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 Ad2 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 Ad2
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.
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.
Second and third term on the right hand side are zero, as x' and y' are centroidal axes. Thus
Thus, the theorem is proved
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,
Ixy= 0 = Iyx
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.