Moment of Inertia of Curves and Surfaces

[Jhenrique]

Greetings!

I enjoyed the definition of moment of inertia for a volume and for an area in the form of matrix. It's very enlightening!

$$I = \int \begin{bmatrix} y^2+z^2 & -xy & -xz\\ -yx & x^2+z^2 & -yz\\ -zx & -zy & x^2+y^2 \end{bmatrix}dxdydz$$

'-> http://mathworld.wolfram.com/MomentofInertia.html

$$J = \int \begin{bmatrix} y^2 & -xy\\ -yx & x^2\\ \end{bmatrix}dxdy$$

'-> http://mathworld.wolfram.com/AreaMomentofInertia.html

So, analogously, I'd like to know how would be the matrices of moment of inertia for curves and for surfaces...

Thx,

Jhenrique

 

[SteamKing]

The second moments of area have a specific usage, particularly in calculating certain stresses for beams.

The second moments of a volume are used in mechanics to describe the motions of a body under the influence of external forces and moments.

I am not aware of a definition of a second moment for a general curve, unless you wish to approximate the curve as a rod of negligible radius. There are second moments defined for surfaces whose thickness is very small. These moments are used for objects which are composed of thin shells and can be derived using the definitions for the I matrix in the OP.

See:

http://en.wikipedia.org/wiki/Second_moment_of_area
http://en.wikipedia.org/wiki/List_of_area_moments_of_inertia

http://en.wikipedia.org/wiki/Mass_moment_of_inertia
http://en.wikipedia.org/wiki/List_of_moments_of_inertia