Moment of inertia formula | Definition for moment of inertia

The formula for moment of inertia

The formula for the moment of inertia is different for different shapes of the object. It depends on geometrical shape of the object like a circular, rectangular, triangular, thin rod, etc. So, I have given some formula for the moment of inertia of different objects having a different geometrical shape. Let’s see what are they:-

Moment of inertia definition

The product of the area ( or mass ) and the square of the distance of the C. G. of the area ( or mass ) from an axis is called the moment of inertia of the area ( or mass ) about that axis. It is represented by I. hence moment of inertia about the X-axis is represented by Iₓₓ whereas about Y-axis represents Iyy.

Physical significance of moment of inertia

If the greater mass is concentrated away from the axis, then moment of inertia will be greater.

For example(1), take a baseball club. If you rotate baseball by holding the handle, you have to put in more effort than when you rotate baseball by holding it at its hitting end.

Example (2)  Consider a disc of radius R and mass M. Moment of inertia is given by,

Now, melt this disc and convert it into ring which is shown in figure;

 The mass remains same and its radius is larger compare to above disc.

Let, the radius be R’ (R<R’). The moment of inertia of this ring is,

                                         I’ = MR’²        ….. (2)

Now, from equation (1) and (2), we get

                  I < I’

Polar moment of inertia

Polar moment of inertia is a quantity used to predict an object’s ability to resist torsion, in objects ( or segments of objects) with an invariant circular cross section. Polar moment of inertia describes the cylindrical object’s resistance to torsional deformation when torque is applied in a plane that is parallel to the cross section area or in a plane that is perpendicular to the object’s central axis.

Polar moment of inertia formula

 Polar moment of inertia is also known as second polar moment of area. It is denoted by I_z . However, it is also denoted by J or J_z. Polar moment of inertia can be mathematically represented by the given formula;

                                              I or J = r²dA

Where, r = distance to the element dA

Units

The SI unit of the polar moment of inertia is metered to the fourth power (m⁴). In the imperial system, the unit is inches to the fourth power (in⁴).

What is the radius of gyration?

The radius of gyration of a body ( or given lamina ) about an axis is a distance such that its square multiplied by the area gives a moment of inertia of the area about the given axis.

Let the whole mass ( or area ) of the body is concentrated at a distance ‘k’ from the axis of reference, then the moment of inertia of the whole area about the given axis will be equal to Ak².

If Ak² = I, then k is known as the radius of gyration about the given axis.

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Theorem of the perpendicular axis

The theorem of the perpendicular axis states that if Ixx and Iyy be the moment of inertia of a plane section about two mutually perpendicular axes X-X and Y-Y in the plane of the section, then the moment of inertia of the section Izz about the Z-Z, perpendicular to the plane and passing through the intersection of X-X and Y-Y is given by

I_ZZ = I_XX + I_YY

The moment of inertia is also known as the polar moment of inertia.

The theorem of parallel axis

It states that if the moment of inertia of a plane area about an axis in the plane of area through the center of gravity of the plane area be represented by I_G, then the moment of inertia of the given plane area about a parallel axis AB In the plane of area at a distance h from the C.G. of the area is given by

         I_AB = I_G + Ah²

Where,

I_AB = Moment of inertia of the given area along AB

I_G = Moment of inertia of the given area about C.G.

A = Area of the section

h = Distance between the C.G. of the section and the axis AB.

Determination of moment of inertia

The moment of inertia of the following sections will be determined by the method of integration:

1. Moment of inertia of a rectangular section
2. Moment of inertia of a circular section
3. Moment of inertia of a triangular section
4. Moment of inertia of a uniform thin rod

For Rectangle section

Moment of inertia along X-axis is given by,

Where, b is the width of the rectangular section

And, d is the depth of the rectangular section.

Moment of inertia along Y-axis is given by,

            For circular section

Moment of inertia along X-axis and Y-axis is given by,

Where, D is the diameter of the circle.

For the triangular section

Moment of inertia along X-axis is given by,

Where, b is the base of the triangular section

           h is the height of the triangular section

 Moment of inertia of a thin rod

Moment of inertia along the Y-Y axis is given by,

Where, Total mass of the rod is M

  And, Length of the rod is L

Q) Find the moment of inertia of the rectangular section of size

2 m * 3 m

Solution,

Given that;

Width of rectangle ( b ) = 2 m

Depth of rectangle ( d ) = 3 m

Moment of inertia =?

We know that,

Numerical to calculate Moment of Inertia

In this way, we can calculate the moment of inertia of different objects having a different geometrical shape. So, friends, I hope you liked my article on “Moment of inertia formula | Definition for moment of inertia” and remains helpful for you.