CHAPTER Solution: The work done is equal to the change in kinetic energy kg 10m

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1 CHPTER. Sluti: 5kg Th wrk d i qual t th chag i kitic rgy U gx, r gx 4/. Sluti: Draw a FD t th frc actig. Th lb wight ha a a f /. lug. Cpar th chag i kitic rgy fr t ft/ with th wrk d agait th frc f fricti. Suig frc i th rtical dircti yild -N 5 i, r N 5 lb. Thu f µn (.)(5) 5 lb. Th frc i th hriztal dircti i f x 5 c - 5. Thu, 5 N f (4. 5). ad 5.5ft/

2 . Sluti: g 4 N f 45 Lt dt th prig cpri. Th t frc actig alg th icli i g i 45 - f whr f i th fricti frc. Suig frc prpdicular t th icli yild N g c 45 ()(9.8)(.77). Thu f µn (.6)()(9.8)(.77). Nw th t frc alg th icli will a ditac (4 ). Th wrk f th cprd prig i -k /. Sttig th wrk d ig th a dw th icli ad cprig th prig t zr yild: (gi 45 f )(4 ) k Writig thi all i tr f yild th quadratic quati.77()(9.8)(-.6)(4 ) 5, r , r. 9

3 .4 Sluti: uig fricti, th ly frc actig th particl ar graity ad th ral frc a idicatd i th FD part f th ktch. gi R N gc St th wrk d by th wight qual t th chag i kitic rgy whr dt th lcity a it la th urfac (ral frc d wrk). g( R R c ) () Suig th frc i th ral dircti f th particl uig ral-tagtial crdiat yild: ΣF a N gc. Lt β b th agl whr N ad uig th xpri fr th ral acclrati yild g c β () R Fr () with β: gr( cβ). Subtitut thi it Eq. () t gt gr cβ gr( cβ) r cβ gr, thu β c gr gr gr

4 .5 Sluti: g R Sktch th gtry f th yt. St th wrk d qual t th chag i kitic rgy t gt: U Slig fr yild U N ( i i ) ( ) gr.6 Sluti: g ( i i ) gr 5kg F Sic th prig i iitially cprd i th dii gi i th drawig, th wrk d crrpdig t th prig k (.5 ) ut b accutd fr i th calculati f th ttal wrk f th particl. Th iitial ad fial kitic rgy i zr, thrfr: [(.5) (.5) ] 5 5g ( ) Hr i th additial cpri f th prig caud by th 5 kg a. Expadig yild:

5 [. ] r Th piti luti fr i.494 Thu th axiu cpri i Th tp prig (k ) cpr a aut whil th btt prig (k 5) cpr a aut ( -.). Thrfr th ttal wrk d i qual t zr: g ( ) ( ) ( 5)(.) Thi fllw th luti f.7 xcpt th frc du t graity act r th ditac (. ) rathr tha. Th ttal wrk bc: g (.) ( ) ( 5)(.) Fr th figur, th wrk d by graity i gri. Lt β dt th alu f fr which th a c t rt that th arc frd by th cprd prig i dfid by β - π/. Th prig i thu cprd R(β - π/). Th ttal wrk i: griβ k R β π Thi tracdtal quati with kg, R., k N/ tak th fr Nurical luti yild: 5.886iβ 9 β π β.76rad.4

6 . Fr Figur P., th wrk d by graity i g l i ad th chag i kitic rgy (fr rt) i (/). Thu g li gli Th u f th frc i th tagtial dircti yild T gi l r T g( i i ) gi. Th lcity will b zr at axiu agl g li. Thrfr 8 (Th rt i th iitial piti.). Th fr-bdy diagra f th car ad trailr ad f th car ar W W P N (a) f (b) N f Fr (a) Σ F y W N r N W that f µ W. Nt 6ph 88 fp, that wrk -Fd - bc:.7wd W. ( 88) Slig fr d yild d 7.9 ft Fr (b) F ( P µ W ). Thu th wrk rlatihip bc Slig yild [ P.7( ) ] P ( )

7 . Sluti: y Wc (a) N (b) Wi f x Fr th drawig (a) a % grad dfi th agl ta - (/) 5.7 FD gi i (b) yild ΣF y N - Wi that N Wi ad f µn.7wc. F ΣF x -f Wi whr dw i piti. Wrk Fd chag i kitic rgy r: Thu W(i.7c)d W 64.4 ( 88), ( 88) d 64.4(.7c i ), d.4ft which i a bit largr tha i prbl. a xpctd..4 Sluti: 5 lb lb f N t (a) N c Figur (a) i a FD f th car-trailr yt. With fricti actig ly th car: ΣF y N r N ad f (.7)(). Th ttal a, hwr, i bad th u f th wight f bth th car ad th trailr (55 lb). Thu Fd (.5) bc:

