How can I determine whether the mass of an object is evenly distributed? How can I determine whether the mass of an object is evenly distributed without doing any permanent damage? Suppose I got all the typical lab equipment. I guess I can calculate its center of mass and compare with experiment result or measure its moment of inertia among other things, but is there a way to be 99.9% sure?
 A: If you have a rigid mass distribution sealed inside a black box, then the only things you can observe about its motion are its velocity vector and its angular velocity vector as functions of time. These can be predicted if you know the total force and total torque that act, plus the mass, center of mass, and moment of inertia tensor. So all that can be determined by any external, mechanical measurements is its mass, its center of mass, the orientation of its principal axes, and the three diagonal elements of the moment of inertia tensor along the principal axes. This is nowhere near enough information to recover the full mass distribution or to determine whether the mass is evenly distributed.
As an example, suppose object A is a spherical mass $m$ of uniform density, with radius $a$. Then you can make object B with the same mass distributed uniformly on a hollow shell of radius $b=\sqrt{3/5}a$, so that B has the same moment of inertia as A.
If you want them to look the same visually you can for example create object C by superposing two objects: (1) a uniform sphere like A, but with half the density, and (2) a concentric shell like B but with half the mass per unit area. Then A and C are indistinguishable.
By the shell theorem, A and C are also not distinguishable by their external gravitational fields.
A: Malicious counter example
The desired object is a sphere of radius $R$ and mass $M$ with uniform density $\rho = \frac{M}{V} = \frac{3}{4} \frac{M}{\pi R^3}$ and moment of inertia $I = \frac{2}{5} M R^2 = \frac{8}{15} \rho \pi R^5$.
Now, we design a false object, also spherically symmetric but consisting of three regions of differing density
$$ \rho_f(r) = \left\{
  \begin{array}{l l}
    2\rho\ ,           &  r \in [0,r_1) \\
    \frac{1}{2}\rho\ , &  r \in [r_1,r_2) \\
    2\rho\ ,           &  r \in [r_2,R) \\
  \end{array} \right.$$
We have two constraints (total mass and total moment of inertia) and two unknowns ($r_1$ and $r_2$), so we can find a solution which perfectly mimics our desired object.
A: It has been pointed out that this cannot simply be done by examining the mass distribution (first and second moment of mass). But there is a way to "look inside" most common objects: Take a CT scan. Not sure if you consider that "typical" lab equipment - but it's equipment I have in my lab...
Of course depending on the size of the object and the material composition, it can be quite hard to get a definitive answer - beam hardening effects need to be taken into account, which means you have to know your Xray spectrum. Reconstructing a dense (high Z) object with the correct density everywhere is actually quite hard.
A: If mass is evenly distributed inside the volume occupied by your mass, and you know your mass theoretical density, then if your mass is evenly distributed inside the volume it occupy, it will be exactly equal to the theoretical density, if there is a mismatch between the theoretical density and the measured density, it means that mass is not evenly distributed or that another kind of mass with different density is in your mass volume (like when there are bubbles inside a solid mass) but it may be the case that the bubbles are also evenly distributed then mass is evenly distributed but theoretical density does not match your one element measured density because mass density averages but it is not evenly distributed.
A: Tie a rope to the object and attach the other end to a scale. Slowly lower the object into some water, recording the force on the scale and the amount of water displaced at many intervals. Using this data, compute the density of the section of the object that is submerged at all intervals. If the mass of the object is uniformly distributed then the density values you get should not vary.
