# CHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 9. Intoduction CHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS In this chapte we show how Keple s laws can be deived fom Newton s laws of motion and gavitation, and consevation of angula momentum, and we deive fomulas fo the enegy and angula momentum in an obit. We show also how to calculate the position of a planet in its obit as a function of time. It would be foolish to embak upon this chapte without familiaity with much of the mateial coveed in Chapte. The discussion hee is limited to two dimensions. The coesponding poblem in thee dimensions, and how to calculate an ephemeis of a planet o comet in the sky, will be teated in Chapte Keple s Laws Keple s law of planetay motion (the fist two announced in 609, the thid in 69) ae as follows:. Evey planet moves aound the Sun in an obit that is an ellipse with the Sun at a focus.. The adius vecto fom Sun to planet sweeps out equal aeas in equal time. 3. The squaes of the peiods of the planets ae popotional to the cubes of thei semi majo axes. The fist law is a consequence of the invese squae law of gavitation. An invese squae law of attaction will actually esult in a path that is a conic section that is, an ellipse, a paabola o a hypebola, although only an ellipse, of couse, is a closed obit. An invese squae law of epulsion (fo example, α-paticles being deflected by gold nuclei in the famous Geige-Masden expeiment) will esult in a hypebolic path. An attactive foce that is diectly popotional to the fist powe of the distance also esults in an elliptical path (a Lissajous ellipse) - fo example a mass whiled at the end of a Hooke s law elastic sping - but in that case the cente of attaction is at the cente of the ellipse, athe than at a focus. We shall deive, in section 9.5, Keple s fist and thid laws fom an assumed invese squae law of attaction. The poblem facing Newton was the opposite: Stating fom Keple s laws, what is the law of attaction govening the motions of the planets? To stat with, he had to invent the diffeential and integal calculus. This is a fa cy fom the popula notion that he discoveed gavity by seeing an apple fall fom a tee. The second law is a consequence of consevation of angula momentum, and would be valid fo any law of attaction (o epulsion) as long as the foce was entiely adial with no tansvese component. We deive it in section 9.3.

2 Although a full teatment of the fist and thid laws awaits section 9.5, the thid law is tivially easy to deive in the case of a cicula obit. Fo example, if we suppose that a planet of mass m is in a cicula obit of adius a aound a Sun of mass M, M being supposed to be so much lage than m that the Sun can be egaded as stationay, we can just equate the poduct of mass and centipetal acceleation of the planet, maω, to the gavitational foce between planet and Sun, GMm/a ; and, with the peiod being given by P π/ω, we immediately obtain the thid law: P 4 π a GM The eade might like to show that, if the mass of the Sun is not so high that the Sun s motion can be neglected, and that planet and Sun move in cicula obits aound thei mutual cente of mass, the peiod is P 4π 3 a. 9.. G( M + m) Hee a is the distance between Sun and planet. Execise. Expess the peiod in tems of a, the adius of the planet s cicula obit aound the cente of mass. 9.3 Keple s Second Law fom Consevation of Angula Momentum δθ δ θ & δθ O FIGURE IX. In figue IX., a paticle of mass m is moving in some sot of tajectoy in which the only foce on it is diected towads o away fom the point O. At some time, its pola coodinates ae (, θ). At a time δt late these coodinates have inceased by δ and δθ.

3 3 Using the fomula one half base times height fo the aea of a tiangle, we see that the aea swept out by the adius vecto is appoximately δ A δθ + δθδ On dividing both sides by δt and taking the limit as δt 0, we see that the ate at which the adius vecto sweeps out aea is A & θ & But the angula momentum is m θ &, and since this is constant, the aeal speed is also constant. The aeal speed, in fact, is half the angula momentum pe unit mass. 9.4 Some Functions of the Masses In section 9.5 I am going to conside the motion of two masses, M and m aound thei mutual cente of mass unde the influence of thei gavitational attaction. I shall pobably want to make use of seveal functions of the masses, which I shall define hee, as follows: Total mass of the system: M M + m Mm Reduced mass m M + m Mass function : 3 M M ( ). M + m m No paticula name: m + m M Mass atio: q m /M Mass faction: µ m /( M + m) The fist fou ae of dimension M; the last two ae dimensionless. When m << M, m m, M M and m + m.

4 4 (Fo those who may be inteested, the fonts I have used ae: M Aial bold m Centuy Gothic M Fench scipt MT) 9.5 Keple s Fist and Thid Laws fom Newton s Law of Gavitation M C ' m FIGURE IX. In figue IX. I illustate two masses (they needn t be point masses as long as they ae spheically symmetic, they act gavitationally as if they wee point masses) evolving about thei common cente of mass C. At some time they ae a distance apat, whee m M +,, 9.5. M + m M + m The equations of motion of m in pola coodinates (with C as pole) ae Radial: & θ & GM / Tansvese: & θ + & θ& Elimination of t between these equations will in pinciple give us the equation, in pola coodinates, of the path. A slightly easie appoach is to wite down expessions fo the angula momentum and the enegy. The angula momentum pe unit mass of m with espect to C is h & θ The speed of m is & + θ &, and the speed of M is m/m times this. Some effot will be equied of the eade to detemine that the total enegy E of the system is E m+ GM µ ( + θ& ). & [It is possible that you may have found this line quite difficult. The eason fo the difficulty is that we ae not making the appoximation of a planet of negligible mass moving aound a stationay Sun, but we ae allowing both bodies to have compaable

