Wednesday, 13 April 2016

newtonian mechanics - 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?



Answer



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.



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