position nd th t incsing. t which th mgnitud of th locity is 4i j k

15 . An musmnt id clld th cokscw tks th pssngs though th upsid-down cu of hoizontl cylindicl hlix. Th locity of th cs s thy pss position A is 15 m/s, nd th componnt of thi ccltion msud long th tngnt to th pth is gcos γ t this point. Th ffcti dius of th cylindicl hlix is 5 m, nd th hlix ngl is γ40 o. Comput th mgnitud of th ccltion of th pssngs s thy pss position A.

16 3. An icft P tks off t A with locity o of 50 km/h nd climbs in th ticl y -z pln t th constnt 15 o ngl with n ccltion long its flight pth of 0.8 m/s. flight pogss is monitod by d t point O. sol th locity of P into cylindicl-coodint componnts 60 s ft tkoff nd find, nd z fo tht instnt. (Suggstion: Dw th ltd x- y nd x-z pojctions of th locity componnts. )

17 4. Th disc A otts bout th ticl z xis with constnt spd ω π/3 d/s. Simultnously, th hingd m OB is ltd t th constnt t of π/3 d/s. At tim t0, both 0 nd 0. Th ngl is msud fom th fixd fnc x-xis. Th smll sph P slids out long th od ccoding to 5000t, wh is in millimts nd t is in sconds. Dtmin th ccltion of P whn t1/ s.

18 5. Th obotic dic otts bout fixd ticl xis whil its m xtnds nd lts. At gin instnt, 30 o, 10 dg/sconstnt, l0.5 m, 0. m/s, -0.3 m/s, nd Ω0 dg/sconstnt. Dtmin th mgnitud of locity nd ccltion of gippd pt P. l l