5 5 masses and the move aound thei common cente of mass. You might fist like to ty the simple poblem of a planet of negligible mass moving aound a stationay Sun. In that case 0 and and m m, M M and m + m.] It is easy to eliminate the time between equations and Thus you can wite d d dθ d &. θ& and then use equation to eliminate θ &. You should dt dθ dt dθ eventually obtain m h d + GM µ. 4 + E dθ This is the diffeential equation, in pola coodinates, fo the path of m. All that is now equied is to integate it to obtain as a function of θ. At fist, integation looks hopelessly difficult, but it poceeds by making one tentative substitution afte anothe to see if we can t make it look a little easie. Fo example, we have (if we multiply out the squae backet) in the denominato thee times in the equation. Let s at least ty the substitution w /. That will suely make it look a little d d dw dw easie. You will have to use, and afte a little algeba, you dθ dw dθ w dθ should obtain dw dθ + GM µ w m h + E m h G M µ m h This may at fist sight not look like much of an impovement, but the ight hand side is just a lot of constants, and, since it is positive, let s call the ight hand side H. (In case you doubt that the ight hand side is positive, the left hand side cetainly is!) Also, make the obvious substitution u GM µ w, m h + and the equation becomes almost tivial: du dθ + u H, fom which we poceed to

6 6. dθ ± H At this stage you can choose eithe the + o the and you can choose to make the next substitution u H sin φ o u H cos φ; you'll get the same esult in the end. I'll choose the plus sign and I ll let u H cos φ, and I get dθ dφ and hence du u θ φ + ω, 9.5. whee ω is the abitay constant of integation. Now you have to go back and emembe what φ was, what u was and what w was and what H was. Thus θ ω φ, âcos (θ ω) cos ( φ) cos φ u/h...and so on. You aim is to get it in the fom function of θ, and, if you pesist, you should eventually get m+ h /( GM µ ) / Eh m cos( θ ω) 4 G M µ You ll immediately ecognize this fom equation.3.37 o.4.6 o.5.8: l ecos ( θ ω) as being the pola equation to a conic section (ellipse, paabola o hypebola). Equation 9.5. is the equation of the path of the mass m about the cente of mass of the two bodies. The eccenticity is / o, if you now ecall what ae meant by µ and m +, (Check the dimensions of this!) Eh m + e +, G M µ / 3 Eh. ( M + m) e G M m The eccenticity is less than, equal to, o geate than (i.e. the path is an ellipse, a paabola o a hypebola) accoding to whethe the total enegy E is negative, zeo o positive.

7 7 The semi latus ectum of the path of m elative to the cente of mass is of length m+ h l, GM µ o, if you now ecall what ae meant by m + and µ (see equations and 9.3.6), l h. ( M + m) G M (Check the dimensions of this!) We can also wite equations o as h G M l At this point it is useful to ecall what we mean by M and by h. M is the mass function, given by equation 9.4.3: 3 M M ( ). M + m Let us suppose that the total enegy is negative, so that the obits ae elliptical. The two masses ae evolving in simila elliptic obits aound the cente of masses; the semi latus ectum of the obit of m is l, and the semi latus ectum of the obit of M is l, whee l l M m Relative to M the mass m is evolving in a lage but still simila ellipse with semi latus ectum l given by l M + m l M I am now going to define h as the angula momentum pe unit mass of m elative to M. In othe wods, we ae woking in a fame in which M is stationay and m is moving aound M in an elliptic obit of semi latus ectum l. Now angula momentum pe unit mass is popotional to the aeal speed, and theefoe it is popotional to the squae of the semi latus ectum. Thus we have

8 8 h h l l M + M m Combining equations 9.5.8, 9.4.3, 9.5.9, and 9.4. we obtain h GMl, 9.5. whee M is the total mass of the system. Once again: The angula momentum pe unit mass of m elative to the cente of mass is GM l, whee l is the semi latus ectum of the obit of m elative to the cente of mass, and it is M. GMl elative to M, whee l is the semi latus ectum of the obit of m elative to If you wee to stat this analysis with the assumption that m << M, and that M emains stationay, and that the cente of mass coincides with M, you would find that eithe equation o 9.5. educes to h G M l The peiod of the elliptic obit is aea + aeal speed. The aea of an ellipse is π ab πa e, and the aeal speed is half the angula momentum pe unit mass (see section 9.3) ( h GM l GMa e ). Theefoe the peiod is π 3/ P a, o GM P 4 π a 3, GM which is Keple s thid law. We might also, while we ae at it, expess the eccenticity (equation) in tems of h athe than h, using equation We obtain: / Eh e +. ( ) G Mm M + m

