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We all know the "Inertia" of a particle is actually related to its "Mass". But when we talk about "Moment of inertia" of a body, we actually are talking about its "Mass distribution" about some point/axis. So depending on how the mass is distributed in the body (what shape a body has), its moment of inertia could be determined.

So my question is can we do it vice-versa? I mean if we had the moment of inertia of a body about some axis, can we construct its shape?

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No you can't reconstruct the shape just from the mass moment of inertia (MMOI) values. At most you can get 3 eigenvalues (the principal moments of inertia) and an orientation matrix which diagonalizes the MMOI matrix.

Finding the MMOI values from a shape involves a triple integral over the volume and integration (like any other summation process) loses information. It is a sort of averaging process. If you know the average weight of two people you can't deduce anything about the individual weights without additional information for example. That is because the averaging process loses information.

From the diagonalized MMOI matrix with principal values $I_1$, $I_2$ and $I_3$ in decreasing order you can find an equivalent solid ellipsoid with the following semi-major radiuses

$$\begin{aligned} r_1 & = \sqrt{ \frac{5}{2 m} | I_2 + I_3 - I_1 | } \\ r_2 & = \sqrt{ \frac{5}{2 m} | I_1 - I_2 - I_3 | } \\ r_3 & = \sqrt{ \frac{5}{2 m} | I_1 + I_2 - I_3 | } \end{aligned}$$

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  • $\begingroup$ Thank you for your answer. Another question: What other information we need to reconstruct the shape? Do such complete set of information even exists? $\endgroup$ Nov 5 '16 at 5:34
  • $\begingroup$ You can never deduce the shape. You can assume a certain shape (like a quadrilateral, or a cylinder or a disk) and then try to fit the measured MMOI about different axes into the theoretical value in order to estimate the shape parameters (length, diameter, etc) $\endgroup$ Nov 6 '16 at 16:10
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Evidently not in the absence of further information. If you know $I$ of the body, regardless its shape, you can always take a point with mass $M$ (the mass of the body) and fix its distance $d$ from the axis such that $Md^2=I$.

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  • $\begingroup$ The interpenetration of MMOI $I$ is as a ring of mass $M$ at a distance $d$. You you place all the mass at a single point you are moving the center of mass disrupting the dynamic behavior. The equivalent system is that of ring of mass for one rotation, or an equivalent ellipsoid for a 3D case. $\endgroup$ Nov 4 '16 at 20:13
  • $\begingroup$ Indeed I wrote, in the absence of further information... $\endgroup$ Nov 4 '16 at 20:25
  • $\begingroup$ However, even in your case you can reduce everything to rings loosing informations about the real shape of the body. $\endgroup$ Nov 4 '16 at 20:26
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If you have the moment of inertia of an object from multiple axis, and the object is simple in geometry, has no holes or discontinuities or at list the holes are within anticipations You can compare the I's and do inverse analysis.

For the same object larger I means it is more elongated and stretched out WRT that axis, mass is more distributed to extremes as opposed to clumped around that axis. Similar algorithms are used in MRI machines to rebuild complex body structures out of reading of a rotating sensor that registers the opacity of one axis to X-ray or magnetic field resonation sensing along that axis.

However even the best MRI machines have to be calibrated for and tuned against erronous interpretation!

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