Hip Hop solutions of the 2N Body problem


 Adela Patrick
 2 years ago
 Views:
Transcription
1 Hip Hop solutions of the N Boy poblem Esthe Baabés Depatament Infomàtica i Matemàtica Aplicaa, Univesitat e Giona. Josep Maia Cos Depatament e Matemàtica Aplicaa III, Univesitat Politècnica e Catalunya. Conxita Pinyol Depatament Economia i Històia Econòmica, Univesitat Autònoma e Bacelona. Jaume Sole Depatament Infomàtica i Matemàtica Aplicaa, Univesitat e Giona. Abstact. Hip Hop solutions of the N boy poblem with equal masses ae shown to exist using an analytic continuation agument. These solutions ae close to plana egula N gon elative equilibia with small vetical oscillations. Fo fixe N, an infinity of these solutions ae thee imensional choeogaphies, with all the boies moving along the same close cuve in the inetial fame. Keywos: Nboy poblem, analytic continuation, Hip Hop, Choeogaphies. Intouction The equal mass n boy poblem has ecently attacte much attention thanks to the wok of Chencine an othe authos on the type of obits calle hiphop solutions, an on the solutions that have eventually been calle choeogaphies. In a hiphop solution, N boies of equal mass stay fo all time in the vetices of a egula otating antipism whose basis, i.e. the egula polygons that efine it, pefom an oscillatoy motion sepaating, eaching a maximum istance, appoaching, cossing each othe, an so on, as sketche in Figue fo N = 3. The N boies can be aange in two goups of N, each goup moving on its plane on a otating egula N gon configuation homogaphic, while the planes ae always pepenicula to the zaxis, oscillate along this axis, an coincie with opposite velocities at egula intevals when they coss the oigin. The othogonal pojection of both N gons on the z = 0 plane is always a egula otating N gon. On the othe han, a choeogaphy is a solution in which n boies move along the same close line in the inetial fame, chasing each othe at equi space intevals of time. It is well known the figue eight choeogaphy in the thee boy poblem, shown by Chencine c 006 Kluwe Acaemic Publishes. Pinte in the Nethelans. hiphop.tex; 6/0/006; 5:44; p.
2 Baabés, Cos, Pinyol, Sole an Montgomey 00 in a most celebate pape. A geat many choeogaphies with n > 3 have been shown numeically to exist by Simo 00. The above esults wee obtaine mostly by means of vaiational methos, which make it possible to fin solutions that o not epen on a small paamete, i.e. fa fom solutions of an integable poblem. See Chencine an Ventuelli, 000, Chencine et al., 00, Chencine an Féjoz, 005, an efeences theein fo etails. In the case of hip hop solutions, the question aises whethe in some simple cases they coul be obtaine though the taitional analytic continuation metho of Poincaé, which woul give families iffeentiable with espect to a paamete of peioic solutions, at least in a otating fame. In this espect, mention shoul be mae of a esult by Meye an Schmit 993 on a simila solution with a lage cental mass an n vey small, equal masses aoun it which was suggeste as a moel fo the baie stuctue of some of Satun s ings. In this pape we show that Poincae s agument of analytic continuation can be use to a vetical oscillations to the cicula motion of N boies of equal mass occupying the vetices of a egula Ngon. In this way, a family of theeimensional obits, peioic in a otating fame, can be shown to exist. This is a Lyapunov family of obits whose peios ten to the peio of the vetical oscillations of the lineaize system aoun the elative equilibium solution. These solutions wee foun numeically by Davies et al Infinitely many of this solutions ae peioic in the inetial fame, povie that the quantity HN given by 7 oes not vanish, an ae thee imensional choeogaphies, in the sense that all boies move at equi space time intevals along a close twiste cuve in the inetial fame. Some solutions foun in ou aticle may coincie with the genealize hip hop solutions obtaine by Chencine in 00. Teacini an Ventuelli 005 ecently showe the existence of hip hop solutions in the same poblem using vaiational methos, aing vetical vaiations to the plana elative equilibium in oe to euce the value of the action functional. The vaiational appoach oes not epen on any small paamete an yiels global existence, while continuation methos give explicit appoximations to solutions in a small neighbouhoo of the elative equilibium. A pecise compaison of both methos fom a puely analytic point of view woul involve eithe estimating the istance fom the vaiational solutions to the elative equilibium o estimating the size of the neighbouhoo in which the family can be continue, but both questions seem fa fom easy. hiphop.tex; 6/0/006; 5:44; p.
