**1. The problem statement, all variables and given/known data**

I have a rectangular prism of wood (stick of lumber) that I intend to float in a bucket of water for a presentation. The lumber can be cut lengthwise (height) but will retain a specific base and width. I'm trying to find the length/height of the piece of lumber at which the metacenter will equal the center of gravity, because I believe that is the point of buoyant equilibrium, correct? Any longer and the piece of lumber will float unstable, and any shorter it will float with stability. Can you please help me find the finite height/length at which the lumber would be at theoretical equilibrium?

specific gravity of lumber = .663

dimensions of lumber = 0.035m * 0.036m * h

lumber assumed to be uniform in distribution of mass

**2. Relevant equations**

metacenter = distance to metacenter from center of buoyancy + center of buoyancy

(MC = MB + CB)

MB = moment of intertia / volume displaced

(MB = I / Vd)

I = (b * w^3)/12

**3. The attempt at a solution**

I eventually got the height as equaling 0.0302m (3.2cm) by using substitutions in multiple derivations of the above equations. I don't honestly remember how I did it at this point, because my work was lost when this forum logged me out while trying to post it. I know that I approached it by trying to set CG = MC, and then using substitution so as that all terms were simplified down so height was the only variable.

I don't think the way I approached it was correct, however. The answer doesn't seem to check out when I run it through the equations for finding the metacenter and the center of gravity.

I'm quite confident in my ability to find a metacenter and center of gravity for a given length/height of this lumber, but I'm not confident in my abilities of algebraic manipulation to find the length/height at which the metacenter equals the center of gravity.

Thank you for your help- I could make my presentation simpler by just using a length of lumber that floats stably and a length that float unstably and showing calculations for each, but it would be more thorough to include an explanation of what the exact height/length at which the lumber shifts from stable to unstable is.