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5.6 part 2 - 1U 11 12 13 14 36 5.6 Applications of Multiple...

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Unformatted text preview: 1U. 11. 12. 13. 14. 36. 5.6 Applications of Multiple Integrals 549 The region between 3; = $2 and y : a: if the density is z + y The disk determined by (a: — 1)2 + 3.3 5 1 if the density is x3 A solid with constant density is bounded above by the plane z = n and below by the cone described in spherical coordinates by ql) = k, where is is a constant 0 < k < a/ 2. Set up an integral for its moment of inertia about the z axis. Do not try to evaluate the integral Find the moment of inertia around the y axis for the ball 3:2 +y2 + Z2 S R2 if the mass density is constant, and the total mass is M. Find the gravitational potential due to a spherical planet with mass M = 3 X 1026 kg, on a mass m at a distance of 2 :1: ll]8 m from its center. Find the gravitational force exerted on a 70-kg obiect at the position in Exercise 13. . A body W in eyz coerdinates is symmetric with respect to a plane if for every particle one one side of the plane there is a particle of equal mass located at its miner image through the plane. (a) Discuss the planes of symmetry for the body of an automobile. (b) Let a plane of symmetry be the my plane, and denote by W+ and W“ the portions of W above and below the plane, respectively. By our assumption, the mass density p(a:, y, 2) satisfies p(:r,y, —z) = ptz, y, z). justify these steps: Z/ffw plus y.z udxdydz — 4/], 2p 1..., ) [flan z”($’y*zld$dydz+ ff] _ zalawlndyn = [ffw zfllx’y’zld‘rdydz‘L [MW empluavr-wlduduaw l] (c) Explain why part (b) proves that if a body is symmetrical with respect to a plane, then its center of mass lies in that plane. (d) Derive this law of mechanics: {fa body is symmetrical in two planes, than its center of mass lies on their line of intersection. As is well kn0wn, the density of a typical planet is not constant throughout the planet. Assame that the planet MAX (including its atmosphere) has a ...
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