3 Hip Hop solutions of the N Boy poblem Figue. Qualitative epesentation of a HipHop motion in the case of 6 boies. Equations of motion Consie N boies with equal mass m moving une thei mutual gavitational attaction an let i, ṙ i, i =,, N, be thei positions an velocities. The equations of motion of the N boy poblem ae i = Gm N,k i k i ki 3, whee ki = k i. Scaling the time t by Gm t the Lagangian function associate to the poblem becomes L = N i= ṙ i + i<j N i j. As we ae looking fo solutions of the N boy poblem such that all the boies stay fo all time on the vetices of an antipism, it will suffice to know the position an velocity of one of the N boies. Given = t, we efine i = R i, ṙ i = R i ṙ, fo i =,, N, whee R is a otation plus a eflection in such a way that all the boies ae on the vetices of an antipism. Since in this configuation N of the boies on a plane an the othe N on a paallel plane we can assume, without loss of geneality, that both planes ae pepenicula to the z axis. In this case, the matix R can be witten hiphop.tex; 6/0/006; 5:44; p.3
4 4 Baabés, Cos, Pinyol, Sole as cos π N sin π N 0 R = sin π N cos π N PROPOSITION.,,..., N =, R,..., R N is a solution of the N boy poblem given by if an only if t satisfies the equation N R k I = R k I 3. Poof. Substituting,,..., N =, R,..., R N in we have R i = N j=,j i R j R i R j R i 3 = N j=,j i R i R j i I R j i I 3, fo i =,..., N. Using the fact that R l = R N l, fo l =,..., N, we get = = = i j= i l= N R j i I R j i I 3 + N j=i+ R N l N i I R N l I 3 + R k I R k I 3. l= R j i I R j i I 3 R l I R l I 3 That is, we get the same equation fom the initial N equations. If = x, y, z is the position of the fist boy, then the equations of motion can be witten as the following iffeential system of oe two ẍ = Ux, y, z, ÿ = x Ux, y, z, z = y whee Ux, y, z is the potential function Ux, y, z = N Ux, y, z, 3 z. 4x + y sin kπ N + k z The poblem state by system 3 can be fomulate in Hamiltonian tems by the Hamiltonian function H = p x + p y + p z Ux, y, z, 4 hiphop.tex; 6/0/006; 5:44; p.4
5 Hip Hop solutions of the N Boy poblem 5 whee p x, p y an p z ae the momenta associate to the x, y, z cooinates. Intoucing cylinical cooinates by means of the canonical change x = cos θ, p x = p cos θ p θ sin θ, y = sin θ, p y = p sin θ + p θ cos θ, z =, p z = p, the Hamiltonian 4 becomes H = p + p θ + p N 4 sin kπ N Then the equations of motion fo the fist boy ae. 5 + k ṙ = p, θ = p θ, = p, p = p N θ 3 p θ = 0, p = N 4 sin kπ N sin kπ N + k 3/, k 4 sin kπ N + k 3/. 6 Since p θ = 0, the angula momentum p θ = Θ is constant an can be calculate fom the initial conitions. Then, once is obtaine, we will get θ fom the secon equation in Symmetic peioic solutions of the euce poblem We call euce poblem the poblem given by consieing in 6 only the equations fo the vaiables an, an complete poblem the whole set of equations 6. Ou aim is to fin peioic solutions of this euce poblem. This solutions will be, in geneal, quasi peioic solutions of the complete poblem. Consie the poblem pose by the Hamiltonian 5 fo a fixe value of the angula momentum p θ = Θ. The equations of motion of the hiphop.tex; 6/0/006; 5:44; p.5
