Vektor basis Vektor satuan i = 1,0,0Ò, j = 0,1,0Ò, dan k = 0,0,1Ò sebagai pembentuk ruang dinamakan vektor basis untuk ruang 3.

Transkripsi

1 1 Vektor Vektor dlh rs grs errh ng dtentkn oleh pnjng dn rhn. D ektor dktkn sm jk pnjng dn rhn sm. Vektor dgmrkn seg rs grs dr ttk pngkl ke ttk jng dengn tnd pnh djng, dn der lmng hrf kecl cetk tel. Pnjng ektor Pnjng ektor dlh jrk dr ttk pngkl ke ttk jngn, dn dtls. Vektor stn Vektor stn dlh ektor ng pnjngn 1 stn. Untk serng ektor dperoleh ektor stn ng pnjngn 1. ttk jng ttk pngkl k z = ( 1,, ) j 1 ( 1,,) = + j + k 1 1 = + + = π Vektor poss Jk ttk pngkl ektor dlh (,,) dn ttk jngn ( 1,, ), mk dnmkn ektor poss, dn dtls = 1,, Ò. Pnjng ektor = 1,, Ò = Vektor ss Vektor stn = 1,,Ò, j =,1,Ò, dn k =,,1Ò seg pementk rng dnmkn ektor ss ntk rng. Vektor dpt dntkn seg = 1+ j+ k.

2 V & FsPr Vektor d dng Vektor poss d dng dlh = 1, Ò, ektor dengn ttk pngkl (,) dn ttk jng ( 1, ). Pnjng ektor n dlh 1 = +. Bss k d dng terdr dr ektor stn = 1,Ò dn j =,1Ò. Vektor = 1, Ò d dng dtls = 1 + j. Vektor nol Vektor nol dlh ektor dengn ttk pngkl ermpt dengn ttk jng, rhn serng. Vektor nol d dlh =,,Ò. Kesmn d ektor poss Vektor = 1,, Ò = 1 + j + k dn = 1,, Ò = 1 + j + k sm, dtls = 1 = 1, =, =. Vektor dr rs grs Jk P dn Q dlh ttk d dng (rng), mk ektor dengn ttk pngkl P dn ttk jng Q dtls PQ. Jk ttk pngkln Q dn ttk jngn P, mk dperoleh ektor QP. Penjmlhn ektor Pengrngn ektor Perkln ektor dengn sklr Penjmlhn ektor Untk ektor dn dengn ttk jng ttk pngkl, jmlh dn (dtls + ) dlh ektor dr ttk pngkl ke ttk jng. Jmlh dr ektor = 1,, Ò dn = 1,, Ò dlh + = 1 + 1, +, + Ò. Perkln ektor dengn sklr Hslkl ektor dengn sklr c (dtls c) dlh ektor ng serh jk c >, erlwnn rh dengn jk c >, dn ektor nol jk c =. Hsl kl sklr dr = 1,, Ò dengn sklr c dlh c = c 1,c,c Ò dn pnjngn c. Pengrngn ektor Selsh dr ektor dn (dtls - ) dlh ektor + (-). Selsh dr ektor = 1,, Ò dn = 1,, Ò dlh - = 1-1, -, - Ò. -

3 V & FsPr A Sft Vektor Untk serng ektor,, w dn sklr, erlk: + = + + (-) = ( + ) = + ( + ) + w = + ( + w) () = () 1 = + = + = ( + ) = + =. Contoh D Q E B P C Pd gmr dperlhtkn jjrgenjng ABCD dengn dgonl AC dn BD ng erpotongn d E, P ttk-tengh BC, dn Q ttk-tengh ED. Jk AB = dn AD =, ntkn rs grs errh AP, AQ, dn CQ dlm ektor dn. Dr sft jjrgenjng dperoleh DC = AB =, BC = AD =, CD = BA=-, dn CB = DA= Kren P ttk-tengh BC, mk AP = AB + BP = + BC = +. Kren Q ttk-tengh ED dn E ttk potong dgonl AC dn BD, mk BQ = BD, sehngg 4 1 AQ = AB + BQ = + BD = + BA+ AD = + (- + ) = +. ( ) Dengn rgments ng sm dperoleh 1 CQ = CB + BQ =- + BD =- + BA+ AD =- + (- + ) =- -. ( ) Contoh Jk = (1,,), = (1,1,), w = (1,1,1), dn = (,-,4), tentkn konstnt,, dn c gr memenh = + + cw. Dr = + + cw dperoleh (,-,4) = (1,,) + (1,1,) + c(1,1,1), t (,-,4) = ( + + c, + c,c) Berdsrkn kesmn d ektor dperoleh + + c =, + c = -, dn c = 4. Aktn = - - c = = -7, dn = - - c = - (-7) - 4 = 5. Jd konstnt,, dn c ng memenh = + + cw dlh = 5, = -7, dn c = 4.

