(Solution) - Los Angeles to Tokyo It can be shown that the -(2025 Original AI-Free Solution)

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Los Angeles to Tokyo It can be shown that the force on an object inside a spherical homogeneous mass is directed towards the center of the sphere with a magnitude proportional to the distance from the center of the sphere. Using this principle, a train starting at rest and traveling in a vacuum without friction on a straight line tunnel from Los Angeles to Tokyo experiences a force in its direction of motion equal to -kr cos (, where
? r is the distance of the train from the center of the earth,
? x is the distance of the train from the center of the tunnel,
? 8 is the angle between r and x,
? 2d is the length of the tunnel between L.A. and Tokyo,
? R is the radius of the earth (4,000 miles), as shown in Fig. 4.1.15.

(a) Show that the train position x can be modeled by the initial-value problem
m + kx = 0, x(0) = d, (0) = 0.
Where x is the distance of the train to the center of [he earth and R is the radius of the earth
(b) How long does it take the train to go from Los Angeles to Tokyo?
(c) Show that if a train Starts at any point on earth and goes to another point on earth in this science fiction scenario, the time will be the same as calculated in part (a)!