6 6 Baabés, Cos, Pinyol, Sole euce poblem ae ṙ = p, = p, p = Θ N 3 p = N sin kπ N 4 sin kπ N + k 3/, k + k 3/, 4 sin kπ N which has a unique equilibium point,, p, p = a, 0, 0, 0, whee a = Θ /KN an is N KN = 4 sin kπ N The matix of the lineaize equations aoun this equilibium point whee an λ M = = 3Θ a 4 λ = 6a 3 S N = λ λ K N a 3 N = K8 N Θ 6, 7. 8, k = S N K6 N sin 3 kπ Θ 6, N N k. 9 sin 3 kπ N The matix M has two pais of imaginay eigenvalues ±iλ an ±iλ. By Lyapunov s cente theoem, Meye an Hall, 99, thee exist two one paamete families of peioic solutions, emanating fom the equilibium point povie that λ /λ is not a ational numbe. That is, it suffices to ensue that λ λ = 4 N N k sin 3 kπ N sin kπ N 0 hiphop.tex; 6/0/006; 5:44; p.6
7 Hip Hop solutions of the N Boy poblem 7 Table I. Values of λ /λ iffeent values of N N λ /λ fo is not the squae of a ational numbe. A numbe of values of this expession fo iffeent values of N ae shown on Table I. The = p = 0 plane is invaiant an the moe solutions lie in this plane with peios appoaching π/λ. These ae actually the homogaphic solutions nea the elative equilibium. The moe solutions ae thee imensional obits whose peios ten to π/λ an this is the family we ae inteeste in. The equations of motion of the euce poblem 7 ae invaiant by the symmeties S : t,,, p, p t,,, p, p, S : t,,, p, p t,,, p, p, an we have the following well known poposition. PROPOSITION. Let qt = t, t, p t, p t be a solution of the equations 7. If qt satisfies that 0, p 0 = 0, 0 an p T, p T = 0, 0, then qt is a oubly symmetic peioic solution of peio 4T. We will show the existence of oubly symmetic peioic obits in system 7. Let q 0 = 0, 0, p 0, p 0 = 0, 0, p 0, 0. The solution of system 7 with these initial conitions is given by t = 0, fo all t, togethe with any solution t of the Keple poblem = Θ 3 K N, with K N given by 8. As is well known, its solutions can be witten as t = a e cos Et, hiphop.tex; 6/0/006; 5:44; p.7
8 8 Baabés, Cos, Pinyol, Sole whee a is the semimajo axis, e the eccenticity an E the eccentic anomaly. The function t is peioic of peio T = πa 3/ /K N = π/λ an a e K N = Θ. These solutions will be calle plana as oppose to the spatial o thee imensional solutions when t the istance fom the fist boy to the z = 0 plane is not ientically zeo. In oe to obtain peioic solutions of the euce poblem, we a petubations to peioic plana obits in the vetical iection. If the petubation is small enough, the motion can be ecouple into a plana plus a vetical motion in a fist appoximation. Substituting by ε on the equations 7, an keeping tems in ɛ, we obtain whee ṙ = p, = p, ṗ ṗ = Θ 3 K N + 3S N 4 ɛ + Oɛ 4, = S N 3 + 3W N 5 3 ɛ + Oɛ 4, WN = N k sin 5 kπ N System can then be witten as whee q = Fq,ɛ = F 0 q + ɛ F q + Oɛ 4 3 F 0 q = p, p, Θ 3 K N, S N 3, F q = 0, 0, 3S N 4, 3W N 5 3. Let q 0 be a vecto of initial conitions. The solution of 3 with initial value q 0 at t = 0 can be expane as a powe seies in ɛ as qt, q 0, ɛ = q 0 t, q 0 + ɛ q t, q 0 + Oɛ 4, whee q 0 t, q 0 is the solution of the unpetube poblem q 0 t, q 0 = F 0 q 0 4 with initial conitions q 0, an q t, q 0 is the solution of q t, q 0 = F q 0 t, q 0 + DF 0 q 0 t, q 0 q t, q 0 with initial conitions q 0, q 0 = 0. The enties of the matix DF ae the patial eivatives of F with espect to the q vaiable, an by the fomula of Lagange we have hiphop.tex; 6/0/006; 5:44; p.8
