A straight tunnel is dug through the earth as shown in Figure. Assume that the walls of the tunnel are frictionless.

(a) The gravitational force exerted by the earth on a particle of mass m at a distance r from the center of the earth when r < R_E is F_r = – (GmM_E/R³_E)r, where M_E is the mass of the earth and R_E is its radius. Show that the net force on a particle of mass m at a distance x from the middle of the tunnel is given by F_x = (GmM_E/R³_E)x, and that the motion of the particle is therefore simple harmonic motion.

(b) Show that the period of the motion is given by T = 2π √RE/g and find its value in minutes. (This is the same period as that of a satellite orbiting near the surface of the earth and is independent of the length of the tunnel.)

Tunnel ME R4