8 Slig fr d yild:.7( )d 55. ( 88) d 4.9ft 5 P N (b) Nxt cidr a FD f jut th trailr gi i (b). Th frc i th x dircti i jut P. Thu Fd (.5) bc Slig fr P yild P ( 4.9) ( 88) P lb 5..5 t th btt f th lid t th wrk d by graity (gh) t th chag i kitic rgy (.5 ) t gt 69.8 ( ) 6, r 6.6 / t th flr, t th wrk d qual t th chag i kitic rgy (tartig fr th rt piti 4 up) i Thu g4 8g 8.86 /

9 .6 Sluti: gi gc g N f N f (a) (b) Fr th FD i (a), N gi that f µgc. Th ttal frc alg th icli i g (i. c ) which a ditac f 4. Fr th FD i (b) N g, f µgd that th ttal wrk d i: g( i.c )4 g.d D (i.c )4...7 Rlat th wrk d (Fd) t th chag i kitic rgy i ach ca. lck : g 4.5i / lck : g5.5i / lck C:

10 C g6.5i6 C.5/ lck D: D g7.5i 6 D.9 /.8 Sluti: P F P g(i.4 c ) P 66W.66kW.9 Thi fllw th luti f.8, xcpt i thi ca P i kw ad ukw: (9.8)(i.4c) i.4 c -.9, which i ld urically t yild:.86 rad Thi i th a a.8 xcpt P ad ar kw ad µ i ukw. (9.9)(i µc), r µ.6. Fr th figur, th right triagl yild th ctrait ( ) ( l ) x whr i th ditac th a fall ad x i th ditac trald by th prig ad a. Th alu f l i fud fr l.6. Th chag i rgy i

11 l l x 5g - x Slig th rgy quati ad th ctrait quati tgthr yild:.75 ad x.7. Diffrtiat ctrait q. i Prb.. t gt ( l ) ( x) Th wrk rgy quati bc that ( l ) ( x) 5g x () () 5 Subtituti f th xpri fr fr th driati f th ctrait yild 5g 5x ( l ) ( x).5 Fr th ctrait, ( l ) i liiatd t yild 9 5g 5x ( x).5 Subtitutig ad l.6 it th ctrait quati yild Slig fr yild.6 9 5g 5x ( x).5 [ ] 9 ( x).6 5g 5x 9 ( x).5 9 ( x)

12 Nt that thi luti chck bcau wh x.7, a prdictd by th luti f Prbl... Pick a fixd rfrc ad dfi x ad x a idicatd i th figur. Th ctrait th bc: Ctrait (x x ) (c x ) l 4N x fixd C x Diffrtiatig th ctrait quati yild Th wrk rgy rlati i (t that th ti i th cabl d wrk) 4(.) ( 5 ) Uig th ctrait rlati t liiat fr th wrk rgy quati yild: Slig fr yild: 5.55/.4 Fr th ctrait quati with x., x.4/. Th frc i 4 (.)()(9.8) which act r a ditac f.4/. Th frc i (.)(5)(9.8) which act r a ditac f.4/. 4 f f Thu th rgy-wrk quati bc

13 ( ) Slig yild.5 Sluti: 5.45 / x dw kg P 5kg Fr th figur thr ar tw rp ( ad l ) l which ctrai th ti accrdig t x xp l ad x x p l. Multiplyig th lat ctrait by ad addig t th firt ctrait yild th igl ctrait quati: x 4 x Ctat ad thu x 4 x ad 4. Fr th prbl tatt x.5 that x.5 Th wrk-rgy quati cbid with th ctrait quati yild: g(.5) 5g(.5) ( 4 ) 5 r , thu.7/.6 Sluti: p( x) x π (.) F( x) p( x) 4 F ( x) πx i dx MPa x 6 N.487,. that.9 /

14 .7 Sic th a f th bullt i ry all, w d t xpct fricti t b a igificat factr Th fr bdy diagra i gi i th figur g F(x) f N N g ad d wrk. f µ k g which d gati wrk. Fr Eq. (.6) U ( Fx ()f)dx (ld i ftwar) Th wrk d by fricti i -.59 N, which d t chag th lcity ad it rai.9 /..8 Sluti: g F kx l l-x U l ( kxdx) gl kl gl T T If lcity f a i zr du t ctrait f ti, th chag i kitic rgy bc