9 9 If we now substitute fo h fom equation 9.5., and invet equation 9.5.4, we obtain, fo the enegy of the system Gm( M + m)( e ) E, l o fo the enegy o the system pe unit mass of m: GM( e ) ) E l Hee M is the mass of the system i.e. M+m. E in equation is the total enegy of the system, which includes the kinetic enegy of both masses as well as the mutual potential enegy of the two, while E in equation is meely E/m. That is, it is, as stated, the enegy of the system pe unit mass of m. Equations 9.5. and apply to any conic section. Fo the diffeent types of conic section they can be witten: Fo an ellipse: h GM GMa( e ), E 9.5.7a,b a Fo a paabola: h GM q, E a,b Fo a hypebola: h GM GM a( e ), E a,b a We see that the enegy of an elliptic obit is detemined by the semi majo axis, wheeas the angula momentum is detemined by the semi majo axis and by the eccenticity. Fo a given semi majo axis, the angula momentum is geatest when the obit is cicula. Still efeing the obit of m with espect to M, we can find the speed V of m by noting that GM E, V and, by making use of the b-pats of equations , we find the following elations between speed of m in an obit vesus distance fom M:

10 0 Ellipse: V GM. a Paabola: G M V Hypebola: V GM +. a GM Cicle: V a Execise: Show that in an elliptic obit, the speeds at peihelion and aphelion ae, espectively, 9 to GM a + e e and + e aphelion speed is, theefoe,. e GM a e + e and that the atio of peihelion Chapte It might be noted at this point, fom the definition of the astonomical unit (Chapte 8, section 8.), that if distances ae expessed in astonomical units, peiods and time intevals in sideeal yeas, GM (whee M is the mass of the Sun) has the value 4π. The mass of a comet o asteoid is much smalle than the mass of the Sun, so that M M + m j M. Thus, using these units, and to this appoximation, equation becomes meely 3 P a. A Delightful Constuction I am much indebted to an e-coespondent, D Bob Rimme, fo the following delightful constuction. D Rimme found it the ecent book Feynmann s Lost Lectue, The Motion of the Planets Aound the Sun, by D.L. and J.R. Goodstein, and Feynman in his tun ascibed it to a passage (Section IV, Lemma XV) in the Pincipia of Si Isaac Newton. It has no doubt changed slightly with each telling, and I pesent it hee as follows. C is a cicle of adius a (Figue IX.3). F is the cente of the cicle, and F' is a point inside the cicle such that the distance FF' ae, whee e <. Join F and F' to a point Q on the cicle. MP' is the pependicula bisecto of F'Q, meeting FQ at P.

11 The eade is invited to show that, as the point Q moves ound the cicle, the point P descibes an ellipse of eccenticity e, with F and F' as foci, and that MP ' is tangent to the ellipse. Q α P' P M C F F' FIGURE IX.3 Hint: It is vey easy no math equied! Daw the line F'P, and let the lengths of FP and F'P be and ' espectively. It will then become vey obvious that + ' is always equal to a, and hence P descibes an ellipse. By looking at an isosceles tiangle, it will also be clea that the angles F'PM and FPP' ae equal, thus satisfying the focus-to-focus eflection popety of an ellipse, so that MP' is tangent to the ellipse. But thee is bette to come. You ae asked to find the length QF' in tems of a, e and ', o a, e and. An easy way to do it is as follows. Let QF' p, so that QM p. Fom the ight-angled tiangle QMP we see that cos α p / '. Apply the cosine ule to tiangle QF'P to find anothe expession fo cos α, and eliminate cos α fom you two equations. You should quickly aive at And, since ' a, this becomes p a ( e ) ' a '

12 a 3/ p a ( e ) a ( e ). a Now the speed at a point P on an elliptic obit, in which a planet of negligible mass is in obit aound a sta of mass M is given by V GM a Thus we aive at the esult that the length of F'Q (o of F'M) is popotional to the speed of a planet P moving aound the Sun F in an elliptic obit, and of couse the diection MP', being tangent to the ellipse, is the diection of motion of the planet. Figue IX.4 shows the ellipse. Q α P' P M C F F' FIGURE IX.4 It is left to the eade to investigate what happens it F' is outside, o on, the cicle 9.6 Position in an Elliptic Obit The eade might like to efe back to Chapte, section.3, especially the pat that deals with the pola equation to an ellipse, to be eminded of the meanings of the angles θ, ω and v, which, in an astonomical context, ae called, espectively, the agument of

