**Branch :**First Year-Engineering Syllabus

**Subject :**Elements of Mechanical Engineering

## Second Moments and Products of Area in the rotated Coordinate System

**Second Moments and Products of Area in the rotated Coordinate System:**** **If we know, the second moment and products of area about a reference xy, we can find these quantities relative to a rotated reference x'y' that has the same origin as xy. Let us consider that reference x'y' is rotated at an angle q from xy as shown in figure.

From figure it can be seen that

Now,

or,

Therefore,

Putting

In the same way,

which gives,

The product of inertia about the inclined axes is

Thus we can obtain I_{x'x'} , I_{y'y'} and I_{x'y'} in terms of I_{xx} , I_{yy} and I_{xy}.