**Subject :**Elements of Mechanical Engineering

## Second Moments and the product of area of a plane

**Second Moments and the product of area of a plane:**** **The second moments of the area *A *about the *x *and *y* axes denoted as I_{xx} and I_{yy} , respectively are defined as:

Note that, (1) The first moment of area can be positive or negative, whereas the second moment of area is positive only.

(2) The element of area that are farthest from the axis contribute most to the second moment of area.

Suppose there is a plane surface of area A. Its second moment about x -axis is

If it is assumed that entire area is concentrated at a distance of K_{y} from the x-axis.Then the moment of inertia is k_{y}^{2}*A*.

If it is equal to the actual moment of inertia, then

Likewise, we can write

The distance k_{x} and k_{y} are called the radii of gyration.

The product of area relates one area directly to a set of axes and is defined as,

Please note that this quantity may be negative. The second moment of area is always positive.

If the area under consideration has an axis of symmetry, the product of area for this axis and any axis orthogonal to this axis must be zero. Also, note that I_{xy} = I_{yx}.