13 3 latitude, the agument of peihelion and the tue anomaly. In this section I shall choose the initial line of pola coodinates to coincide with the majo axis of the ellipse, so that ω is zeo and θ v. The equation to the ellipse is then l ecosv FIGURE IX.5 * v I ll suppose that a planet is at peihelion at time t T, and the aim of this section will be to find v as a function of t. The semi majo axis of the ellipse is a, elated to the semi latus ectum by l a( e ) 9.6. and the peiod is given by P 4π 3 a GM Hee the planet, of mass m is supposed to be in obit aound the Sun of mass M, and the oigin, o pole, of the pola coodinates descibed by equation 9.6. is the Sun, athe than the cente of mass of the system. As usual, M M + m. The adius vecto fom Sun to planet does not move at constant speed (indeed Keple s second law states how it moves), but we can say that, ove a complete obit, it moves at π an aveage angula speed of π/p. The angle ( t T ) P is called the mean anomaly of the planet at a time t T afte peihelion passage. It is geneally denoted by the lette M, which is aleady ovewoked in this chapte fo vaious masses and functions of the masses. Fo mean anomaly, I ll ty Coppeplate Gothic Bold italic font, M. Thus

14 4 M π ( t T ) P The fist step in ou effot to find v as a function of t is to calculate the eccentic anomaly E fom the mean anomaly. This was defined in figue II. of Chapte, and it is epoduced below as figue IX.6. In time t T, the aea swept out by the adius vecto is the aea FBP, and, because the adius vecto sweeps out equal aeas in equal times, this aea is equal to the faction ( t T ) πab ( t T ) / P of the aea of the ellipse. In othe wods, this aea is. Now look at P the aea FBP'. Evey odinate of that aea is equal to b/a times the coesponding ( t T ) πa odinate of FBP, and theefoe the aea of FBP' is. The aea FBP' is also P equal to the secto OP'B minus the tiangle OP'F. The aea of the secto OP'B is E πa Ea, and the aea of the tiangle OP'F is ae a sin E a esin. π E ( t T ) πa P Ea a esin E. P' P O E F v B FIGURE IX.6 Multiply both sides by /a, and ecall equation 9.6.4, and we aive at the equied elation between the mean anomaly M and the eccentic anomaly E:

15 5 M E esin E This is Keple s equation. The fist step, then, is to calculate the mean anomaly M fom equation 9.6.4, and then calculate the eccentic anomaly E fom equation This is a tanscendental equation, so I ll say a wod o two about solving it in a moment, but let s pess on fo the time being. We now have to calculate the tue anomaly v fom the eccentic anomaly. This is done fom the geomety of the ellipse, with no dynamics, and the elation is given in Chapte, equations.3.6 and.3.7c, which ae epoduced hee: cose e cosv. ecose.3.6 Fom tigonometic identities, this can also be witten e sin E sinv,.3.7a e cos E o e sin E tanv.3.7b cos E e + e o tan v tan E. e.3.7c If we can just solve equation (Keple s equation), we shall have done what we want namely, find the tue anomaly as a function of the time. The solution of Keple s equation is in fact vey easy. We wite it as f ( E) E esin E M fom which f '( E) ecos E, and then, by the usual Newton-Raphson pocess: E ( E cos E sin E). e M e cos E The computation is then extaodinaily apid (especially if you stoe cos E and don t calculate it twice!).

16 6 Example: Suppose e 0.95 and that M 45 o. Calculate E. Since the eccenticity is vey lage, one might expect the convegence to be slow, and also E is likely to be vey diffeent fom M, so it is not easy to make a fist guess fo E. You might as well ty 45 o fo a fist guess fo E. You should find that it conveges to ten significant figues in a mee fou iteations. Even if you make a mindlessly stupid fist guess of E 0 o, it conveges to ten significant figues in only nine iteations. Thee ae a few exceptional occasions, hadly eve encounteed in pactice, and only fo eccenticities geate than about 0.99, when the Newton-Raphson method will not convege when you make you fist guess fo E equal to M. Chales and Tatum (CelestialMechanics and Dynamical Astonomy 69, 357 (998)) have shown that the Newton-Raphson method will always convege if you make you fist guess E π. Nevetheless, the situations whee Newton-Raphson will not convege with a fist guess of E M ae unlikely to be encounteed except in almost paabolic obits, and usually a fist guess of E M is faste than a fist guess of E π. Τhe chaotic behaviou of Keple s equation on these exceptional occasions is discussed in the above pape and also by Stumpf (Cel. Mechs. and Dyn. Aston. 74, 95 (999)) and efeences theein. Execise: Show that a good fist guess fo E is E M + x( x ), esin M whee x ecosm Execise: Wite a compute pogam in the language of you choice fo solving Keple s equation. The pogam should accept e and M as input, and etun E as output. The Newton-Raphson iteation should be teminated when ( Enew Eold ) / Eold is less than some small faction to be detemined by you. Execise: An asteoid is moving in an elliptic obit of semi majo axis 3 AU and eccenticity 0.6. It is at peihelion at time 0. Calculate its distance fom the Sun and its tue anomaly one sideeal yea late. You may take the mass of the asteoid and the mass of Eath to be negligible compaed with the mass of the Sun. In that case, equation is meely P π G M 4 3 a, whee M is the mass of the Sun, and, if P is expessed in sideeal yeas and a in AU, this becomes just P a 3. Thus you can immediately calculate the peiod in yeas and hence, fom equation you can find the mean anomaly. Fom thee, you have to