15 kl Slig fr yild kl gl gl.9 Th figur idicat th frc actig th a with th prig cprd. t th axiu dflcti th a i at rt, that T T. Lt δ b th dflcti f th prig at thi pit. Th th ttal wrk bc g( δ) δ g With kg, th cubic quati i ld (uig cputatial ftwar) t gi th axiu dflcti a: δ x F x.97. Th iitial cpri f prig yild k δ g. Th wrk rgy xpri bc (zr kitic rgy at pit f cpri). which i quadratic i Thi ha luti: U g( ) g k ( δ ) kδ g( ) k ( δ δ ) kδ k g δ k g 4g k 6g k

16 4g k ± g k ± r g k ± 4g k 4 8g k g k 4g k 6 g k 4 g k. Writ that th chag i kitic rgy f th car fr lcity t a dad tp, ut qual th wrk d by cpri f th bupr car F -kx bupr k. Slig fr k yild k. Th cart i aurd t ha a. l T Th ti frc d wrk. O th thr had th a wig up dig gati wrk. Thrfr U T T bc g( l l c ) whr i th iitial lcity f th cart ad it fial lcity i zr. Slig fr yild c r c gl gl.

17 Nt that th alu f th ci i alway l tha r qual t that (i.. c ) gl Thi ipli < g l. If th lcity i largr tha g l th th a cat b tppd ad will wig i a cplt circl.. Sic th ttal wrk d th ball i ( gh gh), th fial lcity f th ball will b..4 t th tp, r hight pit, th lcity g fr piti (up) t iu (dw) ad ut pa thrugh zr. Thr at th tp, tp. Th wrk rgy quati bc: U U T T, T gh Slig fr h yild h g T gh.5 Sic th wir i th, ly th wight d wrk. Lt W dt th wight. Th wrk rgy rlati bc: U Wh W g Slig yild 6.55ft/ gh with g. ad h 8, th lcity i

18 .6 Thi i th a a th priu prbl, xcpt th particl tart with kitic rgy. Th wrk d i tillu Wh. Nw th wrk rgy rlati bc Wh W g ( g ) r f gh f gh. Slig fr yild 6.85ft/.7 Thi wir i th ly th wight d wrk. Th wrk d i U gh Th chag i kitic rgy i T T ( ) Slig fr yild gh whr h, / Thu 4.5/.8 Firt, w ut xai th ctrait t th ti t dtri th rlatihip btw th lcity f ad. Thi ca b btaid fr th figur. Th lgth f cabl x x x - x. Cbiig yild x x l. Diffrtiatig thi lat xpri yild th lcity ctrait. Nw th wrk-rgy quati bc U T T r (.) ( 5 4), whr th wrk d i frc (N) ti ditac (.). Slig fr th lcity yild.98/. x x (x -x ) x x x

19 .9 cau f th lcity ctrait dlpd i.8, bth cllar will ha a iitial lcity f 5 / (i ppit dircti). Thr th wrk rgy rlatihip bc Slig fr yild (.) 5.8 / ( ) 9 ( 5).4 Fr Prbl.8 th ctrait yild that x x Thrfr th prbl i uchagd xcpt fr additi f fricti. Th fricti frc actig ach cllar i µg ad fricti alway d gati wrk. Thu th wrk rgy rlati U T T bc (.)(F µ ( )g) ( ) r ( ). () 9. Slig yild.75/ Nt that th fricti frc i gati hr a it pp ti..4 t th tart f hr fall h i at rt ad at th pit f axiu dflcti h i agai at rt. Thu th wrk rgy rlati i U -. Th frc ti ditac yild δ ( 5 δ) 6x dx 55 δ 5δ 4 Slig th quati uig cputati ad ftwar yild δ 4.47 ft Fr th lwt rt thrfr, t huld b utd 5-6 ft ab th flr t b af..4 Th fr-bdy diagra hw th frc actig th car. Th iiu lcity that th car ca ha at th tp f th lp ccur wh N. Writig F a at th tp yild g R r i gr. N g Th wrk rgy rlati yild U T T Fr th iiu iitial lcity g( R) i gr which yild 5gR iiu

20 .4 Sluti: U T T bc 5 x x dx () 5 Slig yild x x 6 Thu.55/..44 Th wrk rgy rlati: U T T bc 5( x x )dx () 5 x x4 4 6./ 66.6 r.45 Sluti: 5 kg Pwr dt dt 5 ( 55) 5kW

