How to Calculate Moment Of Inertia? (original) (raw)

Last Updated : 23 Jul, 2025

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.

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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 m 2 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 i r i 2

where,
ri is the distance between the axis and the ith particle,
mi is the mass of ith particle.

How to Calculate Moment Of Inertia?

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

**I = ∑m i r i 2

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.

Moment Of Inertia of various objects

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

**Related Resources:

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 m 2

**Example 2: Calculate the mass of the uniform disc when its moment of inertia is 110 kg m 2 **and 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 m 2

**Example 4: When the uniform hollow right circular cone has a moment of inertia of 98 kg m 2 **and a mass of 20 kg, determine the radius of the cone.

**Answer:

Given:
I = 98 kg m2, M = 20 kg

We have for right circular cone, MOI (I) = 1/2 MR2

R2 = 2I / M

R2 = 2 × 98 / 20

R2 = 9.8

R = √9.8

**R = 3.13 m

**Example 5: If the mass is 10 kg and the radius is 7 m, determine the hollow cylinder's moment of inertia.

**Answer:

Given:
M = 10 kg, R = 7 m

We have for hollow cylinder, MOI (I) = MR2

I = 10 × 49

**I = 490 kg m 2

**Example 6: When r 1 is 10 m, r 2 is 20 m, and the mass of the annular ring is 14 kg, calculate the moment of inertia of the ring.

**Answer:

Given: r1 = 10 m, r2 = 20 m, M = 14 kg

We have for Annular ring (I) = 1/2 M(r22 + r12)

I = 1/2 × 14 × (400 + 100)

I = 7000 / 2

**I = 3500 kg m 2

Conclusion of Moment of Inertia

Moment of Inertia is pivotal in rotational mechanics, defining an object's resistance to rotation. Its application spans engineering, physics, and various fields, guiding the design and optimization of rotating systems like gears and flywheels. Understanding moment of inertia facilitates the enhancement of system performance and reliability, driving innovation across diverse applications.