4 V & FsPr 4 Contoh Jk = 8,1,-4Ò dn = 6,-,-Ò, tentkn pnjng ektor,, dn -. Pnjng ektor dlh = (- 4) = 81= 9. Pnjng ektor dlh = + (- ) + (- 6) = 49 = 7. Kren - = 8,1,-4Ò - 6,-,-Ò = 8,1,-4Ò - 1,-4,-6Ò = 4, 5,-Ò, mk pnjng ektor - dlh - = (- 4) = 45 = w 45 Contoh Pd gmr dperlhtkn seh end dengn ert newton ng dgntng d kwt ersdt 6 dn 45 dengn horsontl. Jk sem g terletk d dlm st dng dn end dlm kedn setmng, tentkn esrn g tegngn pd setp kwt. Mslkn g tegngn pd kwt kr dlh ektor, pd kwt knn dlh ektor, dn g ert end dlh ektor w. Urkn g tegngn dn ts komponen horsontl dn ertkl. Dlm kedn setmng esrn g horsontl ke rh kr dn knn hrs sm, ktn cos6 = cos45. Dr sn dperoleh 1 1 =, sehngg =. Dlm kedn setmng esrn g horsontl ke rh ts dn wh hrs sm, ktn sn 6 + sn 45 = w =. Seleskn persmn n dengn dt sol dn =, dperoleh dn =, ( ) = = = ª 1,5 newton. ( ) ( ) = = = - 1 ª 146,4 newton.

5 V & FsPr 5 Perkln ttk Hslkl ttk dr ektor dn, dtls, ddefnskn seg erkt. Untk ektor d dng: = 1, Ò 1, Ò= Untk ektor d rng : = 1,, Ò 1,, Ò= Sft Perkln ttk Untk ektor,, w dn sklr c erlk = ( + w) = + w c( ) = ( c) = =, > π, = = Ktn hslkl ttk dengn sdt ntr d ektorn Jk, π dn q = sdt terkecl dr dn, mk = cos q. Krter d ektor slng tegk lrs ^ = (D ektor slng tegk lrs jk dn hn jk hslkl ttkn nol) D ektor ng slng tegk lrs dnmkn ortogonl.. - Bkt dr sft = cos q. Rms kosns dr segtg pd gmr memerkn - = + - cos q. q Dr sft perkln ttk dperoleh - = ( -) ( - ) = ( -) -( -) = = + - Smkn ked entk dr - n, dperoleh = cos q. Contoh Tentkn sdt ntr ektor = 8,4,-1Ò dn = 4,-4,-Ò. Jk, π dn q = sdt terkecl dr dn, mk cosq =. Untk sol n, = (- 1) = 81= 9, = 4 + (- 4) + (- ) = 6 = 6, dn = 8(4) + 4( - 4) + (-1)( - ) = 18, sehngg cosq = =. Aktn 11 sdt ntr ektor dn dlh q = cos ª 71. Ilstrs Vektor = 8,-1,-4Ò dn = 1,-4,Ò slng tegk lrs kren = 8, -1, -4Ò 1, -4,Ò= =. -

6 V & FsPr 6 Contoh Tentkn sem ektor stn ng tegk lrs = 1,6,4Ò dn = 1,,Ò. Mslkn w =,,cò dlh st ektor ng tegk lrs dn, mk,,cò 1,6,4Ò = dn,,cò 1,,Ò =. Dr sn dperoleh persmn c = dn + + c =. Selsh d persmn n memerkn 4+ c =, sehngg c =- dn =-- c =- + 4 =. Jd w =,,cò =,,-Ò =,1,-Ò dn w = ( ) =, sehngg sem ektor stn ng tegk lrs dn dlh w,1, -Ò 1 s = = = ±,1, - Ò. w Sdt rh dn kosns rh z k g j Sdt tk negtf terkecl ntr ektor rng π dengn ektor ss, j, k dnmkn sdt rh dr, dntkn dengn,, dn g ; d sn = (,), = (,j), dn g = (,j). Dlm ktn n, cos, cos, dn cos g dnmkn kosns rh dr. Jk = 1+ j+ k, mk 1 cos = = j j, cos = = k k, dn cosg = = 1 Cttlh hw cos + cos + cos g = + + = 1 dn ektor (cos, cos, cos g ) dlh st ektor stn ng serh dengn. Contoh Tentkn sdt rh ektor =,,-6Ò. Kren = = 7, mk cos =, 7 sehngg sdt rh dr ektor =,,-6Ò dlh cos =, dn 7 ª 7,4, ª 64,6 dn g ª cosg =-, 7