17 7 solve Keple s equation to get the eccentic anomaly, and the tue anomaly fom equation.3.6 o 7. Just make sue that you get the quadant ight. Execise: Wite a compute pogam that will give you the tue anomaly and heliocentic distance as a function of time since peihelion passage fo an asteoid whose elliptic obit is chaacteized by a, e. Run the pogam fo the asteoid of the pevious execise fo evey day fo a complete peiod. You ae now making some eal pogess towads ephemeis computation! 9.7 Position in a Paabolic Obit When a long-peiod comet comes in fom the Oot belt, it typically comes in on a highly eccentic obit, of which we can obseve only a vey shot ac. Consequently, it is often impossible to detemine the peiod o semi majo axis with any degee of eliability o to distinguish the obit fom a paabola. Thee is theefoe fequent occasion to have to undestand the dynamics of a paabolic obit. We have no mean o eccentic anomalies. We must ty to get v diectly as a function of t without going though these intemediaies. The angula momentum pe unit mass is given by equation 9.5.8a: h v& GM q, 9.7. whee v is the tue anomaly and q is the peihelion distance. But the equation to the paabola (see equation.4.6) is q, cosv o (see section 3.8 of Chapte 3), by making use of the identity u, cosv whee u tan v, + u 9.7.3a,b the equation to the paabola can be witten q sec v Thus, by substitution of equation into 9.7. and integating, we obtain

18 8 v ( ) d G q v t q sec 4 v M dt T 0 Upon integation (dop me an if you get stuck!) this becomes u + GM ( t ). 3/ q 3 3 u T This equation, when solved fo u (which, emembe, is tan v ), gives us v as a function of t. As explained at the end of section 9.5, if q is in astonomical units and t T is in sideeal yeas, and if the mass of the comet is negligible compaed with the mass of the Sun, this becomes u ( t T ) 3 π + 3 u / q 3 π 8( t T ) o 3u + u C 0, whee C a,b 3/ q Thee is a choice of methods available fo solving equation 9.7.8, so it might be that the only difficulty is to decide which of the seveal methods you want to use! The constant C is sometimes called the paabolic mean anomaly. 3 Method : Just solve it by Newton-Raphson iteation. Thus f 3u + u 3 f ' 3( + u ), so that the Newton-Raphson u u f / f ' becomes C 0 and u 3 u + C, ( + u ) which should convege quickly. Fo economy, calculate u only once pe iteation. Method : Let u x / x and C c / c a,b Then equation 9.7.8a becomes x c / Thus, as soon as c is found, x, u and v can be calculated fom equations 9.7., 0a, and 3a o b, and the poblem is finished as soon as c is found! So, how do we find c? We have to solve equation 9.7.0b.

19 9 Method a: Equation 9.7.0b can be witten as a quadatic equation: c Cc Just be caeful that you choose the coect oot; you should end with v having the same sign as t T. Method b: Let C cot φ and calculate φ. But by a tigonometic identity, cot φ cot φ / cot φ so that, by compaison with equation 9.6.0b, we see that c cot φ Again, just make sue that you choose the ight quadant in calculating φ fom equation 9.7.3, so as to be sue that you end with v having the same sign as t T. Method 3. I am told that equation has the exact analytic solution 3 3 u w w, whee w C C I haven t veified this fo myself, so you might like to have a go. Example: Solve the equation 3u + u 3 3.).6 by all fou methods. (Methods, a, b and Example: A comet is moving in an elliptic obit with peihelion distance 0.9 AU. Calculate the tue anomaly and heliocentic distance 0 days afte peihelion passage. (A sideeal yea is days.)

20 0 Execise: Wite a compute pogam that will etun the tue and anomaly as a function of time, given the peihelion distance of a paabolic obit. Test it with you answe fo the pevious example. 9.8 Position in a Hypebolic Obit If an intestella comet wee to encounte the sola system fom intestella space, it would pusue a hypebolic obit aound the Sun. To date, no such comet with an oiginal hypebolic obit has been found, although thee is no paticula eason why we might not find one some night. Howeve, a comet with a nea-paabolic obit fom the Oot belt may appoach Jupite on its way in to the inne sola system, and its obit may be petubed into a hypebolic obit. This will esult in its ultimate loss fom the sola system. Seveal examples of such cometay obits ae known. Thee is evidence, fom ada studies of meteos, of meteooidal dust encounteing Eath at speeds that ae hypebolic with espect to the Sun, although whethe these ae on obits that ae oiginally hypebolic (and ae theefoe fom intestella space) o whethe they ae of sola system oigin and have been petubed by Jupite into hypebolic obits is not known. I must admit to not having actually caied out a calculation fo a hypebolic obit, but I think we can just poceed in a manne simila to an ellipse o a paabola. Thus we can stat with the angula momentum pe unit mass: whee h v& GMl, 9.8. l ecosv and l a( e ) If we use astonomical units fo distance and mass, we obtain v 0 dv π ( + ecosv ) a 3/ ( e ) 3/ t T dt Hee I am using astonomical units of distance and mass and have theefoe substituted 4π fo G M. I m going to wite this as

