### 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 = r^{2}dA

Where, r = distance to the element dA

### Units

The SI unit of the polar moment of inertia is metered to the fourth power (m^{4}). In the imperial system, the unit is inches to the fourth power (in^{4}).

### 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:

- Moment of inertia of a rectangular section
- Moment of inertia of a circular section
- Moment of inertia of a triangular section
- 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.

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