Will it do S.H.M. for a small angular displacement? In an experiment at home I dipped a cubical block of pure wood in a beaker of water.
Like this,

It was made up of oak I think. So dipped approximately this height as indicated in figure. I displaced it slightly angularly about $z$-axis. It did a small oscillation and then stopped after few seconds. I think it was due to viscosity of water.

Assume $z$-axis perpendicular to the diagram. And passing through Center of mass of block.

But I wonder to know if it was an ideal condition (neglecting viscosity and surface tension of water & assuming uniform density of liquid and block & No cohesive and adhesive forces) will it do S.H.M. for a small angular displacement? Will it depend on the density of wood?
 A: The buoyant force on an object is given by
$$F_{b}=g\rho_lV(z)$$
where $\rho_l$ is the density of the liquid and $V(z)$ is the volume of the object that is submersed in the liquid, or in other words the volume that is under the water level. It depends on the vertical position $z$ of the object: if I raise the cube, less volume will be submersed.
The gravitational force on the object is
$$F_g=-\rho_b gV$$
where $\rho_b$ is the density of the block and $V$ is the total volume of the block. The net force becomes
$$m\ddot z=g(\rho_lV(z)-\rho_bV)$$
For a cube we can find a simple expression for the submersed volume
$$V(z)=\cases{0&$0<z$\\-Az&$-L\leq z<0$\\V&$z\leq -L$}$$
where $A$ is the area of the bottom surface and $L$ is the height of the cube. It is a nice exercise to check that this expression is correct (at least I hope it is). If the cube is partially submersed, we get back the equation for a harmonic oscillator (ignoring friction):
$$m\ddot z=-g(\rho_lAz+\rho_bV)$$
we can redefine $z$ such that the resting position is at $z=0$. This makes it more apparent that it is the equation for a HO.
$$m\ddot z=-g\rho_lAz$$
What is the natural frequency of this HO? What happens when the object is no longer a cube? I.e. the area at a given height is not constant but $A=A(z)$. Do you still get a HO?