9 Hip Hop solutions of the N Boy poblem 9 whee t q t, q 0 = Qt, q 0 Q τ, q 0 F q 0 τ, q 0 τ, 5 0 Qt, q 0 = q0 t, ξ ξ 6 ξ=q0 THEOREM. Let T0 = k+ π λ, a = Θ an q KN 0 = a, 0, 0, p 0. Assume that λ /λ given by 0 is not a ational numbe. Then thee exist 0, T such that the solution qt, q 0, ɛ of system with initial conitions q 0 = a+ 0, 0, 0, p 0 is a oubly symmetic peioic solution of peio 4T0 + T. The functions 0 an T ae given by 0 = ɛ 3 p 0 Sn a 4 λ λ 4 λ + Oɛ 4 T = ɛ 9 p 0 a 5 λ 3 λ 4 λ whee an B k, N λ B k, N + Oɛ 4 B k, N = S N λ λ sin + k π λ λ, B k, N = + k π W N λ 3 λ 4 λ + SN λ B, 7 B k, N = λ π+kλ 4λ 3λ 8λ +3λ 3 λ λ sin + kπ λ. λ Note that fo ɛ 0 the peios of the solutions given by the theoem ten to π/λ an they belong to a symmetic Lyapunov family. Poof. Notice that the solution of the unpetube poblem 4 with initial conition q 0 is This solution satisfies q 0 t, q 0 = a, p 0 λ sinλ t, 0, p 0 cosλ t. 8 q 0 T 0, q 0 = a, p 0 k λ, 0, 0, 9 an q 0 t, q 0 is a oubly symmetic peioic solution of the unpetube system of peio 4T0. hiphop.tex; 6/0/006; 5:44; p.9
10 0 Baabés, Cos, Pinyol, Sole We must fin initial conitions q 0 = a + 0, 0, 0, p 0 an T = T0 + T such that the solution qt, q 0, ɛ of system satisfies { p T0 + T, q 0, ɛ = p 0 T0 + T, q 0 + ɛ p T0 + T, q 0 + Oɛ 4 = 0 p T 0 + T, q 0, ɛ = p 0 T 0 + T, q 0 + ɛ p T 0 + T, q 0 + Oɛ 4 = 0 0 By Poposition, qt, q 0, ɛ will be a oubly symmetic peioic solution of peio 4T0 + T. Fo a fixe value p 0, an k = 0,, we consie p k T0 + T, q 0 an p k T 0 + T, q 0 as functions of T, 0. Expaning 0 as powe seies in the s we get 0 p 0 T0 =, q 0 0 p 0 T + 0, q 0 +ɛ p T0, q 0 p T + 0, q 0 +Oɛ 4 p 0 p 0 p 0 p 0 p p T 0,q 0 p p T 0 + O T, 0 + T 0,q 0 T 0 + O T, 0 Now we have fom 9 that 0 p T0, q 0 0 p 0 T 0, q 0 = 0 p 0 p 0 so that if p 0 p 0 0, the system 0 can be solve fo T 0,q 0 T, 0 in a neigbouhoo of 0, 0 by means of the implicit function theoem. An appoximation to T, 0 can be easily compute fom p 0 T = ɛ p 0 p T0, q 0 0 p T + O T, 0 +Oɛ 4 0, q 0 p 0 The functions p0 the tems p0 p 0 p 0 p 0 p 0 an p T0, q 0, p0 T 0,q 0 =, p p 0, p0 T 0,q 0 can be compute fom 8. In oe to get we must compute 6. Then 0 λ sin k+ λ λ π p 0λ k 3 p 0 λ k +k π λ sin λ λ λ a λ λ 4 λ T 0, q 0 ae the last two components of q T 0, q 0 = QT 0, q 0 T 0 0 Q τ, q 0 F q 0 τ, q 0 τ, hiphop.tex; 6/0/006; 5:44; p.0
11 Hip Hop solutions of the N Boy poblem Table II. Numeical values of T, 0 fo Θ =, N = 3 an k = 0. p 0 = p 0 = p 0 = 0.5 ɛ = , , , ɛ = , , , ɛ = , , , ɛ = , , , which can be easily compute an ae given by p z 0 = p z 0 = 3 p 0 S N a 4 λ λ 4 λ + kπ sin 9 k p a 5 λ 3 λ 4 λ 4 λ B, k, N λ, 3 λ whee B k, N is given by 7. Finally we can substitute 3 an in an we obtain the appoximation to T, 0. Theoem gives an appoximation to initial conitions fo oubly symmetic peioic obits fo ɛ sufficiently small. This esults have been checke numeically an a goo ageement has been obtaine. Fo fixe values of N, k, Θ, p 0 an ɛ, we compute numeically the values T, nea to T0, a such that the obit with initial conitions q =, 0, 0, p 0 is a oubly symmetic peioic obit of peio 4T. The integation of the iffeential equations has been one by means of a RungeKutta RK78 algoithm. Then we compute T = T T0 an 0 = a. Table II shows the numeically compute values of T, 0, fo Θ = an iffeent values of ɛ. hiphop.tex; 6/0/006; 5:44; p.