7 V & FsPr 7 Vektor Proeks Proeks ektor pd dlh ektor pr = dn pr dnmkn proeks sklr dr pd. q pr = cos q pr = - cos q q 1 q p = (proeks pd ) = k, k > = cos q = k = cos q = k k = pr k \ = = = 1 p < q p = (proeks pd ) = -k, k > = - cos q = k - = - cos q = k k =- pr k \ = = - = Contoh Jk =,,-1Ò dn = 1,1,Ò, tentkn pr dn pr. Untk contoh n, = 6, =, dn = =, sehngg pr = = 1,1, Ò=,, Ò dn pr = =,, -1 Ò=,,- Ò. tl 5 Contoh Jk sdt ntr tnjkn jln dn horsontl dlh 5, tentkn g tegngn tl gr dpt menhn mol seert ton dlm kedn setmng. T tn 5-5 W = ton Btlh sstem koordnt o dengn ttk sl seg ttk pst mss mol. Dlm sstem koordnt n, W =,-Ò dn T = -, tn 5 Ò, >. G tegngn tl ntk menhn mol dlm kedn setmng dlh pnjng proeks dr w pd T, t TW TW tn5 T T T 1+ tn 5 pr W = = = ª,845 ton. T Cr ln pr T W = W sn 5 =,466=,845 ton.

8 V & FsPr 8 Contoh Jk g + + c =, dn tk sem, tnjkkn ektor + + c,ò tegk lrs g dn jrk A(, ) ke g dlh d =. Untk dn tk sem, terdpt tg kss ng mngkn terjd c = dn π : g + c = g =- g // s-,ò ^ g. π dn = : g + c = c ( ) c + g =- g // s-,ò ^ g. c ( ) c c Ò π dn π : -, Œ g dn, - Œ g,- Œg c c c c,- Ò, Ò=,- ^, Ò A(, ) d ( 1, 1 ) pr n n Ò,Ò ^ g. Mslkn ( 1, 1 ) Œ g dn n =,Ò dlh ektor ng tegk lrs grs g + + c =. Btlh ektor dr ( 1, 1 ) ke (, ), mk = - 1, - 1 Ò. Kren ( 1, 1 ) Œ g, mk c =, sehngg = -c. Dengn menggnkn proeks ektor dperoleh n n 1 1, Ò A g n n n n, Ò ) d= jrk (, ) = pr = = = -, - Ò ( ) ( ( ) c = = = Perkln slng Hslkl slng dr ektor rng = 1,, Ò dn = 1,, Ò, dtls, ddefnskn seg ektor rng = -, -, -Ò; t j k j k (entk determnn) 1 1 = 1 = Sft perkln slng Untk ektor rng dn erlk ^ dn ^ ( ( ) = = ( )),, dn mementk sstem tngn-knn = sn (,) // = (,)

9 V & FsPr 9 Ilstrs Jk = 1,6,4Ò dn = 1,,Ò, mk j k = = 4 4 4,, 4 - j 1 + k 1 = + j - k = - Ò. 1 O t P q F q w w snq w Tors Pd gmr kr dperlhtkn seh end dengn ttk tetp O dn P ttk ln pd end. D P ekerj g F ng memtr end terhdp sm ng mell O dn tegk lrs dng (OP,F). Vektor t = OP F dnmkn tors, ng serh dengn sm ptr dn esrn OP F sn q, q = ( OP, F). Art geometr perkln slng Pd gmr tengh dperlhtkn seh jjrgenjng ng dentk oleh ektor dn w dengn q = (,w). Als dn tngg jjr genjng n dlh dn w sn q, sehngg lsn dlh L = w sn q = w. (sft perkln slng) Perkln trpel sklr Perkln trpel sklr dr ektor,, dn w ddefnskn seg sklr ( w). Jk = 1,, Ò, = 1,, Ò, dn w = w 1,w,w Ò, mk Ê 1 1 ˆ ( w) = 1,, ÒÁ,-, Ë w w w1 w w1 w = = 1. w w w1 w w1 w w1 w w Art geometr Perkln trpel sklr Pd gmr knn dperlhtkn seh prlel eppedm ng dentk ektor,, dn w. Ls lsn dlh w dn tnggn dlh cos g, dengn g = (, w). Volme prlel eppedm n dlh V = w cos g = w cos g = ( w). g g cosg w

