Mass Moment of Inertia

The mass moment of inertia, also called the moment of inertia, is a fundamental concept of rotational motion. It is similar to the mass in linear motion, just like mass resists changes in linear motion, moment of inertia resists changes in rotational motion.

Definition

The moment of inertia (I) of a body about an axis is defined as the sum of the product of the mass of its particles and the square of their perpendicular distances from the axis of rotation.

Formula –

Formula

For a continuous body,

Formula

Where:

I – moment of inertia
m_i – mass of the i-th particle
r_i – distance of the i-th particle from the axis
r – distance variable
dm – small mass element

SI Unit – kg m²
Dimensions – [M][L]²

Factors Affecting Moment of Inertia

Mass of the body: The greater the mass of the object, the greater the inertia
Distribution of mass: The farther the mass is from the axis, the greater the inertia
Axis of rotation: Moment of inertia changes with the axis

Moment of Inertia in Shapes

Shape	Axis of Rotation	Moment of Inertia
Thin Rod (length L)	Center	ML²
Thin Rod (about one end)	End	ML²
Solid Cylinder (radius R)	Central Axis	MR²
Solid Sphere (radius R)	Diameter	MR²
Hollow Sphere	Diameter	MR²
Thin Circular Ring	Diameter	MR²

Where M – mass and R – radius

Parallel Axis Theorem

I = I_cm + Md²

I – moment of inertia about a new axis
I_cm – moment of inertia about the centre of mass
M – mass of the object
d – distance between the two axes

Perpendicular Axis Theorem (for planar objects)

I_z = I_x + I_y

Where I_z is the moment of inertia perpendicular to the plane, and I_x and I_y are along two perpendicular directions in the plane.

Moment of Inertia Used in Other Concepts

Torque

The moment of inertia determines the rotation of an object around an axis. An object with more moment of inertia requires more torque to gain the same angular acceleration.

Formula:

Formula

τ – torque
I – moment of inertia
α – angular acceleration

Rotational Kinetic Energy

This shows the effect of mass distribution in rotational motion. The kinetic energy of a rotating object is:

Formula

where

ω – angular velocity

Some Applications and Real Life Examples

Rotational motion: Used to calculate angular acceleration and energy
Engineering: Design of gears and machines
Sports: Gymnasts and ice skaters change their moment of inertia to spin faster
Astronomy: Used to calculate rotational dynamics of planets and stars
Bicycle wheel: Higher inertia at rim makes rotation harder to start or stop
Flywheels: High moment of inertia stores rotational energy efficiently

Summing Up

The mass moment of inertia measures an object’s resistance to rotational motion. The moment of inertia depends on mass and distance from the axis. It varies with shape and axis. Parallel and perpendicular axis theorems help calculate it. It has major applications in engineering, mechanics, and rotating systems.

Frequently Asked Questions

Q1. Is the moment of inertia a scalar or a vector quantity?

Moment of inertia is a scalar quantity.

Q2. What is the difference between mass and moment of inertia?

Mass resists linear motion, whereas moment of inertia resists angular or rotational motion.

Q3. Find the moment of inertia of a thin rod of mass 2 kg and length 1.5 m about its centre.

Here mass is 2 kg and length is 1.5 m

Formula

Q4. A disc has a moment of inertia of 0.5 kg·m² about its centre. What is the moment of inertia about an axis 0.3 m away from the centre (mass = 4 kg)?

Here, the moment of inertia about the centre is 0.5, the distance from the axis is 0.3, and the mass is 4kg
Formula