**Subject :**Elements of Mechanical Engineering

## second moment of area:Parallel axis theorem for the product of area

__ second moment of area:Parallel axis theorem for the product of area: __The second moment of area about any axis is the sum of the second moment of the area about a parallel axis at centroid and is Ad

^{2}where

*d*is the perpendicular distance between the axis for which

*I*is being computed about the paralell centroid axis. A is the area. Mathematically,

*I *_{about any axis} =* I* _{about a parallel axis at centroid} Ad^{2}

Let *x* be the axis parallel to and at a distance *d* from an axis x' going through the centroid of an area. The *x*' is the centroidal axis.

The second moment of area about the x-axis is

as y = y' d

Simplifying the above expression

The second term on the right hand side is zero, as *x'* is the centroidal axis.

Hence,

Proof is completed

**Parallel axis theorem for the product of area:**** **The product of area *A* for any set of axes is equal to the sum of product of area for a parallel set of axes at centroid and A d c , where *d* and *c* are the distances from the given axes to parallel set of axes passing through centroid.

Simplifying,

Second and third term on the right hand side are zero, as x' and *y'* are centroidal axes. Thus

Thus, the theorem is proved

An example:

Let *x* and y be a set of orthogonal axes passing through the centroid. *x-y* axes are also the axes of symmetry.

Because of this,

*I _{xy}= 0 = I_{yx} *

as is apperant from the following figure.

If we want to find out the moment about the bottom edge, we can use the parallel axis theorem.

Product of area will be zero.