# 1st and 2nd moment of interia of curves and surfaces

1. ### Jhenrique

686
If I can to calculate the 1st and 2st moment of inertia of areas and volumes, I can compute for curves and surfaces too?

2. ### SteamKing

8,789
Staff Emeritus
First, you must define what you mean by the moment of inertia of a curve or surface.

The moment of inertia concept is useful in two areas: the second moment of area of a plane region or the mass moment of inertia of a three-dimensional body. The first quantity is useful in analyzing the bending response of beams, while the second quantity is useful in calculating the motion of objects.

3. ### Jhenrique

686
In a book of mechanical projects (in portuguese, my natural idiom) I found an very surface explanation about first moment of curves and surfaces (see the anex), and I'm curious because I'd like to know if really exist a formula for 1st and 2nd moments of curves and surfaces.

I understand the 1st moment as the follows formula:
$$M=\begin{bmatrix} M_{yz}\\ M_{zx}\\ M_{xy}\\ \end{bmatrix} = \int \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} dV$$
And the 2st like:
$$I = \begin{bmatrix} I_{xx} & I_{xy} &I_{xz} \\ I_{yx} & I_{yy} &I_{yz} \\ I_{zx} & I_{zy} &I_{zz} \end{bmatrix} = \int \begin{bmatrix} y^2+z^2 & -xy & -xz\\ -yx & z^2+x^2 & -yz\\ -zz & -zy & x^2+y^2\\ \end{bmatrix} dV$$

Edit: the book this that the static moment (1st moment) is useful to calculate the geometric center.

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Last edited: Jan 26, 2014