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Let's say we have a planar mass system (all the patches of area covered lie in the same plane)

If we know the center of mass coordinates, the moment of inertia about an axis at any point perpendicular to the plane as a function of $(x,y)$ and the total mass of the body, can we tell the distribution of mass (density at any point) as a function of $(x,y)$?

I think a good way to build upon this is to do this for one dimension.

We know the following integrals of following functions from $-\infty$ to $\infty$:

  1. $t f(t) $
  2. $(t-x)^2 f(t)$
  3. $f(t)$

[For one dimensional body, but same can be extended to two dimensions by replacing it with a double integral]

Is this information sufficient to know the exact mass distribution? And exactly how many details or constraints are required to plot an exact mass distribution of a system?

And lastly, how do we extend this to 3 dimensions?

PS: I'm new here so was unable to use Latex, apologise for that.

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    $\begingroup$ Useful: MathJax tutorial $\endgroup$ Commented Dec 20, 2024 at 16:54

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