21 v dv ( + cosv) 0 π( t T ) 3/ a ( e ) 3/ ( e Q ) 3/, π( t T ) whee Q. Now we have to integate this. 3/ a Method. Guided by the elliptical case, but beaing in mind that we ae now dealing with a hypebola, I m going to ty the substitution If you ty this, I think you ll end up with This is just the analogy of Keple s equation. e cosh E cosv ecosh E e sinh E E Q The pocedue, then, would be to calculate Q fom equation Then calculate E fom equation This could be done, fo example, by a Newton-Raphson iteation in quite the same way as was done fo Keple s equation in the elliptic case, the iteation now taking the fom ( E cosh E sinh E). Q + e E ecosh E Then v is found fom equation 9.8.6, and the heliocentic distance is found fom the pola equation to a hypebola: a( e ) ecosv Method. Method should wok all ight, but it has the disadvantage that you may not be as familia with sinh and cosh as you ae with sin and cos, o thee may not be a sinh o cosh button you calculato. I believe thee ae SINH and COSH functions in FORTRAN, and thee may well be in othe computing languages. Ty it and see. But maybe we d like to ty to avoid hypebolic functions, so let s ty the billiant substitution

22 u( u e) + cosv u( eu ) + e You may have noticed, when you wee leaning calculus, that often the pofesso would make a billiant substitution, and you could see that it woked, but you could neve undestand what made the pofesso think of the substitution. I don t want to tell you what made me think of this substitution, because, when I do, you ll see that it isn t eally vey billiant at all. I emembeed that and then I let e E u, so E E ( e + ) cosh E e 9.8. cosh E ( u + / / u ), 9.8. and I just substituted this into equation and I got equation Now if you put the expession fo cos v into equation 9.8.5, you eventually, afte a few lines, get something that you can integate. Please do wok though it. In the end, on integation of equation 9.8.5, you should get e ( u / u ) lnu Q You aleady know fom Chapte how to solve the equation f (x) 0, so thee is no difficulty in solving equation fo u. Newton-Raphson iteation esults in [ e u( Q ln u) ], u u u( eu ) + and this should convege in the usual apid fashion. So the pocedue in method is to calculate Q fom equation 9.8.5, then calculate u fom equation 9.8.4, and finally v fom equation all vey staightfowad. Execise: Set youself a poblem to make sue that you can cay though the calculation. Then wite a compute pogam that will geneate v and as a function of t. 9.9 Obital Elements and Velocity Vecto In two dimensions, an obit can be completely specified by fou obital elements. Thee of them give the size, shape and oientation of the obit. They ae, espectively, a, e and ω. We ae familia with the semi majo axis a and the eccenticity e. The angle ω, the agument of peihelion, was illustated in figue II.9, which is epoduced hee as figue IX.7. It is the angle that the majo axis makes with the initial line of the pola

23 3 coodinates. Figue II.9 eminds us of the elation between the agument of peihelion ω, the agument of latitude θ and the tue anomaly v. We emind ouselves hee of the equation to a conic section l l + e cosv + e cos( θ ω), 9.9. whee the semi latus ectum l is a( e ) fo an ellipse, and a(e ) fo a hypebola. Fo a hypebola, the paamete a is usually called the semi tansvese axis. Fo a paabola, the size is geneally descibed by the peihelion distance q, and l q. The fouth element is needed to give infomation about whee the planet is in its obit at a paticula time. Usually this is T, the time of peihelion passage. In the case of a cicula obit this cannot be used. One could instead give the time when θ 0, o the value of θ at some specified time. θ v + ω v ω FIGURE IX.7 Refe now to figue IX.8. We ll suppose that at some time t we know the coodinates (x, y) o (, θ) of the planet, and also the velocity that is to say the speed and diection, o the x- and y- o the adial and tansvese components of the velocity. That is, we know fou quantities. The subsequent path of the planet is then detemined. In othe wods, given the fou quantities (two components of the position vecto and two components of the velocity vecto), we should be able to detemine the fou elements a, e, ω and T. Let us ty.

### 2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

### Revision Guide for Chapter 11

Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams

### Ch. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth

Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

### 2. Orbital dynamics and tides

2. Obital dynamics and tides 2.1 The two-body poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body

### mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !

Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!

### PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0

### Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,

### Gravitation. AP Physics C

Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

### Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

### FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

### So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)

Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing

### Determining solar characteristics using planetary data

Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation

### The Role of Gravity in Orbital Motion

! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

### Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43

Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.

### Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

### Trigonometric Functions of Any Angle

Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An

### Physics 235 Chapter 5. Chapter 5 Gravitation

Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

### Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

### A) 2 B) 2 C) 2 2 D) 4 E) 8

Page 1 of 8 CTGavity-1. m M Two spheical masses m and M ae a distance apat. The distance between thei centes is halved (deceased by a facto of 2). What happens to the magnitude of the foce of gavity between

### Chapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom

Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in

### Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

### Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

### Mechanics 1: Motion in a Central Force Field

Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

### Analytical Proof of Newton's Force Laws

Analytical Poof of Newton s Foce Laws Page 1 1 Intouction Analytical Poof of Newton's Foce Laws Many stuents intuitively assume that Newton's inetial an gavitational foce laws, F = ma an Mm F = G, ae tue

### Chapter 13. Vector-Valued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates

13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. Vecto-Valued Functions and Motion in Space 13.6. Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along

### 2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

### Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

### 4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

### Resources. Circular Motion: From Motor Racing to Satellites. Uniform Circular Motion. Sir Isaac Newton 3/24/10. Dr Jeff McCallum School of Physics

3/4/0 Resouces Cicula Motion: Fom Moto Racing to Satellites D Jeff McCallum School of Physics http://www.gap-system.og/~histoy/mathematicians/ Newton.html http://www.fg-a.com http://www.clke.com/clipat

### Exam 3: Equation Summary

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

### Voltage ( = Electric Potential )

V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

### 2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90

. Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal

### Coordinate Systems L. M. Kalnins, March 2009

Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

### est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

### Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

### Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

### Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and

Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon

### General Physics (PHY 2130)

Geneal Physics (PHY 130) Lectue 11 Rotational kinematics and unifom cicula motion Angula displacement Angula speed and acceleation http://www.physics.wayne.edu/~apetov/phy130/ Lightning Review Last lectue:

### Section 5-3 Angles and Their Measure

5 5 TRIGONOMETRIC FUNCTIONS Section 5- Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,

Chapte 5. Foce and Motion In this chapte we study causes of motion: Why does the windsufe blast acoss the wate in the way he does? The combined foces of the wind, wate, and gavity acceleate him accoding

### STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

### 8-1 Newton s Law of Universal Gravitation

8-1 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,

### LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

### Exam I. Spring 2004 Serway & Jewett, Chapters 1-5. Fill in the bubble for the correct answer on the answer sheet. next to the number.

Agin/Meye PART I: QUALITATIVE Exam I Sping 2004 Seway & Jewett, Chaptes 1-5 Assigned Seat Numbe Fill in the bubble fo the coect answe on the answe sheet. next to the numbe. NO PARTIAL CREDIT: SUBMIT ONE

### 6.2 Orbits and Kepler s Laws

Eath satellite in unstable obit 6. Obits and Keple s Laws satellite in stable obit Figue 1 Compaing stable and unstable obits of an atificial satellite. If a satellite is fa enough fom Eath s suface that

### F G r. Don't confuse G with g: "Big G" and "little g" are totally different things.

G-1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just

### Problem Set 6: Solutions

UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 16-4 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente

### Deflection of Electrons by Electric and Magnetic Fields

Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

### Continuous Compounding and Annualization

Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

### Experiment 6: Centripetal Force

Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

### In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

Radians At school we usually lean to measue an angle in degees. Howeve, thee ae othe ways of measuing an angle. One that we ae going to have a look at hee is measuing angles in units called adians. In

### The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

### Orbital Motion & Gravity

Astonomy: Planetay Motion 1 Obital Motion D. Bill Pezzaglia A. Galileo & Fee Fall Obital Motion & Gavity B. Obits C. Newton s Laws Updated: 013Ma05 D. Einstein A. Galileo & Fee Fall 3 1. Pojectile Motion

### Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

### CHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL

CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The

### Lab #7: Energy Conservation

Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

### 1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

### Multiple choice questions [70 points]

Multiple choice questions [70 points] Answe all of the following questions. Read each question caefull. Fill the coect bubble on ou scanton sheet. Each question has exactl one coect answe. All questions

### Multiple choice questions [60 points]

1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions

### Carter-Penrose diagrams and black holes

Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

### INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in

### 2008 Quarter-Final Exam Solutions

2008 Quate-final Exam - Solutions 1 2008 Quate-Final Exam Solutions 1 A chaged paticle with chage q and mass m stats with an initial kinetic enegy K at the middle of a unifomly chaged spheical egion of

### Algebra and Trig. I. A point is a location or position that has no size or dimension.

Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite

### Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning

Gavitational Mechanics of the Mas-Phobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This

### Gravitation and Kepler s Laws

3 Gavitation and Keple s Laws In this chapte we will ecall the law of univesal gavitation and will then deive the esult that a spheically symmetic object acts gavitationally like a point mass at its cente

### PY1052 Problem Set 8 Autumn 2004 Solutions

PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

### PY1052 Problem Set 3 Autumn 2004 Solutions

PY1052 Poblem Set 3 Autumn 2004 Solutions C F = 8 N F = 25 N 1 2 A A (1) A foce F 1 = 8 N is exeted hoizontally on block A, which has a mass of 4.5 kg. The coefficient of static fiction between A and the

### BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR. John R. Graham Adapted from S. Viswanathan FUQUA SCHOOL OF BUSINESS

BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR John R. Gaham Adapted fom S. Viswanathan FUQUA SCHOOL OF BUSINESS DUKE UNIVERSITY 1 In this lectue we conside the effect of govenment

### L19 Geomagnetic Field Part I

Intoduction to Geophysics L19-1 L19 Geomagnetic Field Pat I 1. Intoduction We now stat the last majo topic o this class which is magnetic ields and measuing the magnetic popeties o mateials. As a way o

