Simple Harmonic Motion, object floating in a liquid

[Jack21222]

(apologies for the formatting, mixing LaTeX and regular text looks ugly, but I don't know how to do it otherwise)

Homework Statement

This is problem 3-7 from Thornton and Marion's Classical Dynamics.

A body of uniform cross-sectional area A = 1 cm² and of mass density $$\rho$$=0.8 g/cm³ floats in a liquid of density $$\rho$$₀=1 g/cm³ and at equilibrium displaces a volume V = 0.8 cm³. Show that the peroid of small oscillations about the equilibrium position is given by:

$$\tau$$ = 2$$\pi$$ $$\sqrt{V/gA}$$

where g is the gravitational field strength. Determine the value of $$\tau$$

Homework Equations

$$\tau$$ = 2$$\pi$$$$\sqrt{k/m}$$ (equation 3.13 in the book)

k$$\equiv$$ -(dF/dx)

The Attempt at a Solution

It appears to me that if I can show V/gA = m/k or k = mgA/V, then the equation in the problem would be equal to equation 3.13 quoted above. I tried coming up with an equation for the restoring force F in terms of mg A and V, and then take the x-derivative of k, but it doesn't equal mgA/V.

I'll be using p = rho from here, because I don't want to mix LaTeX and regular text, and p has no other meaning.

The restoring force I came up with was F = Vp₀g - mg where m is the mass of the block.

In terms of x, I get F = Axpg - mg where x is the depth of the submerged block.

Taking the x-derivative of F, from the definition of k, I get Apg, which isn't equal to mgA/V

Any suggestions?

 

[Jack21222]

I'm an idiot, p= m/V by definition, and everything falls together.

I can't believe I got stuck on such a trivial step. I'm embarrassed that I even posted this thread.