Moment of Inertia

Overview

Moment of inertia describes how strongly an object resists a change in its rotational motion. It is also called rotational inertia.

Mass measures resistance to linear acceleration, while moment of inertia measures resistance to angular acceleration.

Learning Outcomes

After studying this topic, students should be able to:

• Define moment of inertia.
• Explain why it is the rotational equivalent of mass.
• Calculate moment of inertia for point masses.
• Calculate the total moment of inertia of several particles.
• State its SI unit and dimensions.
• Explain how mass distribution affects rotational inertia.
• Use standard formulas for uniform objects.

1. Definition

It is represented by:

I

2. Moment of Inertia of a Point Mass

Consider a small object of mass m located at a perpendicular distance r from the axis of rotation.

Its moment of inertia is:

I = mr²

Where:

• I = moment of inertia
• m = mass of the particle
• r = perpendicular distance from the axis of rotation

Key Logic

The formula contains r². Therefore, distance has a stronger effect than mass.

If mass doubles:

I′ = 2mr² I′ = 2I

If distance doubles:

I′ = m(2r)² I′ = 4mr² I′ = 4I

Therefore, doubling the distance makes the moment of inertia four times greater.

3. Moment of Inertia of a Rigid Body

A rigid body can be treated as a large collection of small masses.

Suppose its particles have masses:

m₁, m₂, m₃, …, mₙ

Their perpendicular distances from the rotation axis are:

r₁, r₂, r₃, …, rₙ

The total moment of inertia is:

I = m₁r₁² + m₂r₂² + m₃r₃² + ⋯ + mₙrₙ²

Using summation notation:

I = Σ mᵢ rᵢ²

Each particle contributes according to its mass and its distance from the axis.

Important Observation

A particle lying exactly on the rotation axis has:

r = 0

Therefore:

I = m(0)² I = 0

That particle does not contribute to the moment of inertia about that axis.

4. Factors Affecting Moment of Inertia

Moment of inertia depends on:

Mass of the object

A greater mass normally produces a greater moment of inertia.

Distribution of mass

Mass located farther from the axis produces a much greater moment of inertia.

Position of the rotation axis

The same object can have different moments of inertia about different axes.

Shape and size of the object

A ring, disc, sphere and rod distribute their masses differently, so their moments of inertia are different.

Moment of Inertia

Searchable formulas: I = mr² · I = Σ mᵢ rᵢ² · Idisc = ½ MR² · Iring = MR²

5. SI Unit and Dimensions

From:

I = mr²

The SI unit is:

kg · m² SI unit of I = kg m²

Dimensional formula:

[I] = [M][L]² [I] = [ML²]

Moment of inertia is a scalar quantity for rotation about a specified fixed axis.

6. Standard Formulas for Uniform Objects

These formulas apply only to the stated rotation axes.

Standard Formulas for Uniform Objects

Here:

• M = total mass
• R = radius
• R₁ = inner radius
• R₂ = outer radius
• L = length
• a, b = sides of the rectangular plate

Always identify the rotation axis before selecting a formula.

7. Comparison Between a Disc and a Ring

For a solid disc:

I_disc = (12) MR²

For a thin ring:

I_ring = MR²

Therefore:

I_ring = 2 I_disc

A ring has a greater moment of inertia because most of its mass is concentrated at the outer radius. In a disc, some mass lies closer to the axis.

Solved Examples

Example 1: Single Point Mass

A 3 kg particle rotates at a distance of 0.40 m from an axis. Calculate its moment of inertia.

Given:

m = 3 kg r = 0.40 m

Formula:

I = mr²

Solution:

I = 3(0.40)² I = 3(0.16) I = 0.48 kg m²

Final answer:

I = 0.48 kg m²

Example 2: Two Point Masses

A 2 kg mass is 0.30 m from an axis, while a 1 kg mass is 0.50 m from it. Find the total moment of inertia.