10 V & FsPr 1 n Q(,,z) Q P( 1, 1,z 1 ) PQ = -, -, z-z Ò Persmn krtess dng d rng Pd gmr dperlhtkn dng ng tegk lrs ektor tknol n =,,cò dn mell ttk P( 1, 1,z 1 ). Jk ttk Q(,,z) pd, mk ektor PQ= -1, -1, z-z1ò terletk pd. Kren PQ ^ n, mk PQ n =, ktn : (- 1 ) + ( - 1 ) + c (z- z 1 ) =. Persmn n dpt dtls dlm entk + + cz = cz 1 = k, k konstnt. Jd persmn dng dlh : + + cz = d;,, dn c tk sem nol. Jrk ttk ke dng Jrk ttk A(,,z ) ke : + + cz = d dlh + + cz-d d = + + c Contoh Tentkn persmn dng ng mell ttk A(,-,-1), B(1,1,), dn C(-,,1). n Vektor ng terletk pd dng dlh AB = (1,1,) -(,-,- 1) = -1,, 4Ò AC =- (,,1) -(,-,- 1) = -4,5, Ò Kren ektor norml dng n memenh n ^ AB dn n ^ AC, mk j k n = AB AC = = -14,-14,7Ò= - 7,, -1Ò Amllh n =,,-1Ò, mk : + - z = k, k dcr. Kren mell A(,-,-1), mk k = = 1. Jd : + - z = 1. Contoh Jk : + - z = 1 dn : + + z = 6, tentkn (, ). Sdt ntr d dng dlh sdt ntr d ektor normln. D sn n =,, -1 Ò, n = 1,,Ò dengn n =, n =, dn n n P Kren A C B n n n n n 9 cos ( n, ) = = = =., mk ( n, n ) ª 77,.

11 V & FsPr 11 Tmpln prmeter kr dng St kr dng dpt dtlskn dlm persmn prmeter = (t), = (t), t ΠI, ked fngs n kontn pd st selng I. Cr penlsn lnn dlh entk ektor r(t) = (t) + (t) j, t ΠI = [,]. Kr ttp dn kr sederhn Pd persmn prmeter = (t), = (t), t Π[,], ttk jng kr dlh P((), ()) dn ttk pngkl kr dlh Q( (), ()). St kr dengn ttk pngkl dn ttk jng ermpt dnmkn kr ttp. St kr ng djln tept st kl (kecl ttk pngkl dn ttk jngn) dnmkn kr sederhn. Contoh Tentkn persmn prmeter ntk lngkrn L: + = (,) t - - = cos t L = sn t Lntsn ttp sederhn + =. Persmn prmeter L dlh = cos t, = sn t, t p, t r(t) = cos t + sn t j, t p. Ttk pngkl L r() = (,) dn ttk jng L r(p) = (,), sehngg L dlh lntsn ttp. Kren L djln tept st kl kecl ttk (,), mk L dlh lntsn ttp sederhn. Dr = cos t dn = sn t dperoleh persmn lngkrn + =. Lntsn tdk ttp dn tdk sederhn Q Lntsn tdk ttp dn sederhn Q Lntsn ttp dn tdk sederhn Lntsn ttp dn sederhn P P Kr dng Persmn krtess Persmn prmeter Elps Hperol 1 + = = cos t, = sn t, t p 1 - =, > 1 1 = sec t, = tn t, - p < t < p = cosh t, = snh t, - < t <

