newtonian mechanics - First and Second Moment of Mass

I recently came across the definition of the Center of Mass of a system as the point about which the first moment of mass is zero.

Further, it defined Moment of Inertia as the second moment of mass.

My question is, What is this 'moment of mass'?

Answer

Given some distribution or density $\rho(x),$ a moment is the 'expectation value' of some power of $x \in \mathbb{R}$. To be precise, the $n$-th moment $M_n$ is given by

$$M_n = \int_{\mathbb{R}} x^n \rho(x) \mathrm{d}x.$$ In the mechanics case, $\rho(x)$ is simply the mass density.

You can extend this to vectors in $\mathbb{R}^d$ in a straightforward way; for example, for the moment of inertia you replace $x^2$ by $\mathbf{x}^2 = x_1^2 + \ldots x_d^2$ to obtain

$$I = M_2 = \int_{\mathbb{R}^d} \mathbf{x}^2 \rho(\mathbf{x}) \mathrm{d}^dx$$ which should match the definition given in your mechanics textbook.

For the first moment of mass, you need to distinguish different directions. As you indicate, you can choose your coordinates such that

$$\int_{\mathbb{R}^d} x_i \,\rho(\mathbf{x}) \mathrm{d}^d x = 0$$ where $i$ runs over the coordinates. In three dimensions, you have $x_1 = x, x_2=y$ and $x_3=z.$