Kinetics. Central Force Motion & Space Mechanics

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Nigel Johnson
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1 Kintics Cntal Foc Motion & Spac Mcanics

2 Outlin Cntal Foc Motion Obital Mcanics Exampls

3 Cntal-Foc Motion If a paticl tavls un t influnc of a foc tat as a lin of action ict towas a fix point, tn t motion is call cntal-foc motion Exampls: plantay motion, lctostatic focs, cntifug

4 Cntal-Foc Motion P Consi a paticl of mass m act on by a foc F, wit cnt O θ O θ F 1 A Using t Equations of Motion fo Cylinical Cooinats, it can b sown tat (ivation omitt, stunt plas viw): θ θ F ma F m t t (13-11a) F F boy iagam F ma m t 0 t t (13-11b)

5 Cntal-Foc Motion W may -wit Eqn 13-11b in t fom 1 t t 0 tn by intgation t w is t constant of intgation (13-1)

6 Cntal-Foc Motion Sinc t paticl swps toug angl θ in tim intval t A t A 1 1 t A/t is call t aal vlocity. It mains constant fo a paticl in cntal-foc motion Tis mans t paticl swps toug qual aas in qual tim as it tavls along t pat

7 Cntal-Foc Motion Lt us iv t pat of motion as a function of θ By t cain ul (Calculus 101!) Lt us not t t t t t 1

8 Cntal-Foc Motion So tat w obtain also t squa of Eqn 13-1 bcoms Substituting t abov in Eqn 13-11a w obtain a iffntial quation wic can b solv to tmin t pat of motion t 4 t 3 m F m F OR (13-14)

9 Obital Mcanics Consi a spac vicl of mass m launc into f-fligt obit wit initial vlocity v o. Assum v o acts paalll to t tangnt to t at sufac Nglct gavitational attactions of sun an moon F-fligt tajctoy v 0 Spac vicl F launc Pow fligt tajctoy

10 Obital Mcanics At t instant just aft las into f fligt t only foc acting on it is t gavitational attaction fom t at Accoing to Nwton s law of gavitation; F G M m To obtain t obital pat, substitut into Eqn GM

11 Obital Mcanics T abov iffntial quation can b solv as t sum of t complmntay an paticula solutions (Rviw you Diffntial Equations ) Solution is: 1 GM C cos( ) (13-16)

12 Obital Mcanics Eqn is t quation of a conic sction [stunt, plas viw you P-Cal matials] By finition, Eccnticity 1 FP PA (PA) [ p cos( )] 1 p o cos( ) 1 p x x D A D ictix p P focus F

13 Obital Motion Compaing wit Eqn 13-16; an p C 1 C GM Povi θ is masu fom t x-axis wic is ppnicula to t ictix (an an axis of symmty), tn Ф = 0, an Eqn ucs to 1 (13-17) (13-18) GM C cos( ) (13-19)

14 Obital Motion T constants C an a tmin fom t bounay conitions at t n of t powfligt tajctoy At t bginning of f-fligt = o, v = v o ; if θ = Ф = 0, tn fom cuvilina motioncylinical componnts v t t o 0 v 0 (13-0)

15 Obital Motion Substituting Eqn 13-0, = o, θ = 0, into Eqn C 1 GM 1 0 0v 0 T quation fo t f-fligt tajctoy tfo bcoms (13-1) 1 1 GM GM 1 cos 0 0v (13-) 0 0v 0

16 Obital Motion T typ of pat tavl by t spac vicl pns on t valu of t ccnticity 0 cicl 1 1 paabola llip s (13-3) 1 ypbola [Stunts, plug in ts valus to t appopiat quations an vify ts conclusions]

17 Obital Motion Paabolic pat: T spaccaft is on t bolin of nv tuning to its stating point. T initial vlocity qui fo a paabolic pat is call t scap vlocity Plugging = 1, Eqns 13-1 an 13- into Eqn 1318; v GM 0 (13-4)

18 Obital Motion Similaly, fo Cicula Motion v GM c (13-5) 0 Not tat v o v will sult in vicl scaping at s gavitational pull On t ot an if v o < v c t vicl will fail to ac obit, nt at atmosp, an cas o bun up in t at of nty

19 Elliptical Obit All plants an most atificial satllits obit in an lliptical pat. Fo t spac caft t minimum istanc to t cnt of t at (wit at at a focus of t llips), p can b foun by plugging θ = 0 into Eqn 13. Using θ = 180 o w gt t max istanc a P O a b b a a

20 Elliptical Obit p 0 p is call t pig (gnally piapsis) a (GM 0 ) 1 a is call t apog (gnally apoapsis) Half t lngt of t majo axis a p a 0 v 0 (13-6) (13-8) (13-7)

21 Elliptical Obit It can also b sown tat b p a (13-9) (Stunts, vify on you own) By intgation, t aa of t llips is A ab ( p a ) p (13-30) a

22 Elliptical Motion T aal vlocity was fin in Eqn as A t o A t w T is t tim to mak on obital volution (aka obital pio). Fom 13-30: A T T ( p a ) pa (13-31)

23 Laws of Plantay Motion T toy vlop in tis capt was fist psnt by Joanns Kpl in 161, 6 cla cas bfo Nwton s Pincipia Kpl vlop t laws of plantay motion ov 0 yas by stuying plantay ata collct by is mnto Tyco Ba

24 Laws of Plantay Motion Plants tavl in lliptical obits wit t sun at a focus of t llips (Eqn 13-) Plants tavl in an obit suc tat ty swp qual aas in qual tim intvals (Eqn 13-13) T squa of t pio of any plant is ictly popotional to t cub of t majo axis of its obit (Eqns 13-31, 13-19, 13-8, 13-9)

25 Qustions & Commnts

GRAVITATION. (d) If a spring balance having frequency f is taken on moon (having g = g / 6) it will have a frequency of (a) 6f (b) f / 6

GRAVITATION. (d) If a spring balance having frequency f is taken on moon (having g = g / 6) it will have a frequency of (a) 6f (b) f / 6 GVITTION 1. Two satllits and o ound a plant P in cicula obits havin adii 4 and spctivly. If th spd of th satllit is V, th spd of th satllit will b 1 V 6 V 4V V. Th scap vlocity on th sufac of th ath is