Suppose a tunnel is excavated across the earth (radius = R ) passing through its centre. Calculate the time a particle takes to traverse the length of the tunnel if it is released from a height R above the tunnel.

Step-by-step solution

Let the mass of the Earth be M and its radius be R . The particle is released from a height R above the surface, so its initial distance from the center of the Earth is 2R . Using the principle of conservation of mechanical energy from the point of release to the surface of the Earth: E_i = E_f - GMm 2R = - GMm R + 1 2 mv^2 1 2 mv^2 = GMm 2R v = GM R = gR This is the velocity of the particle as it enters the tunnel. Inside the Earth, the gravitational force at a distance x from the center is F = - GMm R^3 x . The a