21 .46 N fricti rgy crd. Thu T V T V ut T T ic it tart fr rt ad i agai at rt wh th prig i cprd. V ( 4 δ ) i g whr kg. Wh th prig i cprd V kδ. Th rgy quati thu bc δ.5δg g. Slig yild δ I th abc f urfac fricti, th chag i pttial rgy i gh i ach ca. Thu gh. Slig yild gh..48 Sluti: (9.8)δ ( )δ ( 5) ( δ.) 5δ 46δ.5 δ..49 Thi i idtical t th priu prbl with h chagd fr δ t δ.:.5 Sluti: V gr which yild kr π kr (th pttial at ( 9.8)(. δ ) 5δ g i δ.8 π 5δ 46.δ 6.74 R ab th btt piti) gr ( i ) 5δ.5 π V gr( i ) kr (Th particl tart at rt ad w ar itrtd i wh it c t rt T T xapl i hw fr (th pttial aftr hittig th prig ad cig t a tp) gr, r r V V )

22 k N/ kg R Slig th tracdtal quati fr yild (u π/ a a iitial gi)..89rad 6.88 Th luti i fud i Mathcad a fllw: k R g 9.8 f... k R π.57, g. i 5 f rt f, dg 6.88 I MTL, th luti ay b fud fr cratig a -fil f th fr: y th R ; k;;g9.8; Tl((/)*k*R*(th-pi/)^-*g*i(th),th);

23 .5 Th axiu ditac i achid wh th particl lcity ad hc kitic rgy i agai zr. Thu th crati f rgy bc V V : V k δ V kδ Sttig V V. ( δ.6 δ ) i ( g) ( ) 5δ (.8 δ ) i ( 9.8) Slig yild δ th pttial with th a cprd agait prig, th yild. i th pttial with prig cprd S th axiu ditac th a tral i Th Mathcad luti fr th dfrati f th uppr prig: f δ. 5 δ... i. dg 9.8 δ.8.. δ, f δ δ δ. rt f δ, δ.

24 .5 Th axiu agl i btaid wh th lcity chag ig, i.. wh pa thrugh zr. Sic th a al tart fr rt, th crati f rgy rquir V V. l β R β R Nt β Firt writ th lgth f th prig i tr f β (ad hc ) fr th gtry: l r cβ rc t 6, l l r Th pttial at i : V gi kr c Th pttial at i : V gi kr c Nw lt f ( ) V V g[ i i ] kr c c which ca b plttd fr 6, r.5, k 8 N/, g 9.8 / ad. Th plt ( cputr cti) ral a zr ar rad. Nurical luti with rad a a iitial gu yild: 7.4 O ca al d thi i Mathcad a fllw:. 6 dg k 8 g 9.8 r.5 f. g. i i... k r. c 6. dg, 6. dg.. 8. dg. c

25 5 f rt f,.9.9 dg Lt F -kx î, a ctr rlati. ccrdig t th dicui fllwig Eq. (.6), a frc i crati if it curl i zr. Th cput th curl f F t gt F F F z y Fx Fz y Fx F i j k y z z x x y Sic F z F y ad F x -kx i ly a fucti f x. Thu F ad F gi hr i crati. Th pttial caud by thi crati frc i fud by iply lig F V fr V. Thi yild r up itgratig V kx V x kx, th pttial fucti aciatd with th gi frc.

26 .54 T if a frc F i crati, cput F t if it aih. Lt F x i yz j y z k ad i x j y k z Th cput F yz i yz i Sic it i crati u F V t cput th pttial. Cparig cfficit f ach uit ctr i thi xpri yild: x V x (a), yz V y (b), y z V z (c) Each f th thr quati ut b itgratd t l fr V(x,y,z) kpig i id that itgratig a partial driati yild a ctat f itgrati that i a fucti f th raiig ariabl (i.. it i ly ctat with rpct t th ariabl f itgrati). Startig with th î quati V x x yild x () V ( x, y, z) f ( y, z) whr f ( y, z) i th fucti f itgrati. Subtituti f thi alu it th ĵ quati yild yz y x f( y,z) r yz f y r f ( yz )y r f( y,z) y z gz () () whr g(z) i th ctat f itgrati. Subtituti f () it () yild that x V ( x, y, z) y z g( z) () whr w w u th third quati, fr th k cfficit, t dtri g(z). Fr (c), y z V z z x y z gz () g( z) Thu y z y z z r dg dz ad g ctat C. Thu fr quati () x V ( x, y, z) y z C

