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)!

Transcribed Image Text:

Center of tunnel Tokyo Center of Earth FIGURE 4.1.15 Tunnel from Los Angeles to Tokyo (Problem 68)