Box floating in a liquid and undergoing simple harmonic motion

[lower8]

Homework Statement

A box is floating in a liquid. It is pushed down then released to oscillate. How do I determine the period of this oscillation?

No idea where to go with this one.

Homework Equations

The Attempt at a Solution

 

[denverdoc]

What form of eqn for SHM usually appear in and on what is it based?

Here the restoring force is related to displacement: First consider the dipping portion of the diaplacement--what happens to buoyancy as a funtion of depth--put in terms of delta h; when bobbing up the distance between center of mass for neutral buoyancy and the actual Y displacement is now the PE, but instead of thinking of as mgy, it might be best to consider an "excess" of buoyancy. In other words, develop an equation that related forse and displacement--from SHO this and mass should lead to period--ie look for an eqn using T, k, m

 

[lower8]

Ok, so I have that the extra buoyancy force would be the density of the liquid times volume of the extra displacement times g. However, I'm not seeing where T or k would come in.

 

[denverdoc]

Here is what comes to mind:

Force= ma=-(pho)*area*(Xo+\Deltax)-mg where mg is the weight of the object.

Let Xo be the displacement leading to neutral buoyancy--ie -(pho)A*Xo + mg = 0

Then a = -Pho*area(\Deltax)/m

d^{2}y/dt^{2}=-k/m*(\Deltax)

Look familiar?