Question

Q2) To reduce travel time between two cities such as San Francisco and Los Angeles, engineers have proposed building a rail tunnel along a straight line connecting the cities. A train, its d unpropelled by any engine and starting from rest, would fall through the first half of the tunnel and then 10 move up the second half. The technical challenges are immense (the tunnel would need to evacuated) but it is an interesting system to consider. Assuming Earth to be a uniform sphere and ignoring air drag and friction, a) show that the travel between cities is half a cycle of simple harmonic motion and b) find the travel time between cities. [Does this time depend on the distance of separation between the two cities?) [Hint: if you are able to write the equation of motion for the train and show that it matches the equation of motion for simple harmonic motion, then you have completed part a). Once you have the equation of motion for SHM, you can directly use the solution that Giancoli arrived at.]

Answer 1

Let the train of mass m be at a distance r from the center of the earth. along the tunnel. The gravitational attaction on it is entirely due to the portion of the earth contained in a sphere of radius r. The mass of this portion of earth is

Mass = volume times density

Newton law of gravitation is

Put the value of M' into F

This is of the form f = -kx which is the equation of simple harmonic motion. Comparing the equation the value of

(Followed as per the hint given in the question above)

If the train has to move from SF to LA then it has to move in one half cycle of simple harmonic motion.

b) The period of oscillation is

put the value of k we get

on solving we get

Taking 2 pi into the sqrt equation

the density of earth is 5.5x10^3 kg /m^3 so the equation is

T = 5050s or 84.2min

The time of travel is 84.2min/2 = 42.1m

Time of travel from SF to LA is 42.1min along the tunnel without any engine.