There have been a lot of questions and discussions regarding determining the mass distribution of various bodies through non-destructive purposes. Here are a few:
The general conclusion that I reached after reading questions like these is that by mechanics alone, there is no way to positively determine the mass distribution of a 3-Dimensional object. The simplest arguments typically involved spherical masses and newton's shell theorem.
My problem is a simplified case of this general problem that I hope can be solved more easily. I have a stick of balsa wood with dimensions 1/8" x 1/8" x 36", and I would like to know its mass distribution. Because it is so slender, it can be safely treated like a 1-Dimensional object with a density function $\rho(x)$ that depends only on variable. I can find the center of mass very easily just by balancing it, but beyond that I am stuck. I would really like to find an estimation for $\rho(x)$ that is accurate to about the 1" or 1/2" scale.
One assumption that can be made is that the young's modulus, $E$, of the wood varies linearly with density. This might be useful for solutions that measure the deflection of the wood under different forces.
First, I am looking for solutions using the simplest methods and tools. I have a scale accurate to 0.01g (the whole stick weighs about 2.60g) and a micrometer that measures to 0.001". If a solution is possible that uses only these tools and other common and accessible ones, that would be best.
If that is not possible, I would still like to hear other more complex approaches that might require advanced equipment or might simply be unrealistic in practice.
The reason I need to solve this problem is that I am building an 18g, 8" tall structure for a competition that must hold over 600 pounds. I know the mass distributions of each specific stick can vary greatly and be quite unpredictable. Since the structure is subject to such extreme forces, small imperfections can significantly affect performance. If one of my 8" columns has 60% of its mass distributed to its lower half, then it will be weaker in the top half. For this reason it is beneficial for me to select wood that is nearly homogenous, and statistical methods (finding variance, etc...) are not particularly useful in this case because I need to be 100% sure that the wood I select is evenly distributed.