12 V & FsPr 1 Keterdferensln fngs prmeter Jk = (t), = (t), t Œ I mempn trnn pertm ng kontn dn (t) π pd selng k I, mk dlh fngs terdferenslkn terhdp dengn d d d/ dt d/ dt =. Ilstrs Pd fngs prmeter = cos t, = sn t, t p ntk lngkrn L: + = dperoleh dlh fngs dr dengn trnn d d/ dt cost d d/ dt -snt = = = -. P R t C sklod M N p p Sklod Sklod dlh kr dng ng merpkn jejk ttk pd rod lngkrn ng dgelndngkn sepnjng grs lrs tnp tergelncr. Gmr n dlh rod lngkrn ng erpst d C dn erjr-jr dgelndngkn sepnjng s- dengn jejk ttk P ml dr (,). Plh prmeter t sdt serh jrm jm ntr CP dengn poss ertkln st P d ttk O. Kren ON = PN = t, mk dn dlh = OM = ON - MN = t - sn t = (t - sn t) = MP = NR = NC + CR = + (- cos t) = (1 - cos t) Persmn prmeter sklod dlh = (t - sn t), = (1 - cos t), t >. Trnn terhdp dlh d d/ dt snt ± - d d/ dt (1- cos t) ( ) = = =, kren cos t = 1- f sn t =± 1- cos t =± 1-1- =± -. Dr sn dperoleh, ttk mnmmn tercp d = k p dn ttk mksmmn tercp d = p + k p, k lngn lt. Ls derh d wh st sr sklod dn d ts s- dlh p p p L = d = (1-cos t) d( ( t- sn t) ) = (1 -cos t) dt Ú Ú Ú p p Ú ( 1 cos cos ) ( sn sn 4 ) = - t+ + t dt = t- t+ t = p.

13 V & FsPr 1 Fngs Prmeter d Bdng dn Rng Fngs prmeter d dng dlh r() t = () t + (), t j t dn d rng dlh r() t = () t + () t j + z() t k, t. Fngs prmeter n ernl ektor dengn peh sklr. Fngs n memt nforms ttk pngkl, ttk jng, rh, dn erp kl kr djln; rhn terlk jk t dgnt dengn (-t). (kektn) St kr dpt dtls seg fngs prmeter dengn leh dr st cr, trnn tdk tnggl. (kelemhn) t Fngs Prmeter d Bdng t = t = j t = ttkpkl r(t) r = r(t) t = ttkjng r() t = () t + (), t j t = (t), = (t) fngs rel Fngs Prmeter d Rng z t = t = ttkpkl r = r(t) t k t = r(t) ttkjng t = j r() t = () t + () t j+ z() t k, t = (t), = (t), z = z(t) fngs rel q (p,q) p + q j L - p L 1 - L1 : r() t = cost+ sn tj, t p f + = L : s () t = ( cos t+ p ) + ( sn t+ q ) j, t p f ( - p) + ( -q) = t = z = () t = cost = () t = snt tng + = helks lngkrn r() t = cost+ sntj + tk tœ, t - < t< Lntsn sprl mellt tng } f + = r() t = cost+ sn tj+ t k,- < t< tng lngkrn z

14 V & FsPr 14 Contoh Tentkn persmn prmeter dr = 4 -, 4, rh, ttk pngkl, ttk jng, gmrkn kr, dn rh terlk dr kr. 4 (,4) t = = 4 - t = t = 4 ttk ttk 4 pngkl jng Persmn prmeter: r() t = t + (4 t - t ) j, t 4. rh: dr ttk pngkl (,) ke ttk jng (4,) t = t = 4 Jk rh kr dlk, trn fngsn: s() t =- t + (-4 t -t ) j,- 4 t. ttk pngkl: r(-4) = 4 + j = (4,) ttk jng: r() = + j = (,) Contoh Tentkn persmn prmeter grs d rng dengn ektor - rh π dn ektor penngg. Tentkn jg persmn krtessn. z Persmn prmeter: = (t) = + t, π, tœ. Jk = (,,), z = ( 1,, ), dn = ( 1,, ), m- (,,) z = (,, ) + t (,, ). Smkn kompo- k 1 1 nenn, dperoleh (,,) z = ( 1+ t1, + t, + t). Elmns t dr = 1+ t1, = + t, dn z= + t, -1 - z- dperoleh = =,,, π. 1 1 Contoh Tentkn persmn kr ng merpkn perpotongn dr tng lngkrn + = dengn dng z = kemdn gmrkn. z tng + = dng z = r = r(t) Kren persmn kr hrs memenh + =, mk = cos t, = sn t, >. Kren persmn kr hrs memenh z =, mk mllh z = sn t, >. Persmn prmeter dr kr dlh r(t) = cos t + sn t j + sn t k, t p. Bentk kr dlh elps ng djln st kl dengn sm pnjng dn sm pendek.

