# Mass distribution of a moving body

Perhaps it is a simple question, but I am unable to solve it. Let us suppose that we have a body confined in $$\textbf{R}^{3}$$ whose mathematical description is given by a bounded and closed domain $$\textbf{D}\subset\textbf{R}^{3}$$. Moreover, let us also suppose that its mass distribution is given by the function $$m:\textbf{D}\rightarrow\textbf{R}_{\geq0}$$ such that \begin{align*} \int_{\textbf{D}}m(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z = M > 0 \end{align*}

My question is: given that its center of mass' trajectory is described by the curve $$\alpha:[0,1]\rightarrow\textbf{R}^{3}$$, how do we describe the mass distribution $$m(x,y,z,t)$$ along time?

It is worth mentioning here that we are not considering any kind of rotation, and we are dealing with a rigid body.

• You does not provide enough information for this to be solved. Even if it is a rigid body (it does not deforms in any way), you need to know how it rotates. – b.Lorenz Dec 21 '18 at 20:19
• Thank you for the comment. I forgot to mention that I am not considering rotations of any kind. Besides, it is a rigid body. – APC89 Dec 21 '18 at 20:20
• Please edit that into question body – b.Lorenz Dec 21 '18 at 20:21
• I think it's really simple if you can set the origin to the center of mass, then it's just a coordinate shift. – Jasper Dec 21 '18 at 20:32
• @Jasper That should have been an answer, not a comment. – Aaron Stevens Dec 23 '18 at 3:34

$$m_{timedependent}(\mathrm{r},t) = m_{initial}( \mathrm{r} -\alpha(t))$$