Tunnel from north pole to equator

[Matija]

Hi all! I'm not so good at physics so i decided to turn to you. So please help me.

Homework Statement

I have to calculate how the object (ball) moves when droped into the tunnel (as seen in the picture), and how long the journey takes. We assume that Earth is round and there is no friction and air resistance and Earth density is constant.

http://www2.arnes.si/~vvztrzin/pic.jpg

Homework Equations

The Attempt at a Solution

I tried going throug with lagrangian, but i found it hard, how to calculate gravity energy, because the mass of the Earth is larger on the left than on right.

I appreciate all your help.

 

[Matterwave]

I would do this using Newtonian mechanics (probably because I'm partial to it).

I would get the force due to gravity at all times during the trip.

The force is always towards the center of the Earth, and is due to the mass of the Earth that is within the sphere that has boundaries at the point of the object (assuming the tunnel wasn't large enough to affect Earth's symmetry). Thus, you can get a force vs distance along the tube, and from there get the motion.