The motion of n planets (Po, P₁,..., Pn-1) is governed by the set of 6n first order ODEs n r₁ = V₁_and_vi' = Σ Gm; j=1ji rj - ri d¾ 3 (5.10) where mi, ri(t) and vi(t) are the mass, position vector and velocity of the planet Pi, respectively. The scalar dij = dji = ||r¡—ri|| is the distance between the two planets Pi and Pj. The gravitational constant, G, can be taken equal to 1 without loss of generality.
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- As we showed, the horizontal displacements ξi of masses i = 1 . . . N satisfy equations of mo- tion md2ξ1 =k(ξ2−ξ1)+F1, dt2 md2ξi =k(ξi+1−ξi)+k(ξi−1−ξi)+Fi, dt2 md2ξN =k(ξN−1−ξN)+FN. dt2 where m is the mass, k is the spring constant, and Fi is the external force on mass i. In Exam- ple 6.2 we showed how these equations could be solved by guessing a form for the solution and using a matrix method. Here we'll solve them more directly. a) Write a python program to solve for the motion of the masses using the fourth-order Runge- Kutta method for the case we studied previously where m = 1 and k = 6, and the driving forces are all zero except for F1 = cos ωt with ω = 2. Plot your solutions for the displace- mentsξi ofallthemassesasafunctionoftimefromt=0tot=20onthesameplot. Write your program to work with general N, but test it out for small values—N = 5 is a reasonable choice. You will need first of all to convert the N second-order equations of motion into 2N first- order equations.…Graph the functions 8n, 4nlogn, 2n2, n3, and 2n using a logarithmic scale for the x- and y-axes. That is, if the function is f(n) is y, plot this as a point with x-coordinate at log n and y-coordinate at log y.Given vectors :u = (5,2) ; v = (-2,5) w = (0,3) ; q = (10,4)4.1 Calculate the following dot products:u.v ; (u.v).w ; u.(3w) , u.(w-v)4.2 Calculate‖?‖ ; d(u,v) ; ‖? − ?‖24.3 u and v are they orthogonal4.4 u and w are they orthogonal4.5 find c real number that satisfy q = c.u4.6 Deduce that q and u are parallel4.7 Normalize vector w
- GF(8) is defined like GF(256) except the polynomials all have degree less than 3 and the modulus is + x + 1. Which of the following is the multiplicative inverse of O10 in GF(8)? 001 011 110 100 None of theseWhat is the image of the point (a, b, c) after the transformation given by the following matrix 100 0 010 0 00-10 000 1 ? (a, b, c) (a, b, c-1) (b, -a, c) O (a, b, -c)Give regular expressions for the i (a) {w : w contains at most two 1s} (b) {w : w contains at least two 1s} (c) {w: every odd position in w is 1} (d) {w : w contains an even number of 0s, or
- Given two objects represented by the tuples (25, 5, 52, 14) and (10, 0, 36, 8):(a) Compute the Euclidean distance between the two objects.(b) Compute the Manhattan distance between the two objects.(c) Compute the Minkowski distance between the two objects, using q=3Simplify the following Boolean functions by means of K-map:a) f(A,B,C) =∑m(0,2,3,4,6)b) f(A,B,C,D) = ∑ m(0,2,3,7,8,10,12,13)c) F(X,Y,Z) = Y’Z+YZ+X’Y’Z’Show me the steps of determine red