Theorems of Moment of Inertia are important tools in the realm of engineering mechanics, specifically when dealing with the calculation of moments of inertia for various composite geometrical shapes.
Parallel Axis Theorem
- The parallel axis theorem is employed to determine the moment of inertia of an object in cases where the object is undergoing rotational motion around an axis that doesn’t pass through its centre of mass. Instead, it rotates around a distinct axis that runs parallel to the one passing through its centre of mass.
- Moment of inertia of an object about any axis parallel to its centre of mass (or centroid) is equal to the sum of the moment of inertia about its centre of mass (IC) and the product of its mass (or area) and the square of the distance between the two axes.
For the object shown, moment of inertia about an axis passing through point c1 is:
To establish this theorem, let’s examine a two-dimensional plane denoted by area A, with its centroid marked as point c, as shown in Figure. We have a Cartesian axis, X. Additionally, we have a parallel axis, X’, that passes through the centroid c of the plane.
Here, d is the distance between X and the X’ axis, and y’ is the distance of elemental area dA from the X’ axis. Let’s focus on the elemental area dA.
We can express the second moment of the area A with respect to the X-axis as follows:
Put the values in above equation (∫dA = A and ∫y′2dA = IX′ or Ic and ∫y′dA=0 )
Perpendicular Axis Theorem
- It states that for planar objects, the sum of the moments of inertia about an axis perpendicular to the plane of the figure is equal to the sum of the moments of inertia about any two perpendicular axes in the same plane intersecting at the point through which the perpendicular axis passes.
To establish this theorem, let’s examine a two-dimensional plane denoted by area A. We have a Cartesian axis, x and y. Here, x, y, and r are the distances of elemental area dA from the X, Y, and Z axes. Let’s focus on the elemental area dA.
We can express the second moment of the area A with respect to the Z-axis as follows:
Put the values in above equation (∫x2dA = IX and ∫y2dA = IY)
Note: This theorem is very helpful to calculate the polar moment of inertia.
Limitation: The limitation of the perpendicular axis theorem is that it can only be applied to planar shapes that lie within a single plane. This theorem is not valid for three-dimensional objects that extend in multiple directions.