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parallel axis parallel plane



In Mechanical Engineering and Applied Physics we encounter two integral expressions related to rotation of a physical object about a straight line whom we refer as the axis of rotation.

The two similar expressions are important in Mechanical Engineering, the rotational tendency of the Cross-sectional area of Beam subjected to a bending moment about an axis known as Neutral Axis. Suppose we take a cross-sectional area of a loaded beam subjected to pure bending, let the total cross-sectional area be A.


MOI is a physical quantity which represents the inertia or resistances shown by the body against the tendecy to rotate under the action external forces on the body. It is a rotational axis dependent function as its magnitude depends upon our selection of rotational axis. Although for any axis, we can derive the expression for MOI with the help of calculus, but still it is a cumbersome process.

Suppose, in a co-ordinate system, we take an elementary differential are (dA) at a point P(X,Y). It means the area dA is at a distance Y from X-axis and X from Y-axis. Therefore, the Moment of the aread (dA) about Y-axis will be XdA, and the Moment of the Moment of Area again about Y-axis will be X2dA, where the moment of the moment of area about Y-axis will be Y2dA.


The Coordinate Axes passing through the Centroid of a given area is called 'Centroidal Axes'. The Moment of Inertia of an area about its Centroidal Axes is the Minimum. MOI of an area about any other axes parallel to the centroidal axes is more than the MOI of the area about its centroidal axes. Hence, the MOI about centroidal axes is an important concept and it has been called as 'THE CENTROIDAL MOMENT OF INERTIA'.

Now suppose we take a different issue. We know MOI of an area about its centroidal axis is easily be obtained by integral calculus, but can we find a general formula by which we can calculate MOI of an area about any axis if we know its CENTROIDAL MOI.

We shall here find that we can indeed derive an expression by which MOI of any area (A) can be calculated about any Axis, if we know its centroidal MOI and the distance of the axis from it's Centroid G.

If (I)gx be the centroidal moment of inertia of an area (A) about X axis, then we can calculate MOI of the Area about a parallel axis (here X axis) at a distance Y from the centroid if we know the value of (I)gx and Y, then (I)px will be
(I)px = (I)gx + A.Y^2

We can explain the principle in the case of a rectangular area or lamina. Suppose we have a rectangular area of breadth = b, and height = h, hence its area will be (bh), and the Moment of Inertia about an X axis passing through its centroid is (bh^3)/12. Now suppose we want to find its Moment of Inertia about a X axis which is at a distance (d) from the X axis through its center. In this case, the MoI about the new axis will be
(I)gx + A.d^2

where (I)gx is the MoI of the area about the Centroidal X axis, where as (A) is the area of the lamina, and (d) is the distance of the new X axis from the centroidal X axis.

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