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 projected into the tunnel with a speed of gR .

Step-by-step solution

The gravitational force on a particle of mass m at a distance x from the centre of the earth is given by F = - GMm R^3 x . The acceleration of the particle is a = - GM R^3 x = - g R x . This shows that the particle executes simple harmonic motion (SHM) with angular frequency = g R . The velocity v of a particle in SHM is related to its position x and amplitude A by the equation v^2 = ^2 (A^2 - x^2) . Given that at the surface x = R , the velocity is v = gR , we can find the amplitude A : gR = g R (A^2 - R^2) R^2 =