12 Baabés, Cos, Pinyol, Sole 4. Hip Hop peioic obits an choeogaphies The question whethe obits which ae peioic in the euce system ae peioic also in the inetial fame is of couse only a question of commensuability between π an the angle otate in the inetial system in a peio. If this angle can be seen to change along the family of peioic solutions, then thee will exist infinitely many peioic solutions in the inetial system wheneve its value is commensuable with π. It suffices then to see that its eivative with espect to ɛ is iffeent fom zeo fo ɛ = 0. Now, a small vaiation on this simple agument shows that thee exist infinitely many choeogaphies as well. Think of the obit as having peio 4T, whee T is the time spent fom the initial plana position as a egula N gon to the maximum sepaation of the planes containing the N gon configuations. Afte a time T, the N boies that at t = 0 whee thown upwas will hit the initial plane with a velocity symmetic to the initial one, which is exactly the initial velocity of the othe N boies, which wee thown ownwas. If at t = T the position of N boies is the same as the position at t = 0 of the othe N boies, they will follow the same path, so we have the kin of motion that has been calle a choeogaphy. We give an outline of the computation of the eivative. As we have seen in the pevious Section, fo small values of ɛ we can obtain peioic solutions qt, q 0, ɛ of the euce poblem fo initial conitions q 0 = a + 0, 0, 0,p 0 an peio 4T = 4T T. Fo a fixe value p 0, the function t is given by t, q 0, ɛ = 0 0 t, a O 0 + q 0 +ɛ t, a O 0 + Oɛ 4 q 0 = a + cosλ t 0 + ɛ a, t + Oɛ 4 4 whee 0 is given in Theoem, an a, t = 3 p 0 S N sint λ λ + + cost λ λ a 4 λ λ 4 4 λ λ is the fist component of the vecto q t as given by 5. hiphop.tex; 6/0/006; 5:44; p.
13 Hip Hop solutions of the N Boy poblem 3 Thus, in oe to fin peioic solutions in the inetial efeence system, fo a fixe Θ we must fin solutions of the equation with θ0 = 0 at such that θ = Θ 5 θ4qt = πp 6 fo some integes p an q. Substituting 4 in 5, we get that θ = Θ a cosλ t 0 + ɛ a, t + Oɛ 4 a Integating an emembeing that T is Oɛ, we have θ4t = Θ a 4T 0 + ɛ 3 + kπp 0 Θ 6 whee 8 a 7 K N 7 S N 5 K N 4 S N HN + Oɛ4 HN = 8 S N K N 4 W N + K N S N 4 S N W N 7 an K N, S N, W N ae efine, espectively, in 8, 9 an. Thus, it is enough to see that the tem HN is iffeent fom zeo to guaantee that thee exist infinitely many values of the paamete ɛ such that 6 hols. This is inee the case fo N 0, an pobably fo infinitely many values of N, although we o not have a fomal poof of this fact. Acknowlegements The fist autho is patially suppote by DGES gant BFM C00. The secon an fouth authos ae patially suppote by DGES gant numbe BFM C00 an by a DURSI gant numbe 00SGR The thi autho is patially suppote by CI CYT gant numbe SEC00305/ECO an by DURSI gant numbe SGR Refeences Chencine, A.: 00, Action minimizing solutions of the newtonian nboy poblem: fom homology to symmety. Poceeings of the Intenational Congess of Mathematicians Vol. III, Beijing, 00, hiphop.tex; 6/0/006; 5:44; p.3
14 4 Baabés, Cos, Pinyol, Sole Chencine, A. an Féjoz, J.: 005, L équation aux vaiations veticales un équilibe elatif comme souce e nouvelles solutions péioiques u poblème es N cops, C. R. Math. Aca. Sci. Pais 3408, Chencine, A., Geve, J., Montgomey, R. an Simó, C.: 00, Simple choeogaphic motions of N boies: a peliminay stuy, Geomety, mechanics, an ynamics, Spinge, New Yok, pp Chencine, A. an Montgomey, R.: 000, A emakable peioic solution of the theeboy poblem in the case of equal masses, Ann. of Math. 53, Chencine, A. an Ventuelli, A.: 000, Minima e l intégale action u poblème newtonien e 4 cops e masses égales ans R 3 : obites hiphop, Celestial Mech. Dynam. Astonom. 77, Davies, I., Tuman, A. an Williams, D.: 983, Classical peioic solution of the equalmass nboy poblem, nion poblem an the nelecton atom poblem, Phys. Lett. A 99, 5 8. Meye, K. R. an Hall, G. R.: 99, Intouction to Hamiltonian ynamical systems an the Nboy poblem, Vol. 90 of Applie Mathematical Sciences, Spinge Velag, New Yok. Meye, K. R. an Schmit, D. S.: 993, Libations of cental configuations an baie Satun ings, Celestial Mech. Dynam. Astonom. 553, Simó, C.: 00, New families of solutions in N boy poblems, Poceeings of the thi Euopean Congess of Mathematics, in Pog. Math. Vol. 0, 0 5. Teacini, S. an Ventuelli, A.: 005, Symmetic tajectoies fo the nboy poblem with equal masses, pepint. hiphop.tex; 6/0/006; 5:44; p.4
