Simple harmonic motion and central forces.

[WhataRecch]

Homework Statement

Assuming Earth to be a solid sphere, show that if a straight hole were drilled from pole to pole that a particel dropped would undergo simple harmonic motion. Show that the period of the oscillation depends only on the Earth's density and is independent of size. What is the period in hours?

Homework Equations

We have already done studying oscillations and this questions mainly pertains to central forces. The obvious equation I have in mind is F = GMm/(r^2) and I know somehow I have to do something that will lead to d²x/dt² = constant*x

The Attempt at a Solution

Something tells me this problem is extremely simple. But I don't understand how to do it. I've reached a bit of a dead end. The equation for the force that is exterted on the particle doesn't display shm.

 

[grzz]

Remember that:
1. a particle inside such a hole will only be attracted by the gravitational pull of that spherical part of the Earth which lies BELOW the particle;
2. use 'mass = volume x density' for that part of the mass of the Earth;
3. take the particle to be at some displacement x in a direction away from the cente of the Earth;
4. use f(net) = ma for the particle in the direction of displacement x;

 

[technician]

If you follow grzz advice you should find that the force on the mass is proportional to r which means the mass will undergo SHM