27 .55 u th bug crd i a prig that F kδ whr δ i th lgati f th crd c ad accrdig t th prbl tatt k. (Nt that fr a latic atrial k l E/ l ) whr i th cr-ctial ara ad E i a atrial prprty. Nw tak th rfrc rgy t b V ad T at th tart f th jup. Th pttial at th pit f axiu fall i k c V g( l δ ) δ gl gδ δ, ic V T T. l Diid V by l t gt c δ g δ l l g δ ad ly i dtrid a th rt f thi quadratic quati (th alu f δ/l l dpd ly c, ad g). Nw lt δ l δ l. Th F ax c l δ c δ l c Thu th frc i idpdt f th lgth f th rp btw th idiidual ad th achr pit. Thrfr th axiu frc actig th idiidual i idpdt f th ditac thy fall. Nt th iprtac f thi t utai clibr wh cct thl with rp that bha i a liitd way lik tiff bug crd..56 Th pttial rgy du t arth graitatial fild i V gr r t lauch E gr E T cap E. Thu E ipli gr E. Thrfr gre with R E 678k, th cap lcity bc cap / Nt that thi lcity cat b btaid at lauch but thrugh tagd rckt bur. E

28 .57 T if th frc i crati cput F t if it i zr whr: x y F i j ( x y ) ( x y ) Sic z i t prt i F, w ha : F F y x F x ( y k x y ) y x yx ( x y ) x y yx k ( x y ) ( x y ) ic y/x ad x/y ar bth zr a x ad y ary idpdtly. Thu th yt i crati ad w ay prcd t cput th pttial fucti V(x,y) fr th frc by itgratig quati.. Th ar: () x ad () y x y y x x y ( ) V ( ) V Fr (): V l ( x y ) f ( y) whr f i th ctat f itgrati. Subtituti f th alu f V(x,y) it quati () yild y x y y ( lx y ) f() y y df ( y) x y dy whr th ttal driati i ud bcau f dpd ly y. Thu d [ f ( y) ] ad f(y) ctat. dy Subtituti f f(y) C it V(x,y) yild Vx,y ( ) ( lx y ) C.57 U th wrk rgy quati a thi i crati. Th U T T (a) (b) (c) µ g% d (88 ft/) ( 88) 7.8 that d ft. % µ % Fr ach ca d 49.5 ft 86. ft 7.8 ft Nt: ctual car cat aitai thi ctat acclrati withut hiftig gar.

29 .58 Sluti: l V, T, V gl i, T Crati f rgy yild gl i.6 Sic th rgy i crd w ca u V T, whr V gh, T, V T Thu gh /.6 Fr th figur w ha: 9 x l ad δ x.. Th wight drp a ditac l l x l l.6 u zr tat f kitic ad pttial rgy at rla, th E E k( x) g( l l) Thi ut b zr a th yt c t rt (T T ), thu ( ) ( x) 59.8 ( ).6 x 9 ( ) which ha luti x.7, a fud urically blw Ditac th kg a i.7.7 I Mathcad u th fllwig;

30 f x 5. x x 9 x, f x x x.8 rt f x, x.7 d.7 d.7.6 Sluti: x 5kg x C C x kg 4 N Ctrait: Th figur yild th ctrait: ( ) ( ) l x x x C x Thrfr x x ad wrk d ( ) () () chag i kitic rgy Slig yild: 8 9 r. 64 thu: 5.55 /

31 .6 Th ctrait ar x x l ad x x x l C C Writig th crati f rgy with th lcity ctrait yild C x C x ad x 4 x 4 kg ( 9.8).5 5( 9.8).5 ( 4 ) Sluti:.7/ V kx g ( x) x.97 5 Th luti i btaid urically a fllw (Mathcad): f x. x x x,... 5kg x f x x x rt f x, x.97

32 .65 Sluti: h g( h ) g k g k g( h ) k g g k k g k k g g k gh Nurical xapl: Lt k N/, 4 kg, h ad u th fllwig cd t l urically. f x. x x x,... f x 5 x x rt f x, x.8

33 .66 u pttial rgy data pla at zr Crati f rgy T T whr T T E E thu.67 Th ly pttial rgy hr i graitatial Crati f rgy rquir: gh gh T u E E ad that,.68 Sluti: ( ) ( ) 4 4 k j i k j i F k j i F yz x yz x z y x z y x y Fx x Fy x Fz z Fx z Fy y Fz z y x yz x z y x Thu F i t a crati frc. h

34 .69 Sluti: ( ) ( ) ( ) c c c c i k k F k k F r r z z r F F r r F r F z F z F F r z r r rz r z r r r z r r z r r Thu F i crati. Nw cput th pttial by itgrati: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ctat. a, ad Thu i :, c c c :, i i : C g z z g dz z dg rz z g rz r z z V rz k z g f f r z f r r V r r z f r r V r V z r r C rz r V i

