__Moment of Inertia:__

The moment of a force (P) about a point is the product of the force and perpendicular distance (x) between the point and the line of action of the force (P.x). This moment is also called the first **moment of force**. SupposeSupposeSupposeSuppose this moment is again multiplied by the perpendicular distance (x) between the point and the line of action of the force i.e, P.x(x) = Px^{2}. In that case, that case, that case, that case, this quantity is called the moment of the moment of a force or second moment of force or **moment of inertia** (M.I).

## Moment of Inertia of a Plane Area:

Consider a plane area whose moment of inertia must be found. Split up the whole area into several small elements.

Let, a_{1}, a_{2}, a_{3},… = Areas of small elements

r_{1}, r_{2}, r_{3},…= Corresponding distances of the elements from the line about which the moment of inertia is required to be found out.

Now, the moment of inertia of the area,

*I* = a_{1}r_{1}^{2} + a_{2}r_{2}^{2} + a_{3}r_{3}^{2} +…

= ∑ar^{2}

## Unit of Moment of Inertia:

the unit of moment of inertia of a plane area depends upon the area’s units and the length.

1. If the area is in m^{2} and the length is also in m, the moment of inertia is expressed in **m ^{4}**.

2. If the area is in mm^{2} and the length is also in mm, the moment of inertia is expressed in **mm ^{4}**.

__Method for Moment of Inertia:__

The moment of inertia of a plane area may be found by any one of the following two methods:

- By Routh’s Rule
- By Integration

### Moment of Inertia by Routh’s Rule:

Routh’s rule states that if a body is symmetrical about three mutually perpendicular axes, then the moment of inertia about anyone axis passing through its centre of gravity is given by:

I = A(M) x S / 3 … For a Square or Rectangular Lamina

I = A(M) x S / 4 … For a Circular or Elliptical Lamina

I = A(M) x S / 5 … For a Spherical Body

where,

A = Area of the plane area

M = Mass of the body

S = Sum of the squares of the two semi-axis, other than the axis about which the moment of inertia is required to be found out.

### Moment of Inertia by Integration:

The moment of inertia of an area may also be found out by the method of integration as given below:

Consider a plane figure, whose moment of inertia is required to be found out about the X-X axis and Y-Y axis as shown below figure. Let us divide the whole area into several strips. Consider one of these strips.

Let, dA = Area of the strip

x = Distance of the centre of gravity of the strip on the X-X axis

y = Distance of the centre of gravity of the strip on the Y-Y axis

We know that the moment of inertia of the strip about the Y-Y axis

= dA.x^{2}

Now the moment of inertia of the whole area may be found by integrating the above equation, i.e,

I_{YY} = ∑dA.x^{2}

Similarly, I_{XX} = ∑dA.y^{2}