Suppose a tunnel is excavated along a chord of the earth at a perpendicular distance R/2 from the earth's centre, where R is the radius of the earth. The wall of the tunnel is frictionless. Determine the normal force exerted by the wall on a particle of mass m placed in the tunnel at a distance x from the centre of the tunnel.

Step-by-step solution

Let the centre of the earth be the origin (0,0) . The tunnel is excavated along a chord at a perpendicular distance R 2 from the centre, so we can assume the tunnel lies along the line y = R 2 . The position vector of the particle at a distance x from the centre of the tunnel is given by r = x i + R 2 j . The gravitational force on a particle of mass m inside the earth at a position r is directed towards the centre of the earth and is given by: F _g = - GMm R^3 r Substituting the position vector of the particle: F