35 .7 Sluti: F (R i φc) R R cφc R φ i F R ( R iφ φ F Ri φ ) ( RF φ) F R R iφ R F R φ Ri φ ( R F Ri φ) R cφi R ( R RF φ) F R φ c φi φ [ Ri φ R i φi R i φi] [ R R cφ c R cφ c] F i crati. Nw itgrat t cput th pttial V R : R iφ c R R V iφ c f ( φ, ) V R φ : cφ c R φ R f φ ( ) ( φ, ) f cφ c R φ R i R g C f g V R φ i R i i R dg R φ d R V iφ c C cφ c i

36 .7 Sluti: k/hr 5g 7.8 / t t Fr Eq..47 Fdt r Fdt ( 7.8) Th [ F F ] t t gra brak ( 5( 9.8) i( 5.7) 7) 5( 4.8) t U ipul t, t it a ctr ad writ F ( ) 5 F x (.) ( ) Fx.9.9lb 5 F y (.) 6. ( 8.7) F y 9.lb t 5.7 t i cpt fr. F F x F y 7.6lb

37 .7 Cidr th yt f f.74 U crati f tu bb T btai th lcity 7 f 9 Th u wrk rgy ( ) b ( ) 5.6ft/ f t btai ditac.( 9)d 9 (.) d. ( )(.) ( 5.6) 8.9 ft.75 U th prjctil ti quati t dtri if it lad i th bx: y dircti : x dircti : a y g a x y gt 4i 45 x 4c45 y gt 4i 45 t x 4c 45 t x.5 4c45 t t.5 y(.5). w x.8 4c45 t yild t.66 ad y(.66).85 Y, it lad i th bx. Nw cput y gt 4i 45 t.577 x (.577).88 / crati f tu p p ( p b ) f f 5 ( 8.88 ).768 /

38 .76 Sluti: 8kg.768 Crati f rgy g( c ) c.59 ( ) 8( 9.8)( c ) c Ha th bjct tart at rt ( ) ( 6). Writ quati (.47): t, tdt, 6 ( ),t,( 6) t Sluti: T T kg 5kg Fr th kg a: ( g T ) t Fr th 5 kg a: ( T 5 g) t 5 ddig yild.45/ ( g 5g) t 5. Thu at t.75,

39 .79 Sluti (whr ) i th lcity f th frry: c c / ( ) 6.96 ft/,.8 Crati f tu: g wrk-rgy aftr cllii ( c5 i i 5 j ) i j g g µ ( )gd ( ).8 Sluti: µgd.7 ( ).( 6) 5ft/ 7.5 ( 5) c ft/ 54.5 ph ( 5) i5 6.9ft/ 4.6 ph.5 Sic d 6 4 5ph 6.7 ft/ 5ft/ ( ) i.5 j 7.5 ( c i ij ) Fr ( ) ( ) j : , 9. Fr i ( 5) c9. 9.ft/ 6.8 ph

40 .8 Firt chag uit 4 ph 58.7 ft/ ad ph 44 ft/ Crati f tu: 4 i.5 j 7.5 c i i j ( ) Which yild th tw quati : ( ) 7.5 c.5( 44) 7.5 i Slig yild :.5 44 ta ( ) ad ( ) that : 7.44ft/ ftr th cllii u th wrk rgy quati (Fd T) t gt µgd that : d. ft.8 Crati f rgy th lid yild 4g 8.86/ Crati f tu yild: 8.86 ( ) ( 5).5 / Wrk rgy aftr th cllii yild.() 7 ( 9.8)d () 7, that d.9.84 Crati f tu with: 5kg yild : C kg C C Slig fr C ( ) yild :./ 7.8/ C C C 9.5 /

41 .85 Fr Prbl Fr wrk rgy: C µgd C d.6.86 Fial rgy (7,) C 7. 6 Jul [ ] Iitial rgy 5,. ( ) ( 7.8) Jul L f rgy.7 6 Jul.87 Cidr th chag i tu f th bu Th F t r F a t ( ) 5 kg (. / 9.5 /) 47,5 N Sluti: 9 k/hr 5.8 / Ft F (.5).6( 5.8) f 6. N.89 Crati f tu Crati f rgy gl( iβ) gl( iβ) gl( iβ) Sic thi i rlad fr th hriztal β ad: gl

42 .9 Fllwig th luti f.89,.9 Firt l th prjctil ti quati: && x && y g x& O c 45 gl( iβ) y& O i 45 gt gt x O c 45 t y O i 45 t gt 4 O c 45 4 O i 45 t Sic i45 c45, w ca cbi th lat tw xpri t gt th ti, th lcity: t t.79 O.9 / Nw u th ipul tu quati: Ft F O (.4).4(.9) F.9 N.9 Durig ti f t wh P i acti Ipul Ipul (.785) ( 6 5.4)(.785) Thi will b axiu lcity ( 4) 77.5 N Fricti ipul gµ t t µ g t 7.5, Ttal ti ic lad applid.5