### 12. Rolling, Torque, and Angular Momentum

12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

### SAMPLE CHAPTERS UNESCO EOLSS THE MOTION OF CELESTIAL BODIES. Kaare Aksnes Institute of Theoretical Astrophysics University of Oslo

THE MOTION OF CELESTIAL BODIES Kaae Aksnes Institute of Theoetical Astophysics Univesity of Oslo Keywods: celestial mechanics, two-body obits, thee-body obits, petubations, tides, non-gavitational foces,

### Chapter 26 - Electric Field. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapte 6 lectic Field A PowePoint Pesentation by Paul. Tippens, Pofesso of Physics Southen Polytechnic State Univesity 7 Objectives: Afte finishing this unit you should be able to: Define the electic field

### 4.1 - Trigonometric Functions of Acute Angles

4.1 - Tigonometic Functions of cute ngles a is a half-line that begins at a point and etends indefinitel in some diection. Two as that shae a common endpoint (o vete) fom an angle. If we designate one

### Review of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction.

Appendi A Review of Vectos This appendi is a summa of the mathematical aspects of vectos used in electicit and magnetism. Fo a moe detailed intoduction to vectos, see Chapte 1. A.1 DESCRIBING THE 3D WORLD:

### Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

### Trigonometry in the Cartesian Plane

Tigonomet in the Catesian Plane CHAT Algeba sec. 0. to 0.5 *Tigonomet comes fom the Geek wod meaning measuement of tiangles. It pimail dealt with angles and tiangles as it petained to navigation astonom

### Structure and evolution of circumstellar disks during the early phase of accretion from a parent cloud

Cente fo Tubulence Reseach Annual Reseach Biefs 2001 209 Stuctue and evolution of cicumstella disks duing the ealy phase of accetion fom a paent cloud By Olusola C. Idowu 1. Motivation and Backgound The

### Physics 505 Homework No. 5 Solutions S5-1. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z.

Physics 55 Homewok No. 5 s S5-. Angula momentum uncetainty elations. A system is in the lm eigenstate of L 2, L z. a Show that the expectation values of L ± = L x ± il y, L x, and L y all vanish. ψ lm

### Forces & Magnetic Dipoles. r r τ = μ B r

Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

### Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

### Valuation of Floating Rate Bonds 1

Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned

### Voltage ( = Electric Potential )

V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

### Ilona V. Tregub, ScD., Professor

Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

### The LCOE is defined as the energy price (\$ per unit of energy output) for which the Net Present Value of the investment is zero.

Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the

### 9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics

Available Online Tutoing fo students of classes 4 to 1 in Physics, 9. Mathematics Class 1 Pactice Pape 1 3 1. Wite the pincipal value of cos.. Wite the ange of the pincipal banch of sec 1 defined on the

### Ch. 14: Gravitation (Beta Version 7/01) 14 Gravitation

Ch. 14: Gavitation (Beta Vesion 7/01) 14 Gavitation The Milky Way galaxy is a disk-shaped collection of dust, planets, and billions of stas, including ou Sun and sola system. The foce that binds it o any

### Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers

Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,

### A r. (Can you see that this just gives the formula we had above?)

24-1 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down - you can pedict (o contol) motion

### Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool

Univesal Cycles 2011 Yu She Wial Gamma School fo Gils Depatment of Mathematical Sciences Univesity of Livepool Supeviso: Pofesso P. J. Giblin Contents 1 Intoduction 2 2 De Buijn sequences and Euleian Gaphs

### An Introduction to Omega

An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

### New proofs for the perimeter and area of a circle

New poofs fo the peimete and aea of a cicle K. Raghul Kuma Reseach Schola, Depatment of Physics, Nallamuthu Gounde Mahalingam College, Pollachi, Tamil Nadu 64001, India 1 aghul_physics@yahoo.com aghulkumak5@gmail.com

### 10. Collisions. Before During After

10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: -F(t)

### Gravity and the figure of the Earth

Gavity and the figue of the Eath Eic Calais Pudue Univesity Depatment of Eath and Atmospheic Sciences West Lafayette, IN 47907-1397 ecalais@pudue.edu http://www.eas.pudue.edu/~calais/ Objectives What is

### 7 Circular Motion. 7-1 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary

7 Cicula Motion 7-1 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o

### 2. An asteroid revolves around the Sun with a mean orbital radius twice that of Earth s. Predict the period of the asteroid in Earth years.

CHAPTR 7 Gavitation Pactice Poblems 7.1 Planetay Motion and Gavitation pages 171 178 page 174 1. If Ganymede, one of Jupite s moons, has a peiod of days, how many units ae thee in its obital adius? Use

### Chapter F. Magnetism. Blinn College - Physics Terry Honan

Chapte F Magnetism Blinn College - Physics 46 - Tey Honan F. - Magnetic Dipoles and Magnetic Fields Electomagnetic Duality Thee ae two types of "magnetic chage" o poles, Noth poles N and South poles S.