How to Calculate Moment Of Inertia?

Last Updated : 15 Oct, 2022

The moment of inertia, referred to as the angular mass or rotational inertia, with respect to the rotation axis is a quantity that determines the amount of torque necessary to achieve a desired angular acceleration or a characteristic of a body that prevents angular acceleration. The moment of inertia is calculated as the sum of each particle’s mass times the square of its distance from the rotational axis.

Moment of Inertia

The term “moment of inertia” refers to the quantity that describes how a body resists angular acceleration and is calculated as the product of the mass of each particle times the square of the particle’s distance from the rotational axis. Or, to put it another way, you could say that it’s a quantity that determines how much torque is required for a certain angular acceleration in a rotating axis. Inertia moment is often referred to as rotational inertia or angular mass. kg m² is the unit of moment of inertia in the SI system.

Moment of Inertia of a System of n Particles

The moment of inertia is the following for a system of point particles rotating around a fixed axis:

I = ∑m_ir_i²

where,
r_i is the distance between the axis and the i_th particle,
m_i is the mass of i_th particle.

How to Calculate Moment Of Inertia?

Several ways are used to calculate the moment of inertia of any rotating object.

For uniform objects, the moment of inertia is calculated by taking the product of its mass with the square of its distance from the axis of rotation (r²).
For non-uniform objects, we calculate the moment of inertia by taking the sum of the product of individual point masses at each different radius for this the formula used is

I = ∑m_ir_i²

Formulas For Calculating Moment Of Inertia

Expressions for the moment of inertia for some symmetric objects along with their axis of rotation are discussed below in this table.

I = \frac{2}{5}M\frac{(r_2^5-r_1^5)}{(r_2^3-r_1^3)}

Object	Axis	Expression of the Moment of Inertia
Hollow Cylinder Thin-walled	Central	I = Mr²
Thin Ring	Diameter	I = 1/2 Mr²
Annular Ring or Hollow Cylinder	Central	I = 1/2 M(r₂² + r₁²)
Solid Cylinder	Central	I = 1/2 Mr²
Uniform Disc	Diameter	I = 1/4 Mr²
Hollow Sphere	Central	I = 2/3 Mr²
Solid Sphere	Central	I = 2/5 Mr²
Uniform Symmetric Spherical Shell	Central	$I = \frac{2}{5}M\frac{(r_2^5-r_1^5)}{(r_2^3-r_1^3)}$
Uniform Plate or Rectangular Parallelepiped	Central	I = 1/12 M(a² + b²)
Thin rod	Central	I = 1/12 Mr²
Thin rod	At the End of Rod	I = 1/3 Mr²

Solved Examples of Moment of Inertia

Example 1: Determine the solid sphere’s moment of inertia at a mass of 22 kg and a radius of 5 m.

Answer:

Given:
M = 22 kg, R = 5 m

We have for solid sphere, MOI (I) = 2/5 MR²

I = 2/5 × 22 × 25

I = 220 kg m²

Example 2: Calculate the mass of the uniform disc when its moment of inertia is 110 kg m²and its radius is 10 m.

Answer:

Given:
I = 110 kg m², R = 10 m

We have for uniform disc (I) = 1/4 MR²

M = 4I / R²

M = 4 × 110 / 10²

M = 440 / 100

M = 4.4 kg

Example 3: If a uniform plate has a mass of 23 kg, a length of 10 m, and a breadth of 7 m, determine its moment of inertia.

Answer:

Given: M = 23 kg, L = 10 m, b = 7 m

We have for uniform plate MOI $I=\frac{1}{12}M(L^2+b^2)$

$I=\frac{1}{12}×23×(10^2+7^2)$

$I=\frac{1}{12}×23×(100+49)$

$I=\frac{1}{12}×23×(149)$

I = 285 kg m²