43 .9 U th fllwig cputr cd t xai th agl by plttig th ariu quatiti f itrt: Nt that fr.9 th frc i 5.4 P ( ) c Th ti t i P ( ) ( ) t Th ipul i I ( ) P ( )( t( )).5( 6 P( )) t ( ) Th lcity i () I( ) 4 ad t i ( ) t 6 (.6) ( 9.8) ) Th ca b plttd uig a cd. I Mathcad :, 5. dg dg T bgi t: ( ) P 5.4 c t P I P. t. 6 P. t. 6 Ti t tart t. t.8 t dg gl f P dg I

44 4 Maxiu Vlcity dg agl f P dg Ttal ti t tp.4 t dg gl f P dg.94 Sluti: k/hr 5.56 / 5 Crati f tu: 5 (a) /, 5.79 / E (b) ( )

45 E E 94J E ( )( ) ( 5)( ) E 4768Jul 646J (c) F (.5) ( ) F 458 N.95 alyz kid firt 7.8 / - µ gd µ gd ( ) i 5.9 / at ipact Fr Prbl.94 lig fr fial lciti: Ergy l ( )( ) J Ergy l i crah: 5.9 / 6.64 / ( )5.9 ( ) [ 5.9 ( ) 5( 6.64) ].5 J.96 Sluti: gh gh.86 h h

46 .97 Firt ipact Scd ipact V " C " C" " V ( ) V ( )V i l tha " C" ( ) 4 ( ) V " ( ) ( 4 )V thr i a third ctact " " ( ) ( ) [ ( ) ] ( 4 )V ( 8 ) ( )V ( ) " [ ( ) ] 4 ( ) V ( 8 ) ( )V.98 Sic C

47 .99 Sl fr gral V V ( )V ( )V a) V b) V c) V 4 4 V. Sluti:. Sluti:.7V.4 ( ) E V.7 ( ) [ ( )] V 4 ( ) 4 %E lt ( ) ( ) a) 5% b) % c) 7.5%. Sluti: h h gh gh gh h h h h

48 . Sluti: β x c y i y x taβ y ta x β ta ( ta) x c y i.4 Sluti: t c65 i i 65 j t k t i65 i c65 j 5 i 6 j / I Mathcad, cput: ( ) ( 5i 65 6c65 ) 7.67 t ( t ) t ( 5c65 6i 65 )c65 i - i65 j.45 i. j ( ) 5 6 t c 65. dg i 65. dg k.7 k t.... t t t p. t p.9

49 .5 Sluti: 45 y x c45 i i 45 j t k i 45 i c45 j ( ) /.5 c i i j.48 / alyi f dircti: ( ) ( ) t ( ).95 ( ) ( t ) t t..8..

50 Scd ipact: " x i y j.6. "..6 Sluti: 5kg t kg 6 8 Nral dircti.6 Sluti c i i j t k ( ) ( ) 7 i i ( ) ( ) [( ) ( ) ]

51 ( ) ( ) c i k t k ( ) ( ) t ( t ) t t ( t ) t [ ] p ( ) ( ) ( ) p p p p t.6 p 47. p p t.4 p Thi i ld dirctly i Mathcad. ral t th urfac i cputd fr Equati (.): T 7 5 t 5 N T t Nxt tr th lcity, cput th uit ral ad tr th alu f : N N.8 Nxt cput th ral ad tagtial cpt f :

52 . t Cput th ral cpt aftr ipact fr Equati (.65): p. Th lcity aftr ipact i th jut: p p t p i / I MTL tr ad ru th fllwig cd: t[-;;5];t[;-7;5];ncr(t,t); [-.5;-.;];.8;N/r(N); dt(,)*;t-;p-*;ppt

53 .8 Thi i ld dirctly i Mathcad fllwig xactly th tp ab. Th MTL cad fllw a wll. T 5 7 t 5 N T t N N... t p. p p t p i / I MTL tr ad ru th fllwig cd: t[-;;5]; T[;-7;5];Ncr(T,t); [-;-.;-];.;N/r(N); dt(,)*;t-;p-*;ppt.9 Sluti: (a) Firt, w will crat a ral t th urfac. t 5.5. T 7 5 N T t N N

54 ( ) p t p p p t I MTL tr ad ru th fllwig cd: t[-;;5]; T[;-7;5];Ncr(T,t); [-.5;-.;];.;N/r(N); dt(,)*;t-;p-*;ppt (b) Firt, w will crat a ral t th urfac: N N t T N T t ( ) p t p p p t I MTL tr ad ru th fllwig cd: t[-;;5]; T[;-7;5];Ncr(T,t); [-.5;-.;];;N/r(N); dt(,)*;t-;p-*;ppt

