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Suppose a comet travels on a parabolic orbit around the Sun. (i) Show that in this case the following holds: 1 k mji? + kro p2 0 T where k = GM,m and M. is the mass of the Sun. (ii) Starting from the result in part (i), show that the radial distance as a function of time from the moment of closest approach to the Sun is given by t(r) = 2 -(r + 2ro) Vr-TO 9GM,
(1 point) The acceleration due to gravity, g, is given by 8= GM r2 where M is the mass of the Earth, r is the distance from the center of the Earth, and G is the uniform gravitational constant. (a) Suppose that we increase from our distance from the center of the Earth by a distance Ar = x. Use a linear approximation to find an approximation to the resulting change in g, as a fraction of the original acceleration:...
achieves its closest approach A particle of mass m moving in the Kepler potential V -k/ to the force center, r-ro, at 0, where r, p denote polar coordinates in the plane of motion of the particle. At φ = π/3, its distance from the force center is r = 5r0/4. Determine the eccentricity e of the orbit, the angular momentum, the energy, and the ratio of speeds v(p /3)/(p 0). Hint: If you're not completely confident in your knowledge...
For the following, quote your numerical answers to two significant figures: a) A binary system consists of two stars orbiting a common center of mass. Star A is a white star of mass MA = 2.02Mo, where Mo = 1.989 x 1030 kg is the mass of the Sun. Its companion, Star B, is a white dwarf with mass Mg = 0.978M. The orbital period of the two objects is observed to be t = 50.09 years. What is the...
A mass of 500 gm is attached to a spring (k = 24.5 N/m) on a horizontal, frictionless surface. A force of 4.90 N pulls the mass to the right, displacing it some distance, x, from its equilibrium position. The mass is then released and oscillates in simple harmonic motion. (A) What is the maximum speed of the mass for this motion? ANSWER: 1.4 m/s (B) What is the position, x, of the mass 0.500 seconds (this is time, t)...
{ <N> : L(M) contains a string starting with a). Rice's theorem can be F 20, L used to prove that LD. T L(M2) >. Rice's theorem can be used to prove T F 21. L that L D. <M,, M2> L(M,) 22. L-( <M,M> : L(M) = L(M2) }, and R is a mapping reduction function from H to L. It is possible that R retur a TM. T F ns <M#>, where M # is the string encoding...
Use Greens theorem (b) Let r(t) = X(t)i+Y(t)j be the position of the planet at the instant t and we suppose that the sun is located at the origin (0,0). Between the times t; and t2, the line joining the sun and the planet sweeps out an area Altı, t2) (see the blue region). Express A(t1, t2) in terms of X(t), Y(t), X(t)' and Y (t)'. (c) We denote by F(t) the force exerted on the planet by the sun...
014 (part 1 of 2) 10.0 points Ever since it was learned that an aster- oid hit the earth 65 million years ago and destroyed most life-forms, including the di- nosaurs, observations of new comets are fol- lowed with great interest! Suppose a new comet is found orbiting the sun as it is pass- ing its point of closest approach at a distance Tmin 0.658 AU (1 AU-semimajor axis (average radius) of the Earth's orbit.) Careful studies determine that its...
Two metal disks, one with radius R1 = 2.50 cm and mass M1 = 0.800 kg and the other with radius R2 = 5.10 cm and mass M2 = 1.70 kg, are welded together and mounted on a frictionless axis through their common center (Figure 1). Part A What is the total moment of inertia of the two disks? Part B A light string is wrapped around the edge of the smaller disk, and a 1.50 kg block is suspended from the free end...
Problem 5. Consider the dynamics of two mass mechanical system captured by d2xi(t) Middt?t2+k(x1(t)-x2(t)) = f(t) d'x2(t) dt2 + k(x2(t)-x where M, , M2, and k are constants. Suppose the input is () and the output is X2 (t), find the transfer function G(s) of the system. Note: Consider all zero initial conditions.