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
More informationPhysics 111 Fall 2007 Electrostatic Forces and the Electric Field  Solutions
Physics 111 Fall 007 Electostatic Foces an the Electic Fiel  Solutions 1. Two point chages, 5 µc an 8 µc ae 1. m apat. Whee shoul a thi chage, equal to 5 µc, be place to make the electic fiel at the
More information2 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
More informationChapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6
Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe
More informationVector 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
More informationDetermining 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
More informationPhysics 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
More informationPRICING MODEL FOR COMPETING ONLINE AND RETAIL CHANNEL WITH ONLINE BUYING RISK
PRICING MODEL FOR COMPETING ONLINE AND RETAIL CHANNEL WITH ONLINE BUYING RISK Vaanya Vaanyuwatana Chutikan Anunyavanit Manoat Pinthong Puthapon Jaupash Aussaavut Dumongsii Siinhon Intenational Institute
More informationExam 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
More informationGravitation. 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
More informationGravitation 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,
More informationCLOSE RANGE PHOTOGRAMMETRY WITH CCD CAMERAS AND MATCHING METHODS  APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT
CLOSE RANGE PHOTOGRAMMETR WITH CCD CAMERAS AND MATCHING METHODS  APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT Tim Suthau, John Moé, Albet Wieemann an Jens Fanzen Technical Univesit of Belin, Depatment
More informationCh. 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
More informationInfinitedimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria.
Infinitedimensional äcklund tansfomations between isotopic and anisotopic plasma equilibia. Infinite symmeties of anisotopic plasma equilibia. Alexei F. Cheviakov Queen s Univesity at Kingston, 00. Reseach
More informationHow many times have you seen something like this?
VOL. 77, NO. 4, OTOR 2004 251 Whee the amea Was KTHRN McL. YRS JMS M. HNL Smith ollege Nothampton, M 01063 jhenle@math.smith.eu How many times have you seen something like this? Then Now Souces: outesy
More informationChapter 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.
More informationPoynting Vector and Energy Flow in a Capacitor Challenge Problem Solutions
Poynting Vecto an Enegy Flow in a Capacito Challenge Poblem Solutions Poblem 1: A paallelplate capacito consists of two cicula plates, each with aius R, sepaate by a istance. A steay cuent I is flowing
More informationPHYSICS 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
More informationToday in Physics 217: multipole expansion
Today in Physics 17: multipole expansion Multipole expansions Electic multipoles and thei moments Monopole and dipole, in detail Quadupole, octupole, Example use of multipole expansion as appoximate solution
More informationUNIT 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: + =   
More informationRevision 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
More informationExplicit, analytical solution of scaling quantum graphs. Abstract
Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT 064590155, USA Email: ydabaghian@wesleyan.edu (Januay 6, 2003)
More informationPhysics 505 Homework No. 5 Solutions S51. 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
More informationmv2. 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 =!
More informationSpirotechnics! 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.
