What is a Moment of Inertia?

The moment of inertia of a rigid body may be defined as the minimum amount of torque required by the body to rotate in the given angular momentum. However, the moment of inertia is calculated for a given axis and changes concerning the rotation axis. For calculation purposes, the moment of inertia is depicted. The moment of inertia formula depends on the mass of the body and the distance between the rotational axis and the center of mass of the rigid body.

Moment Of Inertia Definition 

When a rigid body is allowed to freely rotate about an axis, a torque is applied to bring that body to a definite angular momentum. This required amount of torque is desired to change the angular acceleration of the body. Furthermore, this torque is equivalent to the moment of inertia of the body. The angular acceleration is counted in the terms of change in angular velocity for a given period.

The SI unit for measuring the moment of inertia is kilogram per meter square (Kg-m²). Similarly, the dimensional formula for the moment of inertia is ML².

Moment Of Inertia Formula 

The moment of inertia formula is given below. Let m be the mass of a rigid body and r be the distance between the center of mass and rotational axis, and I will be the moment of inertia.

I = mr² 

Also, L is the angular momentum,w is the angular velocity, p is the linear momentum, and v is the linear velocity

if, v = w x r

p = m x v

Then L= r x p

Also, L= r x m x (r x w)

L = m x r² x w

From (1),

L = I x w

Moment Of Inertia Examples

Some of the common and daily life examples of the moment of inertia are shared here and some important points regarding them. The most common and simple example is simple pendulums. Let’s study them one by one.

Simple pendulum

The moment of inertia of a simple pendulum is always calculated from its pivot point. According to the mathematical derivation, the simple pendulum’s moment of inertia is equal to the ratio of weight due to gravity concerning the angular acceleration. However, the whole calculation is taken from the pivot point.

Let us understand this with the help of the formula

For a simple pendulum, let m be the mass of the pendulum and r be the distance between the pivot point and the simple pendulum, and I will be the moment of inertia.

I = mr²

Factors Affecting Moment of Inertia

The moment of inertia of a rigid body is mainly dependent on the below-mentioned aspects: 

• Firstly, the moment of inertia is dependent on the mass of the body.
• Secondly, the shape and size are key features that decide the magnitude of the moment of inertia.
• The axis of rotation of the body is the main governing factor for MOI.

 Moment Of Inertia Formula 

The moment of inertia formula for discrete mass systems is given below:

I = Σ m .r²

Where I is the moment of inertia, m is the mass of the particle and r is the distance between the rotational axis and center of mass.

The moment of inertia formula for a continuous mass system is given below:

I = ∫ dI = ∫0m r² dm

Where I is the moment of inertia, r is the distance between origin and the particles, dm is the point-sized mass.

Moment Of Inertia in different bodies

We have categorized some of the moment of inertia of some rigid bodies according to the axis of rotation.

Parallel Axis Theorem

According to the parallel axis theorem, the moment of inertia of a rigid body rotating on the axis which is parallel to the axis passing through the center of the body is equivalent to the moment of inertia on the axis passing through the center along with the product of the square of the distance between the axes and its mass. 

Hence, the moment of inertia formula for the parallel axis is given below:

I = Icentre + ML²

Where Icenter is the moment of the axis about the central axis

M is the mass of the object

L is the distance between the central and rotational axis.

Perpendicular axis theorem

According to the perpendicular axis theorem, the moment of inertia about an axis perpendicular to the plane of the body is equivalent to the sum of the moment of inertia about any two axes perpendicular to the plane and intersects the previous axis plane. 

Iz = Ix + Iy

Where Iz is the moment of inertia about a perpendicular axis (z- plane)

Ix is the moment of inertia in x- plane

Iy is the moment of inertia in y- plane

Conclusion 

The moment of inertia is the amount of minimum torque required by the body to rotate around an axis. The moment of inertia mainly depends on the shape, size, mass distribution, and rotation of the axis. Also, according to the moment of inertia definition, it is equivalent to the product of mass and the distance between the center of mass and the rotational axis. We can evaluate the moment of inertia using the parallel and perpendicular axis theorem.