An entrepreneur plans to link two cities on the Earth\'s surface that lie direct
ID: 2042464 • Letter: A
Question
An entrepreneur plans to link two cities on the Earth's surface that lie directly opposite each other on a straight line through the center of the Earth by means of a direct, straight tunnel through the Earth. A train will travel through the tunnel powered by gravity. Assuming the Earth to be non-rotating uniform sphere:
A) Show that a train released at one city executes simple horamonic motion if damping effects are negligble
B) Calculate the time for a journey through the tunnel if the Earth's density is 5.5 x 103 kg/m3
Explanation / Answer
= 5.5 x 103 kg/m3, radius of earth = R, mass of train = m
a) at the radius r < R, the gravitational force on the train = -Gm(*4r3/3)/r2 = -Gm(*4/3)*r
acceleration a = -G(*4/3)*r = -2r
where 2 = 4G/3
so it is SHM
b) time = period/2 = / = [3/(4G)] = 2534 s = 42.2 min