Moment of Inertia Summary | Additional Study Material for Mechanical Engineering PDF Download

Moment of Inertia 

The Second moment of area, also known as moment of inertia of plane area is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis.

Moment of Inertia is given as the sum of the products of each area in the lamina with the square of its distance from the axis of rotation.

It is denoted with I for an axis that lies in the plane or with a J for an axis perpendicular to the plane (referred as polar moment of inertia).

Parallel Axis Theorem

Parallel Axis Theorem states that the moment of inertia of a plane lamina about any axis is equal to the sum of moment of inertia about a parallel axis passing through the centroid and the product of area and square of the distance between the two parallel axes.

Proof

In Figure, the x₀-y₀ axes pass through the centroid C of the area. If I’_x and I’_y are the moments of inertia of the area about the centroidal axes x₀-y₀ and I_x and I_y the moments of inertia of the area about the parallel axes x-y.

By definition, the moment of inertia of the element dA about the x-axis is given by

Since

Similarly

Perpendicular Axis Theorem 

Perpendicular Axis Theorem states that the moment of inertia of a plane lamina about an axis perpendicular to the lamina and passing through the centroid is equal to the sum of moment of inertias of the lamina about two mutually perpendicular axes in the same plane and passing through the centroid.

OR

Perpendicular Axis Theorem states that the polar moment of inertia is equal to the sum of moment of inertias of the two centroidal axis in the plane of the lamina

Proof

In Figure, the x0-y0 axes pass through the centroid C of the area. If I’x and I’y are the moments of inertia of the area about the centroidal axes

We know that by defination

Moment Of Inertia Of Regular Sections

A. MOMENT OF INERTIA OF COMPOSITE SECTIONS

Polar Moment of Inertia 

The moment of inertia of a plane lamina about an axis perpendicular to the lamina and passing through the centroid is called polar moment of inertia

Radius of Gyration

The radius of gyration of an area about an axis is the distance to a long narrow strip whose area is equal to the area of the lamina and whose moment of inertia remain the same as that of the original area

Consider a lamina of area A whose moment of Inertia is Ix, Iy and Iz with respect to x, y and z axis.
Imagine the area is concentrated into a thin strip parallel to the axis and has the same moment of inertia as that of the lamina

Product of Inertia

The product of inertia is the sum of the product of each differential area and its moment arms about the two perpendicular reference axis.

Rotation of Moments of Inertia

The moment of inertia and product of inertia of a lamina with respect to an inclined axis u and v rotated an angle θ with respect to the reference axis x- y is given by

Principal Axes and Principal moments of Inertia 

The inclined set of axis u-v about which the product of inertia Ixy is zero is called the Principal Axes.
The angle made by the principal axes with respect to the reference axes is called principal angle θp.
The moments of inertia about the principal axes are called Principal Moment of Inertia and yield the maximum and minimum values of moment of inertia of the lamina.

Mass moment of Inertia Mass

moment of inertia or rotational inertia of a body is a measure of the body’s resistance to angular acceleration. Mass Moment of Inertia is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation.
The mass moment of inertia about the axis O-O is given as

Mass Moment of Inertia of Circular Thin Disc

Mass Moment of Inertia of Cylinder