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationContinuous 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
More informationMULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION
MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe and subsolution method with
More informationPY1052 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 ighthand end. If H 6.0 m and h 2.0 m, what
More informationDiscussion on Fuzzy Logic Operation of Impedance Control for Upper Limb Rehabilitation Robot 1,a Zhai Yan
Intenational Confeence on Automation, Mechanical Contol an Computational Engineeing (AMCCE 05) Discussion on Fuzzy Logic Opeation of Impeance Contol fo Uppe Limb Rehabilitation Robot,a Zhai Yan,b Guo Xiaobo
More information!( r) =!( r)e i(m" + kz)!!!!. (30.1)
3 EXAMPLES OF THE APPLICATION OF THE ENERGY PRINCIPLE TO CYLINDRICAL EQUILIBRIA We now use the Enegy Pinciple to analyze the stability popeties of the cylinical! pinch, the Zpinch, an the Geneal Scew
More informationSamples 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
More informationDeflection 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
More informationCoordinate 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
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More informationChapter 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,
More informationSeshadri constants and surfaces of minimal degree
Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth
More informationSeventh Edition DYNAMICS 15Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University
Seenth 15Fedinand P. ee E. Russell Johnston, J. Kinematics of Lectue Notes: J. Walt Ole Texas Tech Uniesity Rigid odies CHPTER VECTOR MECHNICS FOR ENGINEERS: YNMICS 003 The McGawHill Companies, Inc. ll
More informationHour 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
More informationFXA 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
More informationConcept and Experiences on using a Wikibased System for Softwarerelated Seminar Papers
Concept and Expeiences on using a Wikibased System fo Softwaeelated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wthaachen.de,
More informationGauss Law. Physics 231 Lecture 21
Gauss Law Physics 31 Lectue 1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationSo 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
More informationNew 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
More informationSkills 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
More informationVoltage ( = Electric Potential )
V1 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
More informationEpisode 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
More information2. Orbital dynamics and tides
2. Obital dynamics and tides 2.1 The twobody 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
More informationProblems on Force Exerted by a Magnetic Fields from Ch 26 T&M
Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuentcaying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to
More informationProblem Set 6: Solutions
UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 164 / 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
More informationPower and Sample Size Calculations for the 2Sample ZStatistic
Powe and Sample Size Calculations fo the Sample ZStatistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.
More informationCarterPenrose diagrams and black holes
CatePenose 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
More informationClassical Lifetime of a Bohr Atom
1 Poblem Classical Lifetime of a Boh Atom James D. Olsen and Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 85 (Mach 7, 5) In the Boh model of the hydogen atom s gound state,
More informationChapter 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
More informationIn the lecture on double integrals over nonrectangular domains we used to demonstrate the basic idea
Double Integals in Pola Coodinates In the lectue on double integals ove nonectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example
More informationEffects of Projectile Motion in a NonUniform Gravitational Field, with Linearly Varying Air Density
Effects of Pojectile Motion in a NonUnifom Gavitational Field, with Linealy Vaying Ai Density Todd Cutche Decembe 2, 2002 Abstact In ode to study Pojectile Motion one needs to have a good woking model
More informationProblem Set # 9 Solutions
Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new highspeed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease
More informationChapter 13. VectorValued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates
13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. VectoValued Functions and Motion in Space 13.6. Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along
More informationSoftware Engineering and Development
I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationCHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS
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
More informationForces & 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
More informationROBERTA FILIPPUCCI. div(a( Du )Du) q( x )f(u) in R n,
ENTIRE RADIAL SOLUTIONS OF ELLIPTIC SYSTEMS AND INEQUALITIES OF THE MEAN CURVATURE TYPE ROBERTA FILIPPUCCI Abstact. In this pape we study non existence of adial entie solutions of elliptic systems of the
More informationQuestions & 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
More informationLAL Update. Letter From the President. Dear LAL User:
LAL Update ASSOCIATES OF CAPE COD, INCORPORATED OCTOBER 00 VOLUME 0, NO. Lette Fom the Pesident Dea LAL Use: This Update will claify some of the statistics used with tubidimetic and chomogenic LAL tests.
More informationThe Supply of Loanable Funds: A Comment on the Misconception and Its Implications
JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently FieldsHat
More informationMechanics 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).
More information1240 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
More informationFast FPTalgorithms for cleaning grids
Fast FPTalgoithms fo cleaning gids Josep Diaz Dimitios M. Thilikos Abstact We conside the poblem that given a gaph G and a paamete k asks whethe the edit distance of G and a ectangula gid is at most k.