Formula:

I = m₁r₁² + m₂r₂²

Solution:

I = 2(0.30)² + 1(0.50)² I = 2(0.09) + 1(0.25) I = 0.18 + 0.25 I = 0.43 kg m²

Final answer:

I = 0.43 kg m²

Example 3: Solid Disc

A solid disc has mass 4 kg and radius 0.25 m. Calculate its moment of inertia about its central axis.

Formula:

I = (12) MR²

Solution:

I = (12)(4)(0.25)² I = 2(0.0625) I = 0.125 kg m²

Final answer:

I = 0.125 kg m²

Example 4: Thin Ring

Suppose the disc in the previous example is replaced by a thin ring having the same mass and radius.

Formula:

I = MR²

Solution:

I = 4(0.25)² I = 4(0.0625) I = 0.25 kg m²

Final answer:

I = 0.25 kg m²

The ring has twice the moment of inertia of the solid disc.

Example 5: Rod About Different Axes

A uniform rod has mass 3 kg and length 2 m.

About its centre

Icentre = (112) ML² Icentre = (112)(3)(2)² Icentre = 1212 Icentre = 1 kg m²

About one end

I_end = (13) ML² I_end = (13)(3)(2)² I_end = 4 kg m²

Changing the axis from the centre to one end makes the moment of inertia four times greater.

Real-Life Applications

Spinning person

A spinning person rotates more easily when their arms are close to the body. Moving the arms outward places more mass farther from the axis and increases the moment of inertia.

Door

A door rotates about its hinges. Pushing near the outer edge is more effective than pushing close to the hinges.

Flywheel

A flywheel has considerable mass distributed away from its axis, giving it a large moment of inertia and helping it maintain smooth rotational motion.

Ring and disc

A ring is more difficult to angularly accelerate than a disc of equal mass and radius because the ring’s mass lies farther from the axis.

Important Exam Facts

• Moment of inertia is the rotational equivalent of mass.
• Its symbol is I.
• Its SI unit is kg m².
• Its dimensions are [ML²].
• For a point mass, I = mr².
• For several particles, I = Σ mᵢ rᵢ².
• Moment of inertia depends on the selected rotation axis.
• Distance is squared, so it strongly affects I.
• A particle on the axis has zero contribution.
• Objects with equal masses can have different moments of inertia.

Common Mistakes

• Using diameter instead of radius.
• Forgetting to square the distance.
• Using total distance instead of perpendicular distance from the axis.
• Ignoring the position of the rotation axis.
• Using the ring formula for a solid disc.
• Writing kg m instead of kg m².
• Adding distances before calculating each particle’s contribution.

MCQs with Answers

Short-Question Preparation

Short-Question Preparation

Long-Question Preparation

For a detailed question about moment of inertia, include:

• Definition of moment of inertia.
• Symbol I.
• Explanation of rotational inertia.
• Comparison with mass in linear motion.
• Point-mass formula I = mr².
• Meaning of m and r.
• Importance of perpendicular distance.
• Rigid-body expression I = Σ mᵢ rᵢ².
• Dependence on total mass.
• Dependence on mass distribution.
• Dependence on the rotation axis.
• SI unit kg m².
• Dimensional formula [ML²].
• Effect of doubling mass.
• Effect of doubling distance.
• Comparison between a disc and ring.
• At least one real-life example.
• A correctly labelled rotation axis where required.

Sticky Notes

Sticky Notes

• Memory trick: Mass tells how difficult translation is; moment of inertia tells how difficult rotation is.
• Formula trick: The farther the mass, the larger the r², and therefore the larger the moment of inertia.
• Exam warning: A moment-of-inertia formula is incomplete unless the rotation axis is stated.

60-Second Revision

I = mr² I = Σ mᵢ rᵢ² SI unit = kg m² [I] = [ML²] Greater mass ⇒ greater I Greater distance from axis ⇒ much greater I Different axis ⇒ different I Ring has greater I than a disc of equal mass and radius Mass on the rotation axis contributes zero