FW Physics 130 G:\130 lctur\ch 13 Elliticl motion.docx g 1 of 7 11/3/010; 6:40 PM; Lst rintd 11/3/010 6:40:00 PM Fig. 1 Elliticl motion, grvity, tc minor xis mjor xis F 1 =A F =B C - D, mjor nd minor xs : distnc from focl oint to cntr of llis = focl distnc rmtr, rndiculr distnc from focl oint to llis rimtr. Th dfinition of th llis is: th st of ll oints th sum of whos distncs to two givn oints (foci F 1 nd F ) is constnt. Thus on cn s tht th distnc from F 1 to th oint rndiculr low F lus th distnc is qul to. ABCB ADB If you construct th llis y mns of string you cn s this sily. Put in into F 1 nd F. Attch string to oth ins nd strtch it. Whn you strtch it horizontlly you covr th distncs + + = W cn sily vrify th following rortis: = (-) -4 = 4-4 + - 4 - = or = ( )/ nd lso from th othr tringl c = which rsults in th rltionshi : 1) = / W lso dfin nw vlu ε, clld ccntricity, which w us furthr low. 0<ε<1
FW Physics 130 G:\130 lctur\ch 13 Elliticl motion.docx g of 7 11/3/010; 6:40 PM; Lst rintd 11/3/010 6:40:00 PM (1.1) 1 1 1 1 (1.) 1 Fig. r 1 r θ 1 x 1 x θ θ F (0,0) 1 F cosθ 1 = x 1 /r 1 cosθ = x /r ; r 1 = /(1- ε cosθ 1 ); r = /(1- ε cosθ )= /(1+ ε cosθ); r = for θ or θ =π/ r 1 +r = (dfinition of th llis) lso x 1 + x = r r with = - cos cos (1.3) 1 1 cos1 1 cos 1 cos
FW Physics 130 G:\130 lctur\ch 13 Elliticl motion.docx g 3 of 7 11/3/010; 6:40 PM; Lst rintd 11/3/010 6:40:00 PM (Polr rrsnttion of th llis; for ε = 0 this is circl, for ε =1 this is rol, for 0< ε < 1 llis, for ε>1 hyrol). Chck tht th rihlion nd th hlion distncs com out corrctly: for =0;r 1 cos 1 1 for =0;r 1 1 cos 1 1 r 1 (1- ε cosθ 1 ) = r 1 (1 εx 1 /r 1 ) = r 1 εx 1 = r (1- ε cosθ ) = r (1 εx /r ) = r εx = If w dd th two qutions w gt: r 1 εx 1 + r εx = r 1 + r + ε(x 1 + x ) = W isolt th rdii on on sid: r 1 + r = - ε(x 1 + x ) = ( ε) = ( / /) nd s = w gt th finl rsult: r 1 + r = which is th dfinition of th llis. Th Crtsin rrsnttion of th llis with th cntr t x=0, y=0 is of cours x y 1 If th x, y coordints r cntrd in th lft focl oint, w gt th formul: x y 1; th cntr of th llis is now (in th lft-shiftd coordint systm) loctd t (,0) (1.4) r ; r x y ; ; ; 1cos r mgnitud of th rdius vctor; xcntricity of th llis x=rcos ; Lt us convrt th olr form of th llis into th stndrd form. Us th olr qution with th origin of r in th lft focl oint. For th hysics of lliticl motion s clcs.xls (1.5) r r 1cos 1 x r
FW Physics 130 G:\130 lctur\ch 13 Elliticl motion.docx g 4 of 7 11/3/010; 6:40 PM; Lst rintd 11/3/010 6:40:00 PM Exml: Find th lngth of r if th ngl θ is 0 dgrs nd =0.5. 3 ; ; 4 (1.6) 3 r 4 1.5 r hlion 1cos 1 cos0 10.5 (1.7) (1.8) x r 1 r x r x r 4 r x x x ; r x y x x x y 4 x cx y divid y 4 x x y x x y x y 0 x y x y 1 Ellis with rdius vctor strting in F (th focl oint to th lft) (1.9) x y r is quivlnt to 1 1cos 1
FW Physics 130 G:\130 lctur\ch 13 Elliticl motion.docx g 5 of 7 11/3/010; 6:40 PM; Lst rintd 11/3/010 6:40:00 PM Angulr momntum in olr coordints: rd circulr rc lngth, dr=rdil chng in lngth of th vctor r rd r = r of rctngl da d r r L r swt ovr y r in th tim dt = constnt dt dt m t L L A L A(t)= dt t; =constnt m m t m 0 Consrvtion of ngulr momntum for n lliticl motion of mss m yilds Klr s scond lw, cus th mgnitud of ngulr momntum, dividd y m is qul to th r swt ovr y th rdius vctor r in th tim dt. (1.10) L r r m ru ru r m ru mr u r z rdθ u θ dr u r r 1 dθ dr dr d rur ru ur r u dt dt dt dr drur rdu θ F (0,0) 1 F Chck lso Angulr momntum Th tringl with ngl dθ hs th s r+dr nd hight rdθ. Th r is thrfor da=1/(r+dr)rdθ which is roximtly 1/r dθ. da 1 L (1.11) r dt m
FW Physics 130 G:\130 lctur\ch 13 Elliticl motion.docx g 6 of 7 11/3/010; 6:40 PM; Lst rintd 11/3/010 6:40:00 PM By intgrting Nwton s lw of motion for lnt with mss m ttrctd y th sun of mss M, w cn rlt th rmtrs of th llis to hysicl constnts: S: ch 13 Grvittionl Enrgy intgrt llis.doc L EL m m mmg; ; 1 (1.1) E ; L mr Th totl nrgy is indndnt of oth L nd ω. In ordr to gt n orit chrctrizd y th ccntricity w nd to djust th vlu for L. On of th wys to do this is to choos n rorit vlocity t th og or rig osition. For xml, to gt n llis with ccntricity 0.5 nd smi-mjor xis w cn clcult th vlu for r s follows: From th gomtricl dfinition (1.1) ihlion: r 1 1 (1.13) rhlion: r 1 1 Th gomtry of th llis is dtrmind in th momnt th smi-mjor xis nd th ccntricity r givn. Th st wy to chiv rticulr sh is to chng th st th vlocity t th rig or og, cus L=mvr t ths loctions. From (1.1) L (1.14) 1 m W know tht L is constnt nd tht t th rig r w hv: L mr v (1.15) mrv v m 1 m mr 1 (1.16) mmg MG v nd v mr m 1 1 1 1 1 MG 1 1 =10.00 m; 0.5; r 1.50010 m; r 0.510 m 8 8 8 v 1537 m/ s; v 3v In contrst to this w would gt for circulr motion with rdius r= simly:
FW Physics 130 G:\130 lctur\ch 13 Elliticl motion.docx g 7 of 7 11/3/010; 6:40 PM; Lst rintd 11/3/010 6:40:00 PM v MG MG (1.17) or v S th othr r ch 13 Grvittionl Enrgy intgrt llis.doc.