More informationAn Efficient Group Key Agreement Protocol for Ad hoc Networks
An Efficient Goup Key Ageement Potocol fo Ad hoc Netwoks Daniel Augot, Raghav haska, Valéie Issany and Daniele Sacchetti INRIA Rocquencout 78153 Le Chesnay Fance {Daniel.Augot, Raghav.haska, Valéie.Issany,
More informationUniversal 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
More informationThe 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
More informationYARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH
nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav
More informationGravitational Mechanics of the MarsPhobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning
Gavitational Mechanics of the MasPhobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This
More informationON THE (Q, R) POLICY IN PRODUCTIONINVENTORY SYSTEMS
ON THE R POLICY IN PRODUCTIONINVENTORY SYSTEMS Saifallah Benjaafa and JoonSeok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poductioninventoy
More informationGravity 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 479071397 ecalais@pudue.edu http://www.eas.pudue.edu/~calais/ Objectives What is
More informationModel Question Paper Mathematics Class XII
Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat
More informationLesson 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
More informationMechanics 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.
More informationCubic Spline Interpolation by Solving a Recurrence Equation Instead of a Tridiagonal Matrix
Matematical Metods in Science and Engineeing Cubic Spline Intepolation by Solving a Recuence Equation Instead of a Tidiagonal Matix Pete Z Revesz Depatment of Compute Science and Engineeing Univesity of
More informationSymmetric polynomials and partitions Eugene Mukhin
Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation
More information81 Newton s Law of Universal Gravitation
81 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,
More informationLearning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds.
Lectue #0 Chapte Atomic Bonding Leaning Objectives Descibe ionic, covalent, and metallic, hydogen, and van de Waals bonds. Which mateials exhibit each of these bonding types? What is coulombic foce of
More informationThe 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
More informationNontrivial lower bounds for the least common multiple of some finite sequences of integers
J. Numbe Theoy, 15 (007), p. 393411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to
More informationEfficient Redundancy Techniques for Latency Reduction in Cloud Systems
Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo
More informationStructure 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
More informationThe advent of ecommerce has prompted many manufacturers to redesign their traditional
Diect Maketing, Iniect Pofits: A Stategic Analysis of DualChannel SupplyChain Design Weiyu Kevin Chiang Dilip Chhaje James D. Hess Depatment of Infomation Systems, Univesity of Maylan at Baltimoe County,
More informationVISCOSITY OF BIODIESEL FUELS
VISCOSITY OF BIODIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More informationChapter 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
More informationTrigonometric 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
More informationarxiv:1012.5438v1 [astroph.ep] 24 Dec 2010
FistOde Special Relativistic Coections to Keple s Obits Tyle J. Lemmon and Antonio R. Mondagon Physics Depatment, Coloado College, Coloado Spings, Coloado 80903 (Dated: Decembe 30, 00) Abstact axiv:0.5438v
More informationIlona 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
More informationChapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.
Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More information2. PROPELLER GEOMETRY
a) Fames of Refeence 2. PROPELLER GEOMETRY 10 th Intenational Towing Tank Confeence (ITTC) initiated the pepaation of a dictionay and nomenclatue of ship hydodynamic tems and this wok was completed in
More informationMultiple 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
More informationInternational Monetary Economics Note 1
36632 Intenational Monetay Economics Note Let me biefly ecap on the dynamics of cuent accounts in small open economies. Conside the poblem of a epesentative consume in a county that is pefectly integated
More informationPhysics 202, Lecture 4. Gauss s Law: Review
Physics 202, Lectue 4 Today s Topics Review: Gauss s Law Electic Potential (Ch. 25Pat I) Electic Potential Enegy and Electic Potential Electic Potential and Electic Field Next Tuesday: Electic Potential
More informationUNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100minute sessions
Name St.No.  Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon,
More informationINVESTIGATION OF FLOW INSIDE AN AXIALFLOW PUMP OF GV IMP TYPE
1 INVESTIGATION OF FLOW INSIDE AN AXIALFLOW PUMP OF GV IMP TYPE ANATOLIY A. YEVTUSHENKO 1, ALEXEY N. KOCHEVSKY 1, NATALYA A. FEDOTOVA 1, ALEXANDER Y. SCHELYAEV 2, VLADIMIR N. KONSHIN 2 1 Depatment of
More information