55 . Sluti: (a) Firt w will crat a ral t th urfac. c t i 5 ( ) ( ) T c i N N ( 45 ) ( 45 ) N T t t p p p ( ) p I MTL tr ad ru th fllwig cd: t[-c(*pi/8);;i(*pi/8)]; T[;-c(45*pi/8);i(45*pi/8)]; Ncr(T,t); [-;-;-5];;N/r(N); dt(,)*;t-;p-*;ppt t (b) Firt w will crat a ral t th urfac. c t i 5 ( ) ( ) T c i N N ( 45 ) ( 45 ) N T t

56 t p p p ( ) p I MTL tr ad ru th fllwig cd: t[-c(*pi/8);;i(*pi/8)]; T[;-c(45*pi/8);i(45*pi/8)]; Ncr(T,t); [-;-;-5];;N/r(N); dt(,)*;t-;p-*;ppt t (c) Firt w will crat a ral t th urfac. c t i 5 ( ) ( ).5 T c i ( 45 ) ( 45 ) N N N T t t p p p ( ) p I MTL tr ad ru th fllwig cd: t[-c(*pi/8);;i(*pi/8)]; T[;-c(45*pi/8);i(45*pi/8)]; Ncr(T,t); [-;-;-5];-.5;N/r(N); dt(,)*;t-;p-*;ppt t

57 . Mtu f th yt i crd ly i th x-dircti: y w w i () t c4 i i 4 j t k i 4 i c4 j j t c βi i βj Dfi ctr t k w wi t 4 x Th ral ad tagtial cpt f th lcity ar: t c 4 t i 4 I th dircti: Fr (): Fr (): w t w t () Ipact ( t ) t () Tagtial dircti w ( t ) i w w [( w ) i ( t ) t i ] w ( i ) w ( )( i ) ( t ) t i ( ) Thi ca all b d i Mathcad r MTL. I Mathcad tr th ctat f th prbl:.7 β. 4 dg w 5

58 Etablih th cary uit ctr: i k t c β i β t k Th tw ctr quati which ut b atifid ar crati f tu i th x-dircti ad th ipact rlatihip i th ral ad tagtial dircti... pt t t i.. wp. w. i.... i. pt i.. p wp.. p pt p.86 wp.55 p.4. Lt th a f th kull b, th a f th brai b b, th iitial lcity, th cfficit f rtituti btw th kull ad th wall b ad btw th kull ad th brai b. Firt ipact: Scd ipact: b b () " " " b b " b ( ) () b " " ( ) b ( ) b b

59 ( ) [ ] [ ] ( ) ( ) ( ) [ ] ( ) [ ] " " " " ) ( b b b b b b b b b b b b Sic th lcity f th brai i th gati dircti i gratr tha that f th kull, i gral, thr will b a cd ipact.. Sluti: b Crati f tu at ipact ( ) b b b b Wrk-rgy aftr ipact ( ) ( )( ) ( ) ( ) b k b k b k b g g d d g µ µ µ Fr th alu g t b µ Th luti fr d i: d.8

60 .4 Sluti: c i β i - i i i ipact β j i.5 Rfr t th luti f Prbl.: y β t t cβi i βj t k i βi c βj x Mtu crd ly i th x-dircti: i Ipact: Cbiig quati: w w w w w w w t w w w ( t ) t ( ) ( ci ij ) i [ ] ( ) ( ) [( ( ))( ) ( )( w i t t i) ] ( i ) i [ β w [ ( )( ) ( )( i i t t i) ] c ( i β c c β i ) i i β ( c β c i β i ) c β ] c i β w [ c i β i( β ) c β ( β )]

61 .6 Sluti: y x β ( ) ( ) ( ) ( ) t i c i c i c i c Ipact j i j i j i j i β β β β β β β β

Lectur 22. RF and Microwave Circuit Design Γ-Plane and Smith Chart Analysis. ECE 303 Fall 2005 Farhan Rana Cornell University

Lectur 22. RF and Microwave Circuit Design Γ-Plane and Smith Chart Analysis. ECE 303 Fall 2005 Farhan Rana Cornell University ctur RF ad Micrwav Circuit Dig -Pla ad Smith Chart Aalyi I thi lctur yu will lar: -pla ad Smith Chart Stub tuig Quartr-Wav trafrmr ECE 33 Fall 5 Farha Raa Crll Uivrity V V Impdac Trafrmati i Tramii i ω