Tunnel through the center of the earth

[DarkWing]

Hello, hopefully some of you physics people can help me out here, because I'm pretty clueless. I've seen this asked before, but I couldn't relate that to my specific problem.
Part I: A tunnel is dug through the center of the earth, and an object of mass m is dropped through the tunnel.

• Show that the force on m is a linear restoring force
• Find the period of oscillation
• Show that the period of an obriting satellite (at Re) is equal to the period of the object dropped through the tunnel

Part II: A frictionless tunnel is dug through the earth, not though the center.

• Show the period is equal to that of the object from Part I, which is dropped through the center of the earth.

I've been trying this all day, but I'm not even sure where to begin. Thanks for the help everyone.
Note: I'm not necessarily asking from someone to do this whole problem, although that would be nice. I just need a little help getting started.

 

[Tide]

HINT: Use Gauss' Law!

 

[James R]

The force due to gravity at any time is

$$F=\frac{GMm}{r^2}$$

but the M in that equation is only that part of the Earth's mass at a distance less than r.

So, your first problem is to determine how M varies with r. The simplest way is to assume a constant-density Earth.

Let me know if you have any further problems.