15 V & FsPr 15 Lmt fngs prmeter Untk fngs prmeter r = r(t), t dn t, lm r( t) = L jk " e> $ d > ' < t- t < d f r( t) - L < e. tæ t (r(t) dpt dt serng dekt ke L dengn cr memt t ckp dekt ke t tetp t π t ) Kekontnn fngs prmeter Fngs prmeter r = r(t) kontn d t, t, jk lm r( t) = r ( t ) dn kontn pd st selng jk fngs tæ t r = r(t) kontn d setp ttk pd selng t. Sft lmt dn kekontnn fngs prmeter Untk fngs prmeter r = r(t) = (t) + (t) j + z(t) k, t, t, dn L = ( 1,, ), lm r( t) = L lm ( t) =, lm ( t) =, dn lm z( t) =, 1 tæt tæt tæt tæt r = r(t) kontn d t = (t), = (t), dn z = z(t) kontn d t. Trnn fngs prmeter Trnn fngs prmeter z C : r = r(t) r(t ) grs snggng r(t + h) - r(t ) r(t + h) grs snggng: s(t) = r(t ) + t r (t ) r = r(t) = (t) + (t) j + z(t) k, t d t Œ (, ), dtls r (t ), ddefnskn seg r( t+ h) -r( t) r ( t ) = lm. h Æ Trnn fngsn d t Œ [, ], ddefnskn seg r( t+ h) -r( t) r () t = lm h Æ Art geometr r (t ) ektor snggng d r (t ) pd kr C: r = r(t), persmn grs snggng d r (t ) pd kr C dlh s(t) = r(t ) + t r (t ). Art fss r (t ) ektor keceptn d r (t ) pd gerk prtkel sepnjng kr C: r = r(t). Sft trnn fngs prmeter Trnn dr fngs prmeter r = r(t) = (t) + (t) j + z(t) k, t dlh r () t = () t + () t j+ z () t k. Jk fngs r = r(t) dn s = s(t) terdferenslkn d t Œ [, ], mk (r + s) (t) = r (t) + s (t) (r s) (t) = r(t) s (t) + r (t) s(t) (r - s) (t) = r (t) - s (t) (r s) (t) = r(t) s (t) + r (t) s(t) ( d ) h h

16 V & FsPr 16 Ú Gerkn prtkel sepnjng kr Jk st prtkel ergerk sepnjng kr rng C: r = r(t), mk ntk setp st t Œ [, ], Vektor poss: r = r(t) Vektor keceptn: = (t) = r (t); lj: = (t) = (t) Vektor perceptn: = (t) = (t) = r (t); perceptn: = (t) = (t) Integrl fngs prmeter Integrl tk tent dr fngs r = r(t) pd selng I ddefnskn seg nt dferensln, r() t dt = s() t + C s (t) = r(t) "t Œ I. Integrl tent dr fngs r = r(t), t ddefnskn seg lmt jmlh Remnn, Ú Â r () tdt = lm r ( c) D t, P st prts ntk P Æ k = 1 [, ], Dt k = t k - t k-1, c k Œ [t k-1, t k ], dn P = mks {Dt k : 1 k n} Sft ntegrl fngs prmeter Jk r(t) = (t) + (t) j + z(t) k, t, Ú Ú Ú Ú Ú = Ú + Ú + Ú mk () tdt= ( tdt () ) + ( tdt () ) + ( ztdt () ) r j k r() tdt Ê tdt () ˆ Ê tdt () ˆ j Ê ztdt () ˆ k Ë Ë Ë C: r = r(t), D = f () D D -1-1 n k k Pnjng sr kr Untk kr C: = f (),, f kontn pd [,], D = sr ke- ª tlsr ke- ( D ) D 1 L = Ú 1 + ( f ( ) ) d D ª D +D = + D. Pnjng sr: Untk kr C: r(t) = (t) + (t) j dengn r (t) = (t) + (t) j kontn pd [, ] dn r () t = ( () t ) + ( () t ) = ( ) + ( ) = ke- dperoleh ( ) ( ) D D Dt Dt Pnjng sr: D ª D +D = + D t., dr D = sr ke- ª tlsr Ú Ú r. L () t () t dt () t dt Rms L= Ú r ( t) dt erlk ntk ntk kr rng r = r(t), t.

17 Contoh Htnglh ln (1 + t) 1/(1 + t) Kren tæ t tæ 1 sn t cos t lm lm tæ t tæ 1 V & FsPr 17 ln (1 + t) 1- e sn lm ( t t + j+ k ) t Æ t t t lm = lm = 1, = =, mk t Æ 1-e -et tæ t tæ 1 lm = lm = - 1, dn lm r( t) = (1, - 1, ) = - j+ k. Contoh Tentkn persmn krtess grs snggng d ttk A(-1,,p) pd kr C: r(t) = cos t + sn t j + t k, tœ. Ttk A(-1,,p) terletk pd C kren r(p) = - + j + p k = (-1,,p). Trnn fngs r = r(t) dlh r (t) = -sn t + cos t j + k, sehngg ektor snggngn dlh r (p) = - j + k = (,-1,1). Kren ttk A tercp pd st t = p, mk persmn grs snggng d A pd kr C dlh s(t) = r(p) + t r (p) = (-1,,p) + t(,-1,1). Untk menentkn persmn krtessn, mslkn s(t) = (,,z), mk = -1, = -t, dn z = p + t. Elmns t menghslkn - = z - p. Jd persmn krtess grs snggngn dlh = -1 dn = p - z. Contoh St prtkel ergerk dengn r(t) = cos t + sn t j + e t k, tœ. Tentkn sdt ntr ektor keceptn dn perceptnn pd st. Vektor keceptn prtkel (t) = r (t) = -sn t + cos t j + e t k, sehngg ektor keceptnn pd st t = () = (,1,1). Vektor perceptn prtkel (t) = r (t) = -cos t - sn t j + e t k, sehngg ektor perceptnn pd st t = () = (-1,,1). Gnkn () () = () () cos ((),()) dengn () () = (,1,1) (-1,,1) = 1, () =, dn () =, dperoleh 1 = cos ((),()), t cos ((),()) = 1. Jd (ektor keceptn, ektor perceptn) d ((),()) = 6. Contoh Htnglh pnjng sr (kellng) lngkrn erjr-jr >. Tlslh lngkrnn dlm entk C: r(t) = cos t + sn t j, t p. Kren r (t) =, mk Ú p p L = kellng C = r ( t) dt= dt= p. Ú

18 V & FsPr 18 Contoh Jk r(t) = sn t + sn t j + sn t k, htnglh Ú r () tdt dn Ú r () tdt. Úr( t) dt = ( Úsnt dt) + ( Úsn t dt) j+ ( Úsn t dt) k =- cost + ( t - sn t 4 ) j+ ( cos t - cost ) k + C p ( ( ) ( ) ) Ú r tdt t t t j t t k j k. p () =- cos + - sn + cos - cos = + p Contoh Htnglh pnjng sr helks lngkrn C: r(t) = cos t + sn t j + t k, t p. Kren r (t) = - sn t + cos t j + k, dengn r (t) = pnjng sr C dlh Contoh lntsn q pelr Ú p p Ú, p + mk L= r ( t) dt= + dt= p +. Setr pelr dtemkkn dr ttk sl O dengn lj wl m/det dn (pelr,s- postf) = q. Jk gesekn dr dkn, tentkn ektor poss ntk gerkn pelr n dn tnjkkn lntsn pelrn erentk prol. Perceptn ng terkt dengn g grts dlh (t) = -9,8 j m/det dengn konds wln r() = dn () = cosq + snq j. Tentkn r = r(t) dengn mengntegrlknn d kl. Ú Dr (t) = -9,8 j dperoleh () t = () t dt = (- 9,8) j dt = - 9,8t j + C 1 dengn C 1 dtentkn dr () = cosq + snq j. Kren () = C 1, mk C1= cosq+ snqj, sehngg () t = ( cos q) + ( snq-9,8) t j. Dr sn dperoleh Ú Ú q ( q ) r() t = () t dt = ( cos ) t+ ( sn ) t- 4,9t j+ C dengn C dtentkn dr r() =. Kren r() =, mk C =. Jd ektor possn dlh r() t = ( cos q) t+ (( sn q) t-4,9t ) j. Dr ektor poss n dperoleh = ( cosq )t dn = ( snq )t - 4,9t. Kren t cos q =, mk ( sn q) 4,9 4,9 (tn q) cosq cos q cos q = - = -. sehngg fngs kdrt dlm dn lntsn pelrn dlh prol.

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